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Fundamental Equations of Mechanics of Materials Axial Load

Normal Stress

s = P A

Displacement

d = L L

0 P(x)dx A (x)E

d = ! PL AE

dT = a "TL

Torsion

Shear stress in circular shaft

t = Tr

where

J =

2 c4 solid cross section

J =

2 (co

4 - ci 4) tubular cross section

Power

P = Tv = 2pf T

Angle of twist

f = L L

0 T(x)dx J(x)G

f = !

TL JG

Average shear stress in a thin-walled tube

tavg =

T 2tA m

Shear Flow

q = tavg t =

T 2Am

Bending

Normal stress

s =

I Unsymmetric bending

s = -

Mz y

Iz +

Myz

Iy , tan a =

Iz Iy

tan u

Shear

Average direct shear stress

tavg = V A

Transverse shear stress

t =

VQ It

Shear flow

q = tt =

VQ I

Stress in Thin-Walled Pressure Vessel

Cylinder

s1 =

t s2 =

Sphere

s1 = s2 =

Stress Transformation Equations

sx# = sx + sy

2 +

sx - sy 2

cos 2u + txy sin 2u

tx#y# = - sx - sy

2 sin 2u + txy cos 2u

Principal Stress

tan 2up = txy

(sx - sy)>2 s1,2 =

sx + sy 2

{ A asx - sy2 b2 + txy2 Maximum in-plane shear stress

tan 2us = - (sx - sy)>2

txy

tmax = A asx - sy2 b2 + t2xy savg =

sx + sy 2

Absolute maximum shear stress

tabs max

= smax

2 for smax, smin same sign

tabs max

= smax - smin

2 for smax, smin opposite signs

Geometric Properties of Area Elements Material Property Relations

Poisson’s ratio

n = - Plat

Plong

Generalized Hooke’s Law

ex =

1 E

3sx - n(sy + sz)4

ey = 1 E

3sy - n(sx + sz)4

ez = 1 E

3sz - n(sx + sy)4

gxy = 1 G

txy, gyz = 1 G

tyz, gzx = 1 G

tzx

where

G =

E 2(1 + n)

Relations Between w, V, M

dV dx

= w(x), dM dx

= V

Elastic Curve

1 r

= M EI

d4v

dx4 = w(x)

d3v

dx3 = V (x)

d2v

dx2 = M(x)

Buckling Critical axial load

Pcr =

p2EI

(KL)2

Critical stress

scr =

p2E

(KL >r)2 , r = 2I>A Secant formula

smax =

P A

c 1 + ec r2

sec a L 2r

A PEA b d Energy Methods Conservation of energy

Ue = Ui

Strain energy Ui =

N2L 2AE

constant axial load

Ui = L

M2dx 2EI

bending moment

Ui = L

fsV 2dx

2GA transverse shear

Ui = L

T2dx 2GJ

torsional moment

y A = bh

Rectangular area

Ix = bh 31

Iy = hb 31

Ix = bh 3x

A = bh

Triangular area

1 36h13

1 2

A = h(a + b)

Trapezoidal area

h13 2a + b a + b

1 2

Ix = pr 4

Semicircular area

1 8

A = pr 2

4r 3p

Iy = pr 41

Ix = pr 4

Circular area

1 4

A = pr2

Iy = pr 41

A = ab

Semiparabolic area

2 3

a25

b38 a zero slope

Exparabolic area

a34 a

b310 zero slope

b C

A = ab3

Average Mechanical Properties of Typical Engineering Materialsa

(SI Units)

Materials Density R (Mg/m 3 )

Moduls of Elasticity E

(GPa)

Modulus of Rigidity G

(GPa)

Yield Strength (MPa) SY

Tens. Comp.b Shear

Ultimate Strength (MPa) Su

Tens. Comp. b Shear

%Elongation in 50 mm specimen

Poisson’s Ratio N

Coef. of Therm. Expansion A

(10–6)/°C

Metallic

Aluminum 2014-T6 Wrought Alloys 6061-T6

2.79 73.1 27 414 414 172 469 469 290 10 0.35 23

2.71 68.9 26 255 255 131 290 290 186 12 0.35 24

Cast Iron Gray ASTM 20 Alloys Malleable ASTM A-197

7.19 67.0 27 – – – 179 669 – 0.6 0.28 12

7.28 172 68 – – – 276 572 – 5 0.28 12

Copper Red Brass C83400 Alloys Bronze C86100

8.74 101 37 70.0 70.0 – 241 241 – 35 0.35 18

8.83 103 38 345 345 – 655 655 – 20 0.34 17

Magnesium [Am 1004-T61]

Alloy 1.83 44.7 18 152 152 – 276 276 152 1 0.30 26

Structural A-36

Steel Structural A992

Alloys Stainless 304

Tool L2

7.85 200 75 250 250 – 400 400 – 30 0.32 12

7.85 200 75 345 345 – 450 450 – 30 0.32 12

7.86 193 75 207 207 – 517 517 – 40 0.27 17

8.16 200 75 703 703 – 800 800 – 22 0.32 12

Titanium [Ti-6Al-4V]

Alloy 4.43 120 44 924 924 – 1,000 1,000 – 16 0.36 9.4

Nonmetallic

Concrete Low Strength

High Strength

2.38 22.1 – – – 12 – – – – 0.15 11

2.37 29.0 – – – 38 – – – – 0.15 11

Plastic Kevlar 49

Reinforced 30% Glass

1.45 131 – – – – 717 483 20.3 2.8 0.34 –

1.45 72.4 – – – – 90 131 – – 0.34 –

Wood Douglas Fir Select Structural White Spruce Grade

0.47 13.1 – – – – 2.1c 26d 6.2 d – 0.29e –

3.60 9.65 – – – – 2.5 c 36 d 6.7 d – 0.31 e –

a Specific values may vary for a particular material due to alloy or mineral composition,mechanical working of the specimen,or heat treatment. For a more exact value reference books for the material should be consulted.

b The yield and ultimate strengths for ductile materials can be assumed equal for both tension and compression. c Measured perpendicular to the grain. d Measured parallel to the grain. e Deformation measured perpendicular to the grain when the load is applied along the grain.

Average Mechanical Properties of Typical Engineering Materialsa

(U.S. Customary Units)

Materials Specific Weight (lb/in 3 )

Moduls of Elasticity E

(103) ksi

Modulus of Rigidity G (103) ksi

Yield Strength (ksi) SY

Tens. Comp.b Shear

Ultimate Strength (ksi) Su

Tens. Comp. b Shear

%Elongation in 2 in. specimen

Poisson’s Ratio N

Coef. of Therm. Expansion A

(10–6)/°F

Metallic

Aluminum 2014-T6 Wrought Alloys 6061-T6

0.101 10.6 3.9 60 60 25 68 68 42 10 0.35 12.8

0.098 10.0 3.7 37 37 19 42 42 27 12 0.35 13.1

Cast Iron Gray ASTM 20 Alloys Malleable ASTM A-197

0.260 10.0 3.9 – – – 26 96 – 0.6 0.28 6.70

0.263 25.0 9.8 – – – 40 83 – 5 0.28 6.60

Copper Red Brass C83400

Alloys Bronze C86100

0.316 14.6 5.4 11.4 11.4 – 35 35 – 35 0.35 9.80

0.319 15.0 5.6 50 50 – 35 35 – 20 0.34 9.60

Magnesium [Am 1004-T61]

Alloy 0.066 6.48 2.5 22 22 – 40 40 22 1 0.30 14.3

Structural A-36

Steel Structural A992

Alloys Stainless 304

Tool L2

0.284 29.0 11.0 36 36 – 58 58 – 30 0.32 6.60

0.284 29.0 11.0 50 50 – 65 65 – 30 0.32 6.60

0.284 28.0 11.0 30 30 – 75 75 – 40 0.27 9.60

0.295 29.0 11.0 102 102 – 116 116 – 22 0.32 6.50

Titanium [Ti-6Al-4V]

Alloy 0.160 17.4 6.4 134 134 – 145 145 – 16 0.36 5.20

Nonmetallic

Concrete Low Strength

High Strength

0.086 3.20 – – – 1.8 – – – – 0.15 6.0

0.086 4.20 – – – 5.5 – – – – 0.15 6.0

Plastic Kevlar 49

Reinforced 30% Glass

0.0524 19.0 – – – – 104 70 10.2 2.8 0.34 –

0.0524 10.5 – – – – 13 19 – – 0.34 –

Wood Douglas Fir Select Structural White Spruce Grade

0.017 1.90 – – – – 0.30c 3.78d 0.90 d – 0.29e –

0.130 1.40 – – – – 0.36 c 5.18 d 0.97 d – 0.31 e –

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS NINTH EDITION

R. C. HIBBELER

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Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on appropriate page within text (or on page xix).

Copyright © 2014, 2011, 2008, 2005, 2003, 2001 by R. C. Hibbeler. Published by Pearson Prentice Hall. All rights reserved. Manufactured in the United States of America. This publication is protected by Copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, 1 Lake Street, Upper Saddle River, NJ 07458.

Many of the designations by manufacturers and seller to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps.

10 9 8 7 6 5 4 3

Library of Congress Cataloging-in-Publication Data on File.

ISBN 10: 0-13-325442-9 ISBN 13: 978-0-13-325442-6

Prentice Hall is an imprint of

www.pearsonhighered.com

To the Student With the hope that this work will stimulate

an interest in Engineering Mechanics and provide an acceptable guide to its understanding.

PREFACE

It is intended that this book provide the student with a clear and thorough presentation of the theory and application of the principles of mechanics of materials. To achieve this objective, over the years this work has been shaped by the comments and suggestions of hundreds of reviewers in the teaching profession, as well as many of the author’s students. The eighth edition has been significantly enhanced from the previous edition, and it is hoped that both the instructor and student will benefit greatly from these improvements.

New to This Edition

• Preliminary Problems. This feature can be found throughout the text, and is given just before the Fundamental Problems. The intent here is to test the student’s conceptual understanding of the theory. Normally the solutions require little or no calculation, and as such, these problems provide a basic understanding of the concepts before they are applied numerically. All the solutions are given in the back of the text.

• Updated Examples. Some portions of the text have been rewritten in order to enhance clarity and be more succinct. In this regard, some new examples have been added and others have been modified to provide more emphasis on the application of important concepts. Included is application of the LRFD method of design, and use of A992 steel for structural applications. Also, the artwork has been improved throughout the book to support these changes.

• New Photos. The relevance of knowing the subject matter is reflected by the real-world applications depicted in over 30 new or updated photos placed throughout the book. These photos generally are used to explain how the relevant principles apply to real-world situations and how materials behave under load.

• Additional Fundamental Problems. These problem sets are located just after each group of example problems. In this edition they have been expanded. They offer students simple applications of the concepts covered in each section and, therefore, provide them with the chance to develop their problem-solving skills before attempting to solve any of the standard problems that follow. The fundamental problems may be considered as extended examples, since the key equations and answers are all listed in the back of the book. Additionally, when assigned, these problems offer students an excellent means of preparing for exams, and they can be used at a later time as a review when studying for the Fundamentals of Engineering Exam.

VI I I PREFACE

• Additional Conceptual Problems. Throughout the text, usually at the end of each chapter, there is a set of problems that involve conceptual situations related to the application of the principles contained in the chapter. These analysis and design problems are intended to engage the students in thinking through a real-life situation as depicted in a photo. They can be assigned after the students have developed some expertise in the subject matter and they work well either for individual or team projects.

• New Problems. There are approximately 31%, or about 460, new problems added to this edition, which involve applications to many different fields of engineering.

Contents The subject matter is organized into 14 chapters. Chapter 1 begins with a review of the important concepts of statics, followed by a formal definition of both normal and shear stress, and a discussion of normal stress in axially loaded members and average shear stress caused by direct shear.

In Chapter 2 normal and shear strain are defined, and in Chapter 3 a discussion of some of the important mechanical properties of materials is given. Separate treatments of axial load, torsion, and bending are presented in Chapters 4 , 5 , and 6 , respectively. In each of these chapters, both linear- elastic and plastic behavior of the material are considered. Also, topics related to stress concentrations and residual stress are included. Transverse shear is discussed in Chapter 7 , along with a discussion of thin-walled tubes, shear flow, and the shear center. Chapter 8 includes a discussion of thin-walled pressure vessels and provides a partial review of the material covered in the previous chapters, where the state of stress results from combined loadings. In Chapter 9 the concepts for transforming multiaxial states of stress are presented. In a similar manner, Chapter 10 discusses the methods for strain transformation, including the application of various theories of failure. Chapter 11 provides a means for a further summary and review of previous material by covering design applications of beams and shafts. In Chapter 12 various methods for computing deflections of beams and shafts are covered. Also included is a discussion for finding the reactions on these members if they are statically indeterminate. Chapter 13 provides a discussion of column buckling, and lastly, in Chapter 14 the problem of impact and the application of various energy methods for computing deflections are considered.

Sections of the book that contain more advanced material are indicated by a star (*). Time permitting, some of these topics may be included in the course. Furthermore, this material provides a suitable reference for basic principles when it is covered in other courses, and it can be used as a basis for assigning special projects.

PREFACE IX

Alternative Method of Coverage. Some instructors prefer to cover stress and strain transformations first, before discussing specific applications of axial load, torsion, bending, and shear. One possible method for doing this would be first to cover stress and its transformation, Chapter 1 and Chapter 9, followed by strain and its transformation, Chapter 2 and the first part of Chapter 10. The discussion and example problems in these later chapters have been styled so that this is possible. Also, the problem sets have been subdivided so that this material can be covered without prior knowledge of the intervening chapters. Chapters 3 through 8 can then be covered with no loss in continuity.

Hallmark Elements Organization and Approach. The contents of each chapter are organized into well-defined sections that contain an explanation of specific topics, illustrative example problems, and a set of homework problems. The topics within each section are placed into subgroups defined by titles. The purpose of this is to present a structured method for introducing each new definition or concept and to make the book convenient for later reference and review.

Chapter Contents. Each chapter begins with a full-page illustration that indicates a broad-range application of the material within the chapter. The “Chapter Objectives” are then provided to give a general overview of the material that will be covered.

Procedures for Analysis. Found after many of the sections of the book, this unique feature provides the student with a logical and orderly method to follow when applying the theory. The example problems are solved using this outlined method in order to clarify its numerical application. It is to be understood, however, that once the relevant principles have been mastered and enough confidence and judgment have been obtained, the student can then develop his or her own procedures for solving problems.

Photographs. Many photographs are used throughout the book to enhance conceptual understanding and explain how the principles of mechanics of materials apply to real-world situations.

Important Points. This feature provides a review or summary of the most important concepts in a section and highlights the most significant points that should be realized when applying the theory to solve problems.

Example Problems. All the example problems are presented in a concise manner and in a style that is easy to understand.

Homework Problems. Apart from the preliminary, fundamental, and conceptual problems, there are numerous standard problems in the

X PREFACE

book that depict realistic situations encountered in engineering practice. It is hoped that this realism will both stimulate the student’s interest in the subject and provide a means for developing the skill to reduce any such problem from its physical description to a model or a symbolic representation to which principles may be applied. Throughout the book there is an approximate balance of problems using either SI or FPS units. Furthermore, in any set, an attempt has been made to arrange the problems in order of increasing difficulty. The answers to all but every fourth problem are listed in the back of the book. To alert the user to a problem without a reported answer, an asterisk (*) is placed before the problem number. Answers are reported to three significant figures, even though the data for material properties may be known with less accuracy. Although this might appear to be a poor practice, it is done simply to be consistent, and to allow the student a better chance to validate his or her solution. A solid square (!) is used to identify problems that require a numerical analysis or a computer application.

Appendices. The appendices of the book provide a source for review and a listing of tabular data. Appendix A provides information on the centroid and the moment of inertia of an area. Appendices B and C list tabular data for structural shapes, and the deflection and slopes of various types of beams and shafts.

Accuracy Checking. The Ninth Edition has undergone a rigorous Triple Accuracy Checking review. In addition to the author’s review of all art pieces and pages, the text was checked by the following individuals:

• Scott Hendricks, Virginia Polytechnic University • Karim Nohra, University of South Florida • Kurt Norlin, LaurelTech Integrated Publishing Services • Kai Beng Yap, Engineering Consultant

Acknowledgments Over the years, this text has been shaped by the suggestions and comments of many of my colleagues in the teaching profession. Their encouragement and willingness to provide constructive criticism are very much appreciated and it is hoped that they will accept this anonymous recognition. A note of thanks is given to the reviewers.

S. Suliman, Penn State C. Valle, Georgia Institute of Tech C. Sulzbach, Colorado School of Mines K. Cook-Chennault, Rutgers University J. Ramirez, Purdue University J. Oyler, University of Pittsburg

PREFACE XI

P. Mokashi, Ohio State Y. Liao, Arizona State University P. Ziehl, University of South Carolina

There are a few people that I feel deserve particular recognition. A long- time friend and associate, Kai Beng Yap, was of great help to me in checking the entire manuscript and helping to prepare the problem solutions. A special note of thanks also goes to Kurt Norlin in this regard. During the production process I am thankful for the assistance of Rose Kernan, my production editor for many years, and to my wife, Conny, and daughter, Mary Ann, for their help in proofreading and typing, that was needed to prepare the manuscript for publication.

I would also like to thank all my students who have used the previous edition and have made comments to improve its contents; including those in the teaching profession who have taken the time to e-mail me their comments, notably S. Alghamdi, A. Atai, S. Larwood, D. Kuemmerle, and J. Love.

I would greatly appreciate hearing from you if at any time you have any comments or suggestions regarding the contents of this edition.

Russell Charles Hibbeler [email protected] Mastering Ad to come

your work...

your answer specific feedback

XIV PREFACE

Resources for Instructors • MasteringEngineering. This online Tutorial Homework program allows you to integrate dynamic homework with automatic grading and adaptive tutoring. MasteringEngineering allows you to easily track the performance of your entire class on an assignment-by- assignment basis, or the detailed work of an individual student. • Instructor’s Solutions Manual. An instructor’s solutions manual was prepared by the author. The manual includes homework assignment lists and was also checked as part of the accuracy checking program. The Instructor Solutions Manual is available at www.pearsonhighered.com . • Presentation Resources. All art from the text is available in PowerPoint slide and JPEG format. These files are available for download from the Instructor Resource Center at www.pearsonhighered.com . If you are in need of a login and password for this site, please contact your local Pearson representative. • Video Solutions. Developed by Professor Edward Berger, University of Virginia, video solutions located on the Companion Website offer step-by-step solution walkthroughs of representative homework problems from each section of the text. Make efficient use of class time and office hours by showing students the complete and concise problem solving approaches that they can access anytime and view at their own pace. The videos are designed to be a flexible resource to be used however each instructor and student prefers. A valuable tutorial resource, the videos are also helpful for student self-evaluation as students can pause the videos to check their understanding and work alongside the video. Access the videos at www.pearsonhighered.com/ hibbeler and follow the links for the Mechanics of Materials text.

Resources for Students • Mastering Engineering. Tutorial homework problems emulate the instrutor’s office-hour environment. • Companion Website —The Companion Website, located at www.pearsonhighered.com/hibbeler includes opportunities for practice and review including: • Video Solutions —Complete, step-by-step solution walkthroughs of representative homework problems from each section. Videos offer: students need it with over 20 hours helpful review.

An access code for the Mechanics of Materials, Ninth Edition companion website was included with this text. To redeem the code and gain access to the site, go to www.pearsonhighered.com/hibbeler and follow the directions on the access code card. Access can also be purchased directly from the site.

CONTENTS

1 Stress 3

Chapter Objectives 3 1.1 Introduction 3 1.2 Equilibrium of a Deformable Body 4 1.3 Stress 22 1.4 Average Normal Stress in an Axially

Loaded Bar 24 1.5 Average Shear Stress 32 1.6 Allowable Stress Design 46 1.7 Limit State Design 48

2 Strain 67

Chapter Objectives 67 2.1 Deformation 67 2.2 Strain 68

3 Mechanical Properties of Materials 83

Chapter Objectives 83 3.1 The Tension and Compression Test 83 3.2 The Stress–Strain Diagram 85 3.3 Stress–Strain Behavior of Ductile and

Brittle Materials 89 3.4 Hooke’s Law 92 3.5 Strain Energy 94 3.6 Poisson’s Ratio 104 3.7 The Shear Stress–Strain Diagram 106 *3.8 Failure of Materials Due to Creep

and Fatigue 109

4 Axial Load 121

Chapter Objectives 121 4.1 Saint-Venant’s Principle 121 4.2 Elastic Deformation of an Axially Loaded

Member 124 4.3 Principle of Superposition 138 4.4 Statically Indeterminate Axially Loaded

Member 139 4.5 The Force Method of Analysis for Axially

Loaded Members 145 4.6 Thermal Stress 153 4.7 Stress Concentrations 160 *4.8 Inelastic Axial Deformation 164 *4.9 Residual Stress 166

5 Torsion 181

Chapter Objectives 181 5.1 Torsional Deformation of a Circular

Shaft 181 5.2 The Torsion Formula 184 5.3 Power Transmission 192 5.4 Angle of Twist 204 5.5 Statically Indeterminate Torque-Loaded

Members 218 *5.6 Solid Noncircular Shafts 225 *5.7 Thin-Walled Tubes Having Closed Cross

Sections 228 5.8 Stress Concentration 238 *5.9 Inelastic Torsion 241 *5.10 Residual Stress 243

XVI CONTENTS

6 Bending 259

Chapter Objectives 259 6.1 Shear and Moment Diagrams 259 6.2 Graphical Method for Constructing Shear

and Moment Diagrams 266 6.3 Bending Deformation of a Straight

Member 285 6.4 The Flexure Formula 289 6.5 Unsymmetric Bending 306 *6.6 Composite Beams 316 *6.7 Reinforced Concrete Beams 319 *6.8 Curved Beams 323 6.9 Stress Concentrations 330 *6.10 Inelastic Bending 339

7 Transverse Shear 363

Chapter Objectives 363 7.1 Shear in Straight Members 363 7.2 The Shear Formula 365 7.3 Shear Flow in Built-Up Members 382 7.4 Shear Flow in Thin-Walled Members 391 *7.5 Shear Center for Open Thin-Walled

Members 396

8 Combined Loadings 409

Chapter Objectives 409 8.1 Thin-Walled Pressure Vessels 409 8.2 State of Stress Caused by Combined

Loadings 416

9 Stress Transformation 441

Chapter Objectives 441 9.1 Plane-Stress Transformation 441 9.2 General Equations of Plane-Stress

Transformation 446 9.3 Principal Stresses and Maximum In-Plane

Shear Stress 449 9.4 Mohr’s Circle—Plane Stress 465 9.5 Absolute Maximum Shear Stress 477

10 Strain Transformation 489

Chapter Objectives 489 10.1 Plane Strain 489 10.2 General Equations of Plane-Strain

Transformation 490 *10.3 Mohr’s Circle—Plane Strain 498 *10.4 Absolute Maximum Shear Strain 506 10.5 Strain Rosettes 508 10.6 Material-Property Relationships 512 *10.7 Theories of Failure 524

11 Design of Beams and Shafts 541

Chapter Objectives 541 11.1 Basis for Beam Design 541 11.2 Prismatic Beam Design 544 *11.3 Fully Stressed Beams 558 *11.4 Shaft Design 562

CONTENTS XVI I

12 Deflection of Beams and Shafts 573

Chapter Objectives 573 12.1 The Elastic Curve 573 12.2 Slope and Displacement by

Integration 577 *12.3 Discontinuity Functions 597 *12.4 Slope and Displacement by the Moment-

Area Method 608 12.5 Method of Superposition 623 12.6 Statically Indeterminate Beams

and Shafts 631 12.7 Statically Indeterminate Beams and

Shafts—Method of Integration 632 *12.8 Statically Indeterminate Beams and

Shafts—Moment-Area Method 637 12.9 Statically Indeterminate Beams and

Shafts—Method of Superposition 643

13 Buckling of Columns 661

Chapter Objectives 661 13.1 Critical Load 661 13.2 Ideal Column with Pin Supports 664 13.3 Columns Having Various Types

of Supports 670 *13.4 The Secant Formula 682 *13.5 Inelastic Buckling 688 *13.6 Design of Columns for Concentric

Loading 696 *13.7 Design of Columns for Eccentric

Loading 707

14 Energy Methods 719

Chapter Objectives 719 14.1 External Work and Strain Energy 719 14.2 Elastic Strain Energy for Various Types

of Loading 724 14.3 Conservation of Energy 737 14.4 Impact Loading 744 *14.5 Principle of Virtual Work 755 *14.6 Method of Virtual Forces Applied

to Trusses 759 *14.7 Method of Virtual Forces Applied

to Beams 766 *14.8 Castigliano’s Theorem 775 *14.9 Castigliano’s Theorem Applied

to Trusses 777 *14.10 Castigliano’s Theorem Applied

to Beams 780

Appendix A. Geometric Properties of an Area 788 B. Geometric Properties of Structural

Shapes 804 C. Slopes and Deflections of Beams 812

Solutions and Answers for Preliminary Problems 814

Fundamental Problems Partial Solutions and Answers 825

Selected Answers 844

Index 866

Cover. Construction cranes. Mushakesa / Shutterstock. Chapter 1 Opener. Structural steel framework on new industrial unit. Copyright by Joe Gough/Shutterstock. Chapter 2 Opener. A copper rod after necking and fracture. sciencephotos / Alamy. Chapter 3 Opener. Earthquake damage on Olfusarbru bridge, South Coast, Iceland. ARCTIC IMAGES / Alamy. Page 63. Photo for Prob. C1-3 is from the Deutsches Museum. Used with permission. Chapter 4 Opener. Oil Rig. George Hammerstein /Corbis. Chapter 5 Opener. Piling machine drilling foundation on building site. HAWKEYE / Alamy. Chapter 6 Opener. Fragment view of the road under reconstruction. Lev Kropotov/Shutterstock. Chapter 7 Opener. Edmonton, Alberta, Canada. Design Pics / LJM Photo/LJM Photo/Newscom. Chapter 8 Opener. Tram on mountain, Snowbird Ski Resort, Wasatch Mountains, Utah, United States. Tetra Images / Alamy. Chapter 9 Opener. Industrial texture of power station generator turbine. Yurchyks/Shutterstock. Chapter 10 Opener. Support for Frederick Douglass-Susan B. Anthony Memorial Bridge, Rochester, NY USA. Russell C. Hibbeler. Chapter 11 Opener. Zinc steel structure of light fl oor. Jarous/Shutterstock. Chapter 12 Opener. Pole vaulter in midair. Corbis Flirt / Alamy. Chapter 13 Opener. Water tower against blue sky. VanHart/Shutterstock Chapter 14 Opener. An Olympic worker cleans a diving board as an Olympic athlete dives in practice at the Aquatics Center ahead of competition in the London Olympics in London, Britain, 24 July 2012. BARBARA WALTON/EPA/Newscom.

All other photos provided by the author.

CREDITS

MECHANICS OF MATERIALS

The bolts used for the connections of this steel framework are subjected to stress. In this chapter we will discuss how engineers design these

connections and their fasteners.

Chapter 1

1.1 Introduction Mechanics of materials is a branch of mechanics that studies the internal effects of stress and strain in a solid body that is subjected to an external loading. Stress is associated with the strength of the material from which the body is made, while strain is a measure of the deformation of the body. In addition to this, mechanics of materials includes the study of the body’s stability when a body such as a column is subjected to compressive loading. A thorough understanding of the fundamentals of this subject is of vital importance because many of the formulas and rules of design cited in engineering codes are based upon the principles of this subject.

CHAPTER OBJECTIVES

■ In this chapter we will review some of the important principles of statics and show how they are used to determine the internal resultant loadings in a body. Afterwards the concepts of normal and shear stress will be introduced, and specific applications of the analysis and design of members subjected to an axial load or direct shear will be discussed.

Stress

4 CHAPTER 1 STRESS

1 Historical Development. The origin of mechanics of materials dates back to the beginning of the seventeenth century, when Galileo performed experiments to study the effects of loads on rods and beams made of various materials. However, at the beginning of the eighteenth century, experimental methods for testing materials were vastly improved, and at that time many experimental and theoretical studies in this subject were undertaken primarily in France, by such notables as Saint-Venant, Poisson, Lamé and Navier.

Over the years, after many of the fundamental problems of mechanics of materials had been solved, it became necessary to use advanced mathematical and computer techniques to solve more complex problems. As a result, this subject expanded into other areas of mechanics, such as the theory of elasticity and the theory of plasticity . Research in these fields is ongoing, in order to meet the demands for solving more advanced problems in engineering.

1.2 Equilibrium of a Deformable Body Since statics has an important role in both the development and application of mechanics of materials, it is very important to have a good grasp of its fundamentals. For this reason we will review some of the main principles of statics that will be used throughout the text.

External Loads. A body is subjected to only two types of external loads; namely, surface forces and body forces, Fig. 1–1 .

Surface Forces. Surface forces are caused by the direct contact of one body with the surface of another. In all cases these forces are distributed over the area of contact between the bodies. If this area is small in comparison with the total surface area of the body, then the surface force can be idealized as a single concentrated force , which is applied to a point on the body. For example, the force of the ground on the wheels of a bicycle can be considered as a concentrated force. If the surface loading is applied along a narrow strip of area, the loading can be idealized as a linear distributed load , w ( s ). Here the loading is measured as having an intensity of force/length along the strip and is represented graphically by a series of arrows along the line s . The resultant force FR of w(s) is equivalent to the area under the distributed loading curve, and this resultant acts through the centroid C or geometric center of this area. The loading along the length of a beam is a typical example of where this idealization is often applied.

w(s)

Concentrated force idealization

Linear distributed load

Surface force

Body force

FR W

Fig. 1–1

1.2 EQUILIBRIUM OF A DEFORMABLE BODY 5

Many machine elements are pin connected in order to enable free rotation at their connections. These supports exert a force on a member, but no moment.

Body Forces. A body force is developed when one body exerts a force on another body without direct physical contact between the bodies. Examples include the effects caused by the earth’s gravitation or its electromagnetic field. Although body forces affect each of the particles composing the body, these forces are normally represented by a single concentrated force acting on the body. In the case of gravitation, this force is called the weight of the body and acts through the body’s center of gravity.

Support Reactions. The surface forces that develop at the supports or points of contact between bodies are called reactions . For two-dimensional problems, i.e., bodies subjected to coplanar force systems, the supports most commonly encountered are shown in Table 1–1 . Note carefully the symbol used to represent each support and the type of reactions it exerts on its contacting member. As a general rule, if the support prevents translation in a given direction, then a force must be developed on the member in that direction. Likewise, if rotation is prevented, a couple moment must be exerted on the member . For example, the roller support only prevents translation perpendicular or normal to the surface. Hence, the roller exerts a normal force F on the member at its point of contact. Since the member can freely rotate about the roller, a couple moment cannot be developed on the member.

Type of connection Reaction

Cable

Roller

One unknown: F

F Smooth support One unknown: F

External pin

Internal pin

Two unknowns: Fx, Fy

FyM

Fixed support Three unknowns: Fx, Fy, M

Two unknowns: Fx, Fy

Type of connection Reaction

u u

TABLE 1–1

6 CHAPTER 1 STRESS

1 Equations of Equilibrium. Equilibrium of a body requires both a balance of forces , to prevent the body from translating or having accelerated motion along a straight or curved path, and a balance of moments , to prevent the body from rotating. These conditions can be expressed mathematically by two vector equations

!F = 0 !MO = 0

(1–1)

Here, ! F represents the sum of all the forces acting on the body, and ! MO is the sum of the moments of all the forces about any point O either on or off the body. If an x, y, z coordinate system is established with the origin at point O , the force and moment vectors can be resolved into components along each coordinate axis and the above two equations can be written in scalar form as six equations, namely,

!Fx = 0 !Fy = 0 !Fz = 0

!Mx = 0 !My = 0 !Mz = 0 (1–2)

Often in engineering practice the loading on a body can be represented as a system of coplanar forces . If this is the case, and the forces lie in the x – y plane, then the conditions for equilibrium of the body can be specified with only three scalar equilibrium equations; that is,

!Fx = 0 !Fy = 0

!MO = 0 (1–3)

Here all the moments are summed about point O and so they will be directed along the z axis.

Successful application of the equations of equilibrium requires complete specification of all the known and unknown forces that act on the body, and so the best way to account for all these forces is to draw the body’s free-body diagram .

In order to design the horizontal members of this building frame, it is first necessary to find the internal loadings at various points along their length.

1.2 EQUILIBRIUM OF A DEFORMABLE BODY 7

Internal Resultant Loadings. In mechanics of materials, statics is primarily used to determine the resultant loadings that act within a body. For example, consider the body shown in Fig. 1–2 a , which is held in equilibrium by the four external forces. * In order to obtain the internal loadings acting on a specific region within the body, it is necessary to pass an imaginary section or “cut” through the region where the internal loadings are to be determined. The two parts of the body are then separated, and a free-body diagram of one of the parts is drawn, Fig. 1–2 b . Notice that there is actually a distribution of internal force acting on the “exposed” area of the section. These forces represent the effects of the material of the top part of the body acting on the adjacent material of the bottom part.

Although the exact distribution of this internal loading may be unknown , we can use the equations of equilibrium to relate the external forces on the bottom part of the body to the distribution’s resultant force and moment , FR and MRO, at any specific point O on the sectioned area, Fig. 1–2 c . It will be shown in later portions of the text that point O is most often chosen at the centroid of the sectioned area, and so we will always choose this location for O , unless otherwise stated. Also, if a member is long and slender, as in the case of a rod or beam, the section to be considered is generally taken perpendicular to the longitudinal axis of the member. This section is referred to as the cross section .

* The body’s weight is not shown, since it is assumed to be quite small, and therefore negligible compared with the other loads.

section

F2 (a)

F1 F2

(b)

F1 F2

MRO

(c)

Fig. 1–2

8 CHAPTER 1 STRESS

Three Dimensions. Later in this text we will show how to relate the resultant loadings, FR and MRO, to the distribution of force on the sectioned area, and thereby develop equations that can be used for analysis and design. To do this, however, the components of FR and MRO acting both normal and perpendicular to the sectioned area must be considered, Fig. 1–2 d . Four different types of resultant loadings can then be defined as follows:

Normal force, N. This force acts perpendicular to the area. It is developed whenever the external loads tend to push or pull on the two segments of the body.

Shear force, V. The shear force lies in the plane of the area, and it is developed when the external loads tend to cause the two segments of the body to slide over one another.

Torsional moment or torque, T. This effect is developed when the external loads tend to twist one segment of the body with respect to the other about an axis perpendicular to the area.

Bending moment, M. The bending moment is caused by the external loads that tend to bend the body about an axis lying within the plane of the area.

In this text, note that graphical representation of a moment or torque is shown in three dimensions as a vector with an associated curl. By the right-hand rule , the thumb gives the arrowhead sense of this vector and the fingers or curl indicate the tendency for rotation (twisting or bending).

(c)

MRO

F1 F2

(d)

F1 F2

M V

Torsional Moment

Bending Moment

Shear Force

MRO

Normal Force

Fig. 1–2 (cont.)

The weight of this sign and the wind loadings acting on it will cause normal and shear forces and bending and torsional moments in the supporting column.

1.2 EQUILIBRIUM OF A DEFORMABLE BODY 9

Coplanar Loadings. If the body is subjected to a coplanar system of forces , Fig. 1–3 a , then only normal-force, shear-force, and bending-moment components will exist at the section, Fig. 1–3 b . If we use the x, y, z coordinate axes, as shown on the left segment, then N can be obtained by applying !Fx = 0, and V can be obtained from !Fy = 0. Finally, the bending moment MO can be determined by summing moments about point O (the z axis), !MO = 0, in order to eliminate the moments caused by the unknowns N and V .

section

F3F2

(a)

V MO

N x

Bending Moment

Shear Force

Normal Force

(b)

Fig. 1–3

Important Points

• Mechanics of materials is a study of the relationship between the external loads applied to a body and the stress and strain caused by the internal loads within the body.

• External forces can be applied to a body as distributed or concentrated surface loadings , or as body forces that act throughout the volume of the body.

• Linear distributed loadings produce a resultant force having a magnitude equal to the area under the load diagram, and having a location that passes through the centroid of this area.

• A support produces a force in a particular direction on its attached member if it prevents translation of the member in that direction, and it produces a couple moment on the member if it prevents rotation .

• The equations of equilibrium !F = 0 and !M = 0 must be satisfied in order to prevent a body from translating with accelerated motion and from rotating.

• When applying the equations of equilibrium, it is important to first draw the free-body diagram for the body in order to account for all the terms in the equations.

• The method of sections is used to determine the internal resultant loadings acting on the surface of the sectioned body. In general, these resultants consist of a normal force, shear force, torsional moment, and bending moment.

10 CHAPTER 1 STRESS

1 Procedure for Analysis

The resultant internal loadings at a point located on the section of a body can be obtained using the method of sections. This requires the following steps.

Support Reactions. • First decide which segment of the body is to be considered. If the

segment has a support or connection to another body, then before the body is sectioned, it will be necessary to determine the reactions acting on the chosen segment. To do this draw the free-body diagram of the entire body and then apply the necessary equations of equilibrium to obtain these reactions.

Free-Body Diagram. • Keep all external distributed loadings, couple moments, torques,

and forces in their exact locations , before passing an imaginary section through the body at the point where the resultant internal loadings are to be determined.

• Draw a free-body diagram of one of the “cut” segments and indicate the unknown resultants N, V, M , and T at the section. These resultants are normally placed at the point representing the geometric center or centroid of the sectioned area.

• If the member is subjected to a coplanar system of forces, only N, V , and M act at the centroid.

• Establish the x, y, z coordinate axes with origin at the centroid and show the resultant internal loadings acting along the axes.

Equations of Equilibrium. • Moments should be summed at the section, about each of the

coordinate axes where the resultants act. Doing this eliminates the unknown forces N and V and allows a direct solution for M (and T ).

• If the solution of the equilibrium equations yields a negative value for a resultant, the directional sense of the resultant is opposite to that shown on the free-body diagram.

The following examples illustrate this procedure numerically and also provide a review of some of the important principles of statics.

1.2 EQUILIBRIUM OF A DEFORMABLE BODY 11

1 EXAMPLE 1.1

Determine the resultant internal loadings acting on the cross section at C of the cantilevered beam shown in Fig. 1–4 a .

(a)

A B

C 3 m 6 m

270 N/m

Fig. 1–4

SOLUTION Support Reactions. The support reactions at A do not have to be determined if segment CB is considered.

Free-Body Diagram. The free-body diagram of segment CB is shown in Fig. 1–4 b . It is important to keep the distributed loading on the segment until after the section is made. Only then should this loading be replaced by a single resultant force. Notice that the intensity of the distributed loading at C is found by proportion, i.e., from Fig. 1–4 a , w >6 m = (270 N>m)>9 m, w = 180 N>m. The magnitude of the resultant of the distributed load is equal to the area under the loading curve (triangle) and acts through the centroid of this area. Thus, F = 12(180 N>m)(6 m) = 540 N, which acts 13(6 m) = 2 m from C as shown in Fig. 1–4 b .

Equations of Equilibrium. Applying the equations of equilibrium we have

S + !Fx = 0; -NC = 0 NC = 0 Ans. + c!Fy = 0; VC - 540 N = 0 VC = 540 N Ans. a+ !MC = 0; -MC - 540 N(2 m) = 0

MC = -1080 N # m Ans. NOTE: The negative sign indicates that MC acts in the opposite direction to that shown on the free-body diagram. Try solving this problem using segment AC , by first obtaining the support reactions at A , which are given in Fig. 1–4 c .

180 N/m

540 N

2 m 4 mVC

(b)

1.5 m 0.5 m

1 m

180 N/m90 N/m

540 N 135 N

(c)

1215 N

3645 N"m CA

12 CHAPTER 1 STRESS

1 EXAMPLE 1.2

The 500-kg engine is suspended from the crane boom in Fig. 1–5 a . Determine the resultant internal loadings acting on the cross section of the boom at point E .

SOLUTION Support Reactions. We will consider segment AE of the boom, so we must first determine the pin reactions at A . Notice that member CD is a two-force member. The free-body diagram of the boom is shown in Fig. 1–5 b . Applying the equations of equilibrium,

a+ !MA = 0; FCD1352(2 m) - [500(9.81) N](3 m) = 0 FCD = 12 262.5 N

S + !Fx = 0; A x - (12 262.5 N)1452 = 0 A x = 9810 N

+ c!Fy = 0; -A y + (12 262.5 N)1352 - 500(9.81) N = 0 A y = 2452.5 N

Free-Body Diagram. The free-body diagram of segment AE is shown in Fig. 1–5 c .

Equations of Equilibrium.

S + !Fx = 0; NE + 9810 N = 0

NE = -9810 N = -9.81 kN Ans.

+ c !Fy = 0; -VE - 2452.5 N = 0

VE = -2452.5 N = -2.45 kN Ans.

a+ !ME = 0; ME + (2452.5 N)( 1 m) = 0

ME = -2452.5 N # m = -2.45 kN # m Ans.

A 1 m1 m1 m

1.5 m

(a)

1 m2 m

500(9.81) N

FCD

(b)

3 4

9810 N

2452.5 N

(c)

1 m

Fig. 1–5

1.2 EQUILIBRIUM OF A DEFORMABLE BODY 13

1 EXAMPLE 1.3

Determine the resultant internal loadings acting on the cross section at G of the beam shown in Fig. 1–6 a . Each joint is pin connected.

(a)

300 lb/ft

2 ft 2 ft 6 ft

1500 lb

G D

3 ft E

3 ft

6 ft (6 ft) ! 4 ft

(6 ft)(300 lb/ft) ! 900 lb

1500 lb

Ey ! 2400 lb

Ex ! 6200 lb

FBC ! 6200 lb

(b)

2 3

1 2

6200 lb

3 4

(c)

FBA ! 7750 lb FBD ! 4650 lb

(d)

MGVG2 ft

3 4

7750 lb1500 lb

A G

SOLUTION Support Reactions. Here we will consider segment AG . The free-body diagram of the entire structure is shown in Fig. 1–6 b . Verify the calculated reactions at E and C . In particular, note that BC is a two-force member since only two forces act on it. For this reason the force at C must act along BC, which is horizontal as shown.

Since BA and BD are also two-force members, the free-body diagram of joint B is shown in Fig. 1–6 c . Again, verify the magnitudes of forces FBA and FBD .

Free-Body Diagram. Using the result for FBA , the free-body diagram of segment AG is shown in Fig. 1–6 d .

Equations of Equilibrium.

S + !Fx = 0; 7750 lb1452 + NG = 0 NG = -6200 lb Ans. + c !Fy = 0; -1500 lb + 7750 lb1352 - VG = 0 VG = 3150 lb Ans.

a+ !MG = 0; MG - (7750 lb)1352(2 ft) + 1500 lb(2 ft) = 0 MG = 6300 lb # ft Ans.

Fig. 1–6

14 CHAPTER 1 STRESS

1 EXAMPLE 1.4

Determine the resultant internal loadings acting on the cross section at B of the pipe shown in Fig. 1–7 a . End A is subjected to a vertical force of 50 N, a horizontal force of 30 N, and a couple moment of 70 N # m . Neglect the pipe’s mass.

0.75 m

50 N

1.25 m

0.5 m

70 N!m

(a)

30 N

1.25 m

70 N·m 30 N

(b)

50 N

0.5 m

(FB)z (MB)z

(MB)x (FB)x

(MB)y

(FB)y

Fig. 1–7

SOLUTION The problem can be solved by considering segment AB , so we do not need to calculate the support reactions at C .

Free-Body Diagram. The x, y, z axes are established at B and the free-body diagram of segment AB is shown in Fig. 1–7 b . The resultant force and moment components at the section are assumed to act in the positive coordinate directions and to pass through the centroid of the cross-sectional area at B .

Equations of Equilibrium. Applying the six scalar equations of equilibrium, we have *

"Fx = 0; (FB)x = 0 Ans.

"Fy = 0; (FB)y + 30 N = 0 (FB)y = -30 N Ans.

"Fz = 0; (FB)z - 50 N = 0 (FB)z = 50 N Ans.

"(MB)x = 0; (MB)x + 70 N # m - 50 N (0.5 m) = 0 (MB)x = -45 N # m Ans. "(MB)y = 0; (MB)y + 50 N (1.25 m) = 0

(MB)y = -62.5 N # m Ans. "(MB)z = 0; (MB)z + (30 N)(1.25) = 0 Ans.

(MB)z = -37.5 N # m NOTE: What do the negative signs for (FB)y, (MB)x , (MB)y and (MB)z indicate? The normal force NB = ! (FB)y ! = 30 N, whereas the shear force is VB = 2(0)2 + (50)2 = 50 N. Also, the torsional moment is TB = ! (MB)y ! = 62.5 N # m and the bending moment is MB = 2(45)2 + (37.5)2 = 58.6 N # m.

* The magnitude of each moment about an axis is equal to the magnitude of each force times the perpendicular distance from the axis to the line of action of the force. The direction of each moment is determined using the right-hand rule, with positive moments (thumb) directed along the positive coordinate axes.

1.2 EQUILIBRIUM OF A DEFORMABLE BODY 15

P1–1. In each case, explain how to find the resultant internal loading acting on the cross section at point A . Draw all necessary free-body diagrams, and indicate the relevant equations of equilibrium. Do not calculate values. The lettered dimensions, angles, and loads are assumed to be known.

B Au

2a aa

(a)

a a

A C

(b)

B CA

(c)

a/2 a/2

O P

(d)

(e)

a a

a 3a

P (f)

PRELIMINARY PROBLEMS

It is suggested that you test yourself on the solutions to these examples, by covering them over and then trying to think about which equilibrium equations must be used and how they are applied in order to determine the unknowns. Then before solving any of the Problems, build your skills by first trying to solve the Preliminary Problems, which actually require little or no calculations, and then do some of the Fundamental Problems given on the following pages. The solutions and answers to all these problems are given in the back of the book . Doing this throughout the book will help immensely in understanding how to apply the theory, and thereby develop your problem-solving skills.

16 CHAPTER 1 STRESS

F1–1. Determine the internal normal force, shear force, and bending moment at point C in the beam.

A B

2 m 2 m1 m

60 kN!m

1 m

10 kN

F1–1

F1–2. Determine the internal normal force, shear force, and bending moment at point C in the beam.

A B C

1.5 m 1.5 m

100 N/m 200 N/m

F1–2

F1–3. Determine the internal normal force, shear force, and bending moment at point C in the beam.

2 m 2 m 2 m

20 kN/m

F1–3

F1–4. Determine the internal normal force, shear force, and bending moment at point C in the beam.

A C

3 m3 m

10 kN/m

F1–4

F1–5. Determine the internal normal force, shear force, and bending moment at point C in the beam.

300 lb/ft

3 ft 3 ft 3 ft

A BC

F1–5

F1–6. Determine the internal normal force, shear force, and bending moment at point C in the beam.

3 m

2 m 2 m 2 m

C B

5 kN/m

F1–6

FUNDAMENTAL PROBLEMS

1.2 EQUILIBRIUM OF A DEFORMABLE BODY 17

1 PROBLEMS

1–1. The shaft is supported by a smooth thrust bearing at B and a journal bearing at C . Determine the resultant internal loadings acting on the cross section at E .

A E DB C

4 ft

400 lb 800 lb

4 ft 4 ft 4 ft

Prob. 1–1

1–2. Determine the resultant internal normal and shear force in the member at (a) section a – a and (b) section b – b , each of which passes through point A . The 500-lb load is applied along the centroidal axis of the member.

30!

b a

500 lb500 lb

Prob. 1–2

1–3. The beam AB is fixed to the wall and has a uniform weight of 80 lb>ft. If the trolley supports a load of 1500 lb, determine the resultant internal loadings acting on the cross sections through points C and D.

5 ft 20 ft

3 ft 10 ft

1500 lb

Prob. 1–3

*1–4. The shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B . Determine the resultant internal loadings acting on the cross section at C .

A DB

900 N

1.5 m

600 N/m

1.5 m1 m1 m1 m

Prob. 1–4

1–5. Determine the resultant internal loadings in the beam at cross sections through points D and E . Point E is just to the right of the 3-kip load.

6 ft 4 ft

4 ft

B CD E

6 ft

3 kip

1.5 kip/ft

Prob. 1–5

1–6. Determine the normal force, shear force, and moment at a section through point C . Take P = 8 kN.

1–7. The cable will fail when subjected to a tension of 2 kN. Determine the largest vertical load P the frame will support and calculate the internal normal force, shear force, and moment at the cross section through point C for this loading.

0.75 m

0.5 m 0.1 m

0.75 m 0.75 m

Probs. 1–6/7

18 CHAPTER 1 STRESS

1 1–11. The forearm and biceps support the 2-kg load at A . If C can be assumed as a pin support, determine the resultant internal loadings acting on the cross section of the bone of the forearm at E. The biceps pulls on the bone along BD.

75!

230 mm 35 mm35 mm

C E B

Prob. 1–11

*1–12. The serving tray T used on an airplane is supported on each side by an arm. The tray is pin connected to the arm at A , and at B there is a smooth pin. (The pin can move within the slot in the arms to permit folding the tray against the front passenger seat when not in use.) Determine the resultant internal loadings acting on the cross section of the arm through point C when the tray arm supports the loads shown.

9 N

500 mm

12 N

15 mm 150 mm

60!

VC MC

100 mm

Prob. 1–12

*1–8. Determine the resultant internal loadings on the cross section through point C . Assume the reactions at the supports A and B are vertical.

1–9. Determine the resultant internal loadings on the cross section through point D . Assume the reactions at the supports A and B are vertical.

0.5 m 0.5 m 1.5 m1.5 m

C A B

3 kN/m 6 kN

Probs. 1–8/9

1–10. The boom DF of the jib crane and the column DE have a uniform weight of 50 lb>ft. If the hoist and load weigh 300 lb, determine the resultant internal loadings in the crane on cross sections through points A , B , and C .

5 ft

7 ft

D F

B A

300 lb

2 ft 8 ft 3 ft

Prob. 1–10

1.2 EQUILIBRIUM OF A DEFORMABLE BODY 19

1 1–17. Determine resultant internal loadings acting on section a – a and section b – b . Each section passes through the centerline at point C .

45!

1.5 m

3 m

45!

b a

a b

5 kN

Prob. 1–17

1–18. The bolt shank is subjected to a tension of 80 lb. Determine the resultant internal loadings acting on the cross section at point C .

A B

90! 6 in.

Prob. 1–18

1–19. Determine the resultant internal loadings acting on the cross section through point C . Assume the reactions at the supports A and B are vertical.

*1–20. Determine the resultant internal loadings acting on the cross section through point D . Assume the reactions at the supports A and B are vertical.

3 ft 3 ft

DC A B

6 ft

6 kip/ft6 kip/ft

Probs. 1–19/20

1–13. The blade of the hacksaw is subjected to a pretension force of F = 100 N. Determine the resultant internal loadings acting on section a – a that passes through point D .

1–14. The blade of the hacksaw is subjected to a pretension force of F = 100 N. Determine the resultant internal loadings acting on section b – b that passes through point D .

A B

F F

b a

30!

225 mm

150 mm

Probs. 1–13/14

1–15. A 150-lb bucket is suspended from a cable on the wooden frame. Determine the resultant internal loadings on the cross section at D.

*1–16. A 150-lb bucket is suspended from a cable on the wooden frame. Determine the resultant internal loadings acting on the cross section at E .

2 ft

3 ft

1 ft1 ft

1 ft E

D C

30! G

Probs. 1–15/16

20 CHAPTER 1 STRESS

1 *1–24. Determine the resultant internal loadings acting on the cross section of the semicircular arch at C .

A B

Prob. 1–24

1–25. Determine the resultant internal loadings acting on the cross section through point B of the signpost. The post is fixed to the ground and a uniform pressure of 7 lb> ft2 acts perpendicular to the face of the sign.

4 ft

6 ft

3 ft

2 ft

3 ft

7 lb/ft2

Prob. 1–25

1–21. The forged steel clamp exerts a force of F = 900 N on the wooden block. Determine the resultant internal loadings acting on section a – a passing through point A .

200 mm

a F ! 900 N

F ! 900 N

30" A

Prob. 1–21

1–22. The metal stud punch is subjected to a force of 120 N on the handle. Determine the magnitude of the reactive force at the pin A and in the short link BC . Also, determine the internal resultant loadings acting on the cross section passing through the handle arm at D .

1–23. Solve Prob. 1–22 for the resultant internal loadings acting on the cross section passing through the handle arm at E and at a cross section of the short link BC .

60! 50 mm

100 mm

200 mm

300 mm B

120 N

50 mm 100 mm

30! A

Probs. 1–22/23

1.2 EQUILIBRIUM OF A DEFORMABLE BODY 21

1 *1–28. The brace and drill bit is used to drill a hole at O . If the drill bit jams when the brace is subjected to the forces shown, determine the resultant internal loadings acting on the cross section of the drill bit at A .

x y

9 in. 6 in.

6 in. 6 in.

9 in.3 in.

Fx ! 30 lb

Fy ! 50 lb

Fz ! 10 lb

Prob. 1–28 1–29. The curved rod AD of radius r has a weight per length of w . If it lies in the vertical plane, determine the resultant internal loadings acting on the cross section through point B . Hint : The distance from the centroid C of segment AB to point O is OC = [2 r sin (u>2)]>u.

u 2

Prob. 1–29 1–30. A differential element taken from a curved bar is shown in the figure. Show that dN >du = V , dV >du = -N, dM >du = -T, and dT >du = M.

M V

N du

M ! dM T ! dT

N ! dN V ! dV

Prob. 1–30

1–26. The shaft is supported at its ends by two bearings A and B and is subjected to the forces applied to the pulleys fixed to the shaft. Determine the resultant internal loadings acting on the cross section located at point C. The 300-N forces act in the -z direction and the 500-N forces act in the +x direction. The journal bearings at A and B exert only x and z components of force on the shaft.

400 mm

150 mm

200 mm

250 mm

300 N 300 N

500 N

Prob. 1–26

1–27. The pipe assembly is subjected to a force of 600 N at B . Determine the resultant internal loadings acting on the cross section at C .

A C

400 mm

150 mm

500 mm

600 N

150 mm

30"

60"

Prob. 1–27

22 CHAPTER 1 STRESS

1 1.3 Stress It was stated in Section 1.2 that the force and moment acting at a specified point O on the sectioned area of the body, Fig. 1–8 , represents the resultant effects of the actual distribution of loading acting over the sectioned area, Fig. 1–9 a . Obtaining this distribution is of primary importance in mechanics of materials. To solve this problem it is necessary to establish the concept of stress.

We begin by considering the sectioned area to be subdivided into small areas, such as !A shown in Fig. 1–9 a . As we reduce !A to a smaller and smaller size, we must make two assumptions regarding the properties of the material. We will consider the material to be continuous , that is, to consist of a continuum or uniform distribution of matter having no voids. Also, the material must be cohesive , meaning that all portions of it are connected together, without having breaks, cracks, or separations. A typical finite yet very small force !F, acting on !A , is shown in Fig. 1–9 a . This force, like all the others, will have a unique direction, but for further discussion we will replace it by its three components , namely, !Fx, !Fy, and !Fz, which are taken tangent, tangent, and normal to the area, respectively. As !A approaches zero, so do !F and its components; however, the quotient of the force and area will, in general, approach a finite limit. This quotient is called stress , and as noted, it describes the intensity of the internal force acting on a specific plane (area) passing through a point.

F1 F2

MRO FR

Fig. 1–8

F1 F2 F1

!F !Fz

yx !Fx !Fy z

(c)x y

(b)

x y(a)x y

tyz

sy tyx

txz

sx txy

Fig. 1–9

1.3 STRESS 23

1 Normal Stress. The intensity of the force acting normal to !A is defined as the normal stress , s (sigma). Since !Fz is normal to the area then

sz = lim!A S0 !Fz !A (1–4)

If the normal force or stress “pulls” on !A as shown in Fig. 1–9 a , it is referred to as tensile stress , whereas if it “pushes” on !A it is called compressive stress .

Shear Stress. The intensity of force acting tangent to !A is called the shear stress , t (tau). Here we have shear stress components,

tzx = lim!A S0 !Fx !A

(1–5)

tzy = lim!A S0 !Fy !A

Note that in this subscript notation z specifies the orientation of the area !A , Fig. 1–10 , and x and y indicate the axes along which each shear stress acts.

General State of Stress. If the body is further sectioned by planes parallel to the x – z plane, Fig. 1–9 b , and the y – z plane, Fig. 1–9 c , we can then “cut out” a cubic volume element of material that represents the state of stress acting around the chosen point in the body. This state of stress is then characterized by three components acting on each face of the element, Fig. 1–11 .

Units. Since stress represents a force per unit area, in the International Standard or SI system, the magnitudes of both normal and shear stress are specified in the basic units of newtons per square meter (N>m2). This unit, called a p ascal (1 Pa = 1 N>m2) is rather small, and in engineering work prefixes such as kilo- (103), symbolized by k, mega- (106), symbolized by M, or giga- (109), symbolized by G, are used to represent larger, more realistic values of stress. * Likewise, in the Foot-Pound-Second system of units, engineers usually express stress in pounds per square inch (psi) or kilopounds per square inch (ksi), where 1 kilopound (kip) = 1000 lb.

x y

Tzx Tzy

Fig. 1–10

x y

zx zy

x xy

s s

t t

Fig. 1–11

* Sometimes stress is expressed in units of N>mm2, where 1 mm = 10-3 m. However, in the SI system, prefixes are not allowed in the denominator of a fraction and therefore it is better to use the equivalent 1 N>mm2 = 1 MN>m2 = 1 MPa.

24 CHAPTER 1 STRESS

1 1.4 Average Normal Stress in an Axially Loaded Bar

In this section we will determine the average stress distribution acting on the cross-sectional area of an axially loaded bar such as the one shown in Fig. 1–12 a . This bar is prismatic since all cross sections are the same throughout its length. When the load P is applied to the bar through the centroid of its cross-sectional area, then the bar will deform uniformly throughout the central region of its length, as shown in Fig. 1–12 b , provided the material of the bar is both homogeneous and isotropic.

Homogeneous material has the same physical and mechanical properties throughout its volume, and isotropic material has these same properties in all directions. Many engineering materials may be approximated as being both homogeneous and isotropic as assumed here. Steel, for example, contains thousands of randomly oriented crystals in each cubic millimeter of its volume, and since most problems involving this material have a physical size that is very much larger than a single crystal, the above assumption regarding its material composition is quite realistic.

Note that anisotropic materials such as wood have different properties in different directions, and although this is the case, if the anisotropy is oriented along the bar’s axis (as for instance in a typical wood rod), then the bar will also deform uniformly when subjected to the axial load P .

Average Normal Stress Distribution. If we pass a section through the bar, and separate it into two parts, then equilibrium requires the resultant normal force at the section to be P , Fig. 1–12 c . Due to the uniform deformation of the material, it is necessary that the cross section be subjected to a constant normal stress distribution , Fig. 1–12 d .

P (a) (b)

Region of uniform deformation of bar

External force

Cross-sectional area

Internal force

!F " s!A

Fig. 1–12

1.4 AVERAGE NORMAL STRESS IN AN AXIALLY LOADED BAR 25

1 As a result, each small area !A on the cross section is subjected to a

force !F = s !A , and the sum of these forces acting over the entire cross-sectional area must be equivalent to the internal resultant force P at the section. If we let !A S dA and therefore !F S dF , then, recognizing s is constant , we have

+ c FRz = "Fz; LdF = LAs dA P = s A

s = P A (1–6)

Here

s = average normal stress at any point on the cross-sectional area

P = internal resultant normal force , which acts through the centroid of the cross-sectional area. P is determined using the method of sections and the equations of equilibrium

A = cross-sectional area of the bar where s is determined

Since the internal load P passes through the centroid of the cross-section, the uniform stress distribution will produce zero moments about the x and y axes passing through this point, Fig. 1–12 d . To show this, we require the moment of P about each axis to be equal to the moment of the stress distribution about the axes, namely,

(MR)x = " Mx; 0 = LA y dF = LA ys dA = sLA y dA

(MR)y = " My; 0 = - LA x dF = - LA xs dA = -sLA x dA

These equations are indeed satisfied, since by definition of the centroid, 1y dA = 0 and 1x dA = 0. (See Appendix A .)

Equilibrium. It should be apparent that only a normal stress exists on any small volume element of material located at each point on the cross section of an axially loaded bar. If we consider vertical equilibrium of the element, Fig. 1–13 , then apply the equation of force equilibrium,

"Fz = 0; s(!A ) - s#(!A ) = 0

s = s#

s¿

Fig. 1–13

26 CHAPTER 1 STRESS

In other words, the two normal stress components on the element must be equal in magnitude but opposite in direction. This is referred to as uniaxial stress .

The previous analysis applies to members subjected to either tension or compression, as shown in Fig. 1–14 . As a graphical interpretation, the magnitude of the internal resultant force P is equivalent to the volume under the stress diagram; that is, P = s A (volume = height * base). Furthermore, as a consequence of the balance of moments, this resultant passes through the centroid of this volume .

Although we have developed this analysis for prismatic bars, this assumption can be relaxed somewhat to include bars that have a slight taper . For example, it can be shown, using the more exact analysis of the theory of elasticity, that for a tapered bar of rectangular cross section, for which the angle between two adjacent sides is 15°, the average normal stress, as calculated by s = P>A , is only 2.2% less than its value found from the theory of elasticity.

Maximum Average Normal Stress. In our analysis both the internal force P and the cross-sectional area A were constant along the longitudinal axis of the bar, and as a result the normal stress s = P>A is also constant throughout the bar’s length. Occasionally, however, the bar may be subjected to several external loads along its axis, or a change in its cross-sectional area may occur. As a result, the normal stress within the bar could be different from one section to the next, and, if the maximum average normal stress is to be determined, then it becomes important to find the location where the ratio P>A is a maximum . To do this it is necessary to determine the internal force P at various sections along the bar. Here it may be helpful to show this variation by drawing an axial or normal force diagram . Specifically, this diagram is a plot of the normal force P versus its position x along the bar’s length. As a sign convention, P will be positive if it causes tension in the member, and negative if it causes compression. Once the internal loading throughout the bar is known, the maximum ratio of P > A can then be identified. This steel tie rod is used as a hanger to suspend a portion of a staircase, and as a result it is subjected to tensile stress.

Tension Compression

P— A!s

P— A

Fig. 1–14

1.4 AVERAGE NORMAL STRESS IN AN AXIALLY LOADED BAR 27

1 Important Points

• When a body subjected to external loads is sectioned, there is a distribution of force acting over the sectioned area which holds each segment of the body in equilibrium. The intensity of this internal force at a point in the body is referred to as stress .

• Stress is the limiting value of force per unit area, as the area approaches zero. For this definition, the material is considered to be continuous and cohesive.

• The magnitude of the stress components at a point depends upon the type of loading acting on the body, and the orientation of the element at the point.

• When a prismatic bar is made from homogeneous and isotropic material, and is subjected to an axial force acting through the centroid of the cross-sectional area, then the center region of the bar will deform uniformly. As a result, the material will be subjected only to normal stress . This stress is uniform or averaged over the cross-sectional area.

Procedure for Analysis

The equation s = P>A gives the average normal stress on the cross-sectional area of a member when the section is subjected to an internal resultant normal force P . For axially loaded members, application of this equation requires the following steps.

Internal Loading. • Section the member perpendicular to its longitudinal axis at the

point where the normal stress is to be determined and use the necessary free-body diagram and force equation of equilibrium to obtain the internal axial force P at the section.

Average Normal Stress. • Determine the member’s cross-sectional area at the section and

calculate the average normal stress s = P>A . • It is suggested that s be shown acting on a small volume element

of the material located at a point on the section where stress is calculated. To do this, first draw s on the face of the element coincident with the sectioned area A. Here s acts in the same direction as the internal force P since all the normal stresses on the cross section develop this resultant. The normal stress s on the other face of the element acts in the opposite direction.

28 CHAPTER 1 STRESS

The bar in Fig. 1–15 a has a constant width of 35 mm and a thickness of 10 mm. Determine the maximum average normal stress in the bar when it is subjected to the loading shown.

(b)

9 kN

12 kN

PAB ! 12 kN

PBC ! 30 kN

PCD ! 22 kN 22 kN

P (kN)

x 12 22 30

(c)

12 kN 22 kN 9 kN

9 kN

4 kN

4 kN 35 mm

A DB C

(a)

EXAMPLE 1.5

(d)

30 kN

85.7 MPa35 mm

10 mm

Fig. 1–15

SOLUTION Internal Loading. By inspection, the internal axial forces in regions AB, BC , and CD are all constant yet have different magnitudes. Using the method of sections, these loadings are determined in Fig. 1–15 b ; and the normal force diagram, which represents these results graphically, is shown in Fig. 1–15 c . The largest loading is in region BC , where PBC = 30 kN. Since the cross-sectional area of the bar is constant , the largest average normal stress also occurs within this region of the bar.

Average Normal Stress. Applying Eq. 1–6 , we have

sBC = PBC A

= 30(103) N

(0.035 m)(0.010 m) = 85.7 MPa Ans.

NOTE: The stress distribution acting on an arbitrary cross section of the bar within region BC is shown in Fig. 1–15 d . Graphically the volume (or “block”) represented by this distribution of stress is equivalent to the load of 30 kN; that is, 30 kN = (85.7 MPa)(35 mm)(10 mm).

1.4 AVERAGE NORMAL STRESS IN AN AXIALLY LOADED BAR 29

1 EXAMPLE 1.6

The 80-kg lamp is supported by two rods AB and BC as shown in Fig. 1–16 a . If AB has a diameter of 10 mm and BC has a diameter of 8 mm, determine the average normal stress in each rod.

60! B

3 4

(a) (b)

60!

FBA FBC

80(9.81) " 784.8 N

B 3

Fig. 1–16

SOLUTION Internal Loading. We must first determine the axial force in each rod. A free-body diagram of the lamp is shown in Fig. 1–16 b . Applying the equations of force equilibrium,

S+ !Fx = 0; FBC1452 - FBA cos 60" = 0 + c !Fy = 0; FBC1352 + FBA sin 60" - 784.8 N = 0 FBC = 395.2 N, FBA = 632.4 N

By Newton’s third law of action, equal but opposite reaction, these forces subject the rods to tension throughout their length.

Average Normal Stress. Applying Eq. 1–6 ,

sBC = FBC A BC

= 395.2 N

p(0.004 m)2 = 7.86 MPa Ans.

sBA = FBA A BA

= 632.4 N

p(0.005 m)2 = 8.05 MPa Ans.

NOTE: The average normal stress distribution acting over a cross section of rod AB is shown in Fig. 1–16 c , and at a point on this cross section, an element of material is stressed as shown in Fig. 1–16 d .

632.4 N

8.05 MPa

(c)(d)

30 CHAPTER 1 STRESS

1 EXAMPLE 1.7

The casting shown in Fig. 1–17 a is made of steel having a specific weight of gst = 490 lb>ft3. Determine the average compressive stress acting at points A and B .

0.75 ft

2.75 ft

x (a)

B0.75 ft 0.4 ft

2.75 ft

(b)

(c)

9.36 psi

Wst

Fig. 1–17

SOLUTION Internal Loading. A free-body diagram of the top segment of the casting where the section passes through points A and B is shown in Fig. 1–17 b . The weight of this segment is determined from Wst = gst Vst . Thus the internal axial force P at the section is

+ c !Fz = 0; P - Wst = 0 P - (490 lb>ft3)(2.75 ft)3p(0.75 ft)24 = 0

P = 2381 lb Average Compressive Stress. The cross-sectional area at the section is A = p(0.75 ft)2, and so the average compressive stress becomes

s = P A

= 2381 lb

p(0.75 ft)2 = 1347.5 lb>ft2

s = 1347.5 lb>ft2 (1 ft2>144 in2) = 9.36 psi Ans. NOTE: The stress shown on the volume element of material in Fig. 1–17 c is representative of the conditions at either point A or B . Notice that this stress acts upward on the bottom or shaded face of the element since this face forms part of the bottom surface area of the section, and on this surface, the resultant internal force P is pushing upward.

1.4 AVERAGE NORMAL STRESS IN AN AXIALLY LOADED BAR 31

1 EXAMPLE 1.8

Member AC shown in Fig. 1–18 a is subjected to a vertical force of 3 kN. Determine the position x of this force so that the average compressive stress at the smooth support C is equal to the average tensile stress in the tie rod AB . The rod has a cross-sectional area of 400 mm2 and the contact area at C is 650 mm2.

200 mm

(a)

3 kN

(b)

3 kN

200 mm

FAB

Fig. 1–18 SOLUTION Internal Loading. The forces at A and C can be related by considering the free-body diagram for member AC , Fig. 1–18 b . There are three unknowns, namely, FAB , FC , and x . To solve this problem we will work in units of newtons and millimeters. + c !Fy = 0; FAB + FC - 3000 N = 0 (1) a+ !MA = 0; -3000 N(x) + FC(200 mm) = 0 (2) Average Normal Stress. A necessary third equation can be written that requires the tensile stress in the bar AB and the compressive stress at C to be equivalent, i.e.,

s = FAB

400 mm2 =

FC 650 mm2

FC = 1.625FAB

Substituting this into Eq. 1 , solving for FAB , then solving for FC , we obtain FAB = 1143 N FC = 1857 N The position of the applied load is determined from Eq. 2 ,

x = 124 mm Ans. NOTE: 0 6 x 6 200 mm, as required.

32 CHAPTER 1 STRESS

1 1.5 Average Shear Stress Shear stress has been defined in Section 1.3 as the stress component that acts in the plane of the sectioned area. To show how this stress can develop, consider the effect of applying a force F to the bar in Fig. 1–19 a . If the supports are considered rigid, and F is large enough, it will cause the material of the bar to deform and fail along the planes identified by AB and CD . A free-body diagram of the unsupported center segment of the bar, Fig. 1–19 b , indicates that the shear force V = F>2 must be applied at each section to hold the segment in equilibrium. The average shear stress distributed over each sectioned area that develops this shear force is defined by

tavg = V A

(1–7)

Here

tavg = average shear stress at the section, which is assumed to be the same at each point located on the section

V = internal resultant shear force on the section determined from the equations of equilibrium

A = area at the section

The distribution of average shear stress acting over the sections is shown in Fig. 1–19 c . Notice that tavg is in the same direction as V , since the shear stress must create associated forces all of which contribute to the internal resultant force V at the section.

The loading case discussed here is an example of simple or direct shear , since the shear is caused by the direct action of the applied load F . This type of shear often occurs in various types of simple connections that use bolts, pins, welding material, etc. In all these cases, however, application of Eq. 1–7 is only approximate . A more precise investigation of the shear-stress distribution over the section often reveals that much larger shear stresses occur in the material than those predicted by this equation. Although this may be the case, application of Eq. 1–7 is generally acceptable for many problems in engineering design and analysis. For example, engineering codes allow its use when considering design sizes for fasteners such as bolts and for obtaining the bonding strength of glued joints subjected to shear loadings.

(b)

(c)

tavg

(a)

B D

A C

Fig. 1–19

B C A

The pin A used to connect the linkage of this tractor is subjected to double shear because shearing stresses occur on the surface of the pin at B and C. See Fig 1–21c.

1.5 AVERAGE SHEAR STRESS 33

Shear Stress Equilibrium. Fig. 1–20 a shows a volume element of material taken at a point located on the surface of a sectioned area which is subjected to a shear stress tzy . Force and moment equilibrium requires the shear stress acting on this face of the element to be accompanied by shear stress acting on three other faces. To show this we will first consider force equilibrium in the y direction. Then

force ƒiiƒ

stress area ƒiƒ ƒiiiƒ

!Fy = 0; tzy1"x "y2 - t=zy "x "y = 0 tzy = t=zy

In a similar manner, force equilibrium in the z direction yields tyz = t=yz . Finally, taking moments about the x axis,

moment ƒiiiƒ

force arm ƒiiƒ ƒ

stress area ƒ ƒiiƒ ƒiiiƒ ƒiƒ

!Mx = 0; -tzy1"x "y2 "z + tyz1"x "z2 "y = 0 tzy = tyz so that

tzy = t=zy = tyz = t=yz = t

In other words, all four shear stresses must have equal magnitude and be directed either toward or away from each other at opposite edges of the element , Fig. 1–20 b . This is referred to as the complementary property of shear , and under the conditions shown in Fig. 1–20 , the material is subjected to pure shear .

Pure shear

(a) (b)

Section plane

!xt¿zy

tzy

t¿yz

tyz

Fig. 1–20

34 CHAPTER 1 STRESS

1 Important Points

• If two parts are thin or small when joined together, the applied loads may cause shearing of the material with negligible bending. If this is the case, it is generally assumed that an average shear stress acts over the cross-sectional area.

• When shear stress t acts on a plane, then equilibrium of a volume element of material at a point on the plane requires associated shear stress of the same magnitude act on three adjacent sides of the element.

Procedure for Analysis

The equation tavg = V>A is used to determine the average shear stress in the material. Application requires the following steps.

Internal Shear. • Section the member at the point where the average shear stress is

to be determined.

• Draw the necessary free-body diagram, and calculate the internal shear force V acting at the section that is necessary to hold the part in equilibrium.

Average Shear Stress. • Determine the sectioned area A , and determine the average

shear stress tavg = V>A. • It is suggested that tavg be shown on a small volume element of

material located at a point on the section where it is determined. To do this, first draw tavg on the face of the element, coincident with the sectioned area A . This stress acts in the same direction as V . The shear stresses acting on the three adjacent planes can then be drawn in their appropriate directions following the scheme shown in Fig. 1–20 .

1.5 AVERAGE SHEAR STRESS 35

1 EXAMPLE 1.9

Determine the average shear stress in the 20-mm-diameter pin at A and the 30-mm-diameter pin at B that support the beam in Fig. 1–21 a .

SOLUTION

Internal Loadings. The forces on the pins can be obtained by considering the equilibrium of the beam, Fig. 1–21 b .

a+ !MA = 0;

FBa 45 b 16 m2 - 30 kN12 m2 = 0 FB = 12.5 kN S+ !Fx = 0; 112.5 kN2 a 35 b - Ax = 0 Ax = 7.50 kN + c !Fy = 0; Ay + 112.5 kN2 a 45 b - 30 kN = 0 Ay = 20 kN Thus, the resultant force acting on pin A is

FA = 2A 2x + A 2y = 2(7.50 kN)2 + (20 kN)2 = 21.36 kN The pin at A is supported by two fixed “leaves” and so the free-body diagram of the center segment of the pin shown in Fig. 1–21 c has two shearing surfaces between the beam and each leaf. The force of the beam (21.36 kN) acting on the pin is therefore supported by shear force on each of these surfaces. This case is called double shear . Thus,

VA = FA 2

= 21.36 kN

2 = 10.68 kN

In Fig. 1–21 a , note that pin B is subjected to single shear, which occurs on the section between the cable and beam, Fig. 1–21 d . For this pin segment,

VB = FB = 12.5 kN

Average Shear Stress.

1tA2avg = V AA A = 10.6811032 Np 4 10.02 m22 = 34.0 MPa Ans.

1tB2avg = V BA B = 12.511032 Np 4 10.03 m22 = 17.7 MPa Ans.

4 m

(a)

2 m

30 kN

A B

3 45

4 m

(b)

2 m

30 kN

3 45

(c)

VA VA

FA ! 21.36 kN

(d)

FB ! 12.5 kN

Fig. 1–21

36 CHAPTER 1 STRESS

1 EXAMPLE 1.10

If the wood joint in Fig. 1–22 a has a width of 150 mm, determine the average shear stress developed along shear planes a – a and b – b . For each plane, represent the state of stress on an element of the material.

0.1 m 0.125 m

(a)

6 kN 6 kN

b b

a a

(b)

6 kN

Fig. 1–22

SOLUTION

Internal Loadings. Referring to the free-body diagram of the member, Fig. 1–22 b ,

S+ !Fx = 0; 6 kN - F - F = 0 F = 3 kN

Now consider the equilibrium of segments cut across shear planes a – a and b – b , shown in Figs. 1–22 c and 1–22 d .

S+ !Fx = 0; Va - 3 kN = 0 Va = 3 kN

S+ !Fx = 0; 3 kN - Vb = 0 Vb = 3 kN

Average Shear Stress.

1ta2avg = VaAa = 311032 N10.1 m2 10.15 m2 = 200 kPa Ans. 1tb2avg = VbAb = 311032 N10.125 m2 10.15 m2 = 160 kPa Ans. The state of stress on elements located on sections a – a and b – b is shown in Figs. 1–22 c and 1–22 d , respectively.

3 kN

(c)

Vata ! 200 kPa

3 kN

(d)

Vbtb = 160 kPa

1.5 AVERAGE SHEAR STRESS 37

1 EXAMPLE 1.11

The inclined member in Fig. 1–23 a is subjected to a compressive force of 600 lb. Determine the average compressive stress along the smooth areas of contact defined by AB and BC , and the average shear stress along the horizontal plane defined by DB .

1 in.

600 lb

1.5 in. 3 in. 2 in.

A C

B D

(a) Fig. 1–23 SOLUTION Internal Loadings. The free-body diagram of the inclined member is shown in Fig. 1–23 b . The compressive forces acting on the areas of contact are

S+ !Fx = 0; FAB - 600 lb1352 = 0 FAB = 360 lb + c !Fy = 0; FBC - 600 lb1452 = 0 FBC = 480 lb Also, from the free-body diagram of the top segment ABD of the bottom member, Fig. 1–23 c , the shear force acting on the sectioned horizontal plane DB is

S+ !Fx = 0; V - 360 lb = 0 V = 360 lb

Average Stress. The average compressive stresses along the horizontal and vertical planes of the inclined member are

sAB = FAB AAB

= 360 lb11 in.2 11.5 in.2 = 240 psi Ans.

sBC = FBC ABC

= 480 lb12 in.2 11.5 in.2 = 160 psi Ans.

These stress distributions are shown in Fig. 1–23 d . The average shear stress acting on the horizontal plane defined

by DB is

tavg = 360 lb13 in.2 11.5 in.2 = 80 psi Ans.

This stress is shown uniformly distributed over the sectioned area in Fig. 1–23 e .

(b)

600 lb

FAB

FBC

360 lb

(d)

600 lb

160 psi

240 psi

(e)

360 lb

80 psi

38 CHAPTER 1 STRESS

1 PRELIMINARY PROBLEMS

P1–2. In each case, determine the largest internal shear force resisted by the bolt. Include all necessary free-body diagrams.

6 kN 2 kN

(a)

5 kNA

10 kN 4 kN

20 kN

(b)

6 kNA

C 8 kN

P1–2

P1–3. Determine the largest internal normal force in the bar.

10 kN 6 kN2 kN5 kN

C B A

P1–3

P1–4. Determine the internal normal force at section A if the rod is subjected to the external uniformally distributed loading along its length of 8 kN>m.

8 kN/mA

2 m 3 m

P1–4

P1–5. The lever is held to the fixed shaft using the pin AB . If the couple is applied to the lever, determine the shear force in the pin between the pin and the lever.

20 N

0.2 m 0.2 m

0.1 m

P1–5

P1–6. The single-V butt joint transmits the force of 5 kN from one plate to the other. Determine the resultant normal and shear force components that this force creates on the face of the weld, section AB .

30! 30!

5 kN

20 mm

100 mm

A B

P1–6

1.5 AVERAGE SHEAR STRESS 39

1 FUNDAMENTAL PROBLEMS

F1–7. The uniform beam is supported by two rods AB and CD that have cross-sectional areas of 10 mm2 and 15 mm2 , respectively. Determine the intensity w of the distributed load so that the average normal stress in each rod does not exceed 300 kPa.

A C

B D

6 m

F1–7

F1–8. Determine the average normal stress developed on the cross section. Sketch the normal stress distribution over the cross section.

300 kN

100 mm

80 mm

F1–8

F1–9. Determine the average normal stress developed on the cross section. Sketch the normal stress distribution over the cross section.

4 in. 1 in.

1 in. 4 in. 1 in.

15 kip

F1–9

F1–10. If the 600-kN force acts through the centroid of the cross section, determine the location y of the centroid and the average normal stress developed on the cross section. Also, sketch the normal stress distribution over the cross section.

80 mm

300 mm

60 mm

–y 80 mm

600 kN

x y60 mm

F1–10

F1–11. Determine the average normal stress developed at points A, B, and C . The diameter of each segment is indicated in the figure.

2 kip3 kip 8 kip9 kip

1 in. 0.5 in. 0.5 in.

A B

F1–11

F1–12. Determine the average normal stress developed in rod AB if the load has a mass of 50 kg. The diameter of rod AB is 8 mm.

8 mm

5 4

F1–12

40 CHAPTER 1 STRESS

FUNDAMENTAL PROBLEMS1 PROBLEMS

1–31. The supporting wheel on a scaffold is held in place on the leg using a 4-mm-diameter pin as shown. If the wheel is subjected to a normal force of 3 kN, determine the average shear stress developed in the pin. Neglect friction between the inner scaffold puller leg and the tube used on the wheel.

3 kN

Prob. 1–31

*1–32. The lever is held to the fixed shaft using a tapered pin AB, which has a mean diameter of 6 mm. If a couple is applied to the lever, determine the average shear stress in the pin between the pin and lever.

20 N 20 N

250 mm 250 mm

12 mm

Prob. 1–32

1–33. The bar has a cross-sectional area A and is subjected to the axial load P . Determine the average normal and average shear stresses acting over the shaded section, which is oriented at u from the horizontal. Plot the variation of these stresses as a function of u 10 … u … 90!2.

Prob. 1–33

1–34. The built-up shaft consists of a pipe AB and solid rod BC . The pipe has an inner diameter of 20 mm and outer diameter of 28 mm. The rod has a diameter of 12 mm. Determine the average normal stress at points D and E and represent the stress on a volume element located at each of these points.

A 4 kN

8 kN B 6 kN

6 kN

Prob. 1–34

1–35. If the turnbuckle is subjected to an axial force of P = 900 lb, determine the average normal stress developed in section a – a and in each of the bolt shanks at B and C. Each bolt shank has a diameter of 0.5 in.

*1–36. The average normal stresses developed in section a – a of the turnbuckle, and the bolts shanks at B and C, are not allowed to exceed 15 ksi and 45 ksi, respectively. Determine the maximum axial force P that can be applied to the turnbuckle. Each bolt shank has a diameter of 0.5 in.

B P PC

1 in.

0.25 in.

Probs. 1–35/36

1.5 AVERAGE SHEAR STRESS 41

1 *1–40. Determine the average normal stress in each of the 20-mm diameter bars of the truss. Set P = 40 kN.

1–41. If the average normal stress in each of the 20-mm diameter bars is not allowed to exceed 150 MPa, determine the maximum force P that can be applied to joint C .

1–42. Determine the average shear stress developed in pin A of the truss. A horizontal force of P = 40 kN is applied to joint C . Each pin has a diameter of 25 mm and is subjected to double shear.

1.5 m

2 m

A B

Probs. 1–40/41/42

1–43. The 150-kg bucket is suspended from end E of the frame. Determine the average normal stress in the 6-mm diameter wire CF and the 15-mm diameter short strut BD .

*1–44. The 150-kg bucket is suspended from end E of the frame. If the diameters of the pins at A and D are 6 mm and 10 mm, respectively, determine the average shear stress developed in these pins. Each pin is subjected to double shear.

0.6 m0.6 m

1.2 m

D E

0.6 m

30!

Probs. 1–43/44

1–37. The plate has a width of 0.5 m. If the stress distri bution at the support varies as shown, determine the force P applied to the plate and the distance d to where it is applied.

4 m

30 MPa

P d

" (15x1/2) MPas

Prob. 1–37

1–38. The two members used in the construction of an aircraft fuselage are joined together using a 30° fish-mouth weld. Determine the average normal and average shear stress on the plane of each weld. Assume each inclined plane supports a horizontal force of 400 lb.

800 lb 800 lb

30!

1 in. 1 in.

1.5 in. 30!

Prob. 1–38

1–39. If the block is subjected to the centrally applied force of 600 kN, determine the averege normal stress in the material. Show the stress acting on a differential volume element of the material.

50 mm

150 mm

150 mm 50 mm

100 mm 100 mm

600 kN150 mm

150 mm

Prob. 1–39

42 CHAPTER 1 STRESS

1 *1–48. The beam is supported by a pin at A and a short link BC. If P = 15 kN, determine the average shear stress developed in the pins at A, B, and C. All pins are in double shear as shown, and each has a diameter of 18 mm.

B A

0.5 m 1 m 1.5 m 1.5 m

0.5 m P 4P 4P 2P

30!

Prob. 1–48

1–49. The joint is subjected to the axial member force of 6 kip. Determine the average normal stress acting on sections AB and BC . Assume the member is smooth and is 1.5-in. thick.

60!

20! 4.5 in.

1.5 in.

6 kip

Prob. 1–49

1–50. The driver of the sports car applies his rear brakes and causes the tires to slip. If the normal force on each rear tire is 400 lb and the coefficient of kinetic friction between the tires and the pavement is µ k = 0.5, determine the average shear stress developed by the friction force on the tires. Assume the rubber of the tires is flexible and each tire is filled with an air pressure of 32 psi.

400 lb

Prob. 1–50

1–45. The pedestal has a triangular cross section as shown. If it is subjected to a compressive force of 500 lb, specify the x and y coordinates for the location of point P ( x , y ), where the load must be applied on the cross section, so that the average normal stress is uniform. Compute the stress and sketch its distribution acting on the cross section at a location removed from the point of load application.

3 in. 6 in.x

500 lb

P(x,y)12 in.

Prob. 1–45

1–46. The 20-kg chandelier is suspended from the wall and ceiling using rods AB and BC , which have diameters of 3 mm and 4 mm, respectively. Determine the angle u so that the average normal stress in both rods is the same.

1–47. The chandelier is suspended from the wall and ceiling using rods AB and BC, which have diameters of 3 mm and 4 mm, respectively. If the average normal stress in both rods is not allowed to exceed 150 MPa, determine the lar gest mass of the chandelier that can be supported if u = 45!.

30!

Probs. 1–46/47

1.5 AVERAGE SHEAR STRESS 43

1 1–54. When the hand is holding the 5-lb stone, the humerus H , assumed to be smooth, exerts normal forces F C and F A on the radius C and ulna A , respectively, as shown. If the smallest cross-sectional area of the ligament at B is 0.30 in 2 , determine the greatest average tensile stress to which it is subjected.

14 in. 2 in.

0.8 in.

C GFB

75!

Prob. 1–54

1–55. The 2-Mg concrete pipe has a center of mass at point G . If it is suspended from cables AB and AC , determine the average normal stress developed in the cables. The diameters of AB and AC are 12 mm and 10 mm, respectively.

*1–56. The 2-Mg concrete pipe has a center of mass at point G . If it is suspended from cables AB and AC , determine the diameter of cable AB so that the average normal stress developed in this cable is the same as in the 10-mm diameter cable AC .

30! 45!

Probs. 1–55/56

1–51. During the tension test, the wooden specimen is subjected to an average normal stress of 2 ksi. Determine the axial force P applied to the specimen. Also, find the average shear stress developed along section a – a of the specimen.

1 in. 2 in.

4 in.

Prob. 1–51

*1–52. If the joint is subjected to an axial force of P = 9 kN, determine the averege shear stress developed in each of the 6-mm diameter bolts between the plates and the members and along each of the four shaded shear planes.

1–53. The average shear stress in each of the 6-mm diameter bolts and along each of the four shaded shear planes is not allowed to exceed 80 MPa and 500 kPa, respectively. Determine the maximum axial force P that can be applied to the joint.

100 mm

Probs. 1–52/53

44 CHAPTER 1 STRESS

1 1–59. The jib crane is pinned at A and supports a chain hoist that can travel along the bottom flange of the beam, 1 ft … x … 12 ft. If the hoist is rated to support a maximum of 1500 lb, determine the maximum average normal stress in the 34 -in. diameter tie rod BC and the maximum average shear stress in the 58 -in. -diamater pin at B.

10 ft

30!

1500 lb

Prob. 1–59

*1–60. If the shaft is subjected to an axial force of 5 kN, determine the bearing stress acting on the collar A .

1–61. If the 60-mm diameter shaft is subjected to an axial force of 5 kN, determine the average shear stress developed in the shear plane where the collar A and shaft are connected.

A 2.5 mm

2.5 mm

100 mm60 mm 5 kN

15 mm

Probs. 1–60/61

1–57. If the concrete pedestal has a specific weight of g , determine the average normal stress developed in the pedestal as a function of z .

2r0 h

Prob. 1–57

1–58. The anchor bolt was pulled out of the concrete wall and the failure surface formed part of a frustum and cylinder. This indicates a shear failure occurred along the cylinder BC and tension failure along the frustum AB . If the shear and normal stresses along these surfaces have the magnitudes shown, determine the force P that must have been applied to the bolt.

30 mm4.5 MPa

3 MPa 3 MPa

50 mm

25 mm 25 mm

45!45!

Prob. 1–58

1.5 AVERAGE SHEAR STRESS 45

1 1–66. Determine the largest load P that can be applied to the frame without causing either the average normal stress or the average shear stress at section a – a to exceed s = 150 MPa and t = 60 MPa , respectively. Member CB has a square cross section of 25 mm on each side.

2 m

A C

1.5 m

Prob. 1–66

1–67. The pedestal in the shape of a frustum of a cone is made of concrete having a specific weight of 150 lb>ft3 . Determine the average normal stress acting in the pedestal at its base. Hint: The volume of a cone of radius r and height h is V = 13pr2h .

*1–68. The pedestal in the shape of a frustum of a cone is made of concrete having a specific weight of 150 lb>ft3. Determine the average normal stress acting in the pedestal at its midheight, z = 4 ft . Hint: The volume of a cone of radius r and height h is V = 13pr2h.

8 ft

z ! 4 ft

1 ft

1.5 ft

Probs. 1–67/68

1–62. The crimping tool is used to crimp the end of the wire E . If a force of 20 lb is applied to the handles, determine the average shear stress in the pin at A . The pin is subjected to double shear and has a diameter of 0.2 in. Only a vertical force is exerted on the wire.

1–63. Solve Prob. 1–62 for pin B . The pin is subjected to double shear and has a diameter of 0.2 in.

20 lb

5 in. 1.5 in. 2 in. 1 in.

E C

B D

Probs. 1–62/63

*1–64. A vertical force of P = 1500 N is applied to the bell crank. Determine the average normal stress developed in the 10-mm diamater rod CD , and the average shear stress developed in the 6-mm diameter pin B that is subjected to double shear.

1–65. Determine the maximum vertical force P that can be applied to the bell crank so that the average normal stress developed in the 10-mm diameter rod CD, and the average shear strees developed in the 6-mm diameter double sheared pin B not exceed 175 MPa and 75 MPa respectively.

450 mm

300 mm 45!

Probs. 1–64/65

46 CHAPTER 1 STRESS

1 1.6 Allowable Stress Design To ensure the safety of a structural member or mechanical element, it is necessary to restrict the applied load to one that is less than the load the member (or element) can fully support. There are many reasons for doing this. For example, the intended measurements of a structure or machine may not be exact, due to errors in fabrication or in the assembly of its component parts. Unknown vibrations, impact, or accidental loadings can occur that may not be accounted for in the design. Atmospheric corrosion, decay, or weathering tend to cause materials to deteriorate during service. And finally, some materials, such as wood, concrete, or fiber-reinforced composites, can show high variability in mechanical properties.

One method of specifying the allowable load for a member is to use a number called the factor of safety. The factor of safety (F.S.) is a ratio of the failure load Ffail to the allowable load Fallow. Here Ffail is found from experimental testing of the material, and the factor of safety is selected based on experience so that all the above mentioned uncertainties are accounted for when the member is used under similar conditions of loading and geometry. Stated mathematically,

F.S. = Ffail

Fallow (1–8)

If the load applied to the member is linearly related to the stress developed within the member, as in the case of using s = P>A and tavg = V>A, then we can also express the factor of safety as a ratio of the failure stress sfail (or tfail ) to the allowable stress sallow (or tallow ). * Here the area A will cancel and so,

F.S. = sfail sallow

(1–9)

F.S. = tfail tallow

(1–10)

* In some cases, such as columns, the applied load is not linearly related to stress and therefore only Eq. 1–8 can be used to determine the factor of safety. See Chapter 13.

Cranes are often supported using bearing pads to give them stability. Care must be taken not to crush the supporting surface, due to the large bearing stress developed between the pad and the surface.

1.6 ALLOWABLE STRESS DESIGN 47

1 In any of these equations, the factor of safety must be greater than 1 in

order to avoid the potential for failure. Specific values depend on the types of materials to be used and the intended purpose of the structure or machine. For example, the F.S. used in the design of aircraft or space- vehicle components may be close to 1 in order to reduce the weight of the vehicle. Or, in the case of a nuclear power plant, the factor of safety for some of its components may be as high as 3 due to uncertainties in loading or material behavior. In many cases, the factor of safety or the allowable stress for a specific case can be found in design codes and engineering handbooks. Once it is obtained, a design that is based on an allowable stress limit is called allowable stress design (ASD). This method is intended to ensure a balance between both public and environmental safety on the one hand and economic considerations on the other.

Simple Connections. By making simplifying assumptions regarding the behavior of the material, the equations s = P>A and tavg = V>A can often be used to analyze or design a simple connection or mechanical element. In particular, if a member is subjected to normal force at a section, its required area at the section is determined from

A = P

sallow (1–11)

On the other hand, if the section is subjected to an average shear force , then the required area at the section is

A = V

tallow (1–12)

As discussed in Sec. 1.6 , the allowable stress used in each of these equations is determined either by applying a factor of safety to the material’s normal or shear failure stress or by finding these stresses directly from an appropriate design code.

Three examples of where the above equations apply are shown in Fig. 1–24 .

V ! P

Assumed uniform shear stress

A !

tallow

P tallow

("b)allow

Assumed uniform normal stress distribution

A ! P

("b)allow

The area of the column base plate B is determined from the allowable bearing stress for the concrete.

Assumed uniform shear stress

tallow

l ! —————P tallowpd

The embedded length l of this rod in concrete can be determined using the allowable shear

stress of the bonding glue.

The area of the bolt for this lap joint is determined from the shear stress which is largest between the plates. Fig. 1–24

48 CHAPTER 1 STRESS

1 1.7 Limit State Design We have stated in the previous section that a properly designed member must account for uncertainties resulting from the variability of both the material’s properties and the loading that must be supported. Since uncertainty can be considered using statistics and probability theory, in structural engineering there has been an increasing trend to separate load uncertainty from material uncertainty.* This method of design is called limit state design (LSD), or more specifically, in the United States it is called load and resistance factor design (LRFD). We will now discuss how this method is applied. Load Factors. Various types of loads R can act on a structure or structural member, and each is multiplied by a load factor g (gamma) that accounts for its variability. The loads include dead load , which is the fixed weight of the structure, and live loads , which involve people or vehicles that move about. Other types of live loads include wind, earthquake, and snow loads. The dead load D is multiplied by a small factor such as g

D = 1.4 ,

since it can be determined with greater certainty than, for example, the live load L caused by people. It can have a load factor of gL = 1.6 .

Building codes often require a structure to be designed to support various combinations of the loads, and when applied in combination, each type of load will have a unique load factor. For example, one load combination of dead (D), live (L), and snow (S) loads is

R = 1.2D + 1.6L + 0.5S The load factors for this combined loading reflect the probability that the total loading R will occur for all the events stated. Here, however, the load factor gS = 0.5 is small because of the low probability that a maximum snow load will occur simultaneously with the maximum dead and live loads. Resistance Factors. Resistance factors f (phi) are determined from the probability of material failure as it relates to the material’s quality and the consistency of its strength. These factors will differ for different types of materials. For example, concrete has smaller factors than steel because engineers have more confidence about the behavior of steel under load than they do about concrete. For example, a typical resistance factor f = 0.9 is used for a steel member in tension. Design Criteria. Once the load and resistance factors g and f have been specified using a code, then proper design of a structural member requires that its strength be greater than the load it supports. Thus, the LRFD criterion can be stated as fPn Ú !gi Ri (1–13) Here Pn is the nominal strength of the member, meaning the load, when applied to the member, causes it either to fail (ultimate load), or deform to a state where it is no longer serviceable or becomes unsuitable for its intended purpose. In summary then, the resistance factor f reduces the nominal strength of the member and requires it to be equal to or greater than the applied load or combination of loads calculated using the load factors g. * ASD combines these uncertainties by using the factor of safety or defining the allowable stress.

1.7 LIMIT STATE DESIGN 49

1 Important Point

• Design of a member for strength is based on selecting either an allowable stress or a factor of safety that will enable it to safely support its intended load (ASD), or using load and resistance factors to modify the strength of the material and the load, respectively (LRFD).

Procedure for Analysis

When solving problems using the average normal and shear stress equations, a careful consideration should first be made as to choosing the section over which the critical stress is acting. Once this section is determined, the member must then be designed to have a sufficient area at the section to resist the stress that acts on it. This area is determined using the following steps.

Internal Loading. • Section the member through the area and draw a free-body

diagram of a segment of the member. The internal resultant force at the section is then determined using the equations of equilibrium.

Required Area. • Provided either the allowable stress or the load and resistance

factors are known or can be determined, then the required area needed to sustain the calculated load or factored load at the section is determined from A = P>s or A = V >t.

Appropriate factors of safety must be considered when d esigning cranes and cables used to transfer heavy loads.

50 CHAPTER 1 STRESS

1 EXAMPLE 1.12

The control arm is subjected to the loading shown in Fig. 1–25 a . Determine to the nearest 14 in. the required diameters of the steel pins at A and C if the allowable shear stress for the steel is tallow = 8 ksi.

(a)

3 5

42 in.3 in.

8 in.

Double shear

3 kip 5 kip

Single shear

3 5

42 in.3 in.

8 in.

3 kip 5 kip

FAB

Cy (b)

Fig. 1–25

SOLUTION

Pin Forces. A free-body diagram of the arm is shown in Fig. 1–25 b . For equilibrium we have

a+ !MC = 0; FAB18 in.2 - 3 kip 13 in.2 - 5 kip 1352 15 in.2 = 0 FAB = 3 kip

S+ !Fx = 0; -3 kip - Cx + 5 kip 1452 = 0 Cx = 1 kip + c !Fy = 0; Cy - 3 kip - 5 kip1352 = 0 Cy = 6 kip The pin at C resists the resultant force at C, which is

FC = 211 kip22 + 16 kip22 = 6.083 kip Pin A. This pin is subjected to single shear, Fig. 1–25c, so that

A = V

tallow ; pa dA

2 b2 = 3 kip

8 kip >in2; dA = 0.691 in. Use dA = 34 in. Ans.

Pin C. Since this pin is subjected to double shear, a shear force of 3.041 kip acts over its cross-sectional area between the arm and each supporting leaf for the pin, Fig. 1–25 d . We have

A = V

tallow ; pa dC

2 b2 = 3.041 kip

8 kip>in2 ; dC = 0.696 in. Use dC = 34 in. Ans.

(d)

3.041 kip

6.082 kip

Pin at C

(c)

3 kip

3 kip Pin at A

1.7 LIMIT STATE DESIGN 51

1 EXAMPLE 1.13

The suspender rod is supported at its end by a fixed-connected circular disk as shown in Fig. 1–26 a . If the rod passes through a 40-mm-diameter hole, determine the minimum required diameter of the rod and the minimum thickness of the disk needed to support the 20-kN load. The allowable normal stress for the rod is sallow = 60 MPa, and the allowable shear stress for the disk is tallow = 35 MPa.

20 kN

(a)

40 mm

20 kN

A allow

(b)

40 mm

Fig. 1–26

SOLUTION Diameter of Rod. By inspection, the axial force in the rod is 20 kN. Thus the required cross-sectional area of the rod is

A = P

sallow ;

4 d2 =

2011032 N 6011062 N>m2

so that

d = 0.0206 m = 20.6 mm Ans.

Thickness of Disk. As shown on the free-body diagram in Fig. 1–26 b , the material at the sectioned area of the disk must resist shear stress to prevent movement of the disk through the hole. If this shear stress is assumed to be uniformly distributed over the sectioned area, then, since V = 20 kN, we have

A = V

tallow ; 2p10.02 m2 1t2 = 2011032 N

3511062 N>m2 t = 4.55(10-32m = 4.55 mm Ans.

52 CHAPTER 1 STRESS

The shaft shown in Fig. 1–27 a is supported by the collar at C , which is attached to the shaft and located on the right side of the bearing at B . Determine the largest value of P for the axial forces at E and F so that the bearing stress on the collar does not exceed an allowable stress of 1sb2allow = 75 MPa and the average normal stress in the shaft does not exceed an allowable stress of 1st2allow = 55 MPa.

EXAMPLE 1.14

80 mm

60 mm P

F P2

(a)

20 mm

P 2P

F E C

(b)

(c)

Axial Force

Position

F E C

Fig. 1–27 SOLUTION To solve the problem we will determine P for each possible failure condition. Then we will choose the smallest value. Why? Normal Stress. Using the method of sections, the axial load within region FE of the shaft is 2 P , whereas the largest axial force, 3 P , occurs within region EC , Fig. 1–27 b . The variation of the internal loading is clearly shown on the normal-force diagram, Fig. 1–27 c . Since the cross- sectional area of the entire shaft is constant, region EC is subjected to the maximum average normal stress. Applying Eq. 1–11 , we have

A = P

sallow ; p10.03 m22 = 3P

5511062N>m2 P = 51.8 kN Ans. Bearing Stress. As shown on the free-body diagram in Fig. 1–27 d , the collar at C must resist the load of 3 P , which acts over a bearing area of A b = [p10.04 m22 - p10.03 m22] = 2.199110-32m2. Thus,

A = P

sallow ; 2.199110 -32m2 = 3P

7511062N>m2 P = 55.0 kN

By comparison, the largest load that can be applied to the shaft is P = 51.8 kN, since any load larger than this will cause the allowable normal stress in the shaft to be exceeded. NOTE: Here we have not considered a possible shear failure of the collar as in Example 1.13 .

(d) C

1.7 LIMIT STATE DESIGN 53

The 400-kg uniform bar AB shown in Fig. 1–28a is supported by a steel rod AC and a roller at B. If it supports a live distributed loading of 3 kN>m, determine the required diameter of the rod. The failure stress for the steel is sfail = 345 MPa. Use the LRFD method, where the resistance factor for tension is f = 0.9 and the load factors for the dead and live loads are gD = 1.2 and gL = 1.6, respectively.

EXAMPLE 1.15

2 m

3 kN/m

(a)

SOLUTION Factored Loads. Here the dead load is the bar’s weight D = 400(9.81) N = 3.924 kN. Therefore, the factored dead load is 1.2D = 4.709 kN. The live load resultant is L = (3 kN>m)(2 m) = 6 kN, so that the factored live load is 1.6L = 9.60 kN.

From the free-body diagram of the bar, Fig. 1–28b, the factored load in the rod can now be determined. a+ !MB = 0; 9.60 kN(1 m) + 4.709 kN(1 m) - FAC (2 m) = 0

FAC = 7.154 kN

Area. The nominal strength of the rod is determined from Pn = sfail A , and since the nominal strength is defined by the resistance factor f = 0.9, we require

fPn Ú FAC; 0.9[345(106) N>m2] A AC = 7.154(103) N A AC = 23.04(10-6) m2 = 23.04 mm2 =

4 dAC

dAC = 5.42 mm Ans.

Fig. 1–28

(b)

1 m 1 m

9.60 kN/m

4.709 kN/m

FAC

54 CHAPTER 1 STRESS

F1–13. Rods AC and BC are used to suspend the 200-kg mass. If each rod is made of a material for which the average normal stress can not exceed 150 MPa, determine the minimum required diameter of each rod to the nearest mm.

A B

60!60!

F1–13

F1–14. The frame supports the loading shown. The pin at A has a diameter of 0.25 in. If it is subjected to double shear, determine the average shear stress in the pin.

3 ft

2 ft

A E C

2 ft

600 lb

F1–14

F1–15. Determine the maximum average shear stress developed in each 3>4@in. -diameter bolt.

10 kip

5 kip

F1–15

F1–16. If each of the three nails has a diameter of 4 mm and can withstand an average shear stress of 60 MPa, determine the maximum allowable force P that can be applied to the board.

F1–16

F1–17. The strut is glued to the horizontal member at surface AB . If the strut has a thickness of 25 mm and the glue can withstand an average shear stress of 600 kPa, determine the maximum force P that can be applied to the strut.

50 mm

A B

60!

F1–17

F1–18. Determine the maximum average shear stress developed in the 30-mm-diameter pin.

30 kN

40 kN

F1–18

FUNDAMENTAL PROBLEMS

1.7 LIMIT STATE DESIGN 55

1 F1–19. If the eyebolt is made of a material having a yield stress of sY = 250 MPa, determine the minimum required diameter d of its shank. Apply a factor of safety F.S. = 1.5 against yielding.

30 kN d

F1–19

F1–20. If the bar assembly is made of a material having a yield stress of sY = 50 ksi, determine the minimum required dimensions h1 and h2 to the nearest 1>8 in. Apply a factor of safety F.S. = 1.5 against yielding. Each bar has a thickness of 0.5 in.

15 kip

30 kip h1

F1–20

F1–21. Determine the maximum force P that can be applied to the rod if it is made of material having a yield stress of sY = 250 MPa. Consider the possibility that failure occurs in the rod and at section a–a. Apply a factor of safety of F.S. = 2 against yielding.

50 mm

60 mm120 mm

P 40 mm

Section a-a

F1–21

F1–22. The pin is made of a material having a failure shear stress of tfail = 100 MPa . Determine the minimum required diameter of the pin to the nearest mm. Apply a factor of safety of F.S. = 2.5 against shear failure.

80 kN

F1–22

F1–23. If the bolt head and the supporting bracket are made of the same material having a failure shear stress of tfail = 120 MPa , determine the maximum allowable force P that can be applied to the bolt so that it does not pull through the plate. Apply a factor of safety of F.S. = 2.5 against shear failure.

40 mm

75 mm

80 mm

30 mm

P F1–23

F1–24. Six nails are used to hold the hanger at A against the column. Determine the minimum required diameter of each nail to the nearest 1>16 in. if it is made of material having tfail = 16 ksi . Apply a factor of safety of F.S. = 2 against shear failure.

A B

9 ft

300 lb/ft

F1–24

56 CHAPTER 1 STRESS

*1–72. The truss is used to support the loading shown Determine the required cross-sectional area of member BC if the allowable normal strees is sallow = 24 ksi.

800 lb

F 400 lb

6 ft 6 ft

45!

30!

60!

Prob. 1–72

1–73. The steel swivel bushing in the elevator control of an airplane is held in place using a nut and washer as shown in Fig. ( a ). Failure of the washer A can cause the push rod to separate as shown in Fig. ( b ). If the maximum average normal stress for the washer is smax = 60 ksi and the maximum average shear stress is tmax = 21 ksi, determine the force F that must be applied to the bushing that will cause this to happen. The washer is 116 in. thick.

(a) (b) A

0.75 in.

Prob. 1–73

1–69. Member B is subjected to a compressive force of 800 lb. If A and B are both made of wood and are 38 in. thick, determine to the nearest 14 in. the smallest dimension h of the horizontal segment so that it does not fail in shear. The average shear stress for the segment is tallow = 300 psi.

800 lbB

h A

5 13

Prob. 1–69

1–70. The lever is attached to the shaft A using a key that has a width d and length of 25 mm. If the shaft is fixed and a vertical force of 200 N is applied perpendicular to the handle, determine the dimension d if the allowable shear stress for the key is tallow = 35 MPa.

500 mm

20 mm

d aa

200 N

Prob. 1–70

1–71. The joint is fastened together using two bolts. Determine the required diameter of the bolts if the failure shear stress for the bolts is tfail = 350 MPa. Use a factor of safety for shear of F.S. = 2.5.

80 kN

40 kN

30 mm

40 kN

Prob. 1–71

PROBLEMS

1.7 LIMIT STATE DESIGN 57

1 1–77. The tension member is fastened together using two bolts, one on each side of the member as shown. Each bolt has a diameter of 0.3 in. Determine the maximum load P that can be applied to the member if the allowable shear stress for the bolts is tallow = 12 ksi and the allowable average normal stress is sallow = 20 ksi .

60!

Prob. 1–77

1–78. The 50-kg flowerpot is suspended from wires AB and BC . If the wires have a normal failure stress of sfail = 350 MPa, determine the minimum diameter of each wire. Use a factor of safety of 2.5.

1–79. The 50-kg flowerpot is suspended from wires AB and BC which have diameters of 1.5 mm and 2 mm, respectively. If the wires have a normal failure stress of sfail = 350 MPa, determine the factor of safety of each wire.

30!

45!

Probs. 1–78/79

1–74. Member B is subjected to a compressive force of 600 lb. If A and B are both made of wood and are 1.5 in. thick, determine to the nearest 18 in. the smallest dimension a of the support so that the average shear stress along the blue line does not exceed tallow = 50 psi . Neglect friction.

600 lb

B4 53

Prob. 1–74

1–75. The hangers support the joist uniformly, so that it is assumed the four nails on each hanger carry an equal portion of the load. If the joist is subjected to the loading shown, determine the average shear stress in each nail of the hanger at ends A and B . Each nail has a diameter of 0.25 in. The hangers only support vertical loads.

*1–76. The hangers support the joists uniformly, so that it is assumed the four nails on each hanger carry an equal portion of the load. Determine the smallest diameter of the nails at A and at B i f the allowable stress for the nails is tallow = 4 ksi. The hangers only support vertical loads.

18 ft A B

30 lb/ft 40 lb/ft

Probs. 1–75/76

58 CHAPTER 1 STRESS

1 *1–80. The thrust bearing consists of a circular collar A fixed to the shaft B . Determine the maximum axial force P that can be applied to the shaft so that it does not cause the shear stress along a cylindrical surface a or b to exceed an allowable shear stress of tallow = 170 MPa.

20 mm

30 mm 58 mm

b 35 mm

B A

a P

Prob. 1–80

1–81. The steel pipe is supported on the circular base plate and concrete pedestal. If the normal failure stress for the steel is (s fail ) st = 350 MPa, determine the minimum thickness t of the pipe if it supports the force of 500 kN. Use a factor of safety against failure of 1.5. Also, find the minimum radius r of the base plate so that the minimum factor of safety against failure of the concrete due to bearing is 2.5. The failure bearing stress for concrete is (s fail ) con = 25 MPa.

100 mm

500 kN

Prob. 1–81

1–82. The steel pipe is supported on the circular base plate and concrete pedestal. If the thickness of the pipe is t = 5 mm and the base plate has a radius of 150 mm, determine the factors of safety against failure of the steel and concrete. The applied force is 500 kN, and the normal failure stresses for steel and concrete are (s fail ) st = 350 MPa and (s fail ) con = 25 MPa, respectively.

100 mm

500 kN

Prob. 1–82

1–83. The 60 mm * 60 mm oak post is supported on the pine block. If the allowable bearing stresses for these materials are soak = 43 MPa and spine = 25 MPa , determine the greatest load P that can be supported. If a rigid bearing plate is used between these materials, determine its required area so that the maximum load P can be supported. What is this load?

Prob. 1–83

1.7 LIMIT STATE DESIGN 59

1 1–86. The two aluminum rods support the vertical force of P = 20 kN. Determine their required diameters if the allowable tensile stress for the aluminum is s allow = 150 MPa.

AC 45!

Prob. 1–86

1–87. The two aluminum rods AB and AC have diameters of 10 mm and 8 mm, respectively. Determine the largest vertical force P that can be supported. The allowable tensile stress for the aluminum is s allow = 150 MPa.

AC 45!

Prob. 1–87

*1–84. The frame is subjected to the load of 4 kN which acts on member ABD at D . Determine the required diameter of the pins at D and C if the allowable shear stress for the material is tallow = 40 MPa. Pin C is subjected to double shear, whereas pin D is subjected to single shear.

1.5 m

4 kN

45! 1.5 m1 m

1.5 m

DC E

Prob. 1–84

1–85. The beam is made from southern pine and is supported by base plates resting on brick work. If the allowable bearing stresses for the materials are (spine)allow = 2.81 ksi and (sbrick)allow = 6.70 ksi, determine the required length of the base plates at A and B to the nearest 14 inch in order to support the load shown. The plates are 3 in. wide.

6 kip 200 lb/ft

5 ft 3 ft5 ft

B A

Prob. 1–85

60 CHAPTER 1 STRESS

1 *1–88. The compound wooden beam is connected together by a bolt at B . Assuming that the connections at A , B , C , and D exert only vertical forces on the beam, determine the required diameter of the bolt at B and the required outer diameter of its washers if the allowable tensile stress for the bolt is 1st2allow = 150 MPa and the allowable bearing stress for the wood is 1sb2allow = 28 MPa. Assume that the hole in the washers has the same diameter as the bolt.

1.5 m1.5 m1.5 m1.5 m2 m2 m

C D A

3 kN 1.5 kN 2 kN

Prob. 1–88

1–89. Determine the required minimum thickness t of member AB and edge distance b of the frame if P = 9 kip and the factor of safety against failure is 2. The wood has a normal failure stress of s fail = 6 ksi, and shear failure stress of t fail = 1.5 ksi.

At 3 in. 3 in.

30!

Prob. 1–89

1–90. Determine the maximum allowable load P that can be safely supported by the frame if t = 1.25 in. and b = 3.5 in. The wood has a normal failure stress of s fail = 6 ksi, and shear failure stress of t fail = 1.5 ksi. Use a factor of safety against failure of 2.

At 3 in. 3 in.

30!

Prob. 1–90

1–91. If the allowable bearing stress for the material under the supports at A and B is 1sb2 allow = 1.5 MPa, determine the size of square bearing plates A! and B! required to support the load. Dimension the plates to the nearest mm. The reactions at the supports are vertical. Take P = 100 kN.

3 m

A¿ B¿A B

40 kN/m

1.5 m 1.5 m

Prob. 1–91

*1–92. If the allowable bearing stress for the material under the supports at A and B is 1sb2allow = 1.5 MPa, determine the maximum load P that can be applied to the beam. The bearing plates A! and B! have square cross sections of 150 mm * 150 mm and 250 mm * 250 mm, respectively.

3 m

A¿ B¿A B

40 kN/m

1.5 m 1.5 m

Prob. 1–92

1.7 LIMIT STATE DESIGN 61

1 1–95. The pin support A and roller support B of the bridge truss are supported on concrete abutments. If the bearing failure stress of the concrete is (s fail ) b = 4 ksi, determine the required minimum dimension of the square bearing plates at C and D to the nearest 116 in. Apply a factor of safety of 2 against failure.

C D

6 ft

150 kip 100 kip

200 kip 300 kip300 kip300 kip 300 kip

6 ft 6 ft 6 ft 6 ft 6 ft

Prob. 1–95

*1–96. The pin support A and roller support B of the bridge truss are supported on the concrete abutments. If the square bearing plates at C and D are 21 in. × 21 in., and the bearing failure stress for concrete is (s fail ) b = 4 ksi, determine the factor of safety against bearing failure for the concrete under each plate.

C D

6 ft

150 kip 100 kip

200 kip 300 kip300 kip300 kip 300 kip

6 ft 6 ft 6 ft 6 ft 6 ft

Prob. 1–96

1–93. The rods AB and CD are made of steel. Determine their smallest diameter so that they can support the dead loads shown. The beam is assumed to be pin connected at A and C . Use the LRFD method, where the resistance factor for steel in tension is f = 0.9, and the dead load factor is gD = 1.4. The failure stress is sfail = 345 MPa.

4 kN

6 kN 5 kN

3 m2 m2 m 3 m

Prob. 1–93

1–94. The aluminum bracket A is used to support the centrally applied load of 8 kip. If it has a constant thickness of 0.5 in., determine the smallest height h in order to prevent a shear failure. The failure shear stress is tfail = 23 ksi. Use a factor of safety for shear of F.S. = 2.5.

8 kip

Prob. 1–94

62 CHAPTER 1 STRESS

CHAPTER REVIEW

The internal loadings in a body consist of a normal force, shear force, bending moment, and torsional moment. They represent the resultants of both a normal and shear stress distribution that acts over the cross section. To obtain these resultants, use the method of sections and the equations of equilibrium.

!Fx = 0 !Fy = 0 !Fz = 0 !Mx = 0 !My = 0 !Mz = 0

F1 F2

M V

Torsional Moment

Bending Moment

Shear Force

Normal Force

If a bar is made from homogeneous isotropic material and it is subjected to a series of external axial loads that pass through the centroid of the cross section, then a uniform normal stress distribution will act over the cross section. This average normal stress can be determined from s = P>A, where P is the internal axial load at the section.

s = P A

V V

tavg ! V A

The average shear stress can be determined using tavg = V >A , where V is the shear force acting on the cross- sectional area A . This formula is often used to find the average shear stress in fasteners or in parts used for connections.

tavg = V A

s s ! P A

The ASD method of design of any simple connection requires that the average stress along any cross section not exceed an allowable stress of sallow or tallow. These values are reported in codes and are considered safe on the basis of experiments or through experience. Sometimes a factor of safety is reported provided the ultimate stress is known.

F.S. = sfail

sallow =

tfail tallow

The LRFD method of design is used for the design of structural members. It modifies the load and the strength of the material separately, using load and resistance factors.

fPn Ú !giRi

1 CONCEPTUAL PROBLEMS

C1–1. Here hurricane winds caused the failure of this highway sign. Assuming the wind creates a uniform pressure on the sign of 2 kPa, use reasonable dimensions for the sign and determine the resultant shear and moment at the two connections where the failure occurred.

C1–1

C1–2. High-heel shoes can often do damage to soft wood or linoleum floors. Using reasonable dimensions for the heel of a regular shoe and a high-heel shoe, determine the bearing stress under each heel if the weight is transferred down only to the heel of one shoe.

C1–2

C1–3. Here is an example of the single shear failure of a bolt. Using appropriate free-body diagrams, explain why the bolt failed along the section between the plates, and not along some intermediate section such as a–a.

C1–3

C1–4. The vertical load on the hook is 1000 lb. Draw the appropriate free-body diagrams and determine the maximum average shear force on the pins at A, B, and C. Note that due to symmetry four wheels are used to support the loading on the railing.

B C

C1–4

CONCEPTUAL PROBLEMS 63

64 CHAPTER 1 STRESS

1 REVIEW PROBLEMS

1–97. The beam AB is pin supported at A and supported by a cable BC . A separate cable CG is used to hold up the frame. If AB weighs 120 lb>ft and the column FC has a weight of 180 lb>ft, determine the resultant internal loadings acting on cross sections located at points D and E . Neglect the thickness of both the beam and column in the calculation.

4 ft

12 ft

4 ft

8 ft 12 ft

6 ft

Prob. 1–97

1–98. The long bolt passes through the 30-mm-thick plate. If the force in the bolt shank is 8 kN, determine the average normal stress in the shank, the average shear stress along the cylindrical area of the plate defined by the section lines a – a , and the average shear stress in the bolt head along the cylindrical area defined by the section lines b – b .

8 kN 18 mm

7 mm

30 mm

8 mm a

Prob. 1–98

1–99. To the nearest 116 in., determine the required thickness of member BC and the diameter of the pins at A and B if the allowable normal stress for member BC is s allow = 29 ksi and the allowable shear stress for the pins is s allow = 10 ksi.

60! 8 ftB A

2 kip/ft

1.5 in.

Prob. 1–99

*1–100. The circular punch B exerts a force of 2 kN on the top of the plate A . Determine the average shear stress in the plate due to this loading.

2 kN

2 mm

4 mm

Prob. 1–100

1 1–103. The yoke-and-rod connection is subjected to a tensile force of 5 kN. Determine the average normal stress in each rod and the average shear stress in the pin A between the members.

25 mm

40 mm

30 mm

5 kN

Prob. 1–103

*1–104. The cable has a specific weight g (weight/volume) and cross-sectional area A . If the sag s is small, so that its length is approximately L and its weight can be distributed uniformly along the horizontal axis, determine the average normal stress in the cable at its lowest point C .

L/2 L/2 C

A B

Prob. 1–104

1–101. Determine the average punching shear stress the circular shaft creates in the metal plate through section AC and BD . Also, what is the bearing stress developed on the surface of the plate under the shaft?

50 mm

10 mm

40 kN

60 mm

120 mm

C D

Prob. 1–101

1–102. The bearing pad consists of a 150 mm by 150 mm block of aluminum that supports a compressive load of 6 kN. Determine the average normal and shear stress acting on the plane through section a – a . Show the results on a differential volume element located on the plane.

30!

150 mm

6 kN

Prob. 1–102

REVIEW PROBLEMS 65

Chapter 2

This specimen exhibits noticeable strain before it fractured. Measurement of this strain is necessary so that the stress in the material can be determined in

cases of complicated loadings.

CHAPTER OBJECTIVES

■ In engineering the deformation of a body is specified using the concepts of normal and shear strain. In this chapter we will define these quantities and show how they can be determined for various types of problems.

2.1 Deformation Whenever a force is applied to a body, it will tend to change the body’s shape and size. These changes are referred to as deformation , and they may be either highly visible or practically unnoticeable. For example, a rubber band will undergo a very large deformation when stretched, whereas only slight deformations of structural members occur when a building is occupied by people walking about. Deformation of a body can also occur when the temperature of the body is changed. A typical example is the thermal expansion or contraction of a roof caused by the weather.

In a general sense, the deformation of a body will not be uniform throughout its volume, and so the change in geometry of any line segment within the body may vary substantially along its length. Hence, to study deformational changes in a more uniform manner, we will consider line segments that are very short and located in the neighborhood of a point. Realize, however, that these changes will also depend on the orientation of the line segment at the point. For example, a line segment may elongate if it is oriented in one direction, whereas it may contract if it is oriented in another direction.

Strain

Note the before and after positions of three different line segments on this rubber membrane which is subjected to tension. The vertical line is lengthened, the horizontal line is shortened, and the inclined line changes its length and rotates.

68 CHAPTER 2 STRAIN

2.2 Strain In order to describe the deformation of a body by changes in length of line segments and the changes in the angles between them, we will develop the concept of strain. Strain is actually measured by experiments, and once the strain is obtained, it will be shown in the next chapter how it can be related to the stress acting within the body.

Normal Strain. If we define the normal strain as the change in length of a line per unit length, then we will not have to specify the actual length of any particular line segment. Consider, for example, the line AB , which is contained within the undeformed body shown in Fig. 2–1 a . This line lies along the n axis and has an original length of !s. After deformation, points A and B are displaced to A" and B", and the line becomes a curve having a length of !s", Fig. 2–1 b . The change in length of the line is therefore !s" - !s. If we define the average normal strain using the symbol P avg (epsilon), then

Pavg = !s"- !s

!s (2–1)

As point B is chosen closer and closer to point A , the length of the line will become shorter and shorter, such that !s S 0. Also, this causes B " to approach A ", such that !s" S 0. Consequently, in the limit the normal strain at point A and in the direction of n is

P = lim BSA along n

!s" - !s

!s (2–2)

Hence, when P (or Pavg ) is positive the initial line will elongate, whereas if P is negative the line contracts.

Note that normal strain is a dimensionless quantity , since it is a ratio of two lengths. Although this is the case, it is sometimes stated in terms of a ratio of length units. If the SI system is used, then the basic unit for length is the meter (m). Ordinarily, for most engineering applications P will be very small, so measurements of strain are in micrometers per meter (mm>m) , where 1 mm = 10-6 m . In the Foot-Pound-Second

Undeformed body

!s A

(a)

Deformed body

A¿

B¿

!s¿

(b)

Fig. 2–1

2.2 STRAIN 69

system, strain is often stated in units of inches per inch ( in.>in. ). Sometimes for experimental work, strain is expressed as a percent (e.g., 0.001 m>m = 0.1%). As an example, a normal strain of 480(10-6) can be reported as 480(10-6) in.>in. , 480 mm>m, or 0.0480%. Also, one can state this answer as simply 480 m (480 “micros”).

Shear Strain. Deformations not only cause line segments to elongate or contract, but they also cause them to change direction. If we select two line segments that are originally perpendicular to one another, then the change in angle that occurs between them is referred to as shear strain . This angle is denoted by g (gamma) and is always measured in radians (rad), which are dimensionless. For example, consider the line segments AB and AC originating from the same point A in a body, and directed along the perpendicular n and t axes, Fig. 2–2 a . After deformation, the ends of both lines are displaced, and the lines themselves become curves, such that the angle between them at A is u!, Fig. 2–2 b . Hence the shear strain at point A associated with the n and t axes becomes

gnt = p

2 - lim u!

(2–3)

Notice that if u! is smaller than p>2 the shear strain is positive, whereas if u! is larger than p>2 the shear strain is negative.

Undeformed body

n t

B C

(a)

Deformed body

B¿

A¿

C¿

(b)

u¿

Fig. 2–2

B S A along n C S A along t

70 CHAPTER 2 STRAIN

Cartesian Strain Components. Using the definitions of normal and shear strain, we will now show how these components can be used to describe the deformation of the body in Fig. 2–3 a . To do so, imagine the body is subdivided into small elements such as the one shown in Figs. 2–3a and 2–3 b . This element is rectangular, has undeformed dimensions !x, !y, and !z, and is located in the neighborhood of a point in the body, Fig. 2–3 a . If the element’s dimensions are very small, then its deformed shape will be a parallelepiped, Fig. 2–3 c , since very small line segments will remain approximately straight after the body is deformed. In order to achieve this deformed shape, we will first consider how the normal strain changes the lengths of the sides of the rectangular element, and then how the shear strain changes the angles of each side. For example, !x elongates Px!x , so its new length is !x + Px!x . Therefore, the approximate lengths of the three sides of the parallelepiped are

(1 + Px) !x (1 + Py) !y (1 + Pz) !z

And the approximate angles between these sides are

2 - gxy

2 - gyz

2 - gxz

Notice that the normal strains cause a change in volume of the element, whereas the shear strains cause a change in its shape . Of course, both of these effects occur simultaneously during the deformation.

In summary, then, the state of strain at a point in a body requires specifying three normal strains, Px, Py , Pz , and three shear strains, gxy , gyz , gxz . These strains completely describe the deformation of a rectangular volume element of material located at the point and oriented so that its sides are originally parallel to the x , y , z axes. Provided these strains are defined at all points in the body, then the deformed shape of the body can be determined.

(a)

Fig. 2–3

(b)

!y !x

Undeformed element

2 p

(1 " Py)!y (1 " Px)!x

(1 " Pz)!z

Deformed element

(

(c)

# gyz)

( # gxz)

( # gxy) p

2.2 STRAIN 71

Small Strain Analysis. Most engineering design involves applications for which only small deformations are allowed. In this text, therefore, we will assume that the deformations that take place within a body are almost infinitesimal. In particular, the normal strains occurring within the material are very small compared to 1, so that P V 1 . This assumption has wide practical application in engineering, and it is often referred to as a small strain analysis. It can be used, for example, to approximate sin u = u, cos u = 1 , and tan u = u , provided u is very small .

The rubber bearing support under this concrete bridge girder is subjected to both normal and shear strain. The normal strain is caused by the weight and bridge loads on the girder, and the shear strain is caused by the horizontal movement of the girder due to temperature changes.

Important Points

• Loads will cause all material bodies to deform and, as a result, points in a body will undergo displacements or changes in position.

• Normal strain is a measure per unit length of the elongation or contraction of a small line segment in the body, whereas shear strain is a measure of the change in angle that occurs between two small line segments that are originally perpendicular to one another.

• The state of strain at a point is characterized by six strain components: three normal strains Px , Py , Pz and three shear strains gxy , gyz , gxz . These components depend upon the original orientation of the line segments and their location in the body.

• Strain is the geometrical quantity that is measured using experimental techniques. Once obtained, the stress in the body can then be determined from material property relations, as discussed in the next chapter .

• Most engineering materials undergo very small deformations, and so the normal strain P V 1. This assumption of “small strain analysis” allows the calculations for normal strain to be simplified, since first-order approximations can be made about their size.

72 CHAPTER 2 STRAIN

The slender rod shown in Fig. 2–4 is subjected to an increase of temperature along its axis, which creates a normal strain in the rod of Pz = 40(10-3)z1>2 , where z is measured in meters. Determine ( a ) the displacement of the end B of the rod due to the temperature increase, and ( b ) the average normal strain in the rod.

200 mm

Fig. 2–4

SOLUTION Part (a). Since the normal strain is reported at each point along the rod, a differential segment dz , located at position z , Fig. 2–4 , has a deformed length that can be determined from Eq. 2–1 ; that is,

dz! = dz + Pz dz dz ! = 31 + 40(10-3)z1>24dz

The sum of these segments along the axis yields the deformed length of the rod, i.e.,

z ! = L 0.2 m

0 31 + 40(10-3)z1>24dz

= 3z + 40(10-3)23 z3>24 ! 00.2 m = 0.20239 m

The displacement of the end of the rod is therefore

" B = 0.20239 m - 0.2 m = 0.00239 m = 2.39 mm T Ans.

Part (b). The average normal strain in the rod is determined from Eq. 2–1 , which assumes that the rod or “line segment” has an original length of 200 mm and a change in length of 2.39 mm. Hence,

Pavg = "s! - "s

"s =

2.39 mm 200 mm

= 0.0119 mm>mm Ans. This strain is called a thermal strain, caused by temperature, not by any load.

EXAMPLE 2.1

2.2 STRAIN 73

When force P is applied to the rigid lever arm ABC in Fig. 2–5 a , the arm rotates counterclockwise about pin A through an angle of 0.05°. Determine the normal strain developed in wire BD.

SOLUTION I Geometry. The orientation of the lever arm after it rotates about point A is shown in Fig. 2–5 b . From the geometry of this figure,

a = tan-1a 400 mm 300 mm

b = 53.1301! Then

f = 90! - a + 0.05! = 90! - 53.1301! + 0.05! = 36.92!

For triangle ABD the Pythagorean theorem gives

LAD = 2(300 mm)2 + (400 mm)2 = 500 mm Using this result and applying the law of cosines to triangle AB’D,

LB"D = 2L2AD + L2AB" - 2(LAD) (LAB") cos f = 2(500 mm)2 + (400 mm)2 - 2(500 mm) (400 mm) cos 36.92! = 300.3491 mm

Normal Strain.

PBD = LB"D - LBD

LBD

= 300.3491 mm - 300 mm

300 mm = 0.00116 mm>mm Ans.

SOLUTION II Since the strain is small, this same result can be obtained by approximating the elongation of wire BD as #LBD , shown in Fig. 2–5 b . Here,

#LBD = uLAB = c a 0.05!180! b (p rad) d (400 mm) = 0.3491 mm Therefore,

PBD = #LBD LBD

= 0.3491 mm 300 mm

= 0.00116 mm>mm Ans.

EXAMPLE 2.2

C 400 mm

(a)

P 300 mm

A B

B¿

C 400 mm

400 mm

(b)

P 300 mm

u ! 0.05"#LBD f

Fig. 2–5

74 CHAPTER 2 STRAIN

The plate shown in Fig. 2–6 a is fixed connected along AB and held in the horizontal guides at its top and bottom, AD and BC . If its right side CD is given a uniform horizontal displacement of 2 mm, determine ( a ) the average normal strain along the diagonal AC , and ( b ) the shear strain at E relative to the x , y axes.

SOLUTION Part (a). When the plate is deformed, the diagonal AC becomes AC!, Fig. 2–6 b . The lengths of diagonals AC and AC! can be found from the Pythagorean theorem. We have

AC = 210.150 m22 + 10.150 m22 = 0.21213 m AC! = 210.150 m22 + 10.152 m22 = 0.21355 m Therefore the average normal strain along the diagonal is

1PAC2avg = AC! - ACAC = 0.21355 m - 0.21213 m0.21213 m = 0.00669 mm>mm Ans.

Part (b). To find the shear strain at E relative to the x and y axes, it is first necessary to find the angle u! after deformation, Fig. 2–6 b . We have

tan a u! 2 b = 76 mm

75 mm

u! = 90.759" = a p 180"

b 190.759"2 = 1.58404 rad Applying Eq. 2–3 , the shear strain at E is therefore

gxy = p

2 - 1.58404 rad = -0.0132 rad Ans.

The negative sign indicates that the angle u! is greater than 90°.

NOTE: If the x and y axes were horizontal and vertical at point E , then the 90° angle between these axes would not change due to the deformation, and so gxy = 0 at point E .

EXAMPLE 2.3

76 mm

(b)

75 mm

E¿

D¿

B C¿

75 mm

76 mm

u¿

150 mm (a)

2 mm

150 mm

B C

Fig. 2–6

2.2 STRAIN 75

P2–1. A loading causes the member to deform into the dashed shape. Explain how to determine the normal strains PCD and PAB. The displacement ≤ and the lettered dimensions are known.

A C B

L 2 L

L L/2

(a)

PRELIMINARY PROBLEMS

P2–2. A loading causes the member to deform into the dashed shape. Explain how to determine the normal strains PCD and PAB. The displacement ≤ and the lettered dimensions are known.

B C

L2 L

L/2 !

(b)

A¿

C B

(c)

P2–4. A loading causes the block to deform into the dashed shape. Explain how to determine the strains PAB, PAC, PBC, (gA)xy. The angles and distances between all lettered points are known.

(d)

B¿

C¿

A D

P2-4

P2–5. A loading causes the block to deform into the dashed shape. Explain how to determine the strains (gA)xy, (gB)xy. The angles and distances between all lettered points are known.

(e)

P2-5

P2–3. A loading causes the wires to elongate into the dashed shape. Explain how to determine the normal strain PAB in wire AB. The displacement B and the distances between all lettered points are known.

P2-1

P2-2

P2-3

76 CHAPTER 2 STRAIN

F2–1. When force P is applied to the rigid arm ABC, point B displaces vertically downward through a distance of 0.2 mm. Determine the normal strain developed in wire CD .

A B C

400 mm 200 mm 300 mm

F2–1

F2–2. If the applied force P causes the rigid arm ABC to rotate clockwise about pin A through an angle of 0.02°, determine the normal strain developed in wires BD and CE .

A B C

600 mm 600 mm

600 mm

400 mm P

F2–2

F2–3. The rectangular plate is deformed into the shape of a parallelogram shown by the dashed line. Determine the average shear strain at corner A with respect to the x and y axes.

x A

D C

300 mm

2 mm

4 mm

400 mm

F2–3

FUNDAMENTAL PROBLEMS

F2–4. The triangular plate is deformed into the shape shown by the dashed line. Determine the normal strain developed along edge BC and the average shear strain at corner A with respect to the x and y axes.

300 mm

3 mm

5 mm 400 mm

F2–4

F2–5. The square plate is deformed into the shape shown by the dashed line. Determine the average normal strain along diagonal AC and the shear strain of point E with respect to the x and y axes.

y x

A B

D C

300 mm

300 mm 3 mm3 mm

4 mm

F2–5

2.2 STRAIN 77

PROBLEMS

2–1. An air-filled rubber ball has a diameter of 6 in. If the air pressure within it is increased until the ball’s diameter becomes 7 in., determine the average normal strain in the rubber.

2–2. A thin strip of rubber has an unstretched length of 15 in. If it is stretched around a pipe having an outer diameter of 5 in., determine the average normal strain in the strip.

2–3. The rigid beam is supported by a pin at A and wires BD and CE . If the load P on the beam causes the end C to be displaced 10 mm downward, determine the normal strain developed in wires CE and BD .

3 m

2 m

4 m

2 m

Prob. 2–3

*2–4. The force applied at the handle of the rigid lever causes the lever to rotate clockwise about the pin B through an angle of 2°. Determine the average normal strain developed in each wire. The wires are unstretched when the lever is in the horizontal position.

A B

G F

200 mm

200 mm 200 mm300 mm 300 mm

Prob. 2–4

2–5. The two wires are connected together at A . If the force P causes point A to be displaced horizontally 2 mm, determine the normal strain developed in each wire.

P 30!

30! A

300 mm

Prob. 2–5

2–6. The rubber band of unstretched length 2r 0 is forced down the frustum of the cone. Determine the average normal strain in the band as a function of z .

2r0

Prob. 2–6

78 CHAPTER 2 STRAIN

2–10. The corners of the square plate are given the displacements indicated. Determine the shear strain along the edges of the plate at A and B .

2–11. The corners of the square plate are given the displacements indicated. Determine the average normal strains along side AB and diagonals AC and DB.

0.3 in.

0.2 in.

0.3 in.

10 in.10 in.

10 in.

0.2 in.

Probs. 2–10/11

*2–12. The piece of rubber is originally rectangular. Determine the average shear strain gxy at A if the corners B and D are subjected to the displacements that cause the rubber to distort as shown by the dashed lines.

2–13. The piece of rubber is originally rectangular and subjected to the deformation shown by the dashed lines. Determine the average normal strain along the diagonal DB and side AD .

300 mm

400 mm

3 mm

2 mm B

Probs. 2–12/13

2–7. The pin-connected rigid rods AB and BC are inclined at u = 30° when they are unloaded. When the force P is applied u becomes 30.2°. Determine the average normal strain developed in wire AC .

C A

600 mm uu

Prob. 2–7

*2–8. Part of a control linkage for an airplane consists of a rigid member CBD and a flexible cable AB. If a force is applied to the end D of the member and causes it to rotate by u = 0.3°, determine the normal strain in the cable. Originally the cable is unstretched.

2–9. Part of a control linkage for an airplane consists of a rigid member CBD and a flexible cable AB. If a force is applied to the end D of the member and causes a normal strain in the cable of 0.0035 mm>mm, determine the displacement of point D. Originally the cable is unstretched.

400 mm

300 mm

D P

300 mm

Probs. 2–8/9

2.2 STRAIN 79

2–17. The plate is deformed uniformly into the shape shown by the dashed lines. If at A , gxy = 0.0075 rad., while P AB = P AF = 0, determine the average shear strain at point G with respect to the x ¿ and y ¿ axes.

y y¿

x¿

F E

G B

600 mm

300 mm 600 mm

300 mm gxy

Prob. 2–17

2–18. The piece of plastic is originally rectangular. Determine the shear strain gxy at corners A and B if the plastic distorts as shown by the dashed lines.

2–19. The piece of plastic is originally rectangular. Determine the shear strain gxy at corners D and C if the plastic distorts as shown by the dashed lines.

*2–20. The piece of plastic is originally rectangular. Determine the average normal strain that occurs along the diagonals AC and DB .

300 mm

400 mm D A

3 mm

2 mm

5 mm

2 mm 4 mm

2 mm

Probs. 2–18/19/20

2–14. The force P applied at joint D of the square frame causes the frame to sway and form the dashed rhombus. Determine the average normal strain developed in wire AC . Assume the three rods are rigid.

2–15. The force P applied at joint D of the square frame causes the frame to sway and form the dashed rhombus. Determine the average normal strain developed in wire AE . Assume the three rods are rigid.

A B

CD E

400 mm

200 mm 200 mm

Probs. 2–14/15

*2–16. The triangular plate ABC is deformed into the shape shown by the dashed lines. If at A, e AB = 0.0075, e AC = 0.01 and g xy = 0.005 rad, determine the average normal strain along edge BC .

300 mm

400 mm

gxy

Prob. 2–16

80 CHAPTER 2 STRAIN

2–25. The square rubber block is subjected to a shear strain of g xy = 40(10 −6 ) x + 20(10 −6 ) y , where x and y are in mm. This deformation is in the shape shown by the dashed lines, where all the lines parallel to the y axis remain vertical after the deformation. Determine the normal strain along edge BC .

D C

A B

400 mm

300 mm

Prob. 2–25

2–26. The square plate is deformed into the shape shown by the dashed lines. If DC has a normal strain P x = 0.004, DA has a normal strain P y = 0.005 and at D , g xy = 0.02 rad, determine the average normal strain along diagonal CA .

2–27. The square plate is deformed into the shape shown by the dashed lines. If DC has a normal strain P x = 0.004, DA has a normal strain P y = 0.005 and at D , g xy = 0.02 rad, determine the shear strain at point E with respect to the x ¿ and y ¿ axes.

y y¿ x¿

A A¿

B B¿

D C C ¿

600 mm

Probs. 2–26/27

2–21. The rectangular plate is deformed into the shape of a parallelogram shown by the dashed lines. Determine the average shear strain g xy at corners A and B .

xA B

D C

300 mm

5 mm

400 mm

Prob. 2–21

2–22. The triangular plate is fixed at its base, and its apex A is given a horizontal displacement of 5 mm. Determine the shear strain, gxy , at A .

2–23. The triangular plate is fixed at its base, and its apex A is given a horizontal displacement of 5 mm. Determine the average normal strain P x along the x axis.

*2–24. The triangular plate is fixed at its base, and its apex A is given a horizontal displacement of 5 mm. Determine the average normal strain Px! along the x¿ axis.

800 mm

x¿

A¿ 5 mm

45"

Probs. 2–22/23/24

2.2 STRAIN 81

*2–32. The rubber block is fixed along edge AB , and edge CD is moved so that the vertical displacement of any point in the block is given by v( x ) = (v0> b 3 ) x 3 . Determine the shear strain g xy at points ( b >2, a >2) and ( b , a ).

A v0

v (x)

Prob. 2–32

2–33. The fiber AB has a length L and orientation u. If its ends A and B undergo very small displacements uA and vB , respectively, determine the normal strain in the fiber when it is in position A! B!.

B¿

B vB

uA A¿

L u

Prob. 2–33

2–34. If the normal strain is defined in reference to the final length, that is,

Pn= = lim pSp!

a "s! - "s "s!

b instead of in reference to the original length, Eq. 2–2 , show that the difference in these strains is represented as a second-order term, namely, Pn - Pn= = Pn Pn=.

*2–28. The wire is subjected to a normal strain that is defined by P = (x >L)e- (x>L)2 . If the wire has an initial length L, determine the increase in its length.

P # (x/L)e–(x/L)2

Probs. 2–28

2–29. The rectangular plate is deformed into the shape shown by the dashed lines. Determine the average normal strain along diagonal AC , and the average shear strain at corner A .

2–30. The rectangular plate is deformed into the shape shown by the dashed lines. Determine the average normal strain along diagonal BD , and the average shear strain at corner B .

xA B

D C

300 mm

400 mm 6 mm

6 mm

2 mm

2 mm 2 mm

3 mm400 mm

Probs. 2–29/30

2–31. The nonuniform loading causes a normal strain in the shaft that can be expressed as P x = k x 2 , where k is a constant. Determine the displacement of the end B . Also, what is the average normal strain in the rod?

A B

Prob. 2–31

Horizontal ground displacements caused by an earthquake produced fracture of this concrete pier. The material properties of the steel and concrete must be

determined so that engineers can properly design the pier to resist the loadings that caused this failure.

Chapter 3

3.1 The Tension and Compression Test The strength of a material depends on its ability to sustain a load without undue deformation or failure. This property is inherent in the material itself and must be determined by experiment . One of the most important tests to perform in this regard is the tension or compression test . Although several important mechanical properties of a material can be determined from this test, it is used primarily to determine the relationship between the average normal stress and average normal strain in many engineering materials such as metals, ceramics, polymers, and composites.

CHAPTER OBJECTIVES

■ Having discussed the basic concepts of stress and strain, we will in this chapter show how stress can be related to strain by using experimental methods to determine the stress–strain diagram for a specific material. The behavior described by this diagram will then be discussed for materials that are commonly used in engineering. Also, mechanical properties and other tests that are related to the development of mechanics of materials will be discussed.

Mechanical Properties of Materials

84 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

load dial

motor and load controls

movable upper

crosshead

tension specimen

Fig. 3–2

To perform a tension or compression test a specimen of the material is made into a “standard” shape and size. It has a constant circular cross section with enlarged ends, so that failure will not occur at the grips. Before testing, two small punch marks are placed along the specimen’s uniform length. Measurements are taken of both the specimen’s initial cross-sectional area, A0, and the gauge-length distance L0 between the punch marks. For example, when a metal specimen is used in a tension test it generally has an initial diameter of d0 = 0.5 in. (13 mm) and a gauge length of L0 = 2 in. (51 mm), Fig. 3–1 . In order to apply an axial load with no bending of the specimen, the ends are usually seated into ball-and-socket joints. A testing machine like the one shown in Fig. 3–2 is then used to stretch the specimen at a very slow, constant rate until it fails. The machine is designed to read the load required to maintain this uniform stretching.

At frequent intervals during the test, data is recorded of the applied load P , as read on the dial of the machine or taken from a digital readout. Also, the elongation d = L - L0 between the punch marks on the specimen may be measured using either a caliper or a mechanical or optical device called an extensometer . This value of d (delta) is then used to calculate the average normal strain in the specimen. Sometimes, however, this measurement is not taken, since it is also possible to read the strain directly by using an electrical-resistance strain gauge , which looks like the one shown in Fig. 3–3 . The operation of this gauge is based on the change in electrical resistance of a very thin wire or piece of metal foil under strain. Essentially the gauge is cemented to the specimen along its length. If the cement is very strong in comparison to the gauge, then the gauge is in effect an integral part of the specimen, so that when the specimen is strained in the direction of the gauge, the wire and specimen will experience the same strain. By measuring the electrical resistance of the wire, the gauge may be calibrated to read values of normal strain directly.

L0 ! 2 in.

d0 ! 0.5 in.

Fig. 3–1

Typical steel specimen with attached strain gauge.

Electrical–resistance strain gauge

Fig. 3–3

3.2 THE STRESS–STRAIN DIAGRAM 85

3.2 The Stress–Strain Diagram It is not feasible to prepare a test specimen to match the size, A0 and L0 , of each structural member. Rather, the test results must be reported so they apply to a member of any size. To achieve this, the load and corresponding deformation data are used to calculate various values of the stress and corresponding strain in the specimen. A plot of the results produces a curve called the stress–strain diagram . There are two ways in which it is normally described.

Conventional Stress–Strain Diagram. We can determine the nominal or engineering stress by dividing the applied load P by the specimen’s original cross-sectional area A0 . This calculation assumes that the stress is constant over the cross section and throughout the gauge length. We have

s = P

A 0 (3–1)

Likewise, the nominal or engineering strain is found directly from the strain gauge reading, or by dividing the change in the specimen’s gauge length, d , by the specimen’s original gauge length L0 . Here the strain is assumed to be constant throughout the region between the gauge points. Thus,

P = d L0

(3–2)

If the corresponding values of s and P are plotted so that the vertical axis is the stress and the horizontal axis is the strain, the resulting curve is called a conventional stress–strain diagram . Realize, however, that two stress–strain diagrams for a particular material will be quite similar, but will never be exactly the same. This is because the results actually depend on variables such as the material’s composition, microscopic imperfections, the way it is manufactured, the rate of loading, and the temperature during the time of the test.

We will now discuss the characteristics of the conventional stress–strain curve as it pertains to steel , a commonly used material for fabricating both structural members and mechanical elements. Using the method described above, the characteristic stress–strain diagram for a steel specimen is shown in Fig. 3–4 . From this curve we can identify four different ways in which the material behaves, depending on the amount of strain induced in the material.

86 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

elastic region

yielding strain hardening

necking

elastic behavior

plastic behavior

elastic limit yield stress

ultimate stress

true fracture stress

fracture stress

Conventional and true stress-strain diagrams for ductile material (steel) (not to scale)

s¿f

sY spl

proportional limit

Fig. 3–4

Elastic Behavior. Elastic behavior of the material occurs when the strains in the specimen are within the light orange region shown in Fig. 3–4 . Here the curve is actually a straight line throughout most of this region, so that the stress is proportional to the strain. The material in this region is said to be linear elastic . The upper stress limit to this linear relationship is called the proportional limit , spl . If the stress slightly exceeds the proportional limit, the curve tends to bend and flatten out as shown. This continues until the stress reaches the elastic limit . Upon reaching this point, if the load is removed the specimen will still return back to its original shape. Normally for steel, however, the elastic limit is seldom determined, since it is very close to the proportional limit and therefore rather difficult to detect.

Yielding. A slight increase in stress above the elastic limit will result in a breakdown of the material and cause it to deform permanently . This behavior is called yielding , and it is indicated by the rectangular dark orange region of the curve. The stress that causes yielding is called the yield stress or yield point , sY , and the deformation that occurs is called plastic deformation . Although not shown in Fig. 3–4 , for low-carbon steels or those that are hot rolled, the yield point is often distinguished by two values. The upper yield point occurs first, followed by a sudden decrease in load-carrying capacity to a lower yield point . Notice that once the yield point is reached, then as shown in Fig. 3–4 , the specimen will continue to elongate (strain) without any increase in load . When the material is in this state, it is often referred to as being perfectly plastic .

3.2 THE STRESS–STRAIN DIAGRAM 87

Strain Hardening. When yielding has ended, an increase in load can be supported by the specimen, resulting in a curve that rises continuously but becomes flatter until it reaches a maximum stress referred to as the ultimate stress , su . The rise in the curve in this manner is called strain hardening , and it is identified in Fig. 3–4 as the region in light green.

Necking. Up to the ultimate stress, as the specimen elongates, its cross-sectional area will decrease. This decrease is fairly uniform over the specimen’s entire gauge length; however, just after, at the ultimate stress, the cross-sectional area will begin to decrease in a localized region of the specimen. As a result, a constriction or “neck” tends to form in this region as the specimen elongates further, Fig. 3–5 a . This region of the curve due to necking is indicated in dark green in Fig. 3–4 . Here the stress–strain diagram tends to curve downward until the specimen breaks at the fracture stress , sf , Fig. 3–5 b .

True Stress–Strain Diagram. Instead of always using the original cross-sectional area and specimen length to calculate the (engineering) stress and strain, we could have used the actual cross-sectional area and specimen length at the instant the load is measured. The values of stress and strain found from these measurements are called true stress and true strain , and a plot of their values is called the true stress–strain diagram . When this diagram is plotted it has a form shown by the light-blue curve in Fig. 3–4 . Note that the conventional and true s9P diagrams are practically coincident when the strain is small. The differences between the diagrams begin to appear in the strain-hardening range, where the magnitude of strain becomes more significant. In particular, there is a large divergence within the necking region. Here it can be seen from the conventional s9P diagram that the specimen actually supports a decreasing load , since A0 is constant when calculating engineering stress, s = P>A0 . However, from the true s9P diagram, the actual area A within the necking region is always decreasing until fracture, sf ! , and so the material actually sustains increasing stress , since s = P>A.

Typical necking pattern which has occurred on this steel specimen just before fracture.

Necking Failure of a ductile material

(a) (b)

Fig. 3–5

This steel specimen clearly shows the necking that occurred just before the specimen failed. This resulted in the formation of a “cup-cone” shape at the fracture location, which is characteristic of ductile materials.

88 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

Although the true and conventional stress–strain diagrams are different, most engineering design is done so that the material supports a stress within the elastic range. This is because the deformation of the material is generally not severe and the material will restore itself when the load is removed. The true strain up to the elastic limit will remain small enough so that the error in using the engineering values of s and P is very small (about 0.1%) compared with their true values. This is one of the primary reasons for using conventional stress–strain diagrams.

The above concepts can be summarized with reference to Fig. 3–6 , which shows an actual conventional stress–strain diagram for a mild steel specimen. In order to enhance the details, the elastic region of the curve has been shown in light blue color using an exaggerated strain scale, also shown in light blue. Tracing the behavior, the proportional limit is reached at spl = 35 ksi (241 MPa), where Ppl = 0.0012 in.>in. This is followed by an upper yield point of 1sY2u = 38 ksi (262 MPa), then suddenly a lower yield point of 1sY2l = 36 ksi (248 MPa). The end of yielding occurs at a strain of PY = 0.030 in.>in. , which is 25 times greater than the strain at the proportional limit! Continuing, the specimen undergoes strain hardening until it reaches the ultimate stress of su = 63 ksi (434 MPa), then it begins to neck down until a fracture occurs, sf = 47 ksi (324 MPa). By comparison, the strain at failure, Pf = 0.380 in.>in. , is 317 times greater than Ppl!

0.10 0.20 0.30 0.40 0.001 0.002 0.003 0.004

0.050

Stress-strain diagram for mild steel

60 su ! 63

s (ksi)

P (in./in.)

Ppl ! 0.0012 PY ! 0.030 Pf ! 0.380

spl ! 35

sf ! 47

(sY)u ! 38 (sY)l ! 36

Fig. 3–6

3.3 STRESS–STRAIN BEHAVIOR OF DUCTILE AND BRITTLE MATERIALS 89

3.3 Stress–Strain Behavior of Ductile and Brittle Materials

Materials can be classified as either being ductile or brittle, depending on their stress–strain characteristics.

Ductile Materials. Any material that can be subjected to large strains before it fractures is called a ductile material . Mild steel, as discussed previously, is a typical example. Engineers often choose ductile materials for design because these materials are capable of absorbing shock or energy, and if they become overloaded, they will usually exhibit large deformation before failing.

One way to specify the ductility of a material is to report its percent elongation or percent reduction in area at the time of fracture. The percent elongation is the specimen’s fracture strain expressed as a percent. Thus, if the specimen’s original gauge length is L0 and its length at fracture is Lf , then

Percent elongation = Lf - L0

L0 (100%) (3–3)

As seen in Fig. 3–6 , since Pf = 0.380, this value would be 38% for a mild steel specimen.

The percent reduction in area is another way to specify ductility. It is defined within the region of necking as follows:

Percent reduction of area = A 0 - A f

A 0 (100%) (3–4)

Here A0 is the specimen’s original cross-sectional area and Af is the area of the neck at fracture. Mild steel has a typical value of 60%.

Besides steel, other metals such as brass, molybdenum, and zinc may also exhibit ductile stress–strain characteristics similar to steel, whereby they undergo elastic stress–strain behavior, yielding at constant stress, strain hardening, and finally necking until fracture. In most metals, however, constant yielding will not occur beyond the elastic range. One metal for which this is the case is aluminum. Actually, this metal often does not have a well-defined yield point , and consequently it is standard practice to define a yield strength using a graphical procedure called the offset method . Normally for structural design a 0.2% strain ( 0.002 in.>in. ) is chosen, and from this point on the P axis, a line parallel to the initial straight-line portion of the stress–strain diagram is drawn. The point where this line intersects the curve defines the yield strength. An example of the construction for determining the yield strength for an aluminum alloy is shown in Fig. 3–7 . From the graph, the yield strength is sYS = 51 ksi (352 MPa). Apart from metals, 0.2 % strain is used as the offset to determine the yield strength of many plastics.

0.005 0.010 0.002

(0.2% offset)

Yield strength for an aluminum alloy

sYS ! 51

P (in./in.)

s (ksi)

Fig. 3–7

90 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

Realize that the yield strength is not a physical property of the material, since it is a stress that causes a specified permanent strain in the material. In this text, however, we will assume that the yield strength, yield point, elastic limit, and proportional limit all coincide unless otherwise stated. An exception would be natural rubber, which in fact does not even have a proportional limit, since stress and strain are not linearly related. Instead, as shown in Fig. 3–8 , this material, which is known as a polymer, exhibits nonlinear elastic behavior .

Wood is a material that is often moderately ductile, and as a result it is usually designed to respond only to elastic loadings. The strength characteristics of wood vary greatly from one species to another, and for each species they depend on the moisture content, age, and the size and arrangement of knots in the wood. Since wood is a fibrous material, its tensile or compressive characteristics will differ greatly when it is loaded either parallel or perpendicular to its grain. Specifically, wood splits easily when it is loaded in tension perpendicular to its grain, and consequently tensile loads are almost always intended to be applied parallel to the grain of wood members.

2.0

1.5

1.0

0.5

2 4 6 8 10

s (ksi)

P (in./in.)

s–P diagram for natural rubber

Fig. 3–8

s–P diagram for gray cast iron

!0.06 !0.05 !0.04 !0.03 !0.02 !0.01

!20

0.01

!40

!60

!80

!100

!120

20 B

sf " 22

s (ksi)

P (in./in.)

Fig. 3–9

Concrete used for structural purposes must be tested in compression to be sure it reaches its ultimate design stress after curing for 30 days.

3.3 STRESS–STRAIN BEHAVIOR OF DUCTILE AND BRITTLE MATERIALS 91

Brittle Materials. Materials that exhibit little or no yielding before failure are referred to as brittle materials . Gray cast iron is an example, having a stress–strain diagram in tension as shown by portion AB of the curve in Fig. 3–9 . Here fracture at sf = 22 ksi (152 MPa) took place initially at an imperfection or microscopic crack and then spread rapidly across the specimen, causing complete fracture. Since the appearance of initial cracks in a specimen is quite random, brittle materials do not have a well-defined tensile fracture stress. Instead the average fracture stress from a set of observed tests is generally reported. A typical failed specimen is shown in Fig. 3–10 a .

Compared with their behavior in tension, brittle materials, such as gray cast iron, exhibit a much higher resistance to axial compression, as evidenced by portion AC of the curve in Fig. 3–9 . For this case any cracks or imperfections in the specimen tend to close up, and as the load increases the material will generally bulge or become barrel shaped as the strains become larger, Fig. 3–10 b .

Like gray cast iron, concrete is classified as a brittle material, and it also has a low strength capacity in tension. The characteristics of its stress–strain diagram depend primarily on the mix of concrete (water, sand, gravel, and cement) and the time and temperature of curing. A typical example of a “complete” stress–strain diagram for concrete is given in Fig. 3–11 . By inspection, its maximum compressive strength is about 12.5 times greater than its tensile strength, 1sc2max = 5 ksi (34.5 MPa) versus 1st2max = 0.40 ksi (2.76 MPa). For this reason, concrete is almost always reinforced with steel bars or rods whenever it is designed to support tensile loads.

It can generally be stated that most materials exhibit both ductile and brittle behavior. For example, steel has brittle behavior when it contains a high carbon content, and it is ductile when the carbon content is reduced. Also, at low temperatures materials become harder and more brittle, whereas when the temperature rises they become softer and more ductile. This effect is shown in Fig. 3–12 for a methacrylate plastic.

(a)

Tension failure ofTT a brittle material

(b)

Compression causes material to bulge out

Fig. 3–10

0.00050

!0.0025!0.0020!0.0015!0.0010!0.0005 P (in./in.)!0.0030

(st)max " 0.4

(sc)max " 5

s!P diagram for typical concrete mix

s (ksi)

Fig. 3–11

Steel rapidly loses its strength when heated. For this reason engineers often require main structural members to be insulated in case of fire.

0.01 0.02 0.03 0.04 0.05 0.06

9 40# F

110# F

160# F

s!P diagrams for a methacrylate plastic

s (ksi)

P (in./in.)

Fig. 3–12

92 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

3.4 Hooke’s Law As noted in the previous section, the stress–strain diagrams for most engineering materials exhibit a linear relationship between stress and strain within the elastic region. Consequently, an increase in stress causes a proportionate increase in strain. This fact was discovered by Robert Hooke in 1676 using springs and is known as Hooke’s law . It may be expressed mathematically as

s = EP (3–5)

Here E represents the constant of proportionality, which is called the modulus of elasticity or Young’s modulus , named after Thomas Young, who published an account of it in 1807.

Equation 3–5 actually represents the equation of the initial straight-lined portion of the stress–strain diagram up to the proportional limit. Furthermore, the modulus of elasticity represents the slope of this line. Since strain is dimensionless, from Eq. 3–5 , E will have the same units as stress, such as psi, ksi, or pascals. As an example of its calculation, consider the stress–strain diagram for steel shown in Fig. 3–6 . Here spl = 35 ksi and Ppl = 0.0012 in.>in., so that

E = spl

Ppl =

35 ksi 0.0012 in.>in. = 2911032 ksi

As shown in Fig. 3–13 , the proportional limit for a particular type of steel alloy depends on its carbon content; however, most grades of steel, from the softest rolled steel to the hardest tool steel, have about the same modulus of

0.002 0.004 0.006 0.008 0.01

160

140

120

100

180

structural steel (0.2% carbon) soft steel (0.1% carbon)

machine steel (0.6% carbon)

hard steel (0.6% carbon) heat treated

spring steel (1% carbon)

s (ksi)

P (in./in.)

Fig. 3–13

3.4 HOOKE’S LAW 93

elasticity, generally accepted to be Est = 2911032 ksi or 200 GPa. Values of E for other commonly used engineering materials are often tabulated in engineering codes and reference books. Representative values are also listed on the inside back cover of this book. It should be noted that the modulus of elasticity is a mechanical property that indicates the stiffness of a material. Materials that are very stiff, such as steel, have large values of E [ Est = 2911032 ksi or 200 GPa], whereas spongy materials such as vulcanized rubber may have low values [ Er = 0.10 ksi or 0.69 MPa].

The modulus of elasticity is one of the most important mechanical properties used in the development of equations presented in this text. It must always be remembered, though, that E can be used only if a material has linear elastic behavior . Also, if the stress in the material is greater than the proportional limit, the stress–strain diagram ceases to be a straight line and so Eq. 3–5 is no longer valid.

Strain Hardening. If a specimen of ductile material, such as steel, is loaded into the plastic region and then unloaded, elastic strain is recovered as the material returns to its equilibrium state. The plastic strain remains , however, and as a result the material is subjected to a permanent set . For example, a wire when bent (plastically) will spring back a little (elastically) when the load is removed; however, it will not fully return to its original position. This behavior can be illustrated on the stress–strain diagram shown in Fig. 3–14 a . Here the specimen is first loaded beyond its yield point A to point A!. Since interatomic forces have to be overcome to elongate the specimen elastically , then these same forces pull the atoms back together when the load is removed, Fig. 3–14 a . Consequently, the modulus of elasticity, E , is the same, and therefore the slope of line O!A! is the same as line OA .

If the load is reapplied, the atoms in the material will again be displaced until yielding occurs at or near the stress A!, and the stress–strain diagram continues along the same path as before, Fig. 3–14 b . It should be noted, however, that this new stress–strain diagram, defined by O!A!B, now has a higher yield point 1A!2, a consequence of strain-hardening. In other words, the material now has a greater elastic region ; however, it has less ductility , a smaller plastic region, than when it was in its original state.

(a)

permanent set

elastic recovery

elastic region

plastic region

load

unload

A¿ B

O¿O

(b) O¿O

elastic region

plastic region

A¿ B

Fig. 3–14

This pin was made from a hardened steel alloy, that is, one having a high carbon content. It failed due to brittle fracture.

94 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

3.5 Strain Energy As a material is deformed by an external load, the load will do external work, which in turn will be stored in the material as internal energy. This energy is related to the strains in the material, and so it is referred to as strain energy . To obtain this strain energy let us consider a volume element of material from a tension test specimen Fig. 3–15 . It is subjected to the uniaxial stress s. This stress develops a force !F = s !A = s1!x !y2 on the top and bottom faces of the element after the element of length !z undergoes a vertical displacement P !z. By definition, work of ! F is determined by the product of a force and the displacement in the direction of the force. Since the force is increased uniformly from zero to its final magnitude !F when the displacement P !z occurs, the work done on the element by the force is then equal to the average force magnitude 1!F>22 times the displacement P !z. The conservation of energy requires this “external work” on the element to be equivalent to the “internal work” or strain energy stored in the element—assuming that no energy is lost in the form of heat. Consequently, the strain energy !U is !U = 112!F2 P !z = 112 s !x !y2 P !z. Since the volume of the element is !V = !x !y !z, then !U = 12 sP !V .

For applications, it is often convenient to specify the strain energy per unit volume of material. This is called the strain-energy density , and it can be expressed as

u = !U !V

= 1 2

sP (3–6)

Finally, if the material behavior is linear elastic , then Hooke’s law applies, s = EP, and therefore we can express the elastic strain-energy density in terms of the uniaxial stress s as

u = 1 2

E (3–7)

Modulus of Resilience. In particular, when the stress s reaches the proportional limit, the strain-energy density, as calculated by Eq. 3–6 or 3–7, is referred to as the modulus of resilience , i.e.,

ur = 1 2

spl Ppl = 1 2

spl

E (3–8)

From the elastic region of the stress–strain diagram, Fig. 3–16 a , notice that ur is equivalent to the shaded triangular area under the diagram. Physically the modulus of resilience represents the largest amount of internal strain energy per unit volume the material can absorb without causing any permanent damage to the material. Certainly this becomes important when designing bumpers or shock absorbers.

Fig. 3–15

(a)

Modulus of resilience ur

PPpl

spl

Fig. 3–16

3.5 STRAIN ENERGY 95

Modulus of Toughness. Another important property of a material is the modulus of toughness , ut. This quantity represents the entire area under the stress–strain diagram, Fig. 3–16 b , and therefore it indicates the maximum amount of strain-energy the material can absorb just before it fractures. This property becomes important when designing members that may be accidentally overloaded. Note that alloying metals can also change their resilience and toughness. For example, by changing the percentage of carbon in steel, the resulting stress–strain diagrams in Fig. 3–17 show how the degrees of resilience ( Fig. 3–16 a ) and toughness ( Fig. 3–16 b ) can be changed.

(b)

Modulus of toughness ut P

Fig. 3–16 (cont.)

soft steel (0.1% carbon) most ductile

hard steel (0.6% carbon) highest strength

structural steel (0.2% carbon) toughest

Fig. 3–17

This nylon specimen exhibits a high degree of toughness as noted by the large amount of necking that has occurred just before fracture.

Important Points

• A conventional stress–strain diagram is important in engineering since it provides a means for obtaining data about a material’s tensile or compressive strength without regard for the material’s physical size or shape.

• Engineering stress and strain are calculated using the original cross-sectional area and gauge length of the specimen.

• A ductile material , such as mild steel, has four distinct behaviors as it is loaded. They are elastic behavior, yielding, strain hardening, and necking .

• A material is linear elastic if the stress is proportional to the strain within the elastic region. This behavior is described by Hooke’s law , s = EP , where the modulus of elasticity E is the slope of the line.

• Important points on the stress–strain diagram are the proportional limit, elastic limit, yield stress, ultimate stress , and fracture stress .

• The ductility of a material can be specified by the specimen’s percent elongation or the percent reduction in area .

• If a material does not have a distinct yield point, a yield strength can be specified using a graphical procedure such as the offset method .

• Brittle materials , such as gray cast iron, have very little or no yielding and so they can fracture suddenly.

• Strain hardening is used to establish a higher yield point for a material. This is done by straining the material beyond the elastic limit, then releasing the load. The modulus of elasticity remains the same; however, the material’s ductility decreases .

• Strain energy is energy stored in a material due to its deformation. This energy per unit volume is called strain-energy density . If it is measured up to the proportional limit, it is referred to as the modulus of resilience , and if it is measured up to the point of fracture, it is called the modulus of toughness . It can be determined from the area under the s-P diagram.

96 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

A tension test for a steel alloy results in the stress–strain diagram shown in Fig. 3–18 . Calculate the modulus of elasticity and the yield strength based on a 0.2% offset. Identify on the graph the ultimate stress and the fracture stress.

EXAMPLE 3.1

sf ! 90

O 0.02

10 20 30 40 50 60 70 80

100 110 120

0.0004

0.04 0.0008

0.06

0.0012

0.08 0.0016

0.10

0.0020

0.12 0.0024

0.14 0.16 0.18 0.20 0.22 0.24

0.2%

E E

A¿ A¿

P (in./in.)

su ! 108

sYS ! 68

Pf ! 0.23

s (ksi)

Fig. 3–18

SOLUTION

Modulus of Elasticity. We must calculate the slope of the initial straight-line portion of the graph. Using the magnified curve and scale shown in blue, this line extends from point O to an estimated point A , which has coordinates of approximately ( 0.0016 in.>in. , 50 ksi). Therefore,

E = 50 ksi

0.0016 in.>in. = 31.211032 ksi Ans. Note that the equation of line OA is thus s = 31.211032P. Yield Strength. For a 0.2% offset, we begin at a strain of 0.2% or 0.0020 in.>in. and graphically extend a (dashed) line parallel to OA until it intersects the s9P curve at A!. The yield strength is approximately sYS = 68 ksi Ans.

Ultimate Stress. This is defined by the peak of the s9P graph, point B in Fig. 3–18 . su = 108 ksi Ans. Fracture Stress. When the specimen is strained to its maximum of Pf = 0.23 in.>in., it fractures at point C . Thus, sf = 90 ksi Ans.

3.5 STRAIN ENERGY 97

EXAMPLE 3.2

The stress–strain diagram for an aluminum alloy that is used for making aircraft parts is shown in Fig. 3–19 . If a specimen of this material is stressed to 600 MPa, determine the permanent strain that remains in the specimen when the load is released. Also, find the modulus of resilience both before and after the load application.

SOLUTION

Permanent Strain. When the specimen is subjected to the load, it strain-hardens until point B is reached on the s9P diagram. The strain at this point is approximately 0.023 mm>mm. When the load is released, the material behaves by following the straight line BC , which is parallel to line OA . Since both lines have the same slope, the strain at point C can be determined analytically. The slope of line OA is the modulus of elasticity, i.e.,

E = 450 MPa

0.006 mm>mm = 75.0 GPa From triangle CBD, we require

E = BD CD

; 75.011092 Pa = 60011062 Pa CD

CD = 0.008 mm>mm This strain represents the amount of recovered elastic strain . The permanent strain, POC, is thus POC = 0.023 mm>mm - 0.008 mm>mm = 0.0150 mm>mm Ans. Note: If gauge marks on the specimen were originally 50 mm apart, then after the load is released these marks will be 50 mm +10.01502 150 mm2 = 50.75 mm apart. Modulus of Resilience. Applying Eq. 3–8 , we have *

1ur2initial = 12 spl Ppl = 12 1450 MPa2 10.006 mm>mm2 = 1.35 MJ>m3 Ans.

1ur2final = 12 spl Ppl = 12 1600 MPa2 10.008 mm>mm2 = 2.40 MJ>m3 Ans.

NOTE: By comparison, the effect of strain-hardening the material has caused an increase in the modulus of resilience; however, note that the modulus of toughness for the material has decreased since the area under the original curve, OABF , is larger than the area under curve CBF .

750

0.040.030.020.01

600

300

150

parallel

0.023

sY ! 450

s (MPa)

P (mm/mm) PY ! 0.006POC

Fig. 3–19

* Work in the SI system of units is measured in joules, where 1 J = 1 N # m.

98 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

An aluminum rod, shown in Fig. 3–20 a , has a circular cross section and is subjected to an axial load of 10 kN. If a portion of the stress–strain diagram is shown in Fig. 3–20 b , determine the approximate elongation of the rod when the load is applied. Take Eal = 70 GPa.

EXAMPLE 3.3

(a)

600 mm 400 mm

15 mm20 mm

A B C 10 kN 10 kN

Fig. 3–20 (b)

O 0.02 0.04 0.06

F 56.6

sY ! 40

s (MPa)

PBC ! 0.0450

SOLUTION For the analysis we will neglect the localized deformations at the point of load application and where the rod’s cross-sectional area suddenly changes. (These effects will be discussed in Sections 4.1 and 4.7 .) Throughout the midsection of each segment the normal stress and deformation are uniform.

In order to find the elongation of the rod, we must first obtain the strain. This is done by calculating the stress, then using the stress–strain diagram. The normal stress within each segment is

sAB = P A

= 1011032 N

p10.01 m22 = 31.83 MPa sBC =

P A

= 1011032 N

p10.0075 m22 = 56.59 MPa From the stress–strain diagram, the material in segment AB is

strained elastically since sAB 6 sY = 40 MPa. Using Hooke’s law,

PAB = sAB

Eal =

31.8311062 Pa 7011092 Pa = 0.0004547 mm>mm

The material within segment BC is strained plastically, since sBC 7 sY = 40 MPa. From the graph, for sBC = 56.59 MPa, PBC ! 0.045 mm>mm . The approximate elongation of the rod is therefore d = !PL = 0.00045471600 mm2 + 0.04501400 mm2 = 18.3 mm Ans.

3.5 STRAIN ENERGY 99

FUNDAMENTAL PROBLEMS

F3–1. Define a homogeneous material.

F3–2. Indicate the points on the stress–strain diagram which represent the proportional limit and the ultimate stress.

B C E

F3–2

F3–3. Define the modulus of elasticity E.

F3–4. At room temperature, mild steel is a ductile material. True or false?

F3–5. Engineering stress and strain are calculated using the actual cross-sectional area and length of the specimen. True or false?

F3–6. As the temperature increases the modulus of elasticity will increase. True or false?

F3–7. A 100-mm long rod has a diameter of 15 mm. If an axial tensile load of 100 kN is applied, determine its change in length. Assume linear-elastic behavior with E = 200 GPa.

F3–8. A bar has a length of 8 in. and cross-sectional area of 12 in 2 . Determine the modulus of elasticity of the material if it is subjected to an axial tensile load of 10 kip and stretches 0.003 in. The material has linear-elastic behavior.

F3–9. A 10-mm-diameter rod has a modulus of elasticity of E = 100 GPa. If it is 4 m long and subjected to an axial tensile load of 6 kN, determine its elongation. Assume linear-elastic behavior.

F3–10. The material for the 50-mm-long specimen has the stress–strain diagram shown. If P = 100 kN, determine the elongation of the specimen.

F3–11. The material for the 50-mm-long specimen has the stress–strain diagram shown. If P = 150 kN is applied and then released, determine the permanent elongation of the specimen.

450

0.00225 0.03 P (mm/mm)

500

20 mms (MPa)

F3–10/11

F3–12. If the elongation of wire BC is 0.2 mm after the force P is applied, determine the magnitude of P . The wire is A-36 steel and has a diameter of 3 mm.

400 mm

200 mm 300 mm P

F3–12

100 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

*3–4. A tension test was performed on a steel specimen having an original diameter of 0.503 in. and a gauge length of 2.00 in. The data is listed in the table. Plot the stress– strain diagram and determine approximately the modulus of elasticity, the ultimate stress, and the rupture stress. Use a scale of 1 in. = 15 ksi and 1 in. = 0.05 in.> in. Redraw the linear-elastic region, using the same stress scale but a strain scale of 1 in. = 0.001 in.

3–5. A tension test was performed on a steel specimen having an original diameter of 0.503 in. and gauge length of 2.00 in. Using the data listed in the table, plot the stress–strain diagram and determine approximately the modulus of toughness.

0 2.50 6.50 8.50 9.20 9.80 12.0 14.0 14.5 14.0 13.2

0 0.0009 0.0025 0.0040 0.0065 0.0098 0.0400 0.1200 0.2500 0.3500 0.4700

Load (kip) Elongation (in.)

0 2.50 6.50 8.50 9.20 9.80 12.0 14.0 14.5 14.0 13.2

0 0.0009 0.0025 0.0040 0.0065 0.0098 0.0400 0.1200 0.2500 0.3500 0.4700

Load (kip) Elongation (in.)

Probs. 3–4/5

3–6. A specimen is originally 1 ft long, has a diameter of 0.5 in., and is subjected to a force of 500 lb. When the force is increased from 500 lb to 1800 lb, the specimen elongates 0.009 in. Determine the modulus of elasticity for the material if it remains linear elastic.

3–7. A structural member in a nuclear reactor is made of a zirconium alloy. If an axial load of 4 kip is to be supported by the member, determine its required cross-sectional area. Use a factor of safety of 3 relative to yielding. What is the load on the member if it is 3 ft long and its elongation is 0.02 in.? Ezr = 14(103) ksi, sY = 57.5 ksi. The material has elastic behavior.

3–1. A tension test was performed on a steel specimen having an original diameter of 0.503 in. and gauge length of 2.00 in. The data is listed in the table. Plot the stress–strain diagram and determine approximately the modulus of elasticity, the yield stress, the ultimate stress, and the rupture stress. Use a scale of 1 in. = 20 ksi and 1 in. = 0.05 in.> in. Redraw the elastic region, using the same stress scale but a strain scale of 1 in. = 0.001 in.> in.

0 1.50 4.60 8.00 11.00 11.80 11.80 12.00 16.60 20.00 21.50 19.50 18.50

0 0.0005 0.0015 0.0025 0.0035 0.0050 0.0080 0.0200 0.0400 0.1000 0.2800 0.4000 0.4600

Load (kip) Elongation (in.)

Prob. 3–1

3–2. Data taken from a stress–strain test for a ceramic are given in the table. The curve is linear between the origin and the first point. Plot the diagram, and determine the modulus of elasticity and the modulus of resilience.

3–3. Data taken from a stress–strain test for a ceramic are given in the table. The curve is linear between the origin and the first point. Plot the diagram, and determine approximately the modulus of toughness. The rupture stress is sr = 53.4 ksi.

0 33.2 45.5 49.4 51.5 53.4

0 0.0006 0.0010 0.0014 0.0018 0.0022

S (ksi) P (in./in.)

Probs. 3–2/3

PROBLEMS

3.5 STRAIN ENERGY 101

3–10. The stress–strain diagram for a metal alloy having an original diameter of 0.5 in. and a gauge length of 2 in. is given in the figure. Determine approximately the modulus of elasticity for the material, the load on the specimen that causes yielding, and the ultimate load the specimen will support.

3–11. The stress–strain diagram for a steel alloy having an original diameter of 0.5 in. and a gauge length of 2 in. is given in the figure. If the specimen is loaded until it is stressed to 90 ksi, determine the approximate amount of elastic recovery and the increase in the gauge length after it is unloaded.

*3–12. The stress–strain diagram for a steel alloy having an original diameter of 0.5 in. and a gauge length of 2 in. is given in the figure. Determine approximately the modulus of resilience and the modulus of toughness for the material.

105

00 0 0.350.05 0.10 0.15 0.20 0.25 0.30

0.0070.001 0.002 0.003 0.004 0.005 0.006

P (in./in.)

s (ksi)

Probs. 3–10/11/12

3–13. A bar having a length of 5 in. and cross-sectional area of 0.7 in.2 is subjected to an axial force of 8000 lb. If the bar stretches 0.002 in., determine the modulus of elasticity of the material. The material has linear-elastic behavior.

8000 lb8000 lb

5 in.

Prob. 3–13

*3–8. The strut is supported by a pin at C and an A-36 steel guy wire AB. If the wire has a diameter of 0.2 in., determine how much it stretches when the distributed load acts on the strut.

9 ft

200 lb/ft

60!

Prob. 3–8

3–9. The s9P diagram for elastic fibers that make up human skin and muscle is shown. Determine the modulus of elasticity of the fibers and estimate their modulus of toughness and modulus of resilience.

21 2.25 11

s (psi)

P (in./in.)

Prob. 3–9

102 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

3–17. A tension test was performed on a magnesium alloy specimen having a diameter 0.5 in. and gauge length 2 in. The resulting stress–strain diagram is shown in the figure. Determine the approximate modulus of elasticity and the yield strength of the alloy using the 0.2% strain offset method.

3–18. A tension test was performed on a magnesium alloy specimen having a diameter 0.5 in. and gauge length of 2 in. The resulting stress–strain diagram is shown in the figure. If the specimen is stressed to 30 ksi and unloaded, determine the permanent elongation of the specimen.

s (ksi)

P (in./in.) 0.002 0.004 0.006 0.008 0.010

Probs. 3–17/18

3–19. The stress–strain diagram for a bone is shown, and can be described by the equation P = 0.45110-62 s + 0.36110-122 s3, where s is in kPa. Determine the yield strength assuming a 0.3% offset.

*3–20. The stress–strain diagram for a bone is shown and can be described by the equation P = 0.45110-62 s + 0.36110-122 s3, where s is in kPa. Determine the modulus of toughness and the amount of elongation of a 200-mm-long region just before it fractures if failure occurs at P = 0.12 mm>mm.

P ! 0.45(10"6)s + 0.36(10"12)s3

Probs. 3–19/20

3–14. The rigid pipe is supported by a pin at A and an A-36 steel guy wire BD. If the wire has a diameter of 0.25 in., determine how much it stretches when a load of P = 600 lb acts on the pipe.

3–15. The rigid pipe is supported by a pin at A and an A-36 guy wire BD. If the wire has a diameter of 0.25 in., determine the load P if the end C is displaced 0.15 in. downward.

3 ft 3 ft

C DA

P4 ft

Probs. 3–14/15

*3–16. The wire has a diameter of 5 mm and is made from A-36 steel. If a 80-kg man is sitting on seat C , determine the elongation of wire DE.

E W

800 mm

600 mm

Prob. 3–16

3.5 STRAIN ENERGY 103

*3–24. The wires AB and BC have original lengths of 2 ft and 3 ft, and diameters of 18 in. and

3 16 in., respectively. If

these wires are made of a material that has the approximate stress–strain diagram shown, determine the elongations of the wires after the 1500-lb load is placed on the platform.

2 ft 45! 30!

3 ft A

0.002 0.01

s (ksi)

P (in./in.)

Prob. 3–24

3–21. The two bars are made of polystyrene, which has the stress–strain diagram shown. If the cross-sectional area of bar AB is 1.5 in 2 and BC is 4 in 2 , determine the largest force P that can be supported before any member ruptures. Assume that buckling does not occur.

3–22. The two bars are made of polystyrene, which has the stress–strain diagram shown. Determine the cross-sectional area of each bar so that the bars rupture simultaneously when the load P = 3 kip. Assume that buckling does not occur.

C B

3 ft

4 ft

P (in./in.)

s (ksi)

0.800.600.400.200

tension

compression

Probs. 3–21/22 3–23. The stress–strain diagram for many metal alloys can be described analytically using the Ramberg-Osgood three parameter equation P = s>E + ksn , where E , k , and n are determined from measurements taken from the diagram, Using the stress–strain diagram shown in the figure, take E = 30(10 3 ) ksi and determine the other two parameters k and n and thereby obtain an analytical expression for the curve.

s (ksi)

P (10– 6) 0.1 0.2 0.3 0.4 0.5

Prob. 3–23

104 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

3.6 Poisson’s Ratio When a deformable body is subjected to an axial tensile force, not only does it elongate but it also contracts laterally. For example, if a rubber band is stretched, it can be noted that both the thickness and width of the band are decreased. Likewise, a compressive force acting on a body causes it to contract in the direction of the force and yet its sides expand laterally.

Consider a bar having an original radius r and length L and subjected to the tensile force P in Fig. 3–21 . This force elongates the bar by an amount d , and its radius contracts by an amount d!. Strains in the longitudinal or axial direction and in the lateral or radial direction are, respectively,

Plong = d

L and Plat =

In the early 1800s, the French scientist S. D. Poisson realized that within the elastic range the ratio of these strains is a constant , since the deformations d and d! are proportional. This constant is referred to as Poisson’s ratio , n (nu), and it has a numerical value that is unique for a particular material that is both homogeneous and isotropic . Stated mathematically it is

n = - Plat

Plong (3–9)

The negative sign is included here since longitudinal elongation (positive strain) causes lateral contraction (negative strain), and vice versa. Notice that these strains are caused only by the axial or longitudinal force P ; i.e., no force or stress acts in a lateral direction in order to strain the material in this direction.

Poisson’s ratio is a dimensionless quantity, and for most nonporous solids it has a value that is generally between 14 and

1 3. Typical values of n

for common engineering materials are listed on the inside back cover. For an “ideal material” having no lateral deformation when it is stretched or compressed, Poisson’s ratio will be 0. Furthermore, it will be shown in Sec. 10.6 that the maximum possible value for Poisson’s ratio is 0.5. Therefore 0 … n … 0.5.

P r

Final Shape

Original Shape

Tension

d/2

d¿

d/2

Fig. 3–21

When the rubber block is compressed (negative strain) its sides will expand (positive strain). The ratio of these strains remains constant.

3.6 POISSON’S RATIO 105

A bar made of A-36 steel has the dimensions shown in Fig. 3–22 . If an axial force of P = 80 kN is applied to the bar, determine the change in its length and the change in the dimensions of its cross section after applying the load. The material behaves elastically.

EXAMPLE 3.4

SOLUTION The normal stress in the bar is

sz = P A

= 8011032 N

10.1 m2 10.05 m2 = 16.011062 Pa From the table on the inside back cover for A-36 steel Est = 200 GPa, and so the strain in the z direction is

Pz = sz

Est =

16.011062 Pa 20011092 Pa = 80110-62 mm>mm

The axial elongation of the bar is therefore

dz = Pz L z = [80110-62]11.5 m2 = 120 mm Ans. Using Eq. 3–9 , where nst = 0.32 as found from the inside back cover,

the lateral contraction strains in both the x and y directions are

Px = Py = -nst Pz = -0.32[80110-62] = -25.6 mm>m Thus the changes in the dimensions of the cross section are

dx = Px L x = -[25.6110-62]10.1 m2 = -2.56 mm Ans. dy = Py L y = -[25.6110-62]10.05 m2 = -1.28 mm Ans.

P ! 80 kN

100 mm

1.5 m 50 mm

Fig. 3–22

106 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

3.7 The Shear Stress–Strain Diagram In Sec. 1.5 it was shown that when a small element of material is subjected to pure shear , equilibrium requires that equal shear stresses must be developed on four faces of the element. These stresses txy must be directed toward or away from diagonally opposite corners of the element, as shown in Fig. 3–23 a . Furthermore, if the material is homogeneous and isotropic , then this shear stress will distort the element uniformly, Fig. 3–23 b . As mentioned in Sec. 2.2, the shear strain gxy measures the angular distortion of the element relative to the sides originally along the x and y axes.

The behavior of a material subjected to pure shear can be studied in a laboratory using specimens in the shape of thin tubes and subjecting them to a torsional loading. If measurements are made of the applied torque and the resulting angle of twist , then by the methods to be explained in Chapter 5 , the data can be used to determine the shear stress and shear strain, and a shear stress–strain diagram plotted. An example of such a diagram for a ductile material is shown in Fig. 3–24 . Like the tension test, this material when subjected to shear will exhibit linear-elastic behavior and it will have a defined proportional limit tpl. Also, strain hardening will occur until an ultimate shear stress tu is reached. And finally, the material will begin to lose its shear strength until it reaches a point where it fractures, tf.

For most engineering materials, like the one just described, the elastic behavior is linear , and so Hooke’s law for shear can be written as

t = Gg (3–10)

Here G is called the shear modulus of elasticity or the modulus of rigidity . Its value represents the slope of the line on the t9g diagram, that is, G = tpl>gpl. Typical values for common engineering materials are listed on the inside back cover. Notice that the units of measurement for G will be the same as those for t (Pa or psi), since g is measured in radians, a dimensionless quantity.

It will be shown in Sec. 10.6 that the three material constants, E, n, and G are actually related by the equation

G = E

211 + n2 (3–11) Provided E and G are known, the value of n can then be determined from this equation rather than through experimental measurement. For example, in the case of A-36 steel, Est = 2911032 ksi and Gst = 1111032 ksi, so that, from Eq. 3–11 , nst = 0.32.

(a)

txy

(b)

2 !

gxy 2

gxy

Fig. 3–23

tu tf

tpl

g grgugpl

Fig. 3–24

3.7 THE SHEAR STRESS–STRAIN DIAGRAM 107

EXAMPLE 3.5

A specimen of titanium alloy is tested in torsion and the shear stress– strain diagram is shown in Fig. 3–25 a . Determine the shear modulus G , the proportional limit, and the ultimate shear stress. Also, determine the maximum distance d that the top of a block of this material, shown in Fig. 3–25 b , could be displaced horizontally if the material behaves elastically when acted upon by a shear force V . What is the magnitude of V necessary to cause this displacement?

SOLUTION Shear Modulus. This value represents the slope of the straight-line portion OA of the t9g diagram. The coordinates of point A are (0.008 rad, 52 ksi). Thus,

G = 52 ksi

0.008 rad = 6500 ksi Ans.

The equation of line OA is therefore t = Gg = 6500g, which is Hooke’s law for shear.

Proportional Limit. By inspection, the graph ceases to be linear at point A . Thus,

tpl = 52 ksi Ans.

Ultimate Stress. This value represents the maximum shear stress, point B . From the graph,

tu = 73 ksi Ans.

Maximum Elastic Displacement and Shear Force. Since the maximum elastic shear strain is 0.008 rad, a very small angle, the top of the block in Fig. 3–25 b will be displaced horizontally:

tan10.008 rad2 ! 0.008 rad = d 2 in.

d = 0.016 in. Ans.

The corresponding average shear stress in the block is tpl = 52 ksi. Thus, the shear force V needed to cause the displacement is

tavg = V A

; 52 ksi = V

13 in.2 14 in.2 V = 624 kip Ans.

4 in. 3 in.

2 in. g

(b)

Fig. 3–25

90 80 70 60 50 40 30 20 10

0.73

(a)

t (ksi)

tu ! 73

gpl ! 0.008 gu ! 0.54 g (rad)

tpl ! 52

108 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

An aluminum specimen shown in Fig. 3–26 has a diameter of d0 = 25 mm and a gauge length of L 0 = 250 mm. If a force of 165 kN elongates the gauge length 1.20 mm, determine the modulus of elasticity. Also, determine by how much the force causes the diameter of the specimen to contract. Take Gal = 26 GPa and sY = 440 MPa.

SOLUTION Modulus of Elasticity. The average normal stress in the specimen is

s = P A

= 16511032 N1p>42 10.025 m22 = 336.1 MPa

and the average normal strain is

P = d L

= 1.20 mm 250 mm

= 0.00480 mm>mm Since s 6 sY = 440 MPa, the material behaves elastically. The modulus of elasticity is therefore

Eal = s

P = 336.111062 Pa

0.00480 = 70.0 GPa Ans.

Contraction of Diameter. First we will determine Poisson’s ratio for the material using Eq. 3–11 .

G = E

211 + n2 26 GPa =

70.0 GPa 211 + n2

n = 0.347

Since Plong = 0.00480 mm>mm, then by Eq. 3–9 , n = -

Plat Plong

0.347 = - Plat

0.00480 mm>mm Plat = -0.00166 mm>mm

The contraction of the diameter is therefore

d! = 10.001662 125 mm2 = 0.0416 mm Ans.

EXAMPLE 3.6

d0 L0

165 kN

Fig. 3–26

3.8 FAILURE OF MATERIALS DUE TO CREEP AND FATIGUE 109

*3.8 Failure of Materials Due to Creep and Fatigue

The mechanical properties of a material have up to this point been discussed only for a static or slowly applied load at constant temperature. In some cases, however, a member may have to be used in an environment for which loadings must be sustained over long periods of time at elevated temperatures, or in other cases, the loading may be repeated or cycled. We will not consider these effects in this book, although we will briefly mention how one determines a material’s strength for these conditions, since they are given special treatment in design.

Creep. When a material has to support a load for a very long period of time, it may continue to deform until a sudden fracture occurs or its usefulness is impaired. This time-dependent permanent deformation is known as creep . Normally creep is considered when metals and ceramics are used for structural members or mechanical parts that are subjected to high temperatures. For some materials, however, such as polymers and composite materials—including wood or concrete—temperature is not an important factor, and yet creep can occur strictly from long-term load application. As a typical example, consider the fact that a rubber band will not return to its original shape after being released from a stretched position in which it was held for a very long period of time. In the general sense, therefore, both stress and/or temperature play a significant role in the rate of creep.

For practical purposes, when creep becomes important, a member is usually designed to resist a specified creep strain for a given period of time. An important mechanical property that is used in this regard is called the creep strength . This value represents the highest stress the material can withstand during a specified time without exceeding an allowable creep strain. The creep strength will vary with temperature, and for design, a given temperature, duration of loading, and allowable creep strain must all be specified. For example, a creep strain of 0.1% per year has been suggested for steel in bolts and piping.

Several methods exist for determining an allowable creep strength for a particular material. One of the simplest involves testing several specimens simultaneously at a constant temperature, but with each subjected to a different axial stress . By measuring the length of time needed to produce either an allowable strain or the fracture strain for each specimen, a curve of stress versus time can be established. Normally these tests are run to a maximum of 1000 hours. An example of the results for stainless steel at a temperature of 1200°F and prescribed creep strain of 1% is shown in Fig. 3–27 . As noted, this material has a yield strength of 40 ksi (276 MPa) at room temperature (0.2% offset) and the creep strength at 1000 h is found to be approximately sc = 20 ksi (138 MPa).

The long-tem application of the cable loading on this pole has caused the pole to deform due to creep.

s (ksi)

0 200 400 600 800 1000 t (h)

s–t diagram for stainless steel at 1200!F and creep strain at 1%

Fig. 3–27

110 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

For longer periods of time, extrapolations from the curves must be made. To do this usually requires a certain amount of experience with creep behavior, and some supplementary knowledge about the creep properties of the material. Once the material’s creep strength has been determined, however, a factor of safety is applied to obtain an appropriate allowable stress for design.

Fatigue. When a metal is subjected to repeated cycles of stress or strain, it causes its structure to break down, ultimately leading to fracture. This behavior is called fatigue , and it is usually responsible for a large percentage of failures in connecting rods and crankshafts of engines; steam or gas turbine blades; connections or supports for bridges, railroad wheels, and axles; and other parts subjected to cyclic loading. In all these cases, fracture will occur at a stress that is less than the material’s yield stress.

The nature of this failure apparently results from the fact that there are microscopic imperfections, usually on the surface of the member, where the localized stress becomes much greater than the average stress acting over the cross section. As this higher stress is cycled, it leads to the formation of minute cracks. Occurrence of these cracks causes a further increase of stress at their tips or boundaries, which in turn causes a further extension of the cracks into the material as the stress continues to be cycled. Eventually the cross-sectional area of the member is reduced to the point where the load can no longer be sustained, and as a result sudden fracture occurs. The material, even though known to be ductile, behaves as if it were brittle .

In order to specify a safe strength for a metallic material under repeated loading, it is necessary to determine a limit below which no evidence of failure can be detected after applying a load for a specified number of cycles. This limiting stress is called the endurance or fatigue limit . Using a testing machine for this purpose, a series of specimens are each subjected to a specified stress and cycled to failure. The results are plotted as a graph representing the stress S (or s ) on the vertical axis and the number of cycles-to-failure N on the horizontal axis. This graph is called an S–N diagram or stress–cycle diagram , and most often the values of N are plotted on a logarithmic scale since they are generally quite large.

Examples of S–N diagrams for two common engineering metals are shown in Fig. 3–28 . The endurance limit is usually identified as the stress for which the S–N graph becomes horizontal or asymptotic. As noted, it has a well-defined value of 1Sel2st = 27 ksi (186 MPa) for steel. For aluminum, however, the endurance limit is not well defined, and so it is normally specified as the stress having a limit of 500 million cycles, 1Sel2al = 19 ksi (131 MPa). Once a particular value is obtained, it is often assumed that for any stress below this value the fatigue life is infinite, and therefore the number of cycles to failure is no longer given consideration.

The design of members used for amusement park rides requires careful consideration of cyclic loadings that can cause fatigue.

Engineers must account for possible fatigue failure of the moving parts of this oil-pumping rig.

3.8 FAILURE OF MATERIALS DUE TO CREEP AND FATIGUE 111

(Sel)st ! 27

(Sel)al ! 19

0 10.1 10 100 1000500

aluminum

steel

N (106)

S (ksi)

S–N diagram for steel and aluminum alloys (N axis has a logarithmic scale)

Fig. 3–28

Important Points

• Poisson’s ratio , n , is a ratio of the lateral strain of a homogeneous and isotropic material to its longitudinal strain. Generally these strains are of opposite signs, that is, if one is an elongation, the other will be a contraction.

• The shear stress–strain diagram is a plot of the shear stress versus the shear strain. If the material is homogeneous and isotropic, and is also linear elastic, the slope of the straight line within the elastic region is called the modulus of rigidity or the shear modulus, G .

• There is a mathematical relationship between G, E, and n .

• Creep is the time-dependent deformation of a material for which stress and/or temperature play an important role. Members are designed to resist the effects of creep based on their material creep strength, which is the largest initial stress a material can withstand during a specified time without exceeding a specified creep strain.

• Fatigue occurs in metals when the stress or strain is cycled. This phenomenon causes brittle fracture of the material. Members are designed to resist fatigue by ensuring that the stress in the member does not exceed its endurance or fatigue limit . This value is determined from an S–N diagram as the maximum stress the material can resist when subjected to a specified number of cycles of loading.

112 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

F3–16. A 20-mm-wide block is bonded to rigid plates at its top and bottom. When the force P is applied the block deforms into the shape shown by the dashed line. If a = 3 mm and P is released, determine the permanent shear strain in the block.

F3–15. A 20-mm-wide block is firmly bonded to rigid plates at its top and bottom. When the force P is applied the block deforms into the shape shown by the dashed line. Determine the magnitude of P . The block’s material has a modulus of rigidity of G = 26 GPa. Assume that the material does not yield and use small angle analysis.

F3–13. A 100-mm long rod has a diameter of 15 mm. If an axial tensile load of 10 kN is applied to it, determine the change in its diameter. E = 70 GPa, n = 0.35.

F3–14. A solid circular rod that is 600 mm long and 20 mm in diameter is subjected to an axial force of P = 50 kN. The elongation of the rod is d = 1.40 mm, and its diameter becomes d! = 19.9837 mm. Determine the modulus of elasticity and the modulus of rigidity of the material. Assume that the material does not yield.

FUNDAMENTAL PROBLEMS

20 mm

P ! 50 kN

600 mm

F3–14

150 mm

0.5 mm

150 mm

F3–15

150 mm a ! 3 mm

150 mm

130

0.005

t (MPa)

g (rad)

F3–16

3.8 FAILURE OF MATERIALS DUE TO CREEP AND FATIGUE 113

3–27. When the two forces are placed on the beam, the diameter of the A-36 steel rod BC decreases from 40 mm to 39.99 mm. Determine the magnitude of each force P .

*3–28. If P = 150 kN, determine the elastic elongation of rod BC and the decrease in its diameter. Rod BC is made of A-36 streel and has a diameter of 40 mm.

1 m 1 m 1 m

0.75 m

1 m

A B

P P C

Probs. 3–27/28

3–29. The friction pad A is used to support the member, which is subjected to an axial force of P = 2 kN. The pad is made from a material having a modulus of elasticity of E = 4 MPa and Poisson’s ratio n = 0.4. If slipping does not occur, determine the normal and shear strains in the pad. The width is 50 mm. Assume that the material is linearly elastic. Also, neglect the effect of the moment acting on the pad.

100 mm

25 mm

60!

Prob. 3–29

3–25. The acrylic plastic rod is 200 mm long and 15 mm in diameter. If an axial load of 300 N is applied to it, determine the change in its length and the change in its diameter. Ep = 2.70 GPa, np = 0.4.

300 N

200 mm

300 N

Prob. 3–25

3–26. The thin-walled tube is subjected to an axial force of 40 kN. If the tube elongates 3 mm and its circumference decreases 0.09 mm, determine the modulus of elasticity, Poisson’s ratio, and the shear modulus of the tube’s material. The material behaves elastically.

40 kN

10 mm

12.5 mm

900 mm

Prob. 3–26

PROBLEMS

114 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

3–33. The aluminum block has a rectangular cross section and is subjected to an axial compressive force of 8 kip. If the 1.5-in. side changed its length to 1.500132 in., determine Poisson’s ratio and the new length of the 2-in. side. E al = 10(10 3 ) ksi.

3 in.

1.5 in.

8 kip 8 kip 2 in.

Prob. 3–33

3–34. A shear spring is made from two blocks of rubber, each having a height h , width b , and thickness a . The blocks are bonded to three plates as shown. If the plates are rigid and the shear modulus of the rubber is G , determine the displacement of plate A if a vertical load P is applied to this plate. Assume that the displacement is small so that d = a tan g ! ag.

A d

Prob. 3–34

3–30. The lap joint is connected together using a 1.25 in. diameter bolt. If the bolt is made from a material having a shear stress–strain diagram that is approximated as shown, determine the shear strain developed in the shear plane of the bolt when P = 75 kip.

3–31. The lap joint is connected together using a 1.25 in. diameter bolt. If the bolt is made from a material having a shear stress-strain diagram that is approximated as shown, determine the permanent shear strain in the shear plane of the bolt when the applied force P = 150 kip is removed.

0.005 0.05

t (ksi)

g (rad)

P 2

Probs. 3–30/31

*3–32. A shear spring is made by bonding the rubber annulus to a rigid fixed ring and a plug. When an axial load P is placed on the plug, show that the slope at point y in the rubber is dy>dr = - tan g = - tan1P> 12phGr22. For small angles we can write dy >dr = -P> 12phGr2. Integrate this expression and evaluate the constant of integration using the condition that y = 0 at r = ro. From the result compute the deflection y = d of the plug.

ro ri

Prob. 3–32

CHAPTER REVIEW 115

CHAPTER REVIEW

One of the most important tests for material strength is the tension test. The results, found from stretching a specimen of known size, are plotted as normal stress on the vertical axis and normal strain on the horizontal axis.

Many engineering materials exhibit initial linear elastic behavior, whereby stress is proportional to strain, defined by Hooke’s law, s = EP. Here E , called the modulus of elasticity, is the slope of this straight line on the stress–strain diagram.

When the material is stressed beyond the yield point, permanent deformation will occur. In particular, steel has a region of yielding, whereby the material will exhibit an increase in strain with no increase in stress. The region of strain hardening causes further yielding of the material with a corresponding increase in stress. Finally, at the ultimate stress, a localized region on the specimen will begin to constrict, forming a neck. It is after this that the fracture occurs.

Ductile materials, such as most metals, exhibit both elastic and plastic behavior. Wood is moderately ductile. Ductility is usually specified by the percent elongation to failure or by the percent reduction in the cross-sectional area.

Percent elongation = Lf - L0

L0 1100%2

Percent reduction of area = A 0 - A f

A 0 1100%2

s = EP

P sE

ductile material

elastic region

yielding strain hardening

necking

elastic behavior

plastic behavior

elastic limit yield stress

ultimate stress fracture

stress

sY spl

proportional limit

116 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

The yield point of a material at A can be increased by strain hardening. This is accomplished by applying a load that causes the stress to be greater than the yield stress, then releasing the load. The larger stress A! becomes the new yield point for the material.

Brittle materials exhibit little or no yielding before failure. Cast iron, concrete, and glass are typical examples.

Modulus of resilience

Ppl

spl

When a load is applied to a member, the deformations cause strain energy to be stored in the material. The strain energy per unit volume or strain energy density is equivalent to the area under the stress–strain curve. This area up to the yield point is called the modulus of resilience. The entire area under the stress–strain diagram is called the modulus of toughness.

Modulus of toughness

permanent

set elastic

recovery

elastic region

plastic region

load

unloadE

A¿

P brittle material

CHAPTER REVIEW 117

n = - PlatPlong

Shear stress versus shear strain diagrams can also be established for a material. Within the elastic region, t = Gg, where G is the shear modulus, found from the slope of the line. The value of n can be obtained from the relationship that exists between G , E and n.

G = E

211 + n2

Poisson’s ratio n is a dimensionless material property that relates the lateral strain to the longitudinal strain. Its range of values is 0 … n … 0.5.

When materials are in service for long periods of time, considerations of creep become important. Creep is the time rate of deformation, which occurs at high stress and/or high temperature. Design requires that the stress in the material not exceed an allowable stress which is based on the material’s creep strength.

Fatigue can occur when the material undergoes a large number of cycles of loading. This effect will cause microscopic cracks to form, leading to a brittle failure. To prevent fatigue, the stress in the material must not exceed a specified endurance or fatigue limit.

P r

Final Shape

Original Shape

Tension

d/2

d¿

d/2

118 CHAPTER 3 MECHANICAL PROPERT IES OF MATER IALS

REVIEW PROBLEMS

3–35. The elastic portion of the tension stress–strain diagram for an aluminum alloy is shown in the figure. The specimen used for the test has a gauge length of 2 in. and a diameter of 0.5 in. When the applied load is 9 kip, the new diameter of the specimen is 0.49935 in. Compute the shear modulus Gal for the aluminum.

*3–36. The elastic portion of the tension stress–strain diagram for an aluminum alloy is shown in the figure. The specimen used for the test has a gauge length of 2 in. and a diameter of 0.5 in. If the applied load is 10 kip, determine the new diameter of the specimen. The shear modulus is Gal = 3.811032 ksi.

0.00614

s (ksi)

P (in./in.)

Probs. 3–35/36

3–37. The rigid beam rests in the horizontal position on two 2014-T6 aluminum cylinders having the unloaded lengths shown. If each cylinder has a diameter of 30 mm. determine the placement x of the applied 80-kN load so that the beam remains horizontal. What is the new diameter of cylinder A after the load is applied? n al = 0.35.

3 m

210 mm220 mm

A B

80 kN

Prob. 3–37

3–38. The wires each have a diameter of 12 in., length of 2 ft, and are made from 304 stainless steel. If P = 6 kip, determine the angle of tilt of the rigid beam AB .

3–39. The wires each have a diameter of 12 in., length of 2 ft, and are made from 304 stainless steel. Determine the magnitude of force P so that the rigid beam tilts 0.015°.

2 ft

1 ft

Probs. 3–38/39

*3–40. The head H is connected to the cylinder of a compressor using six steel bolts. If the clamping force in each bolt is 800 lb, determine the normal strain in the bolts. Each bolt has a diameter of 3 16 in. If sY = 40 ksi and Est = 2911032 ksi, what is the strain in each bolt when the nut is unscrewed so that the clamping force is released?

Prob. 3–40

REVIEW PROBLEMS 119

3–41. The stress–strain diagram for polyethylene, which is used to sheath coaxial cables, is determined from testing a specimen that has a gauge length of 10 in. If a load P on the specimen develops a strain of P = 0.024 in.> in., determine the approximate length of the specimen, measured between the gauge points, when the load is removed. Assume the specimen recovers elastically.

P 5

0 0.008 0.016 0.024 0.032 0.040 0.048

s (ksi)

0 P (in./in.)

Prob. 3–41

3–42. The pipe with two rigid caps attached to its ends is subjected to an axial force P . If the pipe is made from a material having a modulus of elasticity E and Poisson’s ratio n, determine the change in volume of the material.

L Section a – a

ri ro

Prob. 3–42

3–43. The 8-mm-diameter bolt is made of an aluminum alloy. It fits through a magnesium sleeve that has an inner diameter of 12 mm and an outer diameter of 20 mm. If the original lengths of the bolt and sleeve are 80 mm and 50 mm, respectively, determine the strains in the sleeve and the bolt if the nut on the bolt is tightened so that the tension in the bolt is 8 kN. Assume the material at A is rigid. Eal = 70 GPa, Emg = 45 GPa.

50 mm

30 mm

Prob. 3–43

*3–44. An acetal polymer block is fixed to the rigid plates at its top and bottom surfaces. If the top plate displaces 2 mm horizontally when it is subjected to a horizontal force P = 2 kN, determine the shear modulus of the polymer. The width of the block is 100 mm. Assume that the polymer is linearly elastic and use small angle analysis.

400 mm

200 mm

P ! 2 kN

Prob. 3–44

The string of drill pipe stacked on this oil rig will be subjected to extremely large axial deformations when it is placed in the hole.

Chapter 4

121

4.1 Saint-Venant’s Principle In the previous chapters, we have developed the concept of stress as a means of measuring the force distribution within a body and strain as a means of measuring a body’s deformation. We have also shown that the mathematical relationship between stress and strain depends on the type of material from which the body is made. In particular, if the material behaves in a linear elastic manner, then Hooke’s law applies, and there is a proportional relationship between stress and strain.

CHAPTER OBJECTIVES

■ In Chapter 1 , we developed the method for finding the normal stress in axially loaded members. In this chapter we will discuss how to determine the deformation of these members, and we will also develop a method for finding the support reactions when these reactions cannot be determined strictly from the equations of equilibrium. An analysis of the effects of thermal stress, stress concentrations, inelastic deformations, and residual stress will also be discussed.

Axial Load

122 CHAPTER 4 AXIAL LOAD

Using this idea, consider the manner in which a rectangular bar will deform elastically when the bar is subjected to a force P applied along its centroidal axis, Fig. 4–1 a . Here the bar is fixed connected at one end, with the force applied through a hole at its other end. Due to the loading, the bar deforms as indicated by the once horizontal and vertical grid lines drawn on the bar. Notice how the localized deformation that occurs at each end tends to even out and become uniform throughout the midsection of the bar.

If the material remains elastic then the strains caused by this deformation are directly related to the stress in the bar. As a result, the stress will be distributed more uniformly throughout the cross-sectional area when the section is taken farther and farther from the point where any external load is applied. For example, consider a profile of the variation of the stress distribution acting at sections a – a , b – b , and c – c , each of which is shown in Fig. 4–1 b . By comparison, the stress tends to reach a uniform value at section c – c , which is sufficiently removed from the end since the localized deformation caused by P vanishes . The minimum distance from the bar’s end where this occurs can be determined using a mathematical analysis based on the theory of elasticity.

It has been found that this distance should at least be equal to the largest dimension of the loaded cross section. Hence, section c – c should be located at a distance at least equal to the width (not the thickness) of the bar. *

* When section c – c is so located, the theory of elasticity predicts the maximum stress to be s max = 1.02s avg.

(a)

a b c

Lines located away from the load and support remain straight

Load distorts lines located near load

Load distorts lines located near support

Fig. 4–1

4.1 SAINT-VENANT’S PRINCIPLE 123

4 In the same way, the stress distribution at the support will also even

out and become uniform over the cross section located the same distance away from the support.

The fact that stress and deformation behave in this manner is referred to as Saint-Venant’s principle , since it was first noticed by the French scientist Barré de Saint-Venant in 1855. Essentially it states that the stress and strain produced at points in a body sufficiently removed from the region of load application will be the same as the stress and strain produced by any applied loadings that have the same statically equivalent resultant, and are applied to the body within the same region . For example, if two symmetrically applied forces P>2 act on the bar, Fig. 4–1 c , the stress distribution at section c – c will be uniform and therefore equivalent to s avg = P>A as in Fig. 4–1 c .

section a–a section b–b

(b)

section c–c

PPP

savg ! P A

section c–c (c)

savg !

P 2

P A

Notice how the lines on this rubber membrane distort after it is stretched. The localized distortions at the grips smooth out as stated by Saint-Venant’s principle.

Fig. 4–1 (cont.)

124 CHAPTER 4 AXIAL LOAD

4.2 Elastic Deformation of an Axially Loaded Member

Using Hooke’s law and the definitions of stress and strain, we will now develop an equation that can be used to determine the elastic displacement of a member subjected to axial loads. To generalize the development, consider the bar shown in Fig. 4–2 a , which has a cross-sectional area that gradually varies along its length L . The bar is subjected to concentrated loads at its ends and a variable external load distributed along its length. This distributed load could, for example, represent the weight of the bar if it is in the vertical position, or friction forces acting on the bar’s surface. Here we wish to find the relative displacement d (delta) of one end of the bar with respect to the other end as caused by this loading. We will neglect the localized deformations that occur at points of concentrated loading and where the cross section suddenly changes. From Saint-Venant’s principle, these effects occur within small regions of the bar’s length and will therefore have only a slight effect on the final result. For the most part, the bar will deform uniformly, so the normal stress will be uniformly distributed over the cross section.

Using the method of sections, a differential element (or wafer) of length dx and cross-sectional area A ( x ) is isolated from the bar at the arbitrary position x . The free-body diagram of this element is shown in Fig. 4–2 b . The resultant internal axial force will be a function of x since the external distributed loading will cause it to vary along the length of the bar. This load, P ( x ), will deform the element into the shape indicated by the dashed outline, and therefore the displacement of one end of the element with respect to the other end is dd. The stress and strain in the element are

s = P (x) A(x)

and P = dd dx

Provided the stress does not exceed the proportional limit, we can apply Hooke’s law; i.e.,

s = E(x)P

P(x) A (x)

= E(x)a dd dx

b dd =

P(x)dx A (x)E(x)

dx dd

(b)

P(x) P(x)P2P1

x dx

(a) d

Fig. 4–2

4.2 ELASTIC DEFORMATION OF AN AXIALLY LOADED MEMBER 125

For the entire length L of the bar, we must integrate this expression to find d . This yields

d = L L

P(x)dx A (x)E(x)

(4–1)

where d = displacement of one point on the bar relative to the other point L = original length of bar P (x) = internal axial force at the section, located a distance x from

one end A (x) = cross-sectional area of the bar expressed as a function of x E(x) = modulus of elasticity for the material expressed as a function of x .

Constant Load and Cross-Sectional Area. In many cases the bar will have a constant cross-sectional area A ; and the material will be homogeneous, so E is constant. Furthermore, if a constant external force is applied at each end, Fig. 4–3 , then the internal force P throughout the length of the bar is also constant. As a result, Eq. 4–1 can be integrated to yield

d = PL AE

(4–2)

If the bar is subjected to several different axial forces along its length, or the cross-sectional area or modulus of elasticity changes abruptly from one region of the bar to the next, the above equation can be applied to each segment of the bar where these quantities remain constant . The displacement of one end of the bar with respect to the other is then found from the algebraic addition of the relative displacements of the ends of each segment. For this general case,

d = a

PL AE

(4–3)

Fig. 4–3

The vertical displacement of the rod at the top floor B only depends upon the force in the rod along length AB. However, the displacement at the bottom floor C depends upon the force in the rod along its entire length, ABC.

126 CHAPTER 4 AXIAL LOAD

Sign Convention. In order to apply Eq. 4–3 , we must develop a sign convention for the internal axial force and the displacement of one end of the bar with respect to the other end. To do so, we will consider both the force and displacement to be positive if they cause tension and elongation , respectively, Fig. 4–4 ; whereas a negative force and displacement will cause compression and contraction , respectively.

For example, consider the bar shown in Fig. 4–5 a . The internal axial forces “ P ,” are determined by the method of sections for each segment, Fig. 4–5 b . They are PAB = +5 kN, P BC = -3 kN, PCD = -7 kN. This variation in axial load is shown on the axial or normal force diagram for the bar, Fig. 4–5 c . Since we now know how the internal force varies throughout the bar’s length, the displacement of end A relative to end D is determined from

dA >D = a PLAE = (5 kN)L ABAE + (-3 kN)L BCAE + (-7 kN)L CDAE If the other data are substituted and a positive answer is calculated, it means that end A will move away from end D (the bar elongates), whereas a negative result would indicate that end A moves toward end D (the bar shortens). The double subscript notation is used to indicate this relative displacement (dA /D); however, if the displacement is to be determined relative to a fixed point , then only a single subscript will be used. For example, if D is located at a fixed support, then the displacement will be denoted as simply dA.

Fig. 4–4

Positive sign convention for P and d

A B

8 kN

(b)

D 7 kN

PAB ! 5 kN

PBC ! 3 kN

PCD ! 7 kN

5 kN

P (kN)

"3 "7

(c)

DA B C

8 kN 4 kN

LAB LBC LCD

(a)

5 kN 7 kN

Fig. 4–5

4.2 ELASTIC DEFORMATION OF AN AXIALLY LOADED MEMBER 127

Procedure for Analysis

The relative displacement between any two points A and B on an axially loaded member can be determined by applying Eq. 4–1 (or Eq. 4–2 ). Application requires the following steps.

Internal Force. • Use the method of sections to determine the internal axial force P within the member. • If this force varies along the member’s length due to an external distributed loading , a section should be

made at the arbitrary location x from one end of the member and the force represented as a function of x , i.e., P ( x ).

• If several constant external forces act on the member, the internal force in each segment of the member, between any two external forces, must be determined.

• For any segment, an internal tensile force is positive and an internal compressive force is negative . For convenience, the results of the internal loading can be shown graphically by constructing the normal- force diagram.

Displacement. • When the member’s cross-sectional area varies along its length, the area must be expressed as a function

of its position x , i.e., A ( x ). • If the cross-sectional area, the modulus of elasticity, or the internal loading suddenly changes , then Eq. 4–2

should be applied to each segment for which these quantities are constant. • When substituting the data into Eqs. 4–1 through 4–3 , be sure to account for the proper sign for the

internal force P . Tensile loadings are positive and compressive loadings are negative. Also, use a consistent set of units. For any segment, if the result is a positive numerical quantity, it indicates elongation ; if it is negative , it indicates a contraction .

Important Points

• Saint-Venant’s principle states that both the localized deformation and stress which occur within the regions of load application or at the supports tend to “even out” at a distance sufficiently removed from these regions.

• The displacement of one end of an axially loaded member relative to the other end is determined by relating the applied internal load to the stress using s = P>A and relating the displacement to the strain using P = dd>dx. Finally these two equations are combined using Hooke’s law, s = EP, which yields Eq. 4–1 .

• Since Hooke’s law has been used in the development of the displacement equation, it is important that no internal load causes yielding of the material, and that the material behaves in a linear elastic manner.

128 CHAPTER 4 AXIAL LOAD

The assembly shown in Fig. 4–6 a consists of an aluminum tube AB having a cross-sectional area of 400 mm2. A steel rod having a diameter of 10 mm is attached to a rigid collar and passes through the tube. If a tensile load of 80 kN is applied to the rod, determine the displacement of the end C of the rod. Take Est = 200 GPa, Eal = 70 GPa.

EXAMPLE 4.1

80 kN

(b)

PAB ! 80 kN

PBC ! 80 kN

400 mm

600 mm

80 kN

(a)

Fig. 4–6

SOLUTION

Internal Force. The free-body diagram of the tube and rod segments in Fig. 4–6 b , shows that the rod is subjected to a tension of 80 kN and the tube is subjected to a compression of 80 kN.

Displacement. We will first determine the displacement of end C with respect to end B . Working in units of newtons and meters, we have

dC>B = PLAE = [+80(103) N] (0.6 m)p (0.005 m)2[200 (109) N>m2] = +0.003056 m S The positive sign indicates that end C moves to the right relative to end B , since the bar elongates.

The displacement of end B with respect to the fixed end A is

dB = PL AE

= [-80(103) N](0.4 m)

[400 mm2(10-6) m2>mm2][70(109) N>m2] = -0.001143 m = 0.001143 m S

Here the negative sign indicates that the tube shortens, and so B moves to the right relative to A .

Since both displacements are to the right, the displacement of C relative to the fixed end A is therefore

( S+ ) d C = d B + d C>B = 0.001143 m + 0.003056 m = 0.00420 m = 4.20 mm S Ans.

4.2 ELASTIC DEFORMATION OF AN AXIALLY LOADED MEMBER 129

Rigid beam AB rests on the two short posts shown in Fig. 4–7 a . AC is made of steel and has a diameter of 20 mm, and BD is made of aluminum and has a diameter of 40 mm. Determine the displacement of point F on AB if a vertical load of 90 kN is applied over this point. Take Est = 200 GPa, Eal = 70 GPa.

EXAMPLE 4.2

200 mm 400 mm

300 mm

90 kN

(a)

200 mm 400 mm

90 kN

60 kN 30 kN (b)

60 kN

PAC ! 60 kN

30 kN

PBD ! 30 kN(c)

Fig. 4–7

400 mm

0.184 mm

0.286 mm

0.102 mm 600 mm

0.102 mm

BA F

(d)

SOLUTION

Internal Force. The compressive forces acting at the top of each post are determined from the equilibrium of member AB , Fig. 4–7 b . These forces are equal to the internal forces in each post, Fig. 4–7 c .

Displacement. The displacement of the top of each post is

Post AC :

dA = PACL AC A ACEst

= [-60(103) N](0.300 m)

p(0.010 m)2[200(109) N>m2] = -286(10-6) m = 0.286 mm T

Post BD :

dB = PBDL BD A BDEal

= [-30(103)N](0.300 m)

p(0.020 m)2[70(109) N>m2] = -102(10-6) m = 0.102 mm T

A diagram showing the centerline displacements at A, B , and F on the beam is shown in Fig. 4–7 d . By proportion of the blue shaded triangle, the displacement of point F is therefore

dF = 0.102 mm + (0.184 mm) a 400 mm600 mmb = 0.225 mm T Ans.

130 CHAPTER 4 AXIAL LOAD

A member is made from a material that has a specific weight g and modulus of elasticity E . If it is in the form of a cone having the dimensions shown in Fig. 4–8 a , determine how far its end is displaced due to gravity when it is suspended in the vertical position.

SOLUTION Internal Force. The internal axial force varies along the member since it is dependent on the weight W ( y ) of a segment of the member below any section, Fig. 4–8 b . Hence, to calculate the displacement, we must use Eq. 4–1 . At the section located a distance y from its free end, the radius x of the cone as a function of y is determined by proportion; i.e.,

x y

= r0 L

; x = r0 L

The volume of a cone having a base of radius x and height y is

V = 1 3

pyx2 = pr0

3L2 y3

Since W = gV , the internal force at the section becomes

+ c !Fy = 0; P(y) = gpr0

3L2 y3

Displacement. The area of the cross section is also a function of position y , Fig. 4–8 b . We have

A (y) = px2 = pr0

L2 y2

Applying Eq. 4–1 between the limits of y = 0 and y = L yields

d = L L

P(y) dy A (y) E

= L L

3(gpr02>3L2) y34 dy3(pr02>L2)y24 E =

g 3E

L L

0 y dy

= gL 2 6E

Ans.

NOTE: As a partial check of this result, notice how the units of the terms, when canceled, give the displacement in units of length as expected.

EXAMPLE 4.3

(a)

W(y)

(b)

P(y) x

Fig. 4–8

4.2 ELASTIC DEFORMATION OF AN AXIALLY LOADED MEMBER 131

P4–1. In each case, determine the internal normal force between lettered points on the bar. Draw all necessary free- body diagrams.

BA C D E

(a)

100 N 200 N 400 N 700 N

BA C D

(b)

400 N 600 N

300 N

P4–1

P4–2. Determine the internal normal force between lettered points on the cable and rod. Draw all necessary free-body diagrams.

400 N

900 N

500 N

A B

P4–2

P4–3. The post weighs 8 kN>m. Determine the internal normal force in the post as a function of x .

2 m

P4–3

P4–4. The rod is subjected to an external axial force of 800 N and a uniform distributed load of 100 N>m along its length. Determine the internal normal force in the rod as a function of x .

100 N/m

800 N A

2 m x

P4–4

P4–5. The rigid beam supports the load of 60 kN. Determine the displacement at B . Take E = 60 GPa, and A BC = 2 (10 −3 ) m 2 .

3 m

2 m

2 m 4 m

60 kN

P4–5

PRELIMINARY PROBLEMS

132 CHAPTER 4 AXIAL LOAD

F4–4. If the 20-mm-diameter rod is made of A-36 steel and the stiffness of the spring is k = 50 MN>m, determine the displacement of end A when the 60-kN force is applied.

k ! 50 MN/m400 mm

400 mm

60 kN

F4–4

F4–5. The 20-mm-diameter 2014-T6 aluminum rod is subjected to the uniform distributed axial load. Determine the displacement of end A .

900 mm

30 kN/m

F4–5

F4–6. The 20-mm-diameter 2014-T6 aluminum rod is subjected to the triangular distributed axial load. Determine the displacement of end A .

900 mm

45 kN/m

F4–6

FUNDAMENTAL PROBLEMS

F4–1. The 20-mm-diameter A-36 steel rod is subjected to the axial forces shown. Determine the displacement of end C with respect to the fixed support at A .

CB A

50 kN 40 kN

50 kN

600 mm 400 mm

F4–1

F4–2. Segments AB and CD of the assembly are solid circular rods, and segment BC is a tube. If the assembly is made of 6061-T6 aluminum, determine the displacement of end D with respect to end A.

10 kN 15 kN

15 kN

20 kN 10 kN

10 kN 400 mm400 mm 400 mm

20 mm 20 mm

aB D

40 mm30 mm

Section a-a

F4–2

F4–3. The 30-mm-diameter A992 steel rod is subjected to the loading shown. Determine the displacement of end C .

A B C

600 mm

90 kN

30 kN

30 kN 400 mm

F4–3

4.2 ELASTIC DEFORMATION OF AN AXIALLY LOADED MEMBER 133

PROBLEMS

4–1. The A992 steel rod is subjected to the loading shown. If the cross-sectional area of the rod is 60 mm 2 , determine the displacement of B and A , Neglect the size of the couplings at B , C , and D .

0.75 m

1.50 m

0.50 m

3.30 kN

60!

2 kN

3 44

3.30 kN

60!

2 kN

8 kN

Prob. 4–1

4–2. The copper shaft is subjected to the axial loads shown. Determine the displacement of end A with respect to end D if the diameters of each segment are d AB = 0.75 in., d BC = 1 in., and d CD = 0.5 in. Take E cu = 18(10 3 ) ksi.

6 kip8 kip

A 5 kip

5 kip

2 kip

B C D

80 in. 150 in. 100 in.

Prob. 4–2

4–3. The composite shaft, consisting of aluminum, copper, and steel sections, is subjected to the loading shown. Determine the displacement of end A with respect to end D and the normal stress in each section. The cross- sectional area and modulus of elasticity for each section are shown in the figure. Neglect the size of the collars at B and C .

*4–4. Determine the displacement of B with respect to C of the composite shaft in Prob. 4–3 .

Aluminum SteelCopper = 10(10Eal

3) ksi = 18(10Ecu 3) ksi = 29(10Est

3) ksi = 0.09 inAAB

2 = 0.12 inABC 2 = 0.06 inACD

1.50 kip2.00 kip 3.50 kip 1.75 kip

12 in. 16 in.

CBA D

18 in.

3.50 kip 1.75 kip

Probs. 4–3/4

4–5. The assembly consists of a steel rod CB and an aluminum rod BA , each having a diameter of 12 mm. If the rod is subjected to the axial loadings at A and at the coupling B , determine the displacement of the coupling B and the end A . The unstretched length of each segment is shown in the figure. Neglect the size of the connections at B and C , and assume that they are rigid. E st = 200 GPa, E al = 70 GPa.

18 kN

2 m3 m

6 kN

B AC

Prob. 4–5

134 CHAPTER 4 AXIAL LOAD

4–9. The assembly consists of two 10-mm diameter red brass C83400 copper rods AB and CD , a 15-mm diameter 304 stainless steel rod EF , and a rigid bar G . If P = 5 kN, determine the horizontal displacement of end F of rod EF .

4–10. The assembly consists of two 10-mm diameter red brass C83400 copper rods AB and CD , a 15-mm diameter 304 stainless steel rod EF, and a rigid bar G . If the horizontal displacement of end F of rod EF is 0.45 mm, determine the magnitude of P .

A B

C D G

450 mm300 mm

Probs. 4–9/10

4–11. The load is supported by the four 304 stainless steel wires that are connected to the rigid members AB and DC. Determine the vertical displacement of the 500-lb load if the members were originally horizontal when the load was applied. Each wire has a cross-sectional area of 0.025 in2.

*4–12. The load is supported by the four 304 stainless steel wires that are connected to the rigid members AB and DC. Determine the angle of tilt of each member after the 500-lb load is applied. The members were originally horizontal, and each wire has a cross-sectional area of 0.025 in2.

1.8 ft 1 ft 2 ft

A B 3 ft 1 ft

5 ft

3 ft

E F G

500 lb

Probs. 4–11/12

4–6. The bar has a cross-sectional area of 3 in2, and E = 35 (103) ksi. Determine the displacement of its end A when it is subjected to the distributed loading.

w ! 500x1/3 lb/in.

4 ft

Prob. 4–6

4–7. If P 1 = 50 kip and P 2 = 150 kip, determine the vertical displacement of end A of the high strength precast concrete column.

*4–8. If the vertical displacements of end A of the high strength precast concrete column relative to B and B relative to C are 0.08 in. and 0.1 in., respectively, determine the magnitudes of P 1 and P 2 .

10 ft

Section a-a

b b

a a 6 in.

6 in.

Section b-b

10 in.

P1P1

P2 P2

Probs. 4–7/8

4.2 ELASTIC DEFORMATION OF AN AXIALLY LOADED MEMBER 135

*4–16. The hanger consists of three 2014-T6 aluminum alloy rods, rigid beams AC and BD , and a spring. If the hook supports a load of P = 60 kN, determine the vertical displacement of F . Rods AB and CD each have a diameter of 10 mm, and rod EF has a diameter of 15 mm. The spring has a stiffness of k = 100 MN>m and is unstretched when P = 0. 4–17. The hanger consists of three 2014-T6 aluminum alloy rods , rigid beams AC and BD , and a spring. If the vertical displacement of end F is 5 mm, determine the magnitude of the load P . Rods AB and CD each have a diameter of 10 mm, and rod EF has a diameter of 15 mm. The spring has a stiffness of k = 100 MN>m and is unstretched when P = 0.

B D

450 mm

Probs. 4–16/17

4–18. Collar A can slide freely along the smooth vertical guide. If the supporting rod AB is made of 304 stainless steel and has a diameter of 0.75 in., determine the vertical displacement of the collar when P = 10 kip.

4–19. Collar A can slide freely along the smooth vertical guide. If the vertical displacement of the collar is 0.035 in. and the supporting 0.75 in. diameter rod AB is made of 304 stainless steel, determine the magnitude of P .

1.5 ft

2 ft

Probs. 4–18/19

4–13. The rigid bar is supported by the pin-connected rod CB that has a cross-sectional area of 14 mm 2 and is made from 6061-T6 aluminum. Determine the vertical deflection of the bar at D when the distributed load is applied.

2 m 2 m

1.5 m

300 N/m

Prob. 4–13

4–14. The post is made of Douglas fir and has a diameter of 60 mm. If it is subjected to the load of 20 kN and the soil provides a frictional resistance that is uniformly distributed along its sides of w = 4 kN>m , determine the force F at its bottom needed for equilibrium. Also, what is the displacement of the top of the post A with respect to its bottom B? Neglect the weight of the post.

4–15. The post is made of Douglas fir and has a diameter of 60 mm. If it is subjected to the load of 20 kN and the soil provides a frictional resistance that is distributed along its length and varies linearly from w = 0 at y = 0 to w = 3 kN>m at y = 2 m, determine the force F at its bottom needed for equilibrium. Also, what is the displacement of the top of the post A with respect to its bottom B? Neglect the weight of the post.

2 m

20 kN

B F

Probs. 4–14/15

136 CHAPTER 4 AXIAL LOAD

4–23. The rod has a slight taper and length L. It is suspended from the ceiling and supports a load P at its end. Show that the displacement of its end due to this load is d = PL >(pEr2r1). Neglect the weight of the material. The modulus of elasticity is E.

Prob. 4–23

*4–24. Determine the relative displacement of one end of the tapered plate with respect to the other end when it is subjected to an axial load P .

Prob. 4–24

4–25. Determine the elongation of the A-36 steel member when it is subjected to an axial force of 30 kN. The member is 10 mm thick. Use the result of Prob. 4–24 .

30 kN 30 kN

0.5 m

20 mm

75 mm

Prob. 4–25

*4–20. The A992 steel drill shaft of an oil well extends 12000 ft into the ground. Assuming that the pipe used to drill the well is suspended freely from the derrick at A , determine the maximum average normal stress in each pipe segment and the elongation of its end D with respect to the fixed end at A . The shaft consists of three different sizes of pipe, AB , BC , and CD , each having the length, weight per unit length, and cross-sectional area indicated.

AAB= 2.50 in.2 wAB= 3.2 lb/ft

ABC = 1.75 in.2 wBC = 2.8 lb/ft

ACD= 1.25 in.2 wCD= 2.0 lb/ft

5000 ft

2000 ft

Prob. 4–20

4–21. A spring-supported pipe hanger consists of two springs which are originally unstretched and have a stiffness of k = 60 kN>m , three 304 stainless steel rods, AB and CD, which have a diameter of 5 mm, and EF, which has a diameter of 12 mm, and a rigid beam GH. If the pipe and the fluid it carries have a total weight of 4 kN, determine the displacement of the pipe when it is attached to the support.

4–22. A spring-supported pipe hanger consists of two springs, which are originally unstretched and have a stiffness of k = 60 kN>m , three 304 stainless steel rods, AB and CD , which have a diameter of 5 mm, and EF , which has a diameter of 12 mm, and a rigid beam GH. If the pipe is displaced 82 mm when it is filled with fluid, determine the combined weight of the pipe and fluid.

A C

G H E

0.25 m0.25 m

0.75 m

k k

0.75 m

Probs. 4–21/22

4.2 ELASTIC DEFORMATION OF AN AXIALLY LOADED MEMBER 137

4–26. Determine the elongation of the tapered A992 steel shaft when it is subjected to an axial force of 18 kip. Hint: Use the result of Prob. 4–23 .

18 kip 18 kip4 in. 4 in.20 in.

r1 ! 0.5 in. r1 ! 0.5 in.r2 ! 2 in.

Prob. 4–26

4–27. The circular bar has a variable radius of r = r0eax and is made of a material having a modulus of elasticity of E . Determine the displacement of end A when it is subjected to the axial force P .

r ! r0 e

A P

Prob. 4–27

*4–28. Bone material has a stress–strain diagram that can be defined by the relation s = E3P>(1 + kEP)4 , where k and E are constants. Determine the compression within the length L of the bone, where it is assumed the cross-sectional area A of the bone is constant.

Prob. 4–28

4–29. The weight of the kentledge exerts an axial force of P = 1500 kN on the 300-mm diameter high-strength concrete bore pile. If the distribution of the resisting skin friction developed from the interaction between the soil and the surface of the pile is approximated as shown, and the resisting bearing force F is required to be zero, determine the maximum intensity p0 kN>m for equilibrium. Also, find the corresponding elastic shortening of the pile. Neglect the weight of the pile.

12 m

Prob. 4–29

4–30. The weight of the kentledge exerts an axial force of P = 1500 kN on the 300-mm diameter high-strength concrete bore pile. If the distribution of the resisting skin friction developed from the interaction between the soil and the surface of the pile is approximated as shown, determine the resisting bearing force F for equilibrium. Take p 0 = 180 kN>m. Also, find the corresponding elastic shortening of the pile. Neglect the weight of the pile.

12 m

Prob. 4–30

138 CHAPTER 4 AXIAL LOAD

4.3 Principle of Superposition The principle of superposition is often used to determine the stress or displacement at a point in a member when the member is subjected to a complicated loading. By subdividing the loading into components, the principle of superposition states that the resultant stress or displacement at the point can be determined by algebraically summing the stress or displacement caused by each load component applied separately to the member.

The following two conditions must be satisfied if the principle of superposition is to be applied.

1. The loading must be linearly related to the stress or displacement that is to be determined . For example, the equations s = P>A and d = PL >AE involve a linear relationship between P and s or d.

2. The loading must not significantly change the original geometry or configuration of the member. If significant changes do occur, the direction and location of the applied forces and their moment arms will change. For example, consider the slender rod shown in Fig. 4–9 a , which is subjected to the load P . In Fig. 4–9 b , P is replaced by two of its components, P = P1 + P2 . If P causes the rod to deflect a large amount, as shown, the moment of the load about its support, Pd, will not equal the sum of the moments of its component loads, Pd ! P1d1 + P2d2 , because d1 ! d2 ! d.

This principle will be used throughout this text whenever we assume Hooke’s law applies and also, the bodies that are considered will be such that the loading will produce deformations that are so small that the change in position and direction of the loading will be insignificant and can be neglected.

(a)

(b)

! d1

" P1

Fig. 4–9

4.4 STATICALLY INDETERMINATE AXIALLY LOADED MEMBER 139

4.4 Statically Indeterminate Axially Loaded Member

Consider the bar shown in Fig. 4–10 a which is fixed supported at both of its ends. From the free-body diagram, Fig. 4–10 b , equilibrium requires

+ c !F = 0; FB + FA - P = 0

This type of problem is called statically indeterminate , since the equilibrium equation(s) are not sufficient to determine the two reactions on the bar.

In order to establish an additional equation needed for solution, it is necessary to consider how points on the bar displace. Specifically, an equation that specifies the conditions for displacement is referred to as a compatibility or kinematic condition . In this case, a suitable compatibility condition would require the displacement of one end of the bar with respect to the other end to be equal to zero, since the end supports are fixed. Hence, the compatibility condition becomes

dA >B = 0 This equation can be expressed in terms of the applied loads by using

a load – displacement relationship , which depends on the material behavior. For example, if linear-elastic behavior occurs, d = PL >AE can be used. Realizing that the internal force in segment AC is +FA , and in segment CB the internal force is -FB , Fig. 4–10 c , the above equation can be written as

FALAC

AE -

FBLCB AE

= 0

Since AE is constant, then FA = FB(LCB>LAC) , so that using the equilibrium equation, the equations for the reactions become

FA = P¢LCBL ≤ and FB = P¢LACL ≤ Since both of these results are positive, the direction of the reactions is shown correctly on the free-body diagram.

LAC

LCB

B (a)

(c)

FA FA

FA FB

FB (b)

Fig. 4–10

140 CHAPTER 4 AXIAL LOAD

Important Points

• The principle of superposition is sometimes used to simplify stress and displacement problems having complicated loadings. This is done by subdividing the loading into components, then algebracially adding the results.

• Superposition requires that the loading be linearly related to the stress or displacement, and the loading does not significantly change the original geometry of the member.

• A problem is statically indeterminate if the equations of equilibrium are not sufficient to determine all the reactions on a member.

• Compatibility conditions specify the displacement constraints that occur at the supports or other points on a member.

Most concrete columns are reinforced with steel rods; and since these two materials work together in supporting the applied load, the forces in each material become statically indeterminate.

Procedure for Analysis

The support reactions for statically indeterminate problems are determined by satisfying equilibrium, compatibility, and force- displacement requirements for the member.

Equilibrium. • Draw a free-body diagram of the member in order to identify all

the forces that act on it. • The problem can be classified as statically indeterminate if the

number of unknown reactions on the free-body diagram is greater than the number of available equations of equilibrium.

• Write the equations of equilibrium for the member.

Compatibility. • Consider drawing a displacement diagram in order to investigate

the way the member will elongate or contract when subjected to the external loads.

• Express the compatibility conditions in terms of the displacements caused by the loading.

Load-Displacement. • Use a load–displacement relation, such as d = PL >AE, to relate

the unknown displacements to the reactions. • Solve the equations for the reactions. If any of the results has a

negative numerical value, it indicates that this force acts in the opposite sense of direction to that indicated on the free-body diagram.

4.4 STATICALLY INDETERMINATE AXIALLY LOADED MEMBER 141

EXAMPLE 4.4

The steel rod shown in Fig. 4–11 a has a diameter of 10 mm. It is fixed to the wall at A , and before it is loaded, there is a gap of 0.2 mm between the wall at B ! and the rod. Determine the reactions at A and B ! if the rod is subjected to an axial force of P = 20 kN as shown. Neglect the size of the collar at C . Take Est = 200 GPa.

SOLUTION Equilibrium. As shown on the free-body diagram, Fig. 4–11 b , we will assume that force P is large enough to cause the rod’s end B to contact the wall at B !. The problem is statically indeterminate since there are two unknowns and only one equation of equilibrium.

S + "F x = 0; -F A - F B + 20(103) N = 0 (1)

Compatibility. The force P causes point B to move to B !, with no further displacement. Therefore the compatibility condition for the rod is

d B>A = 0.0002 m Load-Displacement. This displacement can be expressed in terms of the unknown reactions using the load–displacement relationship, Eq. 4–2 , applied to segments AC and CB , Fig. 4–11 c . Working in units of newtons and meters, we have

d B>A = 0.0002 m = FA L AC AE - FB L CB AE 0.0002 m =

FA(0.4 m)

p(0.005 m)2 [200(109 ) N>m2] -

F B (0.8 m)

p(0.005 m) 2 [200(10 9 ) N>m 2] or

FA (0.4 m) - F B (0.8 m) = 3141.59 N~m (2)

Solving Eqs. 1 and 2 yields FA = 16.0 kN F B = 4.05 kN Ans.

Since the answer for F B is positive , indeed end B contacts the wall at B ! as originally assumed.

NOTE: If F B were a negative quantity, the problem would be statically determinate, so that FB = 0 and FA = 20 kN.

400 mm 800 mm

A B¿ C B

P ! 20 kN

(a)

0.2 mm

FA FB

(b)

P ! 20 kN

(c)

FA FA

Fig. 4–11

142 CHAPTER 4 AXIAL LOAD

The aluminum post shown in Fig. 4–12 a is reinforced with a brass core. If this assembly supports an axial compressive load of P = 9 kip, applied to the rigid cap, determine the average normal stress in the aluminum and the brass. Take Eal = 10(103) ksi and Ebr = 15(103) ksi.

SOLUTION Equilibrium. The free-body diagram of the post is shown in Fig. 4–12 b . Here the resultant axial force at the base is represented by the unknown components carried by the aluminum, Fal, and brass, Fbr . The problem is statically indeterminate. Why?

Vertical force equilibrium requires

+ c !F y = 0; -9 kip + F al + F br = 0 (1)

Compatibility. The rigid cap at the top of the post causes both the aluminum and brass to displace the same amount. Therefore,

dal = dbr

Load-Displacement. Using the load–displacement relationships,

Fal L

A al Eal =

F br L A br Ebr

Fal = Fbr a A al A br b a Eal Ebr b Fal = Fbr J p[(2 in.) 2 - (1 in.) 2]

p(1 in.)2 R J10(103) ksi

15(103) ksi R

Fal = 2Fbr (2)

Solving Eqs. 1 and 2 simultaneously yields

Fal = 6 kip Fbr = 3 kip

Since the results are positive, indeed the stress will be compressive. The average normal stress in the aluminum and brass is therefore

sal = 6 kip

p[(2 in.)2 - (1 in.)2] = 0.637 ksi Ans.

sbr = 3 kip

p(1 in.)2 = 0.955 ksi Ans.

NOTE: Using these results, the stress distributions are shown in Fig. 4–12 c .

EXAMPLE 4.5

(c)

sal ! 0.637 ksi

sbr ! 0.955 ksi

Fig. 4–12

P ! 9 kip

(b)

Fbr

Fal

1.5 ft

P ! 9 kip

(a)

1 in.2 in.

4.4 STATICALLY INDETERMINATE AXIALLY LOADED MEMBER 143

The three A992 steel bars shown in Fig. 4–13 a are pin connected to a rigid member. If the applied load on the member is 15 kN, determine the force developed in each bar. Bars AB and EF each have a cross- sectional area of 50 mm2, and bar CD has a cross-sectional area of 30 mm2.

SOLUTION Equilibrium. The free-body diagram of the rigid member is shown in Fig. 4–13 b . This problem is statically indeterminate since there are three unknowns and only two available equilibrium equations.

+ c !F y = 0; F A + F C + F E - 15 kN = 0 (1)

a+ !MC = 0; -F A(0.4 m) + 15 kN(0.2 m) + F E (0.4 m) = 0 (2)

Compatibility. The applied load will cause the horizontal line ACE shown in Fig. 4–13 c to move to the inclined line A"C"E". The displacements of points A , C , and E can be related by similar triangles. Thus the compatibility equation that relates these displacements is

dA - dE

0.8 m =

dC - dE 0.4 m

dC = 1 2

dA + 1 2

Load-Displacement. Using the load–displacement relationship, Eq. 4–2 , we have

F C L

(30 mm2 )E st =

1 2

c F A L (50 mm2 )E st

d + 1 2

c F E L (50 mm2 )E st

d F C = 0.3F A + 0.3F E (3)

Solving Eqs. 1 – 3 simultaneously yields

F A = 9.52 kN Ans.

F C = 3.46 kN Ans.

F E = 2.02 kN Ans.

EXAMPLE 4.6

0.4 m

(c)

C 0.4 m

A E

A¿ C ¿

E ¿

dA ! dE

dC ! dE

dC dE

Fig. 4–13

15 kN

0.4 m

(a)

0.5 m

0.2 m 0.2 m

15 kN

0.2 m 0.2 m 0.4 m

(b)

FA FC FE

144 CHAPTER 4 AXIAL LOAD

The bolt shown in Fig. 4–14 a is made of 2014-T6 aluminum alloy and is tightened so it compresses a cylindrical tube made of Am 1004-T61 magnesium alloy. The tube has an outer radius of 12 in., and it is assumed that both the inner radius of the tube and the radius of the bolt are 14 in. The washers at the top and bottom of the tube are considered to be rigid and have a negligible thickness. Initially the nut is hand tightened snugly; then, using a wrench, the nut is further tightened one-half turn. If the bolt has 20 threads per inch, determine the stress in the bolt.

SOLUTION

Equilibrium. The free-body diagram of a section of the bolt and the tube, Fig. 4–14 b , is considered in order to relate the force in the bolt F b to that in the tube, F t . Equilibrium requires

+ c !F y = 0; F b - F t = 0 (1)

Compatibility. When the nut is tightened on the bolt, the tube will shorten d t , and the bolt will elongate d b , Fig. 4–14 c . Since the nut undergoes one-half turn, it advances a distance of (12) (

1 20 in.) = 0.025 in.

along the bolt. Thus, the compatibility of these displacements requires

(+ c ) d t = 0.025 in. - d b Load-Displacement. Taking the moduli of elasticity from the table on the inside back cover, and applying the load-displacement relationship, Eq. 4–2 , yields

F t (3 in.)

p[(0.5 in.)2 - (0.25 in.)2] [6.48(103) ksi] =

0.025 in. - F b (3 in.)

p(0.25 in.)2 [10.6(103) ksi]

0.78595F t = 25 - 1.4414F b (2)

Solving Eqs. 1 and 2 simultaneously, we get F b = F t = 11.22 kip

The stresses in the bolt and tube are therefore

s b = F b A b

= 11.22 kip

p(0.25 in.)2 = 57.2 ksi Ans.

s t = F t A t

= 11.22 kip

p[(0.5 in.)2 - (0.25 in.)2] = 19.1 ksi

These stresses are less than the reported yield stress for each material, (s Y) al = 60 ksi and (s Y) mg = 22 ksi (see the inside back cover), and therefore this “elastic” analysis is valid.

EXAMPLE 4.7

(c)

0.025 in.

Initial position

Final position

Fig. 4–14

3 in.

in.

(a)

in.12 1 4

(b)

4.5 THE FORCE METHOD OF ANALYSIS FOR AXIALLY LOADED MEMBERS 145

4.5 The Force Method of Analysis for Axially Loaded Members

It is also possible to solve statically indeterminate problems by writing the compatibility equation using the principle of superposition. This method of solution is often referred to as the flexibility or force method of analysis . To show how it is applied, consider again the bar in Fig. 4–15 a . If we choose the support at B as “redundant” and temporarily remove its effect on the bar, then the bar will become statically determinate as in Fig. 4–15 b . By using the principle of superposition, we must add back the unknown redundant load FB , as shown in Fig. 4–15 c .

If load P causes B to be displaced downward by an amount dP, the reaction FB must displace end B of the bar upward by an amount dB , such that no displacement occurs at B when the two loadings are superimposed. Thus,

(+ T ) 0 = d P - d B

This equation represents the compatibility equation for displacements at point B , for which we have assumed that displacements are positive downward.

Applying the load–displacement relationship to each case, we have d P = PL AC >AE and d B = F B L >AE. Consequently,

0 = PL AC

AE -

F B L AE

F B = P a L AC L b From the free-body diagram of the bar, Fig. 4–15 d , the reaction at A

can now be determined from the equation of equilibrium,

+ c !F y = 0; P a L AC L b + F A - P = 0 Since L CB = L - L AC, then

F A = P a L CB L b These results are the same as those obtained in Sec. 4.4 , except that here we have applied the condition of compatibility to obtain one reaction and then the equilibrium condition to obtain the other.

No displacement at B

(a)

Displacement at B when redundant force at B

is removed

(b)

Displacement at B when only the redundant force

at B is applied

(c)

C L

(d)

! "

LAC

LCB

Fig. 4–15

146 CHAPTER 4 AXIAL LOAD

EXAMPLE 4.8

The A-36 steel rod shown in Fig. 4–16 a has a diameter of 10 mm. It is fixed to the wall at A , and before it is loaded there is a gap between the wall at B ! and the rod of 0.2 mm. Determine the reactions at A and B !. Neglect the size of the collar at C. Take E st = 200 GPa.

SOLUTION Compatibility. Here we will consider the support at B ! as redundant. Using the principle of superposition, Fig. 4–16 b , we have

(S) + 0.0002 m = d P - d B (1)

Load-Displacement. The deflections d P and d B are determined from Eq. 4–2 .

d P = PL AC AE

= [20(103 ) N](0.4 m)

p(0.005 m)2 [200(109 ) N>m2] = 0.5093(10-3) m d B =

F B L AB AE

= F B (1.20 m)

p(0.005 m)2 [200(10 9) N>m2] = 76.3944(10-9)FB Substituting into Eq. 1 , we get

0.0002 m = 0.5093(10-3 ) m - 76.3944(10-9 )F B F B = 4.05(103 ) N = 4.05 kN Ans.

Equilibrium. From the free-body diagram, Fig. 4–16 c , S + "F x = 0; -F A + 20 kN - 4.05 kN = 0 F A = 16.0 kN Ans.

Procedure for Analysis

The force method of analysis requires the following steps.

Compatibility.

• Choose one of the supports as redundant and write the equation of compatibility. To do this, the known displacement at the redundant support, which is usually zero, is equated to the displacement at the support caused only by the external loads acting on the member plus (vectorially) the displacement at this support caused only by the redundant reaction acting on the member.

• Express the external load and redundant displacements in terms of the loadings by using a load– displacement relationship, such as d = PL >AE.

• Once established, the compatibility equation can then be solved for the magnitude of the redundant force.

Equilibrium. • Draw a free-body diagram and write the appropriate equations of equilibrium for the member using the

calculated result for the redundant. Solve these equations for any other reactions.

(b)

0.2 mm

" FB

P ! 20 kN

P ! 20 kN Initial

position

Final position

400 mm 800 mm

C P ! 20 kN

(a)

0.2 mm

A B¿

20 kNFA 4.05 kN

(c)

Fig. 4–16

4.5 THE FORCE METHOD OF ANALYSIS FOR AXIALLY LOADED MEMBERS 147

4–31. The concrete column is reinforced using four steel reinforcing rods, each having a diameter of 18 mm. Determine the stress in the concrete and the steel if the column is subjected to an axial load of 800 kN. E st = 200 GPa, E c = 25 GPa.

*4–32. The column is constructed from high-strength concrete and four A-36 steel reinforcing rods. If it is subjected to an axial force of 800 kN, determine the required diameter of each rod so that one-fourth of the load is carried by the steel and three-fourths by the concrete. E st = 200 GPa, E c = 25 GPa.

800 kN

300 mm300 mm

Probs. 4–31/32

4–33. The steel pipe is filled with concrete and subjected to a compressive force of 80 kN. Determine the average normal stress in the concrete and the steel due to this loading. The pipe has an outer diameter of 80 mm and an inner diameter of 70 mm. E st = 200 GPa, E c = 24 GPa.

500 mm

80 kN

Prob. 4–33

PROBLEMS

4–34. If column AB is made from high strength pre-cast concrete and reinforced with four 34 in. diameter A-36 steel rods, determine the average normal stress developed in the concrete and in each rod. Set P = 75 kip.

4–35. If column AB is made from high strength pre-cast concrete and reinforced with four 34 in. diameter A-36 steel rods, determine the maximum allowable floor loadings P . The allowable normal stress for the high strength concrete and the steel are (s allow ) con = 2.5 ksi and (s allow ) st = 24 ksi, respectively.

10 ft

Section a-a

9 in.

Probs. 4–34/35

*4–36. Determine the support reactions at the rigid supports A and C . The material has a modulus of elasticity of E .

4–37. If the supports at A and C are flexible and have a stiffness k , determine the support reactions at A and C . The material has a modulus of elasticity of E .

3 4 d

Probs. 4–36/37

148 CHAPTER 4 AXIAL LOAD

4–41. The 2014-T6 aluminum rod AC is reinforced with the firmly bonded A992 steel tube BC . If the assembly fits snugly between the rigid supports so that there is no gap at C, determine the support reactions when the axial force of 400 kN is applied. The assembly is attached at D .

4–42. The 2014-T6 aluminum rod AC is reinforced with the firmly bonded A992 steel tube BC . When no load is applied to the assembly, the gap between end C and the rigid support is 0.5 mm. Determine the support reactions when the axial force of 400 kN is applied.

400 mm 400 kN

800 mm

a a 50 mm

25 mm

Section a–a

A992 steel

2014–T6 aluminum alloy

Probs. 4–41/42

4–43. The assembly consists of two red brass C83400 copper rods AB and CD of diameter 30 mm, a stainless 304 steel alloy rod EF of diameter 40 mm, and a rigid cap G . If the supports at A , C and F are rigid, determine the average normal stress developed in rods AB , CD and EF .

40 kN

300 mm 450 mm

30 mm

40 mm

A B

C D

E F

Prob. 4–43

4–38. The load of 2800 lb is to be supported by the two essentially vertical A-36 steel wires. If originally wire AB is 60 in. long and wire AC is 40 in. long, determine the force developed in each wire after the load is suspended. Each wire has a cross-sectional area of 0.02 in 2 .

4–39. The load of 2800 lb is to be supported by the two essentially vertical A-36 steel wires. If originally wire AB is 60 in. long and wire AC is 40 in. long, determine the cross- sectional area of AB if the load is to be shared equally between both wires. Wire AC has a cross-sectional area of 0.02 in 2 .

60 in.

40 in.

Probs. 4–38/39

*4–40. The rigid member is held in the position shown by three A-36 steel tie rods. Each rod has an unstretched length of 0.75 m and a cross-sectional area of 125 mm2. Determine the forces in the rods if a turnbuckle on rod EF undergoes one full turn. The lead of the screw is 1.5 mm. Neglect the size of the turnbuckle and assume that it is rigid. Note: The lead would cause the rod, when unloaded , to shorten 1.5 mm when the turnbuckle is rotated one revolution.

0.5 m

C0.5 m

0.75 m

Prob. 4–40

4.5 THE FORCE METHOD OF ANALYSIS FOR AXIALLY LOADED MEMBERS 149

*4–44. The assembly consists of two red brass C83400 copper rods AB and CD having a diameter of 30 mm, a 304 stainless steel rod EF having a diameter of 40 mm, and a rigid member G . If the supports at A , C and F each have a stiffness of k = 200 MN>m, determine the average normal stress developed in the rods when the load is applied.

40 kN

300 mm 450 mm

30 mm

40 mm

A B

C D

E F

Prob. 4–44

4–45. The bolt has a diameter of 20 mm and passes through a tube that has an inner diameter of 50 mm and an outer diameter of 60 mm. If the bolt and tube are made of A-36 steel, determine the normal stress in the tube and bolt when a force of 40 kN is applied to the bolt. Assume the end caps are rigid.

40 kN

150 mm

160 mm

40 kN

Prob. 4–45

4–46. If the gap between C and the rigid wall at D is initially 0.15 mm, determine the support reactions at A and D when the force P = 200 kN is applied. The assembly is made of A-36 steel.

B C

50 mm

600 mm 600 mm 0.15 mm

25 mm

Prob. 4–46

4–47. The support consists of a solid red brass C83400 copper post surrounded by a 304 stainless steel tube. Before the load is applied the gap between these two parts is 1 mm. Given the dimensions shown, determine the greatest axial load that can be applied to the rigid cap A without causing yielding of any one of the materials.

0.25 m

80 mm 60 mm

10 mm

A 1 mm

Prob. 4–47

*4–48. The specimen represents a filament-reinforced matrix system made from plastic (matrix) and glass (fiber). If there are n fibers, each having a cross-sectional area of A f and modulus of E f , embedded in a matrix having a cross- sectional area of A m and modulus of E m , determine the stress in the matrix and each fiber when the force P is imposed on the specimen.

Prob. 4–48

150 CHAPTER 4 AXIAL LOAD

4–53. Each of the three A-36 steel wires has the same diameter. Determine the force in each wire needed to support the 200-kg load .

4–54. The 200-kg load is suspended from three A-36 steel wires each having a diameter of 4 mm. If wire BD has a length of 800.25 mm before the load is applied, determine the average normal stress developed in each wire.

D C

800 mm

600 mm 600 mm

Probs. 4–53/54

4–55. The three suspender bars are made of A992 steel and have equal cross-sectional areas of 450 mm2 . Determine the average normal stress in each bar if the rigid beam is subjected to the loading shown.

2 m 50 kN 80 kN

1 m 1 m 1 m 1 m

Prob. 4–55

4–49. The tapered member is fixed connected at its ends A and B and is subjected to a load P = 7 kip at x = 30 in. Determine the reactions at the supports. The material is 2 in. thick and is made from 2014-T6 aluminum.

4–50. The tapered member is fixed connected at its ends A and B and is subjected to a load P . Determine the greatest possible magnitude for P without exceeding an average normal stress of s allow = 4 ksi anywhere in the member, and determine the location x at which P would need to be applied. The member is 2 in. thick.

60 in.

3 in.

A B

6 in. P

Probs. 4–49/50

4–51. The rigid bar supports the uniform distributed load of 6 kip>ft. Determine the force in each cable if each cable has a cross-sectional area of 0.05 in2, and E = 31(103) ksi.

*4–52. The rigid bar is originally horizontal and is supported by two cables each having a cross-sectional area of 0.05 in2, and E = 31(103) ksi. Determine the slight rotation of the bar when the uniform load is supplied.

3 ft

A D

B 3 ft

6 kip/ft

3 ft

6 ft

Probs. 4–51/52

4.5 THE FORCE METHOD OF ANALYSIS FOR AXIALLY LOADED MEMBERS 151

*4–60. The assembly consists of two posts AD and CF made of A-36 steel and having a cross-sectional area of 1000 mm2, and a 2014-T6 aluminum post BE having a cross- sectional area of 1500 mm2 If a central load of 400 kN is applied to the rigid cap, determine the normal stress in each post. There is a small gap of 0.1 mm between the post BE and the rigid member ABC.

D E F

A B

400 kN

0.5 m 0.5 m

0.4 m

Prob. 4–60

4–61. The three suspender bars are made of the same material and have equal cross-sectional areas A . Determine the average normal stress in each bar if the rigid beam ACE is subjected to the force P .

L P

d— 2

Prob. 4–61

*4–56. The rigid bar supports the 800-lb load. Determine the normal stress in each A-36 steel cable if each cable has a cross-sectional area of 0.04 in2 .

4–57. The rigid bar is originally horizontal and is supported by two A-36 steel cables each having a cross- sectional area of 0.04 in2 . Determine the rotation of the bar when the 800-lb load is applied.

5 ft 5 ft 6 ftA D

800 lb

12 ft

Probs. 4–56/57

4–58. Two identical rods AB and CD each have a length L and diameter d , and are used to support the rigid beam, which is pinned at F . If a vertical force P is applied at the end of the beam, determine the normal stress developed in each rod. The rods are made of material that has a modulus of elasticity of E .

4–59. Two identical rods AB and CD each have a length L and diameter d , and are used to support the rigid beam, which is pinned at F . If a vertical force P is applied at the end of the beam, determine the angle of rotation of the beam. The rods are made of material that has a modulus of elasticity of E .

a A

F C

a 2a

Probs. 4–58/59

152 CHAPTER 4 AXIAL LOAD

4–65. Initially the A-36 bolt shank fits snugly against the rigid caps E and F on the 6061-T6 aluminum sleeve. If the thread of the bolt shank has a lead of 1 mm, and the nut is tightened 34 of a turn, determine the average normal stress developed in the bolt shank and the sleeve. The diameter of bolt shank is d = 60 mm.

4–66. Initially the A-36 bolt shank fits snugly against the rigid caps E and F on the 6061-T6 aluminum sleeve. If the thread of the bolt shank has a lead of 1 mm, and the nut is tightened 34 of a turn, determine the required diameter d of the shank and the force developed in the shank and sleeve so that the normal stress developed in the shank is four times that of the sleeve.

450 mm

250 mm

500 mm

15 mm

Probs. 4–65/66 4–67. The assembly consists of a 6061-T6-aluminum member and a C83400-red-brass member that rest on the rigid plates. Determine the distance d where the vertical load P should be placed on the plates so that the plates remain horizontal when the materials deform. Each member has a width of 8 in. and they are not bonded together.

30 in.

P d

6 in. 3 in.

Red brass

Aluminum

Prob. 4–67

4–62. If the 2-in. diameter supporting rods are made from A992 steel, determine the average normal stress developed in each rod when P = 100 kip.

4–63. If the supporting rods of equal diameter are made from A992 steel, determine the required diameter to the nearest 18 in. of each rod when P = 100 kip. The allowable normal stress of the steel is s allow = 24 ksi.

B D

2 ft2 ft 30! 30!

Probs. 4–62/63

*4–64. The center post B of the assembly has an original length of 124.7 mm, whereas posts A and C have a length of 125 mm. If the caps on the top and bottom can be considered rigid, determine the average normal stress in each post. The posts are made of aluminum and have a cross-sectional area of 400 mm2. E al = 70 GPa.

125 mm 100 mm 100 mm

A B C

800 kN/m

Prob. 4–64

4.6 THERMAL STRESS 153

4.6 Thermal Stress A change in temperature can cause a body to change its dimensions. Generally, if the temperature increases, the body will expand, whereas if the temperature decreases, it will contract * . Ordinarily this expansion or contraction is linearly related to the temperature increase or decrease that occurs. If this is the case, and the material is homogeneous and isotropic, it has been found from experiment that the displacement of a member having a length L can be calculated using the formula

d T = a !TL (4–4)

where

a = a property of the material, referred to as the linear coefficient of thermal expansion . The units measure strain per degree of temperature. They are 1>"F (Fahrenheit) in the FPS system, and 1>"C (Celsius) or 1>K (Kelvin) in the SI system. Typical values are given on the inside back cover

!T = the algebraic change in temperature of the member L = the original length of the member d T = the algebraic change in the length of the member

The change in length of a statically determinate member can easily be calculated using Eq. 4–4 , since the member is free to expand or contract when it undergoes a temperature change. However, in a statically indeterminate member, these thermal displacements will be constrained by the supports, thereby producing thermal stresses that must be considered in design. Determining these thermal stresses is possible using the methods outlined in the previous sections. The following examples illustrate some applications.

Most traffic bridges are designed with expansion joints to accommodate the thermal movement of the deck and thus avoid any thermal stress.

* There are some materials, like Invar, an iron-nickel alloy, and scandium trifluoride, that behave in the opposite way, but we will not consider these here.

Long extensions of ducts and pipes that carry fluids are subjected to variations in climate that will cause them to expand and contract. Expansion joints, such as the one shown, are used to mitigate thermal stress in the material.

154 CHAPTER 4 AXIAL LOAD

The A-36 steel bar shown in Fig. 4–17 a is constrained to just fit between two fixed supports when T 1 = 60!F. If the temperature is raised to T 2 = 120!F, determine the average normal thermal stress developed in the bar.

SOLUTION

Equilibrium. The free-body diagram of the bar is shown in Fig. 4–17 b . Since there is no external load, the force at A is equal but opposite to the force at B ; that is,

+ c "F y = 0; F A = F B = F

The problem is statically indeterminate since this force cannot be determined from equilibrium.

Compatibility. Since d A >B = 0, the thermal displacement d T at A that occurs, Fig. 4–17 c , is counteracted by the force F that is required to push the bar d F back to its original position. The compatibility condition at A becomes

(+ c ) d A >B = 0 = d T - d F Load-Displacement. Applying the thermal and load–displacement relationships, we have

0 = a#TL - FL AE

Thus, from the data on the inside back cover,

F = a#TAE = [6.60(10-6)>!F](120!F - 60!F)(0.5 in.)2 [29(103) kip>in 2] = 2.871 kip

Since F also represents the internal axial force within the bar, the average normal compressive stress is thus

s = F A

= 2.871 kip

(0.5 in.) 2 = 11.5 ksi Ans.

NOTE: From the magnitude of F , it should be apparent that changes in temperature can cause large reaction forces in statically indeterminate members.

EXAMPLE 4.9

2 ft

0.5 in.

(a)

(b)

(c)

Fig. 4–17

4.6 THERMAL STRESS 155

The rigid beam shown in Fig. 4–18 a is fixed to the top of the three posts made of A992 steel and 2014-T6 aluminum. The posts each have a length of 250 mm when no load is applied to the beam, and the temperature is T 1 = 20!C. Determine the force supported by each post if the bar is subjected to a uniform distributed load of 150 kN>m and the temperature is raised to T 2 = 80!C.

SOLUTION

Equilibrium. The free-body diagram of the beam is shown in Fig. 4–18 b . Moment equilibrium about the beam’s center requires the forces in the steel posts to be equal. Summing forces on the free-body diagram, we have

+ c "F y = 0; 2F st + F al - 90(10 3) N = 0 (1)

Compatibility. Due to load, geometry, and material symmetry, the top of each post is displaced by an equal amount. Hence,

(+ T ) d st = d al (2)

The final position of the top of each post is equal to its displacement caused by the temperature increase, plus its displacement caused by the internal axial compressive force, Fig. 4–18 c . Thus, for the steel and aluminum post, we have (+ T ) d st = -(d st) T + (d st) F (+ T ) d al = -(d al) T + (d al) F Applying Eq. 2 gives

-(d st) T + (d st) F = -(d al) T + (d al) F Load-Displacement. Using Eqs. 4–2 and 4–4 and the material properties on the inside back cover, we get

-[12(10 -6)>!C](80!C - 20!C)(0.250 m) + F st (0.250 m) p(0.020 m) 2 [200(10 9) N>m 2]

= -[23(10 -6)>!C](80!C - 20!C)(0.250 m) + F al (0.250 m) p(0.030 m) 2 [73.1(10 9) N>m 2]

Fst = 1.216F al - 165.9(103) (3)

To be consistent , all numerical data has been expressed in terms of newtons, meters, and degrees Celsius. Solving Eqs. 1 and 3 simultaneously yields

F st = -16.4 kN F al = 123 kN Ans.

The negative value for F st indicates that this force acts opposite to that shown in Fig. 4–18 b . In other words, the steel posts are in tension and the aluminum post is in compression.

EXAMPLE 4.10

150 kN/m300 mm 300 mm

(a)

SteelAluminumSteel

250 mm 40 mm

60 mm

40 mm

(b)

Fst Fal Fst

90 kN

(c)

Initial Position

Final Position

(dst)T (dal)T

(dal)F

(dst)Fdst ! dal

Fig. 4–18

156 CHAPTER 4 AXIAL LOAD

A 2014-T6 aluminum tube having a cross-sectional area of 600 mm2 is used as a sleeve for an A-36 steel bolt having a cross-sectional area of 400 mm2, Fig. 4–19 a . When the temperature is T1 = 15!C, the nut holds the assembly in a snug position such that the axial force in the bolt is negligible. If the temperature increases to T2 = 80!C, determine the force in the bolt and sleeve.

SOLUTION

Equilibrium. The free-body diagram of a top segment of the assembly is shown in Fig. 4–19 b . The forces Fb and Fs are produced since the sleeve has a higher coefficient of thermal expansion than the bolt, and therefore the sleeve will expand more when the temperature is increased. It is required that

+ c "F y = 0; Fs = Fb (1)

Compatibility. The temperature increase causes the sleeve and bolt to expand (ds )T and (db)T , Fig. 4–19 c . However, the redundant forces Fb and Fs elongate the bolt and shorten the sleeve. Consequently, the end of the assembly reaches a final position, which is not the same as its initial position. Hence, the compatibility condition becomes

(+ T ) d = (d b) T + (d b) F = (d s) T - (d s) F

Load-Displacement. Applying Eqs. 4–2 and 4–4 , and using the mechanical properties from the table on the inside back cover, we have

[12(10 -6 )>!C](80!C - 15!C)(0.150 m) +

F b (0.150 m)

(400 mm2)(10-6 m2>mm2)[200(109) N>m2] = [23(10 -6)>!C](80!C - 15!C)(0.150 m)

- F s (0.150 m)

(600 mm2)(10-6 m2>mm2)[73.1(109) N>m2] Using Eq. 1 and solving gives

F s = F b = 20.3 kN Ans.

NOTE: Since linear elastic material behavior was assumed in this analysis, the average normal stresses should be checked to make sure that they do not exceed the proportional limits for the material.

EXAMPLE 4.11

(ds)F

(ds)T (db)T (db)Fd

Initial position

Final position

(c)

Fig. 4–19

(b)

150 mm

(a)

4.6 THERMAL STRESS 157

*4–68. The C83400-red-brass rod AB and 2014-T6- aluminum rod BC are joined at the collar B and fixed connected at their ends. If there is no load in the members when T 1 = 50°F, determine the average normal stress in each member when T 2 = 120°F. Also, how far will the collar be displaced? The cross-sectional area of each member is 1.75 in 2 .

3 ft 2 ft

A B C

Prob. 4–68

4–69. The assembly has the diameters and material make- up indicated. If it fits securely between its fixed supports when the temperature is T 1 = 70°F, determine the average normal stress in each material when the temperature reaches T 2 = 110°F.

12 in.A D

2014-T6 Aluminum C 86100 Bronze

304 Stainless steel

4 ft 3 ft6 ft

8 in.

4 in.

Prob. 4–69

4–70. The rod is made of A992 steel and has a diameter of 0.25 in. If the rod is 4 ft long when the springs are compressed 0.5 in. and the temperature of the rod is T = 40!F , determine the force in the rod when its temperature is T = 160!F.

4 ft

k ! 1000 lb/ in. k ! 1000 lb/ in.

Prob. 4–70

4–71. If the assembly fits snugly between two rigid supports A and C when the temperature is at T 1 , determine the normal stress developed in both rod segments when the temperature rises to T 2 . Both segments are made of the same material, having a modulus of elasticity of E and coefficient of thermal expansion of a .

*4–72. If the assembly fits snugly between the two supports A and C when the temperature is at T 1 , determine the normal stress developed in both segments when the temperature rises to T 2 . Both segments are made of the same material having a modulus of elasticity of E and coefficient of the thermal expansion of a. The flexible supports at A and C each have a stiffness k .

A B C

L 2

d d 1 2

Probs. 4–71/72

4–73. The pipe is made of A992 steel and is connected to the collars at A and B . When the temperature is 60°F, there is no axial load in the pipe. If hot gas traveling through the pipe causes its temperature to rise by "T = (40 + 15x)!F, where x is in feet, determine the average normal stress in the pipe. The inner diameter is 2 in., the wall thickness is 0.15 in.

4–74. The bronze C86100 pipe has an inner radius of 0.5 in. and a wall thickness of 0.2 in. If the gas flowing through it changes the temperature of the pipe uniformly from T A = 200!F at A to T B = 60!F at B , determine the axial force it exerts on the walls. The pipe was fitted between the walls when T = 60!F.

8 ft

A B

Probs. 4–73/74

PROBLEMS

158 CHAPTER 4 AXIAL LOAD

4–75. The 40-ft-long A-36 steel rails on a train track are laid with a small gap between them to allow for thermal expansion. Determine the required gap d so that the rails just touch one another when the temperature is increased from T 1 = -20!F to T 2 = 90!F. Using this gap, what would be the axial force in the rails if the temperature were to rise to T 3 = 110!F? The cross-sectional area of each rail is 5.10 in2.

40 ft

d d

Prob. 4–75

*4–76. The device is used to measure a change in temperature. Bars AB and CD are made of A-36 steel and 2014-T6 aluminum alloy, respectively. When the temperature is at 75°F, ACE is in the horizontal position. Determine the vertical displacement of the pointer at E when the temperature rises to 150°F.

A C E

B D

1.5 in.

0.25 in. 3 in.

Prob. 4–76

4–77. The bar has a cross-sectional area A , length L , modulus of elasticity E , and coefficient of thermal expansion a. The temperature of the bar changes uniformly along its length from T A at A to T B at B so that at any point x along the bar T = T A + x(T B - T A )>L. Determine the force the bar exerts on the rigid walls. Initially no axial force is in the bar and the bar has a temperature of TA.

TA TB

A B

Prob. 4–77

4–78. When the temperature is at 30°C, the A-36 steel pipe fits snugly between the two fuel tanks. When fuel flows through the pipe, the temperatures at ends A and B rise to 130°C and 80°C, respectively. If the temperature drop along the pipe is linear, determine the average normal stress developed in the pipe. Assume each tank provides a rigid support at A and B .

6 m

150 mm

10 mm

Section a - a

x a

aA B

Prob. 4–78

4–79. When the temperature is at 30°C, the A-36 steel pipe fits snugly between the two fuel tanks. When fuel flows through the pipe, the temperatures at ends A and B rise to 130°C and 80°C, respectively. If the temperature drop along the pipe is linear, determine the average normal stress developed in the pipe. Assume the walls of each tank act as a spring, each having a stiffness of k = 900 MN>m. *4–80. When the temperature is at 30°C, the A-36 steel pipe fits snugly between the two fuel tanks. When fuel flows through the pipe, it causes the temperature to vary along the pipe as T = (53 x 2 − 20 x + 120)°C, where x is in meters. Determine the normal stress developed in the pipe. Assume each tank provides a rigid support at A and B .

6 m

150 mm

10 mm

Section a - a

x a

aA B

Probs. 4–79/80

4.6 THERMAL STRESS 159

*4–84. The center rod CD of the assembly is heated from T 1 = 30°C to T 2 = 180°C using electrical resistance heating. At the lower temperature T 1 the gap between C and the rigid bar is 0.7 mm. Determine the force in rods AB and EF caused by the increase in temperature. Rods AB and EF are made of steel, and each has a cross-sectional area of 125 mm 2 . CD is made of aluminum and has a cross-sectional area of 375 mm 2 . E st = 200 GPa, E al = 70 GPa, and a al = 23(10 −6 )/°C.

4–85. The center rod CD of the assembly is heated from T 1 = 30°C to T 2 = 180°C using electrical resistance heating. Also, the two end rods AB and EF are heated from T1 = 30°C to T2 = 50°C. At the lower temperature T 1 the gap between C and the rigid bar is 0.7 mm. Determine the force in rods AB and EF caused by the increase in temperature. Rods AB and EF are made of steel, and each has a cross-sectional area of 125 mm 2 . CD is made of aluminum and has a cross-sectional area of 375 mm 2 . E st = 200 GPa, E al = 70 GPa, a st = 12(10 −6 )/°C and a al = 23(10 −6 )/°C.

300 mm240 mm

0.7 mm

–

Probs. 4–84/85

4–86. The metal strap has a thickness t and width w and is subjected to a temperature gradient T 1 to T 2 ( T 1 < T 2 ). This causes the modulus of elasticity for the material to vary linearly from E 1 at the top to a smaller amount E 2 at the bottom. As a result, for any vertical position y , E = [( E 2 − E 1 )> w ] y + E 1 . Determine the position d where the axial force P must be applied so that the bar stretches uniformly over its cross section.

w d

Prob. 4–86

4–81. The 50-mm-diameter cylinder is made from Am 1004-T61 magnesium and is placed in the clamp when the temperature is T 1 = 20° C. If the 304-stainless-steel carriage bolts of the clamp each have a diameter of 10 mm, and they hold the cylinder snug with negligible force against the rigid jaws, determine the force in the cylinder when the temperature rises to T 2 = 130°C.

4–82. The 50-mm-diameter cylinder is made from Am 1004- T61 magnesium and is placed in the clamp when the temp- erature is T 1 = 15°C. If the two 304-stainless-steel carriage bolts of the clamp each have a diameter of 10 mm, and they hold the cylinder snug with negligible force against the rigid jaws, determine the temperature at which the average normal stress in either the magnesium or the steel first becomes 12 MPa.

100 mm 150 mm

Probs. 4–81/82

4–83. The wires AB and AC are made of steel, and wire AD is made of copper. Before the 150-lb force is applied, AB and AC are each 60 in. long and AD is 40 in. long. If the temperature is increased by 80!F, determine the force in each wire needed to support the load. Take E st = 29(10 3 ) ksi, E cu = 17(10 3 ) ksi, a st = 8(10 -6 )>!F, a cu = 9.60(10 -6 )>!F. Each wire has a cross-sectional area of 0.0123 in2.

45! 45! 60 in.60 in.

150 lb

40 in.

CDB

Prob. 4–83

160 CHAPTER 4 AXIAL LOAD

4.7 Stress Concentrations In Sec. 4.1 , it was pointed out that when an axial force is applied to a member, it creates a complex stress distribution within the localized region of the point of load application. Not only do complex stress distributions arise just under a concentrated loading, they can also arise at sections where the member’s cross-sectional area changes. For example, consider the bar in Fig. 4–20 a , which is subjected to an axial force P . Here the once horizontal and vertical grid lines deflect into an irregular pattern around the hole centered in the bar. The maximum normal stress in the bar occurs on section a – a , which is taken through the bar’s smallest cross-sectional area. Provided the material behaves in a linear-elastic manner, the stress distribution acting on this section can be determined either from a mathematical analysis, using the theory of elasticity, or experimentally by measuring the strain normal to section a – a and then calculating the stress using Hooke’s law, s = EP. Regardless of the method used, the general shape of the stress distribution will be like that shown in Fig. 4–20 b . In a similar manner, if the bar has a reduction in its cross section, achieved using shoulder fillets as in Fig. 4–21 a , then again the maximum normal stress in the bar will occur at the smallest cross-sectional area, section a – a , and the stress distribution will look like that shown in Fig. 4–21 b .

This saw blade has grooves cut into it in order to relieve both the dynamic stress that develops within it as it rotates and the thermal stress that develops as it heats up. Note the small circles at the end of each groove. These serve to reduce the stress concentrations that develop at the end of each groove.

Distorted (a)

Actual stress distribution (b)

Average stress distribution (c)

smax

savg

Undistorted

Fig. 4–20

4.7 STRESS CONCENTRATIONS 161

In both of these cases, force equilibrium requires the magnitude of the resultant force developed by the stress distribution to be equal to P . In other words,

P = L

A s dA (4–5)

This integral graphically represents the total volume under each of the stress-distribution diagrams shown in Fig. 4–20 b or Fig. 4–21 b . The resultant P must act through the centroid of each volume .

In engineering practice, the actual stress distributions in Fig. 4–20 b and Fig. 4–21 b do not have to be determined. Instead, only the maximum stress at these sections must be known, and the member is then designed to resist this stress when the axial load P is applied. Specific values of this maximum normal stress can be determined by experimental methods or by advanced mathematical techniques using the theory of elasticity. The results of these investigations are usually reported in graphical form using a stress-concentration factor K . We define K as a ratio of the maximum stress to the average normal stress acting at the cross section; i.e.,

K = smax savg

(4–6)

Provided K is known, and the average normal stress has been calculated from savg = P>A , where A is the smallest cross-sectional area, Figs. 4–20 c and 4–21 c , then the maximum normal stress at the cross section is smax = K(P>A ).

Distorted (a)

Actual stress distribution (b)

smax

savg

Average stress distribution (c)

Undistorted

Fig. 4–21

Stress concentrations often arise at sharp corners on heavy machinery. Engineers can mitigate this effect by using stiffeners welded to the corners.

162 CHAPTER 4 AXIAL LOAD

Specific values of K are generally reported in handbooks related to stress analysis. Examples are given in Figs. 4–23 and 4–24 . Note that K is independent of the bar’s material properties; rather, it depends only on the bar’s geometry and the type of discontinuity. As the size r of the discontinuity is decreased , the stress concentration is increased. For example, if a bar requires a change in cross section, it has been determined that a sharp corner, Fig. 4–22 a , produces a stress-concentration factor greater than 3. In other words, the maximum normal stress will be three times greater than the average normal stress on the smallest cross section. However, this can be reduced to, say, 1.5 by introducing a fillet, Fig. 4–22 b . A further reduction can be made by means of small grooves or holes placed at the transition, Fig. 4–22 c and 4–22 d . In all of these cases these designs help to reduce the rigidity of the material surrounding the corners, so that both the strain and the stress are more evenly spread throughout the bar.

The stress-concentration factors given in Figs. 4–23 and 4–24 were determined on the basis of a static loading, with the assumption that the stress in the material does not exceed the proportional limit. If the material is very brittle , the proportional limit may be at the fracture stress, and so for this material, failure will begin at the point of stress concentration. Essentially a crack begins to form at this point, and a higher stress concentration will develop at the tip of this crack. This, in turn, causes the crack to propagate over the cross section, resulting in sudden fracture. For this reason, it is very important to use stress- concentration factors in design when using brittle materials. On the other hand, if the material is ductile and subjected to a static load, it is often not necessary to use stress-concentration factors since any stress that exceeds the proportional limit will not result in a crack. Instead, the material will have reserve strength due to yielding and strain- hardening. In the next section we will discuss the effects caused by this phenomenon.

Stress concentrations are also responsible for many failures of structural members or mechanical elements subjected to fatigue loadings . For these cases, a stress concentration will cause the material to crack if the stress exceeds the material’s endurance limit, whether or not the material is ductile or brittle. Here, the material localized at the tip of the crack remains in a brittle state , and so the crack continues to grow, leading to a progressive fracture. As a result, one must seek ways to limit the amount of damage that can be caused by fatigue.

P P

(a)

Fig. 4–22

P P

(b)

P P

(c)

P P

(d)

4.7 STRESS CONCENTRATIONS 163

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

P P

savg !

! 4.0w h

! 3.0w h ! 2.0

w h

! 1.5w h

! 1.2w h

! 1.1w h

r h

P ht

Fig. 4–23

3.2

3.0

2.8

2.6

2.4

2.2

2.0 0 0.1 0.2 0.3 0.4 0.5

P P

2r w

P (w ! 2r)t

savg "

Fig. 4–24

Failure of this steel pipe in tension occurred at its smallest cross-sectional area, which is through the hole. Notice how the material yielded around the fractured surface.

Important Points

• Stress concentrations occur at sections where the cross-sectional area suddenly changes. The more severe the change, the larger the stress concentration.

• For design or analysis, it is only necessary to determine the maximum stress acting on the smallest cross-sectional area. This is done using a stress concentration factor, K , that has been determined through experiment and is only a function of the geometry of the specimen.

• Normally the stress concentration in a ductile specimen that is subjected to a static loading will not have to be considered in design; however, if the material is brittle , or subjected to fatigue loadings, then stress concentrations become important.

164 CHAPTER 4 AXIAL LOAD

*4.8 Inelastic Axial Deformation

Up to this point we have only considered loadings that cause the material of a member to behave elastically. Sometimes, however, a member may be designed so that the loading causes the material to yield and thereby permanently deform. Such members are often made from a highly ductile metal such as annealed low-carbon steel having a stress–strain diagram that is similar to that of Fig. 3–6 and for simplicity can be modeled as shown in Fig. 4–25 b . A material that exhibits this behavior is referred to as being elastic perfectly plastic or elastoplastic.

To illustrate physically how such a material behaves, consider the bar in Fig. 4–25 a , which is subjected to the axial load P . If the load causes an elastic stress s = s1 to be developed in the bar, then applying Eq. 4–5 , equilibrium requires P = 1s1 dA = s1 A. Furthermore, the stress s1 causes the bar to strain e1 as indicated on the stress–strain diagram, Fig. 4–25 b . If P is now increased to Pp such that it causes yielding of the material, that is, s = sY , then again Pp = 1sY dA = sYA . The load Pp is called the plastic load since it represents the maximum load that can be supported by an elastoplastic material. For this case, the strains are not uniquely defined. Instead, at the instant sY is attained, the bar is first subjected to the yield strain PY , Fig. 4–25 b , after which the bar will continue to yield (or elongate) such that the strains P2 , then P3 , etc., are generated. Since our “model” of the material exhibits perfectly plastic material behavior, this elongation will continue indefinitely with no increase in load. In reality, however, the material will, after some yielding, actually begin to strain-harden so that the extra strength it attains will stop any further straining. As a result, any design based on this behavior will be safe, since strain-hardening provides the potential for the material to support an additional load if necessary.

(a) s

(b)

P1 P2 P3 PPY

Fig. 4–25

4.8 INELASTIC AXIAL DEFORMATION 165

Consider now the case of a bar having a hole through it as shown in Fig. 4–26 a . As the magnitude of P is increased, a stress concentration occurs in the material at the edge of the hole, on section a – a . The stress here will reach a maximum value of, say, smax = s1 and have a corresponding elastic strain of P1 , Fig. 4–26 b . The stresses and corresponding strains at other points on the cross section will be smaller, as indicated by the stress distribution shown in Fig. 4–26 c . Equilibrium requires P = 1s dA . In other words, P is geometrically equivalent to the “volume” contained within the stress distribution. If the load is now increased to P!, so that smax = sY , then the material will begin to yield outward from the hole, until the equilibrium condition P! = 1s dA is satisfied, Fig. 4–26 d . As shown, this produces a stress distribution that has a geometrically greater “volume” than that shown in Fig. 4–26 c . A further increase in load will cause the material over the entire cross section to yield eventually. When this happens, no greater load can be sustained by the bar. This plastic load Pp is shown in Fig. 4–26 e . It can be calculated from the equilibrium condition

Pp = L AsY dA = sYA where A is the bar’s cross-sectional area at section a – a .

The following examples illustrate numerically how these concepts apply to other types of problems for which the material has elastoplastic behavior.

(c)

s1 s1

(d)

P ¿

sY sY

(e)

sY sY

Fig. 4–26

(a) P

(b)

PP1 PY

166 CHAPTER 4 AXIAL LOAD

*4.9 Residual Stress

If an axially loaded member or group of such members forms a statically indeterminate system that can support both tensile and compressive loads, then excessive external loadings, which cause yielding of the material, will create residual stresses in the members when the loads are removed. The reason for this has to do with the elastic recovery of the material that occurs during unloading. To show this, consider a prismatic member made from an elastoplastic material having the stress–strain diagram shown in Fig. 4–27 . If an axial load produces a stress sY in the material and a corresponding strain PC , then when the load is removed , the material will respond elastically and follow the line CD in order to recover some of the strain. A recovery to zero stress at point O! will be possible only if the member is statically determinate, since the support reactions for the member must be zero when the load is removed. Under these circumstances the member will be permanently deformed so that the permanent set or strain in the member is PO!.

If the member is statically indeterminate , however, removal of the external load will cause the support forces to respond to the elastic recovery CD . Since these forces will constrain the member from full recovery, they will induce residual stresses in the member. To solve a problem of this kind, the complete cycle of loading and then unloading of the member can be considered as the superposition of a positive load (loading) on a negative load (unloading). The loading, O to C , results in a plastic stress distribution, whereas the unloading, along CD , results only in an elastic stress distribution. Superposition requires the loads to cancel; however, the stress distributions will not cancel, and so residual stresses will remain.

PO¿

CA B

O¿ PC

Fig. 4–27

4.9 RESIDUAL STRESS 167

The bar in Fig. 4–28 a is made of steel that is assumed to be elastic perfectly plastic, with sY = 250 MPa. Determine (a) the maximum value of the applied load P that can be applied without causing the steel to yield and (b) the maximum value of P that the bar can support. Sketch the stress distribution at the critical section for each case.

SOLUTION Part (a). When the material behaves elastically, we must use a stress-concentration factor determined from Fig. 4–23 that is unique for the bar’s geometry. Here

r h

= 4 mm

(40 mm - 8 mm) = 0.125

w h

= 40 mm

(40 mm - 8 mm) = 1.25

From the figure K ! 1.75. The maximum load, without causing yielding, occurs when smax = sY . The average normal stress is savg = P>A . Using Eq. 4–6 , we have smax = Ksavg; sY = K aPYA b

250(106) Pa = 1.75 c PY (0.002 m)(0.032 m)

d PY = 9.14 kN Ans.

This load has been calculated using the smallest cross section. The resulting stress distribution is shown in Fig. 4–28 b . For equilibrium, the “volume” contained within this distribution must equal 9.14 kN.

Part (b). The maximum load sustained by the bar will cause all the material at the smallest cross section to yield. Therefore, as P is increased to the plastic load Pp , it gradually changes the stress distribution from the elastic state shown in Fig. 4–28 b to the plastic state shown in Fig. 4–28 c . We require

sY = Pp A

250(106) Pa = Pp

(0.002 m)(0.032 m)

Pp = 16.0 kN Ans.

Here Pp equals the “volume” contained within the stress distribution, which in this case is Pp = sYA .

EXAMPLE 4.12

2 mm

40 mm

4 mm

(a)

(b)

(c)

Fig. 4–28

168 CHAPTER 4 AXIAL LOAD

The rod shown in Fig. 4–29 a has a radius of 5 mm and is made of an elastic perfectly plastic material for which sY = 420 MPa, E = 70 GPa, Fig. 4–29 c . If a force of P = 60 kN is applied to the rod and then removed, determine the residual stress in the rod.

SOLUTION The free-body diagram of the rod is shown in Fig. 4–29 b . Application of the load P will cause one of three possibilities, namely, both segments AC and CB remain elastic, AC is plastic while CB is elastic, or both AC and CB are plastic. *

An elastic analysis , similar to that discussed in Sec. 4.4 , will produce FA = 45 kN and FB = 15 kN at the supports. However, this results in a stress of

s AC = 45 kN

p(0.005 m)2 = 573 MPa (compression) 7 sY = 420 MPa

sCB = 15 kN

p(0.005 m)2 = 191 MPa (tension)

Since the material in segment AC will yield, we will assume that AC becomes plastic, while CB remains elastic.

For this case, the maximum possible force developed in AC is

(FA)Y = sYA = 420(103) kN>m2 [p(0.005 m)2] = 33.0 kN and from the equilibrium of the rod, Fig. 4–29 b ,

FB = 60 kN - 33.0 kN = 27.0 kN

The stress in each segment of the rod is therefore

sAC = sY = 420 MPa (compression)

sCB = 27.0 kN

p(0.005 m)2 = 344 MPa (tension) 6 420 MPa (OK)

EXAMPLE 4.13

C P ! 60 kNA B FA FB

(b)

Fig. 4–29

100 mm 300 mm

CA BP ! 60 kN

(a)

* The possibility of CB becoming plastic before AC will not occur because when point C moves, the strain in AC (since it is shorter) will always be larger than the strain in CB .

4.9 RESIDUAL STRESS 169

Residual Stress. In order to obtain the residual stress, it is also necessary to know the strain in each segment due to the loading. Since CB responds elastically,

dC = FBL CB AE

= (27.0 kN)(0.300 m)

p(0.005 m)2 [70(106) kN>m2] = 0.001474 m PCB =

dC LCB

= 0.001474 m

0.300 m = +0.004913

PAC = dC

LAC = - 0.001474 m

0.100 m = -0.01474

Here the yield strain is

PY = sY

E =

420(106) N>m2 70(109) N>m2 = 0.006

Therefore, when P is applied , the stress–strain behavior for the material in segment CB moves from O to A !, Fig. 4–29 c , and the stress- strain behavior for the material in segment AC moves from O to B ! If the load P is applied in the reverse direction, in other words, the load is removed, then an elastic response occurs and a reverse force of FA = 45 kN and FB = 15 kN must be applied to each segment. As calculated previously, these forces now produce stresses sAC = 573 MPa (tension) and sCB = 191 MPa (compression), and as a result the residual stress in each member is

(sAC)r = -420 MPa + 573 MPa = 153 MPa Ans.

(sCB)r = 344 MPa - 191 MPa = 153 MPa Ans.

This residual stress is the same for both segments, which is to be expected. Also note that the stress–strain behavior for segment AC moves from B ! to D! in Fig. 4–29 c , while the stress–strain behavior for the material in segment CB moves from A ! to C! when the load is removed.

420

!420

PAC " !0.01474

PCB " 0.004913

344 153

D¿

B¿

A¿

C¿O

s (MPa)

(c)

P (mm/mm)

Fig. 4–29 (cont.)

170 CHAPTER 4 AXIAL LOAD

Two steel wires are used to lift the weight of 3 kip, Fig. 4–30 a . Wire AB has an unstretched length of 20.00 ft and wire AC has an unstretched length of 20.03 ft. If each wire has a cross- sectional area of 0.05 in2, and the steel can be considered elastic perfectly plastic as shown by the s9P graph in Fig. 4–30 b , determine the force in each wire and its elongation.

SOLUTION Once the weight is supported by both wires, then the stress in the wires depends on the corresponding strain.

There are three possibilities, namely, the strains in both wires are elastic, wire AB is plastically strained while wire AC is elastically strained, or both wires are plastically strained. We will assume that AC remains elastic and AB is plastically strained.

Investigation of the free-body diagram of the suspended weight, Fig. 4–30 c , indicates that the problem is statically indeterminate. The equation of equilibrium is

+ c !Fy = 0; TAB + TAC - 3 kip = 0 (1) Since AB becomes plastically strained then it must support its maximum load.

TAB = sYA AB = 50 ksi (0.05 in2) = 2.50 kip Ans. Therefore, from Eq. 1 ,

TAC = 0.500 kip Ans. Note that wire AC remains elastic as assumed since the stress in the

wire is sAC = 0.500 kip>0.05 in2 = 10 ksi 6 50 ksi. The corresponding elastic strain is determined by proportion, Fig. 4–30 b ; i.e.,

P AC

10 ksi =

0.0017 50 ksi

PAC = 0.000340

The elongation of AC is thus

dAC = (0.000340)(20.03 ft) = 0.00681 ft Ans.

And from Fig. 4–30 d , the elongation of AB is then

dAB = 0.03 ft + 0.00681 ft = 0.0368 ft Ans.

EXAMPLE 4.14

20.03 ft20.00 ft

(a)

Final position

B C

dAB " 0.03 ft # dAC

(d)

Initial position

20.00 ft 20.03 ft

dAC

0.0017

(b)

P (in./in.)

s (ksi)

TACTAB

3 kip

(c)

Fig. 4–30

PROBLEMS

4.9 RESIDUAL STRESS 171

4–87. Determine the maximum normal stress developed in the bar when it is subjected to a tension of P = 8 kN.

*4–88. If the allowable normal stress for the bar is s allow = 120 MPa, determine the maximum axial force P that can be applied to the bar.

r ! 10 mm

40 mm 20 mm

P P

20 mm

5 mm

Probs. 4–87/88

4–89. The steel bar has the dimensions shown. Determine the maximum axial force P that can be applied so as not to exceed an allowable tensile stress of sallow = 150 MPa.

r ! 15 mm

60 mm 60 mm30 mm

P P

24 mm

20 mm

Prob. 4–89

4–90. Determine the maximum axial force P that can be applied to the steel plate. The allowable stress is sallow = 21 ksi.

0.25 in.

5 in.

0.25 in.

2.5 in.

1 in.

Prob. 4–90

PROBLEMS

4–91. Determine the maximum axial force P that can be applied to the bar. The bar is made from steel and has an allowable stress of sallow = 21 ksi.

*4–92. Determine the maximum normal stress developed in the bar when it is subjected to a tension of P = 2 kip.

P P

1.25 in. 1.875 in. 0.125 in.

0.75 in. r ! 0.25 in.

Probs. 4–91/92

4–93. Determine the maximum normal stress developed in the bar when it is subjected to a tension of P = 8 kN.

r = 15 mm

60 mm 30 mm

P P

12 mm

5 mm

Prob. 4–93

4–94. The resulting stress distribution along section AB for the bar is shown. From this distribution, determine the approximate resultant axial force P applied to the bar. Also, what is the stress-concentration factor for this geometry?

1 in.

4 in.

0.5 in.

3 ksi 12 ksi

Prob. 4–94

172 CHAPTER 4 AXIAL LOAD

4–98. The bar has a cross-sectional area of 0.5 in2 and is made of a material that has a stress–strain diagram that can be approximated by the two line segments shown. Determine the elongation of the bar due to the applied loading.

5 kip8 kipA B C

5 ft 2 ft

0.001 0.021 P (in./in.)

s (ksi)

Prob. 4–98

4–99. The rigid beam is supported by a pin at A and two steel wires, each having a diameter of 4 mm. If the yield stress for the wires is sY = 530 MPa, and Est = 200 GPa, determine the intensity of the distributed load w that can be placed on the beam and will just cause wire EB to yield. What is the displacement of point G for this case? For the calculation, assume that the steel is elastic perfectly plastic.

*4–100. The rigid beam is supported by a pin at A and two steel wires, each having a diameter of 4 mm. If the yield stress for the wires is sY = 530 MPa, and Est = 200 GPa, determine (a) the intensity of the distributed load w that can be placed on the beam that will cause only one of the wires to start to yield and (b) the smallest intensity of the distributed load that will cause both wires to yield. For the calculation, assume that the steel is elastic perfectly plastic.

400 mm 250 mm

150 mm

800 mm

C G

Probs. 4–99/100

4–95. The 10-mm-diameter shank of the steel bolt has a bronze sleeve bonded to it. The outer diameter of this sleeve is 20 mm. If the yield stress for the steel is ( sY ) st = 640 MPa, and for the bronze ( sY ) br = 520 MPa, determine the largest possible value of P that can be applied to the bolt. Assume the materials to be elastic perfectly plastic. E st = 200 GPa, E br = 100 GPa.

*4–96. The 10-mm-diameter shank of the steel bolt has a bronze sleeve bonded to it. The outer diameter of this sleeve is 20 mm. If the yield stress for the steel is ( sY ) st = 640 MPa, and for the bronze ( sY ) br = 520 MPa, determine the magnitude of the largest elastic load P that can be applied to the assembly. E st = 200 GPa, E br = 100 GPa.

10 mm

20 mm

Probs. 4–95/96

4–97. The weight is suspended from steel and aluminum wires, each having the same initial length of 3 m and cross- sectional area of 4 mm 2 . If the materials can be assumed to be elastic perfectly plastic, with (s Y ) st = 120 MPa and ( s Y ) al = 70 MPa, determine the force in each wire if the weight is (a) 600 N and (b) 720 N. E al = 70 GPa, E st = 200 GPa.

SteelAluminum

Prob. 4–97

4.9 RESIDUAL STRESS 173

*4–104. The rigid beam is supported by three 25-mm diameter A-36 steel rods. If the beam supports the force of P = 230 kN , determine the force developed in each rod. Consider the steel to be an elastic perfectly-plastic material.

4–105. The rigid beam is supported by three 25-mm diameter A-36 steel rods. If the force of P = 230 kN is applied on the beam and removed, determine the residual stresses in each rod. Consider the steel to be an elastic perfectly-plastic material.

D E

B C

400 mm

600 mm

400 mm 400 mm

Probs. 4–104/105

4–106. A material has a stress–strain diagram that can be described by the curve s = cP1>2 . Determine the deflection of the end of a rod made from this material if it has a length L , cross-sectional area A , and a specific weight g.

4–107. Solve Prob. 4–106 if the stress–strain diagram is defined by s = cP3>2 .

L A

d P

Probs. 4–106/107

4–101. The rigid lever arm is supported by two A-36 steel wires having the same diameter of 4 mm. If a force of P = 3 kN is applied to the handle, determine the force developed in both wires and their corresponding elongations. Consider A-36 steel as an elastic-perfectly plastic material.

4–102. The rigid lever arm is supported by two A-36 steel wires having the same diameter of 4 mm. Determine the smallest force P that will cause (a) only one of the wires to yield; (b) both wires to yield. Consider A-36 steel as an elastic-perfectly plastic material.

B D

C E

450 mm

150 mm

30!

150 mm

300 mm

Probs. 4–101/102

4–103. Two steel wires, each having a cross-sectional area of 2 mm 2 are tied to a ring at C , and then stretched and tied between the two pins A and B .The initial tension in the wires is 50 N. If a horizontal force P is applied to the ring, determine the force in each wire if P = 20 N. What is the smallest force P that must be applied to the ring to reduce the force in wire CB to zero? Take s Y = 300 MPa. E st = 200 GPa.

A BC

2 m 3 m

Prob. 4–103

174 CHAPTER 4 AXIAL LOAD

4–111. The bar having a diameter of 2 in. is fixed connected at its ends and supports the axial load P . If the material is elastic perfectly plastic as shown by the stress–strain diagram, determine the smallest load P needed to cause segment CB to yield. If this load is released, determine the permanent displacement of point C .

*4–112. Determine the elongation of the bar in Prob. 4–111 when both the load P and the supports are removed.

2 ft

A C

0.001

3 ft

P (in./in.)

s (ksi)

Probs. 4–111/112

4–113. The three bars are pinned together and subjected to the load P. If each bar has a cross-sectional area A, length L, and is made from an elastic perfectly plastic material, for which the yield stress is sY , determine the largest load (ultimate load) that can be supported by the bars, i.e., the load P that causes all the bars to yield. Also, what is the horizontal displacement of point A when the load reaches its ultimate value? The modulus of elasticity is E.

P u

Prob. 4–113

*4–108. The rigid beam is supported by the three posts A, B , and C of equal length. Posts A and C have a diameter of 75 mm and are made of a material for which E = 70 GPa and sY = 20 MPa. Post B has a diameter of 20 mm and is made of a material for which E! = 100 GPa and sY! = 590 MPa. Determine the smallest magnitude of P so that (a) only rods A and C yield and (b) all the posts yield.

4–109. The rigid beam is supported by the three posts A, B , and C . Posts A and C have a diameter of 60 mm and are made of a material for which E = 70 GPa and sY = 20 MPa. Post B is made of a material for which E! = 100 GPa and sY! = 590 MPa. If P = 130 kN , determine the diameter of post B so that all three posts are about to yield. (Do not assume that the three posts have equal uncompressed lengths.)

2 m 2 m 2 m

A B

P P

2 m

Probs. 4–108/109

4–110. The wire BC has a diameter of 0.125 in. and the material has the stress–strain characteristics shown in the figure. Determine the vertical displacement of the handle at D if the pull at the grip is slowly increased and reaches a magnitude of (a) P = 450 lb, (b) P = 600 lb. Assume the bar is rigid.

A B

50 in. 30 in.

40 in.

0.007 0.12

70 80

s (ksi)

P (in./in.)

Prob. 4–110

CHAPTER REVIEW 175

CHAPTER REVIEW

When a loading is applied at a point on a body, it tends to create a stress distribution within the body that becomes more uniformly distributed at regions removed from the point of application of the load. This is called Saint-Venant’s principle.

savg ! P A

The relative displacement at the end of an axially loaded member relative to the other end is determined from

d = L L

P(x)dx A (x)E(x)

x dx

P2P1

L d

If a series of concentrated external axial forces are applied to a member and AE is also piecewise constant, then

d = ! PL AE

For application, it is necessary to use a sign convention for the internal load P and displacement d. We considered tension and elongation as positive values. Also, the material must not yield, but rather it must remain linear elastic.

Superposition of load and displacement is possible provided the material remains linear elastic and no significant changes in the geometry of the member occur after loading.

The reactions on a statically indeterminate bar can be determined using the equilibrium equations and compatibility conditions that specify the displacement at the supports. These displacements are related to the loads using a load–displacement relationship such as d = PL >AE.

P4P1 P2 P3

L d

176 CHAPTER 4 AXIAL LOAD

A change in temperature can cause a member made of homogeneous isotropic material to change its length by

d = a!TL

If the member is confined, this change will produce thermal stress in the member.

Holes and sharp transitions at a cross section will create stress concentrations. For the design of a member made of brittle material one obtains the stress concentration factor K from a graph, which has been determined from experiment. This value is then multiplied by the average stress to obtain the maximum stress at the cross section.

smax = Ksavg

If the loading in a bar made of ductile material causes the material to yield, then the stress distribution that is produced can be determined from the strain distribution and the stress–strain diagram. Assuming the material is perfectly plastic, yielding will cause the stress distribution at the cross section of a hole or transition to even out and become uniform.

If a member is constrained and an external loading causes yielding, then when the load is released, it will cause residual stress in the member.

s1 s1

sY sY

CONCEPTUAL PROBLEMS 177

CONCEPTUAL PROBLEMS

C4–1. In each photo the concrete footings A were poured when the column was already in place. Later the concrete slab was poured. Explain why the 45! cracks formed in the slab at each corner of the square footing and not for the circular footing.

C4–1

C4–2. The row of bricks, along with mortar and an internal steel reinforcing rod, was intended to serve as a lintel beam to support the bricks above this ventilation opening on an exterior wall of a building. Explain what may have caused the bricks to fail in the manner shown.

C4–2

178 CHAPTER 4 AXIAL LOAD

4–114. The assembly consists of two A992 steel bolts AB and EF and an 6061-T6 aluminum rod CD. When the temperature is at 30° C, the gap between the rod and rigid member AE is 0.1 mm. Determine the normal stress developed in the bolts and the rod if the temperature rises to 130° C. Assume BF is also rigid.

4–115. The assembly shown consists of two A992 steel bolts AB and EF and an 6061-T6 aluminum rod CD . When the temperature is at 30° C, the gap between the rod and rigid member AE is 0.1 mm. Determine the highest temperature to which the assembly can be raised without causing yielding either in the rod or the bolts. Assume BF is also rigid.

300 mm400 mm

25 mm25 mm

50 mm

A C E

B D F

0.1 mm

Probs. 4–114/115

*4–116. The rods each have the same 25-mm diameter and 600-mm length. If they are made of A992 steel, determine the forces developed in each rod when the temperature increases by 50° C.

600 mm

60!

Prob. 4–116

REVIEW PROBLEMS

4–117. Two A992 steel pipes, each having a cross- sectional area of 0.32 in2, are screwed together using a union at B as shown. Originally the assembly is adjusted so that no load is on the pipe. If the union is then tightened so that its screw, having a lead of 0.15 in., undergoes two full turns, determine the average normal stress developed in the pipe. Assume that the union at B and couplings at A and C are rigid. Neglect the size of the union. Note: The lead would cause the pipe, when unloaded , to shorten 0.15 in. when the union is rotated one revolution.

2 ft3 ft

B CA

Prob. 4–117

4–118. The force P is applied to the bar, which is composed of an elastic perfectly plastic material. Construct a graph to show how the force in each section AB and BC (ordinate) varies as P (abscissa) is increased. The bar has cross-sectional areas of 1 in 2 in region AB and 4 in 2 in region BC , and s Y = 30 ksi.

6 in. 2 in.

A B C

Prob. 4–118

REVIEW PROBLEMS 179

4–121. The rigid link is supported by a pin at A and two A-36 steel wires, each having an unstretched length of 12 in. and cross-sectional area of 0.0125 in2. Determine the force developed in the wires when the link supports the vertical load of 350 lb.

5 in.

4 in.

12 in.

6 in.

350 lb

Prob. 4–121

4–122. The joint is made from three A992 steel plates that are bonded together at their seams. Determine the displacement of end A with respect to end B when the joint is subjected to the axial loads shown. Each plate has a thickness of 5 mm.

600 mm 200 mm 800 mm

100 mm

23 kN

23 kN46 kN

A B

Prob. 4–122

4–119. The 2014-T6 aluminum rod has a diameter of 0.5 in and is lightly attached to the rigid supports at A and B when T 1 = 70°F. If the temperature becomes T 2 = -10°F, and an axial force of P = 16 lb is applied to the rigid collar as shown, determine the reactions at A and B .

5 in. 8 in.

P/2

P/2 A B

Prob. 4–119

*4–120. The 2014-T6 aluminum rod has a diameter of 0.5 in. and is lightly attached to the rigid supports at A and B when T 1 = 70°F. Determine the force P that must be applied to the collar so that, when T = 0°F, the reaction at B is zero.

5 in. 8 in.

P/2

P/2 A B

Prob. 4–120

Chapter 5

The torsional stress and angle of twist of this soil auger depend upon the output of the machine turning the bit as well as the resistance of the soil

in contact with the shaft.

181

5.1 Torsional Deformation of a Circular Shaft

Torque is a moment that tends to twist a member about its longitudinal axis. Its effect is of primary concern in the design of drive shafts used in vehicles and machinery. We can illustrate physically what happens when a torque is applied to a circular shaft by considering the shaft to be made of a highly deformable material such as rubber, Fig. 5–1 a . When the torque is applied, the circles and longitudinal grid lines originally marked on the shaft tend to distort into the pattern shown in Fig. 5–1 b . Note that twisting causes the circles to remain circles , and each longitudinal grid line deforms into a helix that intersects the circles at equal angles. Also, the cross sections at the ends of the shaft will remain flat — that is, they do not warp or bulge in or out — and radial lines remain straight during the deformation, Fig. 5–1 b . From these observations we can assume that if the angle of twist is small , the length of the shaft and its radius will remain unchanged .

CHAPTER OBJECTIVES ■ In this chapter we will discuss the effects of applying a torsional

loading to a long straight member such as a shaft or tube. Initially we will consider the member to have a circular cross section. We will show how to determine both the stress distribution within the member and the angle of twist when the material behaves in a linear elastic manner and also when it is inelastic. Statically indeterminate analysis of shafts and tubes will also be discussed, along with special topics that include those members having noncircular cross sections. Lastly, stress concentrations and residual stress caused by torsional loadings will be given special consideration.

Torsion

182 CHAPTER 5 TORS ION

If the shaft is fixed at one end and a torque is applied to its other end, the dark green shaded plane in Fig. 5–2 will distort into a skewed form as shown. Here a radial line located on the cross section at a distance x from the fixed end of the shaft will rotate through an angle f(x). The angle f(x), so defined, is called the angle of twist . It depends on the position x and will vary along the shaft as shown.

In order to understand how this distortion strains the material, we will now isolate a small element located at a radial distance r (rho) from the axis of the shaft, Fig. 5–3 . Due to the deformation as noted in Fig. 5–2 , the front and rear faces of the element will undergo a rotation—the back face by f(x), and the front face by f(x) + !f. As a result, the difference in these rotations, !f, causes the element to be subjected to a shear strain . To calculate this strain, note that before deformation the angle between the edges AB and AC is 90°; after deformation, however, the edges of the element are AD and AC and the angle between them is u". From the definition of shear strain, Eq. 2–3 , we have

g = p 2

- u"

After deformation (b)

Longitudinal lines become

twisted

Circles remain circular

Radial lines remain straight

Fig. 5–1

Before deformation (a)

The angle of twist f(x) increases as x increases.

Undeformed plane

Deformed plane

f(x)

Fig. 5–2

Notice the deformation of the rectangular element when this rubber bar is subjected to a torque.

5.1 TORSIONAL DEFORMATION OF A CIRCULAR SHAFT 183

This angle, g, is indicated on the element. It can be related to the length !x of the element and the angle !f between the shaded planes by considering the length of arc BD, that is

BD = r!f = !x g

Therefore, if we let !x S dx and !f S df,

g = r df

dx (5–1)

Since dx and df are the same for all elements located at points on the cross section at x , then df>dx is constant over the cross section, and Eq. 5–1 states that the magnitude of the shear strain for any of these elements varies only with its radial distance r from the axis of the shaft. In other words, the shear strain within the shaft varies linearly along any radial line, from zero at the axis of the shaft to a maximum gmax at its outer boundary, Fig. 5–4 . Since df>dx = g>r = gmax>c, then g = ar

c bgmax (5–2)

The results obtained here are also valid for circular tubes. They depend only on the assumptions regarding the deformations mentioned above.

x ! "x

Shear strain of element

D B

Deformed plane

Undeformed plane

"x g

"f f(x)

u¿

Fig. 5–3

The shear strain at points on the cross section increases linearly

with r, i.e., g#(r/c)gmax.

gmax

Fig. 5–4

184 CHAPTER 5 TORS ION

5.2 The Torsion Formula When an external torque is applied to a shaft, it creates a corresponding internal torque within the shaft. In this section, we will develop an equation that relates this internal torque to the shear stress distribution on the cross section of a circular shaft or tube.

If the material is linear-elastic, then Hooke’s law applies, t = Gg, and consequently a linear variation in shear strain , as noted in the previous section, leads to a corresponding linear variation in shear stress along any radial line on the cross section. Hence, t will vary from zero at the shaft’s longitudinal axis to a maximum value, tmax , at its outer surface. This variation is shown in Fig. 5–5 on the front faces of a selected number of elements, located at an intermediate radial position r and at the outer radius c . Due to the proportionality of triangles, we can write

t = ar c btmax (5–3)

This equation expresses the shear-stress distribution over the cross section in terms of the radial position r of the element. Using it, we can now apply the condition that requires the torque produced by the stress distribution over the entire cross section to be equivalent to the resultant internal torque T at the section, which holds the shaft in equilibrium, Fig. 5–5 .

tmax

tmaxtt

Shear stress varies linearly along each radial line of the cross section.

c dA

Fig. 5–5

5.2 THE TORSION FORMULA 185

Specifically, each element of area dA , located at r, is subjected to a force of dF = t dA . The torque produced by this force is dT = r(t dA ). We therefore have for the entire cross section

T = LAr(t dA ) = LArarc btmax dA (5–4) Since tmax>c is constant, T =

tmax c L

A r2 dA (5–5)

The integral depends only on the geometry of the shaft. It represents the polar moment of inertia of the shaft’s cross-sectional area about the shaft’s longitudinal axis. We will symbolize its value as J , and therefore the above equation can be rearranged and written in a more compact form, namely,

tmax = Tc J

(5–6)

Here

tmax = the maximum shear stress in the shaft, which occurs at the outer surface

T = the resultant internal torque acting at the cross section. Its value is determined from the method of sections and the equation of moment equilibrium applied about the shaft’s longitudinal axis

J = the polar moment of inertia of the cross-sectional area c = the outer radius of the shaft

Combining Eqs. 5–3 and 5–6 , the shear stress at the intermediate distance r can be determined from

t = Tr

J (5–7)

Either of the above two equations is often referred to as the torsion formula . Recall that it is used only if the shaft is circular and the material is homogeneous and behaves in a linear elastic manner, since the derivation is based on Hooke’s law.

The shaft attached to the center of this wheel is subjected to a torque, and the maximum stress it creates must be resisted by the shaft to prevent failure.

186 CHAPTER 5 TORS ION

Solid Shaft. If the shaft has a solid circular cross section, the polar moment of inertia J can be determined using an area element in the form of a differential ring or annulus having a thickness dr and circumference 2pr, Fig. 5–6 . For this ring, dA = 2pr dr, and so

J = LAr2 dA = L c

0 r212pr dr2 = 2pL c0 r3 dr = 2pa 14 br4 ` c0

J = p

2 c4 (5–8)

Note that J is a geometric property of the circular area and is always positive. Common units used for its measurement are mm4 or in4.

The shear stress has been shown to vary linearly along each radial line of the cross section of the shaft. However, if an element of material on the cross section is isolated, then due to the complementary property of shear, equal shear stresses must also act on four of its adjacent faces as shown in Fig. 5–7 a . Hence, not only does the internal torque T develop a linear distribution of shear stress along each radial line in the plane of the cross-sectional area, but also an associated shear-stress distribution is developed along an axial plane , Fig. 5–7 b . It is interesting to note that because of this axial distribution of shear stress, shafts made from wood tend to split along the axial plane when subjected to excessive torque, Fig. 5–8 . This is because wood is an anisotropic material. Its shear resistance parallel to its grains or fibers, directed along the axis of the shaft, is much less than its resistance perpendicular to the fibers, directed in the plane of the cross section.

c r

Fig. 5–6

(a)

t tmax

tmax

Shear stress varies linearly along each radial line of the cross section.

(b)

tmax

Fig. 5–7

T T Failure of a wooden shaft due to torsion.

Fig. 5–8

5.2 THE TORSION FORMULA 187

Tubular Shaft. If a shaft has a tubular cross section, with inner radius ci and outer radius co , then from Eq. 5–8 we can determine its polar moment of inertia by subtracting J for a shaft of radius ci from that determined for a shaft of radius co . The result is

J = p

2 (c4o - c4i ) (5–9)

Like the solid shaft, the shear stress distributed over the tube’s cross-sectional area varies linearly along any radial line, Fig. 5–9 a . Furthermore, the shear stress varies along an axial plane in this same manner, Fig. 5–9 b .

Absolute Maximum Torsional Stress. If the absolute maximum torsional stress is to be determined, then it becomes important to find the location where the ratio Tc>J is a maximum. In this regard, it may be helpful to show the variation of the internal torque T at each section along the axis of the shaft by drawing a torque diagram , which is a plot of the internal torque T versus its position x along the shaft’s length. Once the internal torque throughout the shaft is determined, the maximum ratio of Tc>J can then be identified.

The tubular drive shaft for this truck was subjected to an excessive torque resulting in failure caused by yielding of the material. Engineers deliberately design drive shafts to fail before torsional damage can occur to parts of the engine or transmission.

(a)

ci co

Shear stress varies linearly along each radial line of the cross section.

(b)

tmax

Fig. 5–9

188 CHAPTER 5 TORS ION

Important Points

• When a shaft having a circular cross section is subjected to a torque, the cross section remains plane while radial lines rotate. This causes a shear strain within the material that varies linearly along any radial line, from zero at the axis of the shaft to a maximum at its outer boundary.

• For linear elastic homogeneous material the shear stress along any radial line of the shaft also varies linearly , from zero at its axis to a maximum at its outer boundary. This maximum shear stress must not exceed the proportional limit.

• Due to the complementary property of shear, the linear shear stress distribution within the plane of the cross section is also distributed along an adjacent axial plane of the shaft.

• The torsion formula is based on the requirement that the resultant torque on the cross section is equal to the torque produced by the shear stress distribution about the longitudinal axis of the shaft. It is required that the shaft or tube have a circular cross section and that it is made of homogeneous material which has linear-elastic behavior.

Procedure for Analysis

The torsion formula can be applied using the following procedure.

Internal Loading. • Section the shaft perpendicular to its axis at the point where the shear stress is to be determined, and use

the necessary free-body diagram and equations of equilibrium to obtain the internal torque at the section.

Section Property. • Calculate the polar moment of inertia of the cross-sectional area. For a solid section of radius c , J = pc4>2,

and for a tube of outer radius co and inner radius ci , J = p1c4o - c4i 2 >2. Shear Stress.

• Specify the radial distance r, measured from the center of the cross section to the point where the shear stress is to be found. Then apply the torsion formula t = Tr>J, or if the maximum shear stress is to be determined use tmax = Tc>J. When substituting the data, make sure to use a consistent set of units.

• The shear stress acts on the cross section in a direction that is always perpendicular to r. The force it creates must contribute a torque about the axis of the shaft that is in the same direction as the internal resultant torque T acting on the section. Once this direction is established, a volume element located at the point where t is determined can be isolated, and the direction of t acting on the remaining three adjacent faces of the element can be shown.

5.2 THE TORSION FORMULA 189

EXAMPLE 5.1

The solid shaft of radius c is subjected to a torque T , Fig. 5–10 a . Determine the fraction of T that is resisted by the material contained within the outer core of the shaft, which has an inner radius of c>2 and outer radius c.

SOLUTION The stress in the shaft varies linearly, such that t = (r>c)tmax , Eq. 5–3 . Therefore, the torque dT ! on the ring (area) located within the outer core, Fig. 5–10 b , is

dT ! = r(t dA ) = r(r>c)tmax(2pr dr) For the entire outer core area the torque is

T ! = 2ptmax

c L c

c>2r3dr =

2ptmax c

1 4

r4 2 c>2 c

So that

T ! = 15p 32

tmaxc 3 (1)

This torque T ! can be expressed in terms of the applied torque T by first using the torsion formula to determine the maximum stress in the shaft. We have

tmax = Tc J

= Tc

(p>2)c4 or

tmax = 2T pc3

Substituting this into Eq. 1 yields

T ! = 15 16

T Ans.

NOTE: Here, approximately 94% of the torque is resisted by the outer core, and the remaining 6% (or 116 ) of T is resisted by the inner core of the shaft, r = 0 to r = c>2 . As a result, the material located at the outer core of the shaft is highly effective in resisting torque, which justifies the use of tubular shafts as an efficient means for transmitting torque, and thereby saving material.

(a)

cc– 2

c– 2

(b)

r tmaxt

Fig. 5–10

190 CHAPTER 5 TORS ION

EXAMPLE 5.2

42.5 kip!in.

30 kip!in.

12.5 kip!in.a

(a)

SOLUTION

Internal Torque. Since the bearing reactions do not offer resistance to shaft rotation, the applied torques satisfy moment equilibrium about the shaft’s axis.

The internal torque at section a – a will be determined from the free-body diagram of the left segment, Fig. 5–11 b . We have

!Mx = 0; 42.5 kip # in. - 30 kip # in. - T = 0 T = 12.5 kip # in. Section Property. The polar moment of inertia for the shaft is

J = p

2 10.75 in.24 = 0.497 in.4

Shear Stress. Since point A is at r = c = 0.75 in.,

tA = Tc J

= 112.5 kip # in.2 10.75 in.210.497 in.42 = 18.9 ksi Ans.

Likewise for point B , at r = 0.15 in., we have

tB = Tr

J =

112.5 kip # in.2 10.15 in.210.497 in.42 = 3.77 ksi Ans. NOTE: The directions of these stresses on each element at A and B , Fig. 5–11 c , are established from the direction of the resultant internal torque T , shown in Fig. 5–11 b . Note carefully how the shear stress acts on the planes of each of these elements.

The shaft shown in Fig. 5–11 a is supported by two bearings and is subjected to three torques. Determine the shear stress developed at points A and B , located at section a – a of the shaft, Fig. 5–11 c .

18.9 ksi

3.77 ksi

12.5 kip·in.

(b)

42.5 kip!in.

30 kip!in.

Fig. 5–11

5.2 THE TORSION FORMULA 191

EXAMPLE 5.3

The pipe shown in Fig. 5–12 a has an inner diameter of 80 mm and an outer diameter of 100 mm. If its end is tightened against the support at A using a torque wrench at B , determine the shear stress developed in the material at the inner and outer walls along the central portion of the pipe when the 80-N forces are applied to the wrench.

SOLUTION Internal Torque. A section is taken at an intermediate location C along the pipe’s axis, Fig. 5–12 b . The only unknown at the section is the internal torque T . We require

!My = 0; 80 N 10.3 m2 + 80 N 10.2 m2 - T = 0 T = 40 N # m

Section Property. The polar moment of inertia for the pipe’s cross-sectional area is

J = p

2 3 10.05 m24 - 10.04 m244 = 5.796110-62 m4

Shear Stress. For any point lying on the outside surface of the pipe, r = co = 0.05 m , we have

to = Tco

J =

40 N # m10.05 m2 5.796110-62m4 = 0.345 MPa Ans.

And for any point located on the inside surface, r = ci = 0.04 m , so that

ti = Tci J

= 40 N # m10.04 m2 5.796110-62m4 = 0.276 MPa Ans.

NOTE: To show how these stresses act at representative points D and E on the cross-section, we will first view the cross section from the front of segment CA of the pipe, Fig. 5–12 a . On this section, Fig. 5–12 c , the resultant internal torque is equal but opposite to that shown in Fig. 5–12 b . The shear stresses at D and E contribute to this torque and therefore act on the shaded faces of the elements in the directions shown. As a consequence, notice how the shear-stress components act on the other three faces. Furthermore, since the top face of D and the inner face of E are in stress-free regions taken from the pipe’s outer and inner walls, no shear stress can exist on these faces or on the other corresponding faces of the elements.

300 mm

200 mm

80 N

(b)

T y

200 mm

80 N

(a)

80 N C

300 mm

tE ! 0.276 MPa tD ! 0.345 MPa

Fig. 5–12

192 CHAPTER 5 TORS ION

5.3 Power Transmission Shafts and tubes having circular cross sections are often used to transmit power developed by a machine. When used for this purpose, they are subjected to a torque that depends on the power generated by the machine and the angular speed of the shaft. Power is defined as the work performed per unit of time. Also, the work transmitted by a rotating shaft equals the torque applied times the angle of rotation. Therefore, if during an instant of time dt an applied torque T causes the shaft to rotate du, then the instantaneous power is

P = T du dt

Since the shaft’s angular velocity is v = du>dt, we can express the power as

P = Tv (5–10)

In the SI system, power is expressed in watts when torque is measured in newton-meters 1N # m2 and v is in radians per second 1rad>s2 11 W = 1 N # m>s2. In the FPS system, the basic units of power are foot-pounds per second 1ft # lb>s2; however, horsepower (hp) is often used in engineering practice, where

1 hp = 550 ft # lb>s For machinery, the frequency of a shaft’s rotation, f , is often reported.

This is a measure of the number of revolutions or cycles the shaft makes per second and is expressed in hertz 11 Hz = 1 cycle>s2 . Since 1 cycle = 2p rad , then v = 2pf, and so the above equation for power becomes

P = 2pfT (5–11)

Shaft Design. When the power transmitted by a shaft and its frequency of rotation are known, the torque developed in the shaft can be determined from Eq. 5–11 , that is, T = P>2pf. Knowing T and the allowable shear stress for the material, tallow , we can determine the size of the shaft’s cross section using the torsion formula, provided the material behavior is linear elastic. Specifically, the design or geometric parameter J>c becomes

J c

= T

tallow (5–12)

For a solid shaft , J = 1p>22c4 , and thus, upon substitution, a unique value for the shaft’s radius c can be determined. If the shaft is tubular , so that J = 1p>22 1co4 - ci42 , design permits a wide range of possibilities for the solution. This is because an arbitrary choice can be made for either co or ci and the other radius can then be determined from Eq. 5–12 .

The belt drive transmits the torque developed by an electric motor to the shaft at A. The stress developed in the shaft depends upon the power transmitted by the motor and the rate of rotation of the shaft. P = Tv.

5.3 POWER TRANSMISSION 193

EXAMPLE 5.4

A solid steel shaft AB , shown in Fig. 5–13 , is to be used to transmit 5 hp from the motor M to which it is attached. If the shaft rotates at v = 175 rpm and the steel has an allowable shear stress of tallow = 14.5 ksi, determine the required diameter of the shaft to the nearest 18 in.

B v

Fig. 5–13

SOLUTION The torque on the shaft is determined from Eq. 5–10 , that is, P = Tv . Expressing P in foot-pounds per second and v in radians>second, we have

P = 5 hp a 550 ft # lb>s 1 hp

b = 2750 ft # lb>s v =

175 rev min

a 2p rad 1 rev

b a 1 min 60 s

b = 18.33 rad>s Thus,

P = Tv; 2750 ft # lb>s = T 118.33 rad>s2 T = 150.1 ft # lb

Applying Eq. 5–12 yields

J c =

2 c4

c =

T tallow

c = ¢ 2T ptallow

≤1>3 = ¢ 21150.1 ft # lb2 112 in.>ft2 p114 500 lb>in22 ≤1>3

c = 0.429 in.

Since 2c = 0.858 in., select a shaft having a diameter of

d = 7 8

in. = 0.875 in. Ans.

194 CHAPTER 5 TORS ION

P5–1. Determine the internal torque at each section and show the torsional stress on differential volume elements located at A, B, C, and D.

C A

B 800 N!m

300 N!m

(a)

P5–1

P5–2. Determine the internal torque at each section and show the torsional stress on differential volume elements located at A, B, C, and D.

400 N!m

600 N!m

(b)

P5–2

P5–3. The solid and hollow shafts are each subjected to the torque T . In each case, sketch the shear stress distribution along the two radial lines.

P5–3

P5–4. The motor delivers 10 hp to the shaft. If it rotates at 1200 rpm, detemine the torque produced by the motor.

P5–4

PRELIMINARY PROBLEMS

5.3 POWER TRANSMISSION 195

F5–3. The shaft is hollow from A to B and solid from B to C . Determine the maximum shear stress developed in the shaft. The shaft has an outer diameter of 80 mm, and the thickness of the wall of the hollow segment is 10 mm.

2 kN!m

4 kN!m

F5–3

F5–4. Determine the maximum shear stress developed in the 40-mm-diameter shaft.

6 kN

10 kN

4 kN

2 kN

150 mm

100 mm

F5–4

F5–1. The solid circular shaft is subjected to an internal torque of T = 5 kN # m. Determine the shear stress developed at points A and B . Represent each state of stress on a volume element.

40 mm

30 mm

F5–1

F5–2. The hollow circular shaft is subjected to an internal torque of T = 10 kN # m. Determine the shear stress developed at points A and B . Represent each state of stress on a volume element.

60 mm

T " 10 kN!m

40 mm

F5–2

FUNDAMENTAL PROBLEMS

196 CHAPTER 5 TORS ION

F5–5. Determine the maximum shear stress developed in the shaft at section a-a .

600 N ! m

1500 N ! m

B 1500 N ! m

Section a–a

40 mm 30 mm

F5–5

F5–6. Determine the shear stress developed at point A on the surface of the shaft. Represent the state of stress on a volume element at this point. The shaft has a radius of 40 mm.

800 mm

5 kN!m/m

F5–6

F5–7. The solid 50-mm-diameter shaft is used to transmit the torques applied to the gears. Determine the absolute maximum shear stress in the shaft.

250 N m!

325 N m!

75 N m!

500 mm

400 mm

500 mm

150 N m!B

F5–7

F5–8. The gear motor can develop 3 hp when it turns at 150 revNmin. If the allowable shear stress for the shaft is t allow = 12 ksi, determine the smallest diameter of the shaft to the nearest 18 in. that can be used.

F5–8

5.3 POWER TRANSMISSION 197

5–1. The solid shaft of radius r is subjected to a torque T . Determine the radius r ! of the inner core of the shaft that resists one-half of the applied torque ( T N2). Solve the problem two ways: (a) by using the torsion formula, (b) by finding the resultant of the shear-stress distribution.

5–2. The solid shaft of radius r is subjected to a torque T . Determine the radius r ! of the inner core of the shaft that resists one-quarter of the applied torque ( T N4). Solve the problem two ways: (a) by using the torsion formula, (b) by finding the resultant of the shear-stress distribution.

r¿

Probs. 5–1/2

5–3. The solid shaft is fixed to the support at C and subjected to the torsional loadings shown. Determine the shear stress at points A and B and sketch the shear stress on volume elements located at these points.

C 75 mm

10 kN!m

75 mm50 mm

A B 4 kN!m

Prob. 5–3

*5–4. The copper pipe has an outer diameter of 40 mm and an inner diameter of 37 mm. If it is tightly secured to the wall at A and three torques are applied to it as shown, determine the absolute maximum shear stress developed in the pipe.

80 N!m

20 N!m

30 N!m

Prob. 5–4

5–5. The copper pipe has an outer diameter of 2.50 in. and an inner diameter of 2.30 in. If it is tightly secured to the wall at C and three torques are applied to it as shown, determine the shear stress developed at points A and B . These points lie on the pipe’s outer surface. Sketch the shear stress on volume elements located at A and B .

600 lb!ft

350 lb!ft

450 lb!ft

Prob. 5–5

PROBLEMS

198 CHAPTER 5 TORS ION

5–9. The solid shaft is fixed to the support at C and subjected to the torsional loadings shown. Determine the shear stress at points A and B on the surface, and sketch the shear stress on volume elements located at these points.

C 35 mm

800 N!m

35 mm 20 mm

300 N!m

Prob. 5–9

5–10. The coupling is used to connect the two shafts together. Assuming that the shear stress in the bolts is uniform , determine the number of bolts necessary to make the maximum shear stress in the shaft equal to the shear stress in the bolts. Each bolt has a diameter d .

Prob. 5–10

5–6. The solid shaft has a diameter of 0.75 in. If it is subjected to the torques shown, determine the maximum shear stress developed in regions BC and DE of the shaft. The bearings at A and F allow free rotation of the shaft.

5–7. The solid shaft has a diameter of 0.75 in. If it is subjected to the torques shown, determine the maximum shear stress developed in regions CD and EF of the shaft. The bearings at A and F allow free rotation of the shaft.

A B

C D

E F

40 lb!ft 25 lb!ft

20 lb!ft

35 lb!ft

Probs. 5–6/7

*5–8. The solid 30-mm-diameter shaft is used to transmit the torques applied to the gears. Determine the absolute maximum shear stress on the shaft.

300 N!m

A 200 N!m

500 N!m

300 mm

400 mm

500 mm

400 N!m

Prob. 5–8

5.3 POWER TRANSMISSION 199

5–11. The assembly consists of two sections of galvanized steel pipe connected together using a reducing coupling at B . The smaller pipe has an outer diameter of 0.75 in. and an inner diameter of 0.68 in., whereas the larger pipe has an outer diameter of 1 in. and an inner diameter of 0.86 in. If the pipe is tightly secured to the wall at C , determine the maximum shear stress developed in each section of the pipe when the couple shown is applied to the handles of the wrench.

15 lb 6 in.

15 lb

8 in.

Prob. 5–11

*5–12. The 150-mm-diameter shaft is supported by a smooth journal bearing at E and a smooth thrust bearing at F . Determine the maximum shear stress developed in each segment of the shaft.

5–13. If the tubular shaft is made from material having an allowable shear stress of t allow = 85 MPa, determine the required minimum wall thickess of the shaft to the nearest millimeter. The shaft has an outer diameter of 150 mm.

E B

C15 kN!m

30 kN!m

75 kN!m

30 kN!m

Probs. 5–12/13

5–14. A steel tube having an outer diameter of 2.5 in. is used to transmit 9 hp when turning at 27 revNmin. Determine the inner diameter d of the tube to the nearest 18 in. if the allowable shear stress is t allow = 10 ksi.

2.5 in. d

Prob. 5–14

5–15. The solid shaft is made of material that has an allowable shear stress of tallow = 10 MPa. Determine the required diameter of the shaft to the nearest millimeter.

*5–16. The solid shaft has a diameter of 40 mm. Determine the absolute maximum shear stress in the shaft and sketch the shear-stress distribution along a radial line of the shaft where the shear stress is maximum.

25 N!m

15 N!m

70 N!m

30 N!m

60 N!m

D E

Probs. 5–15/16

200 CHAPTER 5 TORS ION

5–21. If the 40-mm-diameter rod is subjected to a uniform distributed torque of t 0 = 1.5 kN # mNm, determine the shear stress developed at point C .

5–22. If the rod is subjected to a uniform distributed torque of t 0 = 1.5 kN # mNm, determine the rod’s minimum required diameter d if the material has an allowable shear stress of t allow = 75 MPa.

5–23. If the 40-mm diameter rod is made from a material having an allowable shear stress of t allow = 75 MPa, determine the maximum allowable intensity t 0 of the uniform distributed torque.

300 mm

A C

Probs. 5–21/22/23

*5–24. The copper pipe has an outer diameter of 2.50 in. and an inner diameter of 2.30 in. If it is tightly secured to the wall at C and a uniformly distributed torque is applied to it as shown, determine the shear stress developed at points A and B. These points lie on the pipe’s outer surface. Sketch the shear stress on volume elements located at A and B.

5–25. The copper pipe has an outer diameter of 2.50 in. and an inner diameter of 2.30 in. If it is tightly secured to the wall at C and it is subjected to the uniformly distributed torque along its entire length, determine the absolute maximum shear stress in the pipe. Discuss the validity of this result.

125 lb!ft/ft 4 in.

9 in.

12 in.

Probs. 5–24/25

5–17. The rod has a diameter of 1 in. and a weight of 10 lb>ft. Determine the maximum torsional stress in the rod at a section located at A due to the rod’s weight.

5–18. The rod has a diameter of 1 in. and a weight of 15 lb>ft. Determine the maximum torsional stress in the rod at a section located at B due to the rod’s weight.

4 ft

1.5 ft

4.5 ft A

B 1.5 ft

Probs. 5–17/18

5–19. The shaft consists of solid 80-mm-diameter rod segments AB and CD , and the tubular segment BC which has an outer diameter of 100 mm and inner diameter of 80 mm. If the material has an allowable shear stress of t allow = 75 MPa, determine the maximum allowable torque T that can be applied to the shaft.

*5–20. The shaft consists of rod segments AB and CD , and the tubular segment BC . If the torque T = 10 kN # m is applied to the shaft, determine the required minimum diameter of the rod and the maximum inner diameter of the tube. The outer diameter of the tube is 120 mm, and the material has an allowable shear stress of t allow = 75 MPa.

Probs. 5–19/20

5.3 POWER TRANSMISSION 201

5–29. The steel shafts are connected together using a fillet weld as shown. Determine the average shear stress in the weld along section a – a if the torque applied to the shafts is T = 60 N · m. Note: The critical section where the weld fails is along section a – a .

12 mm 12 mm

12 mm

12 mm 50 mm

T = 60 N!m

45"

Prob. 5–29

5–30. The shaft has a diameter of 80 mm. Due to friction at its surface within the hole, it is subjected to a variable torque described by the function t = 125xex22 N # mNm, where x is in meters. Determine the minimum torque T 0 needed to overcome friction and cause it to twist. Also, determine the absolute maximum stress in the shaft.

2 m

t = (25x ex 2 ) N!m/m

80 mm

Prob. 5–30

5–26. A cylindrical spring consists of a rubber annulus bonded to a rigid ring and shaft. If the ring is held fixed and a torque T is applied to the shaft, determine the maximum shear stress in the rubber.

T h

ro ri

Prob. 5–26

5–27. The assembly consists of the solid rod AB , tube BC, and the lever arm. If the rod and the tube are made of material having an allowable shear stress of t allow = 12 ksi, determine the maximum allowable torque T that can be applied to the end of the rod and from there the couple forces P that can be applied to the lever arm. The diameter of the rod is 2 in., and the outer and inner diameters of the tube are 4 in. and 2 in., respectively.

*5–28. The assembly consists of the solid rod AB , tube BC , and the lever arm. If a torque of T = 20 kip # in. is applied to the rod and couple forces of P = 5 kip are applied to the lever arm, determine the required diameter for the rod, and the outer and inner diameters of the tube, if the ratio of the inner diameter d i , to outer diameter d o , is required to be d i N d o = 0.75. The rod and the tube are made of material having an allowable shear stress of t allow = 12 ksi.

12 in. 12 in.

Probs. 5–27/28

202 CHAPTER 5 TORS ION

5–33. The motor M is connected to the speed reducer C by the tubular shaft and coupling. If the motor supplies 20 hp and rotates the shaft at a rate of 600 rpm, determine the minimum inner and outer diameters di and d o of the shaft if d i N d o = 0.75. The shaft is made from a material having an allowable shear stress of t allow = 12 ksi.

5–34. The motor M is connected to the speed reducer C by the tubular shaft and coupling. The shaft has an outer and inner diameter of 1 in. and 0.75 in., respectively, and is made from a material having an allowable shear stress of t allow = 12 ksi, when the motor supplies 20 hp of power. Determine the smallest allowable angular velocity of the shaft.

A B

Probs. 5–33/34

5–35. The 25-mm-diameter shaft on the motor is made of a material having an allowable shear stress of t allow = 75 MPa. If the motor is operating at its maximum power of 5 kW, determine the minimum allowable rotation of the shaft.

*5–36. The drive shaft of the motor is made of a material having an allowable shear stress of t allow = 75 MPa. If the outer diameter of the tubular shaft is 20 mm and the wall thickness is 2.5 mm, determine the maximum allowable power that can be supplied to the motor when the shaft is operating at an angular velocity of 1500 rev>min.

Probs. 5–35/36

5–31. The solid steel shaft AC has a diameter of 25 mm and is supported by smooth bearings at D and E . It is coupled to a motor at C , which delivers 3 kW of power to the shaft while it is turning at 50 rev>s. If gears A and B remove 1 kW and 2 kW, respectively, determine the maximum shear stress developed in the shaft within regions AB and BC . The shaft is free to turn in its support bearings D and E .

A CB E

2 kW 3 kW 25 mm1 kW

Prob. 5–31

*5–32. The pump operates using the motor that has a power of 85 W. If the impeller at B is turning at 150 rev>min, determine the maximum shear stress developed in the 20-mm-diameter transmission shaft at A .

A B150 rev/min

Prob. 5–32

5.3 POWER TRANSMISSION 203

5–37. A ship has a propeller drive shaft that is turning at 1500 rev>min while developing 1800 hp. If it is 8 ft long and has a diameter of 4 in., determine the maximum shear stress in the shaft caused by torsion.

5–38. The motor A develops a power of 300 W and turns its connected pulley at 90 rev>min. Determine the required diameters of the steel shafts on the pulleys at A and B if the allowable shear stress is tallow = 85 MPa.

60 mm

150 mm

90 rev/min

BBB

Prob. 5–38

5–39. The solid steel shaft DF has a diameter of 25 mm and is supported by smooth bearings at D and E . It is coupled to a motor at F , which delivers 12 kW of power to the shaft while it is turning at 50 rev>s. If gears A, B , and C remove 3 kW, 4 kW, and 5 kW respectively, determine the maximum shear stress developed in the shaft within regions CF and BC . The shaft is free to turn in its support bearings D and E .

*5–40. Determine the absolute maximum shear stress developed in the shaft in Prob. 5–39 .

FC ED

4 kW 5 kW 12 kW

25 mm3 kW

Probs. 5–39/40

5–41. The A-36 steel tubular shaft is 2 m long and has an outer diameter of 50 mm. When it is rotating at 40 rad>s, it transmits 25 kW of power from the motor M to the pump P . Determine the smallest thickness of the tube if the allowable shear stress is tallow = 80 MPa .

5–42. The A-36 solid steel shaft is 2 m long and has a diameter of 60 mm. It is required to transmit 60 kW of power from the motor M to the pump P. Determine the smallest angular velocity the shaft can have if the allowable shear stress is tallow = 80 MPa .

Probs. 5–41/42

5–43. The solid shaft has a linear taper from r A at one end to r B at the other. Derive an equation that gives the maximum shear stress in the shaft at a location x along the shaft’s axis.

Prob. 5–43

*5–44. The rod has a diameter of 0.5 in. and weight of 5 lbNft. Determine the maximum torsional stress in the rod at a section located at A due to the rod’s weight.

5–45. Solve Prob. 5–44 for the maximum torsional stress at B .

3 ft

1 ft

3 ft

B 1 ft

Probs. 5–44/45

5–46. A motor delivers 500 hp to the shaft, which is tubular and has an outer diameter of 2 in. If it is rotating at 200 radNs, determine its largest inner diameter to the nearest 18 in. if the allowable shear stress for the material is t allow = 25 ksi.

6 in.

A B

Prob. 5–46

204 CHAPTER 5 TORS ION

5.4 Angle of Twist Occasionally the design of a shaft depends on restricting the amount of rotation or twist that may occur when the shaft is subjected to a torque. Furthermore, being able to compute the angle of twist for a shaft is important when analyzing the reactions on statically indeterminate shafts.

In this section we will develop a formula for determining the angle of twist f (phi) of one end of a shaft with respect to its other end. The shaft is assumed to have a circular cross section that can gradually vary along its length, Fig. 5–14 a . Also, the material is assumed to be homogeneous and to behave in a linear-elastic manner when the torque is applied. As with the case of an axially loaded bar, we will neglect the localized deformations that occur at points of application of the torques and where the cross section changes abruptly. By Saint-Venant’s principle, these effects occur within small regions of the shaft’s length and generally they will have only a slight effect on the final result.

Using the method of sections, a differential disk of thickness dx , located at position x , is isolated from the shaft, Fig. 5–14 b . The internal resultant torque is T ( x ), since the external loading may cause it to vary along the axis of the shaft. Due to T ( x ), the disk will twist, such that the relative rotation of one of its faces with respect to the other face is df, Fig. 5–14 b . As a result an element of material located at an arbitrary radius r within the disk will undergo a shear strain g. The values of g and df are related by Eq. 5–1 , namely,

df = g dx r

(5–13)

Long shafts subjected to torsion can, in some cases, have a noticeable elastic twist.

(a) (b)

T(x)

dxdf

g g

gmax

Fig. 5–14

5.4 ANGLE OF TWIST 205

Since Hooke’s law, g = t>G , applies and the shear stress can be expressed in terms of the applied torque using the torsion formula t = T(x)r>J(x) , then g = T(x)r>J(x)G(x) . Substituting this into Eq. 5–13 , the angle of twist for the disk is

df = T 1x2

J1x2G(x) dx Integrating over the entire length L of the shaft, we obtain the angle of twist for the entire shaft, namely,

f = L L

T(x) dx J(x)G(x)

(5–14)

Here f = the angle of twist of one end of the shaft with respect to the

other end, measured in radians T(x) = the internal torque at the arbitrary position x , found from the

method of sections and the equation of moment equilibrium applied about the shaft’s axis

J(x) = the shaft’s polar moment of inertia expressed as a function of x . G(x) = the shear modulus of elasticity for the material expressed as a

function of x .

Constant Torque and Cross-Sectional Area. Usually in engineering practice the material is homogeneous so that G is constant. Also, the shaft’s cross-sectional area and the external torque are constant along the length of the shaft, Fig. 5–15 . If this is the case, the internal torque T 1x2 = T, the polar moment of inertia J1x2 = J, and Eq. 5–14 can be integrated, which gives

f = TL JG

(5–15)

The similarities between the above two equations and those for an axially loaded bar ( d = 1P1x2dx >A 1x2 E(x) and d = PL >AE ) should be noted.

When computing both the stress and the angle of twist of this soil auger, it is necessary to consider the variable torsional loading which acts along its length.

Fig. 5–15

206 CHAPTER 5 TORS ION

Equation 5–15 is often used to determine the shear modulus of elasticity G of a material. To do so, a specimen of known length and diameter is placed in a torsion testing machine like the one shown in Fig. 5–16 . The applied torque T and angle of twist f are then measured along the length L . Using Eq. 5–15 , G = TL >Jf. Usually, to obtain a more reliable value of G , several of these tests are performed and the average value is used.

Multiple Torques. If the shaft is subjected to several different torques, or the cross-sectional area or shear modulus changes abruptly from one region of the shaft to the next, Eq. 5–15 can be applied to each segment of the shaft where these quantities are all constant. The angle of twist of one end of the shaft with respect to the other is then found from the vector addition of the angles of twist of each segment. For this case,

f = a

TL JG

(5–16)

Sign Convention. In order to apply this equation, we must develop a sign convention for both the internal torque and the angle of twist of one end of the shaft with respect to the other end. To do this, we will use the right-hand rule, whereby both the torque and angle will be positive , provided the thumb is directed outward from the shaft when the fingers curl to give the tendency for rotation, Fig. 5–17 .

To illustrate the use of this sign convention, consider the shaft shown in Fig. 5–18 a . The angle of twist of end A with respect to end D is to be determined. Three segments of the shaft must be considered, since the

Load dial

Motor controls

Load range

selector

Torque strain

recorder

Fixed head

Specimen

Motor Turning

head

Movable unit on rails

Fig. 5–16

5.4 ANGLE OF TWIST 207

5 internal torque will change at B and at C . Using the method of sections, the internal torques are found for each segment, Fig. 5–18 b . By the right-hand rule, with positive torques directed away from the sectioned end of the shaft, we have TAB = +80 N # m, TBC = -70 N # m, and TCD = -10 N # m. These results are also shown on the torque diagram for the shaft, Fig. 5–18 c . Applying Eq. 5–16 , we have

fA >D = 1+80 N # m2 L AB JG + 1-70 N # m2 L BC JG + 1-10 N # m2 L CD JG If the other data is substituted and the answer is found as a positive quantity, it means that end A will rotate as indicated by the curl of the right-hand fingers when the thumb is directed away from the shaft, Fig. 5–18 a . The double subscript notation is used to indicate this relative angle of twist 1fA >D2; however, if the angle of twist is to be determined relative to a fixed support , then only a single subscript will be used. For example, if D is a fixed support, then the angle of twist will be denoted as fA.

Positive sign convention for T and f.

!T(x)

!f(x)

Fig. 5–17

(a)

LAB

LBC

LCD

B C

80 N"m

150 N"m 60 N"m

10 N"m

(b)

80 N"m

150 N"m TBC # 70 N"m

80 N"m TAB # 80 N"m

TCD # 10 N"m

10 N"m

T (N"m)

$70

$10

208 CHAPTER 5 TORS ION

Important Point

• When applying Eq. 5–14 to determine the angle of twist, it is important that the applied torques do not cause yielding of the material and that the material is homogeneous and behaves in a linear elastic manner.

Procedure for Analysis

The angle of twist of one end of a shaft or tube with respect to the other end can be determined using the following procedure.

Internal Torque. • The internal torque is found at a point on the axis of the shaft by

using the method of sections and the equation of moment equilibrium, applied along the shaft’s axis.

• If the torque varies along the shaft’s length, a section should be made at the arbitrary position x along the shaft and the internal torque represented as a function of x , i.e., T ( x ).

• If several constant external torques act on the shaft between its ends, the internal torque in each segment of the shaft, between any two external torques, must be determined. The results can be represented graphically as a torque diagram.

Angle of Twist. • When the circular cross-sectional area of the shaft varies along

the shaft’s axis, the polar moment of inertia must be expressed as a function of its position x along the axis, J ( x ).

• If the polar moment of inertia or the internal torque suddenly changes between the ends of the shaft, then f = 1 1T(x)>J(x)G(x)2 dx or f = TL >JG must be applied to each segment for which J , G , and T are continuous or constant.

• When the internal torque in each segment is determined, be sure to use a consistent sign convention for the shaft, such as the one discussed in Fig. 5–17 . Also make sure that a consistent set of units is used when substituting numerical data into the equations.

5.4 ANGLE OF TWIST 209

EXAMPLE 5.5

The gears attached to the fixed-end steel shaft are subjected to the torques shown in Fig. 5–19 a . If the shear modulus of elasticity is 80 GPa and the shaft has a diameter of 14 mm, determine the displacement of the tooth P on gear A . The shaft turns freely within the bearing at B .

SOLUTION Internal Torque. By inspection, the torques in segments AC , CD , and DE are different yet constant throughout each segment. Free-body diagrams of appropriate segments of the shaft along with the calculated internal torques are shown in Fig. 5–19 b . Using the right-hand rule and the established sign convention that positive torque is directed away from the sectioned end of the shaft, we have

TAC = +150 N # m TCD = -130 N # m TDE = -170 N # m These results are also shown on the torque diagram, Fig. 5–19 c .

Angle of Twist. The polar moment of inertia for the shaft is

J = p

2 10.007 m24 = 3.771110-92 m4

Applying Eq. 5–16 to each segment and adding the results algebraically, we have

fA = a

TL JG

= 1+150 N # m2 10.4 m2

3.771110-92m4 [8011092N>m2] +

1-130 N # m2 10.3 m2 3.771110-92m4 [8011092N>m2]

+ 1-170 N # m2 10.5 m2

3.771110-92m4 [8011092 N>m2] = -0.2121 rad Since the answer is negative, by the right-hand rule the thumb is directed toward the end E of the shaft, and therefore gear A will rotate as shown in Fig. 5–19 d .

The displacement of tooth P on gear A is

sP = fAr = 10.2121 rad2 1100 mm2 = 21.2 mm Ans. NOTE: Remember that this analysis is valid only if the shear stress does not exceed the proportional limit of the material.

280 N!m

150 N!m

TCD " 130 N!m

TAC " 150 N!m 150 N!m

TDE " 170 N!m

(b)

40 N!m

280 N!m

150 N!m

(c)

T (N!m)

x (m)

150

#130

0.4 0.7 1.2

#170

(d)

fA " 0.212 rad

100 mm P sP

Fig. 5–19

(a)

40 N!m

280 N!m

0.4 m

0.3 m

0.5 m

100 mm

150 N!m

210 CHAPTER 5 TORS ION

EXAMPLE 5.6

The two solid steel shafts shown in Fig. 5–20 a are coupled together using the meshed gears. Determine the angle of twist of end A of shaft AB when the torque T = 45 N # m is applied. Take G = 80 GPa. Shaft AB is free to rotate within bearings E and F , whereas shaft DC is fixed at D . Each shaft has a diameter of 20 mm. Ignore lateral flexing of the shafts.

SOLUTION

Internal Torque. Free-body diagrams for each shaft are shown in Fig. 5–20 b and 5–20 c . Summing moments along the x axis of shaft AB yields the tangential reaction between the gears of F = 45 N # m>0.15 m = 300 N. Summing moments about the x axis of shaft DC , this force then creates a torque of 1TD2x = 300 N 10.075 m2 = 22.5 N # m on shaft DC . Angle of Twist. To solve the problem, we will first calculate the rotation of gear C due to the torque of 22.5 N # m in shaft DC , Fig. 5–20 c . This angle of twist is

fC = TLDC

JG =

1+22.5 N # m2 11.5 m21p>22 10.010 m24 [8011092N>m2] = +0.0269 rad Since the gears at the end of the shaft are in mesh, the rotation fC of

gear C causes gear B to rotate fB , Fig. 5–20 d , where

fB10.15 m2 = 10.0269 rad2 10.075 m2 fB = 0.0134 rad

We will now determine the angle of twist of end A with respect to end B of shaft AB caused by the 45 N # m torque, Fig. 5–20 b . We have

fA >B = TABL ABJG = 1+45 N # m2 12 m21p>22 10.010 m24 [8011092 N>m2] = +0.0716 rad The rotation of end A is therefore determined by adding fB and

fA >B , since both angles are in the same direction , Fig. 5–20 b . We have fA = fB + fA >B = 0.0134 rad + 0.0716 rad = +0.0850 rad Ans.

T ! 45 N"m

E F

(a)

2 m

75 mm

150 mm

1.5 m C

Fig. 5–20

A Fy

T ! 45 N"m

Fz Ey

Ez (b)

F ! 300 N 0.150 m

fB ! 0.0134 rad

Dy Dz(TD)x ! 22.5 N"m

0.075 m

F ! 300 N

(c)

(MD)y

(MD)z

(d)

0.075 m

0.150 m

5.4 ANGLE OF TWIST 211

The 2-in.-diameter solid cast-iron post shown in Fig. 5–21 a is buried 24 in. in soil. If a torque is applied to its top using a rigid wrench, determine the maximum shear stress in the post and the angle of twist at its top. Assume that the torque is about to turn the post, and the soil exerts a uniform torsional resistance of t lb # in.>in. along its 24-in. buried length. G = 5.511032 ksi. SOLUTION

Internal Torque. The internal torque in segment AB of the post is constant. From the free-body diagram, Fig. 5–21 b , we have

!Mz = 0; TAB = 25 lb 112 in.2 = 300 lb # in. The magnitude of the uniform distribution of torque along the buried segment BC can be determined from equilibrium of the entire post, Fig. 5–21 c . Here

!Mz = 0 25 lb 112 in.2 - t124 in.2 = 0 t = 12.5 lb # in.>in. Hence, from a free-body diagram of a section of the post located at the position x , Fig. 5–21 d , we have

!Mz = 0; TBC - 12.5x = 0 TBC = 12.5x

Maximum Shear Stress. The largest shear stress occurs in region AB , since the torque is largest there and J is constant for the post. Applying the torsion formula, we have

tmax = TAB c

J =

1300 lb # in.2 11 in.21p>22 11 in.24 = 191 psi Ans. Angle of Twist. The angle of twist at the top can be determined relative to the bottom of the post, since it is fixed and yet is about to turn. Both segments AB and BC twist, and so in this case we have

fA = TAB LAB

JG + L

LBC

TBC dx JG

= 1300 lb # in.236 in.

JG + L

24 in.

12.5x dx JG

= 10 800 lb # in2

JG +

12.5[12422>2] lb # in2 JG

= 14 400 lb # in21p>22 11 in.24 550011032 lb>in2 = 0.00167 rad Ans.

EXAMPLE 5.7

t ! 12.5 lb"in./ in.

(d)

TBC

TAB (b)

6 in.

6 in. 25 lb25 lb

2 in.

36 in.

24 in.

(a)

6 in.

6 in. 25 lb25 lb

24 in.

24t

(c)

36 in.

6 in.

6 in. 25 lb25 lb

Fig. 5–21

212 CHAPTER 5 TORS ION

F5–12. A series of gears are mounted on the 40-mm- diameter steel shaft. Determine the angle of twist of gear B relative to gear A . Take G = 75 GPa.

200 mm

600 N!m

500 N!m

300 N!m

500 N!m

900 N!m

F5–12

F5–13. The 80-mm-diameter shaft is made of steel. If it is subjected to the uniform distributed torque, determine the angle of twist of end A with respect to B . Take G = 75 GPa.

800 mm

5 kN!m/m

F5–13

F5–14. The 80-mm-diameter shaft is made of steel. If it is subjected to the triangular distributed load, determine the angle of twist of end A with respect to C . Take G = 75 GPa.

400 mm

600 mm

15 kN!m/m

F5–14

F5–9. The 60-mm-diameter steel shaft is subjected to the torques shown. Determine the angle of twist of end A with respect to C . Take G = 75 GPa.

600 mm

400 mm

3 kN!m

2 kN!m

F5–9

F5–10. Determine the angle of twist of wheel B with respect to wheel A . The shaft has a diameter of 40 mm and is made of steel for which G = 75 GPa.

4 kN 10 kN

6 kN

2 kN

150 mm

100 mm

150 mm

450 mm

F5–10

F5–11. The hollow 6061-T6 aluminum shaft has an outer and inner radius of c o = 40 mm and c i = 30 mm, respectively. Determine the angle of twist of end A . The flexible support at B has a torsional stiffness of k = 90 kN # m>rad.

900 mm

3 kN!m

F5–11

FUNDAMENTAL PROBLEMS

5.4 ANGLE OF TWIST 213

5–50. The hydrofoil boat has an A992 steel propeller shaft that is 100 ft long. It is connected to an in-line diesel engine that delivers a maximum power of 2500 hp and causes the shaft to rotate at 1700 rpm. If the outer diameter of the shaft is 8 in. and the wall thickness is 38 in. , determine the maximum shear stress developed in the shaft. Also, what is the “wind up,” or angle of twist in the shaft at full power?

100 ft

Prob. 5–50

5–51. The 60-mm-diameter shaft is made of 6061-T6 aluminum having an allowable shear stress of t allow = 80 MPa. Determine the maximum allowable torque T . Also, find the corresponding angle of twist of disk A relative to disk C .

*5–52. The 60-mm-diameter shaft is made of 6061-T6 aluminum. If the allowable shear stress is t allow = 80 MPa, and the angle of twist of disk A relative to disk C is limited so that it does not exceed 0.06 rad, determine the maximum allowable torque T .

T23

T13

1.20 m

Probs. 5–51/52

5–47. The propellers of a ship are connected to an A-36 steel shaft that is 60 m long and has an outer diameter of 340 mm and inner diameter of 260 mm. If the power output is 4.5 MW when the shaft rotates at 20 rad>s, determine the maximum torsional stress in the shaft and its angle of twist.

*5–48. The solid shaft of radius c is subjected to a torque T at its ends. Show that the maximum shear strain developed in the shaft is g max = Tc > JG . What is the shear strain on an element located at point A , c >2 from the center of the shaft? Sketch the strain distortion of this element.

T Tc

/2c

Prob. 5–48

5–49. The A-36 steel axle is made from tubes AB and CD and a solid section BC. It is supported on smooth bearings that allow it to rotate freely. If the gears, fixed to its ends, are subjected to 85@N # m torques, determine the angle of twist of gear A relative to gear D. The tubes have an outer diameter of 30 mm and an inner diameter of 20 mm. The solid section has a diameter of 40 mm.

400 mm

250 mm 85 N m!

85 N m!

Prob. 5–49

PROBLEMS

214 CHAPTER 5 TORS ION

*5–56. The A-36 steel axle is made from tubes AB and CD and a solid section BC . It is supported on smooth bearings that allow it to rotate freely. If the gears, fixed to its ends, are subjected to 85@N # m torques, determine the angle of twist of the end B of the solid section relative to end C . The tubes have an outer diameter of 30 mm and an inner diameter of 20 mm. The solid section has a diameter of 40 mm.

400 mm

250 mm 85 N m!

85 N m!

Prob. 5–56

5–57. The turbine develops 150 kW of power, which is transmitted to the gears such that C receives 70% and D receives 30%. If the rotation of the 100-mm-diameter A-36 steel shaft is v = 800 rev>min., determine the absolute maximum shear stress in the shaft and the angle of twist of end E of the shaft relative to B . The journal bearing at E allows the shaft to turn freely about its axis.

5–58. The turbine develops 150 kW of power, which is transmitted to the gears such that both C and D receive an equal amount. If the rotation of the 100-mm-diameter A-36 steel shaft is v = 500 rev>min., determine the absolute maximum shear stress in the shaft and the rotation of end B of the shaft relative to E . The journal bearing at E allows the shaft to turn freely about its axis.

3 m

B C

E4 m

2 m

Probs. 5–57/58

5–53. The 20-mm-diameter A-36 steel shaft is subjected to the torques shown. Determine the angle of twist of the end B.

80 N!m

20 N!m

30 N!m B

800 mm

600 mm 200 mm

Prob. 5–53

5–54. The shaft is made of A992 steel with the allowable shear stress of t allow = 75 MPa. If gear B supplies 15 kW of power, while gears A , C and D withdraw 6 kW, 4 kW and 5 kW, respectively, determine the required minimum diameter d of the shaft to the nearest millimeter. Also, find the corresponding angle of twist of gear A relative to gear D . The shaft is rotating at 600 rpm.

5–55. Gear B supplies 15 kW of power, while gears A , C and D withdraw 6 kW, 4 kW and 5 kW, respectively. If the shaft is made of steel with the allowable shear stress of t allow = 75 MPa, and the relative angle of twist between any two gears cannot exceed 0.05 rad, determine the required minimum diameter d of the shaft to the nearest millimeter. The shaft is rotating at 600 rpm.

C600 mm

600 mm

Probs. 5–54/55

5.4 ANGLE OF TWIST 215

5–59. The shaft is made of A992 steel. It has a diameter of 1 in. and is supported by bearings at A and D , which allow free rotation. Determine the angle of twist of B with respect to D.

*5–60. The shaft is made of A-36 steel. It has a diameter of 1 in. and is supported by bearings at A and D , which allow free rotation. Determine the angle of twist of gear C with respect to B .

60 lb!ft

2 ft

2.5 ft

3 ft D

Probs. 5–59/60

5–61. The two shafts are made of A992 steel. Each has a diameter of 1 in., and they are supported by bearings at A, B, and C, which allow free rotation. If the support at D is fixed, determine the angle of twist of end B when the torques are applied to the assembly as shown.

5–62. The two shafts are made of A992 steel. Each has a diameter of 1 in., and they are supported by bearings at A, B, and C, which allow free rotation. If the support at D is fixed, determine the angle of twist of end A when the torques are applied to the assembly as shown.

A 40 lb!ft

80 lb!ft

8 in.

10 in.

12 in. 4 in.

10 in.

30 in.

6 in. B

Probs. 5–61/62

5–63. If the shaft is made of red brass C83400 copper with an allowable shear stress of t allow = 20 MPa, determine the maximum allowable torques T 1 and T 2 that can be applied at A and B . Also, find the corresponding angle of twist of end A . Set L = 0.75 m.

*5–64. If the shaft is made of red brass C83400 copper and is subjected to torques T 1 = 20 kN # m and T 2 = 50 kN # m, determine the distance L so that the angle of twist at end A is zero.

1 m

Section a–a

L a

C B

100 mm

80 mm

Probs. 5–63/64

5–65. The 8-mm·diameter A-36 steel bolt is screwed tightly into a block at A. Determine the couple forces F that should be applied to the wrench so that the maximum shear stress in the bolt becomes 18 MPa. Also, compute the corresponding displacement of each force F needed to cause this stress. Assume that the wrench is rigid.

80 mm

150 mm

Prob. 5–65

216 CHAPTER 5 TORS ION

5–69. The A-36 steel bolt is tightened within a hole so that the reactive torque on the shank AB can be expressed by the equation t = ( kx 2 ) N # m>m, where x is in meters. If a torque of T = 50 N # m is applied to the bolt head, determine the constant k and the amount of twist in the 50-mm length of the shank. Assume the shank has a constant radius of 4 mm.

5–70. Solve Prob. 5–69 if the distributed torque is t = (kx 2>3) N # m>m .

T = 50 N!m

Ax 50 mm

Probs. 5–69/70

■5–71. Consider the general problem of a circular shaft made from m segments, each having a radius of cm and shear modulus Gm . If there are n torques on the shaft as shown, write a computer program that can be used to determine the angle of twist of its end A. Show an application of the program using the values L1 = 0.5 m, c1 = 0.02 m , G1 = 30 GPa , L2 = 1.5 m , c2 = 0.05 m, G2 = 15 GPa , T1 = -450 N # m , d1 = 0.25 m, T2 = 600 N # m , d2 = 0.8 m.

d1 d2

Prob. 5–71

5–66. The A-36 hollow steel shaft is 2 m long and has an outer diameter of 40 mm. When it is rotating at 80 rad>s, it transmits 32 kW of power from the engine E to the generator G . Determine the smallest thickness of the shaft if the allowable shear stress is t allow = 140 MPa and the shaft is restricted not to twist more than 0.05 rad.

5–67. The A-36 solid steel shaft is 3 m long and has a diameter of 50 mm. It is required to transmit 35 kW of power from the engine E to the generator G . Determine the smallest angular velocity the shaft can have if it is restricted not to twist more than 1°.

E G

Probs. 5–66/67

*5–68. If the shaft is subjected to a uniform distributed torque t0, determine the angle of twist at A . The material has a shear modulus G . The shaft is hollow for exactly half its length.

A c t0

c 2

Prob. 5–68

5.4 ANGLE OF TWIST 217

5–75. The A992 steel posts are “drilled” at constant angular speed into the soil using the rotary installer. If the post has an inner diameter of 200 mm and an outer diameter of 225 mm, determine the relative angle of twist of end A of the post with respect to end B when the post reaches the depth indicated. Due to soil friction, assume the torque along the post varies linearly as shown, and a concentrated torque of 80 kN · m acts at the bit.

4 m

3 m

15 kN!m/m

80 kN!m

Prob. 5–75

*5–76. A cylindrical spring consists of a rubber annulus bonded to a rigid ring and shaft. If the ring is held fixed and a torque T is applied to the rigid shaft, determine the angle of twist of the shaft. The shear modulus of the rubber is G . Hint: As shown in the figure, the deformation of the element at radius r can be determined from rdu = drg. Use this expression, along with t = T > 12pr 2h2 from Prob. 5–26 , to obtain the result.

g du

gdr ! rdu

T h

Prob. 5–76

*5–72. The 80-mm diameter shaft is made of 6061-T6 aluminum alloy and subjected to the torsional loading shown. Determine the angle of twist at end A.

0.6 m

2 kN"m

10 kN"m/ m

Prob. 5–72

5–73. The contour of the surface of the shaft is defined by the equation y = e ax , where a is a constant. If the shaft is subjected to a torque T at its ends, determine the angle of twist of end A with respect to end B . The shear modulus is G .

y = eax

Prob. 5–73

5–74. The rod ABC of radius c is embedded into a medium where the distributed torque reaction varies linearly from zero at C to t 0 at B . If couple forces P are applied to the lever arm, determine the value of t 0 for equilibrium. Also, find the angle of twist of end A . The rod is made from material having a shear modulus of G .

P P

L 2

d 2

t0C

Prob. 5–74

218 CHAPTER 5 TORS ION

5.5 Statically Indeterminate Torque-Loaded Members

A torsionally loaded shaft may be classified as statically indeterminate if the moment equation of equilibrium, applied about the axis of the shaft, is not adequate to determine the unknown torques acting on the shaft. An example of this situation is shown in Fig. 5–22 a . As shown on the free-body diagram, Fig. 5–22 b , the reactive torques at the supports A and B are unknown. We require that

!Mx = 0; T - TA - TB = 0

In order to obtain a solution, we will use the method of analysis discussed in Sec. 4.4 . The necessary condition of compatibility, or the kinematic condition, requires the angle of twist of one end of the shaft with respect to the other end to be equal to zero, since the end supports are fixed. Therefore,

fA >B = 0 Provided the material is linear elastic, we can apply the load–displacement relation f = TL >JG to express the compatibility condition in terms of the unknown torques. Realizing that the internal torque in segment AC is +TA and in segment CB it is -TB , Fig. 5–22 c , we have

TALAC

JG -

TBLBC JG

= 0

(b)

(c)

Fig. 5–22

LAC

LBC

L C

(a)

5.5 STATICALLY INDETERMINATE TORQUE-LOADED MEMBERS 219

Solving the above two equations for the reactions, realizing that L = LAC + LBC , we get

TA = T ¢LBCL ≤ and TB = T ¢LACL ≤ Procedure for Analysis

The unknown torques in statically indeterminate shafts are determined by satisfying equilibrium, compatibility, and torque-displacement requirements for the shaft.

Equilibrium. • Draw a free-body diagram of the shaft in order to identify all the

external torques that act on it. Then write the equation of moment equilibrium about the axis of the shaft.

Compatibility. • Write the compatibility equation between two points along the

shaft. Give consideration as to how the supports constrain the shaft when it is twisted.

Load-Displacement. • Express the angles of twist in the compatibility condition in terms

of the torques, using a torque-displacement relation, such as f = TL >JG.

• Solve the equations for the unknown reactive torques. If any of the magnitudes have a negative numerical value, it indicates that this torque acts in the opposite sense of direction to that shown on the free-body diagram.

The shaft of this cutting machine is fixed at its ends and subjected to a torque at its center, allowing it to act as a torsional spring.

220 CHAPTER 5 TORS ION

EXAMPLE 5.8

The solid steel shaft shown in Fig. 5–23 a has a diameter of 20 mm. If it is subjected to the two torques, determine the reactions at the fixed supports A and B .

(c)

300 ! TB

800 ! TBTB

800 N"m

500 N"m

Fig. 5–23

(a)

B 0.2 m

1.5 m

0.3 m

800 N·m

500 N·m

(b)

x TB

800 N"m

500 N"m

SOLUTION

Equilibrium. By inspection of the free-body diagram, Fig. 5–23 b , it is seen that the problem is statically indeterminate since there is only one available equation of equilibrium and there are two unknowns. We require

!Mx = 0; -TB + 800 N # m - 500 N # m - TA = 0 (1) Compatibility. Since the ends of the shaft are fixed, the angle of twist of one end of the shaft with respect to the other must be zero. Hence, the compatibility equation becomes

f A >B = 0 Load-Displacement. This condition can be expressed in terms of the unknown torques by using the load-displacement relationship, f = TL >JG. Here there are three regions of the shaft where the internal torque is constant. On the free-body diagrams in Fig. 5–23 c we have shown the internal torques acting on the left segments of the shaft which are sectioned in each of these regions. This way the internal torque is only a function of TB . Using the sign convention established in Sec. 5.4 , we have

-TB10.2 m2

JG +

1800 - TB2 11.5 m2 JG

+ (300 - TB)10.3 m2

JG = 0

so that

TB = 645 N # m Ans. Using Eq. 1 ,

TA = -345 N # m Ans. The negative sign indicates that T A acts in the opposite direction of that shown in Fig. 5–23 b .

5.5 STATICALLY INDETERMINATE TORQUE-LOADED MEMBERS 221

EXAMPLE 5.9

The shaft shown in Fig. 5–24 a is made from a steel tube, which is bonded to a brass core. If a torque of T = 250 lb # ft is applied at its end, plot the shear-stress distribution along a radial line of its cross-sectional area. Take Gst = 11.411032 ksi, Gbr = 5.2011032 ksi. SOLUTION

Equilibrium. A free-body diagram of the shaft is shown in Fig. 5–24 b . The reaction at the wall has been represented by the unknown amount of torque resisted by the steel, Tst , and by the brass, Tbr . Working in units of pounds and inches, equilibrium requires -Tst - Tbr + 1250 lb # ft2 112 in.>ft2 = 0 (1) Compatibility. We require the angle of twist of end A to be the same for both the steel and brass since they are bonded together. Thus,

f = fst = fbr Load-Displacement. Applying the load–displacement relationship, f = TL >JG,

TstL1p>22[11 in.24 - 10.5 in.24] 11.4(103) kip>in2 =

TbrL1p>22 10.5 in.24 5.20110 32 kip>in2 Tst = 32.88 Tbr (2) Solving Eqs. 1 and 2 , we get

Tst = 2911.5 lb # in. = 242.6 lb # ft Tbr = 88.5 lb # in. = 7.38 lb # ft

The shear stress in the brass core varies from zero at its center to a maximum at the interface where it contacts the steel tube. Using the torsion formula,

1tbr2max = 188.5 lb # in.2 10.5 in.21p>22 10.5 in.24 = 451 psi For the steel, the minimum and maximum shear stresses are

1tst2min = 12911.5 lb # in.2 10.5 in.21p>22[11 in.24 - 10.5 in.24] = 989 psi 1tst2max = 12911.5 lb # in.2 11 in.21p>22[11 in.24 - 10.5 in.24] = 1977 psi The results are plotted in Fig. 5–24 c . Note the discontinuity of shear

stress at the brass and steel interface. This is to be expected, since the materials have different moduli of rigidity; i.e., steel is stiffer than brass 1Gst 7 Gbr2 and thus it carries more shear stress at the interface. Although the shear stress is discontinuous here, the shear strain is not. Rather, the shear strain is the same for both the brass and the steel.

(a)

4 ft

A T ! 250 lb"ft

1 in.

0.5 in.

(b) x

250 lb"ft

Tbr

Tst

Shear–stress distribution

(c)

1977 psi 989 psi

1 in.

0.5 in.

451 psi

Fig. 5–24

222 CHAPTER 5 TORS ION

5–77. The steel shaft has a diameter of 40 mm and is fixed at its ends A and B . If it is subjected to the couple determine the maximum shear stress in regions AC and CB of the shaft. G st = 75 GPa.

400 mm

600 mm

3 kN

50 mm

Prob. 5–77

5–78. The A992 steel shaft has a diameter of 60 mm and is fixed at its ends A and B . If it is subjected to the torques shown, determine the absolute maximum shear stress in the shaft.

D 1 m

1 m

1.5 m

200 N!m

500 N!m

Prob. 5–78

5–79. The steel shaft is made from two segments: AC has a diameter of 0.5 in, and CB has a diameter of 1 in. If the shaft is fixed at its ends A and B and subjected to a torque of 500 lb· ft, determine the maximum shear stress in the shaft. G st = 10.8(10 3 ) ksi.

5 in.

8 in.

12 in.

1 in.

0.5 in. A

D 500 lb!ft

Prob. 5–79

*5–80. The shaft is made of A-36 steel and is fixed at its ends A and D . Determine the torsional reactions at these supports.

5–81. The shaft is made of A-36 steel and is fixed at end D , while end A is allowed to rotate 0.005 rad when the torque is applied. Determine the torsional reactions at these supports.

1.5 ft

2 ft

6 in.

1.5 ft

40 kip!ft

20 kip!ft

Probs. 5–80/81

5–82. The shaft is made from a solid steel section AB and a tubular portion made of steel and having a brass core. If it is fixed to a rigid support at A, and a torque of T = 50 lb # ft is applied to it at C, determine the angle of twist that occurs at C and compute the maximum shear stress and maximum shear strain in the brass and steel. Take Gst = 11.511032 ksi, Gbr = 5.611032 ksi.

0.5 in.

1 in.

2 ft

3 ft

C T " 50 lb!ft

Prob. 5–82

PROBLEMS

5.5 STATICALLY INDETERMINATE TORQUE-LOADED MEMBERS 223

5–86. The two shafts are made of A-36 steel. Each has a diameter of 25 mm and they are connected using the gears fixed to their ends. Their other ends are attached to fixed supports at A and B. They are also supported by journal bearings at C and D, which allow free rotation of the shafts along their axes. If a torque of 500 N # m is applied to the gear at E as shown, determine the reactions at A and B.

5–87. Determine the rotation of the gear at E in Prob. 5–86 .

50 mm

100 mm

1.5 m

0.75 m

500 N!m

Probs. 5–86/87

*5–88. A rod is made from two segments: AB is steel and BC is brass. It is fixed at its ends and subjected to a torque of T = 680 N · m. If the steel portion has a diameter of 30 mm, determine the required diameter of the brass portion so the reactions at the walls will be the same. G st = 75 GPa, G br = 39 GPa.

5–89. Determine the absolute maximum shear stress in the shaft of Prob. 5–88.

680 N!m B

1.60 m

0.75 m

Probs. 5–88/89

5–83. The motor A develops a torque at gear B of 450 lb # ft, which is applied along the axis of the 2-in.-diameter steel shaft CD . This torque is to be transmitted to the pinion gears at E and F . If these gears are temporarily fixed, determine the maximum shear stress in segments CB and BD of the shaft. Also, what is the angle of twist of each of these segments? The bearings at C and D only exert force reactions on the shaft and do not resist torque. G st = 1211032 ksi.

4 ft 3 ft

E F

450 lb ft·

Prob. 5–83

*5–84. The Am1004-T61 magnesium tube is bonded to the A-36 steel rod. If the allowable shear stresses for the magnesium and steel are (t allow ) mg = 45 MPa and (t allow ) st = 75 MPa, respectively, determine the maximum allowable torque that can be applied at A . Also, find the corresponding angle of twist of end A .

5–85. The Am1004-T61 magnesium tube is bonded to the A-36 steel rod. If a torque of T = 5 kN # m is applied to end A , determine the maximum shear stress in each material. Sketch the shear stress distribution.

900 mm

80 mm 40 mm

Probs. 5–84/85

224 CHAPTER 5 TORS ION

*5–92. If the shaft is subjected to a uniform distributed torque of t = 20 kN # m>m, determine the maximum shear stress developed in the shaft. The shaft is made of 2014-T6 aluminum alloy and is fixed at A and C.

Section a–a

80 mm

60 mm a

a 600 mm

400 mm

20 kN!m/m

Prob. 5–92

5–93. The tapered shaft is confined by the fixed supports at A and B . If a torque T is applied at its mid-point, determine the reactions at the supports.

L/2

A 2c

L/2

Prob. 5–93

5–94. The shaft of radius c is subjected to a distributed torque t , measured as torque> length of shaft. Determine the reactions at the fixed supports A and B .

L A

) t0

2t0

)((t " t0 1 # 2xL

Prob. 5–94

5–90. The composite shaft consists of a mid-section that includes the 1-in. diameter solid shaft and a tube that is welded to the rigid flanges at A and B . Neglect the thickness of the flanges and determine the angle of twist of end C of the shaft relative to end D . The shaft is subjected to a torque of 800 lb # ft. The material is A-36 steel.

0.5 ft

800 lb!ft 1 in. 3 in. 0.25 in.

0.75 ft 0.5 ft

800 lb!ft

Prob. 5–90

5–91. The A992 steel shaft is made from two segments. AC has a diameter of 0.5 in. and CB has a diameter of 1 in. If the shaft is fixed at its ends A and B and subjected to a uniform distributed torque of 60 lb # in.> in. along segment CB, determine the absolute maximum shear stress in the shaft.

5 in.

20 in. 1 in.

0.5 in. A

60 lb!in./ in.

Prob. 5–91

5.6 SOLID NONCIRCULAR SHAFTS 225

*5.6 Solid Noncircular Shafts It was demonstrated in Sec. 5.1 that when a torque is applied to a shaft having a circular cross section—that is, one that is axisymmetric—the shear strains vary linearly from zero at its center to a maximum at its outer surface. Furthermore, due to the uniformity of the shear strain at all points on the same radius, the cross sections do not deform, but rather remain plane after the shaft has twisted. Shafts that have a noncircular cross section, however, are not axisymmetric, and so their cross sections will bulge or warp when the shaft is twisted. Evidence of this can be seen from the way grid lines deform on a shaft having a square cross section when the shaft is twisted, Fig. 5–25 . As a consequence of this deformation the torsional analysis of noncircular shafts becomes considerably more complicated and will not be considered in this text.

Using a mathematical analysis based on the theory of elasticity, however, it is possible to determine the shear-stress distribution within a shaft of square cross section. Examples of how this shear stress varies along two radial lines of the shaft are shown in Fig. 5–26 a . Because these shear-stress distributions vary in a complex manner, the shear strains they create will warp the cross section as shown in Fig. 5–26 b . In particular notice that the corner points of the shaft must be subjected to zero shear stress and therefore zero shear strain. The reason for this can be shown by considering an element of material located at one of these points, Fig. 5–26 c . One would expect the top face of this element to be subjected to a shear stress in order to aid in resisting the applied torque T. This, however, cannot occur since the complementary shear stresses t and t!, acting on the outer surface of the shaft, must be zero.

Undeformed

Deformed

Fig. 5–25

tmax T

(c)

t¿ ! 0

t ! 0

t¿ ! 0 t ! 0

Fig. 5–26

Warping of cross-sectional area

(b)

tmax

Shear stress distribution along two radial lines

(a)

226 CHAPTER 5 TORS ION

Notice the deformation of the square element when this rubber bar is subjected to a torque.

The results of the analysis for square cross sections, along with other results from the theory of elasticity, for shafts having triangular and elliptical cross sections, are reported in Table 5–1 . In all cases the maximum shear stress occurs at a point on the edge of the cross section that is closest to the center axis of the shaft. In Table 5–1 these points are indicated as “dots” on the cross sections. Also given are formulas for the angle of twist of each shaft. By extending these results to a shaft having an arbitrary cross section, it can also be shown that the most efficient shaft has a circular cross section , since it is subjected to both a smaller maximum shear stress and a smaller angle of twist than one having the same cross sectional area, but not circular, and subjected to the same torque.

The drill shaft is connected to the soil auger using a shaft having a square cross section.

TABLE 5–1

Shape of cross section

Tmax

Ellipse

a a

2 T pa3b3G

(a2 + b2)TL

Square

T a3

4.81 T TL a4G

7.10 TL

Equilateral triangle

a3 20 T

a4G 46 TLa

pab2

EXAMPLE 5.10

The 6061-T6 aluminum shaft shown in Fig. 5–27 has a cross-sectional area in the shape of an equilateral triangle. Determine the largest torque T that can be applied to the end of the shaft if the allowable shear stress is tallow = 8 ksi and the angle of twist at its end is restricted to fallow = 0.02 rad. How much torque can be applied to a shaft of circular cross section made from the same amount of material?

SOLUTION By inspection, the resultant internal torque at any cross section along the shaft’s axis is also T . Using the formulas for tmax and f in Table 5–1 , we require

tallow = 20T a3

; 811032 lb>in2 = 20T11.5 in.23 T = 1350 lb # in.

Also,

fallow = 46TL a4Gal

; 0.02 rad = 46T 14 ft2 112 in.>ft211.5 in.24 [3.711062 lb>in2]

T = 170 lb # in. Ans. By comparison, the torque is limited due to the angle of twist.

Circular Cross Section. If the same amount of aluminum is to be used in making the same length of shaft having a circular cross section, then the radius of the cross section can be calculated. We have

A circle = A triangle; pc2 = 1 2

11.5 in.2 11.5 sin 60!2 c = 0.557 in.

The limitations of stress and angle of twist then require

tallow = Tc J

; 811032 lb>in2 = T 10.557 in.21p>22 10.557 in.24 T = 2170 lb # in.

fallow = TL JGal

; 0.02 rad = T 14 ft2 112 in.>ft21p>22 10.557 in.24 [3.7110 62 lb>in2]

T = 233 lb # in. Ans. Again, the angle of twist limits the applied torque.

NOTE: Comparing this result 1233 lb # in.2 with that given above 1170 lb # in.2, it is seen that a shaft of circular cross section can support 37% more torque than the one having a triangular cross section.

60!

1.5 in.

4 ft

Fig. 5–27

5.6 SOLID NONCIRCULAR SHAFTS 227

228 CHAPTER 5 TORS ION

dx s

(b)

Fig. 5–28

* The terminology "flow" is used since q is analogous to water flowing through a tube of rectangular cross section having a constant depth and variable width w Although the water’s velocity v at each point along the tube will be different (like tavg ), the flow q = vw will be constant.

*5.7 Thin-Walled Tubes Having Closed Cross Sections

Thin-walled tubes of noncircular cross section are often used to construct light-weight frameworks such as those used in aircraft. In some applications, they may be subjected to a torsional loading. In this section we will analyze the effects of applying a torque to a thin-walled tube having a closed cross section, that is, a tube that does not have any breaks or slits along its length. Such a tube, having a constant yet arbitrary cross-sectional shape, and variable thickness t, is shown in Fig. 5–28 a . Since the walls are thin, we will obtain the average shear stress by assuming that this stress is uniformly distributed across the thickness of the tube at any given point. Before we do this, however, we will first discuss some preliminary concepts regarding the action of shear stress over the cross section.

Shear Flow. Shown in Figs. 5–28 a and 5–28 b is a small element of the tube having a finite length s and differential width dx . At one end the element has a thickness tA , and at the other end the thickness is tB. Due to the internal torque T , shear stress is developed on the front face of the element. Specifically, at end A the shear stress is tA , and at end B it is tB. These stresses can be related by noting that equivalent shear stresses tA and tB must also act on the longitudinal sides of the element. Since these sides have a constant width dx, the forces acting on them are dFA = tA 1tA dx2 and dFB = tB 1tB dx2 . Equilibrium requires these forces to be of equal magnitude but opposite direction, so that

tA tA = tB tB

This important result states that the product of the average shear stress times the thickness of the tube is the same at each point on the tube’s cross-sectional area . This product is called shear flow , * q , and in general terms we can express it as

q = tavgt (5–17)

Since q is constant over the cross section, the largest average shear stress must occur where the tube’s thickness is the smallest .

(a)

t O

5.7 THIN-WALLED TUBES HAVING CLOSED CROSS SECTIONS 229

Now if a differential element having a thickness t , length ds , and width dx is isolated from the tube, Fig. 5–28 c , it is seen that the front face over which the average shear stress acts is dA = t ds. Hence, dF = tavg (t ds) = q ds, or q = dF>ds. In other words, the shear flow measures the force per unit length along the tube’s cross-sectional area .

It is important to realize that the shear-stress components shown in Fig. 5–28 c are the only ones acting on the tube. Components acting in the other direction, as shown in Fig. 5–28 d , cannot exist. This is because the top and bottom faces of the element are at the inner and outer walls of the tube, and these boundaries must be free of stress. Instead, as noted above, the applied torque causes the shear flow and the average stress to always be directed tangent to the wall of the tube, such that it contributes to the resultant internal torque T .

Average Shear Stress. The average shear stress can be related to the torque T by considering the torque produced by this shear stress about a selected point O within the tube’s boundary, Fig. 5–28 e . As shown, the shear stress develops a force dF = tavg dA = tavg 1t ds2 on an element of the tube. This force acts tangent to the centerline of the tube’s wall, and if the moment arm is h , the torque is

dT = h1dF2 = h1tavg t ds2 For the entire cross section, we require

T = Chtavg t ds

Here the “line integral” indicates that integration must be performed around the entire boundary of the area. Since the shear flow q = tavg t is constant , it can be factored out of the integral, so that

T = tavgtCh ds

A graphical simplification can be made for evaluating the integral by noting that the mean area , shown by the blue colored triangle in Fig. 5–28 e , is dA m = 11>22h ds. Thus,

T = 2tavg tL

dA m = 2tavg tA m

(e) x

T O

(f)

(c)

tavg

dsdx

(d)

Stress-free boundary (bottom)

Stress-free boundary (top)

t¿ ! t¿¿ ! 0

Fig. 5–28 (cont.)

230 CHAPTER 5 TORS ION

Solving for tavg, we have

tavg = T

2tA m (5–18)

Here

tavg = the average shear stress acting over a particular thickness of the tube

T = the resultant internal torque at the cross section t = the thickness of the tube where tavg is to be determined

A m = the mean area enclosed within the boundary of the centerline of the tube’s thickness. A m is shown shaded in Fig. 5–28 f

Since q = tavg t , then the shear flow throughout the cross section becomes

q = T

2A m (5–19)

Angle of Twist. The angle of twist of a thin-walled tube of length L can be determined using energy methods, and the development of the necessary equation is given as a problem later in the text. * If the material behaves in a linear elastic manner and G is the shear modulus, then this angle f, given in radians, can be expressed as

f = TL

4Am 2 G C

ds t

(5–20)

Here again the integration must be performed around the entire boundary of the tube’s cross-sectional area.

* See Prob. 14–14 .

Important Points

• Shear flow q is the product of the tube’s thickness and the average shear stress. This value is the same at all points along the tube’s cross section. As a result, the largest average shear stress on the cross section occurs where the thickness is smallest .

• Both shear flow and the average shear stress act tangent to the wall of the tube at all points and in a direction so as to contribute to the resultant internal torque.

EXAMPLE 5.11

Calculate the average shear stress in a thin-walled tube having a circular cross section of mean radius rm and thickness t , which is subjected to a torque T , Fig. 5–29 a . Also, what is the relative angle of twist if the tube has a length L ?

SOLUTION

Average Shear Stress. The mean area for the tube is A m = prm2 . Applying Eq. 5–18 gives

tavg = T

2tA m =

T 2ptrm

2 Ans.

We can check the validity of this result by applying the torsion formula. In this case, using Eq. 5–9 , we have

J = p

2 1ro4 - ri 42

= p

2 1ro2 + ri22 1ro2 - ri 22

= p

2 1ro2 + ri22 1ro + ri2 1ro - ri2

Since rm ! ro ! ri and t = ro - ri , J = p

2 12rm2 2 12rm2t = 2prm3 t

tavg = Trm

J =

Trm 2prm

3 t =

T 2ptrm

2 Ans.

which agrees with the previous result. The average shear-stress distribution acting throughout the tube’s

cross section is shown in Fig. 5–29 b . Also shown is the shear-stress distribution acting on a radial line as calculated using the torsion formula. Notice how each t avg acts in a direction such that it contributes to the resultant torque T at the section. As the tube’s thickness decreases, the shear stress throughout the tube becomes more uniform.

Angle of Twist. Applying Eq. 5–20 , we have

f = TL

4A m 2 G

C ds t

= TL

41prm2 22Gt Cds The integral represents the length around the centerline boundary, which is 2prm . Substituting, the final result is

f = TL

2prm 3 Gt

Ans.

Show that one obtains this same result using Eq. 5–15 .

T rm

(a)

Actual shear-stress distribution

(torsion formula)

Average shear-stress distribution

(thin-wall approximation)

(b)

tavg

tmax

Fig. 5–29

5.7 THIN-WALLED TUBES HAVING CLOSED CROSS SECTIONS 231

232 CHAPTER 5 TORS ION

The tube is made of C86100 bronze and has a rectangular cross section as shown in Fig. 5–30 a . If it is subjected to the two torques, determine the average shear stress in the tube at points A and B . Also, what is the angle of twist of end C ? The tube is fixed at E .

0.5 m

1.5 m

25 N!m

60 N!m

A 5 mm

3 mm

60 mm

40 mm (a)

Fig. 5–30

SOLUTION

Average Shear Stress. If the tube is sectioned through points A and B , the resulting free-body diagram is shown in Fig. 5–30 b . The internal torque is 35 N # m. As shown in Fig. 5–30 d , the mean area is

A m = 10.035 m2 10.057 m2 = 0.00200 m2 Applying Eq. 5–18 for point A , tA = 5 mm, so that

tA = T

2tA m =

35 N # m 210.005 m2 10.00200 m22 = 1.75 MPa Ans.

And for point B , tB = 3 mm, and therefore

tB = T

2tA m =

35 N # m 210.003 m2 10.00200 m22 = 2.92 MPa Ans.

These results are shown on elements of material located at points A and B , Fig. 5–30 e . Note carefully how the 35@N # m torque in Fig. 5–30 b creates these stresses on the back sides of each element.

EXAMPLE 5.12

60 N!m

25 N!m

35 N!m

(b) 60 N!m

60 N!m

(c)

Angle of Twist. From the free-body diagrams in Fig. 5–30 b and 5–30 c , the internal torques in regions DE and CD are 35 N # m and 60 N # m, respectively. Following the sign convention outlined in Sec. 5.4 , these torques are both positive. Thus, Eq. 5–20 becomes

f = a

4Am 2G C

ds t

= 60 N # m 10.5 m2

410.00200 m222 13811092 N>m22 c 2a 57 mm5 mm b + 2a 35 mm3 mm b d +

35 N # m 11.5 m2 410.00200 m222 13811092 N>m22 c 2a 57 mm5 mm b + 2a 35 mm3 mm b d

= 6.29110-32 rad = 0.360! Ans.

57 mm

35 mm

(d) (e)

2.92 MPa

1.75 MPa

Fig. 5–30 (cont.)

5.7 THIN-WALLED TUBES HAVING CLOSED CROSS SECTIONS 233

234 CHAPTER 5 TORS ION

5–95. The aluminum rod has a square cross section of 10 mm by 10 mm. If it is 8 m long, determine the torque T that is required to rotate one end relative to the other end by 90°. G al = 28 GPa, (t Y ) al = 240 MPa.

8 m

10 mm 10 mm

Prob. 5–95

*5–96. The shafts have elliptical and circular cross sections and are to be made from the same amount of a similar material. Determine the percent of increase in the maximum shear stress and the angle of twist for the elliptical shaft compared to the circular shaft when both shafts are subjected to the same torque and have the same length.

Prob. 5–96

5–97. It is intended to manufacture a circular bar to resist torque; however, the bar is made elliptical in the process of manufacturing, with one dimension smaller than the other by a factor k as shown. Determine the factor by which the maximum shear stress is increased.

kd d

Prob. 5–97

5–98. The shaft is made of red brass C83400 and has an elliptical cross section. If it is subjected to the torsional loading shown, determine the maximum shear stress within regions AC and BC , and the angle of twist f of end B relative to end A .

5–99. Solve Prob. 5–98 for the maximum shear stress within regions AC and BC , and the angle of twist f of end B relative to C .

20 mm

50 mm 2 m

1.5 m

20 N!m

30 N!m

50 N!m

Probs. 5–98/99

*5–100. If end B of the shaft, which has an equilateral triangle cross section, is subjected to a torque of T = 900 lb # ft, determine the maximum shear stress developed in the shaft. Also, find the angle of twist of end B . The shaft is made from 6061-T1 aluminum.

5–101. If the shaft has an equilateral triangle cross section and is made from an alloy that has an allowable shear stress of t allow = 12 ksi, determine the maximum allowable torque T that can be applied to end B . Also, find the corresponding angle of twist of end B .

2 ft

3 in.

Probs. 5–100/101

PROBLEMS

5–105. If the shaft is subjected to the torque of 3 kN·m, determine the maximum shear stress developed in the shaft. Also, find the angle of twist of end B . The shaft is made from A-36 steel. Set a = 50 mm.

5–106. If the shaft is made from A-36 steel having an allowable shear stress of t allow = 75 MPa, determine the minimum dimension a for the cross-section to the nearest millimeter. Also, find the corresponding angle of twist at end B .

600 mm

3 kN!m a

a a

Probs. 5–105/106

5–107. If the solid shaft is made from red brass C83400 copper having an allowable shear stress of t allow = 4 ksi, determine the maximum allowable torque T that can be applied at B .

*5–108. If the solid shaft is made from red brass C83400 copper and it is subjected to a torque T = 6 kip # ft at B , determine the maximum shear stress developed in segments AB and BC .

4 ft

4 in.

2 ft

2 in. 2 in.

Probs. 5–107/108

5–102. The aluminum strut is fixed between the two walls at A and B . If it has a 2 in. by 2 in. square cross section, and it is subjected to the torque of 80 lb # ft at C , determine the reactions at the fixed supports. Also, what is the angle of twist at C ? Gal = 3.811032 ksi.

2 ft

3 ft

80 lb!ft

Probs. 5–102

5–103. A torque of 2 kip · in. is applied to the tube. If the wall thickness is 0.1 in., determine the average shear stress in the tube.

2 in.

1.90 in.

Probs. 5–103

*5–104. The 6061-T6 aluminum bar has a square cross section of 25 mm by 25 mm. If it is 2 m long, determine the maximum shear stress in the bar and the rotation of one end relative to the other end.

1.5 m

25 mm 25 mm

0.5 m

20 N!m

60 N·m 80 N!m

Prob. 5–104

5.7 THIN-WALLED TUBES HAVING CLOSED CROSS SECTIONS 235

236 CHAPTER 5 TORS ION

*5–112. Due to a fabrication error the inner circle of the tube is eccentric with respect to the outer circle. By what percentage is the torsional strength reduced when the eccentricity e is one-fourth of the difference in the radii?

a b

e 2

a ! b 2

Prob. 5–112

5–113. Determine the constant thickness of the rectangular tube if average stress is not to exceed 12 ksi when a torque of T = 20 kip · in. is applied to the tube. Neglect stress concentrations at the corners. The mean dimensions of the tube are shown.

5–114. Determine the torque T that can be applied to the rectangular tube if the average shear stress is not to exceed 12 ksi. Neglect stress concentrations at the corners. The mean dimensions of the tube are shown and the tube has a thickness of 0.125 in.

2 in.

4 in.

Probs. 5–113/114

5–109. For a given maximum average shear stress, deter- mine the factor by which the torque carrying capacity is increased if the half-circular section is reversed from the dashed-line position to the section shown. The tube is 0.1 in. thick.

1.20 in.

0.5 in.

0.6 in.

1.80 in.

Prob. 5–109

5–110. For a given maximum average shear stress, determine the factor by which the torque-carrying capacity is increased if the half-circular sections are reversed from the dashed-line positions to the section shown. The tube is 0.1 in. thick.

1.80 in.

1.20 in.

0.5 in.

0.6 in.

Prob. 5–110

5–111. A torque T is applied to two tubes having the cross sections shown. Compare the shear flow developed in each tube.

Prob. 5–111

5–117. The mean dimensions of the cross section of the leading edge and torsion box of an airplane wing can be approximated as shown. If the wing is made of 2014-T6 aluminum alloy having an allowable shear stress of tallow = 125 MPa and the wall thickness is 10 mm, determine the maximum allowable torque and the corresponding angle of twist per meter length of the wing.

5–118. The mean dimensions of the cross section of the leading edge and torsion box of an airplane wing can be approximated as shown. If the wing is subjected to a torque of 4.5 MN # m and the wall thickness is 10 mm, determine the average shear stress developed in the wing and the angle of twist per meter length of the wing. The wing is made of 2014-T6 aluminum alloy.

2 m

10 mm 0.25 m

10 mm

0.25 m

0.5 m

Probs. 5–117/118

5–119. The symmetric tube is made from a high-strength steel, having the mean dimensions shown and a thickness of 5 mm. If it is subjected to a torque of T = 40 N # m, deter- mine the average shear stress developed at points A and B . Indicate the shear stress on volume elements located at these points.

60 mm

20 mm 30 mm

40 N!m

A B

Prob. 5–119

5–115. The steel tube has an elliptical cross section of mean dimensions shown and a constant thickness of t = 0.2 in. If the allowable shear stress is t allow = 8 ksi, and the tube is to resist a torque of T = 250 lb · ft, determine the necessary dimension b . The mean area for the ellipse is A m = p b (0.5 b ).

b 0.5b

250 lb!ft

Prob. 5–115

*5–116. The tube is made of plastic, is 5 mm thick, and has the mean dimensions shown. Determine the average shear stress at points A and B if the tube is subjected to the torque of T = 500 N · m. Show the shear stress on volume elements located at these points. Neglect stress concen- trations at the corners.

50 mm

30 mm

20 mm 20 mm

Prob. 5–116

5.7 THIN-WALLED TUBES HAVING CLOSED CROSS SECTIONS 237

238 CHAPTER 5 TORS ION

5.8 Stress Concentration The torsion formula, t max = Tc>J, cannot be applied to regions of a shaft having a sudden change in the cross section. Here the shear-stress and shear-strain distributions in the shaft become complex and can be obtained only by using experimental methods or possibly by a mathematical analysis based on the theory of elasticity. Three common discontinuities of the cross section that occur in practice are shown in Fig. 5–31 . They are at couplings , which are used to connect two collinear shafts together, Fig. 5–31 a , keyways , used to connect gears or pulleys to a shaft, Fig. 5–31 b , and a step shaft which is fabricated or machined from a single shaft, Fig. 5–31 c . In each case the maximum shear stress will occur at the point (dot) indicated on the cross section.

The necessity to perform a complex stress analysis at a shaft discontinuity to obtain the maximum shear stress can be eliminated by using a torsional stress-concentration factor , K . As in the case of axially loaded members, Sec. 4.7 , K is usually taken from a graph based on experimental data. An example, for the shoulder-fillet shaft, is shown in Fig. 5–32 . To use this graph, one first finds the geometric ratio D>d to define the appropriate curve, and then once the abscissa r>d is calculated, the value of K is found along the ordinate.

(a)

(b)

(c)

Fig. 5–31

0.300.250.200.150.100.050.00 1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0 r

T T

D d

K D/d ! 2.5

2.0

1.67

1.25

1.11

r d

Fig. 5–32

5.8 STRESS CONCENTRATION 239

The maximum shear stress is then determined from

t max = K Tc J

(5–21)

Here the torsion formula is applied to the smaller of the two connected shafts, since t max occurs at the base of the fillet, Fig. 5–31 c .

Note from the graph that an increase in radius r causes a decrease in K . Hence the maximum shear stress in the shaft can be reduced by increasing the radius. Also, if the diameter of the larger section is reduced, the D / d ratio will be lower and so the value of K and therefore t max will be lower.

Like the case of axially loaded members, torsional stress concentration factors should always be used when designing shafts made from brittle materials , or when designing shafts that will be subjected to fatigue or cyclic torsional loadings . These conditions give rise to the formation of cracks at the stress concentration, and this can often lead to a sudden fracture. On the other hand, if large static torsional loadings are applied to a shaft made from ductile material , then inelastic strains will develop within the shaft. Yielding of the material will cause the stress distribution to become more evenly distributed throughout the shaft, so that the maximum stress will not be limited to the region of stress concentration. This phenomenon will be discussed further in the next section.

Stress concentrations can arise at the coupling of these shafts, and this must be taken into account when the shaft is designed.

Important Points

• Stress concentrations in shafts occur at points of sudden cross-sectional change, such as couplings, keyways, and step shafts. The more severe the change in geometry, the larger the stress concentration.

• For design or analysis, it is not necessary to know the exact shear- stress distribution on the cross section. Instead, it is possible to obtain the maximum shear stress using a stress concentration factor, K , that has been determined through experiment, and is only a function of the geometry of the shaft.

• Normally a stress concentration in a ductile shaft subjected to a static torque will not have to be considered in design; however, if the material is brittle , or subjected to fatigue loadings, then stress concentrations become important.

240 CHAPTER 5 TORS ION

EXAMPLE 5.13

The stepped shaft shown in Fig. 5–33 a is supported by bearings at A and B . Determine the maximum stress in the shaft due to the applied torques. The shoulder fillet at the junction of each shaft has a radius of r = 6 mm.

30 N!m

60 N!m

20 mm

40 mm

(a)

Figure 5–33

30 N!m T " 30 N!m

(b)

Shear-stress distribution predicted by

torsion formula

Actual shear-stress distribution caused

by stress concentration

(c)

tmax = 3.10 MPa

SOLUTION Internal Torque. By inspection, moment equilibrium about the axis of the shaft is satisfied. Since the maximum shear stress occurs at the rooted ends of the smaller diameter shafts, the internal torque 130 N # m2 can be found there by applying the method of sections, Fig. 5–33 b .

Maximum Shear Stress. The stress-concentration factor can be determined by using Fig. 5–32 . From the shaft geometry we have

D d

= 2140 mm2 2120 mm2 = 2

r d

= 6 mm

2120 mm2 = 0.15 Thus, the value of K = 1.3 is obtained.

Applying Eq. 5–21 , we have

tmax = K Tc J

; tmax = 1.3 c 30 N # m (0.020 m)(p>2)(0.020 m)4 d = 3.10 MPa Ans. NOTE: From experimental evidence, the actual stress distribution along a radial line of the cross section at the critical section looks similar to that shown in Fig. 5–33 c . Notice how this compares with the linear stress distribution found from the torsion formula.

5.9 INELASTIC TORSION 241

*5.9 Inelastic Torsion If the torsional loadings applied to the shaft are excessive, then the material may yield, and, consequently, a “plastic analysis” must be used to determine the shear-stress distribution and the angle of twist. To perform this analysis, then as before, it is necessary to meet the conditions of both deformation and equilibrium for the shaft.

It was shown in Sec. 5.1 that regardless of the material behavior, the shear strains that develop in a circular shaft will vary linearly , from zero at the center of the shaft to a maximum at its outer boundary, Fig. 5–34 a . Also, the resultant internal torque at the section must be equivalent to the torque caused by the entire shear-stress distribution over the cross section. This condition can be expressed mathematically by considering the shear stress t acting on an element of area dA located a distance r from the center of the shaft, Fig. 5–34 b . The force produced by this stress is dF = t dA , and the torque produced is dT = r dF = r(t dA ). For the entire shaft we require

T = L

A rt dA (5–22)

If the area dA over which t acts can be defined as a differential ring having an area of dA = 2pr dr, Fig. 5–34 c , then the above equation can be written as

T = 2pL c

0 tr 2 dr (5–23)

These conditions of geometry and loading will now be used to determine the shear-stress distribution in a shaft when the shaft is subjected to two types of torque.

Elastic-Plastic Torque. Let us consider the material in the shaft to exhibit an elastic-perfectly plastic behavior. As shown in Fig. 5–35 a , this is characterized by a shear stress–strain diagram for which the material undergoes an increasing amount of shear strain when the shear stress reaches the yield point t Y .

Severe twist of an aluminum specimen caused by the application of a plastic torque.

(c)

dA ! 2pr dr

dr r

Fig. 5–34

(a)

Linear shear–strain distribution

gmax

(b)

T t

242 CHAPTER 5 TORS ION

If the internal torque produces the maximum elastic shear strain gY , at the outer boundary of the shaft, then the maximum elastic torque TY that produces this distribution can be found from the torsion formula, tY = TY c> 3(p>2)c 44 , so that TY =

2 tY c

3 (5–24)

Furthermore, the angle of twist can be determined from Eq. 5–13 , namely,

df = g dx r

(5–25)

If the applied torque increases in magnitude above TY, it will begin to cause yielding. It will begin at the outer boundary of the shaft, r = c, and then, as the maximum shear strain increases to, say, g! in Fig. 5–35 a , the yielding boundary will progress inward toward the shaft’s center, Fig. 5–35 b . As shown, this produces an elastic core , where, by proportion, the radius of the core is rY = 1gY >g!2c. Also, the outer portion of the material forms a plastic annulus or ring, since the shear strains g within this region are greater than gY . The corresponding shear-stress distribution along a radial line of the shaft is shown in Fig. 5–35 c . It is established by taking successive points on the shear-strain distribution in Fig. 5–35 b and finding the corresponding value of shear stress from the t9g diagram, Fig. 5–35 a . For example, at r = c, g! gives tY, and at r = rY, gY also gives tY ; etc.

Since t in Fig. 5–35 c can now be expressed as a function of r, we can apply Eq. 5–23 to determine the torque. We have

T = 2pL c

0 tr2 dr

= 2pL r Y

0 ¢tY r r Y ≤r2 dp + 2pL cr Y t Y r2 dr

= 2p r Y

tY L r Y

0 r3 dr + 2pt Y L

r Y r2 dr

= p

2rY tYrY

4 + 2p 3

tY 1c3 - rY3 2 =

ptY

6 14c3 - rY3 2 (5–26)

(c)

Shear–stress distribution

tY tY

Fig. 5–35

(b)

Shear–strain distribution

Plastic annulus

Elastic core

g¿

(a)

gY g¿ g

5.10 RESIDUAL STRESS 243

(d)

Fully plastic torque

c TY

Figure 5–35 (cont.)

*5.10 Residual Stress When a shaft is subjected to plastic shear strains caused by torsion, removal of the torque will cause some shear stress to remain in the shaft. This stress is referred to as residual stress , and its distribution can be calculated using superposition and elastic recovery. (See Sec. 4.9 .)

For example, if Tp causes the material at the outer boundary of the shaft to be strained to g1 , shown as point C on the t-g curve in Fig. 5–36 , the release of Tp will cause a reverse shear stress, such that the material behavior will follow the straight-lined segment CD , creating some elastic recovery of the shear strain g1. This line is parallel to the initial straight-lined portion AB of the t9g diagram, and thus both lines have a slope G as indicated.

B C

G G

Maximum elastic recovery is 2gY

Reversed elastic material behavior

Elastic-plastic material behavior

!tY

g1 g

Fig. 5–36

Plastic Torque. Further increases in T tend to shrink the radius of the elastic core until all the material yields, i.e., r Y S 0, Fig. 5–35 b . The material of the shaft will then be subjected to perfectly plastic behavior and the shear-stress distribution becomes uniform, so that t = tY , Fig. 5–35 d . We can now apply Eq. 5–23 to determine the plastic torque Tp , which represents the largest possible torque the shaft will support.

T p = 2pL c

0 tY r

2dr

= 2p 3

tY c 3 (5–27)

Compared with the maximum elastic torque TY , Eq. 5–24 , it can be seen that

Tp = 4 3

In other words, the plastic torque is 33% greater than the maximum elastic torque.

Unfortunately, the angle of twist f for the shear-stress distribution cannot be uniquely defined. This is because t = tY does not correspond to any unique value of shear strain g Ú gY . As a result, once T p is applied, the shaft will continue to deform or twist with no corresponding increase in shear stress.

244 CHAPTER 5 TORS ION

Since elastic recovery occurs, we can superimpose on the plastic torque stress distribution in Fig. 5–37 a a linear stress distribution caused by applying the plastic torque Tp in the opposite direction, Fig. 5–37 b . Here the maximum shear stress tr, for this stress distribution, is called the modulus of rupture for torsion. It is determined from the torsion formula,* which gives

tr = Tpc

J =

Tpc1p>22c4 Using Eq. 5–27 ,

tr = 3 12>32ptYc34c1p>22c4 = 43 tY

Note that reversed application of Tp using the linear shear-stress distribution in Fig. 5–37 b is possible here, since the maximum recovery for the elastic shear strain is 2gY , as noted in Fig. 5–37 . This corresponds to a maximum applied shear stress of 2tY , which is greater than the maximum shear stress of 43 tY calculated above. Hence, by superimposing the stress distributions involving applications and then removal of the plastic torque, we obtain the residual shear-stress distribution in the shaft as shown in Fig. 5–37 c . It should be noted from this diagram that the shear stress at the center of the shaft, shown as tY , must actually be zero, since the material along the axis of the shaft is never strained. The reason this is not zero is that we assumed all the material of the shaft to have been strained beyond the yield point in order to determine the plastic torque, Fig. 5–37 a . To be more realistic, an elastic–plastic torque should be considered when modeling the material behavior. Doing so leads to the superposition of the stress distribution shown in Fig. 5–37 d .

*The torsion formula is valid only when the material behaves in a linear elastic manner; however, the modulus of rupture is so named because it assumes that the material behaves elastically and then suddenly ruptures at the proportional limit.

Plastic torque applied causing plastic shear strains

throughout the shaft (a)

Plastic torque reversed causing elastic shear strains

throughout the shaft (b)

Residual shear–stress distribution in shaft

(c)

tr ! tY

Fig. 5–37

Elastic–plastic torque applied Elastic–plastic torque reversed Residual shear–stress distribution in shaft

(d)

Tep

tmax ! tr

tmax " tY "

Ultimate Torque. In the general case, most engineering materials will have a shear stress–strain diagram as shown in Fig. 5–38 a . Consequently, if T is increased so that the maximum shear strain in the shaft becomes g = gu , Fig. 5–38 b , then, by proportion gY occurs at rY = 1gY >gu2c. Likewise, the shear strains at, say, r = r1 and r = r2, can be found by proportion, i.e., g1 = 1r1>c2gu and g2 = 1r2>c2gu. If the corresponding values of t1, tY , t2 , and tu are taken from the t9g diagram and plotted, we obtain the shear-stress distribution, which acts along a radial line on the cross section, Fig. 5–38 c . The torque produced by this stress distribution is called the ultimate torque , Tu.

The magnitude of Tu can be determined by “graphically” integrating Eq. 5–23 . To do this, the cross-sectional area of the shaft is segmented into a finite number of rings, such as the one shown shaded in Fig. 5–38 d . The area of this ring, !A = 2pr !r , is multiplied by the shear stress t that acts on it, so that the force !F = t !A can be determined. The torque created by this force is then !T = r !F = r1t !A 2 . The addition of all the torques for the entire cross section, as determined in this manner, gives the ultimate torque Tu; that is, Eq. 5–23 becomes Tu ! 2p"tr2 !r . If, however, the stress distribution can be expressed as an analytical function, t = f 1r2, as in the elastic and plastic torque cases, then the integration of Eq. 5–23 can be carried out directly.

(d)

TuT

!A " 2pr!r

Fig. 5–38

(a)

g g1 g2gY gu

(b)

Ultimate shear-strain distribution

c g1

g2gY gu

(c)

Ultimate shear-stress distribution

cTu

TY T1

T2 Tu

Important Points

• The shear - strain distribution along a radial line on the cross section of a shaft is based on geometric considerations, and it is found to always vary linearly along the radial line. Once it is established, the shear-stress distribution can then be determined using the shear stress–strain diagram.

• If the shear-stress distribution for the shaft is established, it produces a torque about the axis of the shaft that is equivalent to the resultant internal torque acting on the cross section.

• Perfectly plastic behavior assumes the shear-stress distribution is constant . When it occurs, the shaft will continue to twist with no increase in torque. This torque is called the plastic torque .

5.10 RESIDUAL STRESS 245

246 CHAPTER 5 TORS ION

The tubular shaft in Fig. 5–39 a is made of an aluminum alloy that is assumed to have an elastic-plastic t9g diagram as shown. Determine the maximum torque that can be applied to the shaft without causing the material to yield, and the maximum torque or plastic torque that can be applied to the shaft. Also, what should the minimum shear strain at the outer wall be in order to develop a fully plastic torque?

SOLUTION Maximum Elastic Torque. We require the shear stress at the outer fiber to be 20 MPa. Using the torsion formula, we have

tY = TYc

J ; 2011062 N>m2 = TY 10.05 m21p>22 3 10.05 m24 - 10.03 m244

TY = 3.42 kN # m Ans. The shear-stress and shear-strain distributions for this case are

shown in Fig. 5–39 b . The values at the tube’s inner wall have been obtained by proportion.

Plastic Torque. The shear-stress distribution in this case is shown in Fig. 5–39 c . Application of Eq. 5–23 requires t = tY . We have

Tp = 2pL 0.05 m

0.03 m 32011062 N>m24r2 dr = 125.66110621

3 r3 2 0.05 m

0.03 m

= 4.11 kN # m Ans. For this tube Tp represents a 20% increase in torque capacity compared with the elastic torque TY .

Outer Radius Shear Strain. The tube becomes fully plastic when the shear strain at the inner wall becomes 0.286110 -32 rad, as shown in Fig. 5–39 c . Since the shear strain remains linear over the cross section, the plastic strain at the outer fibers of the tube in Fig. 5–39 c is determined by proportion.

50 mm =

0.286110-32 rad 30 mm

go = 0.477110-32 rad Ans.

EXAMPLE 5.14

(b)

Elastic shear–stress distribution

Elastic shear–strain distribution

20 MPa

12 MPa

0.286 (10!3) rad

0.172 (10!3) rad

50 mm

30 mm

Fig. 5–39

50 mm

30 mm

(a)

0.286 (10!3)

t (MPa)

g (rad)

20 MPa

Initial plastic shear–strain distribution

0.286 (10!3) rad

0.477 (10!3) rad

A solid circular shaft has a radius of 20 mm and length of 1.5 m. The material has an elastic-plastic t9g diagram as shown in Fig. 5–40 a . Determine the torque needed to twist the shaft f = 0.6 rad.

EXAMPLE 5.15

(a)

t (MPa)

0.0016 0.008 g (rad)

Shear–strain distribution

(b)

gY ! 0.0016 rad

gmax ! 0.008 rad

20 mm

(c)

Shear–stress distribution

20 mm tY ! 75 MPa

rY ! 4 mm

SOLUTION We will first obtain the shear-strain distribution, then establish the shear-stress distribution. Once this is known, the applied torque can be determined.

The maximum shear strain occurs at the surface of the shaft, r = c. Since the angle of twist is f = 0.6 rad for the entire 1.5-m length of the shaft, then using Eq. 5–25 , for the entire length we have

f = g L r

; 0.6 = gmax11.5 m2

0.02 m

gmax = 0.008 rad

The shear-strain distribution is shown in Fig. 5–40 b . Note that yielding of the material occurs since gmax 7 gY = 0.0016 rad in Fig. 5–40 a . The radius of the elastic core, rY , can be obtained by proportion. From Fig. 5–40 b ,

0.0016 =

0.02 m 0.008

rY = 0.004 m = 4 mm

Based on the shear-strain distribution, the shear-stress distribution, plotted over a radial line segment, is shown in Fig. 5–40 c . The torque can now be obtained using Eq. 5–26 . Substituting in the numerical data yields

T = ptY

6 14c3 - rY3 2

= p37511062 N>m24

6 [410.02 m23 - 10.004 m23]

= 1.25 kN # m Ans.

Fig. 5–40

5.10 RESIDUAL STRESS 247

248 CHAPTER 5 TORS ION

A tube in Fig. 5–41 a has a length of 5 ft and the material has an elastic- plastic t9g diagram, also shown in Fig. 5–41 a . Determine the plastic torque Tp . What is the residual shear-stress distribution if Tp is removed just after the tube becomes fully plastic?

EXAMPLE 5.16

Residual shear–stress distribution

2.93 ksi

4.53 ksi (d)

Fig. 5–41

(c)

Plastic torque reversed

7.47 ksi

tr ! 14.93 ksi

(b)

Plastic torque applied

12 ksi

(a) 0.002

co ! 2 in.

ci ! 1 in.

g (rad)

t (ksi)

SOLUTION

Plastic Torque. The plastic torque T p will strain the tube such that all the material yields. Hence the stress distribution will appear as shown in Fig. 5–41 b . Applying Eq. 5–23 , we have

Tp = 2pL co

tYr 2 dr =

2p 3

tY 1co3 - ci32 =

2p 3

11211032 lb>in22[12 in.23 - 11 in.23] = 175.9 kip # in. Ans. When the tube just becomes fully plastic, yielding has started at the

inner wall, i.e., at ci = 1 in. , gY = 0.002 rad , Fig. 5–41 a . The angle of twist that occurs can be determined from Eq. 5–25 , which for the entire tube becomes

fp = gY L ci

= 10.0022 15 ft2 112 in.>ft211 in.2 = 0.120 radd

When T p is removed, or in effect reapplied in the opposite direction, then the “fictitious” linear shear-stress distribution shown in Fig. 5–41 c must be superimposed on the one shown in Fig. 5–41 b . In Fig. 5–41 c the maximum shear stress or the modulus of rupture is found from the torsion formula

tr = Tpco

J =

1175.9 kip # in.2 12 in.21p>22[12 in.24 - 11 in.24] = 14.93 ksi Also, at the inner wall of the tube the shear stress is

ti = 114.93 ksi2 a 1 in.2 in. b = 7.47 ksi Ans. The resultant residual shear-stress distribution is shown in Fig. 5–41 d .

5.10 RESIDUAL STRESS 249

5–123. The transition at the cross sections of the step shaft has a radius of 2.8 mm. Determine the maximum shear stress developed in the shaft.

50 mm

20 mm 100 N!m

60 N!m

40 N!m

Prob. 5–123

*5–124. The steel used for the step shaft has an allowable shear stress of tallow = 8 MPa . If the radius at the transition between the cross sections is r = 2.25 mm, determine the maximum torque T that can be applied.

30 mm 30 mm 15 mm

T T 2

T 2

Prob. 5–124

*5–120. The steel step shaft has an allowable shear stress of tallow = 8 MPa. If the transition between the cross sections has a radius r = 4 mm, determine the maximum torque T that can be applied.

20 mm 20 mm

50 mm

T 2

Prob. 5–120

5–121. The step shaft is to be designed to rotate at 720 rpm while transmitting 30 kW of power. Is this possible? The allowable shear stress is tallow = 12 MPa and the radius at the transition on the shaft is 7.5 mm.

5–122. The built-up shaft is designed to rotate at 540 rpm. If the radius at the transition on the shaft is r = 7.2 mm, and the allowable shear stress for the material is tallow = 55 MPa , determine the maximum power the shaft can transmit.

75 mm

60 mm

Probs. 5–121/122

PROBLEMS

250 CHAPTER 5 TORS ION

5–129. The solid shaft is made of an elastic-perfectly plastic material as shown. Determine the torque T needed to form an elastic core in the shaft having a radius of rY = 20 mm. If the shaft is 3 m long, through what angle does one end of the shaft twist with respect to the other end? When the torque is removed, determine the residual stress distribution in the shaft and the permanent angle of twist.

160

0.004 g (rad)

t (MPa)

T T

80 mm

Prob. 5–129

5–130. The shaft is subjected to a maximum shear strain of 0.0048 rad. Determine the torque applied to the shaft if the material has strain hardening as shown by the shear stress–strain diagram.

2 in.

0.0006 g (rad)

t (ksi)

0.0048

Prob. 5–130

5–125. The step shaft is subjected to a torque of 710 lb # in. If the allowable shear stress for the material is tallow = 12 ksi, determine the smallest radius at the junction between the cross sections that can be used to transmit the torque.

710 lb!ft

1.5 in.

0.75 in.

710 lb!in.A

Prob. 5–125

5–126. A solid shaft has a diameter of 40 mm and length of 1 m. It is made from an elastic-plastic material having a yield stress of t Y = 100 MPa. Determine the maximum elastic torque T Y and the corresponding angle of twist. What is the angle of twist if the torque is increased to T = 1.2 T Y ? G = 80 GPa.

5–127. Determine the torque needed to twist a short 2-mm-diameter steel wire through several revolutions if it is made from steel assumed to be elastic-plastic and having a yield stress of t Y = 50 MPa. Assume that the material becomes fully plastic.

*5–128. A bar having a circular cross section of 3 in.- diameter is subjected to a torque of 100 in. · kip. If the material is elastic-plastic, with t Y = 16 ksi, determine the radius of the elastic core.

5–134. The solid shaft is made from an elastic-plastic material as shown. Determine the torque T needed to form an elastic core in the shaft having a radius of r Y = 23 mm. If the shaft is 2 m long, through what angle does one end of the shaft twist with respect to the other end? When the torque is removed, determine the residual stress distribution in the shaft and the permanent angle of twist.

40 mm

! = 23 mm

150

0.005 " (rad)

# (MPa)

Prob. 5–134

5–135. A 1.5-in.-diameter shaft is made from an elastic-plastic material as shown. Determine the radius of its elastic core if it is subjected to a torque of T = 200 lb # ft. If the shaft is 10 in. long, determine the angle of twist.

10 in.

0.006 " (rad)

# (ksi)

Prob. 5–135

5–131. An 80-mm-diameter solid circular shaft is made of an elastic-perfectly plastic material having a yield shear stress of tY = 125 MPa . Determine (a) the maximum elastic torque TY ; and (b) the plastic torque Tp.

*5–132. The hollow shaft has the cross section shown and is made of an elastic-perfectly plastic material having a yield shear stress of tY . Determine the ratio of the plastic torque Tp to the maximum elastic torque TY .

c 2

Prob. 5–132

5–133. If the step shaft is elastic-plastic as shown, determine the largest torque T that can be applied to the shaft. Also, draw the shear-stress distribution over a radial line for each section. Neglect the effect of stress concentration.

1 in.

0.75 in.

0.005 g (rad)

t (ksi)

Prob. 5–133

5.10 RESIDUAL STRESS 251

252 CHAPTER 5 TORS ION

5–138. The tube is made of elastic-perfectly plastic material, which has the t–g diagram shown. Determine the torque T that just causes the inner surface of the shaft to yield. Also, find the residual shear-stress distribution in the shaft when the torque is removed.

t (ksi)

g (rad)

0.004

3 ft

6 in.

3 in.

Prob. 5–138

5–139. The shear stress–strain diagram for a solid 50-mm- diameter shaft can be approximated as shown in the figure. Determine the torque required to cause a maximum shear stress in the shaft of 125 MPa. If the shaft is 3 m long, what is the corresponding angle of twist?

0.0025 ! (rad)

" (MPa)

125

0.010

Prob. 5–139

*5–136. The tubular shaft is made of a strain-hardening material having a t–g diagram as shown. Determine the torque T that must be applied to the shaft so that the maximum shear strain is 0.01 rad.

0.75 in.

0.005 g (rad)

t (ksi)

0.01

0.5 in.

Prob. 5–136

5–137. The shaft is made from a strain-hardening material having a t–g diagram as shown. Determine the torque T that must be applied to the shaft in order to create an elastic core in the shaft having a radius of r c = 0.5 in.

0.6 in.

0.005 g (rad)

t (ksi)

0.01

Prob. 5–137

5–142. The 2-m-long tube is made from an elastic-plastic material as shown. Determine the applied torque T , which subjects the material of the tube’s outer edge to a shearing strain, of g max = 0.008 rad. What would be the permanent angle of twist of the tube when the torque is removed? Sketch the residual stress distribution of the tube.

*5–140. The 2-m-long tube is made of an elastic-perfectly plastic material as shown. Determine the applied torque T that subjects the material at the tube’s outer edge to a shear strain of gmax = 0.006 rad. What would be the permanent angle of twist of the tube when this torque is removed? Sketch the residual stress distribution in the tube.

35 mm

30 mm

210

0.003 g (rad)

t (MPa)

Prob. 5–140

5–141. A steel alloy core is bonded firmly to the copper alloy tube to form the shaft shown. If the materials have the t-g diagrams shown, determine the torque resisted by the core and the tube.

t (MPa)

180

0.0024

g (rad)

0.002

450 mm

100 mm

60 mm

Steel Alloy

Copper Alloy

15 kN!m

Prob. 5–141

5.10 RESIDUAL STRESS 253

45 mm

40 mm

240

0.003 ! (rad)

" (MPa)

Prob. 5–142

254 CHAPTER 5 TORS ION

CHAPTER REVIEW

Torque causes a shaft having a circular cross section to twist, such that the shear strain in the shaft is proportional to its radial distance from the center of the shaft. Provided the material is homogeneous and linear elastic, then the shear stress is determined from the torsion formula,

t = Tr

The design of a shaft requires finding the geometric parameter,

J c =

T tallow

Often the power P supplied to a shaft rotating at v is reported, in which case the torque is determined from P = Tv.

tmax

co ci

tmax

The angle of twist of a circular shaft is determined from

f = L L

T 1x2 dx J(x)G(x)

If the internal torque and JG are constant within each segment of the shaft then

f = a

TL JG

For application, it is necessary to use a sign convention for the internal torque and to be sure the material remains linear elastic.

T ! T(x)

If the shaft is statically indeterminate, then the reactive torques are determined from equilibrium, compatibility of twist, and a torque-twist relationship, such as f = TL >JG.

CHAPTER REVIEW 255

Solid non-circular shafts tend to warp out of plane when subjected to a torque. Formulas are available to determine the maximum elastic shear stress and the twist for these cases.

T tmax

cT tY tY

The average shear stress in thin-walled tubes is determined by assuming the shear flow across each thickness t of the tube is constant. The average shear

stress value is determined from tavg = T

2t A m .

Stress concentrations occur in shafts when the cross section suddenly changes. The maximum shear stress is determined using a stress concentration factor K , which is determined from experiment and represented

in graphical form. Once obtained, tmax = K aTcJ b . If the applied torque causes the material to exceed the elastic limit, then the stress distribution will not be proportional to the radial distance from the centerline of the shaft. Instead, the internal torque is related to the stress distribution using the shear-stress–shear-strain diagram and equilibrium.

If a shaft is subjected to a plastic torque, which is then released, it will cause the material to respond elastically, thereby causing residual shear stress to be developed in the shaft.

256 CHAPTER 5 TORS ION

REVIEW PROBLEMS

5–143. The shaft is made of A992 steel and has an allowable shear stress of t allow = 75 MPa. When the shaft is rotating at 300 rpm, the motor supplies 8 kW of power, while gears A and B withdraw 5 kW and 3 kW, respectively. Determine the required minimum diameter of the shaft to the nearest millimeter. Also, find the rotation of gear A relative to C .

*5–144. The shaft is made of A992 steel and has an allowable shear stress of t allow = 75 MPa. When the shaft is rotating at 300 rpm, the motor supplies 8 kW of power, while gears A and B withdraw 5 kW and 3 kW, respectively. If the angle of twist of gear A relative to C is not allowed to exceed 0.03 rad, determine the required minimum diameter of the shaft to the nearest millimeter.

300 mm

Probs. 5–143/144

5–145. The A-36 steel circular tube is subjected to a torque of 10 kN # m. Determine the shear stress at the mean radius r = 60 mm and compute the angle of twist of the tube if it is 4 m long and fixed at its far end. Solve the problem using Eqs. 5–7 and 5–15 and by using Eqs. 5–18 and 5–20 .

4 m

t ! 5 mm

r ! 60 mm

10 kN"m

Prob. 5–145

5–146. A portion of an airplane fuselage can be approxi- mated by the cross section shown. If the thickness of its 2014-T6-aluminum skin is 10 mm, determine the maximum wing torque T that can be applied if t allow = 4 MPa. Also, in a 4-m long section, determine the angle of twist.

2 m

0.75 m

Prob. 5–146

5–147. The material of which each of three shafts is made has a yield stress of t Y and a shear modulus of G . Determine which shaft geometry will resist the largest torque without yielding. What percentage of this torque can be carried by the other two shafts? Assume that each shaft is made of the same amount of material and that it has the same cross- sectional area A .

A A A 60#

60#60#

Prob. 5–147

REVIEW PROBLEMS 257

5–150. The tapered shaft is made from 2014-T6 aluminum alloy, and has a radius which can be described by the function r = 0.02(1 + x 3/2 ) m, where x is in meters. Determine the angle of twist of its end A if it is subjected to a torque of 450 N # m.

4 m

450 N!m

r = 0.02(1 + x3/2) m

Prob. 5–150

5–151. The 60-mm-diameter shaft rotates at 300 rev>min. This motion is caused by the unequal belt tensions on the pulley of 800 N and 450 N. Determine the power transmitted and the maximum shear stress developed in the shaft.

800 N

450 N

100 mm

300 rev/min

Prob. 5–151

*5–148. Segments AB and BC of the assembly are made from 6061-T6 aluminum and A992 steel, respectively. If couple forces P = 3 kip are applied to the lever arm, determine the maximum shear stress developed in each segment. The assembly is fixed at A and C .

2.5 ft

4 ft

4 in.

4 ft

2.5 ft

Prob. 5–148

5–149. Segments AB and BC of the assembly are made from 6061-T6 aluminum and A992 steel, respectively. If the allowable shear stress for the aluminum is (t allow ) al = 12 ksi and for the steel (t allow ) st = 10 ksi, determine the maximum allowable couple forces P that can be applied to the lever arm. The assembly is fixed at A and C .

2.5 ft

4 ft

4 in.

4 ft

2.5 ft

Prob. 5–149

The deck of this bridge has been designed on the basis of its ability to resist bending stress.

Chapter 6

259

6.1 Shear and Moment Diagrams Members that are slender and support loadings that are applied perpendicular to their longitudinal axis are called beams. In general, beams are long, straight bars having a constant cross-sectional area. Often they are classified as to how they are supported. For example, a simply supported beam is pinned at one end and roller supported at the other, Fig. 6–1 , a cantilevered beam is fixed at one end and free at the other, and an overhanging beam has one or both of its ends freely extended over the supports. Beams are considered among the most important of all structural elements. They are used to support the floor of a building, the deck of a bridge, or the wing of an aircraft. Also, the axle of an automobile, the boom of a crane, even many of the bones of the body act as beams.

CHAPTER OBJECTIVES

■ Beams and shafts are important structural and mechanical elements in engineering. In this chapter we will determine the stress in these members caused by bending. The chapter begins with a discussion of how to establish the shear and moment diagrams for a beam or shaft. Like the normal-force and torque diagrams, the shear and moment diagrams provide a useful means for determining the largest shear and moment in a member, and they specify where these maximums occur. Once the internal moment at a section is determined, the bending stress can then be calculated. First we will consider members that are straight, have a symmetric cross section, and are made of homogeneous linear elastic material. Afterward we will discuss special cases involving unsymmetric bending and members made of composite materials. Consideration will also be given to curved members, stress concentrations, inelastic bending, and residual stresses.

Bending

Simply supported beam

Cantilevered beam

Overhanging beam

Fig. 6–1

260 CHAPTER 6 BENDING

Because of the applied loadings, beams develop an internal shear force and bending moment that, in general, vary from point to point along the axis of the beam. In order to properly design a beam it therefore becomes necessary to determine the maximum shear and moment in the beam. One way to do this is to express V and M as functions of their arbitrary position x along the beam’s axis. These shear and moment functions can then be plotted and represented by graphs called shear and moment diagrams. The maximum values of V and M can then be obtained from these graphs. Also, since the shear and moment diagrams provide detailed information about the variation of the shear and moment along the beam’s axis, they are often used by engineers to decide where to place reinforcement materials within the beam or how to proportion the size of the beam at various points along its length.

In order to formulate V and M in terms of x we must choose the origin and the positive direction for x. Although the choice is arbitrary, most often the origin is located at the left end of the beam and the positive direction is to the right.

In general, the internal shear and moment functions of x will be discontinuous, or their slope will be discontinuous, at points where a distributed load changes or where concentrated forces or couple moments are applied. Because of this, the shear and moment functions must be determined for each region of the beam between any two discontinuities of loading. For example, coordinates x1, x2, and x3 will have to be used to describe the variation of V and M throughout the length of the beam in Fig. 6–2 . These coordinates will be valid only within the regions from A to B for x1, from B to C for x2, and from C to D for x3.

Beam Sign Convention. Before presenting a method for determining the shear and moment as functions of x and later plotting these functions (shear and moment diagrams), it is first necessary to establish a sign convention so as to define “positive” and “negative” values for V and M . Although the choice of a sign convention is arbitrary, here we will use the one often used in engineering practice and shown in Fig. 6–3 . The positive directions are as follows: the distributed load acts upward on the beam; the internal shear force causes a clockwise rotation of the beam segment on which it acts; and the internal moment causes compression in the top fibers of the segment such that it bends the segment so that it “holds water”. Loadings that are opposite to these are considered negative.

DB C

x1 x2

Fig. 6–2

V Positive external distributed load

Positive internal shear

Positive internal moment

Beam sign convention

w(x)

Fig. 6–3

6.1 SHEAR AND MOMENT DIAGRAMS 261

Important Points

• Beams are long straight members that are subjected to loads perpendicular to their longitudinal axis. They are classified according to the way they are supported, e.g., simply supported, cantilevered, or overhanging.

• In order to properly design a beam, it is important to know the variation of the internal shear and moment along its axis in order to find the points where these values are a maximum.

• Using an established sign convention for positive shear and moment, the shear and moment in the beam can be determined as a function of its position x on the beam, and then these functions can be plotted to form the shear and moment diagrams.

Procedure for Analysis

The shear and moment diagrams for a beam can be constructed using the following procedure.

Support Reactions. • Determine all the reactive forces and couple moments acting on the beam, and resolve all the forces into

components acting perpendicular and parallel to the beam’s axis.

Shear and Moment Functions. • Specify separate coordinates x having an origin at the beam’s left end and extending to regions of the

beam between concentrated forces and/or couple moments, or where there is no discontinuity of distributed loading.

• Section the beam at each distance x, and draw the free-body diagram of one of the segments. Be sure V and M are shown acting in their positive sense, in accordance with the sign convention given in Fig. 6–3 .

• The shear is obtained by summing forces perpendicular to the beam’s axis.

• To eliminate V , the moment is obtained directly by summing moments about the sectioned end of the segment.

Shear and Moment Diagrams. • Plot the shear diagram ( V versus x ) and the moment diagram ( M versus x ). If numerical values of the

functions describing V and M are positive, the values are plotted above the x axis, whereas negative values are plotted below the axis.

• Generally it is convenient to show the shear and moment diagrams below the free-body diagram of the beam.

262 CHAPTER 6 BENDING

EXAMPLE 6.1

Draw the shear and moment diagrams for the beam shown in Fig. 6–4 a .

SOLUTION Support Reactions. The support reactions are shown in Fig. 6–4 c .

Shear and Moment Functions. A free-body diagram of the left segment of the beam is shown in Fig. 6–4 b . The distributed loading on this segment, wx, is represented by its resultant force only after the segment is isolated as a free-body diagram. This force acts through the centroid of the area comprising the distributed loading, a distance of x >2 from the right end. Applying the two equations of equilibrium yields

+ c !Fy = 0; wL 2

- wx - V = 0

V = w aL 2

- x b (1) a+ !M = 0; - awL

2 bx + (wx)a x

2 b + M = 0

M = w 2

(Lx - x 2) (2)

Shear and Moment Diagrams. The shear and moment diagrams shown in Fig. 6–4 c are obtained by plotting Eqs. 1 and 2 . The point of zero shear can be found from Eq. 1 :

V = w aL 2

- x b = 0 x =

L 2

NOTE: From the moment diagram, this value of x represents the point on the beam where the maximum moment occurs, since by Eq. 6–2 (see Sec. 6.2 ) the slope V = dM >dx = 0. From Eq. 2 , we have

Mmax = w 2

JL aL 2 b - aL

2 b2R

= wL2

(a)

x V

M A

(b)

wx x 2

wL 2

(c)

Mmax !

wL 2

L 2

wL 2

wL2

Fig. 6–4

6.1 SHEAR AND MOMENT DIAGRAMS 263

EXAMPLE 6.2

Draw the shear and moment diagrams for the beam shown in Fig. 6–5 a .

(a)

(b)

w0L 2

2 3

(d)

(c)

w "

w0L 2

w0 x L w0 x

x1 3

1 2

Fig. 6–5

SOLUTION Support Reactions. The distributed load is replaced by its resultant force and the reactions have been determined as shown in Fig. 6–5 b .

Shear and Moment Functions. A free-body diagram of a beam segment of length x is shown in Fig. 6–5 c . Note that the intensity of the triangular load at the section is found by proportion, that is, w >x = w0>L or w = w0x >L. With the load intensity known, the resultant of the distributed loading is determined from the area under the diagram. Thus,

+ c !Fy = 0; w0 L

2 - 1

2 ¢w0 x

L ≤x - V = 0

V = w0 2L

(L2 - x 2) (1)

a+ !M = 0; w0L

3 -

w0L

2 (x) + 1

2 ¢w0x

L ≤ x a 1

3 x b + M = 0

M = w0 6L

(-2L3 + 3L2x - x 3) (2)

These results can be checked by applying Eqs. 6–1 and 6–2 of Sec. 6.2 , that is,

w = dV dx

= w0 2L

(0 - 2x) = - w0x

L OK

V = dM dx

= w0 6L

(0 + 3L2 - 3x 2) = w0 2L

(L2 - x 2) OK

Shear and Moment Diagrams. The graphs of Eqs. 1 and 2 are shown in Fig. 6–5 d .

264 CHAPTER 6 BENDING

EXAMPLE 6.3

(a) 18 ft

6 kip/ ft

2 kip/ ft

(d)

(b)

9 ft

42 kip30 kip

12 ft 18 ft

36 kip36 kip 4 kip/ft 2 kip/ft

M (kip!ft)

V (kip)

(ft)

42 kip30 kip

"42

9.735 ft

6 kip/ft

2 kip/ft

{

30 kip

(c)

2 kip/ft{

Mmax # 163 kip!ft

x– 2

x– 3

1– 2

4 xx— 18

4 x—18 kip/ft

Fig. 6–6

Draw the shear and moment diagrams for the beam shown in Fig. 6–6 a .

SOLUTION Support Reactions. The distributed load is divided into triangular and rectangular component loadings and these loadings are then replaced by their resultant forces. The reactions have been determined as shown on the beam’s free-body diagram, Fig. 6–6 b .

Shear and Moment Functions. A free-body diagram of the left segment is shown in Fig. 6–6 c . As above, the trapezoidal loading is replaced by rectangular and triangular distributions. Note that the intensity of the triangular load at the section is found by proportion. The resultant force and the location of each distributed loading are also shown. Applying the equilibrium equations, we have

+ c $Fy = 0; 30 kip - (2 kip>ft)x - 12 (4 kip>ft)a x18 ft bx - V = 0 V = ¢30 - 2x - x 2

9 ≤ kip (1)

a+ $M = 0;

-30 kip(x) + (2 kip>ft)x a x 2 b + 1

2 (4 kip>ft)a x

18 ft bx a x

3 b + M = 0

M = ¢30x - x 2 - x 3 27 ≤ kip # ft (2)

Equation 2 may be checked by noting that dM >dx = V , that is, Eq. 1 . Also, w = dV >dx = -2 - 29 x. This equation checks, since when x = 0, w = -2 kip>ft, and when x = 18 ft, w = -6 kip>ft, Fig. 6–6 a . Shear and Moment Diagrams. Equations 1 and 2 are plotted in Fig. 6–6 d . Since the point of maximum moment occurs when dM >dx = V = 0 ( Eq. 6–2 ), then, from Eq. 1 ,

V = 0 = 30 - 2x - x 2

Choosing the positive root,

x = 9.735 ft

Thus, from Eq. 2 ,

Mmax = 30(9.735) - (9.735)2 - (9.735)3

27 = 163 kip # ft

6.1 SHEAR AND MOMENT DIAGRAMS 265

EXAMPLE 6.4

Draw the shear and moment diagrams for the beam shown in Fig. 6–7 a .

80 kN!m

15 kN 5 kN/m

A C

5 m 5 m (a) (b)

M 80 kN!m

5.75 kN

(c)

5 m

80 kN!m

15 kN

5.75 kN

5(x2 " 5)

x2 " 5 2

(d)

V (kN)

M (kN!m)

x (m)

B 5 m 5 m

5.75 kN 34.25 kN

108.75

5.75

"9.25

"34.25

A C

80 kN!m

15 kN 5 kN/m

Fig. 6–7

SOLUTION Support Reactions. The reactions at the supports have been determined and are shown on the free-body diagram of the beam, Fig. 6–7 d .

Shear and Moment Functions. Since there is a discontinuity of distributed load and also a concentrated load at the beam’s center, two regions of x must be considered in order to describe the shear and moment functions for the entire beam.

0 … x1 6 5 m, Fig. 6–7 b :

+ c !Fy = 0; 5.75 kN - V = 0 V = 5.75 kN (1)

a+ !M = 0; -80 kN # m - 5.75 kN x1 + M = 0 M = (5.75x1 + 80) kN # m (2)

5 m 6 x2 … 10 m, Fig. 6–7 c :

+ c !Fy = 0; 5.75 kN - 15 kN - 5 kN>m(x2 - 5 m) - V = 0 V = (15.75 - 5x2) kN (3)

a+ !M = 0; -80 kN # m - 5.75 kN x2 + 15 kN(x2 - 5 m) + 5 kN>m(x2 - 5 m)¢ x2 - 5 m2 ≤ + M = 0

M = (-2.5x2 2 + 15.75x2 + 92.5) kN # m (4) These results can be checked in part by noting that w = dV >dx and V = dM >dx. Also, when x1 = 0, Eqs. 1 and 2 give V = 5.75 kN and M = 80 kN # m; when x2 = 10 m, Eqs. 3 and 4 give V = -34.25 kN and M = 0. These values check with the support reactions shown on the free-body diagram, Fig. 6–7 d .

Shear and Moment Diagrams. Equations 1 through 4 are plotted in Fig. 6–7 d .

266 CHAPTER 6 BENDING

6.2 Graphical Method for Constructing Shear and Moment Diagrams

In cases where a beam is subjected to several different loadings, determining V and M as functions of x and then plotting these equations can become quite tedious. In this section a simpler method for constructing the shear and moment diagrams is discussed—a method based on two differential relations, one that exists between distributed load and shear, and the other between shear and moment.

Regions of Distributed Load. For purposes of generality, consider the beam shown in Fig. 6–8 a , which is subjected to an arbitrary loading. A free-body diagram for a very small segment !x of the beam is shown in Fig. 6–8 b . Since this segment has been chosen at a position x where there is no concentrated force or couple moment, the results to be obtained will not apply at these points of concentrated loading.

Notice that all the loadings shown on the segment act in their positive directions according to the established sign convention, Fig. 6–3 . Also, both the internal resultant shear and moment, acting on the right face of the segment, must be changed by a small amount in order to keep the segment in equilibrium. The distributed load, which is approximately constant over !x, has been replaced by a resultant force w(x)!x that acts at 12( !x ) from the right side. Applying the equations of equilibrium to the segment, we have

Failure of this table occurred at the brace support on its right side. If drawn, the bending moment diagram for the table loading would indicate this to be the point of maximum internal moment.

(a)

x !x

w(x)

(b)

M ! "M M

w(x)"x

w(x)

("x)

Free-body diagram of segment"x

V ! "V

1 2

Fig. 6–8

6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS 267

+ c !Fy = 0; V + w(x) "x - (V + "V ) = 0 "V = w(x) "x

a+ !MO = 0; -V "x - M - w(x) "x 312("x)4 + (M + "M) = 0 "M = V "x + w(x) 12("x)2

Dividing by "x and taking the limit as "x S 0, the above two equations become

dV dx

= w(x)

slope of

= distributed

shear diagram load intensity at each point at each point

(6–1)

dM dx

= V (x)

slope of

= shear

moment diagram at each at each point point

(6–2)

These two equations provide a convenient means for quickly obtaining the shear and moment diagrams for a beam. Equation 6–1 states that at a point the slope of the shear diagram equals the intensity of the distributed loading. For example, consider the beam in Fig. 6–9 a . The distributed loading is negative and increases from zero to wB . Therefore, the shear diagram will be a curve that has a negative slope, increasing from zero to -wB . Specific slopes wA = 0, -wC, -wD, and -wB are shown in Fig. 6–9 b .

In a similar manner, Eq. 6–2 states that at a point the slope of the moment diagram is equal to the shear. Notice that the shear diagram in Fig. 6–9 b starts at +VA, decreases to zero, and then becomes negative and decreases to -VB . The moment diagram will then have an initial slope of +VA which decreases to zero, then the slope becomes negative and decreases to -VB . Specific slopes VA, VC, VD, 0, and -VB are shown in Fig. 6–9 c .

w ! w(x) wB

A C D

w = negative increasing slope = negative increasing

V = positive decreasing slope = positive decreasing

"wB

VC VD

"VB

"wC "wD

Fig. 6–9

268 CHAPTER 6 BENDING

Equations 6–1 and 6–2 may also be rewritten in the form dV = w(x)dx and dM = Vdx. Noting that w(x) dx and V dx represent differential areas under the distributed loading and shear diagram, respectively, we can integrate these areas between any two points C and D on the beam, Fig. 6–9 d , and write

!V = Lw(x)dx

change in =

area under shear distributed loading

(6–3)

!M = LV (x)dx

change in =

area under moment shear diagram

(6–4)

Equation 6–3 states that the change in shear between C and D is equal to the area under the distributed-loading curve between these two points, Fig. 6–9 d . In this case the change is negative since the distributed load acts downward. Similarly, from Eq. 6–4 , the change in moment between C and D, Fig. 6–9 f , is equal to the area under the shear diagram within the region from C to D. Here the change is positive.

Since the above equations do not apply at points where a concentrated force or couple moment acts, we will now consider each of these cases.

Regions of Concentrated Force and Moment. A free- body diagram of a small segment of the beam in Fig. 6–8 a taken from under the force is shown in Fig. 6–10 a . Here it can be seen that force equilibrium requires

+ c "Fy = 0; V + F - (V + !V ) = 0 !V = F (6–5)

Thus, when F acts upward on the beam, !V is positive so the shear will “jump” upward. Likewise, if F acts downward, the jump (!V ) will be downward.

When the beam segment includes the couple moment M0 , Fig. 6–10 b , then moment equilibrium requires the change in moment to be

a+ "MO = 0; M + !M - M0 - V !x - M = 0

Letting !x S 0, we get

!M = M0 (6–6)

In this case, if M0 is applied clockwise, !M is positive so the moment diagram will “jump” upward. Likewise, when M0 acts counterclockwise, the jump (!M) will be downward.

V ! "V

M ! "M

(a)

(b)

V ! "V

M ! "M

Fig. 6–10

(d) DC

(e)

(f)

Fig. 6–9 (cont.)

6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS 269

Procedure for Analysis

The following procedure provides a method for constructing the shear and moment diagrams for a beam based on the relations among distributed load, shear, and moment.

Support Reactions. • Determine the support reactions and resolve the forces acting on

the beam into components that are perpendicular and parallel to the beam’s axis.

Shear Diagram. • Establish the V and x axes and plot the known values of the shear

at the two ends of the beam.

• Notice how the values of the distributed load vary along the beam, and realize that each of these values indicates the way the shear diagram will slope ( dV>dx = w ). Here w is positive when it acts upward.

• If a numerical value of the shear is to be determined at a point, one can find this value either by using the method of sections and the equation of force equilibrium, or by using !V = 1w(x) dx, which states that the change in the shear between any two points is equal to the area under the load diagram between the two points.

Moment Diagram. • Establish the M and x axes and plot the known values of the

moment at the ends of the beam.

• Notice how the values of the shear diagram vary along the beam, and realize that each of these values indicates the way the moment diagram will slope ( dM>dx = V ).

• At the point where the shear is zero, dM >dx = 0, and therefore this would be a point of maximum or minimum moment.

• If a numerical value of the moment is to be determined at the point, one can find this value either by using the method of sections and the equation of moment equilibrium, or by using !M = 1V (x) dx, which states that the change in moment between any two points is equal to the area under the shear diagram between the two points.

• Since w(x) must be integrated to obtain !V , and V (x) is integrated to obtain M(x), then if w(x) is a curve of degree n, V (x) will be a curve of degree n + 1 and M(x) will be a curve of degree n + 2 . For example, if w(x) is uniform, V (x) will be linear and M(x) will be parabolic.

270 CHAPTER 6 BENDING

Draw the shear and moment diagrams for the beam shown in Fig. 6–11 a .

SOLUTION Support Reactions. The reaction at the fixed support is shown on the free-body diagram, Fig. 6–11 b .

Shear Diagram. The shear at each end of the beam is plotted first, Fig. 6–11 c . Since there is no distributed loading on the beam, the slope of the shear diagram is zero as indicated. Note how the force P at the center of the beam causes the shear diagram to jump downward an amount P , since this force acts downward.

Moment Diagram. The moments at the ends of the beam are plotted, Fig. 6–11 d . Here the moment diagram consists of two sloping lines, one with a slope of +2 P and the other with a slope of + P .

The value of the moment in the center of the beam can be determined by the method of sections, or from the area under the shear diagram. If we choose the left half of the shear diagram,

M 0 x = L = M 0 x = 0 + !M M 0 x = L = -3PL + (2P)(L) = -PL

EXAMPLE 6.5

(a)

3PL

P x

!3PL

!PL

w " 0 slope " 0

V " positive constant slope " positive constant

(b)

(c)

(d)

downward force P downward jump P

Fig. 6–11

6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS 271

EXAMPLE 6.6

Draw the shear and moment diagrams for the beam shown in Fig. 6–12 a .

(a)

L L

(b)

(c)

(d)

L L

M0 /2

–

w " 0 slope " 0

clockwise moment M0 positive jump M0

V " negative constant slope " negative constant

M0 2L

Fig. 6–12

SOLUTION Support Reactions. The reactions are shown on the free-body diagram in Fig. 6–12 b .

Shear Diagram. The shear at each end is plotted first, Fig. 6–12 c . Since there is no distributed load on the beam, the shear diagram has zero slope and is therefore a horizontal line.

Moment Diagram. The moment is zero at each end, Fig. 6–12 d . The moment diagram has a constant negative slope of −M 0 >2 L since this is the shear in the beam at each point. Note that the couple moment M0 causes a jump in the moment diagram at the beam’s center, but it does not affect the shear diagram at this point.

272 CHAPTER 6 BENDING

Draw the shear and moment diagrams for each of the beams shown in Figs. 6–13 a and 6–14 a .

SOLUTION Support Reactions. The reactions at the fixed support are shown on each free-body diagram, Figs. 6–13 b and 6–14 b .

Shear Diagram. The shear at each end point is plotted first, Figs. 6–13 c and 6–14 c . The distributed loading on each beam indicates the slope of the shear diagram and thus produces the shapes shown.

Moment Diagram. The moment at each end point is plotted first, Figs. 6–13 d and 6–14 d . Various values of the shear at each point on the beam indicate the slope of the moment diagram at the point. Notice how this variation produces the curves shown.

NOTE: Observe how the degree of the curves from w to V to M increases by one due to the integration of dV = w dx and dM = Vdx . For example, in Fig. 6–14 , the linear distributed load produces a parabolic shear diagram and cubic moment diagram.

EXAMPLE 6.7

(b)

(c)

V positive decreasing M slope positive decreasing

(d)

w0L w0

w0L2

w0L

w0L2

2 !

w negative constant V slope negative constant

(a) L

Fig. 6–13

(b)

(c)

(d)

w negative decreasing V slope negative decreasing

V positive decreasing M slope positive decreasing

w0L 2

6 w0L

w0 L 2

(a) L

Fig. 6–14

6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS 273

EXAMPLE 6.8

Draw the shear and moment diagrams for the cantilever beam in Fig. 6–15 a .

2 kN 1.5 kN/m

(a)

A B

2 m 2 m

SOLUTION Support Reactions. The support reactions at the fixed support B are shown in Fig. 6–15 b .

Shear Diagram. The shear at end A is −2 kN. This value is plotted at x = 0, Fig. 6–15 c . Notice how the shear diagram is constructed by following the slopes defined by the loading w . The shear at x = 4 m is −5 kN, the reaction on the beam. This value can be verified by finding the area under the distributed loading, Eq. 6–3 .

V 0 x = 4 m = V 0 x = 2 m + !V = -2 kN - (1.5 kN>m)(2 m) = -5 kN Moment Diagram. The moment of zero at x = 0 is plotted in Fig. 6–15 d . Notice how the moment diagram is constructed based on knowing its slope, which is equal to the shear at each point. The change of moment from x = 0 to x = 2 m is determined from the area under the shear diagram. Hence, the moment at x = 2 m is

M 0 x = 2 m = M 0 x = 0 + !M = 0 + [-2 kN(2 m)] = -4 kN # m This same value can be determined from the method of sections, Fig. 6–15 e .

(d)

(e)

(c)

(b)

2 kN

2 m

2 4

2 m

M " 4 kN#m

V " 2 kN

x (m)

V (kN)

2 0

!11

x (m)

M (kN#m)

w " 0 V slope " 0

w negative constant V slope negative constant

V negative constant M slope negative constant

V negative increasing M slope negative increasing

2 kN

2 m

By " 5 kN

MB " 11 kN#m1.5 kN/m

Fig. 6–15

274 CHAPTER 6 BENDING

EXAMPLE 6.9

Draw the shear and moment diagrams for the overhang beam in Fig. 6–16 a .

4 m

2 kN

(e)

V ! 2 kN

M ! 8 kN"m

SOLUTION Support Reactions. The support reactions are shown in Fig. 6–16 b .

Shear Diagram. The shear of −2 kN at end A of the beam is plotted at x = 0, Fig. 6–16 c . The slopes are determined from the loading and from this the shear diagram is constructed, as indicated in the figure. In particular, notice the positive jump of 10 kN at x = 4 m due to the force B y , as indicated in the figure.

Moment Diagram. The moment of zero at x = 0 is plotted, Fig. 6–16 d . Then following the behavior of the slope found from the shear diagram, the moment diagram is constructed. The moment at x = 4 m is found from the area under the shear diagram.

M 0 x = 4 m = M 0 x = 0 + !M = 0 + [-2 kN(4 m)] = -8 kN # m We can also obtain this value by using the method of sections, as shown in Fig. 6–16 e .

4 kN/m

4 m 2 m

(a)

A B

4 m 2 m

Ay ! 2 kN By ! 10 kN

0 x (m)

V (kN)

0 x (m)

M (kN#m)

w ! 0 V slope ! 0

V positive decreasing

M slope positive decreasing

V negative constant

M slope negative constant

w negative constant V slope negative constant

(d)

(c)

(b)

4 kN/m

slope ! 0

Fig. 6–16

6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS 275

EXAMPLE 6.10

The shaft in Fig. 6–17 a is supported by a thrust bearing at A and a journal bearing at B . Draw the shear and moment diagrams.

12 ft

(a)

120 lb/ft

12 ft

Ay = 240 lb

Ay = 240 lb By ! 480 lb

126.93

6.93

240

" 480

V (lb)

x (ft)

12 0

M (lb#ft)

x (ft)

10 x

w negative increasing

V slope negative increasing

V negative increasing M slope negative increasing

V positive decreasing

M slope positive decreasing

(d)

(e)

(c)

(b)

V ! 0 M slope ! 0

1109

[10 x] x x 3

1 2

Fig. 6–17

SOLUTION Support Reactions. The support reactions are shown in Fig. 6–17 b .

Shear Diagram. As shown in Fig. 6–17 c , the shear at x = 0 is +240 lb. Following the slope defined by the loading, the shear diagram is constructed, where at B its value is −480 lb. Since the shear changes sign, the point where V = 0 must be located. To do this we will use the method of sections. The free-body diagram of the left segment of the shaft, sectioned at an arbitrary position x, is shown in Fig. 6–17 e . Notice that the intensity of the distributed load at x is w = 10 x , which has been found by proportional triangles, i.e., 120>12 = w>x .

Thus, for V = 0,

+ c !Fy = 0; 240 lb - 12(10x)x = 0 x = 6.93 ft

Moment Diagram. The moment diagram starts at 0 since there is no moment at A ; then it is constructed based on the slope as determined from the shear diagram. The maximum moment occurs at x = 6.93 ft, where the shear is equal to zero, since dM >dx = V = 0, Fig. 6–17 d ,

a+ !M = 0; Mmax + 12[(10)(6.93)] 6.93 113(6.93)2 - 240(6.93) = 0

Mmax = 1109 lb # ft Finally, notice how integration, first of the loading w which is linear, produces a shear diagram which is parabolic, and then a moment diagram which is cubic.

NOTE: Having studied these examples, test yourself by covering over the shear and moment diagrams in Examples 6.1 through 6.4 and see if you can construct them using the concepts discussed here.

276 CHAPTER 6 BENDING

PRELIMINARY PROBLEMS

P6–1. In each case, the beam is subjected to the loadings shown. Draw the free-body-diagram of the beam, and sketch the general shape of the shear and moment diagrams. The loads and geometry are assumed to be known.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

P6–1

6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS 277

In each case, draw the shear and moment diagrams for the beam.

F6–5.

A B

4 kN/m 4 kN/m

3 m 1.5 m1.5 m

F6–5

F6–6.

A B

3 m

10 kN/m 10 kN/m

3 m

F6–6

F6–7.

A B

600 lb 200 lb/ft

6 ft 3 ft 3 ft

C D

F6–7

F6–8.

2 m

20 kN

4 m

A B

20 kN/m

F6–8

F6–1.

A B

6 m

30 kN·m

F6–1

F6–2.

3 m

9 kN

F6–2

F6–3.

9 ft x

2 kip/ft

18 kip·ft

F6–3

F6–4.

3 m

12 kN/m

F6–4

FUNDAMENTAL PROBLEMS

In each case, express the shear and moment functions in terms of x, and then draw the shear and moment diagrams for the beam.

278 CHAPTER 6 BENDING

6–1. The load binder is used to support a load. If the force applied to the handle is 50 lb, determine the tensions T 1 and T 2 in each end of the chain and then draw the shear and moment diagrams for the arm ABC .

12 in. 50 lb

3 in.

Prob. 6–1

6–2. Draw the shear and moment diagrams for the shaft The bearings at A and D exert only vertical reaction on the shaft. The loading is applied to the pulleys at B and C and E.

14 in. 20 in. 15 in. 12 in.

80 lb 110 lb

35 lb

C D

Prob. 6–2

6–3. The engine crane is used to support the engine, which has a weight of 1200 lb. Draw the shear and moment diagrams of the boom ABC when it is in the horizontal position shown.

5 ft3 ft

4 ft

Prob. 6–3

*6–4. Draw the shear and moment diagrams for the beam.

4 ft4 ft4 ft4 ft4 ft

2 kip 2 kip 2 kip 2 kip

Prob. 6–4

•6–5. Draw the shear and moment diagrams for the beam.

2 m 3 m

10 kN 8 kN

15 kN!m

Prob. 6–5

PROBLEMS

6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS 279

6–6. Express the internal shear and moment in terms of x and then draw the shear and moment diagrams.

A B

L 2

Prob. 6–6

6–7. Draw the shear and moment diagrams for the compound beam which is pin connected at B . (This structure is not fully stable. But with the given loading, it is balanced and will remain as shown if not disturbed.)

4 ft

6 kip 8 kip

A C

6 ft 4 ft 4 ft

Prob. 6–7

*6–8. Express the internal shear and moment in terms of x and then draw the shear and moment diagrams for the beam.

A B

900 lb

400 lb/ft

6 ft 3 ft x

Prob. 6–8

6–9. Express the internal shear and moment in terms of x and then draw the shear and moment diagrams for the overhanging beam.

A B

6 kN/m

4 m 2 m

Prob. 6–9

6–10. Members ABC and BD of the counter chair are rigidly connected at B and the smooth collar at D is allowed to move freely along the vertical slot. Draw the shear and moment diagrams for member ABC.

B C

P ! 150 lb

1.5 ft1.5 ft 1.5 ft

Prob. 6–10

6–11. Draw the shear and moment diagrams for the pipe. The end screw is subjected to a horizontal force of 5 kN. Hint: The reactions at the pin C must be replaced by an equivalent loading at point B on the axis of the pipe.

80 mm

400 mm

5 kN

Prob. 6–11

280 CHAPTER 6 BENDING

■ 6–15. Consider the general problem of the beam subjected to n concentrated loads. Write a computer program that can be used to determine the internal shear and moment at any specified location x along the beam, and plot the shear and moment diagrams for the beam. Show an application of the program using the values P1 = 500 lb, d1 = 5 ft,P2 = 800 lb, d2 = 15 ft, L1 = 10 ft, L = 15 ft .

P1 P2 Pn

d1 d2

dn L1

Prob. 6–15

*6–16. Determine the placement distance a of the roller support so that the largest absolute value of the moment is a minimum. Draw the shear and moment diagrams for this condition.

L– 2

Prob. 6–16

6–17. Express the internal shear and moment in the cantilevered beam as a function of x and then draw the shear and moment diagrams.

300 lb 200 lb/ft

Ax 6 ft

Prob. 6–17

*6–12. A reinforced concrete pier is used to support the stringers for a bridge deck. Draw the shear and moment diagrams for the pier when it is subjected to the stringer loads shown. Assume the columns at A and B exert only vertical reactions on the pier.

1 m 1 m 1 m 1 m1.5 m 60 kN 60 kN35 kN 35 kN 35 kN

1.5 m

A B

Prob. 6–12 6–13. Draw the shear and moment diagrams for the rod. It is supported by a pin at A and a smooth plate at B . The plate slides within the groove and so it cannot support a vertical force, although it can support a moment.

4 m

2 m

15 kN

Prob. 6–13 6–14. The industrial robot is held in the stationary position shown. Draw the shear and moment diagrams of the arm ABC if it is pin connected at A and connected to a hydraulic cylinder (two-force member) BD. Assume the arm and grip have a uniform weight of 1.5 lb>in. and support the load of 40 lb at C.

10 in. 4 in.

50 in. A B C

120!

Prob. 6–14

6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS 281

6–18. Draw the shear and moment diagrams for the beam, and determine the shear and moment throughout the beam as functions of x.

6 ft 4 ft

2 kip/ft 8 kip

10 kip

40 kip!ft

Prob. 6–18

6–19. Draw the shear and moment diagrams for the beam.

30 kip!ft

5 ft 5 ft

2 kip/ ft

5 ft

Prob. 6–19

*6–20. Draw the shear and moment diagrams for the overhanging beam.

B A

12 ft

3 kip/ft

6 ft

Prob. 6–20

6–21. The 150-lb man sits in the center of the boat, which has a uniform width and a weight per linear foot of 3 lb>ft. Determine the maximum bending moment exerted on the boat. Assume that the water exerts a uniform distributed load upward on the bottom of the boat.

7.5 ft 7.5 ft

Prob. 6–21

6–22. Draw the shear and moment diagrams for the overhang beam.

4 kN/m

3 m 3 m

A B

Prob. 6–22

6–23. The footing supports the load transmitted by the two columns. Draw the shear and moment diagrams for the footing if the reaction of soil pressure on the footing is assumed to be uniform.

6 ft 12 ft 6 ft

14 kip14 kip

Prob. 6–23

*6–24. Express the shear and moment in terms of x and then draw the shear and moment diagrams for the simply supported beam.

A B

3 m 1.5 m

300 N/m

Prob. 6–24

6–25. Draw the shear and moment diagrams for the beam and determine the shear and moment in the beam as functions of x , where 4 ft < x < 10 ft.

200 lb!ft

4 ft 4 ft

150 lb/ft

6 ft

200 lb!ft

Prob. 6–25

282 CHAPTER 6 BENDING

6–29. Draw the shear and moment diagrams for the double overhanging beam.

1.5 m 1.5 m3 m

8 kN/m

Prob. 6–29

6–30. The beam is bolted or pinned at A and rests on a bearing pad at B that exerts a uniform distributed loading on the beam over its 2-ft length. Draw the shear and moment diagrams for the beam if it supports a uniform loading of 2 kip>ft.

8 ft 1 ft

2 ft A

2 kip/ft

Prob. 6–30

6–31. The support at A allows the beam to slide freely along the vertical guide so that it cannot support a vertical force. Draw the shear and moment diagrams for the beam.

Prob. 6–31

*6–32. The smooth pin is supported by two leaves A and B and subjected to a compressive load of 0.4 kN>m caused by bar C . Determine the intensity of the distributed load w 0 of the leaves on the pin and draw the shear and moment diagram for the pin.

20 mm

0.4 kN/m

w0 w0

20 mm 60 mm

A B

Prob. 6–32

■ 6–26. Consider the general problem of a cantilevered beam subjected to n concentrated loads and a constant distributed loading w . Write a computer program that can be used to determine the internal shear and moment at any specified location x along the beam, and plot the shear and moment diagrams for the beam. Show an application of the program using the values P1 = 4 kN, d1 = 2 m, w = 800 N>m, a1 = 2 m, a2 = 4 m, L = 4 m.

d1 d2

P2 Pn

Prob. 6–26

6–27. Draw the shear and moment diagrams for the beam.

A 2L 3

L 3

Prob. 6–27

*6–28. Draw the shear and moment diagrams for the beam.

A B

L 3

Prob. 6–28

6.2 GRAPHICAL METHOD FOR CONSTRUCTING SHEAR AND MOMENT DIAGRAMS 283

6–33. The shaft is supported by a smooth thrust bearing at A and smooth journal bearing at B . Draw the shear and moment diagrams for the shaft.

A B

1 m 1 m 1 m

900 N

400 N!m

Prob. 6–33

6–34. Draw the shear and moment diagrams for the cantilever beam.

2 kN

3 kN!m

1.5 m 1.5 m

Prob. 6–34

6–35. Draw the shear and moment diagrams for the beam and determine the shear and moment as functions of x .

3 m 3 m x

A B

200 N/m

400 N/m

Prob. 6–35

*6–36. Thae shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B . Draw the shear and moment diagrams for the shaft.

A B

900 N

600 N!m

0.8 m 0.8 m 0.8 m

Prob. 6–36

6–37. Draw the shear and moment diagrams for the beam.

4.5 m 4.5 m

50 kN/m

Prob. 6–37

6–38. The beam is used to support a uniform load along CD due to the 6-kN weight of the crate. If the reaction at bearing support B can be assumed uniformly distributed along its width, draw the shear and moment diagrams for the beam.

2.75 m 2 m 0.75 m0.5 m

B A

Prob. 6–38

6–39. Draw the shear and moment diagrams for the double overhanging beam.

3 ft 3 ft

200 lb/ft 400 lb

6 ft

400 lb

Prob. 6–39

*6–40. Draw the shear and moment diagrams for the simply supported beam.

A B

2 m 2 m

10 kN 10 kN

15 kN!m

2 m

Prob. 6–40

284 CHAPTER 6 BENDING

*6–44. Draw the shear and moment diagrams for the beam.

8 ft

A B x

w = 1 x2

8 kip/ft 1– 8

Prob. 6–44

6–45. A short link at B is used to connect beams AB and BC to form the compound beam shown. Draw the shear and moment diagrams for the beam if the supports at A and B are considered fixed and pinned, respectively.

A CB

4.5 m 1.5 m 1.5 m

15 kN

3 kN/m

Prob. 6–45

6–46. Determine the placement b of the hooks to minimize the largest moment when the concrete member is being hoisted. Draw the shear and moment diagrams. The member has a square cross section of dimension a on each side. The specific weight of concrete is g .

L b b

60! 60!

Prob. 6–46

6–41. The compound beam is fixed at A , pin connected at B , and supported by a roller at C . Draw the shear and moment diagrams for the beam.

A B C

2 m2 m2 m

400 N/m

600 N

Prob. 6–41

6–42. Draw the shear and moment diagrams for the compound beam.

BA C D

2 m 1 m 1 m

5 kN/m

Prob. 6–42

6–43. The compound beam is fixed at A , pin connected at B , and supported by a roller at C . Draw the shear and moment diagrams for the beam.

A B C

3 kN/m 2 kN

3 m 3 m

Prob. 6–43

6.3 BENDING DEFORMATION OF A STRAIGHT MEMBER 285

6.3 Bending Deformation of a Straight Member

In this section, we will discuss the deformations that occur when a straight prismatic beam, made of homogeneous material, is subjected to bending. The discussion will be limited to beams having a cross-sectional area that is symmetrical with respect to an axis, and the bending moment is applied about an axis perpendicular to this axis of symmetry as shown in Fig. 6–18 . The behavior of members that have unsymmetrical cross sections, or are made from several different materials, is based on similar observations and will be discussed separately in later sections of this chapter.

By using a highly deformable material such as rubber, we can illustrate what happens when a straight prismatic member is subjected to a bending moment. Consider, for example, the undeformed bar in Fig. 6–19 a , which has a square cross section and is marked with longitudinal and transverse grid lines. When a bending moment is applied, it tends to distort these lines into the pattern shown in Fig. 6–19 b . Notice that the longitudinal lines become curved and the vertical transverse lines remain straight and yet undergo a rotation.

The bending moment causes the material within the bottom portion of the bar to stretch and the material within the top portion to compress . Consequently, between these two regions there must be a surface, called the neutral surface, in which longitudinal fibers of the material will not undergo a change in length, Fig. 6–18 .

z M

Axis of symmetry

Longitudinal axis

Neutral surface

Fig. 6–18

Before deformation

(a)

After deformation

(b)

Horizontal lines become curved

Vertical lines remain straight, yet rotate

Fig. 6–19

286 CHAPTER 6 BENDING

From these observations we will make the following three assumptions regarding the way the stress deforms the material. First, the longitudinal axis x , which lies within the neutral surface, Fig. 6–20 a , does not experience any change in length. Rather the moment will tend to deform the beam so that this line becomes a curve that lies in the x – y plane of symmetry, Fig. 6–20 b . Second, all cross sections of the beam remain plane and perpendicular to the longitudinal axis during the deformation. And third, any deformation of the cross section within its own plane, as noticed in Fig. 6–19 b , will be neglected. In particular, the z axis, lying in the plane of the cross section and about which the cross section rotates, is called the neutral axis, Fig. 6–20 b .

In order to show how this distortion will strain the material, we will isolate a small segment of the beam located a distance x along the beam’s length and having an undeformed thickness !x, Fig. 6–20 a . This element, taken from the beam, is shown in profile view in the undeformed and

Note the distortion of the lines due to bending of this rubber bar. The top line stretches, the bottom line compresses, and the center line remains the same length. Furthermore the vertical lines rotate and yet remain straight.

neutral axis

(b)

longitudinal axis

neutral surface

(a)

Fig. 6–20

6.3 BENDING DEFORMATION OF A STRAIGHT MEMBER 287

6deformed positions in Fig. 6–21 . Notice that any line segment !x, located on the neutral surface, does not change its length, whereas any line segment !s, located at the arbitrary distance y above the neutral surface, will contract and become !s" after deformation. By definition, the normal strain along !s is determined from Eq. 2–2 , namely,

P = lim !sS0

!s" - !s !s

We will now represent this strain in terms of the location y of the segment and the radius of curvature r of the longitudinal axis of the element. Before deformation, !s = !x, Fig. 6–21 a . After deformation !x has a radius of curvature r, with center of curvature at point O", Fig. 6–21 b . Since !u defines the angle between the sides of the element, !x = !s = r!u. In the same manner, the deformed length of !s becomes !s" = (r - y)!u. Substituting into the above equation, we get

P = lim !uS0

(r - y)!u - r!u r!u

P = - y r

(6–7)

This important result indicates that the longitudinal normal strain of any element within the beam depends on its location y on the cross

longitudinal axis

Undeformed element

(a)

!s " !x

longitudinal axis y

Deformed element

(b)

!s¿

O¿

Fig. 6–21

288 CHAPTER 6 BENDING

section and the radius of curvature of the beam’s longitudinal axis at the point. In other words, for any specific cross section, the longitudinal normal strain will vary linearly with y from the neutral axis. A contraction (-P) will occur in fibers located above the neutral axis (+y), whereas elongation (+P) will occur in fibers located below the axis (-y). This variation in strain over the cross section is shown in Fig. 6–22 . Here the maximum strain occurs at the outermost fiber, located a distance of y = c from the neutral axis. Using Eq. 6–7 , since Pmax = c>r, then by division,

Pmax = - a y >r

c>r b So that

P = - a y c bPmax (6–8)

This normal strain depends only on the assumptions made with regards to the deformation. When a moment is applied to the beam, therefore, it will only cause a normal stress in the longitudinal or x direction. All the other components of normal and shear stress will be zero. It is this uniaxial state of stress that causes the material to have the longitudinal normal strain component Px, defined by Eq. 6–8 . Furthermore, by Poisson’s ratio, there must also be associated strain components Py = -nPx and Pz = -nPx, which deform the plane of the cross-sectional area, although here we have neglected these deformations. Such deformations will, however, cause the cross-sectional dimensions to become smaller below the neutral axis and larger above the neutral axis. For example, if the beam has a square cross section, it will actually deform as shown in Fig. 6–23 .

Fig. 6–23

y c

Normal strain distribution

"Pmax

PmaxP#" y c

Fig. 6–22

6.4 THE FLEXURE FORMULA 289

6.4 The Flexure Formula In this section, we will develop an equation that relates the stress distribution in a beam to the internal resultant bending moment acting on the beam’s cross section. To do this we will assume that the material behaves in a linear-elastic manner and therefore a linear variation of normal strain, Fig. 6–24 a , must then be the result of a linear variation in normal stress, Fig. 6–24 b . Hence, like the normal strain variation, s will vary from zero at the member’s neutral axis to a maximum value, smax, a distance c farthest from the neutral axis. Because of the proportionality of triangles, Fig. 6–23 b , or by using Hooke’s law, s = EP, and Eq. 6–8 , we can write

s = - a y c bsmax (6–9)

This equation describes the stress distribution over the cross- sectional area. The sign convention established here is significant. For positive M, which acts in the +z direction, positive values of y give negative values for s, that is, a compressive stress since it acts in the negative x direction. Similarly, negative y values will give positive or tensile values for s. If a volume element of material is selected at a specific point on the cross section, only these tensile or compressive normal stresses will act on it. For example, the element located at +y is shown in Fig. 6–24 c .

We can locate the position of the neutral axis on the cross section by satisfying the condition that the resultant force produced by the stress distribution over the cross-sectional area must be equal to zero. Noting that the force dF = s dA acts on the arbitrary element dA in Fig. 6–24 c , we require

FR = !Fx; 0 = LA dF = LAs dA

= LA - a yc bsmax dA =

-smax c LAy dA

c y

Normal strain variation (profile view)

(a)

Pmax

x y

Bending stress variation (profile view)

(b)

smax

This wood specimen failed in bending due to its fibers being crushed at its top and torn apart at its bottom.

Fig. 6–24

290 CHAPTER 6 BENDING

Since smax>c is not equal to zero, then LAy dA = 0 (6–10)

In other words, the first moment of the member’s cross-sectional area about the neutral axis must be zero. This condition can only be satisfied if the neutral axis is also the horizontal centroidal axis for the cross section. * Consequently, once the centroid for the member’s cross- sectional area is determined, the location of the neutral axis is known.

We can determine the stress in the beam from the requirement that the resultant internal moment M must be equal to the moment produced by the stress distribution about the neutral axis. The moment of d F in Fig. 6–24 c about the neutral axis is dM = y dF. Since dF = s dA , using Eq. 6–9 , we have for the entire cross section,

(MR)z = !Mz; M = LAy dF = LAy(s dA ) = LAy ¢ yc smax≤ dA or

M = smax

c LAy 2 dA (6–11)

* Recall that the location y for the centroid of the cross-sectional area is defined from the equation y = 1y dA > 1dA . If 1y dA = 0, then y = 0, and so the centroid lies on the reference (neutral) axis. See Appendix A .

y c

Bending stress variation

(c)

z dA

M dF

smax

Fig. 6–24 (cont.)

6.4 THE FLEXURE FORMULA 291

The integral represents the moment of inertia of the cross-sectional area about the neutral axis. We will symbolize its value as I. Hence, Eq. 6–11 can be solved for smax and written as

smax = Mc

I (6–12)

Here

smax = the the maximum normal stress in the member, which occurs at a point on the cross-sectional area farthest away from the neutral axis

M = the resultant internal moment, determined from the method of sections and the equations of equilibrium, and calculated about the neutral axis of the cross section

c = perpendicular distance from the neutral axis to a point farthest away from the neutral axis. This is where smax acts.

I = moment of inertia of the cross-sectional area about the neutral axis

Since smax>c = -s>y, Eq. 6–9 , the normal stress at the location distance y can be determined from an equation similar to Eq. 6–12 . We have

s = - My

I (6–13)

Note that the negative sign is necessary since it agrees with the established x, y, z axes. By the right-hand rule, M is positive along the +z axis, y is positive upward, and s therefore must be negative (compressive) since it acts in the negative x direction, Fig. 6–24 c .

Either of the above two equations is often referred to as the flexure formula. It is used to determine the normal stress in a straight member, having a cross section that is symmetrical with respect to an axis, and the moment is applied perpendicular to this axis. Although we have assumed that the member is prismatic, we can in most cases of engineering design also use the flexure formula to determine the normal stress in members that have a slight taper. For example, using a mathematical analysis based on the theory of elasticity, a member having a rectangular cross section and a length that is tapered 15° will have an actual maximum normal stress that is about 5.4% less than that calculated using the flexure formula.

292 CHAPTER 6 BENDING

Important Points

• The cross section of a straight beam remains plane when the beam deforms due to bending. This causes tensile stress on one portion of the cross section and compressive stress on the other portion. In between these portions, there exists the neutral axis which is subjected to zero stress.

• Due to the deformation, the longitudinal strain varies linearly from zero at the neutral axis to a maximum at the outer fibers of the beam. Provided the material is homogeneous and linear elastic, then the stress also varies in a linear fashion over the cross section.

• The neutral axis passes through the centroid of the cross-sectional area. This result is based on the fact that the resultant normal force acting on the cross section must be zero.

• The flexure formula is based on the requirement that the resultant internal moment on the cross section is equal to the moment produced by the normal stress distribution about the neutral axis.

Procedure for Analysis

In order to apply the flexure formula, the following procedure is suggested.

Internal Moment. • Section the member at the point where the bending or normal stress is to be determined, and obtain the

internal moment M at the section. The centroidal or neutral axis for the cross section must be known, since M must be calculated about this axis.

• If the absolute maximum bending stress is to be determined, then draw the moment diagram in order to determine the maximum moment in the member.

Section Property. • Determine the moment of inertia of the cross-sectional area about the neutral axis. Methods used for its

calculation are discussed in Appendix A , and a table listing values of I for several common shapes is given on the inside front cover.

Normal Stress. • Specify the location y, measured perpendicular to the neutral axis to the point where the normal stress is

to be determined. Then apply the equation s = -My >I, or if the maximum bending stress is to be calculated, use smax = Mc>I. When substituting the data, make sure the units are consistent.

• The stress acts in a direction such that the force it creates at the point contributes a moment about the neutral axis that is in the same direction as the internal moment M, Fig. 6–24 c . In this manner the stress distribution acting over the entire cross section can be sketched, or a volume element of the material can be isolated and used to graphically represent the normal stress acting at the point.

6.4 THE FLEXURE FORMULA 293

EXAMPLE 6.11

A beam has a rectangular cross section and is subjected to the stress distribution shown in Fig. 6–25 a . Determine the internal moment M at the section caused by the stress distribution (a) using the flexure formula, (b) by finding the resultant of the stress distribution using basic principles.

SOLUTION Part (a). The flexure formula is smax = Mc>I. From Fig. 6–25 a , c = 6 in. and smax = 2 ksi. The neutral axis is defined as line NA, because the stress is zero along this line. Since the cross section has a rectangular shape, the moment of inertia for the area about NA is determined from the formula for a rectangle given on the inside front cover; i.e.,

I = 1

12 bh3 =

1 12

(6 in.)(12 in.)3 = 864 in4

Therefore,

smax = Mc

I ; 2 kip>in2 = M(6 in.)

864 in4

M = 288 kip # in. = 24 kip # ft Ans.

Part (b). The resultant force for each of the two triangular stress distributions in Fig. 6–25 b is graphically equivalent to the volume contained within each stress distribution. Thus, each volume is

F = 1 2

(6 in.)(2 kip>in2)(6 in.) = 36 kip These forces, which form a couple, act in the same direction as the stresses within each distribution, Fig. 6–25 b . Furthermore, they act through the centroid of each volume, i.e., 23(6 in.) = 4 in. from the neutral axis of the beam. Hence the distance between them is 8 in. as shown. The moment of the couple is therefore

M = 36 kip(8 in.) = 288 kip # in. = 24 kip # ft Ans.

NOTE: This result can also be obtained by choosing a horizontal strip of area dA = (6 in.) dy and using integration by applying Eq. 6–11 .

6 in.

2 ksi

2 ksi (a)

6 in.

(b)

6 in.

4 in.

Fig. 6–25

294 CHAPTER 6 BENDING

EXAMPLE 6.12

The simply supported beam in Fig. 6–26 a has the cross-sectional area shown in Fig. 6–26 b . Determine the absolute maximum bending stress in the beam and draw the stress distribution over the cross section at this location.

6 m

5 kN/m

(a)

M (kN!m)

x (m)

22.5

3 6

(c)

20 mm

N A

20 mm

250 mm

150 mm

(b)

20 mm

12.7 MPa

11.2 MPa

11.2 MPaB

D M ! 22.5 kN"m

12.7 MPa

(d)

Fig. 6–26

SOLUTION Maximum Internal Moment. The maximum internal moment in the beam, M = 22.5 kN # m, occurs at the center. Section Property. By reasons of symmetry, the neutral axis passes through the centroid C at the midheight of the beam, Fig. 6–26 b . The area is subdivided into the three parts shown, and the moment of inertia of each part is calculated about the neutral axis using the parallel-axis theorem. (See Eq. A–5 of Appendix A .) Choosing to work in meters, we have

I = !(I + Ad2)

= 2 c 1 12

(0.25 m)(0.020 m)3 + (0.25 m)(0.020 m)(0.160 m)2 d + c 1

12 (0.020 m)(0.300 m)3 d

= 301.3(10-6) m4

smax = Mc

I ; smax =

22.5(103) N # m(0.170 m) 301.3(10-6) m4

= 12.7 MPa Ans.

A three-dimensional view of the stress distribution is shown in Fig. 6–26 d . Notice how the stress at points B and D on the cross section develops a force that contributes a moment about the neutral axis that has the same direction as M . Specifically, at point B , yB = 150 mm, and so as shown in Fig. 6–26 d ,

sB = - MyB

I ; sB = -

22.5(103) N # m(0.150 m) 301.3(10-6) m4

= -11.2 MPa

6.4 THE FLEXURE FORMULA 295

EXAMPLE 6.13

2 m 1 m

2.6 kN

12 5 a

a (a)

2 m

M N

(c)

2.4 kN

1.0 kN 0.05909 m

Fig. 6–27

250 mm

200 mm AN

15 mm

20 mm

(b)

15 mm

_ y ! 59.09 mm

The beam shown in Fig. 6–27 a has a cross-sectional area in the shape of a channel, Fig. 6–27 b . Determine the maximum bending stress that occurs in the beam at section a – a.

SOLUTION Internal Moment. Here the beam’s support reactions do not have to be determined. Instead, by the method of sections, the segment to the left of section a – a can be used, Fig. 6–27 c . In particular, note that the resultant internal axial force N passes through the centroid of the cross section. Also, realize that the resultant internal moment must be calculated about the beam’s neutral axis at section a – a.

To find the location of the neutral axis, the cross-sectional area is subdivided into three composite parts as shown in Fig. 6–27 b . Using Eq. A–2 of Appendix A , we have

y = !y!A

!A =

2[0.100 m](0.200 m)(0.015 m) + [0.010 m](0.02 m)(0.250 m) 2(0.200 m)(0.015 m) + 0.020 m(0.250 m)

= 0.05909 m = 59.09 mm

This dimension is shown in Fig. 6–27 c . Applying the moment equation of equilibrium about the neutral

axis, we have

a+ !MNA = 0; 2.4 kN(2 m) + 1.0 kN(0.05909 m) - M = 0 M = 4.859 kN # m

Section Property. The moment of inertia about the neutral axis is determined using I = π ( I + Ad 2 ) applied to each of the three composite parts of the cross-sectional area. Working in meters, we have

I = c 1 12

(0.250 m)(0.020 m)3 + (0.250 m)(0.020 m)(0.05909 m - 0.010 m)2 d + 2 c 1

12 (0.015 m)(0.200 m)3 + (0.015 m)(0.200 m)(0.100 m - 0.05909 m)2 d

= 42.26(10-6) m4

Maximum Bending Stress. The maximum bending stress occurs at points farthest away from the neutral axis. This is at the bottom of the beam, c = 0.200 m - 0.05909 m = 0.1409 m. Thus,

smax = Mc

I =

4.859(103) N # m(0.1409 m) 42.26(10-6) m4

= 16.2 MPa Ans.

Show that at the top of the beam the bending stress is s" = 6.79 MPa.

NOTE: The normal force of N = 1 kN and shear force V = 2.4 kN will also contribute additional stress on the cross section. The superposition of all these effects will be discussed in Chapter 8 .

296 CHAPTER 6 BENDING

The member having a rectangular cross section, Fig. 6–28 a , is designed to resist a moment of 40 N # m. In order to increase its strength and rigidity, it is proposed that two small ribs be added at its bottom, Fig. 6–28 b . Determine the maximum normal stress in the member for both cases.

SOLUTION Without Ribs. Clearly the neutral axis is at the center of the cross section, Fig. 6–28 a , so y = c = 15 mm = 0.015 m. Thus,

I = 1

12 bh3 =

1 12

(0.060 m)(0.030 m)3 = 0.135(10-6) m4

Therefore the maximum normal stress is

smax = Mc

I =

(40 N # m)(0.015 m) 0.135(10-6) m4

= 4.44 MPa Ans.

With Ribs. From Fig. 6–28 b , segmenting the area into the large main rectangle and the bottom two rectangles (ribs), the location y of the centroid and the neutral axis is determined as follows:

y = !y!A

= [0.015 m](0.030 m)(0.060 m) + 2[0.0325 m](0.005 m)(0.010 m)

(0.03 m)(0.060 m) + 2(0.005 m)(0.010 m) = 0.01592 m

This value does not represent c. Instead

c = 0.035 m - 0.01592 m = 0.01908 m Using the parallel-axis theorem, the moment of inertia about the

neutral axis is

EXAMPLE 6.14

60 mm

30 mm

_ y40 N·m

(a)

40 N!m

30 mm _y

10 mm

5 mm

(b)

Fig. 6–28

I = c 1 12

(0.060 m)(0.030 m)3 + (0.060 m)(0.030 m)(0.01592 m - 0.015 m)2 d + 2 c 1

12 (0.010 m)(0.005 m)3 + (0.010 m)(0.005 m)(0.0325 m - 0.01592 m)2 d

= 0.1642(10-6) m4

Therefore, the maximum normal stress is

smax = Mc

I =

40 N # m(0.01908 m) 0.1642(10-6) m4

= 4.65 MPa Ans.

NOTE: This surprising result indicates that the addition of the ribs to the cross section will increase the maximum normal stress rather than decrease it, and for this reason they should be omitted.

6.4 THE FLEXURE FORMULA 297

P6–2. Determine the moment of inertia of the cross section about the neutral axis.

0.2 m

0.1 m

0.2 m

N A

0.1 m

P6–2

P6–3. Determine the location of the centroid, y and the moment of inertia of the cross section about the neutral axis.

0.3 m

0.1 m

0.2 m

0.1 m

N A

P6–3

P6–4. In each case, show how the bending stress acts on a differential volume element located at point A and point B .

(a)

(b)

P6–4

P6–5. Sketch the bending stress distribution over each cross section.

(a)

(b)

P6–5

PRELIMINARY PROBLEMS

298 CHAPTER 6 BENDING

FUNDAMENTAL PROBLEMS

F6–9. If the beam is subjected to a bending moment of M = 20 kN # m, determine the maximum bending stress in the beam.

300 mm

20 mm

20 mm 20 mm

200 mm

F6–9

F6–10. If the beam is subjected to a bending moment of M = 50 kN # m, sketch the bending stress distribution over the beam’s cross section.

150 mm

300 mm

F6–10

F6–11. If the beam is subjected to a bending moment of M = 50 kN # m, determine the maximum bending stress in the beam.

200 mm

300 mm

20 mm

20 mm 20 mm

F6–11

F6–12. If the beam is subjected to a bending moment of M = 10 kN # m, determine the bending stress in the beam at points A and B, and sketch the results on a differential element at each of these points.

200 mm

150 mm

50 mm

30 mm

50 mm

30 mm

F6–12

F6–13. If the beam is subjected to a bending moment of M = 5 kN # m, determine the bending stress developed at point A and sketch the result on a differential element at this point.

150 mm

25 mm

150 mm

50 mm

25 mm

F6–13

6.4 THE FLEXURE FORMULA 299

6–47. If the A-36 steel sheet roll is supported as shown and the allowable bending stress is 165 MPa, determine the smallest radius r of the spool if the steel sheet has a width of 1 m and a thickness of 1.5 mm. Also, find the corresponding maximum internal moment developed in the sheet.

Prob. 6–47

*6–48. Determine the moment M that will produce a maximum stress of 10 ksi on the cross section.

6–49. Determine the maximum tensile and compressive bending stress in the beam if it is subjected to a moment of M = 4 kip # ft.

3 in.

A B

0.5 in.

3 in. C

10 in.

0.5 in.0.5 in.

Probs. 6–48/49

6–50. A member has the triangular cross section shown. Determine the largest internal moment M that can be applied to the cross section without exceeding allowable tensile and compressive stresses of (s allow ) t = 22 ksi and (s allow ) c = 15 ksi, respectively.

6–51. A member has the triangular cross section shown. If a moment of M = 800 lb · ft is applied to the cross section, determine the maximum tensile and compressive bending stresses in the member. Also, sketch a three-dimensional view of the stress distribution action over the cross section.

2 in. 2 in.

4 in.

M 4 in.

Probs. 6–50/51

*6–52. If the beam is subjected to an internal moment of M = 30 kN # m, determine the maximum bending stress in the beam. The beam is made from A992 steel. Sketch the bending stress distribution on the cross section.

6–53. If the beam is subjected to an internal moment of M = 30 kN # m, determine the resultant force caused by the bending stress distribution acting on the top flange A .

50 mm

150 mm

15 mm

10 mm

15 mm

50 mm

Probs. 6–52/53

PROBLEMS

300 CHAPTER 6 BENDING

6–58. The aluminum machine part is subjected to a moment of M = 75 N # m. Determine the bending stress created at points B and C on the cross section. Sketch the results on a volume element located at each of these points.

6–59. The aluminum machine part is subjected to a moment of M = 75 N # m. Determine the maximum tensile and compressive bending stresses in the part.

M ! 75 N"m 40 mm

10 mm

10 mm 10 mm

20 mm

20 mm 10 mm

10 mm

B A

N C

Probs. 6–58/59

*6–60. The beam is subjected to a moment of 15 kip # ft. Determine the resultant force the bending stress produces on the top flange A and bottom flange B . Also compute the maximum bending stress developed in the beam.

6–61. The beam is subjected to a moment of 15 kip # ft. Determine the percentage of this moment that is resisted by the web D of the beam.

3 in.

5 in.

1 in.

8 in.

M ! 15 kip"ft

1 in.

Probs. 6–60/61

6–54. If the built-up beam is subjected to an internal moment of M = 75 kN # m, determine the maximum tensile and compressive stress acting in the beam.

6–55. If the built-up beam is subjected to an internal moment of M = 75 kN # m, determine the amount of this internal moment resisted by plate A .

300 mm A

20 mm

10 mm

150 mm

150 mm 150 mm

Probs. 6–54/55

*6–56. The aluminum strut has a cross-sectional area in the form of a cross. If it is subjected to the moment M = 8 kN # m, determine the bending stress acting at points A and B , and show the results acting on volume elements located at these points.

6–57. The aluminum strut has a cross-sectional area in the form of a cross. If it is subjected to the moment M = 8 kN # m, determine the maximum bending stress in the beam, and sketch a three-dimensional view of the stress distribution acting over the entire cross-sectional area.

20 mm

100 mm

50 mm 50 mm

100 mm

M ! 8 kN"m

Probs. 6–56/57

6.4 THE FLEXURE FORMULA 301

6–65. A shaft is made of a polymer having an elliptical cross-section. If it resists an internal moment of M = 50 N # m, determine the maximum bending stress developed in the material (a) using the flexure formula, where I z = 14 p(0.08 m)(0.04 m) 3 , (b) using integration. Sketch a three-dimensional view of the stress distribution acting over the cross-sectional area.

6–66. Solve Prob. 6–65 if the moment M = 50 N # m is applied about the y axis instead of the x axis. Here I y = 14 p (0.04 m)(0.08 m)3.

z x

M ! 50 N"m

80 mm

160 mm

y ——— (40)

2 z

——— (80)

2 # ! 1

Probs. 6–65/66

6–67. The shaft is supported by smooth journal bearings at A and B that only exert vertical reactions on the shaft. If d = 90 mm, determine the absolute maximum bending stress in the beam, and sketch the stress distribution acting over the cross section.

*6–68. The shaft is supported by smooth journal bearings at A and B that only exert vertical reactions on the shaft. Determine its smallest diameter d if the allowable bending stress is sallow = 180 MPa.

3 m 1.5 m

12 kN/m

Probs. 6–67/68

6–62. A box beam is constructed from four pieces of wood, glued together as shown. If the moment acting on the cross section is 10 kN # m, determine the stress at points A and B and show the results acting on volume elements located at these points.

20 mm 20 mm

250 mm

M ! 10 kN"m

160 mm

25 mm

25 mm B

Prob. 6–62

6–63. The beam is subjected to a moment of M = 30 lb · ft. Determine the bending stress acting at point A and B . Also, stetch a three-dimensional view of the stress distribution acting over the entire cross-sectional area.

3 in.

1 in.

M ! 30 lb"ft

Prob. 6–63

*6–64. The axle of the freight car is subjected to wheel loading of 20 kip. If it is supported by two journal bearings at C and D , determine the maximum bending stress developed at the center of the axle, where the diameter is 5.5 in.

C DA B

20 kip 20 kip 10 in. 10 in.

60 in.

Prob. 6–64

302 CHAPTER 6 BENDING

*6–72. Determine the absolute maximum bending stress in the 1.5-in.-diameter shaft which is subjected to the concentrated forces. The sleeve bearings at A and B support only vertical forces.

6–73. Determine the smallest allowable diameter of the shaft which is subjected to the concentrated forces. The sleeve bearings at A and B support only vertical forces, and the allowable bending stress is s allow = 22 ksi.

12 in.

18 in.

400 lb

15 in.

300 lb

Probs. 6–72/73

6–74. The pin is used to connect the three links together. Due to wear, the load is distributed over the top and bottom of the pin as shown on the free-body diagram. If the diameter of the pin is 0.40 in., determine the maximum bending stress on the cross-sectional area at the center section a – a . For the solution it is first necessary to determine the load intensities w 1 and w 2 .

800 lb

400 lb 400 lb

1 in.

0.40 in.

1.5 in.

1 in.

w2 w2

Prob. 6–74

6–75. The shaft is supported by a smooth thrust bearing at A and smooth journal bearing at D. If the shaft has the cross section shown, determine the absolute maximum bending stress in the shaft.

A C DB

3 kN 3 kN

0.75 m 0.75 m1.5 m

40 mm 25 mm

Prob. 6–75

6–69. Two designs for a beam are to be considered. Determine which one will support a moment of M = 150 kN # m with the least amount of bending stress. What is that stress?

200 mm

300 mm

(a) (b)

15 mm

30 mm

15 mm

200 mm

300 mm

30 mm

15 mm

30 mm

Prob. 6–69 6–70. The simply supported truss is subjected to the central distributed load. Neglect the effect of the diagonal lacing and determine the absolute maximum bending stress in the truss. The top member is a pipe having an outer diameter of 1 in. and thickness of 316 in., and the bottom member is a solid rod having a diameter of 12 in.

6 ft

5.75 in.

6 ft 6 ft

100 lb/ft

Prob. 6–70

6–71. The boat has a weight of 2300 lb and a center of gravity at G . If it rests on the trailer at the smooth contact A and can be considered pinned at B , determine the absolute maximum bending stress developed in the main strut of the trailer. Consider the strut to be a box-beam having the dimensions shown and pinned at C .

1 ft

B C

3 ft 1 ft

5 ft 4 ft D

1.75 in.

3 in. 1.75 in.

1.5 in.

Prob. 6–71

6.4 THE FLEXURE FORMULA 303

*6–80. If the beam is subjected to an internal moment of M = 100 kN # m, determine the bending stress developed at points A , B and C . Sketch the bending stress distribution on the cross section.

6–81. If the beam is made of material having an allowable tensile and compressive stress of (s allow ) t = 125 MPa and (s allow ) c = 150 MPa, respectively, determine the maximum allowable internal moment M that can be applied to the beam.

M 300 mm

150 mm

30 mm

150 mm

30 mm

Probs. 6–80/81

6–82. The shaft is supported by a smooth thrust bearing at A and smooth journal bearing at C . If d = 3 in., determine the absolute maximum bending stress in the shaft.

6–83. The shaft is supported by a smooth thrust bearing at A and smooth journal bearing at C . If the material has an allowable bending stress of s allow = 24 ksi, determine the required minimum diameter d of the shaft to the nearest 1>16 in.

A C D d

3 ft 3 ft

3600 lb

1800 lb3 ft

Probs. 6–82/83

*6–76. Determine the moment M that must be applied to the beam in order to create a maximum stress of 80 MPa. Also sketch the stress distribution acting over the cross section.

260 mm

20 mm 30 mm

300 mm

30 mm

20 mm

Prob. 6–76

6–77. If the beam is subjected to an internal moment of M = 2 kip # ft, determine the maximum tensile and compressive stress in the beam. Also, sketch the bending stress distribution on the cross section.

6–78. If the allowable tensile and compressive stress for the beam are (s allow ) t = 2 ksi and (s allow ) c = 3 ksi, respectively, determine the maximum allowable internal moment M that can be applied on the cross section.

6–79. If the beam is subjected to an internal moment of M = 2 kip # ft, determine the resultant force of the bending stress distribution acting on the top vertical board A .

4 in.

1 in.

3 in.

3 in. 3 in.

1 in.

Probs. 6–77/78/79

304 CHAPTER 6 BENDING

6–89. If the compound beam in Prob. 6–42 has a square cross section of side length a, determine the minimum value of a if the allowable bending stress is sallow = 150 MPa.

6–90. If the beam in Prob. 6–28 has a rectangular cross section with a width b and a height h, determine the absolute maximum bending stress in the beam.

6–91. Determine the absolute maximum bending stress in the 80-mm-diameter shaft which is subjected to the concentrated forces. The journal bearings at A and B only support vertical forces.

*6–92. Determine, to the nearest millimeter, the smallest allowable diameter of the shaft which is subjected to the concentrated forces. The journal bearings at A and B only support vertical forces. The allowable bending stress is sallow = 150 MPa.

0.5 m 0.6 m0.4 m

20 kN

A B

12 kN

Probs. 6–91/92

6–93. The wing spar ABD of a light plane is made from 2014–T6 aluminum and has a cross-sectional area of 1.27 in. 2 , a depth of 3 in., and a moment of inertia about its neutral axis of 2.68 in 4 . Determine the absolute maximum bending stress in the spar if the anticipated loading is to be as shown. Assume A , B , and C are pins. Connection is made along the central longitudinal axis of the spar.

2 ft DBA

C 3 ft 6 ft

80 lb/in.

Prob. 6–93

*6–84. If the intensity of the load w = 15 kN>m, determine the absolute maximum tensile and compressive stress in the beam.

6–85. If the material of the beam has an allowable bending stress of s allow = 150 MPa, determine the maximum allowable intensity w of the uniform distributed load.

6 m

150 mm

300 mm

A B

Probs. 6–84/85

6–86. Determine the absolute maximum bending stress in the 2-in.-diameter shaft which is subjected to the concentrated forces. The journal bearings at A and B only support vertical forces.

6–87. Determine the smallest allowable diameter of the shaft which is subjected to the concentrated forces. The journal bearings at A and B only support vertical forces. The allowable bending stress is sallow = 22 ksi.

15 in. 15 in. B

A 800 lb

30 in.

600 lb

Probs. 6–86/87

*6–88. If the beam has a square cross section of 9 in. on each side, determine the absolute maximum bending stress in the beam.

A B

8 ft 8 ft

800 lb/ft 1200 lb

Prob. 6–88

305

6–99. If the beam has a square cross section of 6 in. on each side, determine the absolute maximum bending stress in the beam.

A B

6 ft 6 ft

400 lb/ft

Prob. 6–99

*6–100. If d = 450 mm, determine the absolute maximum bending stress in the overhanging beam.

6–101. If wood used for the beam has an allowable bending stress of s allow = 6 MPa, determine the minimum dimension d of the beam’s cross sectional area to the nearest mm.

4 m

8 kN/m

2 m

12 kN

75 mm

25 mm 125 mm

25 mm

75 mm A

Probs. 6–100/101

6–94. The beam has a rectangular cross section as shown. Determine the largest load P that can be supported on its overhanging ends so that the bending stress does not exceed s max = 10 MPa.

6–95. The beam has the rectangular cross section shown. If P = 12 kN, determine the absolute maximum bending stress in the beam. Sketch the stress distribution acting over the cross section.

P P

1.5 m 1.5 m 1.5 m

150 mm

250 mm

Probs. 6–94/95

*6–96. A log that is 2 ft in diameter is to be cut into a rectangular section for use as a simply supported beam. If the allowable bending stress for the wood is s allow = 8 ksi, determine the required width b and height h of the beam that will support the largest load possible. What is this load?

6–97. A log that is 2 ft in diameter is to be cut into a rectangular section for use as a simply supported beam. If the allowable bending stress for the wood is s allow = 8 ksi, determine the largest load P that can be supported if the width of the beam is b = 8 in.

8 ft

2 ft

8 ft

Probs. 6–96/97

6–98. If the beam in Prob. 6–18 has a rectangular cross section with a width of 8 in. and a height of 16 in., determine the absolute maximum bending stress in the beam.

6–102. If the concentrated force P = 2 kN is applied at the free end of the overhanging beam, determine the absolute maxi mum tensile and compressive stress developed in the beam.

6–103. If the overhanging beam is made of wood having the allowable tensile and compressive stresses of (s allow ) t = 4 MPa and (s allow ) c = 5 MPa, determine the maximum concentrated force P that can applied at the free end.

2 m 1 m

150 mm

200 mm

25 mm

25 mm 25 mm

B A

Probs. 6–102/103

6.4 THE FLEXURE FORMULA

306 CHAPTER 6 BENDING

6.5 Unsymmetric Bending When developing the flexure formula we imposed a condition that the cross-sectional area be symmetric about an axis perpendicular to the neutral axis; furthermore, the resultant internal moment M acts along the neutral axis. Such is the case for the “T” or channel sections shown in Fig. 6–29 . These conditions, however, are unnecessary, and in this section we will show that the flexure formula can also be applied either to a beam having a cross-sectional area of any shape or to a beam having a resultant internal moment that acts in any direction.

Moment Applied About Principal Axis. Consider the beam’s cross section to have the unsymmetrical shape shown in Fig. 6–30 a . As in Sec. 6.4 , the right-handed x, y, z coordinate system is established such that the origin is located at the centroid C on the cross section, and the resultant internal moment M acts along the +z axis. We require the stress distribution acting over the entire cross-sectional area to have a zero force resultant, the resultant internal moment about the y axis to be zero, and the resultant internal moment about the z axis to equal M . * These three conditions can be expressed mathematically by considering the force acting on the differential element dA located at (0, y, z), Fig. 6–30 a . This force is dF = s dA , and therefore we have

FR = !Fx; 0 = - LAs dA (6–14) (MR)y = !My; 0 = - LAzs dA (6–15) (MR)z = !Mz; M = LAys dA (6–16)

(a)

dF ! sdA

x M

c y M

Bending-stress distribution (profile view)

(b)

smax

Fig. 6–30

M x

Neutral axis

Axis of symmetry

M x

Neutral axis

Axis of symmetry

Fig. 6–29

* The condition that moments about the y axis be equal to zero was not considered in Sec. 6.4 , since the bending-stress distribution was symmetric with respect to the y axis and such a distribution of stress automatically produces zero moment about the y axis. See Fig. 6–24 c .

6.5 UNSYMMETRIC BENDING 307

As shown in Sec. 6.4 , Eq. 6–14 is satisfied since the z axis passes through the centroid of the area. Also, since the z axis represents the neutral axis for the cross section, the normal stress will vary linearly from zero at the neutral axis, to a maximum at 0 y 0 = c, Fig. 6–30 b . Hence the stress distribution is defined by s = -(y >c)smax . When this equation is substituted into Eq. 6–16 and integrated, it leads to the flexure formula smax = Mc>I. When it is substituted into Eq. 6–15 , we get

0 = -smax

c LAyz dA which requires

LAyz dA = 0 This integral is called the product of inertia for the area. As indicated

in Appendix A , it will indeed be zero provided the y and z axes are chosen as principal axes of inertia for the area. For an arbitrarily shaped area, the orientation of the principal axes can always be determined, using either the inertia transformation equations or Mohr’s circle of inertia as explained in Appendix A , Secs. A.4 and A.5 . If the area has an axis of symmetry, however, the principal axes can easily be established since they will always be oriented along the axis of symmetry and perpendicular to it.

For example, consider the members shown in Fig. 6–31 . In each of these cases, y and z must define the principal axes of inertia for the cross section in order to satisfy Eqs. 6–14 through 6–16 . In Fig. 6–31 a the principal axes are located by symmetry, and in Figs. 6–31 b and 6–31 c their orientation is determined using the methods of Appendix A . Since M is applied about one of the principal axes ( z axis), the stress distribution is determined from the flexure formula, s = -My >Iz, and is shown for each case.

(a)

(b)

Fig. 6–31

Zee sectioned members are often in light- gage metal building construction to support roofs. To design them to support bending loads, it is necessary to determine their principal axes of inertia.

308 CHAPTER 6 BENDING

Moment Arbitrarily Applied. Sometimes a member may be loaded such that M does not act about one of the principal axes of the cross section. When this occurs, the moment should first be resolved into components directed along the principal axes, then the flexure formula can be used to determine the normal stress caused by each moment component. Finally, using the principle of superposition, the resultant normal stress at the point can be determined.

To show this, consider the beam to have a rectangular cross section and to be subjected to the moment M, Fig. 6–32 a . Here M makes an angle u with the maximum principal z axis, i.e., the axis of maximum moment of inertia for the cross section. We will assume u is positive when it is directed from the +z axis toward the +y axis, the minimum pricipal axis, as shown. Resolving M into components along the z and y axes, we have Mz = M cos u and My = M sin u, as shown in Figs. 6–32 b and 6–32 c . The normal-stress distributions that produce M and its components Mz and My are shown in Figs. 6–32 d , 6–32 e , and 6–32 f , where it is assumed that (sx)max 7 (s!x)max. By inspection, the maximum tensile and compressive stresses [(sx)max + (s!x)max] occur at two opposite corners of the cross section, Fig. 6–32 d .

Applying the flexure formula to each moment component in Figs. 6–32 b and 6–32 c , and adding the results algebraically, the resultant normal stress at any point on the cross section, Fig. 6–32 d , is then

s = - Mzy

Iz +

Myz

Iy (6–17)

Here,

s = the normal stress at the point y, z = the coordinates of the point measured from x, y, z axes

having their origin at the centroid of the cross-sectional area and forming a right-handed coordinate system The x axis is directed outward from the cross-section and the y and z axes represent respectively the principal axes of minimum and maximum moment of inertia for the area

My, Mz = the resultant internal moment components directed along the maximum z and minimum y principal axes. They are positive if directed along the +y and +z axes, otherwise they are negative. Or, stated another way, My = M sin u and Mz = M cos u, where u is measured positive from the +z axis toward the +y axis

Iy, Iz = the maximum and minimum principal moments of inertia calculated about the z and y axes, respectively. See Appendix A .

x z

(c)

x z

Mz ! Mcosu

My ! Msinu

(b)

(a)

! "

Fig. 6–32

6.5 UNSYMMETRIC BENDING 309

The x, y, z axes form a right-handed system, and the proper algebraic signs must be assigned to the moment components and the coordinates when applying this equation. When this is the case, the resulting stress will be tensile if it is positive and compressive if it is negative.

Orientation of the Neutral Axis. The angle a of the neutral axis in Fig. 6–32 d can be determined by applying Eq. 6–17 with s = 0, since by definition no normal stress acts on the neutral axis. We have

y = MyIz MzIy

Since Mz = M cos u and My = M sin u, then

y = ¢ Iz Iy

tan u≤z (6–18) This equation defines the neutral axis for the cross section. Since the slope of this line is tan a = y >z, then tan a =

Iz Iy

tan u (6–19)

Here it can be seen that unless Iz = Iy the angle u, defining the direction of the moment M, Fig. 6–32 a , will not equal a, the angle defining the inclination of the neutral axis, Fig. 6–32 d .

Important Points

• The flexure formula can be applied only when bending occurs about axes that represent the principal axes of inertia for the cross section. These axes have their origin at the centroid and are oriented along an axis of symmetry, if there is one, and perpendicular to it.

• If the moment is applied about some arbitrary axis, then the moment must be resolved into components along each of the principal axes, and the stress at a point is determined by superposition of the stress caused by each of the moment components.

(d)

(e)

(f)

[(sx)max ! (s¿x)max]

[(sx)max " (s¿x)max]

(sx)max

(s¿x)max

Fig. 6–32 (cont.)

310 CHAPTER 6 BENDING

EXAMPLE 6.15

The rectangular cross section shown in Fig. 6–33 a is subjected to a bending moment of M = 12 kN # m. Determine the normal stress developed at each corner of the section, and specify the orientation of the neutral axis.

SOLUTION Internal Moment Components. By inspection it is seen that the y and z axes represent the principal axes of inertia since they are axes of symmetry for the cross section. As required we have established the z axis as the principal axis for maximum moment of inertia. The moment is resolved into its y and z components, where

My = - 4 5

(12 kN # m) = -9.60 kN # m Mz =

3 5

(12 kN # m) = 7.20 kN # m Section Properties. The moments of inertia about the y and z axes are

Iy = 1

12 (0.4 m)(0.2 m)3 = 0.2667(10-3) m4

Iz = 1

12 (0.2 m)(0.4 m)3 = 1.067(10-3) m4

Bending Stress. Thus,

s = - Mzy

Iz +

Myz

sB = - 7.20(103) N # m(0.2 m)

1.067(10-3) m4 +

-9.60(103) N # m(-0.1 m) 0.2667(10-3) m4

= 2.25 MPa Ans.

sC = - 7.20(103) N # m(0.2 m)

1.067(10-3) m4 +

-9.60(103) N # m(0.1 m) 0.2667(10-3) m4

= -4.95 MPa Ans.

sD = - 7.20(103) N # m(-0.2 m)

1.067(10-3) m4 +

-9.60(103) N # m(0.1 m) 0.2667(10-3) m4

= -2.25 MPa Ans.

sE = - 7.20(103) N # m(-0.2 m)

1.067(10-3) m4 +

-9.60(103) N # m(-0.1 m) 0.2667(10-3) m4

= 4.95 MPa Ans.

The resultant normal-stress distribution has been sketched using these values, Fig. 6–33 b . Since superposition applies, the distribution is linear as shown.

6.5 UNSYMMETRIC BENDING 311

Orientation of Neutral Axis. The location z of the neutral axis ( NA ), Fig. 6–33 b , can be established by proportion. Along the edge BC, we require

2.25 MPa

z =

4.95 MPa (0.2 m - z)

0.450 - 2.25z = 4.95z z = 0.0625 m

In the same manner this is also the distance from D to the neutral axis in Fig. 6–33 b .

We can also establish the orientation of the NA using Eq. 6–19 , which is used to specify the angle a that the axis makes with the z or maximum principal axis. According to our sign convention, u must be measured from the +z axis toward the +y axis. By comparison, in Fig. 6–33 c , u = - tan-1 43 = -53.1! (or u = +306.9! ). Thus,

tan a = Iz Iy

tan u

tan a = 1.067(10-3) m4

0.2667(10-3) m4 tan(-53.1!)

a = -79.4! Ans.

This result is shown in Fig. 6–33 c . Using the value of z calculated above, verify, using the geometry of the cross section, that one obtains the same answer.

M ! 12 kN"m

(a)

0.2 m

0.1 m

3 4

(b)

0.2 m

2.25 MPa

4.95 MPa

2.25 MPa

(c)

B C

N y

M ! 12 kN"m

a ! #79.4$ a

u ! #53.1$

Fig. 6–33

312 CHAPTER 6 BENDING

EXAMPLE 6.16

The Z -section shown in Fig. 6–34 a is subjected to the bending moment of M = 20 kN # m. Using the methods of Appendix A (see Example A.4 or A.5 ), the principal axes y and z are oriented as shown, such that they represent the minimum and maximum principal moments of inertia, Iy = 0.960(10-3) m4 and Iz = 7.54(10-3) m4 , respectively. Determine the normal stress at point P and the orientation of the neutral axis.

SOLUTION For use of Eq. 6–19 , it is important that the z axis represent the principal axis for the maximum moment of inertia. (Note that most of the area is located furthest from this axis.)

Internal Moment Components. From Fig. 6–34 a ,

My = 20 kN # m sin 57.1! = 16.79 kN # m Mz = 20 kN # m cos 57.1! = 10.86 kN # m

Bending Stress. The y and z coordinates of point P must be determined first. Note that the y ", z " coordinates of P are (-0.2 m, 0.35 m). Using the colored triangles from the construction shown in Fig. 6–34 b , we have

yP = -0.35 sin 32.9! - 0.2 cos 32.9! = -0.3580 m

zP = 0.35 cos 32.9! - 0.2 sin 32.9! = 0.1852 m

Applying Eq. 6–17 ,

sP = - Mz yP

Iz +

My zP Iy

= - (10.86(103) N # m)(-0.3580 m)

7.54(10-3) m4 +

(16.79(103) N # m)(0.1852 m) 0.960(10-3) m4

= 3.76 MPa Ans.

Orientation of Neutral Axis. The angle u = 57.1! is shown in Fig. 6–34 a . Thus,

tan a = J 7.54(10-3) m4 0.960(10-3) m4

R tan 57.1! a = 85.3! Ans.

The neutral axis is oriented as shown in Fig. 6–34 b .

(a)

400 mm

100 mm

300 mm

M ! 20 kN"m

100 mm

32.9#

z¿

y¿ u ! 57.1#

zz¿

(b)

0.350 m0.200 m 32.9!

32.9!

y¿

a " 85.3!

Fig. 6–34

6.5 UNSYMMETRIC BENDING 313

250 mm

125 mmB

M! 850 N"m C

125 mm

Prob. 6–104

*6–104. The member has a square cross section and is subjected to a resultant internal bending moment of M = 850 N # m as shown. Determine the stress at each corner and sketch the stress distribution produced by M . Set u = 45°.

F6–14. Determine the bending stress developed at corners A and B . What is the orientation of the neutral axis?

4 35

150 mm

100 mm

50 kN!m

F6–14

F6–15. Determine the maximum stress in the beam’s cross section.

6 in.

4 in. 50 lb!ft30"

F6–15

FUNDAMENTAL PROBLEMS

6–105. The member has a square cross section and is subjected to a resultant internal bending moment of M = 850 N # m as shown. Determine the stress at each corner and sketch the stress distribution produced by M . Set u = 30°.

250 mm

125 mmB

M! 850 N"m C

125 mm

Prob. 6–105

PROBLEMS

314 CHAPTER 6 BENDING

6–109. The box beam is subjected to the internal moment of M = 4 kN # m, which is directed as shown. Determine the maximum bending stress developed in the beam and the orientation of the neutral axis.

6–110. If the wood used for the box beam has an allowable bending stress of (s allow ) = 6 MPa, determine the maximum allowable internal moment M that can be applied to the beam.

150 mm

25 mm 50 mm

50 mm

45!

25 mm

Probs. 6–109/110

6–111. If the beam is subjected to the internal moment of M = 1200 kN # m, determine the maximum bending stress acting on the beam and the orientation of the neutral axis.

*6–112. If the beam is made from a material having an allowable tensile and compressive stress of (s allow ) t = 125 MPa and (s allow ) c = 150 MPa, respectively, determine the maximum allowable internal moment M that can be applied to the beam.

150 mm

300 mm

150 mm

30!

6–106. Consider the general case of a prismatic beam subjected to bending-moment components My and Mz , as shown, when the x , y , z axes pass through the centroid of the cross section. If the material is linear-elastic, the normal stress in the beam is a linear function of position such that s = a + by + cz . Using the equilibrium conditions 0 = 1As dA , My = 1Azs dA , Mz = 1A - ys dA , determine the constants a , b , and c , and show that the normal stress can be determined from the equation s = [- (MzIy + MyIyz)y + (MyIz + MzIyz)z]>(IyIz - Iyz 2), where the moments and products of inertia are defined in Appendix A .

z x

dA My

Prob. 6–106

6–107. If the beam is subjected to the internal moment of M = 2 kN # m, determine the maximum bending stress developed in the beam and the orientation of the neutral axis.

*6–108. If the wood used for the T-beam has an allowable tensile and compressive stress of (s allow ) t = 4 MPa and (s allow ) c = 6 MPa, respectively, determine the maximum allowable internal moment M that can be applied to the beam.

200 mm

50 mm

100 mm

60°

Probs. 6–107/108 Probs. 6–111/112

6.5 UNSYMMETRIC BENDING 315

*6–116. For the section, Iy! = 31.7(10-6) m4, Iz! = 114(10-6) m4, Iy!z! = 15.1(10-6) m4. Using the techniques outlined in Appendix A , the member’s cross-sectional area has principal moments of inertia of Iy = 29.0(10-6) m4 and Iz = 117(10-6) m4 , computed about the principal axes of inertia y and z , respectively. If the section is subjected to a moment of M = 2500 N # m directed as shown, determine the stress produced at point A , using Eq. 6–17 .

6–117. Solve Prob. 6–116 using the equation developed in Prob. 6–106.

60 mm

60 mm 60 mm

140 mm

80 mm

z¿

y¿

10.10"

M # 2500 N$m C

6–118. If the applied distributed loading of w = 4 kN>m can be assumed to pass through the centroid of the beam’s cross sectional area, determine the absolute maximum bending stress in the joist and the orientation of the neutral axis. The beam can be considered simply-supported at A and B .

6–119. Determine the maximum allowable intensity w of the uniform distributed load that can be applied to the beam. Assume w passes through the centroid of the beam’s cross sectional area and the beam is simply supported at A and B . The beam is made of material having an allowable bending stress of s allow = 165 MPa.

6 m (6 m)

15!

100 mm 100 mm

100 mm

15 mm

15 mm 10 mm

6–113. The board is used as a simply supported floor joist. If a bending moment of M = 800 lb · ft is applied 3° from the z axis, determine the stress developed in the board at the corner A . Compare this stress with that developed by the same moment applied along the z axis (u = 0°). What is the angle a for the neutral axis when u = 3°? Comment: Normally, floor boards would be nailed to the top of the beams so that u ≈ 0° and the high stress due to misalignment would not occur.

M # 800 lb$ft 6 in.

2 in.

u # 3"

6–114. The T-beam is subjected to a bending moment of M = 150 kip # in. directed as shown. Determine the maximum bending stress in the beam and the orientation of the neutral axis. The location y of the centroid, C , must be determined.

8 in.

2 in.

6 in.

2 in.

–y

M # 150 kip$in.

60"

6 in.

6–115. The beam has a rectangular cross section. If it is subjected to a bending moment of M = 3500 N # m directed as shown, determine the maximum bending stress in the beam and the orientation of the neutral axis.

150 mm

150 mmz

M # 3500 N$m

30"

150 mm

Prob. 6–113

Prob. 6–114

Prob. 6–115

Probs. 6–116/117

Probs. 6–118/119

316 CHAPTER 6 BENDING

*6.6 Composite Beams Beams constructed of two or more different materials are referred to as composite beams. For example, a beam can be made of wood with straps of steel at its top and bottom, Fig. 6–35 . Engineers purposely design beams in this manner in order to develop a more efficient means for supporting loads.

Since the flexure formula was developed only for beams having homogeneous material, this formula cannot be applied directly to determine the normal stress in a composite beam. In this section, however, we will develop a method for modifying or “transforming” a composite beam’s cross section into one made of a single material. Once this has been done, the flexure formula can then be used for the stress analysis.

To explain how to do this we will consider a composite beam made of two materials, 1 and 2, bonded together and having the cross-sectional areas shown in Fig. 6–36 a . If a bending moment is applied to this beam, then, like one that is homogeneous, the total cross-sectional area will remain plane after bending, and hence the displacement is continuous so that normal strains will vary linearly from zero at the neutral axis to a maximum in the material located farthest from this axis, Fig. 6–36 b . Provided the material is linear elastic, then at any point the normal stress in material 1 is determined from s = E1P , and for material 2 the stress distribution is found from s = E2P. Assuming material 1 is stiffer than material 2, then E1 7 E2 and so the stress distribution will look like that shown in Fig. 6–36 c or 6–36 d . In particular, notice the jump in stress that occurs at the juncture of the two materials. Here the strain is the same, but since the modulus of elasticity for the materials suddenly changes, so does the stress. It is possible to determine the location of the neutral axis and the maximum stress based on a trial-and-error procedure.

Steel plates

Fig. 6–35

6.6 COMPOSITE BEAMS 317

This requires satisfying the conditions that the stress distribution produces a zero resultant force on the cross section and the moment of the stress distribution about the neutral axis must be equal to M.

A simpler way to satisfy these two conditions is to use the transformed section method, which transforms the beam into one made of a single material. For example, if the beam is thought to consist entirely of the less stiff material 2, then the cross section will look like that shown in Fig. 6–36 e . Here the height h of the beam remains the same, since the strain distribution in Fig. 6–36 b must be preserved. However, the upper portion of the beam must be widened in order to carry a load equivalent to that carried by the stiffer material 1 in Fig. 6–36 d . The necessary width can be determined by considering the force d F acting on an area dA = dz dy of the beam in Fig. 6–36 a . It is dF = s dA = (E1P) dz dy. Assuming the width of a corresponding element of height dy in Fig. 6–36 e is n dz, then dF! = s!dA ! = (E2P)n dz dy. Equating these forces, so that they produce the same moment about the z (neutral) axis, we have

E1P dz dy = E2Pn dz dy or

n = E1 E2

(6–20)

Less stiff material

M dy

y h

Stiff material

(a)

M x

Normal strain variation (profile view)

(b)

M x

Bending stress variation (profile view)

(c)

z x

Bending stress variation

(d)

Fig. 6–36

318 CHAPTER 6 BENDING

This dimensionless number n is called the transformation factor. It indicates that the cross section, having a width b on the original beam, Fig. 6–36 a , must be increased in width to b2 = nb in the region where material 1 is being transformed into material 2, Fig. 6–36 e .

In a similar manner, if the less stiff material 2 is transformed into the stiffer material 1, the cross section will look like that shown in Fig. 6–36 f . Here the width of material 2 has been changed to b1 = n!b, where n! = E2>E1. In this case the transformation factor n! will be less than one since E1 7 E2 . In other words, we need less of the stiffer material to support the moment.

Once the beam has been transformed into one having a single material, the normal-stress distribution over the transformed cross section will be linear as shown in Fig. 6–36 g or Fig. 6–36 h . Consequently, the centroid (neutral axis) and moment of inertia for the transformed area can be determined and the flexure formula applied in the usual manner to determine the stress at each point on the transformed beam. The stress in the transformed beam will be equivalent to the stress in the same material of the actual beam; however, the stress found on the transformed material has to be multiplied by the transformation factor n (or n! ), since the area of the transformed material, dA ! = n dz dy, is n times the area of actual material dA = dz dy. That is,

dF = s dA = s!dA !

s dz dy = s!n dz dy s = ns! (6–21)

Example 6.17 numerically illustrates application of the transformed section method.

(e) Beam transformed to material 2

2 ndz

b2 ! nb

b1 ! n¿b

(f)

Beam transformed to material 1

(g)

Bending-stress variation for beam transformed to material 2

(h)

Bending-stress variation for beam transformed to material 1

Fig. 6–36 (cont.)

Important Points

• Composite beams are made from different materials in order to efficiently carry a load. Application of the flexure formula requires the material to be homogeneous, and so the cross section of the beam must be transformed into a single material if this formula is to be used to calculate the bending stress.

• The transformation factor n is a ratio of the moduli of the different materials that make up the beam. Used as a multiplier, it converts the width of the cross section of the composite beam into a beam made from a single material so that this beam has the same strength as the composite beam. Stiff material will thus be replaced by more of the softer material and vice versa.

• Once the stress in the transformed material is determined, then it must be multiplied by the transformation factor to obtain the stress in the actual beam.

6.7 REINFORCED CONCRETE BEAMS 319

*6.7 Reinforced Concrete Beams All beams subjected to pure bending must resist both tensile and compressive stresses. Concrete, however, is very susceptible to cracking when it is in tension, and therefore by itself it will not be suitable for resisting a bending moment. * In order to circumvent this shortcoming, engineers place steel reinforcing rods within a concrete beam at a location where the concrete is in tension, Fig. 6–37 a . To be most effective, these rods are located farthest from the beam’s neutral axis, so that the moment created by the forces developed in them is greatest about the neutral axis. Furthermore, the rods are required to have some concrete coverage to protect them from corrosion or loss of strength in the event of a fire. Codes used for actual reinforced concrete design assume the concrete will not support any tensile loading, since the possible cracking of concrete is unpredictable. As a result, the normal stress distribution acting on the cross-sectional area of a reinforced concrete beam is assumed to look like that shown in Fig. 6–37 b .

The stress analysis requires locating the neutral axis and determining the maximum stress in the steel and concrete. To do this, the area of steel A st is first transformed into an equivalent area of concrete using the transformation factor n = Est>Econc . This ratio, which gives n 7 1, requires a “greater” amount of concrete to replace the steel. The transformed area is nA st and the transformed section looks like that shown in Fig. 6–37 c . Here d represents the distance from the top of the beam to the thin strip of (transformed) steel, b is the beam’s width, and h! is the yet unknown distance from the top of the beam to the neutral axis. To obtain h! , we require the centroid C of the cross-sectional area of the transformed section to lie on the neutral axis, Fig. 6–37 c . With reference to the neutral axis, therefore, the moment of the two areas together, "y!A , must be zero, since y = "y!A >"A = 0. Thus,

bh!a h! 2 b - nA st (d - h!) = 0

b 2

h!2 + nA st h! - nA st d = 0

Once h! is obtained from this quadratic equation, the solution proceeds in the usual manner for obtaining the stress in the beam. Example 6.18 numerically illustrates application of this method.

(a)

(b)

Concrete assumed cracked within this region.

(c)

b dh¿

n Ast

Fig. 6–37

* Inspection of its particular stress-strain diagram in Fig. 3–11 reveals that concrete can be 12.5 times stronger in compression than in tension.

320 CHAPTER 6 BENDING

EXAMPLE 6.17

A composite beam is made of wood and reinforced with a steel strap located on its bottom side. It has the cross-sectional area shown in Fig. 6–38 a . If the beam is subjected to a bending moment of M = 2 kN # m, determine the normal stress at points B and C. Take Ew = 12 GPa and Est = 200 GPa.

(a)

150 mm

20 mm

B 150 mm

M ! 2 kN"m

_ y

9 mm

(b)

150 mm

20 mm

B¿

150 mm

SOLUTION Section Properties. Although the choice is arbitrary, here we will transform the section into one made entirely of steel. Since steel has a greater stiffness than wood (Est 7 Ew), the width of the wood is reduced to an equivalent width for steel. For this to be the case, n = Ew>Est, so that

bst = nbw = 12 GPa 200 GPa

(150 mm) = 9 mm

The transformed section is shown in Fig. 6–38 b . The location of the centroid (neutral axis), calculated from a

reference axis located at the bottom of the section, is

y = !yA

!A =

[0.01 m](0.02 m)(0.150 m) + [0.095 m](0.009 m)(0.150 m) 0.02 m(0.150 m) + 0.009 m(0.150 m) = 0.03638 m

The moment of inertia about the neutral axis is therefore

Fig. 6–38

INA = c 112 (0.150 m)(0.02 m)3 + (0.150 m)(0.02 m)(0.03638 m - 0.01 m)2 d + c 1

12 (0.009 m)(0.150 m)3 + (0.009 m)(0.150 m)(0.095 m - 0.03638 m)2 d

= 9.358(10-6) m4

6.7 REINFORCED CONCRETE BEAMS 321

7.78 MPa

28.6 MPa

3.50 MPa

B¿

(c)

M ! 2 kN"m

7.78 MPa

3.50 MPa

0.210 MPa

1.71 MPa

(d)

M ! 2 kN"m

Fig. 6–38 (cont.)

Normal Stress. Applying the flexure formula, the normal stress at B ! and C is

sB! = 2(103) N # m(0.170 m - 0.03638 m)

9.358(10-6) m4 = 28.6 MPa

sC = 2(103) N # m(0.03638 m)

9.358(10-6) m4 = 7.78 MPa Ans.

The normal-stress distribution on the transformed (all steel) section is shown in Fig. 6–38 c .

The normal stress in the wood at B in Fig. 6–38 a , is determined from Eq. 6–21 ; that is,

sB = nsB! = 12 GPa 200 GPa

(28.56 MPa) = 1.71 MPa Ans.

Using these concepts, show that the normal stress in the steel and the wood at the point where they are in contact is sst = 3.50 MPa and sw = 0.210 MPa, respectively. The normal-stress distribution in the actual beam is shown in Fig. 6–38 d .

322 CHAPTER 6 BENDING

The reinforced concrete beam has the cross-sectional area shown in Fig. 6–39 a . If it is subjected to a bending moment of M = 60 kip # ft, determine the normal stress in each of the steel reinforcing rods and the maximum normal stress in the concrete. Take Est = 29(103) ksi and Econc = 3.6(103) ksi.

SOLUTION Since the beam is made from concrete, in the following analysis we will neglect its strength in supporting a tensile stress.

Section Properties. The total area of steel, A st = 2[p(0.5 in.)2] = 1.571 in2 will be transformed into an equivalent area of concrete, Fig. 6–39 b . Here

A ! = nA st = 29(103) ksi

3.6(103) ksi (1.571 in2) = 12.65 in2

We require the centroid to lie on the neutral axis. Thus "y!A = 0, or

12 in. (h!) h! 2

- 12.65 in2(16 in. - h!) = 0

h!2 + 2.11h! - 33.7 = 0

Solving for the positive root,

h! = 4.85 in.

Using this value for h!, the moment of inertia of the transformed section about the neutral axis is

I = c 1 12

(12 in.)(4.85 in.)3 + 12 in. (4.85 in.)a 4.85 in. 2

b2 d + 12.65 in2(16 in. - 4.85 in.)2 = 2029 in4

Normal Stress. Applying the flexure formula to the transformed section, the maximum normal stress in the concrete is

(sconc)max = [60 kip # ft(12 in.>ft)](4.85 in.)

2029 in4 = 1.72 ksi Ans.

The normal stress resisted by the “concrete” strip, which replaced the steel, is

s!conc = [60 kip # ft(12 in.>ft)](16 in. - 4.85 in.)

2029 in4 = 3.96 ksi

The normal stress in each of the two reinforcing rods is therefore

sst = ns!conc = ¢ 29(103) ksi3.6(103) ksi≤3.96 ksi = 31.9 ksi Ans. The normal-stress distribution is shown graphically in Fig. 6–39 c .

EXAMPLE 6.18

12 in.

18 in.

2 in.1-in.-diameter bars

60 kip!ft

(a)

C AN

12 in.

h¿ 16 in.

A¿ " 12.65 in2

(b)

(c)

1.72 ksi

31.9 ksi

4.85 in.

Fig. 6–39

6.8 CURVED BEAMS 323

*6.8 Curved Beams The flexure formula applies to a straight member, since it was shown that the normal strain within the member varies linearly from the neutral axis. If the member is curved, however, the strain will not be linear, and so we must develop another method to describe the stress distribution. In this section we will consider the analysis of a curved beam, that is, a member that has a curved axis and is subjected to bending. Typical examples include hooks and chain links. In all these cases, the members are not slender, but rather have a sharp curve, and their cross-sectional dimensions are large compared with their radius of curvature.

The following analysis assumes that the cross section is constant and has an axis of symmetry that is perpendicular to the direction of the applied moment M, Fig. 6–40 a . Also, the material is homogeneous and isotropic, and it behaves in a linear-elastic manner when the load is applied. Like the case of a straight beam, we will also assume that the cross sections of the member remain plane after the moment is applied. Furthermore, any distortion of the cross section within its own plane will be neglected.

To perform the analysis, three radii, extending from the center of curvature O! of the member, are identified in Fig. 6–40 a . Here r references the known location of the centroid for the cross-sectional area, R references the yet unspecified location of the neutral axis, and r locates the arbitrary point or area element dA on the cross section.

A A

This crane hook represents a typical example of a curved beam.

M MR

eN A

(a)

Centroid Neutral axis

Area element dA _ r

_ r

O ¿

Fig. 6–40

324 CHAPTER 6 BENDING

If we isolate a differential segment of the beam, Fig. 6–40 b , the stress tends to deform the material such that each cross section will rotate through an angle du>2 . The normal strain P in the strip (or line) of material located at r will now be determined. This strip has an original length r du, Fig. 6–40 b . However, due to the rotations du>2 the strip’s total change in length is du(R - r). Consequently, P = du(R - r)>r du . If we let k = du>du, which is the same for any particular strip, we have P = k(R - r)>r. Unlike the case of straight beams, here it can be seen that the normal strain is a nonlinear function of r, in fact it varies in a hyperbolic fashion. This occurs even though the cross section of the beam remains plane after deformation. If the material remains linearly elastic then s = EP and so

s = Ek aR - r r

b (6–22) This variation is also hyperbolic, and since it has now been established, we can determine the location of the neutral axis and relate the stress distribution to the resultant internal moment M.

To obtain the location R of the neutral axis, we require the resultant internal force caused by the stress distribution acting over the cross section to be equal to zero; i.e.,

FR = !Fx; LAs dA = 0

LAEk aR - rr b dA = 0 Since Ek and R are constants, we have

R LA dA r

- LA dA = 0 Solving for R yields

R = A

LA dA r

(6–23)

Here

R = the location of the neutral axis, specified from the center of curvature O" of the member

A = the cross-sectional area of the member r = the arbitrary position of the area element dA on the cross section,

specified from the center of curvature O" of the member

The integral in Eq. 6–23 has been evaluated for various cross-sectional geometries, and the results for some common cross sections are listed in Table 6–1 .

Shape

_ r

! c 22p

_ r

_ r2!

2pb a

!bln r2 r1

b r2 (r2 ! r1)

b ln r2 r1

! a 2_r

_ r2!

dA r1A

(b)

M (R ! r)

(R ! r)

rdu

2 du

2 (R ! r)

O¿

Fig. 6–40 (cont.)

TABLE 6–1

6.8 CURVED BEAMS 325

In order to relate the stress distribution to the resultant bending moment, we require the resultant internal moment to be equal to the moment of the stress distribution calculated about the neutral axis. From Fig. 6–40 a , the stress s, acting on the area element dA and located a distance y from the neutral axis, creates a moment about the neutral axis of dM = y(s dA ). For the entire cross section, we require M = 1ys dA . Since y = R - r, and s is defined by Eq. 6–22 , we have

M = LA(R - r)Ek aR - rr b dA Expanding, realizing that Ek and R are constants, then

M = Ek ¢R 2L A dAr - 2R LA dA + LAr dA ≤ The first integral is equivalent to A >R as determined from Eq. 6–23 , and the second integral is simply the cross-sectional area A. Realizing that the location of the centroid of the cross section is determined from r = 1r dA >A , the third integral can be replaced by rA . Thus,

M = EkA (r - R)

Finally, solving for Ek in Eq. 6–22 , substituting into the above equation, and solving for s, we have

s = M(R - r) Ar (r - R)

(6–24)

Here

s = the normal stress in the member M = the internal moment, determined from the method of sections

and the equations of equilibrium and calculated about the neutral axis for the cross section. This moment is positive if it tends to increase the member’s radius of curvature, i.e., it tends to straighten out the member.

A = the cross-sectional area of the member R = the distance measured from the center of curvature to the

neutral axis, determined from Eq. 6–23 r = the distance measured from the center of curvature to the

centroid of the cross section r = the distance measured from the center of curvature to the point

where the stress s is to be determined

326 CHAPTER 6 BENDING

From Fig. 6–40 a , r = R - y. Also, the constant and usually very small distance between the neutral axis and the centroid is e = r - R. When these results are substituted into Eq. 6–24 , we can also write

s = My

Ae(R - y) (6–25)

These two equations represent two forms of the so-called curved-beam formula, which like the flexure formula can be used to determine the normal-stress distribution in a curved member. This distribution is, as previously stated, hyperbolic; an example is shown in Figs. 6–40 c and 6–40 d . Since the stress acts along the circumference of the beam, it is sometimes called circumferential stress. Note that due to the curvature of the beam, the circumferential stress will create a corresponding component of radial stress, so called since this component acts in the radial direction. To show how it is developed, consider the free-body diagram of the segment shown in Fig. 6–40 e . Here the radial stress sr is necessary since it creates the force dFr that is required to balance the two components of circumferential forces dF which act along the line O!B.

Sometimes the radial stresses within curved members may become significant, especially if the member is constructed from thin plates and has, for example, the shape of an I-section. In this case the radial stress can become as large as the circumferential stress, and consequently the member must be designed to resist both stresses. For most cases, however, these stresses can be neglected, especially if the member has a solid section. Here the curved-beam formula gives results that are in very close agreement with those determined either by experiment or by a mathematical analysis based on the theory of elasticity.

The curved-beam formula is normally used when the curvature of the member is very pronounced, as in the case of hooks or rings. However, if the radius of curvature is greater than five times the depth of the member, the flexure formula can normally be used to determine the stress. For example, for rectangular sections for which this ratio equals 5, the maximum normal stress, when determined by the flexure formula, will be about 7% less than its value when determined by the curved-beam formula. This error is further reduced when the radius of curvature-to-depth ratio is more than 5. *

* See, for example, Boresi, A. P., et al., Advanced Mechanics of Materials , John Wiley & Sons, New York.

Bending stress variation (profile view)

(c)

smax

(d)

smax

(e)

s s dF

dFr

O¿

Fig. 6–40 (cont.)

6.8 CURVED BEAMS 327

Important Points

• The curved-beam formula should be used to determine the circumferential stress in a beam when the radius of curvature is less than five times the depth of the beam.

• Due to the curvature of the beam, the normal strain in the beam does not vary linearly with depth as in the case of a straight beam. As a result, the neutral axis generally does not pass through the centroid of the cross section.

• The radial stress component caused by bending can generally be neglected, especially if the cross section is a solid section and not made from thin plates.

Procedure for Analysis

In order to apply the curved-beam formula the following procedure is suggested.

Section Properties. • Determine the cross-sectional area A and the location of the

centroid, r, measured from the center of curvature. • Find the location of the neutral axis, R, using Eq. 6–23 or

Table 6–1 . If the cross-sectional area consists of n “composite” parts, determine 1dA >r for each part. Then, from Eq. 6–23 , for the entire section, R = !A >!(1dA >r). In all cases, R 6 r.

Normal Stress. • The normal stress located at a point r away from the center of

curvature is determined from Eq. 6–24 . If the distance y to the point is measured from the neutral axis, then find e = r - R and use Eq. 6–25 .

• Since r - R generally produces a very small number , it is best to calculate r and R with sufficient accuracy so that the subtraction leads to a number e having at least four significant figures.

• If the stress is positive it will be tensile, whereas if it is negative it will be compressive.

• The stress distribution over the entire cross section can be graphed, or a volume element of the material can be isolated and used to represent the stress acting at the point on the cross section where it has been calculated.

328 CHAPTER 6 BENDING

EXAMPLE 6.19

The curved bar has a cross-sectional area shown in Fig. 6–41 a . If it is subjected to bending moments of 4 kN # m, determine the maximum normal stress developed in the bar.

SOLUTION Internal Moment. Each section of the bar is subjected to the same resultant internal moment of 4 kN # m. Since this moment tends to decrease the bar’s radius of curvature, it is negative. Thus, M = -4 kN # m.

Section Properties. Here we will consider the cross section to be composed of a rectangle and triangle. The total cross-sectional area is

!A = (0.05 m)2 + 1 2

(0.05 m)(0.03 m) = 3.250(10-3) m2

The location of the centroid is determined with reference to the center of curvature, point O", Fig. 6–41 a .

r = ! r!A !A

= [0.225 m](0.05 m)(0.05 m) + [0.260 m] 12(0.050 m)(0.030 m)

3.250(10-3) m2

= 0.23308 m

200 mm

250 mm

200 mm

50 mm

30 mm

50 mm 280 mm

4 kN·m 4 kN·m

–r

(a)

O ¿

Fig. 6–41

6.8 CURVED BEAMS 329

We can find 1A dA >r for each part using Table 6–1 . For the rectangle, LA

dA r

= 0.05 ma ln 0.250 m 0.200 m

b = 0.011157 m And for the triangle,

LA dA r

= (0.05 m)(0.280 m)

(0.280 m - 0.250 m) a ln 0.280 m0.250 mb - 0.05 m = 0.0028867 m Thus the location of the neutral axis is determined from

R = !A

! LAdA >r = 3.250(10-3) m2

0.011157 m + 0.0028867 m = 0.23142 m

Note that R 6 r as expected. Also, the calculations were performed with sufficient accuracy so that (r - R) = 0.23308 m - 0.23142 m = 0.00166 m is now accurate to three significant figures.

Normal Stress. The maximum normal stress occurs either at A or B. Applying the curved-beam formula to calculate the normal stress at B, rB = 0.200 m, we have

sB = M(R - rB) ArB(r - R)

= (-4 kN # m)(0.23142 m - 0.200 m) 3.250(10-3) m2(0.200 m)(0.00166 m)

= -116 MPa

At point A, rA = 0.280 m and the normal stress is

sA = M(R - rA) ArA(r - R)

= (-4 kN # m)(0.23142 m - 0.280 m) 3.250(10-3) m2(0.280 m)(0.00166 m)

= 129 MPa Ans.

By comparison, the maximum normal stress is at A. A two-dimensional representation of the stress distribution is shown in Fig. 6–41 b .

(b)

129 MPa

116 MPa

4 kN!m

Fig. 6–41 (cont.)

330 CHAPTER 6 BENDING

6.9 Stress Concentrations The flexure formula cannot be used to determine the stress distribution within regions of a member where the cross-sectional area suddenly changes, since the normal-stress and strain distributions at the section become nonlinear. The results can only be obtained through experiment or, in some cases, by using the theory of elasticity. Common discontinuities include members having notches on their surfaces, Fig. 6–42 a , holes for passage of fasteners or other items, Fig. 6–42 b , or abrupt changes in the outer dimensions of the member’s cross section, Fig. 6–42 c . The maximum normal stress at each of these discontinuities occurs at the section taken through the smallest cross-sectional area.

For design, it is generally important to only know the maximum normal stress developed at these sections, not the actual stress distribution. As in the previous cases of axially loaded bars and torsionally loaded shafts, we can obtain the maximum normal stress due to bending using a stress-concentration factor K. For example, Fig. 6–43 gives values of K for a flat bar that has a change in cross section using shoulder fillets. To use this graph simply find the geometric ratios w >h

(a)

(b)

(c)

Fig. 6–42

0 0.1 0.2

w h

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2.0

1.9

1.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

r h

w h

! 4

w h

! 3

w h

! 1.5

w h

! 1.1

w h

! 1.25

Fig. 6–43

w h t

0.1 0.2 0.3 0.4 0.5

3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

r h

b r ! 4

b r ! 2

b r ! 1

b r ! 0.5

Fig. 6–44

6.9 STRESS CONCENTRATIONS 331

and r>h and then find the corresponding value of K for a particular geometry. Once K is obtained, the maximum bending stress shown in Fig. 6–45 is determined using

smax = K Mc

I (6–26)

In the same manner, Fig. 6–44 can be used if the discontinuity consists of circular grooves or notches.

Like axial load and torsion, stress concentration for bending should always be considered when designing members made of brittle materials or those that are subjected to fatigue or cyclic loadings. Also, realize that stress-concentration factors apply only when the material is subjected to elastic behavior. If the applied moment causes yielding of the material, as is the case with ductile materials, the stress becomes redistributed throughout the member, and the maximum stress that results will be lower than that determined using stress-concentration factors. This phenomenon is discussed further in the next section.

smax

Fig. 6–45

Stress concentrations caused by bending occur at the sharp corners of this window lintel and are responsible for the crack at the corner.

Important Points

• Stress concentrations occur at points of sudden cross-sectional change, caused by notches and holes, because here the stress and strain become nonlinear. The more severe the change, the larger the stress concentration.

• For design or analysis, the maximum normal stress occurs on the smallest cross-sectional area. This stress can be obtained by using a stress concentration factor, K, that has been determined through experiment and is only a function of the geometry of the member.

• Normally, the stress concentration in a ductile material subjected to a static moment will not have to be considered in design; however, if the material is brittle, or subjected to fatigue loading, then stress concentrations become important.

332 CHAPTER 6 BENDING

EXAMPLE 6.20

The transition in the cross-sectional area of the steel bar is achieved using shoulder fillets as shown in Fig. 6–46 a . If the bar is subjected to a bending moment of 5 kN # m, determine the maximum normal stress developed in the steel. The yield stress is sY = 500 MPa.

SOLUTION The moment creates the largest stress in the bar at the base of the fillet, where the cross-sectional area is smallest. The stress-concentration factor can be determined by using Fig. 6–43 . From the geometry of the bar, we have r = 16 mm, h = 80 mm, w = 120 mm. Thus,

r h

= 16 mm 80 mm

= 0.2 w h

= 120 mm 80 mm

= 1.5

These values give K = 1.45. Applying Eq. 6–26 , we have

smax = K Mc

I = (1.45)

(5(103) N # m)(0.04 m)3 112(0.020 m)(0.08 m)34 = 340 MPa Ans. This result indicates that the steel remains elastic since the stress is below the yield stress (500 MPa).

NOTE: The normal-stress distribution is nonlinear and is shown in Fig. 6–46 b . Realize, however, that by Saint-Venant’s principle, Sec. 4.1 , these localized stresses smooth out and become linear when one moves (approximately) a distance of 80 mm or more to the right of the transition. In this case, the flexure formula gives smax = 234 MPa, Fig. 6–46 c . Also note that the choice of a larger-radius fillet will significantly reduce smax, since as r increases in Fig. 6–43 , K will decrease.

120 mm r ! 16 mm

80 mm

20 mm (a)

5 kN"m

(b)

340 MPa

5 kN"m

(c)

234 MPa

5 kN"m

Fig. 6–46

6.9 STRESS CONCENTRATIONS 333

PROBLEMS

*6–120. The composite beam is made of steel ( A ) bonded to brass ( B ) and has the cross section shown. If it is subjected to a moment of M = 6.5 kN # m, determine the maximum bending stress in the brass and steel. Also, what is the stress in each material at the seam where they are bonded together? E br = 100 GPa, E st = 200 GPa.

6–121. The composite beam is made of steel ( A ) bonded to brass ( B ) and has the cross section shown. If the allowable bending stress for the steel is (s allow ) st = 180 MPa, and for the brass (s allow ) br = 60 MPa, determine the maximum moment M that can be applied to the beam. E br = 100 GPa, E st = 200 GPa.

175 mm

200 mm

50 mm

Probs. 6–120/121

6–122. Segment A of the composite beam is made from 2014-T6 aluminum alloy and segment B is A-36 steel. If w = 0.9 kip>ft, determine the absolute maximum bending stress developed in the aluminum and steel. Sketch the stress distribution on the cross section.

6–123. Segment A of the composite beam is made from 2014-T6 aluminum alloy and segment B is A-36 steel. The allowable bending stress for the aluminum and steel are (s allow ) al = 15 ksi and (s allow ) st = 22 ksi. Determine the maximum allowable intensity w of the uniform distributed load.

15 ft

3 in.

Probs. 6–122/123

*6–124. Using the techniques outlined in Appendix A, Example A.5 or A.6, the Z section has principal moments of inertia of I y = 0.060(10–3) m4 and I z = 0.471(10–3) m4, computed about the principal axes of inertia y and z , respectively. If the section is subjected to an internal moment of M = 250 N # m directed horizontally as shown, determine the stress produced at point B . Solve the problem using Eq. 6–17.

200 mm

50 mm

300 mm z¿

250 N!m

y¿

32.9"

50 mm 200 mm

Prob. 6–124

6–125. The wooden section of the beam is reinforced with two steel plates as shown. Determine the maximum internal moment M that the beam can support if the allowable stresses for the wood and steel are (s allow ) w = 6 MPa, and (s allow ) st = 150 MPa, respectively. Take E w = 10 GPa and E st = 200 GPa.

6–126. The wooden section of the beam is reinforced with two steel plates as shown. If the beam is subjected to an internal moment of M = 30 kN # m, determine the maximum bending stresses developed in the steel and wood. Sketch the stress distribution over the cross section. Take E w = 10 GPa and E st = 200 GPa.

150 mm

15 mm

100 mm

Probs. 6–125/126

334 CHAPTER 6 BENDING

6–130. The beam is made from three types of plastic that are identified and have the moduli of elasticity shown in the figure. Determine the maximum bending stress in the PVC.

PVC EPVC ! 450 ksi Escon EE ! 160 ksi Bakelite EB ! 800 ksi

3 ft 4 ft

500 lb 500 lb

3 ft

3 in.

1 in. 2 in. 2 in.

Prob. 6–130

6–131. The concrete beam is reinforced with three 20-mm- diameter steel rods. Assume that the concrete cannot support tensile stress. If the allowable compressive stress for concrete is (sallow)con = 12.5 MPa and the allowable tensile stress for steel is (sallow)st = 220 MPa , determine the required dimension d so that both the concrete and steel achieve their allowable stress simultaneously. This condition is said to be ‘balanced’. Also, compute the corresponding maximum allowable internal moment M that can be applied to the beam. The moduli of elasticity for concrete and steel are Econ = 25 GPa and Est = 200 GPa , respectively.

200 mm

Prob. 6–131

6–127. The member has a brass core bonded to a steel casing. If a couple moment of 8 kN # m is applied at its end, determine the maximum bending stress in the member. Ebr = 100 GPa, Est = 200 GPa.

3 m

100 mm

20 mm 100 mm

20 mm

20 mm 20 mm

8 kN"m

Prob. 6–127

*6–128. The steel channel is used to reinforce the wood beam. Determine the maximum stress in the steel and in the wood if the beam is subjected to a moment of M = 850 lb # ft. Est = 29(103) ksi, Ew = 1600 ksi.

0.5 in.

4 in.

0.5 in.

15 in. M ! 850 lb"ft

Prob. 6–128

6–129. A wood beam is reinforced with steel straps at its top and bottom as shown. Determine the maximum bending stress developed in the wood and steel if the beam is subjected to a bending moment of M = 5 kN # m. Sketch the stress distribution acting over the cross section. Take E w = 11 GPa, E st = 200 GPa.

200 mm

20 mm

300 mm

M ! 5 kN"m

Prob. 6–129

6.9 STRESS CONCENTRATIONS 335

6–135. The Douglas fir beam is reinforced wizth A-36 straps at its center and sides. Determine the maximum stress developed in the wood and steel if the beam is subjected to a bending moment of M z = 7.50 kip # ft. Sketch the stress distribution acting over the cross section.

z6 in.

0.5 in. 0.5 in.

2 in. 2 in.

0.5 in.

Prob. 6–135

*6–136. For the curved beam in Fig. 6–40 a , show that when the radius of curvature approaches infinity, the curved-beam formula, Eq. 6–24 , reduces to the flexure formula, Eq. 6–13 .

6–137. The curved member is subjected to the internal moment of M = 50 kN # m. Determine the percentage error introduced in the computation of maximum bending stress using the flexure formula for straight members.

6–138. The curved member is made from material having an allowable bending stress of s allow = 100 MPa. Determine the maximum allowable internal moment M that can be applied to the member.

MM 200 mm

200 mm

100 mm

Probs. 6–137/138

*6–132. The wide-flange section is reinforced with two wooden boards as shown. If this composite beam is subjected to an internal moment of M = 100 kN # m, determine the maximum bending stress developed in the steel and the wood. Take E w = 10 GPa and E st = 200 GPa.

6–133. The wide-flange section is reinforced with two wooden boards as shown. If the steel and wood have an allow able bending stress of (s allow ) st = 150 MPa and (s allow ) w = 6 MPa, determine the maximum allowable internal moment M that can be applied to the beam. Take E w = 10 GPa and E st = 200 GPa.

300 mm

10 mm

100 mm

Probs. 6–132/133

6–134. If the beam is subjectd to an internal moment of M = 45 kN # m, determine the maximum bending stress developed in the A-36 steel section A and the 2014-T6 aluminum alloy section B.

150 mm

15 mm

50 mm

Prob. 6–134

336 CHAPTER 6 BENDING

6–143. The elbow of the pipe has an outer radius of 0.75 in. and an inner radius of 0.63 in. If the assembly is subjected to the moments of M = 25 lb # in., determine the maximum stress developed at section a-a .

! 25 lb"in.M

= 25 lb"in.M

1 in. 30#

0.75 in.

0.63 in.

Prob. 6–143

*6–144. The curved member is symmetric and is subjected to a moment of M = 600 lb # ft. Determine the bending stress in the member at points A and B . Show the stress acting on volume elements located at these points.

8 in.

2 in.

1.5 in.

0.5 in.

Prob. 6–144

6–139. The curved beam is subjected to a bending moment of M = 40 lb # ft. Determine the maximum bending stress in the beam. Also, sketch a two-dimensional view of the stress distribution acting on section a – a .

*6–140. The curved beam is made from material having an allowable bending stress of s allow = 24 ksi. Determine the maximum moment M that can be applied to the beam.

3 in. a

M M

2 in.

0.5 in.

2 in.

0.5 in.

5 in.

5.5 in.

Probs. 6–139/140

6–141. If P = 3 kN, determine the bending stress developed at points A , B and C of the cross section at section a – a . Using these results, sketch the stress distribution on section a – a .

6–142. If the maximum bending stress at section a – a is not allowed to exceed s allow = 150 MPa, determine the maximum allowable force P that can be applied to the end E .

600 mm

300 mm

B C

50 mm

25 mm 25 mm 25 mm

20 mm

Section a – a

Probs. 6–141/142

6.9 STRESS CONCENTRATIONS 337

6–147. If the curved member is subjected to the internal moment of M = 600 lb # ft, determine the bending stress developed at points A , B and C . Using these results, sketch the stress distribution on the cross section.

*6–148. If the curved member is made from material having an allowable bending stress of s allow = 15 ksi, determine the maximum allowable internal moment M that can be applied to the member.

C C

A A B

6 in.

2 in.

1.25 in.

1 in.

0.75 in. B

Probs. 6–147/148

6–149. A 100-mm-diameter circular rod is bent into an S shape. If it is subjected to the applied moments M = 125 N # m at its ends, determine the maximum tensile and compressive stress developed in the rod.

M !125 N"m

400 mm

Prob. 6–149

6–145. The curved bar used on a machine has a rectangular cross section. If the bar is subjected to a couple as shown, determine the maximum tensile and compressive stress acting at section a-a. Sketch the stress distribution on the section in three dimensions.

75 mm

50 mm

150 mm

162.5 mm

60! 60!

250 N

75 mm

Prob. 6–145

6–146. The fork is used as part of a nosewheel assembly for an airplane. If the maximum wheel reaction at the end of the fork is 840 lb, determine the maximum bending stress in the curved portion of the fork at section a – a . There the cross-sectional area is circular, having a diameter of 2 in.

30#

6 in.

840 lb

10 in.

Prob. 6–146

338 CHAPTER 6 BENDING

6–155. The simply supported notched bar is subjected to the two loads, each having a magnitude of P = 100 lb. Determine the maximum bending stress developed in the bar, and sketch the bending-stress distribution acting over the cross section at the center of the bar. Each notch has a radius of r = 0.125 in.

20 in. 20 in.

1.75 in.

0.5 in.

P P

1.25 in.

20 in. 20 in.

Prob. 6–155

*6–156. Determine the length L of the center portion of the bar so that the maximum bending stress at A , B , and C is the same. The bar has a thickness of 10 mm.

200 mm 200 mm

7 mm 40 mm60 mm

350 N

BA C

7 mm

L 2

Prob. 6–156

6–157. The stepped bar has a thickness of 15 mm. Determine the maximum moment that can be applied to its ends if it is made of a material having an allowable bending stress of sallow = 200 MPa .

10 mm

30 mm 45 mm

3 mm 6 mm

Prob. 6–157

6–150. The bar is subjected to a moment of M = 153 N # m. Determine the smallest radius r of the fillets so that an allowable bending stress of sallow = 120 MPa is not exceeded.

6–151. The bar is subjected to a moment of M = 17.5 N # m. If r = 6 mm determine the maximum bending stress in the material.

60 mm40 mm 7 mm

M M

Probs. 6–150/151

*6–152. The bar is subjected to a moment of M = 40 N # m. Determine the smallest radius r of the fillets so that an allowable bending stress of sallow = 124 MPa is not exceeded.

6–153. The bar is subjected to a moment of M = 17.5 N # m. If r = 5 mm , determine the maximum bending stress in the material.

80 mm

20 mm 7 mm

M M r

Probs. 6–152/153

6–154. The simply supported notched bar is subjected to two forces P . Determine the largest magnitude of P that can be applied without causing the material to yield. The material is A-36 steel. Each notch has a radius of r = 0.125 in.

20 in. 20 in.

1.75 in.

0.5 in.

P P

1.25 in.

20 in. 20 in.

Probs. 6–154

*6.10 Inelastic Bending The equations for determining the normal stress due to bending that have previously been developed are valid only if the material behaves in a linear-elastic manner. If the applied moment causes the material to yield, a plastic analysis must then be used to determine the stress distribution. For bending of straight members three conditions must be met.

Linear Normal-Strain Distribution. Based only on geometric considerations, it was shown in Sec. 6.3 that the normal strains always vary linearly from zero at the neutral axis of the cross section to a maximum at the farthest point from the neutral axis.

Resultant Force Equals Zero. Since there is only a resultant internal moment acting on the cross section, the resultant force caused by the stress distribution must be equal to zero. Since s creates a force on the area dA of dF = s dA, Fig. 6–47 , then for the entire cross-sectional area A , we have

FR = !Fx; LAs dA = 0 (6–27)

This equation provides a means for obtaining the location of the neutral axis.

Resultant Moment. The resultant moment at the section must be equivalent to the moment caused by the stress distribution about the neutral axis. Since the moment of the force dF = s dA about the neutral axis is dM = y(s dA ), Fig 6 –47, then summing the results over the entire cross section, we have,

(MR)z = !Mz; M = LAy(s dA ) (6–28)

These conditions of geometry and loading will now be used to show how to determine the stress distribution in a beam when it is subjected to a resultant internal moment that causes yielding of the material. Throughout the discussion we will assume that the material has a stress–strain diagram that is the same in tension as it is in compression. For simplicity, we will begin by considering the beam to have a cross-sectional area with two axes of symmetry; in this case, a rectangle of height h and width b , as shown in Fig. 6–48 a . Two cases of loading that are of special interest will be considered.

y dA

Fig. 6–47

6.10 INELASTIC BENDING 339

340 CHAPTER 6 BENDING

Plastic Moment. Some materials, such as steel, tend to exhibit elastic-perfectly plastic behavior when the stress in the material reaches sY . If the applied moment M = MY is just sufficient to produce yielding in the top and bottom fibers of the beam as shown in Fig. 6–48 b , then we can determine MY using the flexure formula sY = MY (h>2)> 3bh3>124 or MY =

1 6

bh2 sY (6–29)

If the internal moment M 7 MY , the material at the top and bottom of the beam will begin to yield, causing a redistribution of stress over the cross section until the required internal moment M is developed. If this causes normal-strain distribution as shown in Fig. 6–48 b , then the corresponding normal-stress distribution is determined from the stress– strain diagram shown in Fig. 6–48 c . Here the strains P1, PY , P2, correspond to stresses s1, sY , sY , respectively. When these and other stresses like them are plotted on the cross section, we obtain the stress distribution shown in Fig. 6–48 d or 6–48e. Here the tension and compression stress “blocks” each consist of component rectangular and triangular blocks. The resultant forces they produce are equivalent to their volumes.

T1 = C1 = 1 2

yY sYb

T2 = C2 = a h2 - yY bsYb Because of the symmetry, Eq. 6–27 is satisfied and the neutral axis passes through the centroid of the cross section as shown. The applied moment M can be related to the yield stress sY using Eq. 6–28 . From Fig. 6–48 e , we require

M = T1a 23 yY b + C1a 23 yY b + T2 c yY + 12 a h2 - yY b d + C2 c yY + 12 a h2 - yY b d

= 2a 1 2

yY sYbb a 23 yY b + 2 c a h2 - yY bsYb d c 12 a h2 + yY b d =

1 4

bh2 sY a1 - 43 yY2 h2 b

Strain distribution (profile view)

(b)

!yY !y1

h 2

Stress–strain diagram (elastic-plastic region)

(c)

P2 PP1 PY

Stress distribution (profile view)

(d)

h 2

!yY

Elastic core

Plastic yielding

(e)

C2 C1

T2 M

h 2

!yY

M " MY

(a)

Fig. 6–48

Or using Eq. 6–29 ,

M = 3 2

MY a1 - 43 y2Yh2 b (6–30) As noted in Fig. 6–48 e , M produces two zones of plastic yielding and an elastic core in the member. The boundary between them is located a distance {yY from the neutral axis. As M increases in magnitude, yY approaches zero. This would render the material entirely plastic and the stress distribution will then look like that shown in Fig. 6–48 f . From Eq. 6–30 with yY = 0, or by finding the moments of the stress “blocks” around the neutral axis, we can write this limiting value as

Mp = 1 4

bh2 sY (6–31)

Using Eq. 6–29 , or Eq. 6–30 with y = 0, we have

Mp = 3 2

MY (6–32)

This moment is referred to as the plastic moment. Its value applies only for a rectangular section, since the analysis depends on the geometry of the cross section.

Beams used in steel buildings are sometimes designed to resist a plastic moment. When this is the case, codes usually list a design property for a beam called the shape factor. The shape factor is defined as the ratio

k = Mp MY

(6–33)

This value specifies the additional moment capacity that a beam can support beyond its maximum elastic moment. For example, from Eq. 6–32 , a beam having a rectangular cross section has a shape factor of k = 1.5. Therefore this section will support 50% more bending moment than its maximum elastic moment when it becomes fully plastic.

Plastic moment (f)

h 2

Fig. 6–48 (cont.)

6.10 INELASTIC BENDING 341

342 CHAPTER 6 BENDING

Residual Stress. When the plastic moment is removed from the beam then it will cause residual stress to be developed in the beam. These stresses are often important when considering fatigue and other types of mechanical behavior, and so we will discuss a method used for their computation. To explain how this is done, we will assume that Mp causes the material at the top and bottom of the beam to be strained to P1 (WPY), as shown by point B on the s9P curve in Fig. 6–49 a . A release of this moment will cause this material to recover some of this strain elastically by following the dashed path BC. Since this recovery is elastic, we can superimpose on the stress distribution in Fig. 6–49 b a linear stress distribution caused by applying the plastic moment in the opposite direction, Fig. 6–49 c . Here the maximum stress, which is called the modulus of rupture for bending, sr, can be determined from the flexure formula when the beam is loaded with the plastic moment. We have

smax = Mc

I =

Mp112 h21 112 bh32 = 114 bh2sY 2112 h21 112 bh32 = 1.5sY

(a)

E E

!0.5 sY

Actual elastic recovery

Elastic-plastic loading

2PY PP1

!sY

Fig. 6–49

This reversed application of the plastic moment is possible here, since maximum elastic recovery strain at the top and bottom of the beam is 2PY as shown in Fig. 6–49 a . This corresponds to a maximum stress of 2sY which is greater than the required stress of 1.5sY as calculated above, Fig. 6–49 c .

The superposition of the plastic moment, Fig. 6–49 b , and its removal, Fig. 6–49 c , gives the residual-stress distribution shown in Fig. 6–49 d . As an exercise, use the component triangular “blocks” that represent this stress distribution and show that it results in a zero-force and zero- moment resultant on the member as required.

Ultimate Moment. Consider now the more general case of a beam having a cross section that is symmetrical only with respect to the vertical axis, while the moment is applied about the horizontal axis, Fig. 6–50 a . We will assume that the material exhibits strain hardening and that its stress– strain diagrams for tension and compression are different, Fig. 6–50 b .

If the moment M produces yielding of the beam, difficulty arises in finding both the location of the neutral axis and the maximum strain that is produced in the beam. This is because the cross section is unsymmetrical about the horizontal axis and the stress–strain behavior of the material is not the same in tension and compression. To solve this problem, a trial-and-error procedure requires the following steps:

1. For a given moment M , assume the location of the neutral axis and the slope of the “linear” strain distribution, Fig. 6–50 c .

2. Graphically establish the stress distribution on the member’s cross section using the s9P curve to plot values of stress corresponding to values of strain. The resulting stress distribution, Fig. 6–50 d , will then have the same shape as the s9P curve.

(b)

Plastic moment applied causing plastic strain

N Mp

+h 2

h 2

(c)

Plastic moment reversal causing elastic strain

Mph 2

h 2

sr ! 1.5sY

(d)

Residual stress distribution in beam

0.5sY

h 6

h 3

Axis of symmetry

Known moment

(a)

(b)

P1 P2 P

Fig. 6–50

Fig. 6–49 (cont.)

6.10 INELASTIC BENDING 343

344 CHAPTER 6 BENDING

3. Determine the volumes enclosed by the tensile and compressive stress “blocks.” (As an approximation, this may require dividing each block into composite regions.) Equation 6–27 requires the volumes of these blocks to be equal , since they represent the resul tant tensile force T and resultant compressive force C on the section, Fig. 6–50 e . If these forces are unequal, an adjustment as to the location of the neutral axis must be made (point of zero strain ) and the process repeated until Eq. 6–27 (T = C) is satisfied.

4. Once T = C, the moments produced by T and C can be calculated about the neutral axis. Here the moment arms for T and C are measured from the neutral axis to the centroids of the volumes defined by the stress distributions, Fig. 6–50 e . Equation 6–28 requires M = Ty ! + Cy ". If this equation is not satisfied, the slope of the strain distribution must be adjusted and the computations for T and C and the moment must be repeated until close agreement is obtained.

This trial-and-error procedure is obviously very tedious, and fortunately it does not occur very often in engineering practice. Most beams are symmetric about two axes, and they are constructed from materials that are assumed to have similar tension-and-compression stress–strain diagrams. Whenever this occurs, the neutral axis will pass through the centroid of the cross section, and the process of relating the stress distribution to the resultant moment is thereby simplified.

Important Points

• The normal strain distribution over the cross section of a beam is based only on geometric considerations and has been found to always remain linear, regardless of the applied load. The normal stress distribution, however, must be determined from the material behavior, or stress–strain diagram once the strain distribution is established.

• The location of the neutral axis is determined from the condition that the resultant force on the cross section is zero.

• The resultant internal moment on the cross section must be equal to the moment of the stress distribution about the neutral axis.

• Perfectly plastic behavior assumes the normal stress distribution is constant over the cross section, and the beam will continue to bend, with no increase in moment. This moment is called the plastic moment.

Assumed location of neutral axis

Strain distribution (profile view)

(c)

Assumed slope of strain distribution

Stress distribution (profile view)

(d)

y¿¿

(e) N y¿ T

Fig. 6–50 (cont.)

EXAMPLE 6.21

The steel wide-flange beam has the dimensions shown in Fig. 6–51 a . If it is made of an elastic perfectly plastic material having a tensile and compressive yield stress of sY = 36 ksi, determine the shape factor for the beam.

SOLUTION In order to determine the shape factor, it is first necessary to calculate the maximum elastic moment MY and the plastic moment Mp.

Maximum Elastic Moment. The normal-stress distribution for the maximum elastic moment is shown in Fig. 6–51 b . The moment of inertia about the neutral axis is

I = c 1 12

(0.5 in.)(9 in.)3 d + 2 c 1 12

(8 in.)(0.5 in.)3 + 8 in. (0.5 in.)(4.75 in.)2 d = 211.0 in4 Applying the flexure formula, we have

smax = Mc

I ; 36 kip>in2 = MY(5 in.)

211.0 in4 MY = 1519.5 kip # in.

Plastic Moment. The plastic moment causes the steel over the entire cross section of the beam to yield, so that the normal-stress distribution looks like that shown in Fig. 6–51 c . Due to symmetry of the cross-sectional area and since the tension and compression stress– strain diagrams are the same, the neutral axis passes through the centroid of the cross section. In order to determine the plastic moment, the stress distribution is divided into four composite rectangular “blocks,” and the force produced by each “block” is equal to the volume of the block. Therefore, we have

C1 = T1 = 36 kip>in2 (0.5 in.)(4.5 in.) = 81 kip C2 = T2 = 36 kip>in210.5 in.)(8 in.) = 144 kip

These forces act through the centroid of the volume for each block. Calculating the moments of these forces about the neutral axis, we obtain the plastic moment:

Mp = 2[(2.25 in.)(81 kip)] + 2[(4.75 in.)(144 kip)] = 1732.5 kip # in. Shape Factor. Applying Eq. 6–33 gives

k = Mp MY

= 1732.5 kip # in. 1519.5 kip # in. = 1.14 Ans.

NOTE: This value indicates that a wide-flange beam provides a very efficient section for resisting an elastic moment. Most of the moment is developed in the flanges, i.e., in the top and bottom segments, whereas the web or vertical segment contributes very little. In this particular case, only 14% additional moment can be supported by the beam beyond that which can be supported elastically.

8 in.

9 in.

0.5 in.

(a)

Fig. 6–51

(b)

36 ksi

(c)

36 ksi

6.10 INELASTIC BENDING 345

346 CHAPTER 6 BENDING

A T-beam has the dimensions shown in Fig. 6–52 a . If it is made of an elastic perfectly plastic material having a tensile and compressive yield stress of sY = 250 MPa, determine the plastic moment that can be resisted by the beam.

EXAMPLE 6.22

120 mm

15 mm

(a)

100 mm

(b)

15 mm

C2 C1

250 MPa

100 mm

15 mm (120 mm ! d)

Fig. 6–52

SOLUTION The “plastic” stress distribution acting over the beam’s cross-sectional area is shown in Fig. 6–52 b . In this case the cross section is not symmetric with respect to a horizontal axis, and consequently, the neutral axis will not pass through the centroid of the cross section. To determine the location of the neutral axis, d , we require the stress distribution to produce a zero resultant force on the cross section. Assuming that d … 120 mm, we have

LAs dA = 0; T - C1 - C2 = 0 250 MPa (0.015 m)(d) - 250 MPa (0.015 m)(0.120 m - d)

- 250 MPa (0.015 m)(0.100 m) = 0 d = 0.110 m 6 0.120 m OK

Using this result, the forces acting on each segment are

T = 250 MN>m2 (0.015 m)(0.110 m) = 412.5 kN C1 = 250 MN>m2 (0.015 m)(0.010 m) = 37.5 kN C2 = 250 MN>m2 (0.015 m)(0.100 m) = 375 kN

Hence the resultant plastic moment about the neutral axis is

Mp = 412.5 kNa 0.110 m2 b + 37.5 kNa 0.01 m2 b + 375 kNa0.01 m + 0.015 m2 b Mp = 29.4 kN # m Ans.

The steel wide-flange beam shown in Fig. 6–53 a is subjected to a fully plastic moment of Mp . If this moment is removed, determine the residual-stress distribution in the beam. The material is elastic perfectly plastic and has a yield stress of sY = 36 ksi.

SOLUTION The normal-stress distribution in the beam caused by Mp is shown in Fig. 6–53 b . When Mp is removed, the material responds elastically. Removal of Mp requires applying Mp in its reverse direction and therefore leads to an assumed elastic stress distribution as shown in Fig. 6–53 c . The modulus of rupture sr is computed from the flexure formula. Using Mp = 1732.5 kip # in. and I = 211.0 in4 from Example 6.21 , we have

smax = Mc

I ;

sr = 1732.5 kip # in. (5 in.)

211.0 in4 = 41.1 ksi

As expected, sr 6 2sY . Superposition of the stresses gives the residual-stress distribution

shown in Fig. 6–53 d . Note that the point of zero normal stress was determined by proportion; i.e., from Figs. 6–53 b and 6–53 c , we require that

41.1 ksi

5 in. =

36 ksi y

y = 4.38 in.

EXAMPLE 6.23

8 in.

9 in.

0.5 in.

(a)

Plastic moment applied (profile view)

36 ksi

(b)

5 in.

Plastic moment reversed (profile view)

(c)

36 ksi

y 5 in.

5 in.

sr ! 41.1 ksi

5.05 ksi

36 ksi

4.38 in.

Residual stress distribution

(d)

5.05 ksi

Fig. 6–53

6.10 INELASTIC BENDING 347

348 CHAPTER 6 BENDING

The beam in Fig. 6–54 a is made of an alloy of titanium that has a stress–strain diagram that can in part be approximated by two straight lines. If the material behavior is the same in both tension and compression, determine the bending moment that can be applied to the beam that will cause the material at the top and bottom of the beam to be subjected to a strain of 0.050 in.>in .

EXAMPLE 6.24

3 in.

2 in.

(a)

0.010

s !

1 5(

10 3 )P

0.050

150

190

s (ksi)

P (in./in.)

s ! 100

0P " 140

SOLUTION I By inspection of the stress–strain diagram, the material is said to exhibit “elastic-plastic behavior with strain hardening.” Since the cross section is symmetric and the tension–compression s9P diagrams are the same, the neutral axis must pass through the centroid of the cross section. The strain distribution, which is always linear, is shown in Fig. 6–54 b . In particular, the point where maximum elastic strain (0.010 in.>in.) occurs has been determined by proportion, such that 0.05>1.5 in. = 0.010>y or y = 0.3 in.

The corresponding normal-stress distribution acting over the cross section is shown in Fig. 6–54 c . The moment produced by this distribution can be calculated by finding the “volume” of the stress blocks. To do so we will subdivide this distribution into two triangular blocks and a rectangular block in both the tension and compression regions, Fig. 6–54 d . Since the beam is 2 in. wide, the resultants and their locations are determined as follows:

T1 = C1 = 1 2

(1.2 in.)(40 kip>in2)(2 in.) = 48 kip y1 = 0.3 in. +

2 3

(1.2 in.) = 1.10 in.

Fig. 6–54

0.05

0.05 Strain distribution

(b)

1.5 in.

0.010

y ! 0.3 in.

1.5 in.

Stress distribution

(c)

150 ksi

190 ksi

y ! 0.3 in.

(d)

C3 T3

T2 T1

C2 C1

150 ksi 40 ksi

1.2 in.

0.3 in.

(e)

2 in.

y dy

Fig. 6–54 (cont.)

T2 = C2 = (1.2 in.)(150 kip>in2)(2 in.) = 360 kip y2 = 0.3 in. +

1 2

(1.2 in.) = 0.90 in.

T3 = C3 = 1 2

(0.3 in.)(150 kip>in2)(2 in.) = 45 kip y3 =

2 3

(0.3 in.) = 0.2 in.

The moment produced by this normal-stress distribution about the neutral axis is therefore

M = 2[48 kip (1.10 in.) + 360 kip (0.90 in.) + 45 kip (0.2 in.)]

= 772 kip # in. Ans.

SOLUTION II Rather than using the above semigraphical technique, it is also possible to find the moment analytically. To do this we must express the stress distribution in Fig. 6–54 c as a function of position y along the beam. Note that s = f(P) has been given in Fig. 6–54 a . Also, from Fig. 6–54 b , the normal strain can be determined as a function of position y by proportional triangles; i.e.,

P = 0.05 1.5

y 0 … y … 1.5 in.

Substituting this into the s9e functions shown in Fig. 6–54 a gives

s = 500y 0 … y … 0.3 in. (1)

s = 33.33y + 140 0.3 in. … y … 1.5 in. (2)

From Fig. 6–54 e , the moment caused by s acting on the area strip dA = 2 dy is

dM = y(s dA ) = ys(2 dy)

Using Eqs. 1 and 2 , the moment for the entire cross section is thus

M = 2J2L0.3 in.0 500y2 dy + 2L1.5 in.0.3 in. (33.3y2 + 140y) dy R = 772 kip # in. Ans.

6.10 INELASTIC BENDING 349

350 CHAPTER 6 BENDING

6–158. Determine the shape factor for the wide-flange beam.

200 mm

15 mm

20 mm

200 mm

Prob. 6–158

6–159. The beam is made of an elastic plastic material for which sY = 250 MPa. Determine the residual stress in the beam at its top and bottom after the plastic moment Mp is applied and then released.

200 mm

15 mm

20 mm

200 mm

Prob. 6–159

*6–160. Determine the shape factor for the cross section of the H-beam.

6–161. The H-beam is made of an elastic-plastic material for which sY = 250 MPa . Determine the residual stress in the top and bottom of the beam after the plastic moment Mp is applied and then released.

200 mm

Mp 20 mm

20 mm

200 mm

20 mm

Probs. 6–160/161

6–162. The box beam is made of an elastic perfectly plastic material for which sY = 250 MPa. Determine the residual stress in the top and bottom of the beam after the plastic moment Mp is applied and then released.

25 mm

150 mm 150 mm

25 mm 25 mm

25 mm

Prob. 6–162

PROBLEMS

6–165. The beam is made of elastic-perfectly plastic material. Determine the maximum elastic moment and the plastic moment that can be applied to the cross section. Take sY = 36 ksi.

3 in.

3 in. 1.5 in.

1.5 in.

6 in.

Prob. 6–165

6–166. Determine the shape factor for the cross section of the beam.

200 mm

10 mm

15 mm

200 mm

Prob. 6–166

6–163. Determine the plastic moment M p that can be supported by a beam having the cross section shown. sY = 30 ksi.

10 in.

1 in.

2 in. 1 in.

Prob. 6–163

*6–164. Determine the shape factor of the beam’s cross section.

3 in.

3 in. 1.5 in.

1.5 in.

6 in.

Prob. 6–164

6.10 INELASTIC BENDING 351

352 CHAPTER 6 BENDING

6–169. The beam is made of elastic-perfectly plastic material. Determine the maximum elastic moment and the plastic moment that can be applied to the cross section. Take sY = 36 ksi.

3 in.

3 in. 3 in. 3 in.

Prob. 6–169

6–170. The box beam is made from an elastic-plastic material for which sY = 36 ksi. Determine the magnitude of each concentrated force P that will cause the moment to be (a) the largest elastic moment and (b) the largest plastic moment.

P P

12 in.10 in.

6 in.

5 in.

6 ft 8 ft 6 ft

Prob. 6–170

6–167. The beam is made of an elastic-plastic material for which sY = 200 MPa. If the largest moment in the beam occurs within the center section a – a , determine the magnitude of each force P that causes this moment to be (a) the largest elastic moment and (b) the largest plastic moment.

200 mm

100 mm

2 m

2 m2 m2 m

Prob. 6–167

*6–168. Determine the shape factor of the cross section.

3 in.

3 in. 3 in. 3 in.

Prob. 6–168

6–173. The member is made from an elastic-plastic material. Determine the maximum elastic moment and the plastic moment that can be applied to the cross section. Take b = 4 in., h = 6 in., sY = 36 ksi.

– 2

Prob. 6–173

6–174. The beam is made of an elastic plastic material for which sY = 30 ksi. If the largest moment in the beam occurs at the center section a – a , determine the intensity of the distributed load w that causes this moment to be (a) the largest elastic moment and (b) the largest plastic moment.

8 in.

10 ft

8 in.

10 ft

Prob. 6–174

6–171. The beam is made from elastic-perfectly plastic material. Determine the shape factor for the thick- walled tube.

Prob. 6–171

*6–172. Determine the shape factor for the member.

– 2

Prob. 6–172

6.10 INELASTIC BENDING 353

354 CHAPTER 6 BENDING

6–177. The beam is made of a polyester that has the stress– strain curve shown. If the curve can be represented by the equation s = [20 tan- 1(15P)] ksi, where tan- 1(15P) is in radians, determine the magnitude of the force P that can be applied to the beam without causing the maximum strain in its fibers at the critical section to exceed Pmax = 0.003 in.> in.

8 ft 8 ft

2 in.

4 in.

P (in./in.)

!s (ksi) s " 20 tan#1(15 P)

Prob. 6–177

6–178. The plexiglass bar has a stress–strain curve that can be approximated by the straight-line segments shown. Determine the largest moment M that can be applied to the bar before it fails.

20 mm

!0.06 !0.04

0.02 0.04

!80

compression

tension

failure

s (MPa)

P (mm/mm)

!100

Prob. 6–178

6–175. The box beam is made from an elastic-plastic material for which sY = 25 ksi . Determine the intensity of the distributed load w 0 that will cause the moment to be (a) the largest elastic moment and (b) the largest plastic moment.

9 ft

16 in.12 in.

8 in.

6 in.

9 ft

Prob. 6–175

*6–176. The wide-flange member is made from an elastic- plastic material. Determine the shape factor .

Prob. 6–176

6–179. The stress–strain diagram for a titanium alloy can be approximated by the two straight lines. If a strut made of this material is subjected to bending, determine the moment resisted by the strut if the maximum stress reaches a value of (a) sA and (b) sB .

3 in.

2 in.

0.040.01

sB ! 180

sA ! 140

P (in./in.)

"s (ksi)

Prob. 6–179

*6–180. A beam is made from polypropylene plastic and has a stress–strain diagram that can be approximated by the curve shown. If the beam is subjected to a maximum tensile and compressive strain of P = 0.02 mm>mm, deter mine the moment M .

P (mm/mm)

M 100 mm

30 mm

!s (Pa)

s" 10(106)P1/4

Prob. 6–180

6–181. The bar is made of an aluminum alloy having a stress–strain diagram that can be approximated by the straight line segments shown. Assuming that this diagram is the same for both tension and compression, determine the moment the bar will support if the maximum strain at the top and bottom fibers of the beam is Pmax = 0.03 .

0.050.006 0.025

P (in./in.)

4 in. M

3 in.

σ (ksi)±

Prob. 6–181

6–182. The bar is made of an aluminum alloy having a stress–strain diagram that can be approximated by the straight line segments shown. Assuming that this diagram is the same for both tension and compression, determine the moment the bar will support if the maximum strain at the top and bottom fibers of the beam is Pmax = 0.05 .

0.050.006 0.025

P (in./in.)

4 in. M

3 in.

σ (ksi)±

Prob. 6–182

6.10 INELASTIC BENDING 355

356 CHAPTER 6 BENDING

Shear and moment diagrams are graphical representations of the internal shear and moment within a beam. They can be constructed by sectioning the beam an arbitrary distance x from the left end, using the equilibrium equations to find V and M as functions of x, and then plotting the results. A sign convention for positive distributed load, shear, and moment must be followed.

It is also possible to plot the shear and moment diagrams by realizing that at each point the slope of the shear diagram is equal to the intensity of the distributed loading at the point.

Likewise, the slope of the moment diagram is equal to the shear at the point.

The area under the distributed loading diagram between the points represents the change in shear.

The area under the shear diagram represents the change in moment.

The shear and moment at any point can be obtained using the method of sections. The maximum (or minimum) moment occurs where the shear is zero.

CHAPTER REVIEW

V Positive external distributed load

Positive internal shear

Positive internal moment

Beam sign convention

w(x)

w ! w(x) wB

A C D

w = negative increasing slope = negative increasing

V = positive decreasing slope = positive decreasing

"wB

VC VD

"VB

"wC "wD

w =

dV dx

V = dM dx

!V = 1w dx

!M = 1V dx

A bending moment tends to produce a linear variation of normal strain within a straight beam. Provided the material is homogeneous, and linear elastic, then equilibrium can be used to relate the inter nal moment in the beam to the stress distribution. The result is the flex ure formula,

smax = Mc

where I and c are determined from the neutral axis that passes through the centroid of the cross section.

If the cross-sectional area of the beam is not symmetric about an axis that is perpendicular to the neutral axis, then unsymmetrical bending will occur. The maximum stress can be determined from formulas, or the problem can be solved by considering the superposition of bending caused by the moment components My and Mz about the principal axes of inertia for the area.

smax

s = - Mzy

Iz +

Myz

n = E1 E2

Beams made from composite materials can be “transformed” so their cross section is considered as if it were made from a single material. To do this, the transformation factor n , which is a ratio of the moduli of elasticity of the materials, is used to change the width b of the beam.

Once the cross section is transformed, then the stress in the transformed material is determined using the flexure formula multiplied by n .

CHAPTER REVIEW 357

358 CHAPTER 6 BENDING

Curved beams deform such that the normal strain does not vary linearly from the neutral axis. Provided the material is homogeneous and linear elastic and the cross section has an axis of symmetry, then the curved beam formula can be used to determine the bending stress.

s = M(R - r) Ar(r - R)

s = My

Ae(R - y)

N Mp h 2

h 2

M Stress concentrations occur in members having a sudden change in their cross section, caused, for example, by holes and notches. The maximum bending stress at these locations is determined using a stress concentration factor K that is found from graphs determined from experiment.

If the bending moment causes the stress in the material to exceed its elastic limit, then the normal strain will remain linear; however, the stress distribution will vary in accordance with the stress–strain diagram. The plastic and ultimate moments supported by the beam can be determined by requiring the resultant force to be zero and the resultant moment to be equivalent to the moment of the stress distribution.

s max = K Mc

If an applied plastic or ultimate moment is released, it will cause the material to respond elastically, thereby inducing residual stresses in the beam.

smax

C6–3. Use reasonable dimensions for this hammer and a loading to show through an analysis why this hammer failed in the manner shown.

C6–3

C6–1. The steel saw blade passes over the drive wheel of the band saw. Using appropriate measurements and data, explain how to determine the bending stress in the blade.

C6–1

C6–2. The crane boom has a noticeable taper along its length. Explain why. To do so, assume the boom is in the horizontal position and in the process of hoisting a load at its end, so that the reaction on the support A becomes zero. Use realistic dimensions and a load, to justify your reasoning.

C6–2

C6–4. These garden shears were manufactured using an inferior material. Using a loading of 50 lb applied normal to the blades, and appropriate dimensions for the shears, determine the absolute maximum bending stress in the material and show why the failure occurred at the critical location on the handle.

CONCEPTUAL PROBLEMS

(b)(a)

C6–4

CONCEPTUAL PROBLEMS 359

360 CHAPTER 6 BENDING

6–183. Determine the shape factor for the wide-flange beam.

*6–184. The beam is made of an elastic plastic material for which sY = 250 MPa. Determine the residual stress in the beam at its top and bottom after the plastic moment Mp is applied and then released.

180 mm

20 mm

30 mm

180 mm

Probs. 6–183/184

6–185. The compound beam consists of two segments that are pinned together at B . Draw the shear and moment diagrams if it supports the distributed loading shown.

2/3 L

A CB

1/3 L

Prob. 6–185

6–186. The composite beam consists of a wood core and two plates of steel. If the allowable bending stress for the wood is (sallow)w = 20 MPa , and for the steel (sallow)st = 130 MPa , determine the maximum moment that can be applied to the beam. Ew = 11 GPa, Est = 200 GPa.

6–187. Solve Prob. 6–186 if the moment is applied about the y axis instead of the z axis as shown.

125 mm

20 mm

20 mm 75 mm

Probs. 6–186/187

*6–188. A shaft is made of a polymer having a parabolic upper and lower cross section. If it resists an internal moment of M = 125 N # m , determine the maximum bending stress developed in the material (a) using the flexure formula and (b) using integration. Sketch a three- dimensional view of the stress distribution acting over the cross-sectional area. Hint: The moment of inertia is determined using Eq. A–3 of Appendix A .

M ! 125 N· m

50 mm

100 mm

50 mm

y ! 100 – z 2/ 25

Prob. 6–188

REVIEW PROBLEMS

*6–192. A wooden beam has a square cross section as shown. Determine which orientation of the beam provides the greatest strength at resisting the moment M . What is the difference in the resulting maximum stress in both cases?

a a

M M

(a) (b)

Prob. 6–192

6–193. Draw the shear and moment diagrams for the shaft if it is subjected to the vertical loadings of the belt, gear, and flywheel. The bearings at A and B exert only vertical reactions on the shaft.

A B

200 mm

450 N

150 N

300 N

200 mm 400 mm 300 mm

Prob. 6–193

6–194. The strut has a square cross section a by a and is subjected to the bending moment M applied at an angle u as shown. Determine the maximum bending stress in terms of a , M , and u . What angle u will give the largest bending stress in the strut? Specify the orientation of the neutral axis for this case.

M x

Prob. 6–194

6–189. Determine the maximum bending stress in the handle of the cable cutter at section a – a . A force of 45 lb is applied to the handles. The cross-sectional area is shown in the figure.

4 in.

45 lb20!

3 in.

5 in.

45 lb

0.75 in.

0.50 in.

Prob. 6–189

6–190. The curved beam is subjected to a bending moment of M = 85 N # m as shown. Determine the stress at points A and B and show the stress on a volume element located at these points.

30!

M " 85 N#m

100 mm

150 mm

20 mm

15 mm 400 mm

Prob. 6–190

6–191. Determine the shear and moment in the beam as functions of x , where 0 … x 6 6 ft , then draw the shear and moment diagrams for the beam.

6 ft 4 ft

2 kip/ ft

50 kip!ft

8 kip

Prob. 6–191

REVIEW PROBLEMS 361

Railroad ties act as beams that support very large transverse shear loadings. As a result, if they are made of wood they will tend to split at their ends,

where the shear loads are the largest.

Chapter 7

363

7.1 Shear in Straight Members In general, a beam will support both shear and moment. The shear V is the result of a transverse shear-stress distribution that acts over the beam’s cross section. Due to the complementary property of shear, however, this stress will create corresponding longitudinal shear stresses which will act along longitudinal planes of the beam as shown in Fig. 7–1 .

Longitudinal shear stress

Transverse shear stress

Fig. 7–1

CHAPTER OBJECTIVES

■ In this chapter, we will develop a method for finding the shear stress in a beam having a prismatic cross section and made from homogeneous material that behaves in a linear-elastic manner. The method of analysis to be developed will be somewhat limited to special cases of cross-sectional geometry. Although this is the case, it has many wide-range applications in engineering design and analysis. The concept of shear flow, along with shear stress, will be discussed for beams and thin-walled members. The chapter ends with a discussion of the shear center.

Transverse Shear

364 CHAPTER 7 TRANSVERSE SHEAR

To illustrate this effect, consider the beam to be made from three boards, Fig. 7–2 a . If the top and bottom surfaces of each board are smooth, and the boards are not bonded together, then application of the load P will cause the boards to slide relative to one another when the beam deflects. However, if the boards are bonded together, then the longitudinal shear stresses acting between the boards will prevent their relative sliding, and consequently the beam will act as a single unit, Fig. 7–2 b .

As a result of the shear stress, shear strains will be developed and these will tend to distort the cross section in a rather complex manner. For example, consider the short bar in Fig. 7–3 a made of a highly deformable material and marked with horizontal and vertical grid lines. When a shear V is applied, it tends to deform these lines into the pattern shown in Fig. 7–3 b . This nonuniform shear-strain distribution will cause the cross section to warp .

Shear connectors are “tack welded” to this corrugated metal floor liner so that when the concrete floor is poured, the connectors will prevent the concrete slab from slipping on the liner surface. The two materials will thus act as a composite slab.

Boards not bonded together (a)

Boards bonded together

(b)

Fig. 7–2

(a) Before deformation

(b) After deformation

Fig. 7–3

7.2 THE SHEAR FORMULA 365

As a result, when a beam is subjected to both bending and shear, the cross section will not remain plane as assumed in the development of the flexure formula. Although this is the case, we can generally assume the cross-sectional warping due to shear is small enough so that it can be neglected . This assumption is particularly true for the most common case of a slender beam ; that is, one that has a small depth compared with its length.

7.2 The Shear Formula Because the strain distribution for shear is not easily defined, as in the case of axial load, torsion, and bending, we will develop the shear formula in an indirect manner. To do this we will consider the horizontal force equilibrium of a portion of the element taken from the beam in Fig. 7–4 a . A free-body diagram of this element is shown in Fig. 7–4 b . This distribution is caused by the bending moments M and M + dM. We have excluded the effects of V , V + dV , and w(x) on the free-body diagram since these loadings are vertical and will therefore not be involved in a horizontal force summation. The element in Fig. 7–4 b will indeed satisfy !Fx = 0 since the stress distribution on each side of the element forms only a couple moment and therefore a zero force resultant.

(a)

F1 F2

M2dxx

Area ! A¿ Section plane

y ¿

_ y ¿

M " dM M

dF ¿¿ dF ¿

#Fx ! 0 satisfied

(b)

Fig. 7–4

366 CHAPTER 7 TRANSVERSE SHEAR

Now consider the shaded top portion of the element that has been sectioned at y ! from the neutral axis, Fig. 7–4 a . This segment has a width t at the section, and the two cross-sectional sides each have an area A !. Because the resultant moments on each side of the element differ by dM , it can be seen in Fig. 7–4 c that "Fx = 0 will not be satisfied unless a longitudinal shear stress t acts over the bottom face of the segment. We will assume this shear stress is constant across the width t of the bottom face. It acts on the area t dx . Applying the equation of horizontal force equilibrium, and using the flexure formula, Eq. 6–13 , we have

d+ "Fx = 0; LA !s! dA ! - LA !s dA ! - t(t dx) = 0

LA !aM + dMI by dA ! - LA !aMI by dA ! - t(t dx) = 0 a dM

I b LA !y dA ! = t(t dx) (7–1)

Solving for t, we get

t = 1 It

a dM dx

b LA !y dA ! This equation can be simplified by noting that V = dM >dx ( Eq. 6–2 ).

Also, the integral represents the moment of the area A ! about the neutral axis. We will denote this by the symbol Q . Since the location of the centroid of the area A ! is determined from y ! = 1A !y dA !>A !, we can also write

Q = LA !y dA ! = y !A ! (7–2)

(a)

Area ! A¿ Section plane

y ¿

_ y ¿

M " dM

y¿

A¿

Three-dimensional view

s s¿s¿ s

Fig. 7–4 (cont.)

7.2 THE SHEAR FORMULA 367

The final result is therefore

t = VQ It

(7–3)

Here, as shown in Fig. 7–5 ,

t = the shear stress in the member at the point located a distance y ! from the neutral axis. This stress is assumed to be constant and therefore averaged across the width t of the member

V = the internal resultant shear force, determined from the method of sections and the equations of equilibrium

I = the moment of inertia of the entire cross-sectional area calculated about the neutral axis

t = the width of the member’s cross-sectional area, measured at the point where t is to be determined

Q = y !A !, where A ! is the area of the top (or bottom) portion of the member’s cross-sectional area, above (or below) the section plane where t is measured, and y ! is the distance from the neutral axis to the centroid of A !

The above equation is referred to as the shear formula . Although in the derivation we considered only the shear stresses acting on the beam’s longitudinal plane, the formula applies as well for finding the transverse shear stress on the beam’s cross-section. Recall that these stresses are complementary and numerically equal.

Also, because the flexure formula was used in the derivation, it is necessary that the material behave in a linear elastic manner and have a modulus of elasticity that is the same in tension as it is in compression.

Area ! A¿

_ y ¿

Fig. 7–5

368 CHAPTER 7 TRANSVERSE SHEAR

Limitations on the Use of the Shear Formula. One of the major assumptions used in the development of the shear formula is that the shear stress is uniformly distributed over the width t at the section. In other words, the average shear stress is calculated across the width. We can test the accuracy of this assumption by comparing it with a more exact mathematical analysis based on the theory of elasticity. For example, if the beam’s cross section is rectangular, the shear-stress distribution across the neutral axis as calculated from the theory of elasticity varies as shown in Fig. 7–6 . The maximum value, t!max, occurs at the sides of the cross section, and its magnitude depends on the ratio b>h ( width>depth ). For sections having a b>h = 0.5, t!max is only about 3% greater than the shear stress calculated from the shear formula, Fig. 7–6 a . However, for flat sections , say b>h = 2, t!max is about 40% greater than tmax, Fig. 7–6 b . The error becomes even greater as the section becomes flatter, or as the b>h ratio increases. Errors of this magnitude are certainly intolerable if one uses the shear formula to determine the shear stress in the flange of the wide-flange beam shown in Fig. 7–7 .

It should also be pointed out that the shear formula will not give accurate results when used to determine the shear stress at the flange— web junction of a wide-flange beam, since this is a point of sudden cross-sectional change and therefore a stress concentration occurs here. Fortunately, these limitations for applying the shear formula to the flanges of a wide-flange beam are not important in engineering practice. Most often engineers must only calculate the average maximum shear stress in the beam, which occurs at the neutral axis, where the b>h ( width>depth ) ratio for the web is very small , and therefore the calculated result is very close to the actual maximum shear stress as explained above.

(a)

b ! 0.5h

h AN

(b)

b ! 2h

t¿max

VQ Ittmax !

Fig. 7–6

Flanges V

Web

Fig. 7–7

7.2 THE SHEAR FORMULA 369

Another important limitation on the use of the shear formula can be illustrated with reference to Fig. 7–8 a , which shows a member having a cross section with an irregular or nonrectangular boundary. If we apply the shear formula to determine the (average) shear stress t along the line AB , it will be directed downward as shown in Fig. 7–8 b . However, consider an element of material taken from the boundary point B , Fig. 7–8 c . Here t on the front face of the element is resolved into components, t! and t" acting perpendicular and parallel to the boundary. By inspection, t! must be equal to zero since its corresponding longitudinal component t!, on the stress-free boundary surface, must be zero. To satisfy this boundary condition, therefore, the shear stress acting on this element must actually be directed tangent to the boundary. As a result, the shear-stress distribution across line AB is directed as shown in Fig. 7–8 d . Here specific values for the shear stress must be obtained using the theory of elasticity. Note, however, that we can apply the shear formula to obtain the shear stress acting across each of the colored lines in Fig. 7–8 a . These lines intersect the tangents to the boundary at right angles , and as shown in Fig. 7–8 e , the transverse shear stress is vertical and constant along each line.

To summarize the above points, the shear formula does not give accurate results when applied to members having cross sections that are short or flat , or at points where the cross section suddenly changes. Nor should it be applied across a section that intersects the boundary of the member at an angle other than 90°. Instead, for these cases the shear stress should be determined using more advanced methods based on the theory of elasticity.

(a)

A B

Stress-free outer surface

(c)

Shear-stress distribution from shear formula

A B

(b)

t¿ ! 0

t¿¿

t¿¿ t¿

(d)

A B

tmax tmax

(e)

Fig. 7–8

370 CHAPTER 7 TRANSVERSE SHEAR

Important Points

• Shear forces in beams cause nonlinear shear-strain distributions over the cross section, causing it to warp . • Due to the complementary property of shear stress, the shear stress developed in a beam acts over the

cross section of the beam and along its longitudinal planes.

• The shear formula was derived by considering horizontal force equilibrium of the longitudinal shear-stress and bending-stress distributions acting on a portion of a differential segment of the beam.

• The shear formula is to be used on straight prismatic members made of homogeneous material that has linear elastic behavior. Also, the internal resultant shear force must be directed along an axis of symmetry for the cross-sectional area.

• The shear formula should not be used to determine the shear stress on cross sections that are short or flat, at points of sudden cross-sectional changes, or at a point on an inclined boundary.

Procedure for Analysis

In order to apply the shear formula, the following procedure is suggested.

Internal Shear. • Section the member perpendicular to its axis at the point where the shear stress is to be determined, and

obtain the internal shear V at the section.

Section Properties. • Determine the location of the neutral axis, and determine the moment of inertia I of the entire cross-

sectional area about the neutral axis. • Pass an imaginary horizontal section through the point where the shear stress is to be determined. Measure

the width t of the cross-sectional area at this section. • The portion of the area lying either above or below this width is A !. Determine Q by using Q = y !A !. Here

y ! is the distance to the centroid of A !, measured from the neutral axis. It may be helpful to realize that A ! is the portion of the member’s cross-sectional area that is being “held onto the member” by the longitudinal shear stresses. See Fig. 7–4 c .

Shear Stress. • Using a consistent set of units, substitute the data into the shear formula and calculate the shear stress t. • It is suggested that the direction of the transverse shear stress t be established on a volume element of

material located at the point where it is calculated. This can be done by realizing that t acts on the cross section in the same direction as V . From this, the corresponding shear stresses acting on the other three planes of the element can then be established.

7.2 THE SHEAR FORMULA 371

The beam shown in Fig. 7–9a is made from two boards. Determine the maximum shear stress in the glue necessary to hold the boards together along the seam where they are joined.

SOLUTION Internal Shear. The support reactions and the shear diagram for the beam are shown in Fig. 7–9b. lt is seen that the maximum shear in the beam is 19.5 kN. Section Properties. The centroid and therefore the neutral axis will be determined from the reference axis placed at the bottom of the cross-sectional area, Fig. 7–9a. Working in units of meters, we have

y = !y!A

EXAMPLE 7.1

4 m 4 m

6.5 kN/m

(a)

N A

30 mm

150 mm

30 mm _ y

6 m

26 kN

(b)

2 m 19.5 kN6.5 kN

6.5

4 5 8

!19.5

V (kN)

x (m)

4.88 MPa

Plane containing glue

(c)

V " 19.5 kN

Fig. 7–9

= [0.075 m](0.150 m)(0.030 m) + [0.165 m](0.030 m)(0.150 m)

(0.150 m)(0.030 m) + (0.030 m)(0.150 m) = 0.120 m

The moment of inertia, about the neutral axis, Fig. 7–9a, is therefore

I = c 1 12

(0.030 m)(0.150 m)3 + (0.150 m)(0.030 m)(0.120 m - 0.075 m)2 d + c 1

12 (0.150 m)(0.030 m)3 + (0.030 m)(0.150 m)(0.165 m - 0.120 m)2 d

= 27.0(10-6) m4

The top board (fl ange) is being held onto the bottom board (web) by the glue, which is applied over the thickness t = 0.03 m. Consequently A " is defi ned as the area of the top board, Fig. 7–9a. We have

Q = y "A" = [0.180 m - 0.015 m - 0.120 m](0.03 m)(0.150 m)

= 0.2025(10-3) m3

Shear Stress. Using the above data and applying the shear formula yields

tmax = VQ It

= 19.5(103) N(0.2025(10-3) m3)

27.0(10-6) m4(0.030 m) = 4.88 MPa Ans.

The shear stress acting at the top of the bottom board is shown in Fig. 7-9 c .

NOTE: It is the glue’s resistance to this longitudinal shear stress that holds the boards from slipping at the right-hand support .

372 CHAPTER 7 TRANSVERSE SHEAR

Determine the distribution of the shear stress over the cross section of the beam shown in Fig. 7–10 a .

EXAMPLE 7.2

(a)

A¿

(b)

h 2

_ y ¿

*The area below y can also be used 3A ! = b(h>2 + y)4 , but doing so involves a bit more algebraic manipulation.

(c)

Tmax

dy y

Shear–stress distribution

Fig. 7–10

SOLUTION The distribution can be determined by finding the shear stress at an arbitrary height y from the neutral axis, Fig. 7–10 b , and then plotting this function. Here, the dark colored area A ! will be used for Q . * Hence

Q = y !A ! = c y + 1 2

a h 2

- y bd a h 2

- y bb = 1 2 a h2

4 - y2bb

Applying the shear formula, we have

t = VQ It

= V 1122 3(h2>4) - y 24b1 112 bh32b = 6Vbh3 a h24 - y 2b (1)

This result indicates that the shear-stress distribution over the cross section is parabolic . As shown in Fig. 7–10 c , the intensity varies from zero at the top and bottom, y = {h>2 , to a maximum value at the neutral axis, y = 0 . Specifically, since the area of the cross section is A = bh , then at y = 0 we have

tmax = 1.5 V A

(2)

7.2 THE SHEAR FORMULA 373

This same value for tmax can be obtained directly from the shear formula, t = VQ>It , by realizing that tmax occurs where Q is largest , since V, I, and t are constant . By inspection, Q will be a maximum when the entire area above (or below) the neutral axis is considered; that is, A ! = bh>2 and y ! = h>4. Thus,

tmax = VQ It

= V (h>4)(bh>2)3 112bh34b = 1.5 VA

By comparison, tmax is 50% greater than the average shear stress determined from Eq. 1–7 ; that is, tavg = V>A .

It is important to realize that tmax also acts in the longitudinal direction of the beam, Fig. 7–10 d . It is this stress that can cause a timber beam to fail as shown Fig. 7–10 e . Here horizontal splitting of the wood starts to occur through the neutral axis at the beam’s ends, since there the vertical reactions subject the beam to large shear stress and wood has a low resistance to shear along its grains, which are oriented in the longitudinal direction.

It is instructive to show that when the shear-stress distribution, Eq. 1 , is integrated over the cross section it yields the resultant shear V . To do this, a differential strip of area dA = b dy is chosen, Fig. 7–10 c , and since t acts uniformly over this strip, we have

LAt dA = L h>2

-h>2 6Vbh3 ¢ h24 - y 2≤b dy =

6V h3

Jh2 4

y - 1 3

y 3R h>2 -h>2

= 6V h3

Jh2 4

a h 2

+ h 2 b - 1

3 ¢ h3

8 + h

8 ≤ R = V

Typical shear failure of this wooden beam occurred at the support and through the approximate center of its cross section.

(d)

tmax

Fig. 7–10 (cont.)

(e)

374 CHAPTER 7 TRANSVERSE SHEAR

EXAMPLE 7.3

A steel wide-flange beam has the dimensions shown in Fig. 7–11 a . If it is subjected to a shear of V = 80 kN, plot the shear-stress distribution acting over the beam’s cross-sectional area.

(a)

300 mm

15 mm

20 mm

100 mm

V ! 80 kN

1.13 MPa

22.6 MPa

B¿

tB¿ ! 1.13 MPa

tB ! 22.6 MPa

tC ! 25.2 MPa

(b)

N A

0.02 m

0.100 m

0.300 m

B B¿ A¿

SOLUTION Since the flange and web are rectangular elements, then like the previous example, the shear-stress distribution will be parabolic and in this case it will vary in the manner shown in Fig. 7–11 b . Due to symmetry, only the shear stresses at points B !, B , and C have to be determined. To show how these values are obtained, we must first determine the moment of inertia of the cross-sectional area about the neutral axis. Working in meters, we have

I = c 1 12

(0.015 m)(0.200 m)3 d + 2 c 1

12 (0.300 m)(0.02 m)3 + (0.300 m)(0.02 m)(0.110 m)2 d

= 155.6(10-6) m4

For point B !, tB ! = 0.300 m, and A ! is the dark shaded area shown in Fig. 7–11 c . Thus,

QB! = y !A ! = [0.110 m](0.300 m)(0.02 m) = 0.660(10-3) m3

so that

tB! = VQB! ItB!

= 80(103) N(0.660(10-3) m3)

155.6(10-6) m4(0.300 m) = 1.13 MPa

For point B , tB = 0.015 m and QB = QB! , Fig. 7–11 c . Hence

tB = VQB ItB

= 80(103) N(0.660(10-3) m3)

155.6(10-6) m4(0.015 m) = 22.6 MPa

Note from the discussion of “Limitations on the Use of the Shear Formula” that the calculated values for both tB! and tB will actually be very misleading. Why?

7.2 THE SHEAR FORMULA 375

For point C , tC = 0.015 m and A ! is the dark shaded area shown in Fig. 7–11 d . Considering this area to be composed of two rectangles, we have

QC = "y !A ! = [0.110 m](0.300 m)(0.02 m)

+ [0.05 m](0.015 m)(0.100 m)

= 0.735(10-3) m3

Thus,

tC = tmax = VQC ItC

= 80(103) N[0.735(10-3) m3]

155.6(10-6) m4(0.015 m) = 25.2 MPa

NOTE: From Fig. 7–11 b , note that most of the shear stress occurs in the web and is almost uniform throughout its depth, varying from 22.6 MPa to 25.2 MPa. It is for this reason that for design, some codes permit the use of calculating the average shear stress on the cross section of the web rather than using the shear formula; that is,

tavg = V

A w =

8011032 N10.015 m2 10.2 m2 = 26.7 MPa This will be discussed further in Chapter 11.

N A

0.02 m

0.100 m

0.300 m

0.015 m A¿

(d)

Fig. 7–11 (cont.)

376 CHAPTER 7 TRANSVERSE SHEAR

P7–1. In each case, calculate the value of Q and t that are used in the shear formula for finding the shear stress at A. Also, show how the shear stress acts on a differential volume element located at point A .

(a)

0.3 m

0.1 m

0.1 m 0.2 m

(b)

0.3 m

0.1 m

0.2 m

(c)

0.1 m

0.3 m

0.5 m 0.1 m

0.1 m

0.1 m 0.1 m A

(d)

0.2 m

0.2 m 0.2 m

0.2 m 0.1 m

0.2 m

(e)

0.2 m 0.4 m

0.2 m

0.1 m

0.2 m

0.3 m

(f)

0.1 m

0.5 m

0.1 m

0.3 m

0.2 m

A V

PRELIMINARY PROBLEMS

7.2 THE SHEAR FORMULA 377

FUNDAMENTAL PROBLEMS

F7–1. If the beam is subjected to a shear force of V = 100 kN, determine the shear stress developed at point A . Represent the state of stress on a volume element at this point.

200 mm

90 mm

300 mm

20 mm

20 mm V

F7–1

F7–2. Determine the shear stress at points A and B on the beam if it is subjected to a shear force of V = 600 kN. Represent the state of stress on a volume element of these points.

100 mm

100 mm B

A V

F7–2

F7–3. Determine the absolute maximum shear stress developed in the beam.

1 ft 3 in.

6 in.

1 ft1 ft

6 kip 3 kip

F7–3

F7–4. If the beam is subjected to a shear force of V = 20 kN, determine the maximum shear stress developed in the beam.

200 mm

150 mm

50 mm

30 mm

30 mm 30 mm

30 mm

50 mm

F7–4

F7–5. If the beam is made from four plates and subjected to a shear force of V = 20 kN, determine the maximum shear stress developed in the beam at point A . Represent the state of stress on a volume element at this point.

150 mm

50 mm

25 mm

A 150 mm

50 mm

F7–5

378 CHAPTER 7 TRANSVERSE SHEAR

7–1. If the wide-flange beam is subjected to a shear of V = 20 kN, determine the shear stress on the web at A . Indicate the shear-stress components on a volume element located at this point.

7–2. If the wide-flange beam is subjected to a shear of V = 20 kN, determine the maximum shear stress in the beam.

7–3. If the wide-flange beam is subjected to a shear of V = 20 kN, determine the shear force resisted by the web of the beam.

B V

20 mm

300 mm

200 mm

Probs. 7–1/2/3

*7–4. If the T-beam is subjected to a vertical shear of V = 12 kip, determine the maximum shear stress in the beam. Also, compute the shear-stress jump at the flange- web junction AB . Sketch the variation of the shear-stress intensity over the entire cross section.

7–5. If the T-beam is subjected to a vertical shear of V = 12 kip, determine the vertical shear force resisted by the flange.

V ! 12 kip

6 in.

3 in.

4 in.

4 in. 4 in.

Probs. 7–4/5

7–6. The wood beam has an allowable shear stress of t allow = 7 MPa. Determine the maximum shear force V that can be applied to the cross section.

50 mm

200 mm

100 mm 50 mm

50 mm

Prob. 7–6

7–7. The shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B . If P = 20 kN, determine the absolute maximum shear stress in the shaft.

*7–8. The shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B . If the shaft is made from a material having an allowable shear stress of t allow = 75 MPa, determine the maximum value for P .

AC DB

P P

1 m 1 m 1 m

40 mm

30 mm

Probs. 7–7/8

PROBLEMS

7.2 THE SHEAR FORMULA 379

7–13. Determine the maximum shear stress in the strut if it is subjected to a shear force of V = 20 kN.

7–14. Determine the maximum shear force V that the strut can support if the allowable shear stress for the material is tallow = 40 MPa.

V 60 mm

12 mm

20 mm

80 mm

12 mm

Probs. 7–13/14

7–15. The strut is subjected to a vertical shear of V = 130 kN. Plot the intensity of the shear-stress distribution acting over the cross-sectional area, and compute the resultant shear force developed in the vertical segment AB .

50 mm 150 mm

150 mm

50 mm

V ! 130 kN

Prob. 7–15

*7–16. The steel rod has a radius of 1.25 in. If it is subjected to a shear of V = 5 kip, determine the maximum shear stress.

V= 5 kip

1.25 in.

Prob. 7–16

7–9. Determine the largest shear force V that the member can sustain if the allowable shear stress is tallow = 8 ksi.

7–10. If the applied shear force V = 18 kip, determine the maximum shear stress in the member.

V 3 in. 1 in.

1 in.

3 in.

Probs. 7–9/10

7–11. The overhang beam is subjected to the uniform distributed load having an intensity of w = 50 kN>m. Determine the maximum shear stress developed in the beam.

3 m 3 m

50 mm

100 mm

A B

Prob. 7–11

*7–12. The beam has a rectangular cross section and is made of wood having an allowable shear stress of tallow = 200 psi. Determine the maximum shear force V that can be developed in the cross section of the beam. Also, plot the shear-stress variation over the cross section.

12 in.

8 in.

Prob. 7–12

380 CHAPTER 7 TRANSVERSE SHEAR

7–21. If the beam is made from wood having an allowable shear stress t allow = 400 psi, determine the maximum magnitude of P . Set d = 4 in.

2 ft 2 ft 2 ft

P 2P

2 in.

A B

Prob. 7–21

7–22. Determine the shear stress at point B on the web of the cantilevered strut at section a – a .

7–23. Determine the maximum shear stress acting at section a – a of the cantilevered strut.

2 kN 4 kN

250 mm 250 mm 300 mm

20 mm 50 mm

70 mm

20 mm

Probs. 7–22/23

*7–24. Determine the maximum shear stress in the T-beam at the critical section where the internal shear force is maximum.

7–25. Determine the maximum shear stress in the T-beam at section C . Show the result on a volume element at this point.

3 m 1.5 m1.5 m

10 kN/m

150 mm

150 mm 30 mm

30 mm

Probs. 7–24/25

7–17. If the beam is subjected to a shear of V = 15 kN, determine the web’s shear stress at A and B . Indicate the shear-stress components on a volume element located at these points. Set w = 125 mm. Show that the neutral axis is located at y = 0.1747 m from the bottom and I NA = 0.2182(10 −3 ) m 4 .

7–18. If the wide-flange beam is subjected to a shear of V = 30 kN, determine the maximum shear stress in the beam. Set w = 200 mm.

7–19. If the wide-flange beam is subjected to a shear of V = 30 kN, determine the shear force resisted by the web of the beam. Set w = 200 mm.

30 mm 25 mm

30 mm

250 mm

200 mm

w Probs. 7–17/18/19

*7–20. The steel rod is subjected to a shear of 30 kip. Determine the maximum shear stress in the rod.

30 kip

2 in.

Prob. 7–20

7.2 THE SHEAR FORMULA 381

■ 7–29. Write a computer program that can be used to determine the maximum shear stress in the beam that has the cross section shown, and is subjected to a specified constant distributed load w and concentrated force P . Show an application of the program using the values L = 4 m, a = 2 m, P = 1.5 kN, d1 = 0, d2 = 2 m, w = 400 N>m, t1 = 15 mm, t2 = 20 mm, b = 50 mm, and h = 150 mm.

P d1

d2 w

A B

Prob. 7–29

7–30. The beam has a rectangular cross section and is subjected to a load P that is just large enough to develop a fully plastic moment Mp = PL at the fixed support. If the material is elastic-plastic, then at a distance x 6 L the moment M = Px creates a region of plastic yielding with an associated elastic core having a height 2y !. This situation has been described by Eq. 6–30 and the moment M is distributed over the cross section as shown in Fig. 6–48 e . Prove that the maximum shear stress developed in the beam is given by tmax = 32(P>A !), where A ! = 2y !b, the cross-sectional area of the elastic core.

b L

P x

Plastic region

2y¿

Elastic region

Prob. 7–30

7–31. The beam in Fig. 6–48 f is subjected to a fully plastic moment Mp . Prove that the longitudinal and transverse shear stresses in the beam are zero. Hint: Consider an element of the beam as shown in Fig. 7–4 c .

7–26. The beam has a square cross section and is made of wood having an allowable shear stress of t allow = 1.4 ksi. If it is subjected to a shear of V = 1.5 kip, determine the smallest dimension a of its sides.

V = 1.5 kip

Prob. 7–26

7–27. The beam is slit longitudinally along both sides as shown. If it is subjected to an internal shear of V = 250 kN, compare the maximum shear stress developed in the beam before and after the cuts were made.

*7–28. The beam is to be cut longitudinally along both sides as shown. If it is made from a material having an allowable shear stress of t allow = 75 MPa, determine the maximum allowable internal shear force V that can be applied before and after the cut is made.

200 mm 25 mm

25 mm

200 mm

25 mm

100 mm

Probs. 7–27/28

382 CHAPTER 7 TRANSVERSE SHEAR

7.3 Shear Flow in Built-Up Members Occasionally in engineering practice, members are “built up” from several composite parts in order to achieve a greater resistance to loads. Examples are shown in Fig. 7–12 . If the loads cause the members to bend, fasteners such as nails, bolts, welding material, or glue may be needed to keep the component parts from sliding relative to one another, Fig. 7–2 . In order to design these fasteners or determine their spacing, it is necessary to know the shear force that must be resisted by the fastener. This loading, when measured as a force per unit length of beam, is referred to as shear flow q . *

The magnitude of the shear flow can be obtained using a development similar to that for finding the shear stress in the beam. To show this, we will consider finding the shear flow along the juncture where the segment in Fig. 7–13 a is connected to the flange of the beam. As shown in Fig. 7–13 b , three horizontal forces must act on this segment. Two of these forces, F and F + dF, are developed by normal stresses caused by the moments M and M + dM, respectively. The third force, which for equilibrium equals dF , acts at the juncture and it is to be supported by the fastener. Realizing that dF is the result of dM , then, like Eq. 7–1 , we have

dF = dM

I LA !y dA ! The integral represents Q , that is, the moment of the segment's area A! in Fig. 7–13 b about the neutral axis for the entire cross section. Since the segment has a length dx , the shear flow, or force per unit length along the beam, is q = dF>dx. Hence dividing both sides by dx and noting that V = dM >dx, Eq. 6–2 , we can write q =

VQ I

(7–4)

Here q = the shear flow, measured as a force per unit length along

the beam V = the internal resultant shear force, determined from the method

of sections and the equations of equilibrium I = the moment of inertia of the entire cross-sectional area computed

about the neutral axis Q = y !A! , where A! is the cross-sectional area of the segment that is

connected to the beam at the juncture where the shear flow is to be calculated, and y ! is the distance from the neutral axis to the centroid of A!

* The use of the word “flow” in this terminology will become meaningful as it pertains to the discussion in Sec. 7.5 .

Fig. 7–12

7.3 SHEAR FLOW IN BUILT-UP MEMBERS 383

M ! dM

(a)

F ! dF (b)

t F

A¿

Fig. 7–13

Application of this equation follows the same “procedure for analysis” as outlined in Sec. 7.2 for the shear formula. It is very important to identify Q correctly when determining the shear flow at a particular junction on the cross section. A few examples should serve to illustrate how this is done. Consider the beam cross sections shown in Fig. 7–14 . The shaded segments are connected to the beam by fasteners and at the planes of connection, identified by the thick black lines, the shear flow q is determined by using a value of Q calculated from A ! and y ! indicated in each figure. This value of q will be resisted by a single fastener in Fig. 7–14 a , by two fasteners in Fig. 7–14 b , and by three fasteners in Fig. 7–14 c . In other words, the fastener in Fig. 7–14 a supports the calculated value of q , and in Figs. 7–14 b and 7–14 c each fastener supports q >2 and q >3, respectively.

A¿

(a)

_ y ¿

N A

(b)

A¿

_ y ¿

A¿

(c)

_ y ¿

Fig. 7–14

Important Point

• Shear flow is a measure of the force per unit length along the axis of a beam. This value is found from the shear formula and is used to determine the shear force developed in fasteners and glue that holds the various segments of a composite beam together.

384 CHAPTER 7 TRANSVERSE SHEAR

The beam is constructed from three boards glued together as shown in Fig. 7–15 a . If it is subjected to a shear of V = 850 kN, determine the shear flow at B and B ! that must be resisted by the glue.

SOLUTION Section Properties. The neutral axis (centroid) will be located from the bottom of the beam, Fig. 7–15 a . Working in units of meters, we have

y = "y!A

"A =

2[0.15 m](0.3 m)(0.01 m) + [0.305 m](0.250 m)(0.01 m) 2(0.3 m)(0.01 m) + 0.250 m(0.01 m)

= 0.1956 m

The moment of inertia about the neutral axis is thus

I = 2 c 1 12

(0.01 m)(0.3 m)3 + (0.01 m)(0.3 m)(0.1956 m - 0.150 m)2 d + c 1

12 (0.250 m)(0.01 m)3 + (0.250 m)(0.01 m)(0.305 m - 0.1956 m)2 d

= 87.42(10-6) m4

Since the glue at B and B ! in Fig. 7–15 a “holds” the top board to the beam, we have

QB = y=BA =B = [0.305 m - 0.1956 m](0.250 m)(0.01 m) = 0.2735(10-3) m3

Shear Flow. For B and B ! we have

q=B = VQB

I =

850(103) N(0.2735(10-3) m3)

87.42(10-6) m4 = 2.66 MN>m

Since two seams are used to secure the board, the glue per meter length of beam at the seam must be strong enough to resist one-half of the calculated value of q! B. Thus,

qB = qB ! = 1.33 MN>m Ans. If the board CC ! is added to the beam, Fig. 7–15b, then y and I have to be recalculated, and the shear flow at C and C! determined from q! C = V y ! C A! C>I. Finally, this value is divided by one-half to obtain qC and q! C .

EXAMPLE 7.4

250 mm 10 mm

300 mm

125 mm 10 mm10 mm

_ y

_ y¿B

V # 850 kN

B B¿

(a)

A¿B

_ y¿C

(b)

C ¿

A¿C

Fig. 7–15

7.3 SHEAR FLOW IN BUILT-UP MEMBERS 385

A box beam is constructed from four boards nailed together as shown in Fig. 7–16 a . If each nail can support a shear force of 30 lb, determine the maximum spacing s of the nails at B and at C so that the beam will support the force of 80 lb.

SOLUTION Internal Shear. If the beam is sectioned at an arbitrary point along its length, the internal shear required for equilibrium is always V = 80 lb, and so the shear diagram is shown in Fig. 7–16 b .

Section Properties. The moment of inertia of the cross-sectional area about the neutral axis can be determined by considering a 7.5-in. * 7.5-in square minus a 4.5-in. * 4.5-in. square.

I = 1

12 (7.5 in.)(7.5 in.)3 - 1

12 (4.5 in.)(4.5 in.)3 = 229.5 in4

The shear flow at B is determined using QB found from the darker shaded area shown in Fig. 7–16 c . It is this “symmetric” portion of the beam that is to be “held” onto the rest of the beam by nails on the left side and by the fibers of the board on the right side. Thus,

QB = y !A ! = [3 in.](7.5 in.)(1.5 in.) = 33.75 in3

Likewise, the shear flow at C can be determined using the “symmetric” shaded area shown in Fig. 7–16 d . We have

QC = y !A ! = [3 in.](4.5 in.)(1.5 in.) = 20.25 in3

Shear Flow.

qB = VQB

I =

80 lb(33.75 in3)

229.5 in4 = 11.76 lb>in.

qC = VQC

I =

80 lb(20.25 in3)

229.5 in4 = 7.059 lb>in.

These values represent the shear force per unit length of the beam that must be resisted by the nails at B and the fibers at B !, Fig. 7–16 c , and the nails at C and the fibers at C!, Fig. 7–16 d , respectively. Since in each case the shear flow is resisted at two surfaces and each nail can resist 30 lb, for B the spacing is

sB = 30 lb

(11.76>2) lb>in. = 5.10 in. Use sB = 5 in. Ans. And for C ,

sC = 30 lb

(7.059>2) lb>in. = 8.50 in. Use sC = 8.5 in. Ans.

EXAMPLE 7.5

(a)

80 lb

6 in. 1.5 in.

6 in.

1.5 in.

(b)

V (lb)

x (ft)

(c)

7.5 in.

B B¿ AN

3 in. 1.5 in.

4.5 in.

C ¿ AN

(d)

3 in. 1.5 in.

Fig. 7–16

386 CHAPTER 7 TRANSVERSE SHEAR

EXAMPLE 7.6

Nails having a total shear strength of 40 lb are used in a beam that can be constructed either as in Case I or as in Case II, Fig. 7–17 . If the nails are spaced at 9 in., determine the largest vertical shear that can be supported in each case so that the fasteners will not fail.

3 in.

1 in. 4 in.

0.5 in.

Case I

A N ! 9 in.s

1 in.

5 in.

0.5 in.

Case II

1 in.1 in.

! 9 in.s

0.5 in.

Fig. 7–17

SOLUTION Since the cross section is the same in both cases, the moment of inertia about the neutral axis is

I = 1

12 (3 in.)(5 in.)3 - 2 c 1

12 (1 in.)(4 in.)3 d = 20.58 in4

Case I. For this design a single row of nails holds the top or bottom flange onto the web. For one of these flanges,

Q = y !A ! = [2.25 in.](3 in.(0.5 in.)) = 3.375 in3 so that

q = VQ

40 lb 9 in.

= V (3.375 in3)

20.58 in4

V = 27.1 lb Ans.

Case II. Here a single row of nails holds one of the side boards onto the web. Thus,

Q = y !A ! = [2.25 in.](1 in.(0.5 in.)) = 1.125 in3

q = VQ

40 lb 9 in.

= V (1.125 in3)

20.58 in4

V = 81.3 lb Ans.

7.3 SHEAR FLOW IN BUILT-UP MEMBERS 387

FUNDAMENTAL PROBLEMS

F7–6. The two identical boards are bolted together to form the beam. Determine the maximum allowable spacing s of the bolts to the nearest mm if each bolt has a shear strength of 15 kN. The beam is subjected to a shear force of V = 50 kN.

100 mm

300 mm

F7–6

F7–7. Two identical 20-mm thick plates are bolted to the top and bottom flange to form the built-up beam. If the beam is subjected to a shear force of V = 300 kN, determine the allowable maximum spacing s of the bolts to the nearest mm. Each bolt has a shear strength of 30 kN.

20 mm

10 mm

200 mm

10 mm

20 mm

300 mm

F7–7

F7–8. The boards are bolted together to form the built-up beam. If the beam is subjected to a shear force of V = 20 kN, determine the allowable maximum spacing of the bolts to the nearest mm. Each bolt has a shear strength of 8 kN.

200 mm

25 mm

50 mm

25 mm

50 mm

150 mm

s s

F7–8

F7–9. The boards are bolted together to form the built-up beam. If the beam is subjected to a shear force of V = 15 kip, determine the allowable maximum spacing of the bolts to the nearest 18 in. Each bolt has a shear strength of 6 kip.

4 in.

3 in.

1 in.

4 in.

1 in.

0.5 in. 0.5 in.

s s

F7–9

388 CHAPTER 7 TRANSVERSE SHEAR

*7–32. The beam is constructed from two boards fastened together at the top and bottom with two rows of nails spaced every 6 in. If each nail can support a 500-Ib shear force, determine the maximum shear force V that can be applied to beam.

7–33. The beam is construced from two boards fastened together at the top and bottom with two rows of nails spaced every 6 in. If an internal shear force of V = 600 lb is applied to the boards, determine the shear force resisted by each nail.

V 2 in.

6 in.

2 in.

*7–36. Three identical boards are bolted together to form the built-up beam. Each bolt has a shear strength of 1.5 kip and the bolts are spaced at a distance of s = 6 in. If the wood has an allowable shear stress of t allow = 450 psi, determine the maximum allowable internal shear V that can act on the beam.

7–37. Three identical boards are bolted together to form the built-up beam. If the wood has an allowable shear stress of t allow = 450 psi, determine the maximum allowable internal shear V that can act on the beam. Also, find the corresponding average shear stress in the 38 in. diameter bolts which are spaced equally at s = 6 in.

2 in.

1 in. 1 in.

1 in.

2 in.

s s

Probs. 7–32/33

7–34. The boards are glued together to form the built-up beam. If the wood has an allowable shear stress of t allow = 3 MPa, and the glue seam at B can withstand a maximum shear stress of 1.5 MPa, determine the maximum allowable internal shear V that can be developed in the beam.

7–35. The boards are glued together to form the built-up beam. If the wood has an allowable shear stress of t allow = 3 MPa, and the glue seam at D can withstand a maximum shear stress of 1.5 MPa, determine the maximum allowable shear V that can be developed in the beam.

50 mm

150 mm

50 mm 50 mm

Probs. 7–34/35

Probs. 7–36/37

7–38. The beam is subjected to a shear of V = 2 kN. Determine the average shear stress developed in each nail if the nails are spaced 75 mm apart on each side of the beam. Each nail has a diameter of 4 mm.

75 mm 75 mm

50 mm

25 mm

200 mm

25 mm V

Prob. 7–38

PROBLEMS

7.3 SHEAR FLOW IN BUILT-UP MEMBERS 389

7–39. A beam is constructed from three boards bolted together as shown. Determine the shear force developed in each bolt if the bolts are spaced s = 250 mm apart and the applied shear is V = 35 kN.

s ! 250 mm

250 mm100 mm

25 mm

25 mm 25 mm

350 mm V

Prob. 7–39

*7–40. The simply-supported beam is built-up from three boards by nailing them together as shown. The wood has an allowable shear stress of t allow = 1.5 MPa, and an allowable bending stress of s allow = 9 MPa. The nails are spaced at s = 75 mm, and each has a shear strength of 1.5 kN. Determine the maximum allowable force P that can be applied to the beam.

7–41. The simply-supported beam is built-up from three boards by nailing them together as shown. If P = 12 kN, determine the maximum allowable spacing s of the nails to support that load, if each nail can resist a shear force of 1.5 kN.

1 m 1 m

s P

100 mm

200 mm

25 mm

A B

Probs. 7–40/41

7–42. The T-beam is constructed as shown. If the nails can each support a shear force of 950 lb, determine the maximum shear force V that the beam can support and the corresponding maximum nail spacing s to the nearest 18 in. The allowable shear stress for the wood is tallow = 450 psi .

12 in.

12 in.2 in.

2 in.

Prob. P7–42

7–43. The box beam is made from four pieces of plastic that are glued together as shown. If the glue has an allowable strength of 400 lb> in 2 , determine the maximum shear the beam will support.

*7–44. The box beam is made from four pieces of plastic that are glued together as shown. If V = 2 kip, determine the shear stress resisted by the seam at each of the glued joints.

5.5 in. 0.25 in.

0.25 in.

4.75 in.

0.25 in.

Probs. 7–43/44

390 CHAPTER 7 TRANSVERSE SHEAR

7–47. The beam is made from four boards nailed together as shown. If the nails can each support a shear force of 100 lb., determine their required spacing s ! and s if the beam is subjected to a shear of V = 700 lb .

1.5 in.

10 in.

2 in.

1 in.

10 in.

1 in.

s s

s¿ s¿

Prob. 7–47

*7–48. The beam is made from three polystyrene strips that are glued together as shown. If the glue has a shear strength of 80 kPa, determine the maximum load P that can be applied without causing the glue to lose its bond.

P 1 — 4 P

1 — 4

0.8 m 0.8 m1 m 1 m

A B

20 mm

40 mm

30 mm

60 mm

40 mm

Prob. 7–48

■ 7–49. The timber T-beam is subjected to a load consisting of n concentrated forces, Pn . If the allowable shear V nail for each of the nails is known, write a computer program that will specify the nail spacing between each load. Show an application of the program using the values L = 15 ft, a1 = 4 ft , P1 = 600 lb , a2 = 8 ft , P2 = 1500 lb, b1 = 1.5 in., h1 = 10 in., b2 = 8 in., h2 = 1 in., and V nail = 200 lb.

P1 P2 Pn

a2 an

s1 s2 s3 sn

A B

Prob. 7–49

7–45. A beam is constructed from four boards which are nailed together. If the nails are on both sides of the beam and each can resist a shear of 3 kN, determine the maximum load P that can be applied to the end of the beam.

2 m 2 m

3 kN

B CA

30 mm

30 mm 30 mm

100 mm

250 mm30 mm

150 mm

Prob. 7–45

7–46. The beam is subjected to a shear of V = 800 N. Determine the average shear stress developed in the nails along the sides A and B if the nails are spaced s = 100 mm apart. Each nail has a diameter of 2 mm.

150 mm

250 mm

30 mm

B 100 mm

100 mm

30 mm V

Prob. 7–46

7.4 SHEAR FLOW IN THIN-WALLED MEMBERS 391

7.4 Shear Flow in Thin-Walled Members In this section we will show how to apply the shear flow equation q = VQ>I to find the shear-flow distribution throughout a member’s cross-sectional area. We will assume that the member has thin walls , that is, the wall thickness is small compared to its height or width. As will be shown in the next section, this analysis has important applications in structural and mechanical design.

Like the shear stress, the shear flow acts on both the longitudinal and transverse planes of the member. To show how to establish its direction on the cross section, consider the segment dx of the wide-flange beam in Fig. 7–18 a . Free-body diagrams of two segments B and C taken from the top flange are shown in Figs. 7–18 b and 7–18 c . The force dF must act on the longitudinal section in order to balance the normal forces F and F + dF created by the moments M and M + dM , respectively. Now, if the corner elements B and C of each segment are removed, then the transverse components q act on the cross section as shown in Figs. 7–18 b and 7–18 c . Using this method, show that the shear flow at the corresponding points B ! and C! on the bottom flange in Fig. 7–18 d is directed as shown.

Although it is also true that V + dV will create vertical shear-flow components on this element, here we will neglect its effects. This is because this component, like the shear stress, is approximately zero throughout the thickness of the element. Here, the flange is thin and the top and bottom surfaces of the element are free of stress, Fig. 7–18 e . To summarize then, only the shear flow component that acts parallel to the sides of the flange will be considered.

(a)

M ! dM

V ! dV

(b)

F ! dF dF

BdA

(c)

F ! dF dx

(d)

B¿

C ¿

q assumed constant throughout flange thickness

q¿ assumed to be zero throughout flange thickness since top and bottom of flange are stress free

(e) Fig. 7–18

392 CHAPTER 7 TRANSVERSE SHEAR

Having determined the direction of the shear flow in each flange, we can now find its distribution along the top right flange of the beam in Fig. 7–19 a . To do this, consider the shear flow q , acting on the colored element dx , located an arbitrary distance x from the centerline of the cross section, Fig. 7–19 b . Here Q = y !A ! = [d>2](b>2 - x)t, so that q =

VQ I

= V [d>2](b>2 - x)t

I =

Vt d 2I

a b 2

- x b (7–5) By inspection, this distribution varies in a linear manner from q = 0 at x = b>2 to (qmax)f = Vt db>4I at x = 0. (The limitation of x = 0 is possible here since the member is assumed to have “thin walls” and so the thickness of the web is neglected.) Due to symmetry, a similar analysis yields the same distribution of shear flow for the other flange segments, so that the results are as shown in Fig. 7–19 d .

The total force developed in each flange segment can be determined by integration. Since the force on the element dx in Fig. 7–19 b is dF = q dx, then

Ff = Lq dx = L b>2

Vt d 2I

a b 2

- x b dx = Vt db2 16I

We can also determine this result by finding the area under the triangle in Fig. 7–19 d . Hence,

Ff = 1 2

(qmax)fa b2 b = Vt db216I All four of these forces are shown in Fig. 7–19 e , and we can see from their direction that horizontal force equilibrium of the cross section is maintained.

(b)

N A

q d 2

b 2

Fig. 7–19

V N

(a)

d 2

(c)

N A

b t

y q

dy d 2

7.4 SHEAR FLOW IN THIN-WALLED MEMBERS 393

A similar analysis can be performed for the web, Fig. 7–19 c . Here q must act downward, and at element dy we have Q = !y "A " = [d>2](bt) + [ y + (1>2)(d >2 - y)] t (d>2 - y ) = bt d>2 + (t>2)(d2>4 - y2), so that q =

VQ I

= Vt I

Jdb 2

+ 1 2

¢ d2 4

- y2≤ R (7–6) For the web, the shear flow varies in a parabolic manner , from q = 2(qmax)f = Vt db>2I at y = d>2 to (qmax)w = (Vt d>I)(b>2 + d>8) at y = 0, Fig. 7–19 d .

Integrating to determine the force in the web, Fw , we have,

Fw = Lq dy = L d>2

-d>2 VtI Jdb2 + 12 ¢ d24 - y2≤ R dy =

Vt I Jdb

2 y + 1

2 ¢ d2

4 y - 1

3 y3≤ R 2

-d>2 d>2

= Vtd2

4I a2b + 1

3 db

Simplification is possible by noting that the moment of inertia for the cross-sectional area is

I = 2J 1 12

bt3 + bta d 2 b2R + 1

12 td3

Neglecting the first term, since the thickness of each flange is small, then

I = td2

4 a2b + 1

3 db

Substituting this into the above equation, we see that Fw = V , which is to be expected, Fig. 7–19 e .

Shear-flow distribution

(d)

(qmax)f

(qmax)w

2(qmax)f

(e)

Ff Ff

FfFf

Fw ! V

Fig. 7–19 (cont.)

394 CHAPTER 7 TRANSVERSE SHEAR

From the foregoing analysis, three important points should be observed. First, the value of q changes over the cross section, since Q will be different for each area segment A ! for which it is determined. In particular, q will vary linearly along segments (flanges) that are perpendicular to the direction of V , and parabolically along segments (web) that are inclined or parallel to V . Second, q will always act parallel to the walls of the member, since the section on which q is calculated is taken perpendicular to the walls. And third, the directional sense of q is such that the shear appears to “ flow ” through the cross section, inward at the beam’s top flange, “combining” and then “flowing” downward through the web, since it must contribute to the shear force V , and then separating and “flowing” outward at the bottom flange. If one is able to “visualize” this “flow” it will provide an easy means for establishing not only the direction of q , but also the corresponding direction of t . Other examples of how q is directed along the segments of thin-walled members are shown in Fig. 7–20 . In all cases, symmetry prevails about an axis that is collinear with V . As a result, q “flows” in a direction such that it will provide the vertical force V and yet also satisfy horizontal force equilibrium for the cross section.

Shear flow q

Fig. 7–20

Important Points

• The shear flow formula q = VQ>I can be used to determine the distribution of the shear flow throughout a thin-walled member provided the shear V acts along an axis of symmetry or principal centroidal axis of inertia for the cross section.

• If a member is made from segments having thin walls, only the shear flow parallel to the walls of the member is important.

• The shear flow varies linearly along segments that are perpendicular to the direction of the shear V .

• The shear flow varies parabolically along segments that are inclined or parallel to the direction of the shear V .

• On the cross section, the shear “flows” along the segments so that it results in the vertical shear force V and yet satisfies horizontal force equilibrium.

7.4 SHEAR FLOW IN THIN-WALLED MEMBERS 395

The thin-walled box beam in Fig. 7–21 a is subjected to a shear of 10 kip. Determine the variation of the shear flow throughout the cross section.

SOLUTION By symmetry, the neutral axis passes through the center of the cross section. For thin-walled members we use centerline dimensions for calculating the moment of inertia.

I = 1

12 (2 in.)(7 in.)3 + 2 [(5 in.)(1 in.)(3.5 in.)2] = 179.7 in.4

Only the shear flow at points B , C , and D has to be determined. For point B , the area A ! ! 0, Fig. 7–21 b , since it can be thought of as being located entirely at point B . Alternatively, A ! can also represent the entire cross-sectional area, in which case QB = y !A ! = 0 since y ! = 0. Because QB = 0, then

qB = 0

For point C , the area A ! is shown dark shaded in Fig. 7–21 c . Here, we have used the mean dimensions since point C is on the centerline of each segment. We have

QC = y !A ! = (3.5 in.)(5 in.)(1 in.) = 17.5 in3

Since there are two points of attachment,

qC = 1 2 aVQC

I b = 1

2 a 10 kip(17.5 in3))

179.7 in4 b = 0.487 kip>in.

The shear flow at D is determined using the three dark-shaded rectangles shown in Fig. 7–21 d . Again, using centerline dimensions

QD = "y !A ! = 2 c3.5 in.2 d(1 in.)(3.5 in.) + [3.5 in.](5 in.)(1 in.) = 29.75 in3 Because there are two points of attachment,

qD = 1 2 aVQD

I b = 1

2 a 10 kip(29.75 in3)

179.7 in4 b = 0.828 kip>in.

Using these results, and the symmetry of the cross section, the shear-flow distribution is plotted in Fig. 7–21 e . The distribution is linear along the horizontal segments (perpendicular to V ) and parabolic along the vertical segments (parallel to V ).

EXAMPLE 7.7

(a)

A N

3 in.

3 in. 1 in.

1 in.

1 in.2 in.2 in. 1 in.

10 kip

(b)

A¿

3.5 in.

4 in.

5 in.

N A

1 in.

1 in. 4 in.

(c)

5 in.

3.5 in. N A

(d)

3.5 in.

Fig. 7–21

N A

(e)

0.828 kip/ in.

0.487 kip/ in.

396 CHAPTER 7 TRANSVERSE SHEAR

*7.5 Shear Center for Open Thin-Walled Members

In the previous section, it was assumed that the internal shear V was applied along a principal centroidal axis of inertia that also represents an axis of symmetry for the cross section. In this section we will consider the effect of applying the shear along a principal centroidal axis that is not an axis of symmetry. As before, only open thin-walled members will be analyzed, so the dimensions to the centerline of the walls of the members will be used. A typical example of this case is the channel section shown in Fig. 7–22 a . Here it is cantilevered from a fixed support and is subjected to the force P . If this force is applied along the once vertical, unsymmetrical axis that passes through the centroid C of the cross section, the channel will not only bend downward, it will also twist clockwise as shown.

(a)

(e)

(qmax)w

(qmax)f

Shear-flow distribution

(b)

dC A

V ! P

(c)

(d)

Fig. 7–22

7.5 SHEAR CENTER FOR OPEN THIN-WALLED MEMBERS 397

To understand why the member twists, it is necessary to show the shear-flow distribution along the channel’s flanges and web, Fig. 7–22 b . When this distribution is integrated over the flange and web areas, it will give resultant forces of Ff in each flange and a force of V = P in the web, Fig. 7–22 c . If the moments of these forces are summed about point A , it can be seen that the couple or torque created by the flange forces is responsible for twisting the member. The actual twist is clockwise when viewed from the front of the beam as shown in Fig. 7–22 a , since reactive internal “equilibrium” forces Ff cause the twisting. In order to prevent this twisting it is therefore necessary to apply P at a point O located an eccentric distance e from the web of the channel, Fig. 7–22 d . We require !MA = Ff d = Pe, or

e = Ff d P

Using the method discussed in the previous section, Ff can be evaluated in terms of P (=V ) and the dimensions of the flanges and web. Once this is done, then P will cancel upon substitution into the above equation, and it becomes possible to express e simply as a function of the cross-sectional geometry (see Example 7.8 ). The point O so located is called the shear center or flexural center . When P is applied at the shear center, the beam will bend without twisting as shown in Fig. 7–22 e . Design handbooks often list the location of this point for a variety of beams having thin- walled cross sections that are commonly used in practice.

From this analysis, it should be noted that the shear center will always lie on an axis of symmetry of a member’s cross-sectional area. For example, if the channel is rotated 90° and P is applied at A , Fig. 7–23 a , no twisting will occur since the shear flow in the web and flanges for this case is symmetrical , and therefore the force resultants in these elements will create zero moments about A , Fig. 7–23 b . Obviously, if a member has a cross section with two axes of symmetry, as in the case of a wide-flange beam, the shear center will then coincide with the intersection of these axes (the centroid).

Notice how this cantilever beam deflects when loaded through the centroid (above) and through the shear center (below).

(a)

A A

(b)

Ff Ff

!V " P2V " P 2

Fig. 7–23

398 CHAPTER 7 TRANSVERSE SHEAR

Important Points

• The shear center is the point through which a force can be applied which will cause a beam to bend and yet not twist.

• The shear center will always lie on an axis of symmetry of the cross section.

• The location of the shear center is only a function of the geometry of the cross section, and does not depend upon the applied loading.

Procedure for Analysis

The location of the shear center for an open thin-walled member for which the internal shear is in the same direction as a principal centroidal axis for the cross section may be determined by using the following procedure.

Shear-Flow Resultants. • By observation, determine the direction of the shear flow

through the various segments of the cross section, and sketch the force resultants on each segment of the cross section. (For example, see Fig. 7–22 c .) Since the shear center is determined by taking the moments of these force resultants about a point ( A ), choose this point at a location that eliminates the moments of as many force resultants as possible.

• The magnitudes of the force resultants that create a moment about A must be calculated. For any segment this is done by determining the shear flow q at an arbitrary point on the segment and then integrating q along the segment’s length. Realize that V will create a linear variation of shear flow in segments that are perpendicular to V , and a parabolic variation of shear flow in segments that are parallel or inclined to V .

Shear Center. • Sum the moments of the shear-flow resultants about point A and

set this moment equal to the moment of V about A . Solve this equation to determine the moment-arm or eccentric distance e , which locates the line of action of V from A .

• If an axis of symmetry for the cross section exists, the shear center lies at the point where this axis intersects the line of action of V .

7.5 SHEAR CENTER FOR OPEN THIN-WALLED MEMBERS 399

Determine the location of the shear center for the thin-walled channel section having the dimensions shown in Fig. 7–24 a .

SOLUTION Shear-Flow Resultants. A vertical downward shear V applied to the section causes the shear to flow through the flanges and web as shown in Fig. 7–24 b . This causes force resultants Ff and V in the flanges and web as shown in Fig. 7–24 c . We will take moments about point A so that only the force Ff on the lower flange has to be determined.

The cross-sectional area can be divided into three component rectangles—a web and two flanges. Since each component is assumed to be thin, the moment of inertia of the area about the neutral axis is

I = 1

12 th3 + 2Jbta h

2 b2R = th2

2 a h

6 + bb

From Fig. 7–24 d , q at the arbitrary position x is

q = VQ

I =

V (h>2)[b - x]t (th2>2)[(h>6) + b] = V (b - x)h[(h>6) + b]

Hence, the force Ff is

Ff = L b

0 q dx =

V h[(h>6) + b] Lb0 (b - x) dx = Vb22h[(h>6) + b]

This same result can also be determined by first finding (qmax)f , Fig. 7–24 b , then determining the triangular area 12 b(qmax)f = Ff .

Shear Center. Summing moments about point A , Fig. 7–24 c , we require

Ve = Ff h = Vb2h

2h[(h>6) + b] Thus,

e = b2

[(h>3) + 2b] Ans. As stated previously, e depends only on the geometry of the

cross section.

EXAMPLE 7.8

(d)

AN q

h 2

Fig. 7–24

(a)

Shear flow distribution

(b)

(qmax)w

(qmax)f

A Ff

P ! V

(c)

400 CHAPTER 7 TRANSVERSE SHEAR

Determine the location of the shear center for the angle having equal legs, Fig. 7–25 a . Also, find the internal shear force resultant in each leg.

EXAMPLE 7.9

(c)

O O!

Fig. 7–25

SOLUTION When a vertical downward shear V is applied at the section, the shear flow and shear-flow resultants are directed as shown in Figs. 7–25 b and 7–25 c , respectively. Note that the force F in each leg must be equal, since for equilibrium the sum of their horizontal components must be equal to zero. Also, the lines of action of both forces intersect point O ; therefore, this point must be the shear center since the sum of the moments of these forces and V about O is zero, Fig. 7–25 c .

The magnitude of F can be determined by first finding the shear flow at the arbitrary location s along the top leg, Fig. 7–25 d . Here

Q = y !A ! = 122 a (b - s) + s2 b ts = 122 ab - s2 bst

45!

(a)

Shear-flow distribution

(b)

qmax

7.5 SHEAR CENTER FOR OPEN THIN-WALLED MEMBERS 401

The moment of inertia of the angle, about the neutral axis, must be determined from “first principles,” since the legs are inclined with respect to the neutral axis. For the area element dA = t ds, Fig. 7–25 e , we have

I = LAy2 dA = 2L b

0 J 122 (b - s)R 2t ds = tab2s - bs2 + 13 s3b 2 0b = tb33

Thus, the shear flow is

q = VQ

I =

V (tb3>3) J 122 ab - s2 bst R

= 3V22b3 sab - s2 b

The variation of q is parabolic, and it reaches a maximum value when s = b as shown in Fig. 7–25 b . The force F is therefore

F = L b

0 q ds =

3V22b3 Lb0 sab - s2 b ds =

3V22b3 ¢b s22 - 16 s3≤ 2 0b =

122 V Ans. NOTE: This result can be easily verified since the sum of the vertical components of the force F in each leg must equal V and, as stated previously, the sum of the horizontal components equals zero.

(d)

q 45!

_ y ¿

45!

(e)

y ds

Fig. 7–25 (cont.)

402 CHAPTER 7 TRANSVERSE SHEAR

7–50. A shear force of V = 300 kN is applied to the box girder. Determine the shear flow at points A and B.

7–51. A shear force of V = 450 kN is applied to the box girder. Determine the shear flow at points C and D.

100 mm

90 mm90 mm

200 mm

180 mm

190 mm

10 mm

A D C

Probs. 7–50/51

*7–52. A shear force of V = 18 kN is applied to the symmetric box girder. Determine the shear flow at A and B .

7–53. A shear force of V = 18 kN is applied to the box girder. Determine the shear flow at C .

150 mm

10 mm

100 mm

10 mm

125 mm

150 mm

10 mm 30 mm

10 mm

10 mm 30 mm

Probs. 7–52/53

7–54. The aluminum strut is 10 mm thick and has the cross section shown. If it is subjected to a shear of, V = 150 N , determine the shear flow at points A and B .

7–55. The aluminum strut is 10 mm thick and has the cross section shown. If it is subjected to a shear of V = 150 N , determine the maximum shear flow in the strut.

30 mm

40 mm 30 mm

A B40 mm

10 mm

10 mm10 mm

Probs. 7–54/55

*7–56. The beam is subjected to a shear force of V = 5 kip . Determine the shear flow at points A and B .

7–57. The beam is constructed from four plates and is subjected to a shear force of V = 5 kip . Determine the maximum shear flow in the cross section.

0.5 in.

5 in.

0.5 in.

2 in. 0.5 in.

8 in.

Probs. 7–56/57

PROBLEMS

7.5 SHEAR CENTER FOR OPEN THIN-WALLED MEMBERS 403

7–62. Determine the shear-stress variation over the cross section of the thin-walled tube as a function of elevation y and show that tmax = 2V >A , where A = 2prt. Hint: Choose a differential area element dA = Rt du . Using dQ = y dA , formulate Q for a circular section from u to (p - u) and show that Q = 2R2t cos u, where cos u = 2R2 - y2 >R.

y du

Prob. 7–62

7–63. Determine the location e of the shear center, point O , for the thin-walled member having the cross section shown. The member segments have the same thickness t .

1 in.

1.80 in.

1 in.

Prob. 7–3

*7–64. Determine the location e of the shear center, point O , for the thin-walled member having the cross section shown. The member segments have the same thickness t .

O h2h1

Prob. 7–64

7–58. The channel is subjected to a shear of V = 75 kN . Determine the shear flow developed at point A .

7–59. The channel is subjected to a shear of V = 75 kN . Determine the maximum shear flow in the channel.

200 mm

30 mm

V ! 75 kN

30 mm

400 mm

Probs. 7–58/59

*7–60. The built-up beam is formed by welding together the thin plates of thickness 5 mm. Determine the location of the shear center O .

200 mm

300 mm

100 mm

100 mme

5 mm

Prob. 7–60

7–61. The assembly is subjected to a vertical shear of V = 7 kip . Determine the shear flow at points A and B and the maximum shear flow in the cross section.

6 in.

0.5 in.

2 in.

2 in. 6 in.

0.5 in.

0.5 in. 0.5 in.

Prob. 7–61

404 CHAPTER 7 TRANSVERSE SHEAR

*7–68. A thin plate of thickness t is bent to form the beam having the cross section shown. Determine the location of the shear center O .

Prob. 7–68

7–69. A thin plate of thickness t is bent to form the beam having the cross section shown. Determine the location of the shear center O .

Prob. 7–69

7–70. Determine the location e of the shear center, point O , for the thin-walled member having the cross section shown.

r O a

Prob. 7–70

7–65. The beam supports a vertical shear of V = 7 kip. Determine the resultant force developed in segment AB of the beam.

5 in.

0.5 in.

10 in. 0.5 in.0.5 in.

Prob. 7–65

7–66. The built-up beam is fabricated from the three thin plates having a thickness t. Determine the location of the shear center O .

a 2

Prob. 7–66

7–67. Determine the location e of the shear center, point O , for the thin-walled member having the cross section shown. The member segments have the same thickness t .

O b1

b1h1

Prob. 7–67

CHAPTER REVIEW 405

CHAPTER REVIEW

Vtmax

Shear–stress distribution

Area ! A¿

_ y ¿

N A y¿¯

A¿

Transverse shear stress in beams is determined indirectly by using the flexure formula and the relationship between moment and shear (V = dM >dx). The result is the shear formula

t = VQ It

In particular, the value for Q is the moment of the area A ! about the neutral axis, Q = y !A ! . This area is the portion of the cross-sectional area that is “held on” to the beam above (or below) the thickness t where t is to be determined.

If the beam has a rectangular cross section, then the shear- stress distribution will be parabolic, having a maximum value at the neutral axis. The maximum shear stress can

be determined using t = 1.5 V A

Fasteners, such as nails, bolts, glue, or welds, are used to connect the composite parts of a “built-up” section. The shear force resisted by these fasteners is determined from the shear flow, q, or force per unit length, that must be carried by the beam. The shear flow is

q = VQ

406 CHAPTER 7 TRANSVERSE SHEAR

Shear-flow distribution

(qmax)f

(qmax)w

2(qmax)f

If the beam is made from thin-walled segments, then the shear-flow distribution along each segment can be determined. This distribution will vary linearly along horizontal segments, and parabolically along inclined or vertical segments.

Provided the shear-flow distribution in each element of an open thin-walled section is known, then using a balance of moments, the location O of the shear center for the cross section can be determined. When a load is applied to the member through this point, the member will bend, and not twist.

REVIEW PROBLEMS 407

*7–72. The T-beam is subjected to a shear of V = 150 kN . Determine the amount of this force that is supported by the web B.

200 mm

40 mm

V = 150 kN

40 mm

200 mm

Prob. 7–72

7–73. The member is subject to a shear force of V = 2 kN. Determine the shear flow at points A, B, and C. The thickness of each thin-walled segment in 15 mm .

300 mm A

100 mm

200 mm

V ! 2 kN

Prob. 7–73

7–74. Determine the shear stress at points B and C on the web of the beam located at section a–a .

7–75. Determine the maximum shear stress acting at section a–a in the beam.

150 lb/ft

8000 lb

2 in. 0.75 in.

0.75 in.4 in.

6 in.

4 ft 1.5 ft1.5 ft

4 ft

0.5 in.

Probs. 7–74/75

REVIEW PROBLEMS

7–71. The beam is fabricated from four boards nailed together as shown. Determine the shear force each nail along the sides C and the top D must resist if the nails are uniformly spaced at s = 3 in. The beam is subjected to a shear of V = 4.5 kip.

1 in.

12 in.

3 in.

1 in.

10 in.

1 in.

Prob. 7–71

The offset hanger supporting this ski gondola is subjected to the combined loadings of axial force and bending moment.

Chapter 8

409

8.1 Thin-Walled Pressure Vessels Cylindrical or spherical vessels are commonly used in industry to serve as boilers or tanks. When under pressure, the material of which they are made is subjected to a loading from all directions. Although this is the case, the vessel can be analyzed in a simple manner provided it has a thin wall. In general, “ thin wall ” refers to a vessel having an inner-radius-to- wall-thickness ratio of 10 or more (r>t Ú 10). Specifically, when r>t = 10 the results of a thin-wall analysis will predict a stress that is approximately 4% less than the actual maximum stress in the vessel. For larger r>t ratios this error will be even smaller.

Provided the vessel wall is “thin,” the stress distribution throughout its thickness will not vary significantly, and so we will assume that it is uniform or constant. Using this assumption, we will now analyze the state of stress in thin-walled cylindrical and spherical pressure vessels. In both cases, the pressure in the vessel is understood to be the gauge pressure, that is, it measures the pressure above atmospheric pressure, since atmospheric pressure is assumed to exist both inside and outside the vessel’s wall before the vessel is pressurized.

CHAPTER OBJECTIVES

■ This chapter serves as a review of the stress analysis that has been developed in the previous chapters regarding axial load, torsion, bending, and shear. We will discuss the solution of problems where several of these internal loads occur simultaneously on a member’s cross section. Before doing this, however, the chapter begins with an analysis of stress developed in thin-walled pressure vessels.

Combined Loadings

Cylindrical pressure vessels, such as this gas tank, have semi-spherical end caps rather than flat ones in order to reduce the stress in the tank.

410 CHAPTER 8 COMBINED LOADINGS

Cylindrical Vessels. Consider the cylindrical vessel in Fig. 8–1 a , having a wall thickness t , inner radius r , and subjected to a gauge pressure p that developed within the vessel by a contained gas. Due to this loading, a small element of the vessel that is sufficiently removed from the ends and oriented as shown in Fig. 8–1 a , is subjected to normal stresses s1 in the circumferential or hoop direction and s2 in the longitudinal or axial direction.

The hoop stress can be determined by considering the vessel to be sectioned by planes a , b , and c . A free-body diagram of the back segment along with the contained gas is shown in Fig. 8–1 b . Here only the loadings in the x direction are shown. These loadings are developed by the uniform hoop stress s1, acting on the vessel’s wall, and the pressure acting on the vertical face of the gas. For equilibrium in the x direction, we require

!Fx = 0; 2[s1(t dy)] - p(2r dy) = 0

s1 = pr

t (8–1)

The longitudinal stress can be determined by considering the left portion of section b of the cylinder, Fig. 8–1 a . As shown in Fig. 8–1 c , s2 acts uniformly throughout the wall, and p acts on the section of the contained gas. Since the mean radius is approximately equal to the vessel’s inner radius, equilibrium in the y direction requires

!Fy = 0; s2(2prt) - p(pr2) = 0

s2 = pr

2t (8–2)

In the above equations,

s1, s2 = the normal stress in the hoop and longitudinal directions, respectively. Each is assumed to be constant throughout the wall of the cylinder, and each subjects the material to tension.

p = the internal gauge pressure developed by the contained gas r = the inner radius of the cylinder t = the thickness of the wall (r>t Ú 10)

(a)

b a c

rs1

(b)

(c)

Fig. 8–1

8.1 THIN-WALLED PRESSURE VESSELS 411

By comparison, note that the hoop or circumferential stress is twice as large as the longitudinal or axial stress. Consequently, when fabricating cylindrical pressure vessels from rolled-formed plates, the longitudinal joints must be designed to carry twice as much stress as the circumferential joints.

Spherical Vessels. We can analyze a spherical pressure vessel in a similar manner. To do this, consider the vessel to have a wall thickness t , inner radius r , and subjected to an internal gauge pressure p , Fig. 8–2 a . If the vessel is sectioned in half, the resulting free-body diagram is shown in Fig. 8–2 b . Like the cylinder, equilibrium in the y direction requires

!Fy = 0; s2(2prt) - p(pr 2) = 0

s2 = pr

2t (8–3)

This is the same result as that obtained for the longitudinal stress in the cylindrical pressure vessel. Furthermore, from the analysis, this stress will be the same regardless of the orientation of the hemispheric free-body diagram. Consequently, a small element of the material is subjected to the state of stress shown in Fig. 8–2 a .

The above analysis indicates that an element of material taken from either a cylindrical or a spherical pressure vessel is subjected to biaxial stress, i.e., normal stress existing in only two directions. Actually, the pressure also subjects the material to a radial stress, s3, which acts along a radial line. This stress has a maximum value equal to the pressure p at the interior wall and it decreases through the wall to zero at the exterior surface of the vessel, since the gauge pressure there is zero. For thin-walled vessels, however, we will ignore this radial-stress component, since our limiting assumption of r>t = 10 results in s2 and s1 being, respectively, 5 and 10 times higher than the maximum radial stress, (s3)max = p. Finally, if the vessel is subjected to an external pressure, the compressive stress developed within the thin wall may cause the vessel to become unstable, and collapse may occur by buckling rather than causing the material to fracture.

This thin-walled pipe was subjected to an excessive gas pressure that caused it to rupture in the circumferential or hoop direction. The stress in this direction is twice that in the axial direction as noted by Eqs. 8–1 and 8–2.

(a)

r y

(b)

Fig. 8–2

412 CHAPTER 8 COMBINED LOADINGS

A cylindrical pressure vessel has an inner diameter of 4 ft and a thickness of 12 in. Determine the maximum internal pressure it can sustain so that neither its circumferential nor its longitudinal stress component exceeds 20 ksi. Under the same conditions, what is the maximum internal pressure that a similar-size spherical vessel can sustain?

SOLUTION Cylindrical Pressure Vessel. The maximum stress occurs in the circumferential direction. From Eq. 8–1 we have

s1 = pr

t ; 20 kip>in2 = p(24 in.)1

2 in.

p = 417 psi Ans.

Note that when this pressure is reached, from Eq. 8–2 , the stress in the longitudinal direction will be s2 = 12 (20 ksi) = 10 ksi. Furthermore, the maximum stress in the radial direction occurs on the material at the inner wall of the vessel and is (s3)max = p = 417 psi. This value is 48 times smaller than the circumferential stress (20 ksi), and as stated earlier, its effects will be neglected.

Spherical Vessel. Here the maximum stress occurs in any two perpendicular directions on an element of the vessel, Fig. 8–2 a . From Eq. 8–3 , we have

s2 = pr

2t ; 20 kip>in2 = p(24 in.)

2112 in.2 p = 833 psi Ans.

NOTE: Although it is more difficult to fabricate, the spherical pressure vessel will carry twice as much internal pressure as a cylindrical vessel.

EXAMPLE 8.1

8.1 THIN-WALLED PRESSURE VESSELS 413

8–5. The open-ended polyvinyl chloride pipe has an inner diameter of 4 in. and thickness of 0.2 in. If it carries flowing water at 60 psi pressure, determine the state of stress in the walls of the pipe.

8–6. If the flow of water within the pipe in Prob. 8–5 is stopped due to the closing of a valve, determine the state of stress in the walls of the pipe. Neglect the weight of the water. Assume the supports only exert vertical forces on the pipe.

Probs. 8–5/6

8–7. A boiler is constructed of 8-mm thick steel plates that are fastened together at their ends using a butt joint consisting of two 8-mm cover plates and rivets having a diameter of 10 mm and spaced 50 mm apart as shown. If the steam pressure in the boiler is 1.35 MPa, determine (a) the circumferential stress in the boiler’s plate apart from the seam, (b) the circumferential stress in the outer cover plate along the rivet line a–a, and (c) the shear stress in the rivets.

8 mm

50 mm a

0.75 m

Prob. 8–7

8–1. A spherical gas tank has an inner radius of r = 1.5 m. If it is subjected to an internal pressure of p = 300 kPa, determine its required thickness if the maximum normal stress is not to exceed 12 MPa.

8–2. A pressurized spherical tank is to be made of 0.5-in.-thick steel. If it is subjected to an internal pressure of p = 200 psi, determine its outer radius if the maximum normal stress is not to exceed 15 ksi.

8–3. The thin-walled cylinder can be supported in one of two ways as shown. Determine the state of stress in the wall of the cylinder for both cases if the piston P causes the internal pressure to be 65 psi. The wall has a thickness of 0.25 in. and the inner diameter of the cylinder is 8 in.

(a) (b)

8 in. 8 in.

Prob. 8–3

*8–4. The tank of the air compressor is subjected to an internal pressure of 90 psi. If the internal diameter of the tank is 22 in., and the wall thickness is 0.25 in., determine the stress components acting at point A. Draw a volume element of the material at this point, and show the results on the element.

Prob. 8–4

PROBLEMS

414 CHAPTER 8 COMBINED LOADINGS

8–11. Two hemispheres having an inner radius of 2 ft and wall thickness of 0.25 in. are fitted together, and the inside gauge pressure is reduced to -10 psi. If the coefficient of static friction is ms = 0.5 between the hemispheres, determine (a) the torque T needed to initiate the rotation of the top hemisphere relative to the bottom one, (b) the vertical force needed to pull the top hemisphere off the bottom one, and (c) the horizontal force needed to slide the top hemisphere off the bottom one.

Prob. 8–11

*8–12. A pressure-vessel head is fabricated by gluing the circular plate to the end of the vessel as shown. If the vessel sustains an internal pressure of 450 kPa, determine the average shear stress in the glue and the state of stress in the wall of the vessel.

450 mm 10 mm

20 mm

Prob. 8–12

*8–8. The steel water pipe has an inner diameter of 12 in. and wall thickness 0.25 in. If the valve A is opened and the flowing water is under a gauge pressure of 250 psi, determine the longitudinal and hoop stress developed in the wall of the pipe.

8–9. The steel water pipe has an inner diameter of 12 in. and wall thickness 0.25 in. If the valve A is closed and the water pressure is 300 psi, determine the longitudinal and hoop stress developed in the wall of the pipe. Draw the state of stress on a volume element located on the wall.

Probs. 8–8/9

8–10. The A-36-steel band is 2 in. wide and is secured around the smooth rigid cylinder. If the bolts are tightened so that the tension in them is 400 lb, determine the normal stress in the band, the pressure exerted on the cylinder, and the distance half the band stretches.

8 in.

in.1—8

Prob. 8–10

8.1 THIN-WALLED PRESSURE VESSELS 415

*8–16. A closed-ended pressure vessel is fabricated by cross winding glass filaments over a mandrel, so that the wall thickness t of the vessel is composed entirely of filament and an expoxy binder as shown in the figure. Consider a segment of the vessel of width w and wrapped at an angle u. If the vessel is subjected to an internal pressure p , show that the force in the segment is Fu = s 0 wt , where s 0 is the stress in the filaments. Also, show that the stresses in the hoop and longitudinal directions are s h = s 0 sin 2 u and s 1 = s 0 cos 2 u, respectively. At what angle u (optimum winding angle) would the filaments have to be would so that the hoop and longitudinal stresses are equivalent?

Prob. 8–16

8–17. In order to increase the strength of the pressure vessel, filament winding of the same material is wrapped around the circumference of the vessel as shown. If the pretension in the filament is T and the vessel is subjected to an internal pressure p , determine the hoop stresses in the filament and in the wall of the vessel. Use the free- body diagram shown, and assume the filament winding has a thickness t ! and width w for a corresponding length L of the vessel.

t ¿

Prob. 8–17

8–13. An A-36-steel hoop has an inner diameter of 23.99 in., thickness of 0.25 in., and width of 1 in. If it and the 24-in.-diameter rigid cylinder have a temperature of 65° F, determine the temperature to which the hoop should be heated in order for it to just slip over the cylinder. What is the pressure the hoop exerts on the cylinder, and the tensile stress in the ring when it cools back down to 65° F?

24 in.

Prob. 8–13

8–14. The ring, having the dimensions shown, is placed over a flexible membrane which is pumped up with a pressure p. Determine the change in the internal radius of the ring after this pressure is applied. The modulus of elasticity for the ring is E.

w ri

Prob. 8–14

8–15. The inner ring A has an inner radius r 1 and outer radius r 2 . Before heating, the outer ring B has an inner radius r 3 and an outer radius r 4 , and r 2 7 r 3 . If the outer ring is heated and then fitted over the inner ring, determine the pressure between the two rings when ring B reaches the temperature of the inner ring. The material has a modulus of elasticity of E and a coefficient of thermal expansion of a .

A B

Prob. 8–15

416 CHAPTER 8 COMBINED LOADINGS

8.2 State of Stress Caused by Combined Loadings

In previous chapters we developed methods for determining the stress distributions in a member subjected to either an internal axial force, a shear force, a bending moment, or a torsional moment. Most often, however, the cross section of a member is subjected to several of these loadings simultaneously. When this occurs, the method of superposition can be used to determine the resultant stress distribution. Recall from Sec. 4.3 that the principle of superposition can be used for this purpose provided a linear relationship exists between the stress and the loads. Also, the geometry of the member should not undergo significant change when the loads are applied. These conditions are necessary in order to ensure that the stress produced by one load is not related to the stress produced by any other load.

This chimney is subjected to the combined loading of wind and weight. It is important to investigate the tensile stress in the chimney since masonry is weak in tension.

Procedure for Analysis

The following procedure provides a general means for establishing the normal and shear stress components at a point in a member when the member is subjected to several different types of loadings simultaneously. It is assumed that the material is homogeneous and behaves in a linear elastic manner. Also, Saint-Venant’s principle requires that the point where the stress is to be determined is far removed from any discontinuities in the cross section or points of applied load.

Internal Loading. • Section the member perpendicular to its axis at the point where

the stress is to be determined and obtain the resultant internal normal and shear force components and the bending and torsional moment components.

• The force components should act through the centroid of the cross section, and the moment components should be computed about centroidal axes , which represent the principal axes of inertia for the cross section.

Stress Components. • Determine the stress component associated with each internal

loading. For each case, represent the effect either as a distribution of stress acting over the entire cross-sectional area, or show the stress on an element of the material located at a specified point on the cross section.

8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS 417

Normal Force. • The internal normal force is developed by a uniform normal-stress

distribution determined from s = P>A . Shear Force.

• The internal shear force in a member is developed by a shear- stress distribution determined from the shear formula, t = VQ>It. Special care, however, must be exercised when applying this equation, as noted in Sec. 7.2 .

Bending Moment. • For straight members the internal bending moment is developed

by a normal-stress distribution that varies linearly from zero at the neutral axis to a maximum at the outer boundary of the member. This stress distribution is determined from the flexure formula, s = -My >I. If the member is curved , the stress distribution is nonlinear and is determined from s = My >[Ae(R - y)].

Torsional Moment. • For circular shafts and tubes the internal torsional moment is

developed by a shear-stress distribution that varies linearly from zero at the central axis of the shaft to a maximum at the shaft’s outer boundary. This stress distribution is determined from the torsional formula, t = Tr>J.

Thin-Walled Pressure Vessels. • If the vessel is a thin-walled cylinder, the internal pressure p will

cause a biaxial state of stress in the material such that the hoop or circumferential stress component is s1 = pr>t and the longitudinal stress component is s2 = pr>2t. If the vessel is a thin-walled sphere, then the biaxial state of stress is represented by two equivalent components, each having a magnitude of s2 = pr>2t.

Superposition. • Once the normal and shear stress components for each loading

have been calculated, use the principle of superposition and determine the resultant normal and shear stress components.

• Represent the results on an element of material located at the point, or show the results as a distribution of stress acting over the member’s cross-sectional area.

Problems in this section, which involve combined loadings, serve as a basic review of the application of the stress equations mentioned above. A thorough understanding of how these equations are applied, as indicated in the previous chapters, is necessary if one is to successfully solve the problems at the end of this section. The following examples should be carefully studied before proceeding to solve the problems.

A B

F F

When a pretension force F is developed in the blade of this coping saw, it will produce both a compressive force F and bending moment M in the region AB of the frame. The material must therefore resist the normal stress produced by both of these loadings.

418 CHAPTER 8 COMBINED LOADINGS

A force of 150 lb is applied to the edge of the member shown in Fig. 8–3 a . Neglect the weight of the member and determine the state of stress at points B and C.

SOLUTION Internal Loadings. The member is sectioned through B and C. For equilibrium at the section there must be an axial force of 150 lb acting through the centroid and a bending moment of 750 lb # in. about the centroidal principal axis, Fig. 8–3 b .

Stress Components. Normal Force. The uniform normal-stress distribution due to the normal force is shown in Fig. 8–3 c . Here

s = P A

= 150 lb

(10 in.)(4 in) = 3.75 psi

Bending Moment. The normal-stress distribution due to the bending moment is shown in Fig. 8–3 d . The maximum stress is

smax = Mc

I =

750 lb # in. (5 in.) 1 12 (4 in.) (10 in.)

3 = 11.25 psi

Superposition. If the above normal-stress distributions are added algebraically, the resultant stress distribution is shown in Fig. 8–3 e .

Elements of material at B and C are subjected only to normal or uniaxial stress as shown in Fig. 8–3 f and 8–3 g . Hence,

sB = - P A

+ Mc I

= -3.75 psi + 11.25 psi = 7.5 psi (tension) Ans.

sC = - P A

- Mc I

= -3.75 psi - 11.25 psi = -15 psi (compression) Ans.

Although it is not needed here, the location of the line of zero stress can be determined by proportional triangles; i.e.,

7.5 psi

x =

15 psi 10 in. - x ; x 5 3.33 in.

EXAMPLE 8.2

(a)

5 in.

2 in. 2 in.

150 lb

C B

5 in.

150 lb

(b) 150 lb

750 lb!in.

C B

11.25 psi 11.25 psi

(d)

Bending Moment

! " C

B 7.5 psi 15 psi

x (10 in. ! x)

(e)

Combined Loading

(f)

7.5 psi

(g)

15 psi

C B

3.75 psi 3.75 psi

(c)

Normal Force

Fig. 8–3

8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS 419

The tank in Fig. 8–4 a has an inner radius of 24 in. and a thickness of 0.5 in. It is filled to the top with water having a specific weight of gw = 62.4 lb>ft3. If it is made of steel having a specific weight of gst = 490 lb>ft 3, determine the state of stress at point A. The tank is open at the top.

SOLUTION Internal Loadings. The free-body diagram of the section of both the tank and the water above point A is shown in Fig. 8–4 b . Notice that the weight of the water is supported by the water surface just below the section, not by the walls of the tank. In the vertical direction, the walls simply hold up the weight of the tank. This weight is

W st = gstV st = (490 lb>ft3) cpa 24.512 ftb2 - pa 2412 ftb2 d (3 ft) = 777.7 lb

The stress in the circumferential direction is developed by the water pressure at level A. To obtain this (gauge) pressure we must use p = gwz , which gives the pressure at a point located a depth z in the water. Consequently, the pressure on the tank at level A is

p = gwz = (62.4 lb>ft3) (3 ft) = 187.2 lb>ft2 = 1.30 psi Stress Components.

Circumferential Stress. Since r>t = 24 in.>0.5 in. = 48 7 10, the tank is a thin-walled vessel. Applying Eq. 8–1 , using the inner radius r = 24 in., we have

s1 = pr

t =

1.30 lb>in2 (24 in.) 0.5 in.

= 62.4 psi Ans.

Longitudinal Stress. Since the weight of the tank is supported uniformly by the walls, we have

s2 = W st A st

= 777.7 lb

p[(24.5 in.)2 - (24 in.)2] = 10.2 psi Ans.

NOTE: Eq. 8–2 , s2 = pr>2t, does not apply here, since the tank is open at the top and therefore, as stated previously, the water cannot develop a loading on the walls in the longitudinal direction.

Point A is therefore subjected to the biaxial stress shown in Fig. 8–4 c .

EXAMPLE 8.3

(a)

t ! 0.5 in.

3 ft

r ! 24 in.

(b)

3 ft

Ww " Wst

ps2

(c)

10.2 psi

62.4 psi A

Fig. 8–4

420 CHAPTER 8 COMBINED LOADINGS

The member shown in Fig. 8–5 a has a rectangular cross section. Determine the state of stress that the loading produces at point C.

EXAMPLE 8.4

(a)

1.5 m

250 mm

50 mm 1.5 m

2 m4 m

125 mm

2.5 m

C C

50 kN/m

125 kN

97.59 kN 3

16.45 kN

21.93 kN

3 45

4 m

1.25 m

(b)

(c)

16.45 kN

21.93 kN

1.5 m

M N

Fig. 8–5

SOLUTION Internal Loadings. The support reactions on the member have been determined and are shown in Fig. 8–5 b . If the left segment AC of the member is considered, Fig. 8–5 c , the resultant internal loadings at the section consist of a normal force, a shear force, and a bending moment. Solving,

N = 16.45 kN V = 21.93 kN M = 32.89 kN # m

8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS 421

Stress Components.

Normal Force. The uniform normal-stress distribution acting over the cross section is produced by the normal force, Fig. 8–5 d . At point C,

sC = P A

= 16.45(103) N

(0.050 m) (0.250 m) = 1.32 MPa

Shear Force. Here the area A ! = 0, since point C is located at the top of the member. Thus Q = y !A ! = 0 and for C, Fig. 8–5 e , the shear stress

tC = 0

Bending Moment. Point C is located at y = c = 0.125 m from the neutral axis, so the normal stress at C, Fig. 8–5 f , is

sC = Mc

I =

(32.89(103) N # m)(0.125 m)3 112 (0.050 m) (0.250 m3)4 = 63.16 MPa Superposition. The shear stress is zero. Adding the normal stresses determined above gives a compressive stress at C having a value of

s C = 1.32 MPa + 63.16 MPa = 64.5 MPa Ans.

This result, acting on an element at C , is shown in Fig. 8–5 g .

sC ! 1.32 MPa

Normal Force

(d)

Shear Force

(e)

tC ! 0

sC ! 63.16 MPa

Bending Moment

(f)

Fig. 8–5 (cont.)

64.5 MPa

(g)

422 CHAPTER 8 COMBINED LOADINGS

The solid rod shown in Fig. 8–6 a has a radius of 0.75 in. If it is subjected to the force of 500 lb, determine the state of stress at point A .

EXAMPLE 8.5

8 in.

10 in.

14 in.

500 lb

x A

(a)

10 in.

14 in.

500 lb

(b)

500 lb (14 in.) ! 7000 lb"in.

500 lb

Bending moment (7000 lb"in.)

(e)

21.13 ksi0.283 ksi

Normal force (500 lb)

(d)

7000 lb"in.

500 lb

(c)

Fig. 8–6

SOLUTION Internal Loadings. The rod is sectioned through point A . Using the free-body diagram of segment AB , Fig. 8–6 b , the resultant internal loadings are determined from the equations of equilibrium. Verify these results. In order to better “visualize” the stress distributions due to these loadings, we can consider the equal but opposite resultants acting on segment AC , Fig. 8–6 c .

Stress Components. Normal Force. The normal-stress distri- bution is shown in Fig. 8–6 d . For point A , we have

(sA)y = P A

= 500 lb

p(0.75 in.)2 = 283 psi = 0.283 ksi

Bending Moment. For the moment, c = 0.75 in., so the normal stress at point A , Fig. 8–6 e , is

(sA)y = Mc

I =

7000 lb # in.(0.75 in.)314p(0.75 in.)44 = 21,126 psi = 21.13 ksi

Superposition. When the above results are superimposed, it is seen that an element of material at A is subjected to the normal stress

(sA)y = 0.283 ksi + 21.13 ksi = 21.4 ksi Ans.

8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS 423

EXAMPLE 8.6

The solid rod shown in Fig. 8–7 a has a radius of 0.75 in. If it is subjected to the force of 800 lb, determine the state of stress at point A .

SOLUTION Internal Loadings. The rod is sectioned through point A . Using the free-body diagram of segment AB , Fig. 8–7 b , the resultant internal loadings are determined from the six equations of equilibrium. Verify these results. The equal but opposite resultants are shown acting on segment AC , Fig. 8–7 c .

Stress Components. Shear Force. The shear-stress distribution is shown in Fig. 8–7 d . For point A , Q is determined from the shaded semi-circular area. Using the table on the inside front cover, we have

Q = y!A ! = 4(0.75 in.)

3p c 1

2 p(0.75 in.)2 d = 0.2813 in3

so that

(tyz)A = VQ It

= 800 lb(0.2813 in3)314p(0.75 in.)442(0.75 in.)

= 604 psi = 0.604 ksi Bending Moment. Since point A lies on the neutral axis, Fig. 8–7 e , the normal stress is

sA = 0 Torque. At point A , rA = c = 0.75 in., Fig. 8–7 f . Thus the shear stress is

(tyz)A = Tc J

= 11 200 lb # in.(0.75 in.)312 p(0.75 in.)44 = 16 901 psi = 16.90 ksi

Superposition. Here the element of material at A is subjected only to a shear stress component, where (tyz)A = 0.604 ksi + 16.90 ksi = 17.5 ksi Ans.

800 lb

8 in.

10 in.

14 in.

x A

(a)

10 in.

14 in. 800 lb

(b)

800 lb (14 in.) ! 11 200 lb"in.

800 lb (10 in.) ! 8000 lb"in.

800 lb

16.90 ksi

Torsional moment (11 200 lb"in.)

(f)

Bending moment (8000 lb"in.)

(e)

0.604 ksi

Shear force (800 lb)

(d)

A¿8000 lb"in.

11 200 lb"in.800 lb

(c)

! " "

Fig. 8–7

424 CHAPTER 8 COMBINED LOADINGS

EXAMPLE 8.7

A rectangular block has a negligible weight and is subjected to a vertical force P, Fig. 8–8 a . ( a ) Determine the range of values for the eccentricity ey of the load along the y axis so that it does not cause any tensile stress in the block. ( b ) Specify the region on the cross section where P may be applied without causing a tensile stress in the block.

SOLUTION Part (a). When P is moved to the centroid of the cross section, Fig. 8–8 b , it is necessary to add a couple moment Mx = Pey in order to maintain a statically equivalent loading. The combined normal stress at any coordinate location y on the cross section caused by these two loadings is

s = - P A

- (Pey)y

Ix = - P

A ¢1 + Aeyy

Ix ≤

Here the negative sign indicates compressive stress. For positive ey, Fig. 8–7 a , the smallest compressive stress will occur along edge AB, where y = -h>2, Fig. 8–8 b . (By inspection, P causes compression there, but Mx causes tension.) Hence,

smin = - P A

¢1 - Aeyh 2Ix ≤

This stress will remain negative, i.e., compressive, provided the term in parentheses is positive; i.e.,

1 7 Aeyh

2Ix

Since A = bh and I x = 112 bh3, then

1 7 6ey h

or ey 6 1 6

h Ans.

In other words, if - 16 h … ey … 1 6 h, the stress in the block along

edge AB or CD will be zero or remain compressive.

NOTE: This is sometimes referred to as the “ middle-third rule. ” It is very important to keep this rule in mind when loading columns or arches having a rectangular cross section and made of material such as stone or concrete, which can support little or no tensile stress. We can extend this analysis in the same way by placing P along the x axis in Fig. 8–8 . The result will produce a shaded parallelogram shown in Fig. 8–8 c . This region is referred to as the core or kern of the section. When P is applied within the kern, the normal stress at the corners of the cross section will be compressive.

(a)

P y

! (b)

P y

Mx ! Pey

xh 2

y ! "

(c)

P y

E G

F Hb 6

b 6

h 6

Fig. 8–8

Here is an example of where combined axial and bending stress can occur.

8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS 425

200 N

500 N 300 N

(d)

1 m

2 m

P8–1

P8–2. The internal loadings act on the section. Show the stress that each of these loads produce on differential elements located at point A and point B .

(a)

T B

(b)

B V

P8–2

P8–1. In each case, determine the internal loadings that act on the indicated section. Show the results on the left segment.

500 N

200 N

0.2 m

(a)

1 m

300 N

2 m

200 N

100 N

(b)

0.4 m

200 N

300 N

0.4 m

0.2 m

(c)

PRELIMINARY PROBLEMS

426 CHAPTER 8 COMBINED LOADINGS

FUNDAMENTAL PROBLEMS

F8–3. Determine the state of stress at point A on the cross section of the beam at section a – a . Show the results in a differential element at the point.

2 m 0.5 m 0.5 m0.5 m

30 kN

50 mm

10 mm

180 mm

100 mm

Section a–a

F8–3

F8–4. Determine the magnitude of the load P that will cause a maximum normal stress of smax = 30 ksi in the link along section a–a.

P 2 in.

2 in. 0.5 in.a a

F8–4

F8–1. Determine the normal stress developed at corners A and B of the column.

300 kN

500 kN

50 mm

100 mm

150 mm150 mm

150 mm 150 mm

y x

F8–1

F8–2. Determine the state of stress at point A on the cross section at section a – a of the cantilever beam. Show the results in a differential element at the point.

300 mm

100 mm

0.5 m

Section a–a

400 kN

F8–2

8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS 427

F8–7. Determine the state of stress at point A on the cross section of the pipe at section a – a . Show the results in a differential element at the point.

a 300 mm

300 mm

Section a – a

50 mm

40 mm

6 kN

F8–7

F8–8. Determine the state of stress at point A on the cross section of the shaft at section a – a . Show the results in a differential element at the point.

100 mm a

300 N

900 N

100 mm

600 mm

400 mm

300 mm

100 mm

Section a – a

25 mm

20 mm

F8–8

F8–5. The beam has a rectangular cross section and is subjected to the loading shown. Determine the state of stress at point B . Show the results in a differential element at the point.

1.5 in. 1.5 in.

10 in.

400 lb x

1 in.

500 lb

2 in.

F8–5

F8–6. Determine the state of stress at point A on the cross section of the pipe assembly at section a – a . Show the results in a differential element at the point.

400 mm

200 mm

Section a – a

20 mm

1500 N

1000 N

F8–6

428 CHAPTER 8 COMBINED LOADINGS

8–21. If the load has a weight of 600 lb, determine the maximum normal stress developed on the cross section of the supporting member at section a – a . Also, plot the normal stress distribution over the cross-section.

1.5 ft 1 in.a a

Section a – a

Prob. 8–21

8–22. The clamp is made from members AB and AC, which are pin connected at A. If it exerts a compressive force at C and B of 180 N, determine the maximum compressive stress in the clamp at section a – a . The screw EF is subjected only to a tensile force along its axis.

8–23. The clamp is made from members AB and AC, which are pin connected at A. If it exerts a compressive force at C and B of 180 N, sketch the stress distribution acting over section a – a . The screw EF is subjected only to a tensile force along its axis.

180 N

180 NB

E A

a a

30 mm 40 mm

15 mm

15 mm Sectiona – a

Probs. 8–22/23

8–18. The vertical force P acts on the bottom of the plate having a negligible weight. Determine the shortest distance d to the edge of the plate at which it can be applied so that it produces no compressive stresses on the plate at section a–a. The plate has a thickness of 10 mm and P acts along the center line of this thickness.

500 mm

300 mm

200 mm

Prob. 8–18

8–19. Determine the maximum and minimum normal stress in the bracket at section a – a when the load is applied at x = 0 .

*8–20. Determine the maximum and minimum normal stress in the bracket at section a – a when the load is applied at x = 300 mm.

100 kN

200 mm 150 mm

15 mm

Probs. 8–19/20

PROBLEMS

8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS 429

*8–28. The cylindrical post, having a diameter of 40 mm, is being pulled from the ground using a sling of negligible thick- ness. If the rope is subjected to a vertical force of P = 500 N, determine the normal stress at points A and B . Show the results on a volume element located at each of these points.

8–29. Determine the maximum load P that can be applied to the sling having a negligible thickness so that the normal stress in the post does not exceed s allow = 30 MPa. The post has a diameter of 50 mm.

Probs. 8–28/29

8–30. The rib-joint pliers are used to grip the smooth pipe C . If the force of 100 N is applied to the handles, determine the state of stress at points A and B on the cross section of the jaw at section a – a . Indicate the results on an element at each point.

250 mm

100 N

Section a – a

25 mm 25 mm

10 mm

20 mm 7.5 mm

45°

Prob. 8–30

*8–24. The bearing pin supports the load of 700 lb. Determine the stress components in the support member at point A. The support is 0.5 in. thick.

8–25. The bearing pin supports the load of 700 lb. Determine the stress components in the support member at point B. The support is 0.5 in. thick.

30!

2 in. A A

B B

3 in.

1.25 in.

700 lb

0.75 in.

0.5 in.

Probs. 8–24/25

8–26. The column is built up by gluing the two identical boards together. Determine the maximum normal stress developed on the cross section when the eccentric force of P = 50 kN is applied.

8–27. The column is built up by gluing the two identical boards together. If the wood has an allowable normal stress of s allow = 6 MPa, determine the maximum allowable eccentric force P that can be applied to the column.

150 mm

250 mm 75 mm

300 mm

50 mm

Probs. 8–26/27

430 CHAPTER 8 COMBINED LOADINGS

8–35. The tubular shaft of the soil auger is subjected to the axial force and torque shown. If the auger is rotating at a constant rate, determine the state of stress at points A and B on the cross section of the shaft at section a – a .

Section a – a

1200 lb

3000 lb!ft

a A

1.5 in.

1 in.

Prob. 8–35

*8–36. The drill is jammed in the wall and is subjected to the torque and force shown. Determine the state of stress at point A on the cross section of drill bit at section a – a .

8–37. The drill is jammed in the wall and is subjected to the torque and force shown. Determine the state of stress at point B on the cross section of drill bit, in back, at section a – a .

150 N

3 4

125 mm

20 N·m

400 mm

5 mm

Section a – a

Probs. 8–36/37

8–31. Determine the smallest distance d to the edge of the plate at which the force P can be applied so that it produces no compressive stresses in the plate at section a–a. The plate has a thickness of 20 mm and P acts along the centerline of this thickness.

*8–32. The horizontal force of P = 80 kN acts at the end of the plate. The plate has a thickness of 10 mm and P acts along the centerline of this thickness such that d = 50 mm. Plot the distribution of normal stress acting along section a–a.

200 mm

300 mm

Probs. 8–31/32

8–33. The control lever is subjected to a horizontal force of 20 lb on the handle. Determine the state of stress at points A and B . Sketch the results on differential elements located at each of these points. The assembly is pin- connected at C and attached to a cable at D .

8–34. The control lever is subjected to a horizontal force of 20 lb on the handle. Determine the state of stress at points E and F . Sketch the results on differential elements located at each of these points. The assembly is pin connected at C and attached to a cable at D .

0.25 in.

A A

B B

0.3 in.

20 lb

1.75 in.

E E 5 in.

2 in.

0.75 in. 0.5 in.

D C

Probs. 8–33/34

8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS 431

8–42. Determine the state of stress at point A on the cross section of the post at section a – a . Indicate the results on a differential element at the point.

8–43. Determine the state of stress at point B on the cross section of the post at section a – a . Indicate the results on a differential element at the point.

a a

5 ft

1.5 ft

300 lb400 lb

2 in.

2.5 in. A

Probs. 8–42/43

*8–44. Determine the normal stress developed at points A and B . Neglect the weight of the block.

8–45. Sketch the normal stress distribution acting over the cross section at section a–a. Neglect the weight of the block.

6 in.

6 kip

12 kip3 in.

A B

Probs. 8–44/45

8–38. The frame supports the distributed load shown. Determine the state of stress acting at point D . Show the results on a differential element at this point.

8–39. The frame supports the distributed load shown. Determine the state of stress acting at point E . Show the results on a differential element at this point.

4 kN/m

D B

1.5 m 1.5 m

20 mm

50 mm

20 mm 60 mm

3 m

5 m

Probs. 8–38/39

*8–40. The 500-kg engine is suspended from the jib crane at the position shown. Determine the state of stress at point A on the cross section of the boom at section a – a.

8–41. The 500-kg engine is suspended from the jib crane at the position shown. Determine the state of stress at point B on the cross section of the boom at section a – a . Point B is just above the bottom flange.

Section a – a

20 mm

150 mm

300 mmA

2 m2 m 2 m

0.4 m

30!a

Probs. 8–40/41

432 CHAPTER 8 COMBINED LOADINGS

8–50. The coiled spring is subjected to a force P . If we assume the shear stress caused by the shear force at any vertical section of the coil wire to be uniform, show that the maximum shear stress in the coil is t max = P>A + PRr>J , where J is the polar moment of inertia of the coil wire and A is its cross-sectional area.

Prob. 8–50

8–51. A post having the dimensions shown is subjected to the bearing load P. Specify the region to which this load can be applied without causing tensile stress to be developed at points A, B, C, and D.

a a a a

D ez

eyB C

Prob. 8–51

8–46. The support is subjected to the compressive load P. Determine the absolute maximum possible and minimum possible normal stress acting in the material, for x Ú 0.

a— 2

Prob. 8–46

8–47. The bent shaft is fixed in the wall at A . If a force F is applied at B , determine the stress components at points D and E . Show the results on a differential element located at each of these points. Take F = 12 lb and u = 0°.

*8–48. The bent shaft is fixed in the wall at A . If a force F is applied at B , determine the stress components at points D and E . Show the results on a differential element located at each of these points. Take F = 12 lb and u = 90°.

8–49. The bent shaft is fixed in the wall at A . If a force F is applied at B , determine the stress components at points D and E . Show the results on a volume element located at each of these points. Take F = 12 lb and u = 45°.

6 in.

8 in.

3 in.

1.25 in. D

y x

Probs. 8–47/48/49

8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS 433

8–55. Determine the state of stress at point A on the cross section of the post at section a – a . Indicate the results on a differential element at the point.

*8–56. Determine the state of stress at point B on the cross section of the post at section a – a . Indicate the results on a differential element at the point.

400 mm

3 kN4 kN

100 mm100 mm

400 mm

50 mm 50 mm

50 mm 50 mm BA

Section a – a

Probs. 8–55/56

8–57. The sign is subjected to the uniform wind loading. Determine the stress components at points A and B on the 100-mm-diameter supporting post. Show the results on a volume element located at each of these points.

8–58. The sign is subjected to the uniform wind loading. Determine the stress components at points C and D on the 100-mm-diameter supporting post. Show the results on a volume element located at each of these points.

1 m

1.5 kPa 3 m

2 m

Probs. 8–57/58

*8–52. The vertebra of the spinal column can support a maximum compressive stress of smax, before undergoing a compression fracture. Determine the smallest force P that can be applied to a vertebra, if we assume this load is applied at an eccentric distance e from the centerline of the bone, and the bone remains elastic. Model the vertebra as a hollow cylinder with an inner radius ri and outer radius ro.

! e

Prob. 8–52

8–53. The 1-in.-diameter rod is subjected to the loads shown. Determine the state of stress at point A , and show the results on a differential element located at this point.

8–54. The 1-in.-diameter rod is subjected to the loads shown. Determine the state of stress at point B , and show the results on a differential element located at this point.

y x

75 lb 8 in.

3 in.

80 lb

100 lb

Probs. 8–53/54

434 CHAPTER 8 COMBINED LOADINGS

8–63. The uniform sign has a weight of 1500 lb and is supported by the pipe AB, which has an inner radius of 2.75 in. and an outer radius of 3.00 in. If the face of the sign is subjected to a uniform wind pressure of p = 150 lb>ft2, determine the state of stress at points C and D. Show the results on a differential volume element located at each of these points. Neglect the thickness of the sign, and assume that it is supported along the outside edge of the pipe.

*8–64. Solve Prob. 8–63 for points E and F.

3 ft

6 ft

12 ft

x z

150 lb/ft2

C D

Probs. 8–63/64

8–65. Determine the state of stress at point A on the cross section of the pipe at section a–a.

8–66. Determine the state of stress at point B on the cross section of the pipe at section a–a.

50 lb B

12 in.10 in.

60°

0.75 in.

1 in. Section a–a

x y

Probs. 8–65/66

8–59. If P = 60 kN, determine the maximum normal stress developed on the cross section of the column.

*8–60. Determine the maximum allowable force P, if the column is made from material having an allowable normal stress of sallow = 100 MPa.

100 mm

15 mm

15 mm 15 mm

75 mm

150 mm

100 mm

Probs. 8–59/60 8–61. The C-frame is used in a riveting machine. If the force at the ram on the clamp at D is P = 8 kN, sketch the stress distribution acting over the section a – a .

8–62. Determine the maximum ram force P that can be applied to the clamp at D if the allowable normal stress for the material is s allow = 180 MPa.

40 mm

10 mm

60 mm 10 mm

200 mm

a P

Probs. 8–61/62

8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS 435

8–70. The 34@in.@diameter shaft is subjected to the loading shown. Determine the stress components at point A. Sketch the results on a volume element located at this point. The journal bearing at C can exert only force components Cy and Cz on the shaft, and the thrust bearing at D can exert force components Dx, Dy, and Dz on the shaft.

8–71. Solve Prob. 8–70 for the stress components at point B.

125 lb

2 in.

8 in.

8 in. 125 lb

20 in.

10 in.

2 in.

Probs. 8–70/71

*8–72. The hook is subjected to the force of 80 lb. Determine the state of stress at point A at section a – a . The cross section is circular and has a diameter of 0.5 in. Use the curved-beam formula to compute the bending stress.

8–73. The hook is subjected to the force of 80 lb. Determine the state of stress at point B at section a – a . The cross section has a diameter of 0.5 in. Use the curved-beam formula to compute the bending stress.

80 lb

1.5 in.

A A

45!

Probs. 8–72/73

8–67. The metal link is subjected to the axial force of P = 7 kN. Its original cross section is to be altered by cutting a circular groove into one side. Determine the distance a the groove can penetrate into the cross section so that the tensile stress does not exceed s allow = 175 MPa. Offer a better way to remove this depth of material from the cross section and calculate the tensile stress for this case. Neglect the effects of stress concentration.

25 mm 40 mm

40 mm P

Prob. 8–67

*8–68. The bar has a diameter of 40 mm. If it is subjected to a force of 800 N as shown, determine the stress components that act at point A and show the results on a volume element located at this point.

8–69. Solve Prob. 8–68 for point B.

150 mm

200 mm z

800 N

30!

Probs. 8–68/69

436 CHAPTER 8 COMBINED LOADINGS

CHAPTER REVIEW

A pressure vessel is considered to have a thin wall provided r> t Ú 10. For a thin-walled cylindrical vessel, the circumferential or hoop stress is

s1 = pr

This stress is twice as great as the longitudinal stress,

s2 = pr

Thin-walled spherical vessels have the same stress within their walls in all directions. It is

s1 = s2 = pr

Superposition of stress components can be used to determine the normal and shear stress at a point in a member subjected to a combined loading. To do this, it is first necessary to determine the resultant axial and shear forces and the internal resultant torsional and bending moments at the section where the point is located. Then the normal and shear stress resultant components at the point are determined by algebraically adding the normal and shear stress components of each loading.

s ! P A

t ! VQ It

M smax

s ! " My

tmax

t ! Tr J

CONCEPTUAL PROBLEMS 437

C8–3. Unlike the turnbuckle at B , which is connected along the axis of the rod, the one at A has been welded to the edges of the rod, and so it will be subjected to additional stress. Use the same numerical values for the tensile load in each rod and the rod’s diameter, and compare the stress in each rod.

C8–3

C8–4. A constant wind blowing against the side of this chimney has caused creeping strains in the mortar joints, such that the chimney has a noticeable deformation. Explain how to obtain the stress distribution over a section at the base of the chimney, and sketch this distribution over the section.

C8–4

C8–1. Explain why failure of this garden hose occurred near its end and why the tear occurred along its length. Use numerical values to explain your result. Assume the water pressure is 30 psi.

C8–1

C8–2. This open-ended silo contains granular material. It is constructed from wood slats and held together with steel bands. Explain, using numerical values, why the bands are not spaced evenly along the height of the cylinder. Also, how would you find this spacing if each band is to be subjected to the same stress?

C8–2

CONCEPTUAL PROBLEMS

438 CHAPTER 8 COMBINED LOADINGS

8–74. The eye hook has the dimensions shown. If it supports a cable loading of 80 kN, determine the maximum normal stress at section a – a and sketch the stress distribution acting over the cross section.

3.75 in. 2.5 in.

800 lb

1.25 in.

800 lb

Prob. 8–74

8–75. The 20-kg drum is suspended from the hook mounted on the wooden frame. Determine the state of stress at point E on the cross section of the frame at section a–a. Indicate the results on an element.

*8–76. The 20-kg drum is suspended from the hook mounted on the wooden frame. Determine the state of stress at point F on the cross section of the frame at section b–b. Indicate the results on an element.

1 m

C B

30!

1 m 0.5 m0.5 m

50 mm

75 mm 25 mm

Section a – a

75 mm

25 mm Section b – b

Probs. 8–75/76

8–77. A bar having a square cross section of 30 mm by 30 mm is 2 m long and is held upward. If it has a mass of 5 kg>m, determine the largest angle u, measured from the vertical, at which it can be supported before it is subjected to a tensile stress along its axis near the grip.

8–78. Solve Prob. 8–77 if the bar has a circular cross section of 30-mm diameter.

2 m u

Probs. 8–77/78

8–79. The gondola and passengers have a weight of 1500 lb and center of gravity at G . The suspender arm AE has a square cross-sectional area of 1.5 in. by 1.5 in., and is pin connected at its ends A and E . Determine the largest tensile stress developed in regions AB and DC of the arm.

4 ft

5.5 ft

1.25 ft

1.5 in.

B C

Prob. 8–79

REVIEW PROBLEMS

REVIEW PROBLEMS 439

8–83. Air pressure in the cylinder is increased by exerting forces P = 2 kN on the two pistons, each having a radius of 45 mm. If the cylinder has a wall thickness of 2 mm, determine the state of stress in the wall of the cylinder.

*8–84. Determine the maximum force P that can be exerted on each of the two pistons so that the circumferential stress component in the cylinder does not exceed 3 MPa. Each piston has a radius of 45 mm and the cylinder has a wall thickness of 2 mm.

47 mm

Probs. 8–83/84

8–85. The wall hanger has a thickness of 0.25 in. and is used to support the vertical reactions of the beam that is loaded as shown. If the load is transferred uniformly to each strap of the hanger, determine the state of stress at points C and D on the strap at A . Assume the vertical reaction F at this end acts in the center and on the edge of the bracket as shown.

8–86. The wall hanger has a thickness of 0.25 in. and is used to support the vertical reactions of the beam that is loaded as shown. If the load is transferred uniformly to each strap of the hanger, determine the state of stress at points C and D on the strap at B . Assume the vertical reaction F at this end acts in the center and on the edge of the bracket as shown.

10 kip

A B

2 kip/ft

2 ft 2 ft 6 ft

2 in.

3.75 in.

2.75 in. 3 in.

1 in.

2 in.

1 in.

Probs. 8–85/86

*8–80. The hydraulic cylinder is required to support a force of P = 100 kN. If the cylinder has an inner diameter of 100 mm and is made from a material having an allowable normal stress of sallow = 150 MPa, determine the required minimum thickness t of the wall of the cylinder.

8–81. The hydraulic cylinder has an inner diameter of 100 mm and wall thickness of t = 4 mm. If it is made from a material having an allowable normal stress of sallow = 150 MPa, determine the maximum allowable force P.

100 mm

Probs. 8–80/81

8–82. If the cross section of the femur at section a–a can be approximated as a circular tube as shown, determine the maximum normal stress developed on the cross section at section a–a due to the load of 75 lb.

a a

2 in. 75 lb

M F

1 in. 0.5 in.

Section a – a

Prob. 8–82

These turbine blades are subjected to a complex pattern of stress. For design it is necessary to determine where and in what direction

the maximum stress occurs.

Chapter 9

441

9.1 Plane-Stress Transformation It was shown in Sec. 1.3 that the general state of stress at a point is characterized by six independent normal and shear stress components, which act on the faces of an element of material located at the point, Fig. 9–1 a . This state of stress, however, is not often encountered in engineering practice. Instead, engineers frequently make approximations or simplifications of the loadings on a body in order that the stress produced in a structural member or mechanical element can be analyzed in a single plane . When this is the case, the material is said to be subjected to plane stress , Fig. 9–1 b . For example, if there is no load on the surface of a body, then the normal and shear stress components will be zero on the face of an element that lies on this surface. Consequently, the corresponding stress components on the opposite face will also be zero, and so the material at the point will be subjected to plane stress. This case occurred throughout the previous chapter.

CHAPTER OBJECTIVES

■ In this chapter, we will show how to transform the stress components that are associated with a particular coordinate system into components associated with a coordinate system having a different orientation. Once the necessary transformation equations are established, we will then be able to obtain the maximum normal and maximum shear stress at a point and find the orientation of elements upon which they act. Plane-stress transformation will be discussed in the first part of the chapter, since this condition is most common in engineering practice. At the end of the chapter we will discuss a method for finding the absolute maximum shear stress at a point when the material is subjected to both plane and three-dimensional states of stress.

Stress

442 CHAPTER 9 STRESS TRANSFORMATION

General state of stress

(a)

Plane stress

(b)

tyz

txy

txy txytxy

txz

sx sx

sy sy

Fig. 9–1

(a)

(b)

y¿

x ¿

sy¿

sx ¿

txy

tx ¿y¿

Fig. 9–2

The general state of plane stress at a point is therefore represented by a combination of two normal-stress components, sx , sy , and one shear- stress component, txy, which act on four faces of the element. For convenience, in this text we will view this state of stress in the x–y plane, with the x , y axes as shown in Fig. 9–2 a . If this state of stress is defined on an element having a different orientation u as in Fig. 9–2 b , then it will be subjected to three different stress components defined as sx!, sy!, tx!y! relative to the x! , y! axes. In other words, the state of plane stress at the point is uniquely represented by two normal stress components and one shear stress component acting on an element. These three components will be different for each specific orientation U of the element at the point .

In this section, we will show how to transform the stress components from the orientation of an element in Fig. 9–2 a to the orientation of the element in Fig. 9–2 b . This is like knowing two force components, say, Fx and Fy, directed along the x , y axes, that produce a resultant force FR , and then trying to find the force components Fx! and Fy! , directed along the x !, y ! axes, so they produce the same resultant. The transformation for force must only account for the force component’s magnitude and direction. The transformation of stress components, however, is more difficult since it must account for the magnitude and direction of each stress component and the orientation of the area upon which each component acts.

9.1 PLANE-STRESS TRANSFORMATION 443

(a)

x face

txy

(b)

y¿

x ¿sy¿ sx ¿

tx ¿y¿ u

x ¿ face

x ¿ y¿

(c)

!A sin u

!A cos u !Au u

(d)

y¿

x ¿

sx ¿

tx ¿y¿

txy

!A sin u

!A cos u

(e)

x ¿

y¿

sy¿

tx ¿y¿

txy

y¿ face

Fig. 9–3

Procedure for Analysis

If the state of stress at a point is known for a given orientation of an element of material, Fig. 9–3 a , then the state of stress in an element having some other orientation, u, Fig. 9–3 b , can be determined using the following procedure.

• To determine the normal and shear stress components sx! , tx!y! acting on the +x ! face of the element, Fig. 9–3 b , section the element in Fig. 9–3 a as shown in Fig. 9–3 c . If the sectioned area is "A , then the adjacent areas of the segment will be "Asin u and "A cos u.

• Draw the free-body diagram of the segment, which requires showing the forces that act on the segment, Fig. 9–3 d . This is done by multiplying the stress components on each face by the area upon which they act.

• When #Fx! = 0 is applied to the free-body diagram the area "A will cancel out of each term and a direct solution for sx! will be possible. Likewise, #Fy! = 0 will yield tx!y! .

• If sy! , acting on the +y ! face of the element in Fig. 9–3 b , is to be determined, then it is necessary to consider a segment of the element as shown in Fig. 9–3 e and follow the same procedure just described. Here, however, the shear stress tx!y! does not have to be determined if it was previously calculated, since it is complementary, that is, it must have the same magnitude on each of the four faces of the element, Fig. 9–3 b .

444 CHAPTER 9 STRESS TRANSFORMATION

The state of plane stress at a point on the surface of the airplane fuselage is represented on the element oriented as shown in Fig. 9–4 a . Represent the state of stress at the point on an element that is oriented 30° clockwise from the position shown.

EXAMPLE 9.1

(a)30!

50 MPa

80 MPa

25 MPa

SOLUTION The rotated element is shown in Fig. 9–4 d . To obtain the stress components on this element we will first section the element in Fig. 9–4 a by the line a – a . The bottom segment is removed, and assuming the sectioned (inclined) plane has an area "A, the horizontal and vertical planes have the areas shown in Fig. 9–4 b . The free-body diagram of this segment is shown in Fig. 9–4 c . Notice that the sectioned x # face is defined by the outward normal x# axis, and the y# axis is along the face.

Equilibrium. Applying the equations of force equilibrium in the x # and y # directions, we have

+ Q$Fx# = 0; sx#"A - (50 "Acos 30!) cos 30! + (25 "Acos 30!) sin 30! + (80 "A sin 30!) sin 30!

+ (25 "Asin 30!) cos 30! = 0 sx# = -4.15 MPa Ans.

+ a$Fy# = 0; tx#y#"A - (50 "A cos 30!) sin 30! - (25 "Acos 30!) cos 30! - (80 "A sin 30!) cos 30!

+ (25 "Asin 30!) sin 30! = 0 t x#y# = 68.8 MPa Ans.

Since sx# is negative, it acts in the opposite direction of that shown in Fig. 9–4 c . The results are shown on the top of the element in Fig. 9–4 d , since this surface is the one considered in Fig. 9–4 c .

30!

(b)

"A "A sin 30!

"A cos 30!

(c)

y¿

x ¿

x 60!

30!

sx ¿ "A

tx ¿y¿ "A

25 "A cos 30!

50 "A cos 30!

80 "A sin 30!

25 "A sin 30!

Fig. 9–4

9.1 PLANE-STRESS TRANSFORMATION 445

We must now repeat the procedure to obtain the stress on the perpendicular plane b – b . Sectioning the element in Fig. 9–4 a along b – b results in a segment having sides with areas shown in Fig. 9–4 e . Orienting the +x ! axis outward, perpendicular to the sectioned face, the associated free-body diagram is shown in Fig. 9–4 f . Thus,

+ R"Fx! = 0; sx!#A - (25 #A cos 30$) sin 30$ + (80 #A cos 30$) cos 30$ - (25 #A sin 30$) cos 30$

- (50 #A sin 30$) sin 30$ = 0 sx! = -25.8 MPa Ans.

+ Q"Fy! = 0; -tx!y! #A + (25 #A cos 30$) cos 30$ + (80 #A cos 30$) sin 30$ - (25 #A sin 30$) sin 30$

+ (50 #A sin 30$) cos 30$ = 0 tx!y! = 68.8 MPa Ans.

Since sx! is a negative quantity, it acts opposite to its direction shown in Fig. 9–4 f . The stress components are shown acting on the right side of the element in Fig. 9–4 d .

From this analysis we may therefore conclude that the state of stress at the point can be represented by choosing an element oriented as shown in Fig. 9–4 a , or by choosing one oriented as shown in Fig. 9–4 d . In other words, these states of stress are equivalent.

(d)

25.8 MPa

68.8 MPa 4.15 MPa

30$

(e)

#A sin 30$

#A cos 30$ #A

y¿

x ¿

30$

(f)

30$

25 #A sin 30$

25 #A cos 30$

80 #A cos 30$

50 #A sin 30$

sx ¿ #A

tx ¿y¿ #A

446 CHAPTER 9 STRESS TRANSFORMATION

9.2 General Equations of Plane-Stress Transformation

The method of transforming the normal and shear stress components from the x , y to the x ! , y ! coordinate axes, as discussed in the previous section, can be developed in a general manner and expressed as a set of stress-transformation equations.

Sign Convention. First we must establish a sign convention for the stress components. To do this the +x and +x ! axes are used to define the outward normal from a side of the element. Then sx and sx! are positive when they act in the positive x and x ! directions, and txy and tx!y! are positive when they act in the positive y and y ! directions, Fig. 9–5 .

The orientation of the plane on which the normal and shear stress components are to be determined will be defined by the angle u , which is measured from the +x axis to the +x ! axis, Fig. 9–5 b . Notice that the unprimed and primed sets of axes in this figure both form right-handed coordinate systems; that is, the positive z (or z ! ) axis is established by the right-hand rule. Curling the fingers from x (or x ! ) toward y (or y ! ) gives the direction for the positive z (or z ! ) axis that points outward, along the thumb. The angle u will be positive provided it follows the curl of the right-hand fingers, i.e., counterclockwise as shown in Fig. 9–5 b .

Normal and Shear Stress Components. Using the established sign convention, the element in Fig. 9–6 a is sectioned along the inclined plane and the segment shown in Fig. 9–6 b is isolated. Assuming the sectioned area is "A, then the horizontal and vertical faces of the segment have an area of "A sin u and "A cos u, respectively.

(a)

#sx

#txy

#sy

(b)

y y¿

x ¿

Positive Sign Convention

sx’

tx’y’

Fig. 9–5

9.2 GENERAL EQUATIONS OF PLANE-STRESS TRANSFORMATION 447

The resulting free-body diagram of the segment is shown in Fig. 9–6 c . Applying the equations of equilibrium to determine the unknown normal and shear stress components sx! and tx!y! , we have

+ Q"Fx! = 0; sx!#A - (txy#A sin u) cos u - (sy#A sin u) sin u - (txy#A cos u) sin u - (sx #A cos u) cos u = 0

sx! = sx cos2 u + sy sin2 u + txy(2 sin u cos u)

+ a"Fy! = 0; tx!y!#A + (txy#A sin u) sin u - (sy#A sin u) cos u - (txy#A cos u) cos u + (sx #A cos u) sin u = 0 tx!y! = (sy - sx) sin u cos u + txy (cos2 u - sin2 u)

These two equations may be simplified by using the trigonometric identities sin 2u = 2 sin u cos u, sin2 u = (1 - cos 2u)>2, and cos2 u = (1 + cos 2u)>2, in which case, sx! =

sx + sy 2

+ sx - sy

2 cos 2u + txy sin 2u (9–1)

t x!y! = - sx - sy

2 sin 2u + txy cos 2u (9–2)

If the normal stress acting in the y ! direction is needed, it can be obtained by simply substituting u + 90$ for u into Eq. 9–1 , Fig. 9–6 d . This yields

s y! = s x + s y

2 -

s x - s y 2

cos 2u - t xy sin 2u (9–3)

If sy! is calculated as a positive quantity, it indicates that it acts in the positive y ! direction as shown in Fig. 9–6 d .

(a)

txy

sxu x

x ¿ y¿

(b)

#A sin u

#A cos u #A

x ¿

y¿

(c)

tx ¿y¿ #A

txy #A sin u

txy #A cos u

sx #A cos u

sy #A sin u

sx ¿ #A

u u

y¿

(d)

x ¿

u ! 90"

tx ¿y¿ sy¿

sx ¿

Fig. 9–6

Procedure for Analysis

To apply the stress transformation Eqs. 9–1 and 9–2 , it is simply necessary to substitute in the known data for sx , sy, txy , and u in accordance with the established sign convention, Fig. 9–5 . Remember that the x ! axis is always directed positive outward from the plane upon which the normal stress is to be determined. The angle u is positive counterclockwise, from the x to the x ! axis. If sx! and tx!y! are calculated as positive quantities, then these stresses act in the positive direction of the x ! and y ! axes. For convenience, these equations can easily be programmed on a pocket calculator.

448 CHAPTER 9 STRESS TRANSFORMATION

The state of plane stress at a point is represented by the element shown in Fig. 9–7 a . Determine the state of stress at the point on another element oriented 30° clockwise from the position shown.

SOLUTION This problem was solved in Example 9.1 using basic principles. Here we will apply Eqs. 9–1 and 9–2 . From the established sign convention, Fig. 9–5 , it is seen that

sx = -80 MPa sy = 50 MPa txy = -25 MPa

Plane CD. To obtain the stress components on plane CD , Fig. 9–7 b , the positive x ! axis must be directed outward, perpendicular to CD , and the associated y ! axis is directed along CD . The angle measured from the x to the x ! axis is u = -30" (clockwise). Applying Eqs. 9–1 and 9–2 yields

sx! = sx + sy

2 +

sx - sy 2

cos 2u + txy sin 2u

= -80 + 50

2 + -80 - 50

2 cos 2(-30") + (-25) sin 2(-30")

= -25.8 MPa Ans.

tx!y! = - sx - sy

2 sin 2u + txy cos 2u

= - -80 - 50 2

sin 2(-30") + (-25) cos 2(-30")

= -68.8 MPa Ans. The negative signs indicate that sx! and tx!y! act in the negative x ! and y ! directions, respectively. The results are shown acting on the element in Fig. 9–7 d .

Plane BC. Establishing the x ! axis outward from plane BC , Fig. 9–7 c , then between the x and x! axes, u = 60" (counterclockwise). Applying Eqs. 9–1 and 9–2 , * we get

sx! = -80 + 50

2 + -80 - 50

2 cos 2(60") + (-25) sin 2(60")

= -4.15 MPa Ans.

tx!y! = - -80 - 50

2 sin 2(60") + (-25) cos 2(60")

= 68.8 MPa Ans.

Here tx!y! has been calculated twice in order to provide a check. The negative sign for sx! indicates that this stress acts in the negative x ! direction, Fig. 9–7 c . The results are shown on the element in Fig. 9–7 d .

EXAMPLE 9.2

50 MPa

(a)

80 MPa

25 MPa

(b)

B y¿

x ¿

30!

u " #30!

(c)

y¿

x¿

30!

u"60!

25.8 MPa

68.8 MPa 4.15 MPa

(d)

Fig. 9–7

* Alternatively, we could apply Eq. 9–3 with u = -30" rather than Eq. 9–1 .

9.3 PRINCIPAL STRESSES AND MAXIMUM IN-PLANE SHEAR STRESS 449

9.3 Principal Stresses and Maximum In-Plane Shear Stress

From Eqs. 9–1 and 9–2 , it can be seen that the magnitudes of sx! and tx!y! depend on the angle of inclination u of the planes on which these stresses act. In engineering practice it is often important to determine the orientation of the element that causes the normal stress to be a maximum and a minimum and the orientation that causes the shear stress to be a maximum. In this section each of these problems will be considered.

In-Plane Principal Stresses. To determine the maximum and minimum normal stress , we must differentiate Eq. 9–1 with respect to u and set the result equal to zero. This gives

dsx! du

= - sx - sy

2 (2 sin 2u) + 2txy cos 2u = 0

Solving this equation we obtain the orientation u = up of the planes of maximum and minimum normal stress.

tan 2up = txy

(sx - sy)>2 (9–4) The solution has two roots, up1 , and up2. Specifically, the values of 2up1 and 2up2 are 180° apart, so up1 and up2 will be 90° apart.

The values of up1 and up2 must be substituted into Eq. 9–1 if we are to obtain the required normal stresses. To do this we can obtain the necessary sine and cosine of 2up1 and 2up2 from the shaded triangles shown in Fig. 9–8 . The construction of these triangles is based on Eq. 9– 4 , assuming that txy and (sx - sy) are both positive or both negative quantities.

!txy

txy

sx ! sy

txy2 2

2up2

2up1

Fig. 9–8

450 CHAPTER 9 STRESS TRANSFORMATION

Substituting these trigonometric values into Eq. 9–1 and simplifying, we obtain

s1,2 = sx + sy

2 { C¢sx - sy2 ≤2 + txy 2 (9–5)

Depending upon the sign chosen, this result gives the maximum or minimum in-plane normal stress acting at a point, where s1 Ú s2 . This particular set of values are called the in-plane principal stresses , and the corresponding planes on which they act are called the principal planes of stress, Fig. 9–9 . Furthermore, if the trigonometric relations for up1 or up2 are substituted into Eq. 9–2 , it can be seen that tx!y! = 0; in other words, no shear stress acts on the principal planes .

The cracks in this concrete beam were caused by tension stress, even though the beam was subjected to both an internal moment and shear. The stress- transformation equations can be used to predict the direction of the cracks, and the principal normal stresses that caused them.

! x

x¿

x–

In-plane principal stresses

txy

up2 ! up1" 90#

up1

Fig. 9–9

Take a piece of blackboard chalk and twist it. Since it is brittle, it will fail in tension, caused by the principal normal stress. See Example 9.5.

9.3 PRINCIPAL STRESSES AND MAXIMUM IN-PLANE SHEAR STRESS 451

Maximum In-Plane Shear Stress. The orientation of an element that is subjected to maximum shear stress on its sides can be determined by taking the derivative of Eq. 9–2 with respect to u and setting the result equal to zero. This gives

tan 2us = -(sx - sy)>2

txy (9–6)

The two roots of this equation, us1 and us2, can be determined from the shaded triangles shown in Fig. 9–10 . By comparison with Eq. 9–4 , tan 2us is the negative reciprocal of tan 2up and so each root 2us is 90° from 2up , and the roots us and up are 45° apart. Therefore, an element subjected to maximum shear stress will be 45° from the position of an element that is subjected to the principal stress .

Using either one of the roots us1 or us2, the maximum shear stress can be found by taking the trigonometric values of sin 2us and cos 2us from Fig. 9–10 and substituting them into Eq. 9–2 . The result is

t max in@plane

= C¢sx - sy2 ≤2 + txy 2 (9–7) The value of t

max in@plane

as calculated from this equation is referred to as the maximum in-plane shear stress because it acts on the element in the x – y plane.

Substituting the values for sin 2us and cos 2us into Eq. 9–1 , we see that there is also an average normal stress on the planes of maximum in-plane shear stress. We get

savg = sx + sy

2 (9–8)

Like the stress-transformation equations, it may be convenient to program Eqs. 9–4 through 9–8 for use on a pocket calculator.

sx ! sy

txy

!txy

2us1

2us2

Fig. 9–10

Important Points

• The principal stresses represent the maximum and minimum normal stress at the point.

• When the state of stress is represented by the principal stresses, no shear stress will act on the element.

• The state of stress at the point can also be represented in terms of the maximum in-plane shear stress . In this case an average normal stress will also act on the element.

• The element representing the maximum in-plane shear stress with the associated average normal stresses is oriented 45° from the element representing the principal stresses.

452 CHAPTER 9 STRESS TRANSFORMATION

The state of plane stress at a failure point on the shaft is shown on the element in Fig. 9–11 a . Represent this stress state in terms of the principal stresses.

SOLUTION From the established sign convention, we have

sx = -20 MPa sy = 90 MPa txy = 60 MPa

Orientation of Element. Applying Eq. 9–4 ,

tan 2up = txy

(sx - sy)>2 = 60(-20 - 90)>2 Solving, and referring to this root as up2, as will be shown below, yields

2up2 = -47.49! up2 = -23.7!

Since the difference between 2up1 and 2up2 is 180°, we have

2up1 = 180! + 2up2 = 132.51! up1 = 66.3!

Recall that u must be measured positive counterclockwise from the x axis to the outward normal ( x " axis) on the face of the element, and so the results are shown in Fig. 9–11 b .

Principal Stress. We have

s1,2 = sx + sy

2 { B ¢sx - sy2 ≤2 + txy 2

= -20 + 90

2 { B a -20 - 902 b2 + (60)2

= 35.0 { 81.4 s1 = 116 MPa Ans. s2 = -46.4 MPa Ans. The principal plane on which each normal stress acts can be

determined by applying Eq. 9–1 with, say, u = up2 = -23.7!. We have

sx" = sx + sy

2 +

sx - sy 2

cos 2u + txy sin 2u

= -20 + 90

2 + -20 - 90

2 cos 2(-23.7!) + 60 sin 2(-23.7!)

= -46.4 MPa

Hence, s2 = -46.4 MPa acts on the plane defined by up2 = -23.7!, whereas s1 = 116 MPa acts on the plane defined by up1 = 66.3!. The results are shown on the element in Fig. 9–11 c . Recall that no shear stress acts on this element.

EXAMPLE 9.3

Notice how the failure plane is at an angle (23.7!) due to tearing of the material, Fig. 9– 11 c .

90 MPa

(a)

60 MPa

20 MPa

(b)

23.7!

66.3!

x¿

y¿y¿

x¿

(c)

up1 " 66.3!

" 46.4 MPa up2 " 23.7!

" 116 MPas1

Fig. 9–11

9.3 PRINCIPAL STRESSES AND MAXIMUM IN-PLANE SHEAR STRESS 453

EXAMPLE 9.4

The state of plane stress at a point on a body is represented on the element shown in Fig. 9–12 a . Represent this stress state in terms of the maximum in-plane shear stress and associated average normal stress.

SOLUTION Orientation of Element. Since sx = -20 MPa, sy = 90 MPa, and txy = 60 MPa, applying Eq. 9–6 , we have

tan 2us = -(sx - sy)>2

txy =

-(-20 - 90)>2 60

2us2 = 42.5! us2 = 21.3!

2us1 = 180! + 2us2 us1 = 111.3!

Note how these angles are formed between the x and x" axes, Fig. 9– 12 b . They happen to be 45° away from the principal planes of stress, which were determined in Example 9.3 .

Maximum In-Plane Shear Stress. Applying Eq. 9–7 ,

t max in@plane

= C¢sx - sy2 ≤2 + txy 2 = B a -20 - 902 b2 + (60)2 = {81.4 MPa Ans. The proper direction of t

max in@plane

on the element can be determined by substituting u = us2 = 21.3! into Eq. 9–2 . We have

tx"y" = - ¢sx - sy2 ≤ sin 2u + txy cos 2u = - a -20 - 90

2 b sin 2(21.3!) + 60 cos 2(21.3!)

= 81.4 MPa

This positive result indicates that t max in@plane

= tx"y" acts in the positive y " direction on this face (u = 21.3!) Fig. 9–12 b . The shear stresses on the other three faces are directed as shown in Fig. 9–12 c .

Average Normal Stress. Besides the maximum shear stress, as calculated above, the element is also subjected to an average normal stress determined from Eq. 9–8 ; that is,

savg = sx + sy

2 =

-20 + 90 2

= 35 MPa Ans.

This is a tensile stress. The results are shown in Fig. 9–12 c .

90 MPa

(a)

60 MPa

20 MPa

(b)

x¿

y¿

x¿

21.3!

111.3!

81.4 MPa

x x

35 MPa

81.4 MPa

21.3!

(c)

Fig. 9–12

454 CHAPTER 9 STRESS TRANSFORMATION

EXAMPLE 9.5

When the torsional loading T is applied to the bar in Fig. 9–13 a , it produces a state of pure shear stress in the material. Determine (a) the maximum in-plane shear stress and the associated average normal stress, and (b) the principal stress.

SOLUTION From the established sign convention,

sx = 0 sy = 0 txy = -t

Maximum In-Plane Shear Stress. Applying Eqs. 9–7 and 9–8 , we have

t max in@plane

= C¢sx - sy2 ≤2 + txy 2 = 2(0)2 + (-t)2 = {t Ans. savg =

sx + sy 2

= 0 + 0

2 = 0 Ans.

Thus, as expected, the maximum in-plane shear stress is represented by the element in Fig. 9–13 a .

NOTE: Through experiment it has been found that materials that are ductile will fail due to shear stress . As a result, if the bar in Fig. 9–13 a is made of mild steel, the maximum in-plane shear stress will cause it to fail as shown in the adjacent photo.

T T

(a)

(b)

45! x

y¿ x¿

s2 " t

s1 " t

Fig. 9–13

Principal Stress. Applying Eqs. 9–4 and 9–5 yields

tan 2up = txy

(sx - sy)>2 = -t(0 - 0)>2 , up2 = 45!, up1 = -45! s1, 2 =

sx + sy 2

{ B asx - sy2 b2 + txy 2 = 0 { 2(0)2 + t2 = {t Ans. If we now apply Eq. 9–1 with up2 = 45!, then

sx" = sx + sy

2 +

sx - sy 2

cos 2u + txy sin 2u

= 0 + 0 + (-t) sin 90! = -t

Thus, s2 = -t acts at up2 = 45! as shown in Fig. 9–13 b , and s1 = t acts on the other face, up1 = -45!.

NOTE: Materials that are brittle fail due to normal stress . Therefore, if the bar in Fig. 9–13 a is made of cast iron it will fail in tension at a 45° inclination as seen in the adjacent photo.

9.3 PRINCIPAL STRESSES AND MAXIMUM IN-PLANE SHEAR STRESS 455

When the axial loading P is applied to the bar in Fig. 9–14 a , it produces a tensile stress in the material. Determine (a) the principal stress and (b) the maximum in-plane shear stress and associated average normal stress.

SOLUTION From the established sign convention,

sx = s sy = 0 txy = 0

Principal Stress. By observation, the element oriented as shown in Fig. 9–14 a illustrates a condition of principal stress since no shear stress acts on this element. This can also be shown by direct substitution of the above values into Eqs. 9–4 and 9–5 . Thus,

s1 = s s2 = 0 Ans.

NOTE: Experiments have shown that brittle materials will fail due to normal stress. Thus if the bar in Fig. 9–14 a is made of cast iron, it will cause failure as shown in the adjacent photo.

Maximum In-Plane Shear Stress. Applying Eqs. 9–6 , 9–7 , and 9–8 , we have

tan 2us = -(sx - sy)>2

txy =

-(s - 0)>2 0

; us1 = 45!, us2 = -45!

t max in@plane

= C¢sx - sy2 ≤2 + txy 2 = B as - 02 b2 + (0)2 = {s2 Ans. savg =

sx + sy 2

= s + 0

2 =

2 Ans.

To determine the proper orientation of the element, apply Eq. 9–2 .

tx"y" = - sx - sy

2 sin 2u + txy cos 2u = -

s - 0 2

sin 90! + 0 = - s 2

This negative shear stress acts on the x " face, in the negative y " direction as shown in Fig. 9–14 b .

NOTE: If the bar in Fig. 9–14 a is made of a ductile material such as mild steel then shear stress will cause it to fail. This can be noted in the adjacent photo, where within the region of necking, shear stress has caused “slipping” along the steel’s crystalline boundaries, resulting in a plane of failure that has formed a cone around the bar oriented at approximately 45° as calculated above.

EXAMPLE 9.6

(a)

(b)

45! x

x¿ y¿

tin-plane " max

s s

2savg "

Fig. 9–14

456 CHAPTER 9 STRESS TRANSFORMATION

PRELIMINARY PROBLEMS

(c)

40!

A A A

B BB

txy

P9–1

P9–2. Given the state of stress shown on the element, find s avg and tmaxin@plane and show the results on a properly oriented element.

4 MPa

P9–2

P9–1. In each case, the state of stress s x , s y , t xy produces normal and shear stress components along section AB of the element that have values of s x ! = -5 kPa and t x!y! = 8 kPa when calculated using the stress transformation equations. Establish the x ! and y ! axes for each segment and specify the angle u, then show these results acting on each segment.

(a)

A txy

20!

(b)

30!

A A

txy

9.3 PRINCIPAL STRESSES AND MAXIMUM IN-PLANE SHEAR STRESS 457

FUNDAMENTAL PROBLEMS

F9–4. Determine the equivalent state of stress on an element at the same point that represents the maximum in-plane shear stress at the point.

100 kPa

400 kPa

700 kPa

F9–4

F9–5. The beam is subjected to the load at its end. Determine the maximum principal stress at point B .

60 mm

2 kN

4 kN

30 mm

2 m

F9–5

F9–6. The beam is subjected to the loading shown. Determine the principal stress at point C .

3 m 3 m

8 kN/m

150 mm

75 mm

F9–6

F9–1. Determine the normal stress and shear stress acting on the inclined plane AB . Sketch the result on the sectioned element.

500 kPa

30!

F9–1

F9–2. Determine the equivalent state of stress on an element at the same point oriented 45° clockwise with respect to the element shown.

300 kPa

400 kPa

F9–2

F9–3. Determine the equivalent state of stress on an element at the same point that represents the principal stresses at the point. Also, find the corresponding orientation of the element with respect to the element shown.

80 kPa

30 kPa

F9–3

458 CHAPTER 9 STRESS TRANSFORMATION

*9–4. Determine the normal stress and shear stress acting on the inclined plane AB . Solve the problem using the method of equilibrium described in Sec. 9.1.

9–5. Determine the normal stress and shear stress acting on the inclined plane AB . Solve the problem using the stress transformation equations. Show the results on the sectional element.

15 ksi

6 ksi A

60!

Probs. 9–4/5

9–6. Determine the normal stress and shear stress acting on the inclined plane AB . Solve the problem using the method of equilibrium described in Sec. 9.1 .

9–7. Determine the normal stress and shear stress acting on the inclined plane AB . Solve the problem using the stress transformation equations. Show the result on the sectioned element.

80 MPa

45 MPa

45!

Probs. 9–6/7

9–1. Prove that the sum of the normal stresses sx + sy = sx! + sy! is constant. See Figs. 9–2 a and 9–2 b .

9–2. The state of stress at a point in a member is shown on the element. Determine the stress components acting on the inclined plane AB . Solve the problem using the method of equilibrium described in Sec. 9.1 .

40"

8 ksi

3 ksi

5 ksi

Prob. 9–2

9–3. The state of stress at a point in a member is shown on the element. Determine the stress components acting on the inclined plane AB . Solve the problem using the method of equilibrium described in Sec. 9.1 .

60!

A 400 psi

650 psi

Prob. 9–3

PROBLEMS

9.3 PRINCIPAL STRESSES AND MAXIMUM IN-PLANE SHEAR STRESS 459

9–13. Determine the equivalent state of stress on an the element if it is oriented 30° clockwise from the element shown. Use the stress-transformation equations.

300 psi

950 psi

Prob. 9–13

9–14. The state of stress at a point is shown on the element. Determine (a) the principal stress and (b) the maximum in-plane shear stress and average normal stress at the point. Specify the orientation of the element in each case. Show the results on each element.

30 ksi

12 ksi

Prob. 9–14

9–15. The state of stress at a point is shown on the element. Determine (a) the principal stress and (b) the maximum in-plane shear stress and average normal stress at the point. Specify the orientation of the element in each case.

60 MPa

30 MPa

45 MPa

Prob. 9–15

*9–8. Determine the equivalent state of stress on an element at the same point oriented 30° clockwise with respect to the element shown. Sketch the results on the element.

9–9. Determine the equivalent state of stress on an element at the same point oriented 30° counterclockwise with respect to the element shown. Sketch the results on the element.

75 MPa

100 MPa

Probs. 9–8/9

9–10. Determine the equivalent state of stress on an element at the same point oriented 60° clockwise with respect to the element shown. Sketch the results on the element.

9–11. Determine the equivalent state of stress on an element at the same point oriented 60° counterclockwise with respect to the element shown. Sketch the results on the element.

100 MPa

75 MPa

150 MPa

Probs. 9–10/11

*9–12. Determine the equivalent state of stress on an element if it is oriented 50° counterclockwise from the element shown. Use the stress-transformation equations.

16 ksi

10 ksi

Prob. 9–12

460 CHAPTER 9 STRESS TRANSFORMATION

9–19. Determine the equivalent state of stress on an element at the same point which represents (a) the principal stress, and (b) the maximum in-plane shear stress and the associated average normal stress. Also, for each case, determine the corresponding orientation of the element with respect to the element shown and sketch the results on the element.

100 MPa

25 MPa

Prob. 9–19

*9–20. Planes AB and BC at a point are subjected to the stresses shown. Determine the principal stresses acting at this point and find sBC.

B45!

5 ksi 15 ksi

6 ksi

sBC

Prob. 9–20

9–21. The stress acting on two planes at a point is indicated. Determine the shear stress on plane a–a and the principal stresses at the point.

80 ksi

60 ksi

90! 45!

60!

Prob. 9–21

*9–16. Determine the equivalent state of stress on an element at the point which represents (a) the principal stresses and (b) the maximum in-plane shear stress and the associated average normal stress. Also, for each case, determine the corresponding orientation of the element with respect to the element shown and sketch the results on the element.

15 MPa

50 MPa

Prob. 9–16

9–17. Determine the equivalent state of stress on an element at the same point which represents (a) the principal stress, and (b) the maximum in-plane shear stress and the associated average normal stress. Also, for each case, determine the corresponding orientation of the element with respect to the element shown. Sketch the results on each element.

50 MPa

125 MPa

75 MPa

Prob. 9–17

9–18. A point on a thin plate is subjected to the two successive states of stress shown. Determine the resultant state of stress represented on the element oriented as shown on the right.

85 MPa

60 MPa

30!

45! 85 MPa

txy

Prob. 9–18

9.3 PRINCIPAL STRESSES AND MAXIMUM IN-PLANE SHEAR STRESS 461

9–25. The wooden block will fail if the shear stress acting along the grain is 550 psi. If the normal stress s x = 400 psi, determine the necessary compressive stress s y that will cause failure.

58! sx " 400 psi

Prob. 9–25

9–26. The bracket is subjected to the force of 3 kip. Determine the principal stress and maximum in-plane shear stress at point A on the cross section at section a–a . Specify the orientation of this state of stress and show the results on elements.

9–27. The bracket is subjected to the force of 3 kip. Determine the principal stress and maximum in-plane shear stress at point B on the cross section at section a–a . Specify the orientation of this state of stress and show the results on elements.

a 3 in. 3 kip3 kip

2 in.

1 in. 0.25 in.

0.25 in.

Section a – a

Probs. 9–26/27

9–22. The grains of wood in the board make an angle of 20! with the horizontal as shown. Determine the normal and shear stress that act perpendicular and parallel to the grains if the board is subjected to an axial load of 250 N.

300 mm

250 N 60 mm

25 mm20! 250 N

Prob. 9–22

9–23. The wood beam is subjected to a load of 12 kN. If a grain of wood in the beam at point A makes an angle of 25° with the horizontal as shown, determine the normal and shear stress that act perpendicular and parallel to the grain due to the loading.

2 m 4 m1 m 12 kN

25! 75 mm

300 mm

200 mm

Prob. 9–23

*9–24. The wood beam is subjected to a load of 12 kN. Determine the principal stress at point A and specify the orientation of the element.

2 m 4 m1 m 12 kN

25! 75 mm

300 mm

200 mm

Prob. 9–24

The following problems involve material covered in Chapter 8.

462 CHAPTER 9 STRESS TRANSFORMATION

9–30. The state of stress at a point in a member is shown on the element. Determine the stress components acting on the plane A B .

50 MPa

30! 28 MPa

100 MPa

Prob. 9–30

9–31. Determine the principal stress at point A on the cross section of the arm at section a–a . Specify the orientation of this state of stress and indicate the results on an element at the point.

*9–32. Determine the maximum in-plane shear stress developed at point A on the cross section of the arm at section a–a . Specify the orientation of this state of stress and indicate the results on an element at the point.

Section a – a

B C

500 N

60!

50 mm

7.5 mm

7.5 mm 20 mm

0.15 m 0.15 m 0.35 m

Probs. 9–31/32

*9–28. The 25-mm thick rectangular bar is subjected to the axial load of 10 kN. If the bar is joined by the weld, which makes an angle of 60° with the horizontal, determine the shear stress parallel to the weld and the normal stress perpendicular to the weld.

80 mm 60!

10 kN 10 kN

Prob. 9–28

9–29. The 3-in. diameter shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B . Determine the principal stresses and maximum in-plane shear stress at a point on the outer surface of the shaft at section a – a .

3 kip

500 lb"ft

500 lb"ft a

Prob. 9–29

9.3 PRINCIPAL STRESSES AND MAXIMUM IN-PLANE SHEAR STRESS 463

9–35. The square steel plate has a thickness of 10 mm and is subjected to the edge loading shown. Determine the maximum in-plane shear stress and the average normal stress developed in the steel.

200 mm

50 N/m

Prob. 9–35

*9–36. The square steel plate has a thickness of 0.5 in. and is subjected to the edge loading shown. Determine the principal stresses developed in the steel.

4 in.

16 lb/in.

Prob. 9–36

9–37. The shaft has a diameter d and is subjected to the loadings shown. Determine the principal stress and the maximum in-plane shear stress that is developed at point A . The bearings only support vertical reactions.

F F

L 2

Prob. 9–37

9–33. The clamp bears down on the smooth surface at E by tightening the bolt. If the tensile force in the bolt is 40 kN, determine the principal stress at points A and B and show the results on elements located at each of these points. The cross- sectional area at A and B is shown in the adjacent figure.

100 mm 50 mm

B B

50 mm

30 mm

25 mm

100 mm

300 mm

Prob. 9–33

9–34. Determine the principal stress and the maximum in-plane shear stress that are developed at point A in the 2-in.-diameter shaft. Show the results on an element located at this point. The bearings only support vertical reactions.

24 in. 12 in. 12 in.

300 lb

3000 lb3000 lb

Prob. 9–34

464 CHAPTER 9 STRESS TRANSFORMATION

9–42. The drill pipe has an outer diameter of 3 in., a wall thickness of 0.25 in., and a weight of 50 lb>ft. If it is subjected to a torque and axial load as shown, determine (a) the principal stresses and (b) the maximum in-plane shear stress at a point on its surface at section a.

800 lb!ft

20 ft

1500 lb

Prob. 9–42

9–43. The nose wheel of the plane is subjected to a design load of 12 kN. Determine the principal stresses acting on the aluminum wheel support at point A .

20 mm

30 mm

12 kN

300 mm

175 mm

150 mm

20 mm A60!

Prob. 9–43

9–38. A paper tube is formed by rolling a paper strip in a spiral and then gluing the edges together as shown. Determine the shear stress acting along the seam, which is at 30° from the vertical, when the tube is subjected to an axial force of 10 N. The paper is 1 mm thick and the tube has an outer diameter of 30 mm.

9–39. Solve Prob. 9–38 for the normal stress acting perpendicular to the seam.

10 N 10 N

30"

30 mm

Probs. 9–38/39

*9–40. The wide-flange beam is subjected to the 50-kN force. Determine the principal stresses in the beam at point A located on the web at the bottom of the upper flange. Although it is not very accurate, use the shear formula to calculate the shear stress.

9–41. Solve Prob. 9–40 for point B located on the web at the top of the bottom flange.

1 m 3 m

50 kN

B 10 mm

12 mm 250 mm 12 mm

200 mm

Probs. 9–40/41

9.4 MOHR’S CIRCLE—PLANE STRESS 465

9.4 Mohr’s Circle—Plane Stress In this section, we will show how to apply the equations for plane stress transformation using a graphical solution that is often convenient to use and easy to remember. Furthermore, this approach will allow us to “visualize” how the normal and shear stress components sx! and tx!y! vary as the plane on which they act is oriented in different directions, Fig. 9–15 a .

If we write Eqs. 9–1 and 9–2 in the form

sx! - ¢sx + sy2 ≤ = ¢sx - sy2 ≤ cos 2u + txy sin 2u (9–9) tx!y! = - ¢sx - sy2 ≤ sin 2u + txy cos 2u (9–10) then the parameter u can be eliminated by squaring each equation and adding the equations together. The result is

Jsx! - ¢sx + sy2 ≤ R 2 + t2x!y! = ¢sx - sy2 ≤2 + t2xy For a specific problem, sx , sy , txy are known constants . Thus the above equation can be written in a more compact form as

(sx! - savg)2 + t2x!y! = R2 (9–11) where

savg = sx + sy

R = C¢sx - sy2 ≤2 + t2xy (9–12) If we establish coordinate axes, s positive to the right and t positive

downward , and then plot Eq. 9–11 , it will be seen that this equation represents a circle having a radius R and center on the s axis at point C(savg, 0) , Fig. 9–15 b . This circle is called Mohr’s circle , because it was developed by the German engineer Otto Mohr.

x ¿

(a)

y¿

tx ¿y¿

txy

sx ¿

Fig. 9–15

(b) t

R !sx

txy

" txy2

2 sx # sy

2 sx " sy

savg !

466 CHAPTER 9 STRESS TRANSFORMATION

Each point on Mohr’s circle represents the two stress components sx! and tx!y! acting on the side of the element defined by the x ! axis, when the axis is in a specific direction u . For example, when x ! is coincident with the x axis as shown in Fig. 9–16 a , then u = 0" and sx! = sx, tx!y! = txy. We will refer to this as the “reference point” A and plot its coordinates A (sx , txy), Fig. 9–16 c .

Now consider rotating the x ! axis 90° counterclockwise, Fig. 9–16 b . Then sx! = sy , tx!y! = -txy . These values are the coordinates of point G(sy,-txy) on the circle, Fig. 9–16 c . Hence, the radial line CG is 180° counterclockwise from the “reference line” CA . In other words, a rotation u of the x ! axis on the element will correspond to a rotation 2u on the circle in the same direction . (If instead the t axis were constructed positive upwards, then the angle 2u on the circle would be measured in the opposite direction to the orientation u of the x ! axis.)

Once constructed, Mohr’s circle can be used to determine the principal stresses, the maximum in-plane shear stress and associated average normal stress, or the stress on any arbitrary plane.

x, x ¿

(a)

y, y¿sy

sx ! sx ¿

txy ! tx ¿y¿ u ! 0"

(c)

txy

savg

!txy

u " 0#

2u " 180#

2 sx ! sy

Fig. 9–16

(b)

x ¿

y¿

txy

u ! 90"

9.4 MOHR’S CIRCLE—PLANE STRESS 467

Procedure for Analysis

The following steps are required to draw and use Mohr’s circle.

Construction of the Circle. • Establish a coordinate system such that the horizontal axis

represents the normal stress s, with positive to the right , and the vertical axis represents the shear stress t, with positive down wards , Fig. 9–17 a. *

• Using the positive sign convention for sx , sy , txy , as shown in Fig. 9–17 b , plot the center of the circle C , which is located on the s axis at a distance savg = (sx + sy)>2 from the origin, Fig. 9–17 a.

• Plot the “reference point” A having coordinates A (sx, txy). This point represents the normal and shear stress components on the element’s right-hand vertical face, and since the x ! axis coincides with the x axis, this represents u = 0", Fig. 9–17 a.

• Connect point A with the center C of the circle and determine CA by trigonometry. This distance represents the radius R of the circle, Fig. 9–17 a.

• Once R has been determined, sketch the circle.

Principal Stress. • The principal stresses s 1 and s2 (s1 Ú s2) are the coordinates

of points B and D where the circle intersects the s axis, i.e., where t = 0, Fig. 9–17 a.

• These stresses act on planes defined by angles up1 and up2 , Fig. 9–17 c. They are represented on the circle by angles 2up1 (shown) and 2up2 (not shown) and are measured from the radial reference line CA to lines CB and CD , respectively.

• Using trigonometry, only one of these angles needs to be calculated from the circle, since up1 and up2 are 90° apart. Remember that the direction of rotation 2up on the circle (here it happens to be counterclockwise) represents the same direction of rotation up from the reference axis (+x) to the principal plane (+x !), Fig. 9–17 c. *

Maximum In-Plane Shear Stress. • The average normal stress and maximum in-plane shear stress

components are determined from the circle as the coordinates of either point E or F , Fig. 9–17 a.

• In this case the angles us1 and us2 give the orientation of the planes that contain these components, Fig. 9–17 d. The angle 2us1 is shown in Fig. 9–17 a and can be determined using trigonometry. Here the rotation happens to be clockwise, from CA to CE, and so us1 must be clockwise on the element, Fig. 9–17 d. *

468 CHAPTER 9 STRESS TRANSFORMATION

Procedure for Analysis (continued)

Stresses on Arbitrary Plane. • The normal and shear stress components sx! and tx!y! acting on a

specified plane or x ! axis, defined by the angle u, Fig. 9–17 e , can be obtained from the circle using trigonometry to determine the coordinates of point P , Fig. 9–17 a.

• To locate P , the known angle u (in this case counterclockwise), Fig. 9–17 e , must be measured on the circle in the same direction 2u (counterclockwise), from the radial reference line CA to the radial line CP, Fig. 9–17 a. *

* If the t axis were constructed positive upwards, then the angle 2u on the circle would be measured in the opposite direction to the orientation u of the x ! axis.

(a)

savg

tx ¿y¿

sx ¿

txy

u ! 0"

2us1

2up1

(c)

x¿ s2

up2 ! up1 " 90#

up1

(b)

txy

x¿

(d)

y¿ savg

savg us1

(tx ¿y¿)max in-plane

x¿

(e)

y¿

tx ¿y¿

txy

sx ¿ sx

Fig. 9–17

9.4 MOHR’S CIRCLE—PLANE STRESS 469

EXAMPLE 9.7

Due to the applied loading, the element at point A on the solid shaft in Fig. 9–18 a is subjected to the state of stress shown. Determine the principal stresses acting at this point.

SOLUTION Construction of the Circle. From Fig. 9–18 a ,

sx = -12 ksi sy = 0 txy = -6 ksi

The center of the circle is at

savg = -12 + 0

2 = -6 ksi

The reference point A (-12, -6) and the center C(-6, 0) are plotted in Fig. 9–18 b . The circle is constructed having a radius of

R = 2(12 - 6)2 + (6)2 = 8.49 ksi Principal Stress. The principal stresses are indicated by the coordinates of points B and D. We have, for s1 7 s2,

s1 = 8.49 - 6 = 2.49 ksi Ans. s2 = -6 - 8.49 = -14.5 ksi Ans.

The orientation of the element can be determined by calculating the angle 2up2 in Fig. 9–18 b , which here is measured counterclockwise from CA to CD. It defines the direction up2 of s2 and its associated principal plane. We have

2up2 = tan -1

6 12 - 6 = 45.0!

u p 2 = 22.5!

The element is oriented such that the x " axis or s2 is directed 22.5° counterclockwise from the horizontal ( x axis) as shown in Fig. 9–18 c .

(a)

12 ksi

6 ksi

A M

(b)

BD C

s (ksi)

t (ksi)

R ! 8.496

2up2

(c)

22.5"

2.49 ksi

x¿14.5 ksi

Fig. 9–18

470 CHAPTER 9 STRESS TRANSFORMATION

EXAMPLE 9.8

The state of plane stress at a point is shown on the element in Fig. 9–19 a . Determine the maximum in-plane shear stress at this point.

SOLUTION Construction of the Circle. From the problem data,

sx = -20 MPa sy = 90 MPa txy = 60 MPa

The s, t axes are established in Fig. 9–19 b . The center of the circle C is located on the s axis, at the point

savg = -20 + 90

2 = 35 MPa

Point C and the reference point A (-20, 60) are plotted. Applying the Pythagorean theorem to the shaded triangle to determine the circle’s radius CA, we have

R = 2(60)2 + (55)2 = 81.4 MPa Maximum In-Plane Shear Stress. The maximum in-plane shear stress and the average normal stress are identified by point E (or F ) on the circle. The coordinates of point E (35, 81.4) give

t max in@plane

= 81.4 MPa Ans.

savg = 35 MPa Ans.

The angle us1 , measured counterclockwise from CA to CE , can be found from the circle, identified as 2us1. We have

2us1 = tan -1a 20 + 35

60 b = 42.5!

us1 = 21.3! Ans.

This counterclockwise angle defines the direction of the x " axis, Fig. 9–19 c . Since point E has positive coordinates, then the average normal stress and the maximum in-plane shear stress both act in the positive x " and y " directions as shown.

90 MPa

(a)

60 MPa

20 MPa

(b)

81.4

R !

81 .4

s (MPa)

t (MPa)

2us1

(c)

81.4 MPa 35 MPa

21.3! x

y¿

x¿

Fig. 9–19

9.4 MOHR’S CIRCLE—PLANE STRESS 471

The state of plane stress at a point is shown on the element in Fig. 9–20 a . Represent this state of stress on an element oriented 30° counterclockwise from the position shown.

SOLUTION Construction of the Circle. From the problem data,

sx = -8 ksi sy = 12 ksi txy = -6 ksi

The s and t axes are established in Fig. 9–20 b . The center of the circle C is on the s axis at

savg = -8 + 12

2 = 2 ksi

The reference point for u = 0! has coordinates A (-8, -6). Hence from the shaded triangle the radius CA is

R = 2(10)2 + (6)2 = 11.66 Stresses on 30° Element. Since the element is to be rotated 30° counterclockwise , we must construct a radial line CP , 2(30!) = 60! counterclockwise , measured from CA (u = 0!), Fig. 9–20 b . The coordinates of point P(sx" , tx"y") must now be obtained. From the geometry of the circle,

f = tan-1 6 10

= 30.96! c = 60! - 30.96! = 29.04!

sx" = 2 - 11.66 cos 29.04! = -8.20 ksi Ans.

tx"y" = 11.66 sin 29.04! = 5.66 ksi Ans.

These two stress components act on face BD of the element shown in Fig. 9–20 c since the x " axis for this face is oriented 30° counterclockwise from the x axis.

The stress components acting on the adjacent face DE of the element, which is 60° clockwise from the positive x axis, Fig. 9–20 c , are represented by the coordinates of point Q on the circle. This point lies on the radial line CQ , which is 180° from CP . The coordinates of point Q are

sx" = 2 + 11.66 cos 29.04! = 12.2 ksi Ans. tx"y" = -(11.66 sin 29.04) = -5.66 ksi (check) Ans.

NOTE: Here tx"y" acts in the -y " direction.

EXAMPLE 9.9

12 ksi

8 ksi

6 ksi

(a)

29.04! 11.6

6 Q

60!

120! R " 11.666

c " 29.04!

s (ksi)

t (ksi)

sx¿

tx¿y¿

8 2

(b)

11.6 6

5.66 ksi

60!

x¿

y¿

30!

8.20 ksi B

12.2 ksi

x¿

(c)

Fig. 9–20

472 CHAPTER 9 STRESS TRANSFORMATION

FUNDAMENTAL PROBLEMS

F9–7. Using Mohr’s circle, determine the normal stress and shear stress acting on the inclined plane AB. Sketch the result on the sectioned element.

500 kPa

30!

F9–7

F9–8. Determine the equivalent state of stress on an element at the same point that represents the principal stresses at the point. Also, find the corresponding orientation of the element with respect to the element shown. Sketch the results on the element.

80 kPa

30 kPa

F9–8

F9–9. Draw Mohr’s circle and determine the principal stresses.

30 MPa

40 MPa

F9–9

F9–10. The hollow circular shaft is subjected to the torque of 4 kN # m. Determine the principal stress developed at a point on the surface of the shaft.

40 mm

4 kN·m

4 kN·m 30 mm

F9–10

F9–11. Determine the principal stress developed at point A on the cross section of the beam at section a–a .

300 mm

30 kN

50 mm

50 mm 150 mm

Section a–a

F9–11

F9–12. Determine the maximum in-plane shear stress developed at point A on the cross section of the beam at section a–a , which is located just to the left of the 60-kN force. Point A is just below the flange.

A B

0.5 m 1 m

60 kN

180 mm

10 mm

100 mm

Section a-a

F9–12

9.4 MOHR’S CIRCLE—PLANE STRESS 473

PROBLEMS

*9–44. Solve Prob. 9–3 using Mohr’s circle.

9–45. Solve Prob. 9–6 using Mohr’s circle.

9–46. Solve Prob. 9–14 using Mohr’s circle.

9–47. Solve Prob. 9–10 using Mohr’s circle.

*9–48. Solve Prob. 9–12 using Mohr’s circle.

9–49. Solve Prob. 9–16 using Mohr’s circle.

9–50. Mohr’s circle for the state of stress in Fig. 9–15 a is shown in Fig. 9–15 b . Show that finding the coordinates of point P(sx!, tx!y!) on the circle gives the same value as the stress-transformation Eqs. 9–1 and 9–2 .

9–51. Determine the equivalent state of stress if an element is oriented 45° clockwise from the element shown.

5 ksi

10 ksi

Prob. 9–51

*9–52. Determine the equivalent state of stress if an element is oriented 20° clockwise from the element shown.

2 ksi

3 ksi

4 ksi

Prob. 9–52

9–53. Determine the equivalent state of stress if an element is oriented 30° clockwise from the element shown.

230 MPa

350 MPa

480 MPa

Prob. 9–53

9–54. Determine the equivalent state of stress which represents (a) the principal stress, and (b) the maximum in-plane shear stress and the associated average normal stress. For each case, determine the corresponding orientation of the element with respect to the element shown.

40 MPa

80 MPa

120 MPa

Prob. 9–54

9–55. Determine the equivalent state of stress which represents (a) the principal stress, and (b) the maximum in-plane shear stress and the associated average normal stress. For each case, determine the corresponding orientation of the element with respect to the element shown.

25 ksi

10 ksi

Prob. 9–55

474 CHAPTER 9 STRESS TRANSFORMATION

9–59. Determine (a) the principal stresses and (b) the maximum in-plane shear stress and average normal stress. Specify the orientation of the element in each case.

8 ksi

12 ksi

4 ksi

Prob. 9–59

*9–60. Determine the principal stresses, the maximum in-plane shear stress, and average normal stress. Specify the orientation of the element in each case.

200 MPa

500 MPa

350 MPa

Prob. 9–60

9–61. Draw Mohr’s circle that describes each of the following states of stress.

5 MPa

(a)

20 ksi

(b)

18 MPa

*9–56. Determine the principal stress, the maximum in-plane shear stress, and average normal stress. Specify the orientation of the element in each case.

5 ksi

15 ksi

Prob. 9–56

9–57. Determine (a) the principal stresses and (b) the maximum in-plane shear stress and average normal stress. Specify the orientation of the element in each case.

50 MPa

30 MPa

Prob. 9–57

9–58. Determine the equivalent state of stress if an element is oriented 25° counterclockwise from the element shown.

550 MPa

Prob. 9–58

9.4 MOHR’S CIRCLE—PLANE STRESS 475

9–62. The grains of wood in the board make an angle of 20° with the horizontal as shown. Using Mohr’s circle, determine the normal and shear stresses that act perpendicular and parallel to the grains if the board is subjected to an axial load of 250 N.

300 mm

250 N 60 mm

25 mm20! 250 N

Prob. 9–62

9–63. The post has a square cross-sectional area. If it is fixed supported at its base and a horizontal force is applied at its end as shown, determine (a) the maximum in-plane shear stress developed at A and (b) the principal stresses at A .

18 in.

y x

60 lb 3 in. 3 in.

1 in.

Prob. 9–63

*9–64. Determine the principal stress, the maximum in-plane shear stress, and average normal stress. Specify the orientation of the element in each case.

20 MPa

30 MPa

80 MPa

Prob. 9–64

9–65. The thin-walled pipe has an inner diameter of 0.5 in. and a thickness of 0.025 in. If it is subjected to an internal pressure of 500 psi and the axial tension and torsional loadings shown, determine the principal stress at a point on the surface of the pipe.

20 lb"ft 20 lb"ft

200 lb200 lb

Prob. 9–65

9–66. Determine the principal stress and maximum in-plane shear stress at point A on the cross section of the pipe at section a – a .

9–67. Determine the principal stress and maximum in-plane shear stress at point B on the cross section of the pipe at section a – a .

300 mm

450 N300 N

300 mm

200 mm Section a – a

30 mm

20 mm

Probs. 9–66/67

*9–68. The rotor shaft of the helicopter is subjected to the tensile force and torque shown when the rotor blades provide the lifting force to suspend the helicopter at midair. If the shaft has a diameter of 6 in., determine the principal stress and maximum in-plane shear stress at a point located on the surface of the shaft.

10 kip!ft

50 kip

Prob. 9–68

476 CHAPTER 9 STRESS TRANSFORMATION

9–74. If the box wrench is subjected to the 50 lb force, determine the principal stress and maximum in-plane shear stress at point A on the cross section of the wrench at section a – a . Specify the orientation of these states of stress and indicate the results on elements at the point.

9–75. If the box wrench is subjected to the 50 lb force, determine the principal stress and maximum in-plane shear stress at point B on the cross section of the wrench at section a – a . Specify the orientation of these states of stress and indicate the results on elements at the point.

2 in.

12 in.

0.5 in.

50 lb

Section a – a

A B

Probs. 9–74/75

*9–76. The ladder is supported on the rough surface at A and by a smooth wall at B . If a man weighing 150 lb stands upright at C , determine the principal stresses in one of the legs at point D . Each leg is made from a 1-in.-thick board having a rectangular cross section. Assume that the total weight of the man is exerted vertically on the rung at C and is shared equally by each of the ladder’s two legs. Neglect the weight of the ladder and the forces developed by the man’s arms.

4 ft

5 ft3 in.

1 in. D

1 in. 1 in.

3 in.

5 ft

12 ft

Prob. 9–76

9–69. The pedal crank for a bicycle has the cross section shown. If it is fixed to the gear at B and does not rotate while subjected to a force of 75 lb, determine the principal stress in the material on the cross section at point C .

B A

75 lb

4 in.

0.3 in. 0.2 in.

0.4 in. 0.4 in.

C 3 in.

Prob. 9–69

9–70. A spherical pressure vessel has an inner radius of 5 ft and a wall thickness of 0.5 in. Draw Mohr’s circle for the state of stress at a point on the vessel and explain the significance of the result. The vessel is subjected to an internal pressure of 80 psi.

9–71. The cylindrical pressure vessel has an inner radius of 1.25 m and a wall thickness of 15 mm. It is made from steel plates that are welded along the 45° seam. Determine the normal and shear stress components along this seam if the vessel is subjected to an internal pressure of 8 MPa.

1.25 m

45!

Probs. 9–70/71 *9–72. Determine the normal and shear stresses at point D that act perpendicular and parallel, respectively, to the grains. The grains at this point make an angle of 30° with the horizontal as shown. Point D is located just to the left of the 10-kN force.

9–73. Determine the principal stress at point D , which is located just to the left of the 10-kN force.

2 m1 m 1 m

100 mm

300 mm

A D

D 100 mm

100 mm 30!

10 kN

Probs. 9–72/73

9.5 ABSOLUTE MAXIMUM SHEAR STRESS 477

9.5 Absolute Maximum Shear Stress When a point in a body is subjected to a general three-dimensional state of stress, an element of material has a normal-stress and two shear-stress components acting on each of its faces, Fig. 9–21 a . As in case of plane stress, it is possible to develop stress-transformation equations that can be used to determine the normal and shear stress components s and t acting on any skewed plane of the element, Fig. 9–21 b . Furthermore, at the point it is also possible to determine the unique orientation of an element having only principal stresses acting on its faces. As shown in Fig. 9–21 c , in general these principal stresses will have magnitudes of maximum, intermediate, and minimum intensity, i.e., smax Ú sint Ú smin. This is a condition known as triaxial stress .

A discussion of the transformation of stress in three dimensions is beyond the scope of this text; however, it is discussed in books related to the theory of elasticity. For our purposes, we will confine our attention only to the case of plane stress. For example, consider the material to be

(a)

(b) (c) triaxial stress

smin

sint

smax

Fig. 9–21

478 CHAPTER 9 STRESS TRANSFORMATION

subjected to the in-plane principal stresses s1 and s2 shown in Fig. 9–22 a , where both of these stresses are tensile. If we view the element in two dimensions, that is, in the y–z , x–z , and x–y planes, Figs. 9–22 b , 9–22 c , and 9–22 d , then we can use Mohr’s circle to determine the maximum in-plane shear for each case and from this, determine the absolute maximum shear stress in the material. For example, the diameter of Mohr’s circle extends between 0 and s2 for the case shown in Fig. 9–22 b . From this circle, Fig. 9–22 e , the maximum in-plane shear stress is ty!z! = s2>2 . For all three circles, it is seen that although the maximum in-plane shear stress is tx!y! = (s1 - s2)>2 , this value is not the absolute maximum shear stress. Instead, from Fig. 9–22 e ,

t max abs =

(9–13)

s1 and s2 have the same sign

x–y plane stress

(a)

(b)

(c)

(d)

s2 s

(e)

Absolute maximum shear stress

Maximum in-plane shear stress

(tx¿y¿)max

(ty¿z¿)max

(tx¿z¿)max

Fig. 9–22

9.5 ABSOLUTE MAXIMUM SHEAR STRESS 479

If one of the in-plane principal stresses has the opposite sign of that of the other, Fig. 9–23 a , then the three Mohr’s circles that describe the state of stress for element orientations about each coordinate axis are shown in Fig. 9–23 b . Clearly, in this case

t max abs =

s1 - s2 2

(9–14)

s1 and s2 have opposite signs

Calculation of the absolute maximum shear stress as indicated here is important when designing members made of a ductile material, since the strength of the material depends on its ability to resist shear stress. This situation will be discussed further in Sec. 10.7 .

s1 s2

x–y plane stress

(a)

(b)

Maximum in-plane and absolute maximum shear stress

s1s2 s

(tx¿y¿)max

(tx¿z¿)max

(ty¿z¿)max

Fig. 9–23

Important Points

• The general three-dimensional state of stress at a point can be represented by an element oriented so that only three principal stresses smax, sint, smin act on it.

• In the case of plane stress, if the in-plane principal stresses both have the same sign , the absolute maximum shear stress will occur out of the plane and has a value of t

max abs = smax>2. This value is

greater than the in-plane shear stress.

• If the in-plane principal stresses are of opposite signs, then the absolute maximum shear stress will equal the maximum in-plane shear stress ; that is, t

max abs = (smax - smin)>2.

480 CHAPTER 9 STRESS TRANSFORMATION

The point on the surface of the cylindrical pressure vessel in Fig. 9–24 a is subjected to the state of plane stress. Determine the absolute maxi- mum shear stress at this point.

(a)

32 MPa

16 MPa

EXAMPLE 9.10

(b)

s (MPa)

t (MPa)

Fig. 9–24

SOLUTION The principal stresses are s1 = 32 MPa, s2 = 16 MPa . If these stresses are plotted along the s axis, the three Mohr’s circles can be constructed that describe the stress state viewed in each of the three perpendicular planes, Fig. 9–24 b . The largest circle has a radius of 16 MPa and describes the state of stress in the plane only containing s1 = 32 MPa, shown shaded in Fig. 9–24 a . An orientation of an element 45° within this plane yields the state of absolute maximum shear stress and the associated average normal stress, namely,

t max abs = 16 MPa Ans.

savg = 16 MPa

This same result for t abs max

can be obtained from direct application of Eq. 9–13 .

t max abs =

2 =

32 2

= 16 MPa Ans.

savg = 32 + 0

2 = 16 MPa

By comparison, the maximum in-plane shear stress can be determined from the Mohr’s circle drawn between s1 = 32 MPa and s2 = 16 MPa, Fig. 9–24 b . This gives a value of

t max in@plane = 32 - 16

2 = 8 MPa

savg = 32 + 16

2 = 24 MPa

9.5 ABSOLUTE MAXIMUM SHEAR STRESS 481

EXAMPLE 9.11

Due to an applied loading, an element at the point on a machine shaft is subjected to the state of plane stress shown in Fig. 9–25 a . Determine the principal stresses and the absolute maximum shear stress at the point.

SOLUTION Principal Stresses. The in-plane principal stresses can be determined from Mohr’s circle. The center of the circle is on the s axis at savg = (-20 + 0)>2 = -10 psi. Plotting the reference point A (-20, -40), the radius CA is established and the circle is drawn as shown in Fig. 9–25 b . The radius is

R = 2 (20 - 10)2 + (40)2 = 41.2 psi The principal stresses are at the points where the circle intersects the s axis; i.e.,

s1 = -10 + 41.2 = 31.2 psi

s2 = -10 - 41.2 = -51.2 psi

From the circle, the counterclockwise angle 2u, measured from CA to the -s axis, is

2u = tan -1 a 40 20 - 10 b = 76.0!

Thus,

u = 38.0!

This counterclockwise rotation defines the direction of the x " axis and s2 and its associated principal plane, Fig. 9–25 c . We have

s1 = 31.2 psi s2 = -51.2 psi Ans.

Absolute Maximum Shear Stress. Since these stresses have opposite signs, applying Eq. 9–14 we have

t max abs =

s1 - s2 2

= 31.2 - (-51.2)

2 = 41.2 psi Ans.

savg = 31.2 - 51.2

2 = -10 psi

NOTE: These same results can also be obtained by drawing Mohr’s circle for each orientation of an element about the x, y, and z axes, Fig. 9–25 d . Since s1 and s2 are of opposite signs , then the absolute maximum shear stress equals the maximum in-plane shear stress.

(a)

40 psi

20 psi

(b)

10 20

R !

41.2

(s2, 0) (s1, 0)

t (psi)

s (psi)

51.2 psi

x 38.0"

x¿31.2 psi

y¿

(c)

(d)

A 2u ! 76.0" # 90" ! 166"

s2 ! $51.2 psi s1 ! 31.2 psi

t (psi)

s (psi)

! 41.2 psitabs max

Fig. 9–25

482 CHAPTER 9 STRESS TRANSFORMATION

PROBLEMS

*9–80. The stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

30 psi

70 psi

yx 120 psi

Prob. 9–80 9–81. The stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

x y

1 ksi

6 ksi

Prob. 9–81

9–82. The stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

120 MPa

150 MPa

Prob. 9–82

9–77. Draw the three Mohr’s circles that describe each of the following states of stress.

5 ksi

3 ksi

(a)

180 MPa

(b)

140 MPa

Prob. 9–77

9–78. Draw the three Mohr’s circles that describe the following state of stress.

400 psi

300 psi

Prob. 9–78

9–79. The stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

x y

60 MPa

90 MPa

20 MPa

Prob. 9–79

9.5 ABSOLUTE MAXIMUM SHEAR STRESS 483

9–83. The state of stress at a point is shown on the element. Determine the principal stresses and the absolute maximum shear stress.

2.5 ksi

4 ksi

5 ksi

Prob. 9–83

■ 9–84. Consider the general case of plane stress as shown. Write a computer program that will show a plot of the three Mohr’s circles for the element, and will also calculate the maximum in-plane shear stress and the absolute maximum shear stress.

txy

Prob. 9–84

9–85. The solid cylinder having a radius r is placed in a sealed container and subjected to a pressure p . Determine the stress components acting at point A located on the center line of the cylinder. Draw Mohr’s circles for the element at this point.

Prob. 9–85

9–86. The plate is subjected to a tensile force P = 5 kip. If it has the dimensions shown, determine the principal stresses and the absolute maximum shear stress. If the material is ductile it will fail in shear. Make a sketch of the plate showing how this failure would appear. If the material is brittle the plate will fail due to the principal stresses. Show how this failure occurs.

12 in.

P ! 5 kip

P ! 5 kip2 in.

2 in.

0.5 in.

Prob. 9–86

9–87. Determine the principal stresses and absolute maximum shear stress developed at point A on the cross section of the bracket at section a – a .

*9–88. Determine the principal stresses and absolute maximum shear stress developed at point B on the cross section of the bracket at section a – a .

6 in.

12 in.

500 lb

1.5 in.1.5 in.

0.25 in.0.25 in.

0.5 in. 0.25 in.

aa 3

A B

Section a – a

Probs. 9–87/88

484 CHAPTER 9 STRESS TRANSFORMATION

CHAPTER REVIEW

Plane stress occurs when the material at a point is subjected to two normal stress components sx and sy and a shear stress txy. Provided these components are known, then the stress components acting on an element having a different orientation u can be determined using the two force equations of equilibrium or the equations of stress transformation.

sx! = sx + sy

2 +

sx - sy 2

cos 2u + txy sin 2u

tx!y! = - sx - sy

2 sin 2u + txy cos 2u

For design, it is important to determine the orientation of the element that produces the maximum principal normal stresses and the maximum in-plane shear stress. Using the stress transformation equations, it is found that no shear stress acts on the planes of principal stress. The principal stresses are

s1,2 = sx + sy

2 { A asx - sy2 b2 + t 2xy

The planes of maximum in-plane shear stress are oriented 45" from this orientation, and on these shear planes there is an associated average normal stress.

t max in@plane

= A asx - sy2 b2 + t 2xy savg =

sx + sy 2

(a)

txy

(b)

y¿

x ¿sy¿

sx ¿

tx ¿y¿

x !

txy

savg

t max in-plane

Mohr’s circle provides a semi-graphical method for finding the stresses on any plane, the principal normal stresses, and the maximum in-plane shear stress. To draw the circle, the s and t axes are established, the center of the circle C[(sx + sy)>2, 0] and the reference point A (sx, txy) are plotted. The radius R of the circle extends between these two points and is determined from trigonometry.

If s1 and s2 are of the same sign, then the absolute maximum shear stress will lie out of plane.

tabs max

= s1

In the case of plane stress, the absolute maximum shear stress will be equal to the maximum in-plane shear provided the principal stresses s1 and s2 have the opposite sign.

tabs max

= s1 - s2

x–y plane stress

CHAPTER REVIEW 485

(b) t

R !sx

txy

" txy2

2 sx # sy

2 sx " sy

savg !

s1 s2

x–y plane stress

486 CHAPTER 9 STRESS TRANSFORMATION

REVIEW PROBLEMS

9–89. The solid propeller shaft on a ship extends outward from the hull. During operation it turns at v = 15 rad>s when the engine develops 900 kW of power. This causes a thrust of F = 1.23 MN on the shaft. If the shaft has an outer diameter of 250 mm, determine the principal stresses at any point located on the surface of the shaft.

9–90. The solid propeller shaft on a ship extends outward from the hull. During operation it turns at v = 15 rad>s when the engine develops 900 kW of power. This causes a thrust of F = 1.23 MN on the shaft. If the shaft has a diameter of 250 mm, determine the maximum in-plane shear stress at any point located on the surface of the shaft.

0.75 m A

Probs. 9–89/90

9–91. The steel pipe has an inner diameter of 2.75 in. and an outer diameter of 3 in. If it is fixed at C and subjected to the horizontal 20-lb force acting on the handle of the pipe wrench at its end, determine the principal stresses in the pipe at point A , which is located on the surface of the pipe.

*9–92. Solve Prob. 9–91 for point B , which is located on the surface of the pipe.

10 in.

20 lb

12 in.

y z

Probs. 9–91/92

9–93. Determine the equivalent state of stress if an element is oriented 40° clockwise from the element shown. Use Mohr’s circle.

10 ksi

6 ksi

Prob. 9–93

9–94. The crane is used to support the 350-lb load. Determine the principal stresses acting in the boom at points A and B . The cross section is rectangular and has a width of 6 in. and a thickness of 3 in. Use Mohr’s circle.

5 ft

45°

3 in.

A B

45°

Prob. 9–94

9–98. The state of stress at a point is shown on the element. Determine (a) the principal stresses and (b) the maximum in-plane shear stress and average normal stress at the point. Specify the orientation of the element in each case.

60 MPa

30 MPa

45 MPa

Prob. 9–98

9–99. The state of stress at a point in a member is shown on the element. Determine the stress components acting on the inclined plane AB . Solve the problem using the method of equilibrium described in Sec. 9.1 .

14 ksi

20 ksi

50!

Prob. 9–99

9–95. Determine the equivalent state of stress on an element at the same point which represents (a) the principal stresses, and (b) the maximum in-plane shear stress and the associated average normal stress. Also, for each case, determine the corresponding orientation of the element with respect to the element shown and sketch the results on the element.

10 ksi

30 ksi

Prob. 9–95

*9–96. The propeller shaft of the tugboat is subjected to the compressive force and torque shown. If the shaft has an inner diameter of 100 mm and an outer diameter of 150 mm, determine the principal stress at a point A located on the outer surface.

2 kN·m

10 kN

Prob. 9–96

9–97. The box beam is subjected to the loading shown. Determine the principal stress in the beam at points A and B .

3 ft 2.5 ft 5 ft2.5 ft

800 lb 1200 lb

6 in. A

B6 in. 8 in.

8 in.

Prob. 9–97

REVIEW PROBLEMS 487

Chapter 10

This pin support for a bridge has been tested with strain gages to ensure that the principle strains in the material do not exceed a failure criterion for the material.

489

10.1 Plane Strain As outlined in Sec. 2.2 , the general state of strain at a point in a body is represented by a combination of three components of normal strain, Px, Py, Pz, and three components of shear strain gxy, gxz, gyz. These six components tend to deform each face of an element of the material, and like stress, the normal and shear strain components at the point will vary according to the orientation of the element. The strains at a point are often determined by using strain gauges, which measure normal strain in specified directions . For both analysis and design, however, engineers must sometimes transform this data in order to obtain the strain in other directions.

CHAPTER OBJECTIVES

■ The transformation of strain at a point is similar to the transformation of stress, and as a result the methods of Chapter 9 will be applied in this chapter. Here we will also discuss various ways for measuring strain and develop some important material-property relationships, including a generalized form of Hooke’s law. At the end of the chapter, a few of the theories used to predict the failure of a material will be discussed.

Strain Transformation

490 CHAPTER 10 STRAIN TRANSFORMATION

To understand how this is done, we will first confine our attention to a study of plane strain . Specifically, we will not consider the effects of the components Pz, gxz, and gyz. In general, then, a plane-strained element is subjected to two components of normal strain, Px, Py, and one component of shear strain, gxy. Although plane strain and plane stress each have three components lying in the same plane, realize that plane stress does not necessarily cause plane strain or vice versa. The reason for this has to do with the Poisson effect discussed in Sec. 3.6 . For example, if the element in Fig. 10–1 is subjected to plane stress sx and sy, not only are normal strains Px and Py produced, but there is also an associated normal strain, Pz. This is obviously not a case of plane strain. In general, then, unless n = 0, the Poisson effect will prevent the simultaneous occurrence of plane strain and plane stress.

10.2 General Equations of Plane-Strain Transformation

It is important in plane-strain analysis to establish transformation equations that can be used to determine the x !, y ! components of normal and shear strain at a point, provided the x , y components of strain are known. Essentially this problem is one of geometry and requires relating the deformations and rotations of line segments, which represent the sides of differential elements that are parallel to each set of axes.

Sign Convention. Before the strain-transformation equations can be developed, we must first establish a sign convention for the strains. With reference to the differential element shown in Fig. 10–2 a , normal strains Px and Py are positive if they cause elongation along the x and y axes, respectively, and the shear strain gxy is positive if the interior angle AOB becomes smaller than 90°. This sign convention also follows the corresponding one used for plane stress, Fig. 9–5 a , that is, positive sx, sy, txy will cause the element to deform in the positive Px, Py, gxy directions, respectively.

The problem here will be to determine at a point the normal and shear strains Px!, Py!, gx!y!, measured relative to the x !, y ! axes, if we know Px, Py, gxy, measured relative to the x , y axes. If the angle between the x and x ! axes is u, then, like the case of plane stress, u will be positive provided it follows the curl of the right-hand fingers, i.e., counterclockwise, as shown in Fig. 10–2 b .

Plane stress, sx, sy, does not cause plane strain in the x–y plane since Pz ≠ 0.

x y

Pydy

Pzdz

Pxdx

sx sy

Fig. 10–1

(a)

O B

!Pydy

dx !Pxdx

gxy

2! gxy

y y ¿

x ¿

(b)

Positive sign convention

Fig. 10–2

10.2 GENERAL EQUATIONS OF PLANE-STRAIN TRANSFORMATION 491

Normal and Shear Strains. In order to develop the strain- transformation equation for Px!, we must determine the elongation of a line segment dx ! that lies along the x ! axis and is subjected to strain components Px, Py, gxy. As shown in Fig. 10–3 a , the components of the line dx ! along the x and y axes are

dx = dx ! cos u

dy = dx ! sin u (10–1)

When the positive normal strain Px occurs, the line dx is elongated Px dx, Fig. 10–3 b , which causes line dx ! to elongate Px dx cos u. Likewise, when Py occurs, line dy elongates Py dy, Fig. 10–3 c , which causes line dx ! to elongate Py dy sin u. Finally, assuming that dx remains fixed in position, the shear strain gxy, which is the change in angle between dx and dy , causes the top of line dy to be displaced gxy dy to the right, as shown in Fig. 10–3 d . This causes dx ! to elongate gxy dy cos u. If all three of these elongations are added together, the resultant elongation of dx ! is then

dx ! = Px dx cos u + Py dy sin u + gxy dy cos u

From Eq. 2–2 , the normal strain along the line dx ! is Px! = dx !>dx !. Using Eq. 10–1 , we therefore have

Px! = Px cos2 u + Py sin2 u + gxy sin u cos u (10–2)

y y ¿

x ¿

Before deformation

(a)

dy dx ¿ u

(b)

Normal strain Px

dx¿

y y ¿ x ¿

Pxdx cosu

Pxdx Pxdx sinu

x ¿y ¿

Normal strain Py

dy dx¿

(c)

Pydy sinu

Pydy cosu

u Pydy

x ¿

dx ¿dy¿

(d)

Shear strain gxy

gxydy cosu gxydy sinu

gxydy

dy gxy

y ¿

Fig. 10–3

The rubber specimen is constrained between the two fi xed supports, and so it will undergo plane strain when loads are applied to it in the horizontal plane.

492 CHAPTER 10 STRAIN TRANSFORMATION

The strain-transformation equation for gx!y! can be developed by considering the amount of rotation each of the line segments dx ! and dy ! undergo when subjected to the strain components Px, Py, gxy. First we will consider the rotation of dx !, which is defined by the counterclockwise angle a shown in Fig. 10–3 e . It can be determined by the displacement caused by dy ! using a = dy !>dx !. To obtain dy !, consider the following three displacement components acting in the y ! direction: one from Px, giving -Px dx sin u, Fig. 10–3 b ; another from Py, giving Py dy cos u, Fig. 10–3 c ; and the last from gxy, giving -gxy dy sin u, Fig. 10–3 d . Thus, dy !, as caused by all three strain components, is

dy ! = -Px dx sin u + Py dy cos u - gxy dy sin u

Dividing each term by dx ! and using Eq. 10–1 , with a = dy !>dx !, we have a = (-Px + Py) sin u cos u - gxy sin2 u (10–3)

As shown in Fig. 10–3 e , the line dy ! rotates by an amount b. We can determine this angle by a similar analysis, or by simply substituting u + 90" for u into Eq. 10–3 . Using the identities sin(u + 90") = cos u, cos(u + 90") = -sin u, we have

b = (-Px + Py) sin(u + 90") cos(u + 90") - gxy sin2(u + 90") = -(-Px + Py) cos u sin u - gxy cos2 u

Since a and b represent the rotation of the sides dx ! and dy ! of a differential element whose sides were originally oriented along the x ! and y ! axes, Fig. 10–3 e , the element is then subjected to a shear strain of

gx!y! = a - b = -2(Px - Py) sin u cos u + gxy(cos2 u - sin2 u) (10–4)

x ¿

dx ¿dy¿

(d)

Shear strain gxy

gxydy cosu gxydy sinu

gxydy

dy gxy

y ¿

x¿

y ¿

dy ¿ b a

udx ¿

dx ¿dy ¿

dy ¿

(e)

Fig. 10–3 (cont.)

10.2 GENERAL EQUATIONS OF PLANE-STRAIN TRANSFORMATION 493

Using the trigonometric identities sin 2u = 2 sin u cos u, cos2 u = (1 + cos 2u)>2, and sin2 u + cos2 u = 1, we can rewrite Eqs. 10–2 and 10–4 in the final form

Px! = Px + Py

2 +

Px - Py 2

cos 2u + gxy

2 sin 2u (10–5)

gx!y!

2 = - ¢ Px - Py

2 ≤ sin 2u + gxy

2 cos 2u (10–6)

These strain-transformation equations give the normal strain Px! in the x ! direction and the shear strain gx!y! of an element oriented at an angle u, as shown in Fig. 10–4 . According to the established sign convention, if Px! is positive , the element elongates in the positive x ! direction, Fig. 10–4 a , and if gx!y! is positive, the element deforms as shown in Fig. 10–4 b .

If the normal strain in the y ! direction is required, it can be obtained from Eq. 10–5 by simply substituting (u + 90") for u. The result is

Py! = Px + Py

2 -

Px - Py 2

cos 2u - gxy

2 sin 2u (10–7)

The similarity between the above three equations and those for plane-stress transformation, Eqs. 9–1 , 9–2 , and 9–3 , should be noted. By comparison, sx, sy, sx!, sy! correspond to Px, Py, Px!, Py!; and txy, tx!y! correspond to gxy>2, gx!y!>2.

x ¿

y ¿

dy ¿ dx ¿

Positive normal strain, Px ¿

(a)

Positive shear strain, gx ¿y¿

(b)

x ¿

y ¿

dy ¿ dx ¿

Fig. 10–4

494 CHAPTER 10 STRAIN TRANSFORMATION

Principal Strains. Like stress, an element can be oriented at a point so that the element’s deformation is caused only by normal strains, with no shear strain. When this occurs the normal strains are referred to as principal strains , and if the material is isotropic, the axes along which these strains occur will coincide with the axes that define the planes of principal stress.

From Eqs. 9–4 and 9–5 , and the correspondence between stress and strain mentioned above, the direction of the x ! axis and the two values of the principal strains P1 and P2 are determined from

tan 2up = gxy

Px - Py (10–8)

P1,2 = Px + Py

2 { C¢ Px - Py2 ≤2 + ¢gxy2 ≤2 (10–9)

Maximum In-Plane Shear Strain. Using Eqs. 9–6 , 9–7 , and 9–8 , the direction of the x ! axis, and the maximum in-plane shear strain and associated average normal strain are determined from the following equations:

tan 2us = - a Px - Pygxy ≤ (10–10)

g max in@plane 2

= B a Px - Py2 b2 + agxy2 b2 (10–11) Pavg =

Px + Py 2

(10–12)

Complex stresses are often developed at the joints where the cylindrical and hemispherical vessels are joined together. The stresses are determined by making measurements of strain.

Important Points

• In the case of plane stress, plane-strain analysis may be used within the plane of the stresses to analyze the data from strain gauges. Remember, though, there will be a normal strain that is perpendicular to the gauges due to the Poisson effect.

• When the state of strain is represented by the principal strains, no shear strain will act on the element. • The state of strain at a point can also be represented in terms of the maximum in-plane shear strain. In

this case an average normal strain will also act on the element.

• The element representing the maximum in-plane shear strain and its associated average normal strains is 45° from the orientation of an element representing the principal strains.

10.2 GENERAL EQUATIONS OF PLANE-STRAIN TRANSFORMATION 495

A differential element of material at a point is subjected to a state of plane strain Px = 500(10-6), Py = -300(10-6), gxy = 200(10-6), which tends to distort the element as shown in Fig. 10–5 a . Determine the equivalent strains acting on an element of the material oriented at the point, clockwise 30° from the original position.

SOLUTION The strain-transformation Eqs. 10–5 and 10–6 will be used to solve the problem. Since u is positive counterclockwise , then for this problem u = -30!. Thus,

Px" = Px + Py

2 +

Px - Py 2

cos 2u + gxy

2 sin 2u

= c 500 + (-300) 2

d (10-6) + c500 - (-300) 2

d (10-6) cos(2(-30!)) + J200(10-6)

2 R sin(2(-30!))

Px" = 213(10-6) Ans.

gx"y"

2 = - ¢ Px - Py

2 ≤ sin 2u + gxy

2 cos 2u

= - c 500 - (-300) 2

d (10-6) sin(2(-30!)) + 200(10-6) 2

cos(2(-30!))

gx"y" = 793(10-6 ) Ans.

The strain in the y " direction can be obtained from Eq. 10–7 with u = -30!. However, we can also obtain Py" using Eq. 10–5 with u = 60!(u = -30! + 90!), Fig. 10–5 b . We have with Py" replacing Px",

Py" = Px + Py

2 +

Px - Py 2

cos 2u + gxy

2 sin 2u

= c 500 + (-300) 2

d (10-6) + c 500 - (-300) 2

d (10-6) cos(2(60!)) +

200(10-6) 2

sin(2(60!))

Py" = -13.4(10-6) Ans.

These results tend to distort the element as shown in Fig. 10–5 c .

EXAMPLE 10.1

(a)

gxy

Pxdx

Pydy

y ¿

x ¿ (b)

u ! 60"

u ! #30"

dy ¿ Py¿dy ¿

Px¿dx ¿

dx ¿

x ¿

y ¿

(c)

gx ¿y¿

Fig. 10–5

496 CHAPTER 10 STRAIN TRANSFORMATION

A differential element of material at a point is subjected to a state of plane strain defined by Px = -350(10-6), Py = 200(10-6), gxy = 80(10-6), which tends to distort the element as shown in Fig. 10–6 a . Determine the principal strains at the point and the associated orientation of the element.

SOLUTION Orientation of the Element. From Eq. 10–8 we have

tan 2up = gxy

Px - Py

= 80(10-6)

(-350 - 200)(10-6)

Thus, 2up = -8.28! and -8.28! + 180! = 171.72!, so that

up = -4.14! and 85.9! Ans.

Each of these angles is measured positive counterclockwise , from the x axis to the outward normals on each face of the element, Fig. 10–6 b .

Principal Strains. The principal strains are determined from Eq. 10–9 . We have

P1,2 = Px + Py

2 { B a Px - Py2 b2 + agxy2 b2

= (-350 + 200)(10-6)

2 { JB a -350 - 2002 b2 + a 802 b2 R (10-6)

= -75.0(10-6) { 277.9(10-6) P1 = 203(10-6) P2 = -353(10-6) Ans.

We can determine which of these two strains deforms the element in the x " direction by applying Eq. 10–5 with u = -4.14!. Thus,

Px" = Px + Py

2 +

Px - Py 2

cos 2u + gxy

2 sin 2u

= a -350 + 200 2

b (10-6) + a -350 - 200 2

b (10-6) cos 2(-4.14!) +

80(10-6) 2

sin 2(-4.14!)

Px" = -353(10-6) Hence Px" = P2. When subjected to the principal strains, the element is distorted as shown in Fig. 10–6 b .

EXAMPLE 10.2

(a)

gxy

Pxdxdx

Pydy

(b)

y ¿

P1dy ¿

P2dx ¿ x ¿

!4.14"

85.9"

Fig. 10–6

10.2 GENERAL EQUATIONS OF PLANE-STRAIN TRANSFORMATION 497

A differential element of material at a point is subjected to a state of plane strain defined by Px = -350(10-6), Py = 200(10-6), gxy = 80(10-6), which tends to distort the element as shown in Fig. 10–7 a . Determine the maximum in-plane shear strain at the point and the associated orientation of the element.

SOLUTION Orientation of the Element. From Eq. 10–10 we have

tan 2us = - a Px - Pygxy ≤ = - (-350 - 200)(10-6)80(10-6) Thus, 2us = 81.72! and 81.72! + 180! = 261.72!, so that

us = 40.9! and 131!

Note that this orientation is 45° from that shown in Fig. 10–6 b in Example 10.2 as expected.

Maximum In-Plane Shear Strain. Applying Eq. 10–11 gives

g max in@plane

2 = B a Px - Py 2 b2 + agxy 2 b2

= JB a -350 - 2002 b2 + a 802 b2 R (10-6) g max in@plane = 556(10-6) Ans.

Due to the square root, the proper sign of g max in@plane can be obtained by applying Eq. 10–6 with us = 40.9!. We have

gx"y"

2 = -

Px - Py 2

sin 2u + gxy

2 cos 2u

= - a -350 - 200 2

b (10-6) sin 2(40.9!) + 80(10-6) 2

cos 2(40.9!)

gx"y" = 556(10-6)

This result is positive and so g max in@plane tends to distort the element so that the right angle between dx " and dy " is decreased (positive sign convention), Fig. 10–7 b .

Also, there are associated average normal strains imposed on the element that are determined from Eq. 10–12 :

Pavg = Px + Py

2 =

-350 + 200 2

(10-6) = -75(10-6)

These strains tend to cause the element to contract, Fig. 10–7 b .

EXAMPLE 10.3

(a) dx

O B

Pxdx

Pydy gxy

2 gxy 2

(b)

x ¿ y ¿

40.9! Pavgdx ¿Pavgdy ¿

dx ¿ dy ¿

(gxy)max 2

Fig. 10–7

498 CHAPTER 10 STRAIN TRANSFORMATION

*10.3 Mohr’s Circle—Plane Strain Since the equations of plane-strain transformation are mathematically similar to the equations of plane-stress transformation, we can also solve problems involving the transformation of strain using Mohr’s circle.

Like the case for stress, the parameter u in Eqs. 10–5 and 10–6 can be eliminated and the result rewritten in the form

(Px! - Pavg)2 + ¢gx!y!2 ≤2 = R2 (10–13) where

Pavg = Px + Py

R = B a Px - Py2 b2 + agxy2 b2 Equation 10–13 represents the equation of Mohr’s circle for strain. It has a center on the P axis at point C(Pavg , 0) and a radius R . Notice how these equations compare with Eqs. 9–11 and 9–12. Here Px!, Pavg, and gxy>2 replace sx!, savg, and txy.

R ! 2 2

"2 Px # Py gxy

2 Px # Py

gxy

Px 2

Px " PyPavg !

u ! 0$

Fig. 10–8

Procedure for Analysis

The procedure for drawing Mohr’s circle for strain follows the same one established for stress.

Construction of the Circle. • Establish a coordinate system such that the abscissa represents

the normal strain P, with positive to the right , and the ordinate represents half the value of the shear strain, g>2, with positive downward , Fig. 10–8 .

• Using the positive sign convention for Px, Py, gxy, as shown in Fig. 10–2 , determine the center of the circle C , which is located on the P axis at a distance Pavg = (Px + Py)>2 from the origin, Fig. 10–8 .

• Plot the reference point A having coordinates A (Px , gxy>2). This point represents the case for which the x ! axis coincides with the x axis. Hence u = 0", Fig. 10–8 .

• Connect point A with the center C of the circle and from the shaded triangle determine the radius R of the circle, Fig. 10–8 .

• Once R has been determined, sketch the circle.

10.3 MOHR’S CIRCLE—PLANE STRAIN 499

Principal Strains. • The principal strains P1 and P2 are determined from the circle as the

coordinates of points B and D , that is where g>2 = 0, Fig. 10–9 a. • The orientation of the plane on which P1 acts can be determined

from the circle by calculating 2up1 using trigonometry. Here this angle happens to be counterclockwise from the radial reference line CA to line CB , Fig. 10–9 a . Remember that the rotation of up1 must be in this same direction , from the element’s reference axis x to the x ! axis, Fig. 10–9 b . *

• When P1 and P2 are indicated as being positive as in Fig. 10–9 a , the element in Fig. 10–9 b will elongate in the x ! and y ! directions as shown by the dashed outline.

Maximum In-Plane Shear Strain. • The average normal strain and half the maximum in-plane shear

strain are determined from the circle as the coordinates of point E or F , Fig. 10–9 a .

• The orientation of the plane on which g max in@plane and Pavg act can be determined from the circle by calculating 2us1 using trigo nometry. Here this angle happens to be clockwise from the radial reference line CA to line CE , Fig. 10–9 a . Remember that the rotation of us1 must be in this same direction , from the element’s reference axis x to the x ! axis, Fig. 10–9 c .*

Strains on Arbitrary Plane. • The normal and shear strain components Px! and gx!y! for a plane

oriented at an angle u, Fig. 10–9 d , can be obtained from the circle using trigonometry to determine the coordinates of point P , Fig. 10–9 a.

• To locate P , the known angle u of the x ! axis is measured on the circle as 2u. This measurement is made from the radial reference line CA to the radial line CP . Remember that measurements for 2u on the circle must be in the same direction as u for the x ! axis.*

• If the value of Py! is required, it can be determined by calculating the P coordinate of point Q in Fig. 10–9 a . The line CQ lies 180° away from CP and thus represents a 90° rotation of the x ! axis.

(a)

gxy

u ! 0"

2up1

2us1 2u

Pavg

P1 P2

(b)

up1

y ¿

x ¿

(1 ! P1)dx ¿

(1 ! P2)dy ¿

(c)

x ¿

y ¿

Pavgdx ¿

Pavgdy ¿

us1

* If the g>2 axis were constructed positive upwards , then the angle 2u on the circle would be measured in the opposite direction to the orientation u of the plane.

(d)

x ¿

y ¿

Px ¿dx ¿

Py¿dy¿

Fig. 10–9

500 CHAPTER 10 STRAIN TRANSFORMATION

EXAMPLE 10.4

The state of plane strain at a point is represented by the components Px = 250(10-6 ), Py = -150(10-6 ), and gxy = 120(10-6 ). Determine the principal strains and the orientation of the element.

SOLUTION Construction of the Circle. The P and g>2 axes are established in Fig. 10–10 a . Remember that the positive g>2 axis must be directed downward so that counterclockwise rotations of the element correspond to counterclockwise rotation around the circle, and vice versa. The center of the circle C is located on the P axis at

Pavg = 250 + (-150)

2 (10-6 ) = 50(10-6 )

Since gxy >2 = 60(10-6 ), the reference point A (u = 0!) has coordinates A (250(10-6 ), 60(10-6 )). From the shaded triangle in Fig. 10–10 a , the radius of the circle is CA ; that is,

R = 32(250 - 50)2 + (60)2 4(10-6 ) = 208.8(10-6 ) Principal Strains. The P coordinates of points B and D represent the principal strains. They are

P1 = (50 + 208.8)(10-6 ) = 259(10-6 ) Ans.

P2 = (50 - 208.8)(10-6 ) = -159(10-6 ) Ans.

The direction of the positive principal strain P1 is defined by the counterclockwise angle 2up1, measured from the radial reference line CA (u = 0!) to the line CB . We have

tan 2up1 = 60

(250 - 50)

up1 = 8.35! Ans.

Hence, the side dx " of the element is oriented counterclockwise 8.35° as shown in Fig. 10–10 b . This also defines the direction of P1. The deformation of the element is also shown in the figure.

(a)

C R ! 208.8

250

B(P1, 0)D("P2, 0) P (10"6)

(10"6)

2up1

(b)

x ¿

dx ¿

y ¿

P1dx ¿

P2dy ¿

dy ¿ up1 ! 8.35"

Fig. 10–10

10.3 MOHR’S CIRCLE—PLANE STRAIN 501

EXAMPLE 10.5

The state of plane strain at a point is represented by the components Px = 250(10-6), Py = -150(10-6 ), and gxy = 120(10-6 ). Determine the maximum in-plane shear strains and the orientation of an element.

SOLUTION The circle has been established in the previous example and is shown in Fig. 10–11 a .

Maximum In-Plane Shear Strain. Half the maximum in-plane shear strain and average normal strain are represented by the coordinates of point E or F on the circle. From the coordinates of point E ,

(gx!y!) max in plane

2 = 208.8(10-6)

(gx!y!)maxin@plane = 418(10 -6 ) Ans.

Pavg = 50(10-6 )

To orient the element, we can determine the clockwise angle 2us1 measured from CA (u = 0") to CE.

2us1 = 90" - 2(8.35")

us1 = 36.7" Ans.

This angle is shown in Fig. 10–11 b . Since the shear strain defined from point E on the circle has a positive value and the average normal strain is also positive, these strains deform the element into the dashed shape shown in the figure.

(a)

60 A

C 50

250

R ! 208.8

u ! 0"

gmax

P (10#6)

2us1

(10#6)g 2

2 in–planePavg,E

(b)

x ¿

y ¿

us1 ! 36.7"

Fig. 10–11

502 CHAPTER 10 STRAIN TRANSFORMATION

EXAMPLE 10.6

The state of plane strain at a point is represented on an element having components Px = -300(10-6 ), Py = -100(10-6 ), and gxy = 100(10-6 ). Determine the state of strain on an element oriented 20° clockwise from this reported position.

SOLUTION Construction of the Circle. The P and g>2 axes are established in Fig. 10–12 a . The center of the circle is on the P axis at

(a)

Q 50

300

R ! 111

200

gx¿y¿

13.43"

40"

c ! 13.43"

Px ¿

Py ¿

P (10#6)

(10#6)g 2

gx¿y¿

Pavg = a -300 - 100 2 b (10-6 ) = -200(10-6 ) The reference point A has coordinates A (-300(10-6 ), 50(10-6 )). The radius CA determined from the shaded triangle is therefore

R = 32(300 - 200)2 + (50)2 4(10-6 ) = 111.8(10-6 ) Strains on Inclined Element. Since the element is to be oriented 20° clockwise , we must establish a radial line CP , 2(20!) = 40! clockwise , measured from CA (u = 0!), Fig. 10–12 a . The coordinates of point P (Px" , gx"y" >2) are obtained from the geometry of the circle. Note that

f = tan-1 a 50 (300 - 200) b = 26.57!, c = 40! - 26.57! = 13.43!

Thus,

Px" = -(200 + 111.8 cos 13.43!)(10-6 )

= -309(10-6 ) Ans.

gx"y"

2 = -(111.8 sin 13.43!)(10-6 )

gx"y" = -52.0(10-6 ) Ans.

The normal strain Py" can be determined from the P coordinate of point Q on the circle, Fig. 10–12 a . Why?

Py" = -(200 - 111.8 cos 13.43!)(10-6 ) = -91.3(10-6 ) Ans.

As a result of these strains, the element deforms relative to the x ", y " axes as shown in Fig. 10–12 b .

20"

(b)

x ¿

y ¿y

Fig. 10–12

10.3 MOHR’S CIRCLE—PLANE STRAIN 503

PROBLEMS

10–1. Prove that the sum of the normal strains in perpendicular directions is constant.

10–2. The state of strain at the point has components of Px = 200 (10-6 ), Py = -300 (10-6 ), and gxy = 400(10-6 ). Use the strain-transformation equations to determine the equivalent in-plane strains on an element oriented at an angle of 30! counterclockwise from the original position. Sketch the deformed element due to these strains within the x – y plane.

Prob. 10–2

10–3. The state of strain at a point on a wrench has components P x = 120(10 −6 ), P y = −180(10 −6 ), gxy = 150(10 −6 ). Use the strain-transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case specify the orientation of the element and show how the strains deform the element within x – y plane.

*10–4. The state of strain at the point on the gear tooth has components P x = 850(10 −6 ), P y = 480(10 −6 ), gxy = 650(10 −6 ). Use the strain-transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case specify the orientation of the element and show how the strains deform the element within the x – y plane.

Prob. 10–4

10–5. The state of strain at the point on the gear tooth has the components P x = 520(10 −6 ), P y = −760(10 −6 ), gxy = −750(10 −6 ). Use the strain-transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case specify the orientation of the element and show how the strains deform the element within the x – y plane.

Prob. 10–5

10–6. A differential elements on the bracket is subjected to plane strain that has the following components: P x = 150(10 −6 ), P y = 200(10 −6 ), gxy = −700(10 −6 ). Use the strain-transformation equations and determine the equivalent in plane strains on an element oriented at an angle of u = 60° counterclockwise from the original position. Sketch the deformed element within the x – y plane due to these strains.

10–7. Solve Prob. 10–6 for an element oriented u = 30° clockwise.

Probs. 10–6/7

504 CHAPTER 10 STRAIN TRANSFORMATION

10–10. The state of strain at the point on the support has components of P x = 350(10 −6 ), P y = 400(10 −6 ), gxy = −675(10 −6 ). Use the strain-transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case specify the orientation of the element and show how the strains deform the element within the x – y plane.

Prob. 10–10

10–11. The state of strain on an element has components P x = −150(10 −6 ), P y = 450(10 −6 ), gxy = 200(10 −6 ). Determine the equivalent state of strain on an element at the same point oriented 30° counterclockwise with respect to the ori ginal element. Sketch the results on this element.

xy 2

xy 2 g

Pydy

Pxdx

Prob. 10–11

*10–8. The state of strain at the point on the bracket has components Px = -200(10-6 ), Py = -650(10-6 ), gxy = -175(10-6 ). Use the strain-transformation equations to determine the equivalent in-plane strains on an element oriented at an angle of u = 20! counterclockwise from the original position. Sketch the deformed element due to these strains within the x – y plane.

Prob. 10–8

10–9. The state of strain at the point has components of Px = 180(10-6 ), Py = -120(10-6 ), and gxy = -100(10-6 ). Use the strain-transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case specify the orientation of the element and show how the strains deform the element within the x–y plane.

Prob. 10–9

10.3 MOHR’S CIRCLE—PLANE STRAIN 505

■10–15. Consider the general case of plane strain where Px, Py, and gxy are known. Write a computer program that can be used to determine the normal and shear strain, Px! and gx!y!, on the plane of an element oriented u from the horizontal. Also, include the principal strains and the element’s orientation, and the maximum in-plane shear strain, the average normal strain, and the element’s orientation.

*10–16. The state of strain on an element has components P x = −300(10 −6 ), P y = 100(10 −6 ), gxy = 150(10 −6 ). Determine the equivalent state of strain, which represents (a) the principal strains, and (b) the maximum in-plane shear strain and the associated average normal strain. Specify the orientation of the corresponding elements for these states of strain with respect to the original element.

xy 2

xy 2 g

Pydy

Pxdx

Prob. 10–16

10–17. Solve part (a) of Prob. 10–3 using Mohr’s circle.

10–18. Solve part (b) of Prob. 10–3 using Mohr’s circle.

10–19. Solve Prob. 10–4 using Mohr’s circle.

*10–20. Solve Prob. 10–5 using Mohr’s circle.

10–21. Solve Prob. 10–7 using Mohr’s circle.

*10–12. The state of strain on an element has components P x = −400(10 −6 ), P y = 0, gxy = 150(10 −6 ). Determine the equivalent state of strain on an element at the same point oriented 30° clockwise with respect to the original element. Sketch the results on this element. 10–13. The state of plane strain on an element is Px = -300(10-6 ), Py = 0, and gxy = 150(10-6 ). Determine the equivalent state of strain which represents (a) the principal strains, and (b) the maximum in-plane shear strain and the associated average normal strain. Specify the orientation of the corresponding elements for these states of strain with respect to the original element.

Pxdx

gxy 2

Probs. 10–12/13

10–14. The state of strain at the point on a boom of an hydraulic engine crane has components of Px = 250(10-6), Py = 300(10-6), and gxy = -180(10-6 ). Use the strain- transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case, specify the orientation of the element and show how the strains deform the element within the x–y plane.

y x

Prob. 10–14

506 CHAPTER 10 STRAIN TRANSFORMATION

*10.4 Absolute Maximum Shear Strain In Sec. 9–5 it was pointed out that in the case of plane stress, the absolute maximum shear stress in an element of material will occur out of the plane when the principal stresses have the same sign, i.e., both are tensile or both are compressive. A similar result occurs for plane strain. For example, if the principal in-plane strains cause elongations, Fig. 10–13 a , then the three Mohr’s circles describing the normal and shear strain components for elements oriented about the x !, y !, and z ! axes are shown in Fig. 10–13 b . By inspection, the largest circle has a radius R = (gx!z! )max >2. Hence,

g max abs = (gx!z! )max = P1

P1 and P2 have the same sign (10–14)

This value gives the absolute maximum shear strain for the material. Note that it is larger than the maximum in-plane shear strain, which is (gx!y! )max = P1 - P2.

Now consider the case where one of the in-plane principal strains is of opposite sign to the other in-plane principal strain, so that P1 causes elongation and P2 causes contraction, Fig. 10–14 a . Mohr’s circles, which describe the strains on each element’s orientation about the x !, y !, z ! axes, are shown in Fig. 10–14 b . Here

g max abs = (gx!y! ) max in@plane = P1 - P2

P1 and P2 have opposite signs (10–15)

We may therefore summarize the above two cases as follows. If the in-plane principal strains both have the same sign , the absolute maximum shear strain will occur out of plane and has a value of g

max abs = Pmax . However, if the in-plane principal strains are of opposite

signs , then the absolute maximum shear strain equals the maximum in-plane shear strain.

x!y plane strain

(a)

(1 " P1)dx(1 " P2)dy

x y

x!y plane strain (a)

(1 – P2)dy (1 " P1)dx

Fig. 10–13 (b)

PP1

(gxz)max 2

(gxy)max 2

(gyz)max 2

Fig. 10–14 (b)

2 (gxz)max

PP1!P2

2 (gyz)max

2 (gxy)max

Important Points

• The absolute maximum shear strain will be larger than the maximum in-plane shear strain whenever the in-plane principal strains have the same sign . When this occurs the absolute maximum shear strain will act out of the plane.

• If the in-plane principal strains are of opposite signs, then the absolute maximum shear strain will equal the maximum in-plane shear strain.

10.4 ABSOLUTE MAXIMUM SHEAR STRAIN 507

EXAMPLE 10.7

The state of plane strain at a point is represented by the strain components Px = -400(10-6), Py = 200(10-6), gxy = 150(10-6). Determine the maximum in-plane shear strain and the absolute maximum shear strain.

400

100

R ! 3 09

A P1 P(10

"6)

(10"6)

2 in–plane

(gx¿y¿)max

Fig. 10–15 SOLUTION Maximum In-Plane Shear Strain. We will solve this problem using Mohr’s circle. From the strain components, the center of the circle is on the P axis at

Pavg = -400 + 200

2 (10-6) = -100(10-6)

Since gxy >2 = 75(10-6 ), the reference point A has coordinates (-400(10-6), 75(10-6)). As shown in Fig. 10–15 , the radius of the circle is therefore

R = 32(400 - 100)2 + (75)2 4(10-6) = 309(10-6) Calculating the in-plane principal strains from the circle, we have

P1 = (-100 + 309)(10-6) = 209(10-6) P2 = (-100 - 309)(10-6) = -409(10-6)

Also, the maximum in-plane shear strain is

gin@planemax = P1 - P2 = [209 - (-409)](10-6) = 618(10-6) Ans.

Absolute Maximum Shear Strain. From the above results, we have P1 = 209(10-6 ), P2 = -409(10-6 ). The three Mohr’s circles, plotted for element orientations about each of the x , y , z axes, are also shown in Fig. 10–15 . It is seen that since the principal in-plane strains have opposite signs , the maximum in-plane shear strain is also the absolute maximum shear strain; i.e., g

max abs = 618(10-6) Ans.

508 CHAPTER 10 STRAIN TRANSFORMATION

10.5 Strain Rosettes When performing a tension test on a specimen as discussed in Sec. 3–1 , the normal strain in the material is measured using an electrical- resistance strain gauge , which consists of a wire grid or piece of metal foil bonded to the specimen. For a general loading on a body, however, the strains at a point on its free surface are determined using a cluster of three electrical- resistance strain gauges, arranged in a specified pattern. This pattern is referred to as a strain rosette , and once the normal strains on the three gauges are measured, the data can then be transformed to specify the state of strain at the point. Since these strains are measured only in the plane of the gauges, and since the body is stress-free on its surface, the gauges may be subjected to plane stress but not plane strain. Although the strain normal to the surface is not measured, realize that the out-of- plane displacement caused by this strain will not affect the in-plane measurements of the gauges.

In the general case, the axes of the three gauges are arranged at the angles ua, ub, uc shown in Fig. 10–16 a . If the readings Pa, Pb, Pc are taken, we can determine the strain components Px, Py, gxy at the point by applying the strain-transformation equation, Eq. 10–2 , for each gauge. We have

Pa = Px cos2 ua + Py sin2 ua + gxy sin ua cos ua Pb = Px cos2 ub + Py sin2 ub + gxy sin ub cos ub

Pc = Px cos2 uc + Py sin2 uc + gxy sin uc cos uc (10–16)

The values of Px, Py, gxy are determined by solving these three equations simultaneously.

Strain rosettes are often arranged in 45° or 60° patterns. In the case of the 45° or “rectangular” strain rosette shown in Fig. 10–16 b , ua = 0!, ub = 45!, uc = 90!, so that Eq. 10–16 gives

Px = Pa Py = Pc

gxy = 2Pb - (Pa + Pc ) And for the 60° strain rosette in Fig. 10–16 c , ua = 0!, ub = 60!, uc = 120!. Here Eq. 10–16 gives

Px = Pa Py =

1 3

(2Pb + 2Pc - Pa ) (10–17)

gxy = 2

13 (Pb - Pc ) Once Px, Py, gxy are determined, the transformation equations of

Sec. 10–2 or Mohr’s circle can then be used to determine the principal in-plane strains and the maximum in-plane shear strain at the point.

c (a)

uc ua

Fig. 10–16

45! 45!

45! strain rosette (b)

60!

60! strain rosette (c)

60!

Typical electrical resistance 45° strain rosette.

10.5 STRAIN ROSETTES 509

EXAMPLE 10.8

The state of strain at point A on the bracket in Fig. 10–17 a is measured using the strain rosette shown in Fig. 10–17 b . Due to the loadings, the readings from the gauges give Pa = 60(10-6 ), Pb = 135(10-6 ), and Pc = 264(10-6 ). Determine the in-plane principal strains at the point and the directions in which they act.

SOLUTION We will use Eqs. 10–16 for the solution. Establishing an x axis as shown in Fig. 10–17 b and measuring the angles counterclockwise from the +x axis to the centerlines of each gauge, we have ua = 0!, ub = 60!, and uc = 120!. Substituting these results, along with the problem data, into the equations gives

60(10-6 ) = Px cos2 0! + Py sin2 0! + gxy sin 0! cos 0! = Px (1)

135(10-6 ) = Px cos2 60! + Py sin2 60! + gxy sin 60! cos 60! = 0.25Px + 0.75Py + 0.433gxy (2)

264(10-6 ) = Px cos2 120! + Py sin2 120! + gxy sin 120! cos 120! = 0.25Px + 0.75Py - 0.433gxy (3)

Using Eq. 1 and solving Eqs. 2 and 3 simultaneously, we get

Px = 60(10-6 ) Py = 246(10-6 ) gxy = -149(10-6 )

These same results can also be obtained in a more direct manner from Eq. 10–17 .

The in-plane principal strains can be determined using Mohr’s circle. The reference point on the circle is at A [60(10-6 ), -74.5(10-6 )] and the center of the circle, C , is on the P axis at Pavg = 153(10-6 ), Fig. 10–17 c . From the shaded triangle, the radius is

R = 32(153 - 60)2 + (74.5)2 4(10-6 ) = 119.1(10-6 ) The in-plane principal strains are thus

P1 = 153(10-6 ) + 119.1(10-6 ) = 272(10-6 ) Ans. P2 = 153(10-6 ) - 119.1(10-6 ) = 33.9(10-6 ) Ans.

2up2 = tan -1

74.5 (153 - 60) = 38.7!

up2 = 19.3! Ans.

NOTE: The deformed element is shown in the dashed position in Fig. 10–17 d . Realize that, due to the Poisson effect, the element is also subjected to an out-of-plane strain, i.e., in the z direction, although this value will not influence the calculated results.

(a)

b a

(b)

60!

120!

(c)

2up2

74.5

153

R " 119.2 P(10#6)P1P2

(10#6)g 2

(d) x

up2 " 19.3!

x¿

y¿

Fig. 10–17

510 CHAPTER 10 STRAIN TRANSFORMATION

*10–24. The steel bar is subjected to the tensile load of 500 lb. If it is 0.5 in. thick determine the three principal strains. E = 29(10 3 ) ksi, n = 0.3.

15 in.

2 in.

500 lb 500 lb

Prob. 10–24

10–25. The 45° strain rosette is mounted on a machine element. The following readings are obtained from each gauge: P a = 650(10 −6 ), P b = −300(10 −6 ), P c = 480(10 −6 ). Determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and associated average normal strain. In each case show the deformed element due to these strains.

45!

a b

Prob. 10–25

10–22. The strain at point A on the bracket has components Px = 300(10-6 ), Py = 550(10-6 ), gxy = -650(10-6 ), Pz = 0 . Determine (a) the principal strains at A in the x9y plane, (b) the maximum shear strain in the x–y plane, and (c) the absolute maximum shear strain.

Prob. 10–22

10–23. The strain at point A on a beam has components P x = 450(10 −6 ), P y = 825(10 −6 ), g xy = 275(10 −6 ), P z = 0. Determine (a) the principal strains at A , (b) the maximum shear strain in the x – y plane, and (c) the absolute maximum shear strain.

Prob. 10–23

PROBLEMS

10.5 STRAIN ROSETTES 511

*10–28. The 60° strain rosette is mounted on a beam. The following readings are obtained from each gauge: P a = 250(10 −6 ), P b = −400(10 −6 ), P c = 280(10 −6 ). Determine (a) the in-plane principal strains and their orientation, and (b) the maximum in-plane shear strain and average normal strain. In each case show the deformed element due to these strains.

60!

c 60! 60!

Prob. 10–28

■10–29. Consider the general orientation of three strain gauges at a point as shown. Write a computer program that can be used to determine the principal in-plane strains and the maximum in-plane shear strain at the point. Show an application of the program using the values ua = 40!, Pa = 160(10-6 ), ub = 125!, Pb = 100(10-6 ), uc = 220!, Pc = 80(10-6 ).

Prob. 10–29

10–26. The 60° strain rosette is attached to point A on the surface of the support. Due to the loading the strain gauges give a reading of P a = 300(10 −6 ), P b = −150(10 −6 ), and P c = −450(10 −6 ). Use Mohr’s circle and determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and the associated average normal strain. Specify the orientation of each element that has these states of strain with respect to the x axis.

60! 60!

Prob. 10–26

10–27. The strain rosette is attached at the point on the surface of the pump. Due to the loading, the strain gauges give a reading of P a = −250(10 −6 ), P b = 300(10 −6 ), and P c = −200(10 −6 ). Determine (a) the in-plane principal strains, and (b) the maximum in-plane shear strain. Specify the orientation of each element that has these states of strain with respect to the x axis.

x 60!

60!

Prob. 10–27

512 CHAPTER 10 STRAIN TRANSFORMATION

10.6 Material-Property Relationships In this section we will present some important relationships involving a material’s properties that are used when the material is subjected to multiaxial stress and strain. To do so we will assume that the material is homogeneous and isotropic and behaves in a linear-elastic manner.

Generalized Hooke’s Law. If the material at a point is subjected to a state of triaxial stress, sx, sy, sz, Fig. 10–18 a , associated normal strains Px, Py, Pz will be developed in the material. The stresses can be related to these strains by using the principle of superposition, Poisson’s ratio, Plat = -nPlong, and Hooke’s law, as it applies in the uniaxial direction, P = s>E. For example, consider the normal strain of the element in the x direction, caused by separate application of each normal stress. When sx is applied, Fig. 10–18 b , the element elongates in the x direction and the strain P!x is

P=x = sx

Application of sy causes the element to contract with a strain P>x , Fig. 10–18 c . Here

P>x = -n sy

Likewise, application of sz, Fig. 10–18 d , causes a contraction such that

P?x = -n sz

(a)

sx sx sy

sz sz

(d)(c)(b)

! " "

Fig. 10–18

10.6 MATERIAL-PROPERTY RELATIONSHIPS 513

When these three normal strains are superimposed, the normal strain Px is determined for the state of stress in Fig. 10–18 a . Similar equations can be developed for the normal strains in the y and z directions. The final results can be written as

Px = 1 E

[sx - n(sy + sz)]

Py = 1 E

[sy - n(sx + sz)] (10–18)

Pz = 1 E

[sz - n(sx + sy)]

These three equations express Hooke’s law in a general form for a triaxial state of stress. For application tensile stresses are considered positive quantities, and compressive stresses are negative. If a resulting normal strain is positive , it indicates that the material elongates , whereas a negative normal strain indicates the material contracts.

If we now apply a shear stress txy to the element, Fig. 10–19 a , experimental observations indicate that the material will deform only due to a shear strain gxy ; that is, txy will not cause other strains in the material. Likewise, tyz and txz will only cause shear strains gyz and gxz, Figs. 10–19 b and Fig. 10–19 c , and so Hooke’s law for shear stress and shear strain can be written as

gxy = 1 G

txy gyz = 1 G

tyz gxz = 1 G

txz (10–19)

(a)

txy

(b)

tyz

(c)

txz

Fig. 10–19

514 CHAPTER 10 STRAIN TRANSFORMATION

Relationship Involving E , N , and G . In Sec. 3–7 it was stated that the modulus of elasticity E is related to the shear modulus G by Eq. 3–11 , namely,

G = E

2(1 + n) (10–20)

One way to derive this relationship is to consider an element of the material to be subjected to pure shear (sx = sy = sz = 0), Fig. 10–20 a . Applying Eq. 9–5 to obtain the principal stresses yields smax = txy and smin = -txy. This element must be oriented up1 = 45! counterclockwise from the x axis as shown in Fig. 10–20 b . If the three principal stresses smax = txy, sint = 0, and smin = -txy are substituted into the first of Eqs. 10–18 , the principal strain Pmax can be related to the shear stress txy. The result is

Pmax = txy

E (1 + n) (10–21)

This strain, which deforms the element along the x " axis, can also be related to the shear strain gxy . To do this, first note that since sx = sy = sz = 0, then from the first and second Eqs. 10–18 , Px = Py = 0. Substituting these results into the strain transformation Eq. 10–9 , we get

P1 = Pmax = gxy

By Hooke’s law, gxy = txy >G, so that Pmax = txy >2G. Substituting into Eq. 10–21 and rearranging terms gives the final result, namely, Eq. 10–20 .

Dilatation and Bulk Modulus. When an elastic material is subjected to normal stress, its volume will change. For example, consider a volume element which is subjected to the principal stresses sx, sy, sz. If the sides of the element are originally dx , dy , dz , Fig. 10–21 a , then after application of the stress they become (1 + Px) dx, (1 + Py) dy, (1 + Pz ) dz, Fig. 10–21 b . The change in volume of the element is therefore

dV = (1 + Px)(1 + Py)(1 + Pz) dx dy dz - dx dy dz

Neglecting the products of the strains since the strains are very small, we have

dV = (Px + Py + Pz) dx dy dz

The change in volume per unit volume is called the “volumetric strain” or the dilatation e . It can be written as

e = dV dV

= Px + Py + Pz (10–22)

By comparison, the shear strains will not change the volume of the element, rather they will only change its rectangular shape.

(a)

txy

smin ! "txy smax ! txy up1 ! 45#

(b)

x¿

Fig. 10–20

(a)

(b)

(1 $ Py)dy (1 $ Px)dx

(1 $ Pz)dz

sy sx

Fig. 10–21

10.6 MATERIAL-PROPERTY RELATIONSHIPS 515

Also, if we use Hooke’s law, as defined by Eq. 10–18 , we can write the dilatation in terms of the applied stress. We have

e = 1 - 2n

E (sx + sy + sz) (10–23)

When a volume element of material is subjected to the uniform pressure p of a liquid, the pressure on the body is the same in all directions and is always normal to any surface on which it acts. Shear stresses are not present , since the shear resistance of a liquid is zero. This state of “hydrostatic” loading requires the normal stresses to be equal in any and all directions, and therefore an element of the body is subjected to principal stresses sx = sy = sz = -p, Fig. 10–22 . Substituting into Eq. 10–23 and rearranging terms yields

p e

= - E 3(1 - 2n) (10–24)

Since this ratio is similar to the ratio of linear elastic stress to strain, which defines E , i.e., s>P = E, the term on the right is called the volume modulus of elasticity or the bulk modulus . It has the same units as stress and will be symbolized by the letter k ; that is,

k = E

3(1 - 2n) (10–25)

Note that for most metals n ! 13 so k ! E. If a material existed that did not change its volume then dV = e = 0, and k would have to be infinite. From Eq. 10–25 the theoretical maximum value for Poisson’s ratio is therefore n = 0.5. During yielding, no actual volume change of the material is observed, and so n = 0.5 is used when plastic yielding occurs.

Hydrostatic stress

sy ! p sx ! p

sz ! p

Fig. 10–22

Important Points

• When a homogeneous isotropic material is subjected to a state of triaxial stress, the strain in each direction is influenced by the strains produced by all the stresses. This is the result of the Poisson effect, and results in the form of a generalized Hooke’s law.

• Unlike normal stress, a shear stress applied to homogeneous isotropic material will only produce shear strain in the same plane.

• The material constants E, G, and n are related mathematically. • Dilatation , or volumetric strain , is caused only by normal strain,

not shear strain.

• The bulk modulus is a measure of the stiffness of a volume of material. This material property provides an upper limit to Poisson’s ratio of n = 0.5, which remains at this value while plastic yielding occurs.

516 CHAPTER 10 STRAIN TRANSFORMATION

The bracket in Example 10.8 , Fig. 10–23 a , is made of steel for which Est = 200 GPa, nst = 0.3. Determine the principal stresses at point A .

EXAMPLE 10.9

(a)

b a

Fig. 10–23

SOLUTION I From Example 10.8 the principal strains have been determined as

P1 = 272(10-6)

P2 = 33.9(10-6)

Since point A is on the surface of the bracket for which there is no loading, the stress on the surface is zero, and so point A is subjected to plane stress. Applying Hooke’s law with s3 = 0, we have

P1 = s1

E - n

E s2 ; 272(10

-6) = s1

200(109) - 0.3

200(109) s2

54.4(106) = s1 - 0.3s2 (1)

P2 = s2

E - n

E s1 ; 33.9(10

-6) = s2

200(109) - 0.3

200(109) s1

6.78(106) = s2 - 0.3s1 (2)

Solving Eqs. 1 and 2 simultaneously yields

s1 = 62.0 MPa Ans.

s2 = 25.4 MPa Ans.

10.6 MATERIAL-PROPERTY RELATIONSHIPS 517

SOLUTION II It is also possible to solve the problem using the given state of strain,

Px = 60(10-6) Py = 246(10-6) gxy = -149(10-6)

as specified in Example 10.8 . Applying Hooke’s law in the x – y plane, we have

Px = sx

E - n

E sy ; 60(10

-6) = sx

200(109) Pa -

0.3sy 200(109) Pa

Py = sy

E - n

E sx ; 246(10

-6) = sy

200(109) Pa -

0.3sx 200(109) Pa

sx = 29.4 MPa sy = 58.0 MPa

The shear stress is determined using Hooke’s law for shear. First, however, we must calculate G .

G = E

2(1 + n) = 200 GPa

2(1 + 0.3) = 76.9 GPa

Thus, txy = Ggxy ; txy = 76.9(109)[-149(10-6)] = -11.46 MPa The Mohr’s circle for this state of plane stress has a reference point A (29.4 MPa, -11.46 MPa) and center at savg = 43.7 MPa, Fig. 10–23 b . The radius is determined from the shaded triangle.

R = 2(43.7 - 29.4)2 + (11.46)2 = 18.3 MPa Therefore,

s1 = 43.7 MPa + 18.3 MPa = 62.0 MPa Ans. s2 = 43.7 MPa - 18.3 MPa = 25.4 MPa Ans.

NOTE: Each of these solutions is valid provided the material is both linear elastic and isotropic, since then the principal planes of stress and strain coincide.

s (MPa)

29.4

11.46

43.7

(b)

s1s2

R ! 18.3

t (MPa)

Fig. 10–23 (cont.)

518 CHAPTER 10 STRAIN TRANSFORMATION

The copper bar in Fig. 10–24 is subjected to a uniform loading along its edges as shown. If it has a length a = 300 mm, width b = 50 mm, and thickness t = 20 mm before the load is applied, determine its new length, width, and thickness after application of the load. Take Ecu = 120 GPa, ncu = 0.34.

EXAMPLE 10.10

500 MPa

800 MPa

Fig. 10–24

SOLUTION By inspection, the bar is subjected to a state of plane stress. From the loading we have

sx = 800 MPa sy = -500 MPa txy = 0 sz = 0

The associated normal strains are determined from the generalized Hooke’s law, Eq. 10–18 ; that is,

Px = sx

E - n

E (sy + sz)

= 800 MPa

120(103) MPa - 0.34

120(103) MPa (-500 MPa + 0) = 0.00808

Py = sy

E - n

E (sx + sz)

= -500 MPa

120(103) MPa - 0.34

120(103) MPa (800 MPa + 0) = -0.00643

Pz = sz

E - n

E (sx + sy)

= 0 - 0.34 120(103) MPa

(800 MPa - 500 MPa) = -0.000850

The new bar length, width, and thickness are therefore

a! = 300 mm + 0.00808(300 mm) = 302.4 mm Ans.

b! = 50 mm + (-0.00643)(50 mm) = 49.68 mm Ans.

t! = 20 mm + (-0.000850)(20 mm) = 19.98 mm Ans.

10.6 MATERIAL-PROPERTY RELATIONSHIPS 519

EXAMPLE 10.11

If the rectangular block shown in Fig. 10–25 is subjected to a uniform pressure of p = 20 psi, determine the dilatation and the change in length of each side. Take E = 600 psi, n = 0.45.

a ! 4 in. b ! 2 in.

c ! 3 in.

Fig. 10–25

SOLUTION Dilatation. The dilatation can be determined using Eq. 10–23 with sx = sy = sz = -20 psi. We have

e = 1 - 2n

E (sx + sy + sz)

= 1 - 2(0.45)

600 psi [3(-20 psi)]

= -0.01 in3>in3 Ans. Change in Length. The normal strain on each side can be determined from Hooke’s law, Eq. 10–18 ; that is,

P = 1 E

[sx - n(sy + sz)]

= 1

600 psi [-20 psi - (0.45)(-20 psi - 20 psi)] = -0.00333 in.>in.

Thus, the change in length of each side is

da = -0.00333(4 in.) = -0.0133 in. Ans.

db = -0.00333(2 in.) = -0.00667 in. Ans.

dc = -0.00333(3 in.) = -0.0100 in. Ans.

The negative signs indicate that each dimension is decreased.

520 CHAPTER 10 STRAIN TRANSFORMATION

*10–36. The spherical pressure vessel has an inner diameter of 2 m and a thickness of 10 mm. A strain gauge having a length of 20 mm is attached to it, and it is observed to increase in length by 0.012 mm when the vessel is pressurized. Determine the pressure causing this deformation, and find the maximum in-plane shear stress, and the absolute maximum shear stress at a point on the outer surface of the vessel. The material is steel, for which Est = 200 GPa and nst = 0.3.

20 mm

Prob. 10–36

10–37. Determine the bulk modulus for each of the following materials: (a) rubber, Er = 0.4 ksi, nr = 0.48, and (b) glass, Eg = 8(103) ksi, ng = 0.24.

10–38. The strain gauge is placed on the surface of a thin- walled steel boiler as shown. If it is 0.5 in. long, determine the pressure in the boiler when the gauge elongates 0.2(10 −3 ) in. The boiler has a thickness of 0.5 in. and inner diameter of 60 in. Also, determine the maximum x , y in-plane shear strain in the material. E st = 29(10 3 ) ksi, n st = 0.3.

60 in.

0.5 in.

Prob. 10–38

10–30. For the case of plane stress, show that Hooke’s law can be written as

sx = E

(1 - n2) (Px + nPy), sy =

(1 - n2) (Py + nPx)

10–31. Use Hooke’s law, Eq. 10–18 , to develop the strain- tranformation equations, Eqs. 10–5 and 10–6 , from the stress-tranformation equations, Eqs. 9–1 and 9–2 .

*10–32. The principal plane stresses and associated strains in a plane at a point are s 1 = 36 ksi, s 2 = 16 ksi, P 1 = 1.02(10 −3 ), P 2 = 0.180(10 −3 ). Determine the modulus of elasticity and Poisson’s ratio.

10–33. A rod has a radius of 10 mm. If it is subjected to an axial load of 15 N such that the axial strain in the rod is P x = 2.75(10 −6 ), determine the modulus of elasticity E and the change in its diameter. n = 0.23.

10–34. The polyvinyl chloride bar is subjected to an axial force of 900 lb. If it has the original dimensions shown determine the change in the angle u after the load is applied. E pvc = 800(10 3 ) psi, n pvc = 0.20.

10–35. The polyvinyl chloride bar is subjected to an axial force of 900 lb. If it has the original dimensions shown determine the value of Poisson’s ratio if the angle u decreases by !u = 0.01° after the load is applied. E pvc = 800(10 3 ) psi.

3 in. 900 lb

6 in. 1 in.

900 lb

Probs. 10–34/35

PROBLEMS

10.6 MATERIAL-PROPERTY RELATIONSHIPS 521

10–43. A strain gauge a is attached in the longitudinal direction (x axis) on the surface of the gas tank. When the tank is pressurized, the strain gauge gives a reading of P a = 100( 10−6) . Determine the pressure p in the tank. The tank has an inner diameter of 1.5 m and wall thickness of 25 mm. It is made of steel having a modulus of elasticity E = 200 GPa and Poisson’s ratio n = 13.

*10–44. Strain gauge b is attached to the surface of the gas tank at an angle of 45° with x axis as shown. When the tank is pressurized, the strain gauge gives a reading of Pb = 250(10 −6 ). Determine the pressure in the tank. The tank has an inner diameter of 1.5 m and wall thickness of 25 mm. It is made of steel having a modulus of elasticity E = 200 GPa and Poisson’s ratio n = 13.

a xb

45!

Probs. 10–43/44

10–45. A material is subjected to principal stresses s x and s y . Determine the orientation u of a strain gauge placed at the point so that its reading of normal strain responds only to s y and not s x . The material constants are E and n .

Prob. 10–45

10–39. The strain in the x direction at point A on the A-36 structural-steel beam is measured and found to be P x = 100(10 −6 ). Determine the applied load P . What is the shear strain g xy at point A ?

*10–40. The strain in the x direction at point A on the A-36 structural-steel beam is measured and found to be P x = 200(10 −6 ). Determine the applied load P . What is the shear strain g xy at point A ?

10–41. If a load of P = 3 kip is applied to the A-36 structural-steel beam, determine the strain P x and g xy at point A.

3 ft 4 ft

2 in.

6 in.

12 in.

2 in.

A x

Probs. 10–39/40/41

10–42. The principal stresses at a point are shown in the figure. If the material is aluminum for which Eal = 10(103) ksi and nal = 0.33, determine the principal strains.

26 ksi

15 ksi 10 ksi

Prob. 10–42

522 CHAPTER 10 STRAIN TRANSFORMATION

10–49. Initially, gaps between the A-36 steel plate and the rigid constraint are as shown. Determine the normal stresses sx and sy developed in the plate if the temperature is increased by !T = 100"F. To solve, add the thermal strain a!T to the equations for Hooke’s law.

6 in.

0.0015 in.

8 in. 0.0025 in.

Prob. 10–49

10–50. The steel shaft has a radius of 15 mm. Determine the torque T in the shaft if the two strain gauges, attached to the surface of the shaft, report strains of P x# = −80(10 −6 ) and P y# = 80(10 −6 ). Also, compute the strains acting in the x and y directions. E st = 200 GPa, n st = 0.3.

10–51. The shaft has a radius of 15 mm and is made of L2 tool steel. Determine the strains in the x# and y# direction if a torque T = 2 kN · m is applied to the shaft.

45!

x¿y¿T

Probs. 10–50/51

10–46. The principal strains in a plane, measured experimentally at a point on the aluminum fuselage of a jet aircraft, are P1 = 630(10-6) and P2 = 350(10-6). If this is a case of plane stress, determine the associated principal stresses at the point in the same plane. Eal = 10(103) ksi and nal = 0.33.

10–47. The principal stresses at a point are shown in the figure. If the material is aluminum for which Eal = 10(103) ksi and nal = 0.33, determine the principal strains.

3 ksi

4 ksi 8 ksi

Prob. 10–47

*10–48. The 6061-T6 aluminum alloy plate fits snugly into the rigid constraint. Determine the normal stresses sx and sy developed in the plate if the temperature is increased by !T = 50"C. To solve, add the thermal strain a!T to the equations for Hooke’s law.

300 mm

400 mmy

Prob. 10–48

10.6 MATERIAL-PROPERTY RELATIONSHIPS 523

*10–56. The thin-walled cylindrical pressure vessel of inner radius r and thickness t is subjected to an internal pressure p . If the material constants are E and n , determine the strains in the circumferential and longitudinal directions. Using these results, compute the increase in both the diameter and the length of a steel pressure vessel filled with air and having an internal gauge pressure of 15 MPa. The vessel is 3 m long, and has an inner radius of 0.5 m and a thickness of 10 mm. E st = 200 GPa, n st = 0.3.

10–57. Estimate the increase in volume of the tank in Prob. 10–56 .

0.5 m

3 m

Prob. 10–57

10–58. A soft material is placed within the confines of a rigid cylinder which rests on a rigid support. Assuming that Px = 0 and Py = 0, determine the factor by which the modulus of elasticity will be increased when a load is applied if n = 0.3 for the material.

Prob. 10–58

*10–52. The metal block is fitted between the fixed supports. If the glued joint can resist a maximum shear stress of tallow = 2 ksi, determine the temperature rise that will cause the joint to fail. Take E = 10(103) ksi, n = 0.2, and a = 6.0(10- 6)>!F. Hint: Use Eq. 10–18 with an additional strain term of a"T ( Eq. 4–4 ).

40!

Prob. 10–52

10–53. Air is pumped into the steel thin-walled pressure vessel at C . If the ends of the vessel are closed using two pistons connected by a rod AB , determine the increase in the diameter of the pressure vessel when the internal gauge pressure is 5 MPa. Also, what is the tensile stress in rod AB if it has a diameter of 100 mm? The inner radius of the vessel is 400 mm, and its thickness is 10 mm. E st = 200 GPa and n st = 0.3.

10–54. Determine the increase in the diameter of the pressure vessel in Prob. 10–53 if the pistons are replaced by walls connected to the ends of the vessel.

400 mm

Probs. 10–53/54

10–55. A thin-walled spherical pressure vessel having an inner radius r and thickness t is subjected to an internal pressure p . Show that the increase in the volume within the vessel is "V = (2ppr 4>Et)(1 - n). Use a small-strain analysis.

524 CHAPTER 10 STRAIN TRANSFORMATION

*10.7 Theories of Failure When an engineer is faced with the problem of design using a specific material, it becomes important to place an upper limit on the state of stress that defines the material’s failure. If the material is ductile , failure is usually specified by the initiation of yielding , whereas if the material is brittle , it is specified by fracture . These modes of failure are readily defined if the member is subjected to a uniaxial state of stress, as in the case of simple tension; however, if the member is subjected to biaxial or triaxial stress, the criterion for failure becomes more difficult to establish.

In this section we will discuss four theories that are often used in engineering practice to predict the failure of a material subjected to a multiaxial state of stress. No single theory of failure, however, can be applied to a specific material at all times , because a material may behave in either a ductile or brittle manner depending on the temperature, rate of loading, chemical environment, or the way the material is shaped or formed. When using a particular theory of failure, it is first necessary to calculate the normal and shear stress at points where they are the largest in the member. Once this state of stress is established, the principal stresses at these critical points are then determined, since each of the following theories is based on knowing the principal stress.

Ductile Materials Maximum-Shear-Stress Theory. The most common type of yielding of a ductile material such as steel is caused by slipping , which occurs along the contact planes of randomly ordered crystals that make up the material. If we make a specimen into a highly polished thin strip and subject it to a simple tension test, we can actually see how this slipping causes the material to yield , Fig. 10–26 . The edges of the planes of slipping as they appear on the surface of the strip are referred to as Lüder’s lines . These lines clearly indicate the slip planes in the strip, which occur at approximately 45° with the axis of the strip.

The slipping that occurs is caused by shear stress. To show this, consider an element of the material taken from a tension specimen, when it is subjected to the yield stress sY , Fig. 10–27 a . The maximum shear stress can be determined by drawing Mohr’s circle for the element, Fig. 10–27 b . The results indicate that

tmax = sY

2 (10–26)

45!

Lüder’s lines on mild steel strip

Fig. 10–26

10.7 THEORIES OF FAILURE 525

Furthermore, this shear stress acts on planes that are 45° from the planes of principal stress, Fig. 10–27 c , and these planes coincide with the direction of the Lüder lines shown on the specimen, indicating that indeed failure occurs by shear.

Using this idea, that ductile materials fail by shear, in 1868 Henri Tresca proposed the maximum-shear-stress theory or Tresca yield criterion . This theory can be used to predict the failure stress of a ductile material subjected to any type of loading. The theory states that yielding of the material begins when the absolute maximum shear stress in the material reaches the shear stress that causes the same material to yield when it is subjected only to axial tension. Therefore, to avoid failure, it is required that t

max abs in the material must be less than or equal to sY >2, where sY is

determined from a simple tension test. For application we will express the absolute maximum shear stress in

terms of the principal stresses . The procedure for doing this was discussed in Sec. 9.5 with reference to a condition of plane stress , that is, where the out-of-plane principal stress is zero. If the two in-plane principal stresses have the same sign , i.e., they are both tensile or both compressive, then failure will occur out of the plane , and from Eq. 9–13 ,

t max abs =

If instead the in-plane principal stresses are of opposite signs , then failure occurs in the plane, and from Eq. 9–14 ,

t max abs =

s1 - s2 2

Using these equations and Eq. 10–26 , the maximum-shear-stress theory for plane stress can be expressed for any two in-plane principal stresses s1 and s2 by the following criteria:

!s1 ! = sY !s2 ! = sY

r s1 , s2 have same signs (10–27)

!s1 - s2 ! = sY} s1, s2 have opposite signs

A graph of these equations is given in Fig. 10–28 . Clearly, if any point of the material is subjected to plane stress, and its in-plane principal stresses are represented by a coordinate ( s1 , s2 ) plotted on the boundary or outside the shaded hexagonal area shown in this figure, the material will yield at the point and failure is said to occur.

Axial tension

(a)

(b)

90! A(0, 0)

savg " sY

2 #max "

s1 " sYs2 " 0

45!

y ¿

(c)

x ¿

tmax " sY

2 savg " sY

Fig. 10–27

Maximum-shear-stress theory

$sY

%sY

$sY

Fig. 10–28

526 CHAPTER 10 STRAIN TRANSFORMATION

Maximum-Distortion-Energy Theory. It was stated in Sec. 3.5 that an external loading will deform a material, causing it to store energy internally throughout its volume. The energy per unit volume of material is called the strain-energy density , and if the material is subjected to a uniaxial stress the strain-energy density, defined by Eq. 3–6 , becomes

u = 1 2

sP (10–28)

If the material is subjected to triaxial stress, Fig. 10–29 a , then each principal stress contributes a portion of the total strain-energy density, so that

u = 1 2

s1 P1 + 1 2

s2 P2 + 1 2

s3 P3

Furthermore, if the material behaves in a linear-elastic manner, then Hooke’s law applies. Therefore, substituting Eq. 10–18 into the above equation and simplifying, we get

u = 1

2E 3s1 2 + s22 + s32 - 2n(s1 s2 + s1 s3 + s3 s2)4 (10–29)

This strain-energy density can be considered as the sum of two parts, one part representing the energy needed to cause a volume change of the element with no change in shape, and the other part representing the energy needed to distort the element. Specifically, the energy stored in the element as a result of its volume being changed is caused by application of the average principal stress, savg = (s1 + s2 + s3)>3, since this stress causes equal principal strains in the material, Fig. 10–29 b . The remaining portion of the stress, (s1 - savg), (s2 - savg), (s3 - savg), causes the energy of distortion, Fig. 10–29 c .

Experimental evidence has shown that materials do not yield when subjected to a uniform (hydrostatic) stress, such as savg discussed above. As a result, in 1904, M. Huber proposed that yielding in a ductile material occurs when the distortion energy per unit volume of the material equals or exceeds the distortion energy per unit volume of the same material when it is subjected to yielding in a simple tension test. This theory is called the maximum-distortion-energy theory , and since it was later redefined independently by R. von Mises and H. Hencky, it sometimes also bears their names.

To obtain the distortion energy per unit volume, we will substitute the stresses (s1 - savg), (s2 - savg), and (s3 - savg) for s1, s2, and s3, respectively, into Eq. 10–29 , realizing that savg = (s1 + s2 + s3)>3. Expanding and simplifying, we obtain

ud = 1 + n

6E 3(s1 - s2)2 + (s2 - s3)2 + (s3 - s1)24

(a)

s2s1

! (b)

savg

savg savg

(c)

(s2 ! savg)

(s3 ! savg)

(s1 ! savg)

Fig. 10–29

10.7 THEORIES OF FAILURE 527

In the case of plane stress , s3 = 0, and this equation reduces to

ud = 1 + n

3E 1s1 2 - s1s2 + s2 22

For a uniaxial tension test, s1 = sY , s2 = s3 = 0, and so

(ud)Y = 1 + n

3E sY

Since the maximum-distortion-energy theory requires ud = (ud)Y , then for the case of plane or biaxial stress, we have

s1 2 - s1s2 + s2 2 = sY 2 (10–30)

This is the equation of an ellipse, Fig. 10–30 . Thus, if a point in the material is stressed such that (s1, s2) is plotted on the boundary or outside the shaded area, the material is said to fail.

A comparison of the above two failure criteria is shown in Fig. 10–31 . Note that both theories give the same results when the principal stresses are equal, i.e., s1 = s2 = sY , or when one of the principal stresses is zero and the other has a magnitude of sY . If the material is subjected to pure shear, t, then the theories have the largest discrepancy in predicting failure. The stress coordinates of these points on the curves can be determined by considering the element shown in Fig. 10–32 a . From the associated Mohr’s circle for this state of stress, Fig. 10–32 b , we obtain principal stresses s1 = t and s2 = -t. Thus, with s1 = -s2 , then from Eq. 10–27 , the maximum-shear-stress theory gives (sY >2, -sY >2), and from Eq. 10–30 , the maximum-distortion-energy theory gives (sY >23 , -sY >23), Fig. 10–31 .

Actual torsion tests, used to develop a condition of pure shear in a ductile specimen, have shown that the maximum-distortion-energy theory gives more accurate results for pure-shear failure than the maximum-shear-stress theory. In fact, since (sY >13)>(sY >2) = 1.15, the shear stress for yielding of the material, as given by the maximum- distortion-energy theory, is 15% more accurate than that given by the maximum-shear-stress theory.

Maximum-distortion-energy theory

!sY

Fig. 10–30

Pure shear s2

!sY

!sY(!sY,!sY)

(sY, sY)

! sY 3

,sY 3

! sY 2

,sY 2

Fig. 10–31

(a)

(b)

90"

A (t, 0)

s2 # !t s1 # t s

Fig. 10–32

528 CHAPTER 10 STRAIN TRANSFORMATION

Brittle Materials Maximum-Normal-Stress Theory. It was previously stated that brittle materials, such as gray cast iron, tend to fail suddenly by fracture with no apparent yielding. In a tension test , the fracture occurs when the normal stress reaches the ultimate stress sult , Fig. 10–33 a . Also, brittle fracture occurs in a torsion test due to tension since the plane of fracture for an element is at 45° to the shear direction, Fig. 10–33 b . The fracture surface is therefore helical as shown. * Experiments have further shown that during torsion the material’s strength is somewhat unaffected by the presence of the associated principal compressive stress being at right angles to the principal tensile stress. Consequently, the tensile stress needed to fracture a specimen during a torsion test is approximately the same as that needed to fracture a specimen in simple tension. Because of this, the maximum-normal-stress theory states that a brittle material will fail when the maximum tensile stress, s1 , in the material reaches a value that is equal to the ultimate normal stress the material can sustain when it is subjected to simple tension.

If the material is subjected to plane stress , we require that

!s1 ! = sult (10–31) !s2 ! = sult

These equations are shown graphically in Fig. 10–34 . Therefore, if the stress coordinates (s1, s2) at a point in the material fall on the boundary or outside the shaded area, the material is said to fracture. This theory is generally credited to W. Rankine, who proposed it in the mid-1800s. Experimentally it has been found to be in close agreement with the behavior of brittle materials that have stress–strain diagrams that are similar in both tension and compression.

Mohr’s Failure Criterion. In some brittle materials tension and compression properties are different . When this occurs a criterion based on the use of Mohr’s circle may be used to predict failure. This method was developed by Otto Mohr and is sometimes referred to as Mohr’s failure criterion . To apply it, one first performs three tests on the material. A uniaxial tensile test and uniaxial compressive test are used to determine the ultimate tensile and compressive stresses (sult)t and (sult)c, respectively. Also a torsion test is performed to determine the material’s ultimate shear stress tult . Mohr’s circle for each of these stress conditions is then plotted as shown in Fig. 10–35 . These three circles are contained

Failure of a brittle material in tension

(a)

45!

Failure of a brittle material in torsion

(b)

45!

Fig. 10–33

Maximum-normal-stress theory

sult

"sult

"sult s1

Fig. 10–34 * A stick of blackboard chalk fails in this way when its ends are twisted with the fingers.

10.7 THEORIES OF FAILURE 529

in a “failure envelope” indicated by the extrapolated colored curve that is drawn tangent to all three circles. If a plane-stress condition at a point is represented by a circle that has a point of tangency with the envelope, or if it extends beyond the envelope’s boundary, then failure is said to occur.

We may also represent this criterion on a graph of principal stresses s1 and s2 . This is shown in Fig. 10–36 . Here failure occurs when the absolute value of either one of the principal stresses reaches a value equal to or greater than (sult)t or (sult)c or in general, if the state of stress at a point defined by the stress coordinates (s1, s2) is plotted on the boundary or outside the shaded area.

Either the maximum-normal-stress theory or Mohr’s failure criterion can be used in practice to predict the failure of a brittle material. However, it should be realized that their usefulness is quite limited. A tensile fracture occurs very suddenly, and its initiation generally depends on stress concentrations developed at microscopic imperfections of the material such as inclusions or voids, surface indentations, and small cracks. Since each of these irregularities varies from specimen to specimen, it becomes difficult to specify fracture on the basis of a single test.

Failure envelope

s (sult)t(sult)c

tult

Fig. 10–35

Mohr’s failure criterion

(sult)t

(sult)c

Fig. 10–36

Important Points

• If a material is ductile , failure is specified by the initiation of yielding , whereas if it is brittle , it is specified by fracture .

• Ductile failure can be defined when slipping occurs between the crystals that compose the material. This slipping is due to shear stress and the maximum-shear-stress theory is based on this idea.

• Strain energy is stored in a material when it is subjected to normal stress. The maximum-distortion-energy theory depends on the strain energy that distorts the material, and not the part that increases its volume.

• The fracture of a brittle material is caused only by the maximum tensile stress in the material, and not the compressive stress. This is the basis of the maximum-normal-stress theory , and it is applicable if the stress-strain diagram is similar in tension and compression.

• If a brittle material has a stress-strain diagram that is different in tension and compression, then Mohr’s failure criterion may be used to predict failure.

• Due to material imperfections, tensile fracture of a brittle material is difficult to predict , and so theories of failure for brittle materials should be used with caution.

530 CHAPTER 10 STRAIN TRANSFORMATION

The solid cast-iron shaft shown in Fig. 10–37 a is subjected to a torque of T = 400 lb # ft. Determine its smallest radius so that it does not fail according to the maximum-normal-stress theory. A specimen of cast iron, tested in tension, has an ultimate stress of (sult)t = 20 ksi.

EXAMPLE 10.12

T ! 400 lb"ft

(a)

s1s2

(b)

tmax

#tmax

Fig. 10–37

SOLUTION The maximum or critical stress occurs at a point located on the surface of the shaft. Assuming the shaft to have a radius r , the shear stress is

tmax = Tc J

= (400 lb # ft)(12 in.>ft)r

(p>2)r 4 = 3055.8 lb # in.r 3 Mohr’s circle for this state of stress (pure shear) is shown in Fig. 10–37 b . Since R = tmax, then

s1 = -s2 = tmax = 3055.8 lb # in.

r 3

The maximum-normal-stress theory, Eq. 10–31 , requires

!s1 ! … sult

3055.8 lb # in.

r 3 … 20 000 lb>in2

Thus, the smallest radius of the shaft is determined from

3055.8 lb # in.

r 3 = 20 000 lb>in2

r = 0.535 in. Ans.

10.7 THEORIES OF FAILURE 531

EXAMPLE 10.13

The solid shaft shown in Fig. 10–38 a has a radius of 0.5 in. and is made of steel having a yield stress of sY = 36 ksi. Determine if the loadings cause the shaft to fail according to the maximum-shear-stress theory and the maximum-distortion-energy theory.

SOLUTION The state of stress in the shaft is caused by both the axial force and the torque. Since maximum shear stress caused by the torque occurs in the material at the outer surface, we have

sx = P A

= -15 kip

p(0.5 in.)2 = -19.10 ksi

txy = Tc J

= 3.25 kip # in. (0.5 in.)

p 2(0.5 in.)

4 = 16.55 ksi

The stress components are shown acting on an element of material at point A in Fig. 10–38 b . Rather than using Mohr’s circle, the principal stresses can also be obtained using the stress-transformation Eq. 9–5 .

s1,2 = sx + sy

2 { B asx - sy 2 b2 + txy2

= -19.10 + 0

2 { B a -19.10 - 02 b2 + (16.55)2

= -9.55 { 19.11 s1 = 9.56 ksi s2 = -28.66 ksi

Maximum-Shear-Stress Theory. Since the principal stresses have opposite signs , then from Sec. 9.5 , the absolute maximum shear stress will occur in the plane, and therefore, applying the second of Eqs. 10–27 , we have

!s1 - s2 ! … sY

! 9.56 - (-28.66) ! … ?

38.2 7 36

Thus, shear failure of the material will occur according to this theory.

Maximum-Distortion-Energy Theory. Applying Eq. 10–30 , we have 1s12 - s1 s2 + s222 … sY 2 3(9.56)2 - (9.56)(-28.66) + (-28.66)24 …? (36)2

1187 … 1296 Using this theory, failure will not occur.

15 kip

(a)

3.25 kip!in.0.5 in.

(b)

16.55 ksi

19.10 ksi

Fig. 10–38

532 CHAPTER 10 STRAIN TRANSFORMATION

*10–68. The principal plane stresses acting on a differential element are shown. If the material is machine steel having a yield stress of sY = 700 MPa, determine the factor of safety with respect to yielding if the maximum-shear-stress theory is considered.

50 MPa

80 MPa

Prob. 10–68

10–69. The short concrete cylinder having a diameter of 50 mm is subjected to a torque of 500 N # m and an axial compressive force of 2 kN. Determine if it fails according to the maximum-normal-stress theory. The ultimate stress of the concrete is sult = 28 MPa.

500 N!m

2 kN

Prob. 10–69

PROBLEMS

10–59. A material is subjected to plane stress. Express the distortion-energy theory of failure in terms of sx, sy, and txy.

*10–60. A material is subjected to plane stress. Express the maximum-shear-stress theory of failure in terms of sx, sy, and txy. Assume that the principal stresses are of different algebraic signs.

10–61. The yield stress for a zirconium-magnesium alloy is s Y = 15.3 ksi. If a machine part is made of this material and a critical point in the material is subjected to in-plane principal stresses s 1 and s 2 = −0.5s 1 , determine the magnitude of s 1 that will cause yielding according to the maximum-shear-stress theory.

10–62. Solve Prob. 10–61 using the maximum-distortion energy theory.

10–63. An aluminum alloy is to be used for a drive shaft such that it transmits 25 hp at 1500 rev>min. Using a factor of safety of 2.5 with respect to yielding, determine the smallest-diameter shaft that can be selected based on the maximum-distortion-energy theory. sY = 3.5 ksi.

*10–64. If a shaft is made of a material for which s Y = 50 ksi, determine the torsional shear stress required to cause yielding using the maximum-distortion-energy theory.

10–65. Solve Prob. 10–64 using the maximum-shear- stress theory.

10–66. Derive an expression for an equivalent torque Te that, if applied alone to a solid bar with a circular cross section, would cause the same energy of distortion as the combination of an applied bending moment M and torque T .

10–67. Derive an expression for an equivalent bending moment Me that, if applied alone to a solid bar with a circular cross section, would cause the same energy of distortion as the combination of an applied bending moment M and torque T .

10.7 THEORIES OF FAILURE 533

10–75. The components of plane stess at a critical point on an A-36 steel shell are shown. Determine if failure (yielding) has occurred on the basis of the maximum-shear-stress theory.

*10–76. The components of plane stress at a critical point on an A-36 steel shell are shown. Determine if failure (yielding) has occurred on the basis of the maximum-distortion-energy theory.

60 MPa

70 MPa

40 MPa

Probs. 10–75/76

10–77. If the A-36 steel pipe has outer and inner diameters of 30 mm and 20 mm, respectively, determine the factor of safety against yielding of the material at point A according to the maximum-shear-stress theory.

10–78. If the A-36 steel pipe has an outer and inner diameter of 30 mm and 20 mm, respectively, determine the factor of safety against yielding of the material at point A according to the maximum-distortion-energy theory.

150 mm

100 mm

200 mm

900 N

Probs. 10–77/78

10–70. Derive an expression for an equivalent bending moment Me that, if applied alone to a solid bar with a circular cross section, would cause the same maximum shear stress as the combination of an applied moment M and torque T . Assume that the principal stresses are of opposite algebraic signs.

10–71. The plate is made of hard copper, which yields at sY = 105 ksi. Using the maximum-shear-stress theory, determine the tensile stress sx that can be applied to the plate if a tensile stress sy = 0.5sx is also applied.

*10–72. Solve Prob. 10–71 using the maximum-distortion energy theory.

sy ! 0.5sx

Probs. 10–71/72

10–73. If the 2-in.-diameter shaft is made from brittle material having an ultimate strength of sult = 50 ksi, for both tension and compression, determine if the shaft fails according to the maximum-normal-stress theory. Use a factor of safety of 1.5 against rupture.

10–74. If the 2-in.-diameter shaft is made from cast iron having tensile and compressive ultimate strengths of 1sult2t = 50 ksi and 1sult2c = 75 ksi, respectively, determine if the shaft fails in accordance with Mohr’s failure criterion.

30 kip

4 kip · ft

Probs. 10–73/74

534 CHAPTER 10 STRAIN TRANSFORMATION

*10–84. A bar with a circular cross-sectional area is made of SAE 1045 carbon steel having a yield stress of sY = 150 ksi. If the bar is subjected to a torque of 30 kip # in. and a bending moment of 56 kip # in., determine the required diameter of the bar according to the maximum-distortion-energy theory. Use a factor of safety of 2 with respect to yielding.

10–85. The state of stress acting at a critical point on a machine element is shown in the figure. Determine the smallest yield stress for a steel that might be selected for the part, based on the maximum-shear-stress theory.

8 ksi

4 ksi

10 ksi

Prob. 10–85

10–86. The principal stresses acting at a point on a thin- walled cylindrical pressure vessel are s1 = pr> t, s2 = pr>2t, and s3 = 0. If the yield stress is sY , determine the maximum value of p based on (a) the maximum-shear-stress theory and (b) the maximum-distortion-energy theory.

10–87. If a solid shaft having a diameter d is subjected to a torque T and moment M , show that by the maximum- shear-stress theory the maximum allowable shear stress is tallow = (16>pd 3)2M2 + T2. Assume the principal stresses to be of opposite algebraic signs.

*10–88. If a solid shaft having a diameter d is subjected to a torque T and moment M , show that by the maximum- normal-stress theory the maximum allowable principal stress is sallow = (16>pd 3)(M + 2M2 + T2).

T T MM

Probs. 10–87/88

10–79. The yield stress for heat-treated beryllium copper is sY = 130 ksi. If this material is subjected to plane stress and elastic failure occurs when one principal stress is 145 ksi, what is the smallest magnitude of the other principal stress? Use the maximum-distortion-energy theory.

*10–80. The yield stress for a uranium alloy is s Y = 160 MPa. If a machine part is made of this material and a critical point in the material is subjected to plane stress, such that the principal stresses are s 1 and s 2 = 0.25s 1 , determine the magnitude of s 1 that will cause yielding according to the maximum-distortion energy theory.

10–81. Solve Prob. 10–80 using the maximum-shear-stress theory.

10–82. The state of stress acting at a critical point on the seat frame of an automobile during a crash is shown in the figure. Determine the smallest yield stress for a steel that can be selected for the member, based on the maximum-shear-stress theory.

10–83. Solve Prob. 10–82 using the maximum-distortion- energy theory.

25 ksi

80 ksi

Probs. 10–82/83

10.7 THEORIES OF FAILURE 535

10–91. The internal loadings at a critical section along the steel drive shaft of a ship are calculated to be a torque of 2300 lb # ft, a bending moment of 1500 lb # ft, and an axial thrust of 2500 lb. If the yield points for tension and shear are sY = 100 ksi and tY = 50 ksi, respectively, determine the required diameter of the shaft using the maximum-shear- stress theory.

2500 lb

2300 lb!ft 1500 lb!ft

Prob. 10–91

*10–92. The shaft consists of a solid segment AB and a hollow segment BC , which are rigidly joined by the coupling at B . If the shaft is made from A-36 steel, determine the maximum torque T that can be applied according to the maximum-shear-stress theory. Use a factor of safety of 1.5 against yielding.

10–93. The shaft consists of a solid segment AB and a hollow segment BC , which are rigidly joined by the coupling at B . If the shaft is made from A-36 steel, determine the maximum torque T that can be applied according to the maximum-distortion-energy theory. Use a factor of safety of 1.5 against yielding.

80 mm

100 mm 80 mm

Probs. 10–92/93

10–89. The gas tank has an inner diameter of 1.50 m and a wall thickness of 25 mm. If it is made from A-36 steel and the tank is pressured to 5 MPa, determine the factor of safety against yielding using (a) the maximum-shear-stress theory, and (b) the maximum-distortion-energy theory.

Prob. 10–89

10–90. The gas tank is made from A-36 steel and has an inner diameter of 1.50 m. If the tank is designed to withstand a pressure of 5 MPa, determine the required minimum wall thickness to the nearest millimeter using (a) the maximum- shear-stress theory, and (b) maximum-distortion-energy theory. Apply a factor of safety of 1.5 against yielding.

Prob. 10–90

536 CHAPTER 10 STRAIN TRANSFORMATION

CHAPTER REVIEW

Px! = Px + Py

2 +

Px - Py 2

cos 2u + gxy

2 sin 2u

Py! = Px + Py

2 -

Px - Py 2

cos 2u - gxy

2 sin 2u

gx!y!

2 = - ¢ Px - Py

2 ≤ sin 2u + gxy

2 cos 2u

P1,2 = Px + Py

2 { B a Px - Py2 b2 + agxy2 b2

in@plane max

2 = B a Px - Py2 b2 + agxy2 b2

Pavg = Px + Py

When an element of material is subjected to deformations that only occur in a single plane, then it undergoes plane strain. If the strain components Px, Py, and gxy are known for a specified orientation of the element, then the strains acting for some other orientation of the element can be determined using the plane-strain transformation equations. Likewise, the principal normal strains and maximum in-plane shear strain can be determined using transformation equations.

Strain transformation problems can also be solved in a semi-graphical manner using Mohr’s circle. To draw the circle, the P and g>2 axes are established and the center of the circle C [(Px + Py)>2, 0] and the “reference point” A (Px , gxy>2) are plotted. The radius of the circle extends between these two points and is determined from trigonometry.

R ! 2 2

"2 Px # Py gxy

2 Px # Py

gxy

Px 2

Px " PyPavg !

u ! 0$

If P1 and P2 have the same sign then the absolute maximum shear strain will be out of plane.

In the case of plane strain, the absolute maximum shear strain will be equal to the maximum in-plane shear strain provided the principal strains P1 and P2 have opposite signs.

g max abs = P1

g max in@plane = P1 - P2

g max abs = P1 - P2

CHAPTER REVIEW 537

If the material is subjected to triaxial stress, then the strain in each direction is influenced by the strain produced by all three stresses. Hooke’s law then involves the material properties E and n .

Px = 1 E

[sx - n(sy + sz)]

Py = 1 E

[sy - n(sx + sz)]

Pz = 1 E

[sz - n(sx + sy)]

G = E

2(1 + n)

e = 1 - 2n

E (sx + sy + sz)

k = E

3(1 - 2n)

If E and n are known, then G can be determined.

If the principal stresses at a critical point in the material are known, then a theory of failure can be used as a basis for design.

Ductile materials fail in shear , and here the maximum-shear-stress theory or the maximum-distortion-energy theory can be used to predict failure. Both of these theories make comparison to the yield stress of a specimen subjected to a uniaxial tensile stress.

Brittle materials fail in tension , and so the maximum-normal-stress theory or Mohr’s failure criterion can be used to predict failure. Here comparisons are made with the ultimate tensile stress developed in a specimen.

The dilatation is a measure of volumetric strain.

The bulk modulus is used to measure the stiffness of a volume of material.

538 CHAPTER 10 STRAIN TRANSFORMATION

REVIEW PROBLEMS

10–97. The components of plane stress at a critical point on a thin steel shell are shown. Determine if failure (yielding) has occurred on the basis of the maximum- distortion-energy theory. The yield stress for the steel is sY = 650 MPa.

340 MPa

55 MPa

65 MPa

Prob. 10–97

10–98. The 60° strain rosette is mounted on a beam. The following readings are obtained for each gauge: Pa = 600(10-6), Pb = -700(10-6), and Pc = 350(10-6). Determine ( a ) the in-plane principal strains and ( b ) the maximum in-plane shear strain and average normal strain. In each case show the deformed element due to these strains.

60!

Prob. 10–98

10–94. In the case of plane stress, where the in-plane principal strains are given by P 1 and P 2 , show that the third principal strain can be obtained from P 3 = −[ n >(1 − n )](P 1 + P 2 ), where n is Poisson’s ratio for the material.

10–95. The plate is made of material having a modulus of elasticity E = 200 GPa and Poisson’s ratio v = 13. Determine the change in width a , height b , and thickness t when it is subjected to the uniform distributed loading shown.

z x

a " 400 mm

t " 20 mm

b " 300 mm

2 MN/m

3 MN/m

Prob. 10–95

*10–96. The principal plane stresses acting at a point are shown in the figure. If the material is machine steel having a yield stress of sY = 500 MPa, determine the factor of safety with respect to yielding if the maximum-shear-stress theory is considered.

100 MPa

150 MPa

Prob. 10–96

REVIEW PROBLEMS 539

10–102. The state of strain at the point on the bracket has components Px = -130(10-6), Py = 280(10-6), gxy = 75(10-6). Use the strain-transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case specify the orientation of the element and show how the strains deform the element within the x – y plane.

Prob. 10–102

10–103. The state of plain strain on an element is Px = 400(10-6), Py = 200(10-6), and gxy = -300(10-6). Determine the equivalent state of strain, which represents (a) the principal strains, and (b) the maximum in-plane shear strain and the associated average normal strain. Specify the orientation of the corresponding element at the point with respect to the original element. Sketch the results on the element.

Pydy

Pxdx

gxy 2

Prob. 10–103

10–99. The state of strain at the point on the bracket has com- ponents Px = 350(10-6), Py = -860(10-6), gxy = 250(10-6). Use the strain-transformation equations to determine the equivalent in-plane strains on an element oriented at an angle of u = 45° clockwise from the original position. Sketch the deformed element within the x – y plane due to these strains.

Prob. 10–99

*10–100. The A-36 steel post is subjected to the forces shown. If the strain gauges a and b at point A give readings of Pa = 300(10-6) and Pb = 175(10-6), determine the magnitudes of P1 and P2.

c c

2 ft

1 in.

2 in.

1 in.

4 in.

a b 45!

Section c–cA

Prob. 10–100

10–101. A differential element is subjected to plane strain that has the following components; P x = 950(10 −6 ), P y = 420(10 −6 ), gxy = -325(10-6). Use the strain- transformation equations and determine (a) the principal strains and (b) the maximum in plane shear strain and the associated average strain. In each case specify the orientation of the element and show how the strains deform the element.

Chapter 11

Beams are important structural members that are used to support roof and floor loadings.

541

11.1 Basis for Beam Design Beams are said to be designed on the basis of strength so that they can resist the internal shear and moment developed along their length. To design a beam in this way requires application of the shear and flexure formulas provided the material is homogeneous and has linear elastic behavior. Although some beams may also be subjected to an axial force, the effects of this force are often neglected in design since the axial stress is generally much smaller than the stress developed by shear and bending.

CHAPTER OBJECTIVES

■ In this chapter, we will discuss how to design a beam so that it is able to resist both bending and shear loads. Specifically, methods used for designing prismatic beams and determining the shape of fully stressed beams will be developed. At the end of the chapter, we will consider the design of shafts based on the resistance of both bending and torsional moments.

Design of Beams and Shafts

542 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

w P

txy sx

Fig. 11–1

Whenever large shear loads occur on a beam it is important to use stiffeners such as at A , in order to prevent any localized failure such as crimping of the beam fl anges.

As shown in Fig. 11–1 , the external loadings on a beam will create additional stresses in the beam directly under the load . Notably, a compressive stress sy will be developed, in addition to the bending stress sx and shear stress txy discussed previously. Using advanced methods of analysis, as treated in the theory of elasticity, it can be shown that sy diminishes rapidly throughout the beam’s depth, and for most beam span-to-depth ratios used in engineering practice, the maximum value of sy generally represents only a small percentage compared to the bending stress sx, that is, sx W sy. Furthermore, the direct application of concentrated loads is generally avoided in beam design. Instead, bearing plates are used to spread these loads more evenly onto the surface of the beam.

Although beams are designed mainly for strength, they must also be braced properly along their sides so that they do not buckle or suddenly become unstable. Furthermore, in some cases beams must be designed to resist a limited amount of deflection , as when they support ceilings made of brittle materials such as plaster. Methods for finding beam deflections will be discussed in Chapter 12 , and limitations placed on beam buckling are often discussed in codes related to structural or mechanical design.

Since the shear and flexure formulas are used for beam design, we will discuss the general results obtained when these equations are applied to various points in a cantilevered beam that has a rectangular cross section and supports a load P at its end, Fig. 11–2 a .

In general, at an arbitrary section a – a along the beam’s axis, Fig. 11–2 b , the internal shear V and moment M are developed from a parabolic shear- stress distribution, and a linear normal-stress distribution, Fig. 11–2 c . As a result, the stresses acting on elements located at points 1 through 5 along the section will be as shown in Fig. 11–2 d . Note that elements 1 and 5 are subjected only to the maximum normal stress, whereas element 3, which is on the neutral axis, is subjected only to the maximum shear stress. The intermediate elements 2 and 4 resist both normal and shear stress.

In each case the state of stress can be transformed into principal stresses , using either the stress-transformation equations or Mohr’s circle. The results are shown in Fig. 11–2 e . Here each successive element, 1 through 5, undergoes a counterclockwise orientation. Specifically, relative to element 1, considered to be at the 0° position, element 3 is oriented at 45° and element 5 is oriented at 90°.

11.1 BASIS FOR BEAM DESIGN 543

If this analysis is extended to many vertical sections along the beam other than a – a , a profile of the results can be represented by curves called stress trajectories . Each of these curves indicated the direction of a principal stress having a constant magnitude. Some of these trajectories are shown for the cantilevered beam in Fig. 11–3 . Here the solid lines represent the direction of the tensile principal stresses and the dashed lines represent the direction of the compressive principal stresses. As expected, the lines intersect the neutral axis at 45° angles (like element 3) and the solid and dashed lines will intersect at 90° because the principal stresses are always 90° apart. Knowing the direction of these lines can help engineers decide where to reinforce a beam if it is made of brittle material so that it does not crack or become unstable.

Stress trajectories for cantilevered beam

Fig. 11–3

(a)

(b)

M2 3 4

Bending stress distribution

Shear stress distribution

(c)

(d)

x–y stress components

Principal stresses

(e)

Fig. 11–2

544 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

11.2 Prismatic Beam Design Most beams are made of ductile materials and when this is the case it is generally not necessary to plot the stress trajectories for the beam. Instead, it is simply necessary to be sure the actual bending stress and shear stress in the beam do not exceed allowable bending and shear stress for the material as defined by structural or mechanical codes. In the majority of cases the suspended span of the beam will be relatively long, so that the internal moments become large. When this occurs the engineer will first consider a design based upon bending and then check the shear strength. A bending design requires a determination of the beam’s section modulus , a geometric property which is the ratio of I and c , that is, S = I>c. Using the flexure formula, s = Mc>I, we have Sreq>d =

Mmax sallow

(11–1)

Here M is determined from the beam’s moment diagram, and the allowable bending stress, sallow, is specified in a design code. In many cases the beam’s as yet unknown weight will be small and can be neglected in comparison with the loads the beam must carry. However, if the additional moment caused by the weight is to be included in the design, a selection for S is made so that it slightly exceeds Sreq>d.

Once Sreq>d is known, if the beam has a simple cross-sectional shape, such as a square, a circle, or a rectangle of known width-to-height proportions, its dimensions can be determined directly from Sreq>d, since Sreq>d = I>c. However, if the cross section is made from several elements, such as a wide-flange section, then an infinite number of web and flange dimensions can be determined that satisfy the value of Sreq>d. In practice, however, engineers choose a particular beam meeting the requirement that S 7 Sreq>d from a handbook that lists the standard shapes available from manufacturers. Often several beams that have the same section modulus can be selected from these tables. If deflections are not restricted, usually the beam having the smallest cross-sectional area is chosen, since it is made of less material and is therefore both lighter and more economical than the others.

A B

The two fl oor beams are connected to the beam AB , which transmits the load to the columns of this building frame. For design, all the connections can be considered to act as pins.

11.2 PRISMATIC BEAM DESIGN 545

Once the beam has been selected, the shear formula tallow Ú VQ>It can then be used to check that the allowable shear stress is not exceeded. Often this requirement will not present a problem. However, if the beam is “short” and supports large concentrated loads, the shear-stress limitation may dictate the size of the beam. This limitation is particularly important in the design of wood beams, because wood tends to split along its grain due to shear (see Fig. 7–10 e ).

Fabricated Beams. Since beams are often made of steel or wood, we will now discuss some of the tabulated properties of beams made from these materials.

Steel Sections. Most manufactured steel beams are produced by rolling a hot ingot of steel until the desired shape is formed. These so-called rolled shapes have properties that are tabulated in the American Institute of Steel Construction (AISC) manual. A representative listing for wide-flange beams taken from this manual is given in Appendix B . As noted in this appendix, the wide-flange shapes are designated by their depth and weight per unit length; for example, W18 * 46 indicates a wide-flange cross section (W) having a depth of 18 in. and a weight of 46 lb > ft, Fig. 11–4 . For any given section, the weight per length, dimensions, cross-sectional area, moment of inertia, and section modulus are reported. Also included is the radius of gyration r , which is a geo metric property related to the section’s buckling strength. This will be discussed in Chapter 13 . Appendix B and the AISC Manual also list data for other members such as channels and angles.

18 in. 0.360 in.

6 in.

W18 ! 46

0.605 in.

Fig. 11–4

Typical profi le view of a steel wide-fl ange beam

Wood Sections. Most beams made of wood have rectangular cross sections because such beams are easy to manufacture and handle. Manuals, such as that of the National Forest Products Association, list the dimensions of lumber often used in the design of wood beams. Often, both the nominal and actual dimensions are reported. Lumber is identified by its nominal dimensions, such as 2 * 4 (2 in. by 4 in.); however, its actual or “dressed” dimensions are smaller, being 1.5 in. by 3.5 in. The reduction in the dimensions occurs in order to obtain a smooth surface from lumber that is rough sawn. Obviously, the actual dimensions must be used whenever stress calculations are performed on wood beams.

The large shear force that occurs at the end reaction of this steel beam can cause localized buckling of the beam’s fl anges. To avoid this, a “stiffener” A is placed along the web to keep the fl anges straight.

546 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

Built-up Sections. A built-up section is constructed from two or more parts joined together to form a single unit. Since Sreq’d = M >sallow , the capacity of the beam to resist a moment will vary directly with its section modulus, and since Sreq’d = I>c, then Sreq’d is increased if I is increased . In order to increase I, most of the material should be placed as far away from the neutral axis as practical. This, of course, is what makes a deep wide-flange beam so efficient in resisting a moment. For a very large load, however, an available rolled-steel section may not have a section modulus great enough to support the load. Rather than using several available beams, instead engineers will usually “build up” a beam made from plates and angles. A deep I-shaped section having this form is called a plate girder . For example, the steel plate girder in Fig. 11–5 has two flange plates that are either welded or, using angles, bolted to the web plate.

Wood beams are also “built up,” usually in the form of a box beam section, Fig. 11–6 a . They may be made having plywood webs and larger boards for the flanges. For very large spans, glulam beams are used. These members are made from several boards glue-laminated together to form a single unit, Fig. 11–6 b .

Just as in the case of rolled sections or beams made from a single piece, the design of built-up sections requires that the bending and shear stresses be checked. In addition, the shear stress in the fasteners, such as weld, glue, nails, etc., must be checked to be certain the beam acts as a single unit. The principles for doing this were outlined in Sec. 7.4 .

Welded Bolted

Steel plate girders

Fig. 11–5

Wooden box beam

(a)

Glulam beam

(b)

Fig. 11–6

Important Points

• Beams support loadings that are applied perpendicular to their axes. If they are designed on the basis of strength, they must resist allowable shear and bending stresses.

• The maximum bending stress in the beam is assumed to be much greater than the localized stresses caused by the application of loadings on the surface of the beam.

11.2 PRISMATIC BEAM DESIGN 547

Procedure for Analysis

Based on the previous discussion, the following procedure provides a rational method for the design of a beam on the basis of strength.

Shear and Moment Diagrams. • Determine the maximum shear and moment in the beam. Often this is done by constructing the beam’s

shear and moment diagrams. • For built-up beams, shear and moment diagrams are useful for identifying regions where the shear and

moment are excessively large and may require additional structural reinforcement or fasteners.

Bending Stress. • If the beam is relatively long, it is designed by finding its section modulus using the flexure formula,

Sreq>d = Mmax>sallow. • Once Sreq>d is determined, the cross-sectional dimensions for simple shapes can then be computed, since

Sreq>d = I>c. • If rolled-steel sections are to be used, several possible values of S may be selected from the tables in

Appendix B . Of these, choose the one having the smallest cross-sectional area, since this beam has the least weight and is therefore the most economical.

• Make sure that the selected section modulus, S , is slightly greater than Sreq>d , so that the additional moment created by the beam’s weight is considered.

Shear Stress. • Normally beams that are short and carry large loads, especially those made of wood, are first designed to

resist shear and then later checked against the allowable-bending-stress requirements. • Using the shear formula, check to see that the allowable shear stress is not exceeded; that is, use

tallow Ú V max Q>It. • If the beam has a solid rectangular cross section, the shear formula becomes tallow Ú 1.5 (V max>A ) (See

Eq. 2 of Example 7.2 .), and if the cross section is a wide flange , it is generally appropriate to assume that the shear stress is constant over the cross-sectional area of the beam’s web so that tallow Ú V max>A web , where A web is determined from the product of the web’s depth and its thickness. (See the note at the end of Example 7.3 .)

Adequacy of Fasteners. • The adequacy of fasteners used on built-up beams depends upon the shear stress the fasteners can

resist. Specifically, the required spacing of nails or bolts of a particular size is determined from the allowable shear flow, qallow = VQ>I, calculated at points on the cross section where the fasteners are located. (See Sec. 7.3 .)

548 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

A beam is to be made of steel that has an allowable bending stress of sallow = 24 ksi and an allowable shear stress of tallow = 14.5 ksi. Select an appropriate W shape that will carry the loading shown in Fig. 11– 7 a .

SOLUTION Shear and Moment Diagrams. The support reactions have been calculated, and the shear and moment diagrams are shown in Fig. 11–7 b . From these diagrams, V max = 30 kip and Mmax = 120 kip # ft. Bending Stress. The required section modulus for the beam is determined from the flexure formula,

Sreq>d = Mmax sallow

= 120 kip # ft (12 in.>ft)

24 kip>in2 = 60 in3 Using the table in Appendix B , the following beams are adequate:

W18 * 40 S = 68.4 in3 W16 * 45 S = 72.7 in3 W14 * 43 S = 62.7 in3 W12 * 50 S = 64.7 in3 W10 * 54 S = 60.0 in3

W8 * 67 S = 60.4 in3

The beam having the least weight per foot is chosen, i.e.,

W18 * 40

The actual maximum moment Mmax, which includes the weight of the beam, can be calculated and the adequacy of the selected beam can be checked. In comparison with the applied loads, however, the beam’s weight, (0.040 kip>ft)(18 ft) = 0.720 kip, will only slightly increase Sreq>d. In spite of this,

Sreq>d = 60 in3 6 68.4 in3 OK

Shear Stress. Since the beam is a wide-flange section , the average shear stress within the web will be considered. (See Example 7.3 .) Here the web is assumed to extend from the very top to the very bottom of the beam. From Appendix B , for a W18 * 40, d = 17.90 in., tw = 0.315 in. Thus,

tavg = V max A w

= 30 kip

(17.90 in.)(0.315 in.) = 5.32 ksi 6 14.5 ksi OK

Use a W18 * 40. Ans.

EXAMPLE 11.1

(a)

40 kip 20 kip

6 ft6 ft6 ft

(b)

10 kip 50 kip

V (kip)

M (kip!ft)

"30

20 x (ft)

x (ft)

"120

8 ft

6 ft6 ft6 ft

40 kip 20 kip

Fig. 11–7

11.2 PRISMATIC BEAM DESIGN 549

The laminated wooden beam shown in Fig. 11–8 a supports a uniform distributed loading of 12 kN>m. If the beam is to have a height-to-width ratio of 1.5, determine its smallest width. The allowable bending stress is sallow = 9 MPa and the allowable shear stress is tallow = 0.6 MPa. Neglect the weight of the beam.

SOLUTION Shear and Moment Diagrams. The support reactions at A and B have been calculated and the shear and moment diagrams are shown in Fig. 11–8 b . Here V max = 20 kN, Mmax = 10.67 kN # m. Bending Stress. Applying the flexure formula,

Sreq>d = Mmax sallow

= 10.67(103) N # m

9(106) N>m2 = 0.00119 m3 Assuming that the width is a , then the height is 1.5a, Fig. 11–8 a . Thus,

Sreq>d = I c

= 0.00119 m3 = 1

12(a)(1.5a) 3

(0.75a) a3 = 0.003160 m3

a = 0.147 m

Shear Stress. Applying the shear formula for rectangular sections (which is a special case of tmax = VQ>It , as shown in Example 7.2 ), we have

tmax = 1.5 V max

A = (1.5)

20(103) N (0.147 m)(1.5)(0.147 m)

= 0.929 MPa 7 0.6 MPa

EQUATION Since the design fails the shear criterion, the beam must be redesigned on the basis of shear.

tallow = 1.5 V max

600 kN>m2 = 1.5 20(103) N (a)(1.5a)

a = 0.183 m = 183 mm Ans.

This larger section will also adequately resist the normal stress.

EXAMPLE 11.2

1 m

(a)

12 kN/m

1.5a

3 mA B

!12

x (m)

(b)

16 kN32 kN

!16

M (kN"m)

V (kN)

10.67

1.33 m

12 kN/m

1 m 3 m

Fig. 11–8

550 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

The wooden T-beam shown in Fig. 11–9 a is made from two 200 mm * 30 mm boards. If the allowable bending stress is sallow = 12 MPa and the allowable shear stress is tallow = 0.8 MPa, determine if the beam can safely support the loading shown. Also, specify the maximum spacing of nails needed to hold the two boards together if each nail can safely resist 1.50 kN in shear.

EXAMPLE 11.3

200 mm

30 mm

_ y

2 m

(a)

1.5 kN 0.5 kN/m

C B

2 m

SOLUTION Shear and Moment Diagrams. The reactions on the beam are shown, and the shear and moment diagrams are drawn in Fig. 11–9 b . Here V max = 1.5 kN, Mmax = 2 kN # m. Bending Stress. The neutral axis (centroid) will be located from the bottom of the beam. Working in units of meters, we have

y = !yA

= (0.1 m)(0.03 m)(0.2 m) + 0.215 m(0.03 m)(0.2 m)

0.03 m(0.2 m) + 0.03 m(0.2 m) = 0.1575 m

Thus,

I = c 1 12

(0.03 m)(0.2 m)3 + (0.03 m)(0.2 m)(0.1575 m - 0.1 m)2 d + c 1

12 (0.2 m)(0.03 m)3 + (0.03 m)(0.2 m)(0.215 m - 0.1575 m)2 d

= 60.125(10-6) m4

Since c = 0.1575 m (not 0.230 m - 0.1575 m = 0.0725 m ), we require

sallow Ú Mmaxc

12(106) Pa Ú 2(103) N # m(0.1575 m)

60.125(10-6) m4 = 5.24(106) Pa OK

1.5

x (m)

x (m) (b)

0.5

M (kN"m)

V (kN)

2 m

1.5 kN

1 kN

0.5 kN/m

1.5 kN

2 m

Fig. 11–9

11.2 PRISMATIC BEAM DESIGN 551

Shear Stress. Maximum shear stress in the beam depends upon the magnitude of Q and t . It occurs at the neutral axis, since Q is a maximum there and the neutral axis is in the web, where the thickness t = 0.03 m is smallest for the cross section. For simplicity, we will use the rectangular area below the neutral axis to calculate Q , rather than a two-part composite area above this axis, Fig. 11–9 c . We have

Q = y !A ! = a 0.1575 m 2

b [(0.1575 m)(0.03 m)] = 0.372(10-3) m3 so that

tallow Ú V maxQ

800(103) Pa Ú 1.5(103) N[0.372(10-3)] m3

60.125(10-6) m4(0.03 m) = 309(103) Pa OK

Nail Spacing. From the shear diagram it is seen that the shear varies over the entire span. Since the nail spacing depends on the magnitude of shear in the beam, for simplicity (and to be conservative), we will design the spacing on the basis of V = 1.5 kN for region BC and V = 1 kN for region CD . Since the nails join the flange to the web, Fig. 11–9 d , we have

Q = y !A ! = (0.0725 m - 0.015 m)[(0.2 m)(0.03 m)] = 0.345(10-3) m3

The shear flow for each region is therefore

qBC = V BC Q

I =

1.5(103) N[0.345(10-3) m3]

60.125(10-6) m4 = 8.61 kN>m

qCD = V CDQ

I =

1(103) N[0.345(10-3) m3]

60.125(10-6) m4 = 5.74 kN>m

One nail can resist 1.50 kN in shear, so the maximum spacing becomes

sBC = 1.50 kN

8.61 kN>m = 0.174 m sCD =

1.50 kN 5.74 kN>m = 0.261 m

For ease of measuring, use

sBC = 150 mm Ans. sCD = 250 mm Ans.

0.1575 m

0.03 m

AN 0.0725 m

(c)

0.2 m

AN 0.0725 m

(d)

0.03 m

Fig. 11–9 (cont.)

552 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

FUNDAMENTAL PROBLEMS

F11–1. Determine the minimum dimension a to the nearest mm of the beam’s cross section to safely support the load. The wood has an allowable normal stress of sallow = 10 MPa and an allowable shear stress of tallow = 1 MPa .

1 m

6 kN6 kN

F11–1

F11–2. Determine the minimum diameter d to the nearest 18 in. of the rod to safely support the load. The rod is made of a material having an allowable normal stress of sallow = 20 ksi and an allowable shear stress of tallow = 10 ksi .

1.5 ft 1.5 ft

3 kip

3 kip!ft

F11–2

F11–3. Determine the minimum dimension a to the nearest mm of the beam’s cross section to safely support the load. The wood has an allowable normal stress of sallow = 12 MPa and an allowable shear stress of tallow = 1 .5 MPa .

1 m0.5 m

15 kN

A B

2 a

F11–3

F11–4. Determine the minimum dimension h to the nearest 1 8 in. of the beam’s cross section to safely support the load. The wood has an allowable normal stress of sallow = 2 ksi and an allowable shear stress of tallow = 200 psi .

6 ft

4 in.

1.5 kip/ft

A B

F11–4

F11–5. Determine the minimum dimension b to the nearest mm of the beam’s cross section to safely support the load. The wood has an allowable normal stress of sallow = 12 MPa and an allowable shear stress of tallow = 1.5 MPa .

A B

1 m 1 m 1 m 1 m

50 kN

5 kN!m 5 kN!m

F11–5

F11–6. Select the lightest W410-shaped section that can safely support the load. The beam is made of steel having an allowable normal stress of sallow = 150 MPa and an allowable shear stress of tallow = 75 MPa .

A B

1 m

150 kN

2 m

F11–6

11.2 PRISMATIC BEAM DESIGN 553

PROBLEMS

11–1. The simply supported beam is made of timber that has an allowable bending stress of s allow = 6.5 MPa and an allowable shear stress of t allow = 500 kPa. Determine its dimensions if it is to be rectangular and have a height-to- width ratio of 1.25 .

2 m 2 m4 m

8 kN/m

Prob. 11–1

11–2. Determine the minimum width of the beam to the nearest 14 in. that will safely support the loading of P = 8 kip. The allowable bending stress is s allow = 24 ksi and the allowable shear stress is t allow = 15 ksi.

11–3. Solve Prob. 11–2 if P = 10 kip.

6 ft 6 ft

A 6 in. B

Probs. 11–2/3

*11–4. The brick wall exerts a uniform distributed load of 1.20 kip> ft on the beam. If the allowable bending stress is s allow = 22 ksi and the allowable shear stress is t allow = 12 ksi, select the lightest wide-flange section with the shortest depth from Appendix B that will safely support the load. If there are several choices of equal weight, choose the one with the shortest height.

4 ft 6 ft10 ft

1.20 kip/ft

Prob. 11–4

11–5. Select the lightest-weight steel wide-flange beam from Appendix B that will safely support the machine loading shown. The allowable bending stress is sallow = 24 ksi and the allowable shear stress is tallow = 14 ksi.

5 kip

2 ft2 ft2 ft2 ft2 ft

5 kip 5 kip 5 kip

Prob. 11–5

11–6. The spreader beam AB is used to slowly lift the 3000-lb pipe that is centrally located on the straps at C and D . If the beam is a W 12 * 45, determine if it can safely support the load. The allowable bending stress is s allow = 22 ksi and the allowable shear stress is t allow = 12 ksi.

3 ft 3 ft6 ft 6 ft

3000 lb

12 in.

5 ft

A B

Prob. 11–6

11–7. Draw the shear and moment diagrams for the W12 * 14 beam and check if the beam will safely support the loading. Take s allow = 22 ksi and t allow = 12 ksi.

50 kip!ft

3 ft 12 ft

1.5 kip/ft

Prob. 11–7

554 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

11–11. The timber beam is to be loaded as shown. If the ends support only vertical forces, determine the greatest magnitude of P that can be applied. s allow = 25 MPa, t allow = 700 kPa.

4 m

150 mm

40 mm

30 mm

120 mm

A B

4 m P

Prob. 11–11

*11–12. The joists of a floor in a warehouse are to be selected using square timber beams made of oak. If each beam is to be designed to carry 90 lb>ft over a simply supported span of 25 ft, determine the dimension a of its square cross section to the nearest 14 in. The allowable bending stress is s allow = 4.5 ksi and the allowable shear stress is t allow = 125 psi.

25 ft

a a

90 lb/ft

Prob. 11–12

11–13. Select the lightest steel wide-flange beam from Appendix B that will safely support the loading shown. The allowable bending stress is sallow = 22 ksi and the allowable shear stress is tallow = 12 ksi. If there are several choices of equal weight, choose the one with the shortest height.

4 ft

4 kip 10 kip

6 kip

4 ft 4 ft 4 ft

Prob. 11–13

*11–8. The simply supported beam is made of timber that has an allowable bending stress of sallow = 1.20 ksi and an allowable shear stress of tallow = 100 psi . Determine its smallest dimensions to the nearest 18 in. if it is rectangular and has a height-to-width ratio of 1.5.

3 ft 3 ft

12 kip/ft

1.5 b

A B

Prob. 11–8

11–9. Select the lightest W360 shape section from Appendix B that can safely support the loading acting on the overhanging beam. The beam is made from steel having an allowable normal stress of s allow = 150 MPa and an allowable shear stress of t allow = 80 MPa.

11–10. Investigate if a W250 * 58 shape section can safely support the loading acting on the overhanging beam. The beam is made from steel having an allowable normal stress of s allow = 150 MPa and an allowable shear stress of t allow = 80 MPa.

50 kN

40 kN/m

4 m 2 m

Probs. 11–9/10

11.2 PRISMATIC BEAM DESIGN 555

11–14. Select the lightest-weight steel wide-flange beam from Appendix B that will safely support the loading shown. The allowable bending stress s allow = 24 ksi and the allowable shear stress of t allow = 14 ksi.

6 ft

15 kip/ft

6 ft

10 kip/ft 10 kip/ft

Prob. 11–14

11–15. Two acetyl plastic members are to be glued together and used to support the loading shown. If the allowable bending stress for the plastic is s allow = 13 ksi and the allowable shear stress is t allow = 4 ksi, determine the greatest load P that can be supported and specify the required shear stress capacity of the glue.

8 in.

6 in. 3 in.

3 in.

5 ft5 ft5 ft

Prob. 11–15

*11–16. If the cable is subjected to a maximum force of P = 50 kN, select the lightest W310 shape that can safely support the load. The beam is made from steel having an allowable normal stress of s allow = 150 MPa and an allowable shear stress of t allow = 85 MPa.

11–17. If the W360 * 45 beam is made from steel having an allowable normal stress of s allow = 150 MPa and an allowable shear stress of t allow = 85 MPa, determine the maximum cable force P that can safely be supported by the beam.

2 m 2 m

Probs. 11–16/17

11–18. If P = 800 lb, determine the minimum dimension a of the beam’s cross section to the nearest 18 in. to safely support the load. The wood species has an allowable normal stress of s allow = 1.5 ksi and an allowable shear stress of t allow = 150 psi.

11–19. If a = 3 in. and the wood has an allowable normal stress of s allow = 1.5 ksi, and an allowable shear stress of t allow = 150 psi, determine the maximum allowable value of P acting on the beam.

3 ft3 ft 3 ft

Probs. 11–18/19

556 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

11–23. The beam is constructed from three boards as shown. If each nail can support a shear force of 50 lb, determine the maximum spacing of the nails, s , s!, s" , for regions AB , BC , and CD , respectively.

1 in.

8 in.

1 in.

6 in.

1200 lb

s s ¿

800 lb

s ¿¿

5 ft5 ft5 ft

B C D

Prob. 11–23

*11–24. The shaft is supported by a smooth thrust bearing at A and smooth journal bearing at B . If P = 10 kN and the shaft is made from steel having an allowable normal stress of s allow = 150 MPa and an allowable shear stress of t allow = 85 MPa, determine the required minimum wall thickness t of the shaft to the nearest millimeter to safely support the load.

11–25. The circular hollow shaft is supported by a smooth thrust bearing at A and smooth journal bearing at B . If the shaft is made from steel having an allowable normal stress of s allow = 150 MPa and allowable shear stress of t allow = 85 MPa, determine the maximum allowable magnitude of the two forces P that can be applied to the shaft. The thickness of the shaft’s wall is t = 5 mm.

B t

1 m 1 m 1 m

P P

80 mm

Probs. 11–24/25

*11–20. The shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B . If P = 5 kN and the shaft is made from steel having an allowable normal stress of s allow = 150 MPa and an allowable shear stress of t allow = 85 MPa, determine the required minimum wall thickness t of the shaft to the nearest millimeter to safely support the load.

11–21. The shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B . If the shaft is made from steel having an allowable normal stress of s allow = 150 MPa and allowable shear stress of t allow = 85 MPa, determine the maximum allowable force P that can be applied to the shaft. The thickness of the shaft’s wall is t = 5 mm.

0.5 m1 m

P40 mm

Probs. 11–20/21

11–22. Determine the minimum depth h of the beam to the nearest 18 in. that will safely support the loading shown. The allowable bending stress is sallow = 21 ksi and the allowable shear stress is tallow = 10 ksi. The beam has a uniform thickness of 3 in.

A B

6 ft12 ft

4 kip/ft

Prob. 11–22

11.2 PRISMATIC BEAM DESIGN 557

11–26. Select the lightest-weight steel wide-flange beam from Appendix B that will safely support the loading shown. The allowable bending stress is sallow = 22 ksi and the allowable shear stress is tallow = 12 ksi .

5 kip

6 ft 12 ft

A B

18 kip ft

Prob. 11–26

11–27. The steel cantilevered T-beam is made from two plates welded together as shown. Determine the maximum loads P that can be safely supported on the beam if the allowable bending stress is s allow = 170 MPa and the allowable shear stress is t allow = 95 MPa.

2 m

150 mm 15 mm

150 mm

15 mm

2 m

Prob. 11–27

*11–28. The joist AB used in housing construction is to be made from 8-in. by 1.5-in. Southern-pine boards. If the design loading on each board is placed as shown, determine the largest room width L that the boards can span. The allowable bending stress for the wood is s allow = 2 ksi and the allowable shear stress is t allow = 180 psi. Assume that the beam is simply supported from the walls at A and B .

8 in.

1.5 in.

150 lb

150 lb 150 lb

B L– 4

L– 4

Prob. 11–28

11–29. The beam is to be used to support the machine, which exerts the forces of 6 kip and 8 kip as shown. If the maximum bending stress is not to exceed s allow = 22 ksi, determine the required width b of the flanges.

6 kip

6 ft 6 ft8 ft

8 kip 0.5 in.

b 0.5 in.

0.5 in. 7 in.

Prob. 11–29

11–30. The simply supported beam supports a load of P = 16 kN. Determine the smallest dimension a of each timber if the allowable bending stress for the wood is s allow = 30 MPa and the allowable shear stress is t allow = 800 kPa. Also, if each bolt can sustain a shear of 2.5 kN, determine the spacing s of the bolts at the calculated dimension a .

3 m

Prob. 11–30

558 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

*11.3 Fully Stressed Beams Since the moment in a beam generally varies along its length, the choice of a prismatic beam is usually inefficient since it is never fully stressed at points where the internal moment is less than the maximum moment in the beam. In order to reduce the weight of the beam, engineers sometimes choose a beam having a variable cross-sectional area, such that at each cross section along the beam, the bending stress reaches its maximum allowable value. Beams having a variable cross-sectional area are called nonprismatic beams . They are often used in machines since they can be readily formed by casting. Examples are shown in Fig. 11–10 a . In structures such beams may be “haunched” at their ends as shown in Fig. 11–10 b . Also, beams may be “built up” or fabricated in a shop using plates. An example is a girder made from a rolled-shaped wide-flange beam and having cover plates welded to it in the region where the moment is a maximum, Fig. 11–10 c .

The stress analysis of a nonprismatic beam is generally very difficult to perform and is beyond the scope of this text. Most often these shapes are analyzed by using a computer or the theory of elasticity. The results obtained from such an analysis, however, do indicate that the assumptions used in the derivation of the flexure formula are approximately correct for predicting the bending stresses in nonprismatic sections, provided the taper or slope of the upper or lower boundary of the beam is not too severe. On the other hand, the shear formula cannot be used for nonprismatic beam design, since the results obtained from it are very misleading.

Although caution is advised when applying the flexure formula to nonprismatic beam design, we will show here, in principle, how this formula can be used as an approximate means for obtaining the beam’s general shape. In this regard, the size of the cross section of a nonprismatic beam that supports a given loading can be determined using the flexure formula written as

S = M sallow

If we express the internal moment M in terms of its position x along the beam, then since sallow is a known constant, the section modulus S or the beam’s dimensions become a function of x . A beam designed in this manner is called a fully stressed beam . Although only bending stresses have been considered in approximating its final shape, attention must also be given to ensure that the beam will resist shear, especially at points where concentrated loads are applied.

(a)

(b) Haunched concrete beam

Fig. 11–10

The beam for this bridge pier has a variable moment of inertia. This design will reduce material weight and save cost.

11.3 FULLY STRESSED BEAMS 559

Determine the shape of a fully stressed, simply supported beam that supports a concentrated force at its center, Fig. 11–11 a . The beam has a rectangular cross section of constant width b , and the allowable stress is sallow.

EXAMPLE 11.4

(a)

L 2

(b)

P 2

SOLUTION The internal moment in the beam, Fig. 11–11 b , expressed as a function of position, 0 … x 6 L >2, is

M = P 2

Hence the required section modulus is

S = M

sallow =

P 2sallow

Since S = I>c, then for a cross-sectional area h by b we have I c

= 1 12 bh

h>2 = P2sallow x h2 =

3P sallowb

If h = h0 at x = L >2, then h0 2 =

3PL 2sallowb

so that

h2 = ¢ 2h0 2 L ≤ x Ans.

By inspection, the depth h must therefore vary in a parabolic manner with the distance x .

NOTE: In practice this shape is the basis for the design of leaf springs used to support the rear-end axles of most heavy trucks or train cars as shown in the adjacent photo. Note that although this result indicates that h = 0 at x = 0, it is necessary that the beam resist shear stress at the supports, and so practically speaking, it must be required that h 7 0 at the supports, Fig. 11–11 a .

Fig. 11–11

560 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

The cantilevered beam shown in Fig. 11–12 a is formed into a trapezoidal shape having a depth h0 at A and a depth 3h0 at B . If it supports a load P at its end, determine the absolute maximum normal stress in the beam. The beam has a rectangular cross section of constant width b .

(a)

3h0

(b)

V ! P

M ! Px

Fig. 11–12

SOLUTION At any cross section, the maximum normal stress occurs at the top and bottom surface of the beam. However, since smax = M >S and the section modulus S increases as x increases, the absolute maximum normal stress does not necessarily occur at the wall B , where the moment is maximum. Using the flexure formula, we can express the maximum normal stress at an arbitrary section in terms of its position x , Fig. 11–12 b . Here the internal moment has a magnitude of M = Px. Since the slope of the bottom of the beam is 2h0>L, Fig. 11–12 a , the depth of the beam at position x is

h = 2h0 L

x + h0 = h0 L

(2x + L)

EXAMPLE 11.5

11.3 FULLY STRESSED BEAMS 561

Applying the flexure formula, we have

s = Mc I

= Px(h>2)1 112 bh32 = 6PL2xbh0 2 (2x + L)2 (1)

To determine the position x where the absolute maximum normal stress occurs, we must take the derivative of s with respect to x and set it equal to zero. This gives

ds dx

= ¢ 6PL2 bh0

2 ≤ 1(2x + L)2 - x(2)(2x + L)(2)(2x + L)4 = 0 Thus,

4x2 + 4xL + L2 - 8x2 - 4xL = 0

L2 - 4x2 = 0

x = 1 2

Substituting into Eq. 1 and simplifying, the absolute maximum normal stress is therefore

sabsmax = 3 4

bh0 2 Ans.

Note that at the wall, B , the maximum normal stress is

(smax)B = Mc

I =

PL(1.5h0)3 112 b(3h0)34 = 23 PLbh0 2 which is 11.1% smaller than sabsmax .

NOTE: Recall that the flexure formula was derived on the basis of assuming the beam to be prismatic . Since this is not the case here, some error is to be expected in this analysis and that of Example 11.4 . A more exact mathematical analysis, using the theory of elasticity, reveals that application of the flexure formula as in the above example gives only small errors in the normal stress if the tapered angle of the beam is small. For example, if this angle is 15°, the stress calculated from the formula will be about 5% greater than that calculated by the more exact analysis. It may also be worth noting that the calculation of (smax)B was done only for illustrative purposes, since, by Saint-Venant’s principle, the actual stress distribution at the support (wall) is highly irregular.

562 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

(a)

B P1

(b)

(P1)z (P1)x

*11.4 Shaft Design Shafts that have circular cross sections are often used in the design of mechanical equipment and machinery. As a result, they can be subjected to cyclic or fatigue stress, which is caused by the combined bending and torsional loads they must transmit or resist. In addition to these loadings, stress concentrations may exist on a shaft due to keys, couplings, and sudden transitions in its cross-sectional area ( Sec. 5.8 ). In order to design a shaft properly, it is therefore necessary to take all of these effects into account.

Here we will discuss some of the important aspects of the design of shafts required to transmit power. These shafts are often subjected to loads applied to attached pulleys and gears, such as the one shown in Fig. 11–13 a . Since the loads can be applied to the shaft at various angles, the internal bending and torsional moments at any cross section can be determined by first replacing the loads by their statically equivalent counterparts and then resolving these loads into components in two perpendicular planes, Fig. 11–13 b . The bending-moment diagrams for the loads in each plane can then be drawn, and the resultant internal moment at any section along the shaft is then determined by vector addition, M = 2 M 2x + M 2z , Fig. 11–13 c . In addition to the moment, segments of the shaft are also subjected to different internal torques, Fig. 11–13 b . To account for this general variation of torque along the shaft, a torque diagram may also be drawn, Fig. 11–13 d .

Moment diagram caused by loads in y-z plane

y Moment diagram caused by

loads in x-y plane

(c)

Mx Mz

Torque diagram caused by torques applied about the shaft’s axis

(d)

Fig. 11–13

11.4 SHAFT DESIGN 563

Once the moment and torque diagrams have been established, it is then possible to investigate certain critical sections along the shaft where the combination of a resultant moment M and a torque T creates the worst stress situation. Since the moment of inertia of the shaft is the same about any diametrical axis, we can apply the flexure formula using the resultant moment to obtain the maximum bending stress. As shown in Fig. 11–13 e , this stress will occur on two elements, C and D , each located on the outer boundary of the shaft. If a torque T is also resisted at this section, then a maximum shear stress is also developed on these elements, Fig. 11–13 f . Furthermore, the external forces will also create shear stress in the shaft determined from t = VQ>It; however, this stress will generally contribute a much smaller stress distribution on the cross section compared with that developed by bending and torsion. In some cases, it must be investigated, but for simplicity, we will neglect its effect in the following analysis. In general, then, the critical element D (or C ) on the shaft is subjected to plane stress as shown in Fig. 11–13 g , where

s = Mc I

and t = Tc J

If the allowable normal or shear stress for the material is known, the size of the shaft is then based on the use of these equations and selection of an appropriate theory of failure. For example, if the material is known to be ductile, then the maximum-shear-stress theory may be appropriate. As stated in Sec. 10.7 , this theory requires the allowable shear stress, which is determined from the results of a simple tension test, to be equal to the maximum shear stress in the element. Using the stress-transformation equation, Eq. 9–7 , for the stress state in Fig. 11–13 g , we have

tallow = B a s 2 b2 + t2 = B a Mc 2I b2 + a Tc J b2

Since I = pc 4>4 and J = pc 4>2, this equation becomes t allow =

2 pc3

2 M2 + T2 Solving for the radius of the shaft, we get

c = ¢ 2 pt allow

2 M2 + T2 ≤ 1>3 (11–2) Application of any other theory of failure will, of course, lead to a

different formulation for c . However, in all cases it may be necessary to apply this formulation at various “critical sections” along the shaft in order to determine the particular combination of M and T that gives the largest value for c .

The following example illustrates the procedure numerically.

N C

(e)

(f)

D t

(g)

t t

Fig. 11–13 (cont.)

564 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

The shaft in Fig. 11–14 a is supported by smooth journal bearings at A and B . Due to the transmission of power to and from the shaft, the belts on the pulleys are subjected to the tensions shown. Determine the smallest diameter of the shaft using the maximum-shear-stress theory, with tallow = 50 MPa.

EXAMPLE 11.6

(a)

B C

0.250 m

0.150 m

400 N

550 N

300 N

200 N

D 0.075 m

0.050 m

(b)

7.5 N!m

950 N

650 N

475 N

150 N

7.5 N!m

0.250 m

0.150 m

500 N

SOLUTION The support reactions have been calculated and are shown on the free-body diagram of the shaft, Fig. 11–14 b . Bending-moment diagrams for Mx and Mz are shown in Figs. 11–14 c and 11–14 d , respectively. The torque diagram is shown in Fig. 11–14 e . By inspection, critical points for bending moment occur either at C or B . Also, just to the right of C and at B the torsional moment is 7.5 N # m. At C , the resultant moment is

MC = 2(118.75 N # m)2 + (37.5 N # m)2 = 124.5 N # m whereas at B it is smaller, namely

MB = 75 N # m

Fig. 11–14

11.4 SHAFT DESIGN 565

Since the design is based on the maximum-shear-stress theory, Eq. 11–2 applies. The radical 2 M2 + T2 will be the largest at a section just to the right of C . We have

c = ¢ 2 ptallow

2 M2 + T2 ≤1>3 = ¢ 2

p(50)(106 ) N>m2 2 (124.5 N # m)2 + (7.5 N # m)2 ≤1>3 = 0.0117 m

Thus, the smallest allowable diameter is

d = 2(0.0117 m) = 23.3 mm Ans.

y (m)

475 N 950 N 475 N

118.75

C B D

0.250 m 0.150 m

(c)

0.250 m

Mx (N!m)

150 N 500 N650 N

75 N!m

C B D

0.250 m 0.250 m 0.150 m

(d)

37.5 N!m

y (m)

Mz (N!m)

–7.5

C B D

0.250 m 0.150 m

(e)

7.5 N!m 7.5 N!m

0.250 m Ty (N!m)

y (m)

Fig. 11–14 (cont.)

566 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

11–31. Determine the variation in the depth w as a function of x for the cantilevered beam that supports a concentrated force P at its end so that it has a maximum bending stress sallow throughout its length. The beam has a constant thickness t .

t L

–– 2

w –– 2

Prob. 11–31

*11–32. The beam is made from a plate that has a constant thickness b . If it is simply supported and carries a uniform load w , determine the variation of its depth as a function of x so that it maintains a constant maximum bending stress sallow throughout its length.

L–– 2

Prob. 11–32

11–33. The simply supported tapered rectangular beam with constant width b supports the concentrated forces P . Determine the absolute maximum normal stress developed in the beam and specify its location.

h0 h0

L 2

L 3

2h0

Prob. 11–33

11–34. The beam is made from a plate that has a constant thickness b . If it is simply supported and carries the distributed loading shown, determine the variation of its depth as a function of x so that it maintains a constant maximum bending stress sallow throughout its length.

L–– 2

h0h

Prob. 11–34

11–35. Determine the variation in the depth d of a cantilevered beam that supports a concentrated force P at its end so that it has a constant maximum bending stress sallow throughout its length. The beam has a constant width b0 .

Prob. 11–35

*11–36. Determine the variation of the radius r of the cantilevered beam that supports the uniform distributed load so that it has a constant maximum bending stress smax throughout its length.

Prob. 11–36

PROBLEMS

11.4 SHAFT DESIGN 567

11–37. Determine the height h of the rectangular cantilever beam of constant width b in terms of h 0 , L , and x so that the maximum normal stress in the beam is constant throughout its length.

Prob. 11–37

11–38. Determine the variation in the width b as a function of x for the cantilevered beam that supports a uniform distributed load along its centerline so that it has the same maximum bending stress sallow throughout its length. The beam has a constant depth t .

b0— 2

b— 2

Prob. 11–38

11–39. The tubular shaft has an inner diameter of 15 mm. Determine to the nearest millimeter its minimum outer diameter if it is subjected to the gear loading. The bearings at A and B exert force components only in the y and z directions on the shaft. Use an allowable shear stress of tallow = 70 MPa , and base the design on the maximum- shear-stress theory of failure.

*11–40. Determine to the nearest millimeter the minimum diameter of the solid shaft if it is subjected to the gear loading. The bearings at A and B exert force components only in the y and z directions on the shaft. Base the design on the maximum- distortion-energy theory of failure with sallow = 150 MPa .

150 mm

A 500 N

100 mm

150 mm

200 mm 500 N

Probs. 11–39/40

11–41. The 50-mm diameter shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B . If the pulleys C and D are subjected to the vertical and horizontal loadings shown, determine the absolute maximum bending stress in the shaft.

400 mm

400 mm 300 N

150 N

150 N300 N

400 mm

Prob. 11–41

568 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

11–42. The shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B . If shaft is made from material having an allowable bending stress of s allow = 150 MPa, determine the minimum diameter of the shaft to the nearest millimeter.

400 mm

400 mm 300 N

150 N

150 N300 N

400 mm

Prob. 11–42

11–43. The two pulleys attached to the shaft are loaded as shown. If the bearings at A and B exert only vertical forces on the shaft, determine the required diameter of the shaft to the nearest 18 in. using the maximum-shear-stress theory. t allow = 12 ksi.

300 lb 120 lb

120 lb 300 lb

D A

C B

0.5 ft 0.5 ft 1 ft 3 ft 2 ft

Prob. 11–43

*11–44. The two pulleys attached to the shaft are loaded as shown. If the bearings at A and B exert only vertical forces on the shaft, determine the required diameter of the shaft to the nearest 18 in. using the maximum-distortion energy theory. s allow = 67 ksi.

300 lb

120 lb 120 lb

300 lb

D A

C B

0.5 ft 0.5 ft 1 ft 3 ft 2 ft

Prob. 11–44

11–45. The bearings at A and D exert only y and z components of force on the shaft. If tallow = 60 MPa , determine to the nearest millimeter the smallest-diameter shaft that will support the loading. Use the maximum-shear- stress theory of failure.

350 mm

400 mm

200 mm

50 mm

75 mm

A Fz ! 2 kN

Fy ! 3 kN

Prob. 11–45

11–46. The bearings at A and D exert only y and z components of force on the shaft. If sallow = 130 MPa, determine to the nearest millimeter the smallest-diameter shaft that will support the loading. Use the maximum- distortion-energy theory of failure.

350 mm

400 mm

200 mm

50 mm

75 mm

A Fz ! 2 kN

Fy ! 3 kN

Prob. 11–46

CHAPTER REVIEW 569

Failure of a beam occurs when the internal shear or moment in the beam is a maximum. To resist these loadings, it is therefore important that the associated maximum shear and bending stress not exceed allowable values as stated in codes. Normally, the cross section of a beam is first designed to resist the allowable bending stress,

sallow = Mmax c

CHAPTER REVIEW

Then the allowable shear stress is checked. For rectangular sections, t allow Ú 1.5(V max>A ) and for wide-flange sections it is appropriate to use t allow Ú V max>A web. In general, use

tallow = VQ It

For built-up beams, the spacing of fasteners or the strength of glue or weld is determined using an allowable shear flow

qallow = VQ I

Fully stressed beams are nonprismatic and designed such that each cross section along the beam will resist the allowable bending stress. This will define the shape of the beam.

A mechanical shaft generally is designed to resist both torsion and bending stresses. Normally, the internal bending moment can be resolved into two planes, and so it is necessary to draw the moment diagrams for each bending- moment component and then select the maximum moment based on vector addition. Once the maximum bending and shear stresses are determined, then depending upon the type of material, an appropriate theory of failure is used to compare the allowable stress to what is required.

B P1

570 CHAPTER 11 DES IGN OF BEAMS AND SHAFTS

11–47. The cantilevered beam has a circular cross section. If it supports a force P at its end, determine its radius y as a function of x so that it is subjected to a constant maximum bending stress s allow throughout its length.

Prob. 11–47

*11–48. Select the lightest-weight steel wide-flange over- hanging beam from Appendix B that will safely support the loading. Assume the support at A is a pin and the support at B is a roller. The allowable bending stress is sallow = 24 ksi and the allowable shear stress is tallow = 14 ksi.

2 kip 2 kip

8 ft 2 ft 4 ft

A B

Prob. 11–48

11–49. The bearings at A and B exert only x and z components of force on the steel shaft. Determine the shaft’s diameter to the nearest millimeter so that it can resist the loadings of the gears without exceeding an allowable shear stress of tallow = 80 MPa. Use the maximum-shear-stress theory of failure.

75 mm

150 mm

350 mm

250 mm

50 mm

Fz ! 7.5 kN

Fx ! 5 kN

Prob. 11–49

11–50. The bearings at A and B exert only x and z components of force on the steel shaft. Determine the shaft’s diameter to the nearest millimeter so that it can resist the loadings of the gears. Use the maximum-distortion-energy theory of failure with sallow = 200 MPa.

75 mm

150 mm

350 mm

250 mm

50 mm

Fz ! 7.5 kN

Fx ! 5 kN

Prob. 11–50

REVIEW PROBLEMS

REVIEW PROBLEMS 571

11–51. Select the lightest-weight steel wide-flange beam from Appendix B that will safely support the loading shown. The allowable bending stress is s allow = 22 ksi and the allowable shear stress is tallow = 22 ksi.

10 ft

8 kip

10 ft

10 kip 8 kip

5 ft 5 ft

Prob. 11–51

*11–52. The simply supported joist is used in the construction of a floor for a building. In order to keep the floor low with respect to the sill beams C and D , the ends of the joists are notched as shown. If the allowable shear stress for the wood is tallow = 350 psi and the allowable bending stress is sallow = 1500 psi, determine the height h that will cause the beam to reach both allowable stresses at the same time. Also, what load P causes this to happen? Neglect the stress concentration at the notch.

15 ft

P 2 in.

10 in.

15 ft

Prob. 11–52

11–53. The simply supported joist is used in the construction of a floor for a building. In order to keep the floor low with respect to the sill beams C and D , the ends of the joists are notched as shown. If the allowable shear stress for the wood is tallow = 350 psi and the allowable bending stress is s allow = 1700 psi, determine the smallest height h so that the beam will support a load of P = 600 lb . Also, will the entire joist safely support the load? Neglect the stress concentration at the notch.

15 ft

P 2 in.

10 in.

15 ft

Prob. 11–53

11–54. The overhang beam is constructed using two 2-in. by 4-in. pieces of wood braced as shown. If the allowable bending stress is s allow = 600 psi, determine the largest load P that can be applied. Also, determine the associated maximum spacing of nails, s , along the beam section AC if each nail can resist a shear force of 800 lb. Assume the beam is pin-connected at A , B , and D . Neglect the axial force developed in the beam along DA .

2 ft

3 ft

4 in.

2 in. 2 in.

Prob. 11–54

If the curvature of this pole is measured, it is then possible to determine the bending stress developed within it.

Chapter 12

573

12.1 The Elastic Curve The deflection of a beam or shaft must often be limited in order to provide integrity and stability of a structure or machine, and prevent the cracking of any attached brittle materials such as concrete or glass. Furthermore, code restrictions often require these members not vibrate or deflect severely in order to safely support their intended loading. Most important, though, deflections at specific points on a beam or shaft must be determined if one is to analyze those that are statically indeterminate.

Before the slope or the displacement at a point on a beam (or shaft) is determined, it is often helpful to sketch the deflected shape of the beam when it is loaded, in order to “visualize” any computed results and thereby partially check these results. The deflection curve of the longitudinal axis that passes through the centroid of each cross-sectional area of a beam is called the elastic curve . For most beams the elastic curve can be sketched without much difficulty. When doing so, however, it is necessary to know how the slope or displacement is restricted at various types of supports. In general, supports that resist a force , such as a pin, restrict displacement , and those that resist a moment , such as a fixed wall, restrict rotation or slope as well as displacement. With this in mind, two typical examples of the elastic curves for loaded beams (or shafts), sketched to an exaggerated scale, are shown in Fig. 12–1 .

CHAPTER OBJECTIVES

■ Often limits must be placed on the amount of deflection a beam or shaft may undergo when it is subjected to a load, and so in this chapter we will discuss various methods for determining the deflection and slope at specific points on beams and shafts. The analytical methods include the integration method, the use of discontinuity functions, and the method of superposition. Also, a semigraphical technique, called the moment-area method, will be presented. At the end of the chapter, we will use these methods to solve for the support reactions on a beam or shaft that is statically indeterminate.

Deflection of Beams and Shafts

Fig. 12–1

574 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

1212121212 If the elastic curve for a beam seems difficult to establish, it is suggested

that the moment diagram for the beam be drawn first. Using the beam sign convention established in Sec. 6.1 , a positive internal moment tends to bend the beam concave upward, Fig. 12–2 a . Likewise, a negative moment tends to bend the beam concave downward, Fig. 12–2 b . Therefore, if the moment diagram is known , it will be easy to construct the elastic curve. For example, consider the beam in Fig. 12–3 a with its associated moment diagram shown in Fig. 12–3 b . Due to the roller and pin supports, the displacement at B and D must be zero. Within the region of negative moment, AC , Fig. 12–3 b , the elastic curve must be concave downward, and within the region of positive moment, CD , the elastic curve must be concave upward. Hence, there must be an inflection point at point C , where the curve changes from concave up to concave down, since this is a point of zero moment. Using these facts, the beam’s elastic curve is sketched in Fig. 12–3 c . It should also be noted that the displacements !A and !E are especially critical. At point E the slope of the elastic curve is zero , and there the beam’s deflection may be a maximum . Whether !E is actually greater than !A depends on the relative magnitudes of P1 and P2 and the location of the roller at B .

Following these same principles, note how the elastic curve in Fig. 12– 4 was constructed. Here the beam is cantilevered from a fixed support at A and therefore the elastic curve must have both zero displacement and zero slope at this point. Also, the largest displacement will occur either at D , where the slope is zero, or at C .

!M !M

Positive internal moment concave upwards

(a)

Negative internal moment concave downwards

(b)

"M "M

Fig. 12–2

Moment diagram

(b)

Inflection point

(c)

Elastic curve

P1 P2

A D EC

B (a)

Fig. 12–3

DInflection point

Elastic curve

(c)

Moment diagram

(b)

#C #D

(a) A CD

Fig. 12–4

12.1 THE ELASTIC CURVE 575

1212121212 Moment-Curvature Relationship. We will now develop an important relationship between the internal moment and the radius of curvature r (rho) of the elastic curve at a point. The resulting equation will be used for establishing each of the methods presented in the chapter for finding the slope and displacement at points on the elastic curve.

The following analysis, here and in the next section, will require the use of three coordinates. As shown in Fig. 12–5 a , the x axis extends positive to the right, along the initially straight longitudinal axis of the beam. It is used to locate the differential element, having an undeformed width dx . The v axis extends positive upward from the x axis. It measures the displacement of the elastic curve. Lastly, a “localized” y coordinate is used to specify the position of a fiber in the beam element. It is measured positive upward from the neutral axis (or elastic curve) as shown in Fig. 12–5 b . Recall that this same sign convention for x and y was used in the derivation of the flexure formula.

To derive the relationship between the internal moment and r, we will limit the analysis to the most common case of an initially straight beam that is elastically deformed by loads applied perpendicular to the beam’s x axis and lying in the x – v plane of symmetry for the beam’s cross-sectional area. Due to the loading, the deformation of the beam is caused by both the internal shear force and bending moment. If the beam has a length that is much greater than its depth, the greatest deformation will be caused by bending, and therefore we will direct our attention to its effects. Deflections caused by shear will be discussed in Chapter 14 .

x dx

(a)

O¿

ds¿

Before deformation

After deformation

(b)

yy dx

M M

Fig. 12–5

576 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12121212 When the internal moment M deforms the element of the beam, the

angle between the cross sections becomes du, Fig. 12–5 b . The arc dx represents a portion of the elastic curve that intersects the neutral axis for each cross section. The radius of curvature for this arc is defined as the distance r, which is measured from the center of curvature O! to dx . Any arc on the element other than dx is subjected to a normal strain. For example, the strain in arc ds , located at a position y from the neutral axis, is P = (ds! - ds)>ds. However, ds = dx = r du and ds! = (r - y) du, and so P = [(r - y) du - r du]>r du or

1 r

= - P y

(12–1)

If the material is homogeneous and behaves in a linear-elastic manner, then Hooke’s law applies, P = s>E. Also, since the flexure formula applies, s = -My >I. Combining these two equations and substituting into the above equation, we have

1 r

= M EI

(12–2)

where

r = the radius of curvature at the point on the elastic curve ( 1>r is referred to as the curvature )

M = the internal moment in the beam at the point E = the material’s modulus of elasticity I = the beam’s moment of inertia about the neutral axis

The product EI in this equation is referred to as the flexural rigidity , and it is always a positive quantity. The sign for r therefore depends on the direction of the moment. As shown in Fig. 12–6 , when M is positive , r extends above the beam, i.e., in the positive v direction; when M is negative , r extends below the beam, or in the negative v direction.

Using the flexure formula, s = -My >I, we can also express the curvature in terms of the stress in the beam, namely,

1 r

= - s Ey

(12–3)

Both Eqs. 12–2 and 12–3 are valid for either small or large radii of curvature. However, the value of r is almost always calculated as a very large quantity . For example, consider an A-36 steel beam made from a W14 * 53 ( Appendix B ), where Est = 29(103) ksi and sY = 36 ksi. When the material at the outer fibers, y = {7 in., is about to yield , then, from Eq. 12–3 , r = {5639 in. Values of r calculated at other points along the beam’s elastic curve may be even larger , since s cannot exceed sY at the outer fibers.

O¿

Inflection point M ! 0

"r #M #M

"M "M

Fig. 12–6

O¿

ds¿

Before deformation

After deformation

(b)

yy dx

M M

Fig 12-5 (cont.)

12.2 SLOPE AND DISPLACEMENT BY INTEGRATION 577

1212121212 12.2 Slope and Displacement by Integration

The equation of the elastic curve for a beam can be expressed mathematically as v = f(x). To obtain this equation, we must first represent the curvature (1>r) in terms of v and x . In most calculus books it is shown that this relationship is

1 r

= d2v>dx2

[1 + (dv>dx)2]3>2 Substituting into Eq. 12–2 , we have

d2v>dx2

[1 + (dv>dx)2]3>2 = MEI (12–4) This equation represents a nonlinear second-order differential equation. Its solution, which is called the elastica , gives the exact shape of the elastic curve, assuming, of course, that beam deflections occur only due to bending. Through the use of higher mathematics, elastica solutions have been obtained only for simple cases of beam geometry and loading.

In order to facilitate the solution of a greater number of deflection problems, Eq. 12–4 can be modified. Most engineering design codes specify limitations on deflections for tolerance or esthetic purposes, and as a result the elastic deflections for the majority of beams and shafts form a shallow curve. Consequently, the slope of the elastic curve, which is determined from dv>dx , will be very small , and its square will be negligible compared with unity. * Therefore the curvature, as defined above, can be approximated by 1>r = d2v>dx2. Using this simplification, Eq. 12–4 can now be written as

d2v dx2

= M EI

(12–5)

It is also possible to write this equation in two alternative forms. If we differentiate each side with respect to x and substitute V = dM >dx ( Eq. 6–2 ), we get

d dx

¢EI d2v dx2 ≤ = V (x) (12–6)

Differentiating again, using w = dV >dx ( Eq. 6–1 ), yields

dx2 ¢EI d2v

dx2 ≤ = w(x) (12–7)

* See Example 12.1 .

578 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

1212121212 For most problems the flexural rigidity ( EI ) will be constant along the

length of the beam. Assuming this to be the case, the above results may be reordered into the following set of three equations:

EI d4v dx4

= w(x) (12–8)

EI d3v dx3

= V (x) (12–9)

EI d2v dx2

= M(x) (12–10)

Solution of any of these equations requires successive integrations to obtain the deflection v of the elastic curve. For each integration it is necessary to introduce a “constant of integration” and then solve for all the constants to obtain a unique solution for a particular problem. For example, if the distributed load w is expressed as a function of x and Eq. 12–8 is used, then four constants of integration must be evaluated; however, if the internal moment M is determined and Eq. 12–10 is used, only two constants of integration must be found. The choice of which equation to start with depends on the problem. Generally, however, it is easier to determine the internal moment M as a function of x , integrate twice, and evaluate only two integration constants.

Recall from Sec. 6.1 that if the loading on a beam is discontinuous, that is, consists of a series of several distributed and concentrated loads, then several functions must be written for the internal moment, each valid within the region between the discontinuities. Also, for convenience in writing each moment expression, the origin for each x coordinate can be selected arbitrarily . For example, consider the beam shown in Fig. 12–7 a . The internal moment in regions AB, BC , and CD can be written in terms of the x1, x2, and x3 coordinates selected, as shown in either Fig. 12–7 b or Fig. 12–7 c , or in fact in any manner that will yield M = f(x) in as simple a form as possible. Once these functions are integrated twice through the use of Eq. 12–10 and the constants of integration determined, the functions will give the slope and deflection (elastic curve) for each region of the beam for which they are valid.

(a) B C

A D

P w

A D

(b)

P w

B C x1

x2 x3

A D

(c)

P w

B C x1 x2 x3

Fig. 12–7

12.2 SLOPE AND DISPLACEMENT BY INTEGRATION 579

1212121212 Sign Convention and Coordinates. When applying Eqs. 12–8 through 12–10 , it is important to use the proper signs for M, V , or w as established by the sign convention that was used in the derivation of these equations. For review, these terms are shown in their positive directions in Fig. 12–8 a . Furthermore, recall that positive deflection , v, is upward , and as a result, the positive slope angle u will be measured counterclockwise from the x axis when x is positive to the right . The reason for this is shown in Fig. 12–8 b . Here positive increases dx and dv in x and v create an increased u that is counterclockwise. If, however, positive x is directed to the left , then u will be positive clockwise , Fig. 12– 8 c .

Realize that by assuming dv>dx to be very small, the original horizontal length of the beam’s axis and the arc of its elastic curve will be about the same. In other words, ds in Figs. 12–8 b and 12–8 c is approximately equal to dx , since ds = 2(dx)2 + (dv)2 = 21 + (dv>dx)2 dx ! dx. As a result, points on the elastic curve are assumed to be displaced vertically , and not horizontally. Also, since the slope angle u will be very small , its value in radians can be determined directly from u ! tan u = dv>dx.

The design of a roof system requires a careful consideration of deflection. For example, rain can accumulate on areas of the roof, which then causes ponding, leading to further deflection, then further ponding, and finally possible failure of the roof.

!V!V

Positive sign convention

(a)

Elastic curve

O¿v

Positive sign convention

(b)

!v !dv

Elastic curve

O¿

dv !v

Positive sign convention

(c)

dsdu !u

Fig. 12–8

580 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

1212121212 Boundary and Continuity Conditions. When solving Eqs. 12–8 , 12–9 , or 12–10 , the constants of integration are determined by evaluating the functions for shear, moment, slope, or displacement at a particular point on the beam where the value of the function is known. These values are called boundary conditions . Several possible boundary conditions that are often used to solve beam (or shaft) deflection problems are listed in Table 12–1 . For example, if the beam is supported by a roller or pin (1, 2, 3, 4), then it is required that the displacement be zero at these points. Furthermore, if these supports are located at the ends of the beam (1, 2), the internal moment in the beam must also be zero. At the fixed support (5), the slope and displacement are both zero, whereas the free-ended beam (6) has both zero moment and zero shear. Lastly, if two segments of a beam are connected by an “internal” pin or hinge (7), the moment must be zero at this connection.

If the elastic curve cannot be expressed using a single coordinate, then continuity conditions must be used to evaluate some of the integration constants. For example, consider the beam in Fig. 12–9 a . Here two x coordinates are chosen with origins at A . Each is valid only within the regions 0 … x1 … a and a … x2 … (a + b). Once the functions for the slope and deflection are obtained, they must give the same values for the slope and deflection at point B so the elastic curve is physically continuous . Expressed mathematically, this requires that u1(a) = u2(a) and v1(a) = v2(a). These conditions can be used to evaluate two constants of integration. If instead the elastic curve is expressed in terms of the coordinates 0 … x1 … a and 0 … x2 … b, shown in Fig. 12–9 b , then the continuity of slope and deflection at B requires u1(a) = -u2(b) and v1(a) = v2(b). In this particular case, a negative sign is necessary to match the slopes at B since x1 extends positive to the right, whereas x2 extends positive to the left. Consequently, u1 is positive counterclockwise, and u2 is positive clockwise. See Figs. 12–8 b and 12–8 c .

A C

a b B

(a)

v1, v2

A C

v1 v2

(b)

P a b

Fig. 12–9

Roller

Fixed end

Internal pin or hinge

! " 0

Free end M " 0

M " 0

V " 0

u " 0

Roller M " 0 ! " 0

Pin M " 0 ! " 0

TABLE 12–1

12.2 SLOPE AND DISPLACEMENT BY INTEGRATION 581

1212121212 Procedure for Analysis

The following procedure provides a method for determining the slope and deflection of a beam (or shaft) using the method of integration.

Elastic Curve. • Draw an exaggerated view of the beam’s elastic curve. Recall that

zero slope and zero displacement occur at all fixed supports, and zero displacement occurs at all pin and roller supports.

• Establish the x and v coordinate axes. The x axis must be parallel to the undeflected beam and can have an origin at any point along the beam, with a positive direction either to the right or to the left. In all cases, the associated positive v axis should be directed upward.

• If several discontinuous loads are present, establish x coordinates that are valid for each region of the beam between the discontinuities. Choose these coordinates so that they will simplify subsequent algebraic work.

Load or Moment Function. • For each region in which there is an x coordinate, express the

loading w or the internal moment M as a function of x . In particular, always assume that M acts in the positive direction when applying the equation of moment equilibrium to determine M = f ( x ).

Slope and Elastic Curve. • Provided EI is constant, apply either the load equation

EI d 4 v / dx 4 = w ( x ), which requires four integrations to get v = v ( x ), or the moment equation EI d 2 v / dx 2 = M ( x ), which requires only two integrations. For each integration it is important to include a constant of integration.

• The constants are evaluated using the boundary conditions for the supports ( Table 12–1 ) and the continuity conditions that apply to slope and displacement at points where two functions meet. Once the constants are evaluated and substituted back into the slope and deflection equations, the slope and displacement at specific points on the elastic curve can then be determined.

• The numerical values obtained can be checked graphically by comparing them with the sketch of the elastic curve. Realize that positive values for slope are counterclockwise if the x axis extends positive to the right , and clockwise if the x axis extends positive to the left . In either of these cases, positive displacement is upward .

582 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.1

The cantilevered beam shown in Fig. 12–10 a is subjected to a vertical load P at its end. Determine the equation of the elastic curve. EI is constant.

SOLUTION I Elastic Curve. The load tends to deflect the beam as shown in Fig. 12–10 a . By inspection, the internal moment can be represented throughout the beam using a single x coordinate.

Moment Function. From the free-body diagram, with M acting in the positive direction , Fig. 12–10 b , we have

M = -Px

Slope and Elastic Curve. Applying Eq. 12–10 and integrating twice yields

EI d2v dx2

= -Px (1)

EI dv dx

= - Px 2

2 + C1 (2)

EIv = - Px 3

6 + C1x + C2 (3)

Using the boundary conditions dv>dx = 0 at x = L and v = 0 at x = L, Eqs. 2 and 3 become

0 = - PL 2

2 + C1

0 = - PL 3

6 + C1L + C2

Thus, C1 = PL2>2 and C2 = -PL3>3. Substituting these results into Eqs. 2 and 3 with u = dv>dx, we get

u = P

2EI (L2 - x2 )

v = P

6EI (-x3 + 3L2x - 2L3) Ans .

Maximum slope and displacement occur at A (x = 0), for which

uA = PL2

2EI (4)

vA = - PL3

3EI (5)

x BA vA

Elastic curve

(a)

M x

(b)

Fig. 12–10

12.2 SLOPE AND DISPLACEMENT BY INTEGRATION 583

12 The positive result for uA indicates counterclockwise rotation and the negative result for vA indicates that vA is downward . This agrees with the results sketched in Fig. 12–10 a .

In order to obtain some idea as to the actual magnitude of the slope and displacement at the end A , consider the beam in Fig. 12–10 a to have a length of 15 ft, support a load of P = 6 kip, and be made of A-36 steel having Est = 29(103) ksi. Using the methods of Sec. 11.2 , if this beam was designed without a factor of safety by assuming the allowable normal stress is equal to the yield stress sallow = 36 ksi; then a W12 * 26 would be found to be adequate (I = 204 in4). From Eqs. 4 and 5 we get

uA = 6 kip(15 ft)2(12 in.>ft)2

2[29(103) kip>in2](204 in4) = 0.0164 rad vA = -

6 kip(15 ft)3(12 in.>ft)3 3[29(103) kip>in2](204 in4) = -1.97 in.

Since uA2 = (dv>dx)2 = 0.000270 rad2 ! 1, this justifies the use of Eq. 12–10 , rather than applying the more exact Eq. 12–4 , for computing the deflection of beams. Also, since this numerical application is for a cantilevered beam , we have obtained larger values for u and v than would have been obtained if the beam were supported using pins, rollers, or other fixed supports.

SOLUTION II This problem can also be solved using Eq. 12–8 , EI d 4v>dx4 = w(x). Here w(x) = 0 for 0 … x … L, Fig. 12–10 a , so that upon integrating once we get the form of Eq. 12–9 , i.e.,

EI d 4v dx4

= 0

EI d3v dx3

= C1= = V

The shear constant C1= can be evaluated at x = 0, since V A = -P (negative according to the beam sign convention, Fig. 12–8 a ). Thus, C1= = -P. Integrating again yields the form of Eq. 12–10 , i.e.,

EI d 3v dx3

= -P

EI d2v dx2

= -Px + C2= = M

Here M = 0 at x = 0, so C2= = 0, and as a result one obtains Eq. 1 and the solution proceeds as before.

584 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.2

The simply supported beam shown in Fig. 12–11 a supports the triangular distributed loading. Determine its maximum deflection. EI is constant.

(a)

Elastic curvex L 2

L 2

x M

(b)

!x x

x 3

2w0 L

w0x 2

w0 L 4

xw ! 2w0 L

1 2

Fig. 12–11

SOLUTION I Elastic Curve. Due to symmetry, only one x coordinate is needed for the solution, in this case 0 … x … L >2. The beam deflects as shown in Fig. 12–11 a . The maximum deflection occurs at the center since the slope is zero at this point.

Moment Function. A free-body diagram of the segment on the left is shown in Fig. 12–11 b . The equation for the distributed loading is

w = 2w0 L

x (1)

Hence,

a+ !MNA = 0; M + w0 x

L a x

3 b - w0L

4 (x) = 0

M = - w0 x

3L +

w0 L

4 x

12.2 SLOPE AND DISPLACEMENT BY INTEGRATION 585

12 Slope and Elastic Curve. Using Eq. 12–10 and integrating twice, we have

EI d2v dx2

= M = - w0 3L

x3 + w0 L

4 x (2)

EI dv dx

= - w0

12L x4 +

w0 L

8 x2 + C1

EIv = - w0

60L x5 +

w0 L 24

x3 + C1x + C2

The constants of integration are obtained by applying the boundary condition v = 0 at x = 0 and the symmetry condition that dv>dx = 0 at x = L >2. This leads to

C1 = - 5w0 L

192 C2 = 0

Hence,

EI dv dx

= - w0

12L x4 +

w0 L

8 x2 -

5w0 L 3

192

EIv = - w0

60L x5 +

w0 L 24

x3 - 5w0 L

192 x

Determining the maximum deflection at x = L >2, we have vmax = -

w0 L 4

120EI Ans .

SOLUTION II Since the distributed loading acts downward, it is negative according to our sign convention. Using Eq. 1 and applying Eq. 12–8 , we have

EI d 4v dx4

= - 2w0 L

EI d3v dx3

= V = - w0 L

x2 + C1=

Since V = +w0L >4 at x = 0, then C1= = w0 L >4. Integrating again yields EI

d3v dx3

= V = - w0 L

x2 + w0L

EI d2v dx2

= M = - w0 3L

x3 + w0L

4 x + C2=

Here M = 0 at x = 0, so C2= = 0. This yields Eq. 2 . The solution now proceeds as before.

586 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.3

The simply supported beam shown in Fig. 12–12 a is subjected to the concentrated force. Determine the maximum deflection of the beam. EI is constant.

A C

6 kN

(a)

2 m 1 m

x1 x2

A C

(b)

D uD ! 0

A x2 V2

6 kN

2 kN

(c)

2 m

(x2 " 2 m)

x1 V1

Fig. 12–12

SOLUTION

Elastic Curve. The beam deflects as shown in Fig. 12–12 b . Two coordinates must be used, since the moment function will change at B . Here we will take x1 and x2, having the same origin at A .

Moment Function. From the free-body diagrams shown in Fig. 12– 12 c ,

M1 = 2x1

M2 = 2x2 - 6(x2 - 2) = 4(3 - x2)

Slope and Elastic Curve. Applying Eq. 12–10 for M1 , for 0 … x1 6 2 m, and integrating twice yields

EI d2v1 dx1

= 2x1

EI dv1 dx1

= x12 + C1 (1)

EIv1 = 1 3

x1 3

+ C1x1 + C2 (2)

Likewise for M2, for 2 m 6 x2 … 3 m,

EI d2v2 dx2

2 = 4(3 - x2)

EI dv2 dx2

= 4¢3x2 - x222 ≤ + C3 (3) EIv2 = 4¢ 32 x22 - x236 ≤ + C3 x2 + C4 (4)

12.2 SLOPE AND DISPLACEMENT BY INTEGRATION 587

12 The four constants are evaluated using two boundary conditions, namely, x1 = 0, v1 = 0 and x2 = 3 m, v2 = 0. Also, two continuity conditions must be applied at B , that is, dv1>dx1 = dv2>dx2 at x1 = x2 = 2 m and v1 = v2 at x1 = x2 = 2 m. Substitution as specified results in the following four equations:

v1 = 0 at x1 = 0; 0 = 0 + 0 + C2

v2 = 0 at x2 = 3 m; 0 = 4¢32 (3)2 - (3)36 ≤ + C3(3) + C4 dv1 dx1

` x = 2 m

= dv2 dx2

` x = 2 m

; (2)2 + C1 = 4¢3(2) - (2)22 ≤ + C3 v1(2 m) = v2(2 m);

1 3

(2)3 + C1(2) + C2 = 4¢ 32 (2)2 - (2)36 ≤ + C3(2) + C4 Solving, we get

C1 = - 8 3

C2 = 0

C3 = - 44 3

C4 = 8

Thus Eqs. 1–4 become

EI dv1 dx1

= x12 - 8 3

(5)

EIv1 = 1 3

x1 3

- 8 3

x1 (6)

EI dv2 dx2

= 12 x2 - 2 x22 - 44 3

(7)

EIv2 = 6x22 - 2 3

x2 3

- 44 3

x2 + 8 (8)

By inspection of the elastic curve, Fig. 12–12 b , the maximum deflection occurs at D , somewhere within region AB . Here the slope must be zero. From Eq. 5 ,

x12 - 8 3

= 0

x1 = 1.633

Substituting into Eq. 6 ,

vmax = - 2.90 kN # m3

EI Ans .

The negative sign indicates that the deflection is downward.

588 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.4

The beam in Fig. 12–13 a is subjected to a load at its end. Determine the displacement at C . EI is constant.

C B

(a)

1 m

4 kN

2 m

(b)

x1 x2

2 kN

4 kN

Fig. 12–13

SOLUTION Elastic Curve. The beam deflects into the shape shown in Fig. 12– 13 a . Due to the loading, two x coordinates will be considered, namely, 0 … x1 6 2 m and 0 … x2 6 1 m, where x2 is directed to the left from C , since the internal moment is easy to formulate.

Moment Functions. Using the free-body diagrams shown in Fig. 12–13 b , we have

M1 = -2x1 M2 = -4x2

Slope and Elastic Curve. Applying Eq. 12–10 ,

For 0 … x1 … 2: EI d2v1 dx1

2 = -2x1

EI dv1 dx1

= -x12 + C1 (1)

EIv1 = - 1 3

x1 3

+ C1x1 + C2 (2)

12.2 SLOPE AND DISPLACEMENT BY INTEGRATION 589

For 0 … x2 … 1 m: EI d 2v2 dx2

= -4x2

EI dv2 dx2

= -2x22 + C3 (3)

EIv2 = - 2 3

x2 3 + C3x2 + C4 (4)

The four constants of integration are determined using three boundary conditions, namely, v1 = 0 at x1 = 0, v1 = 0 at x1 = 2 m, and v2 = 0 at x2 = 1 m , and one continuity equation. Here the continuity of slope at the roller requires dv1>dx1 = -dv2>dx2 at x1 = 2 m and x2 = 1 m. Why is there a negative sign in this equation? (Note that continuity of displacement at B has been indirectly considered in the boundary conditions, since v1 = v2 = 0 at x1 = 2 m and x2 = 1 m. ) Applying these four conditions yields

v1 = 0 at x1 = 0; 0 = 0 + 0 + C2

v1 = 0 at x1 = 2 m; 0 = - 1 3

(2)3 + C1(2) + C2

v2 = 0 at x2 = 1 m; 0 = - 2 3

(1)3 + C3(1) + C4

dv1 dx1

` x = 2 m

= dv2 dx2

` x = 1 m

; -(2)2 + C1 = -(-2(1)2 + C3)

Solving, we obtain

C1 = 4 3 C2 = 0 C3 =

14 3 C4 = -4

Substituting C3 and C4 into Eq. 4 gives

EIv2 = - 2 3

x2 3 + 14

3 x2 - 4

The displacement at C is determined by setting x2 = 0. We get

vC = - 4 kN # m3

EI Ans .

590 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 PRELIMINARY PROBLEMS

P12–1. In each case, determine the internal bending moment as a function of x, and state the necessary boundary and/or continuity conditions used to determine the elastic curve for the beam.

(a)

8 kN

4 m x

(b)

2 m x

10 kN!m

(c)

4 m x

2 kN/m

(d)

4 m

2 m

8 kN!m

(e)

4 m

2 m

4 kN2 kN/m

(f)

4 m

2 m

3 kN/m

Prob. 12–1

12.2 SLOPE AND DISPLACEMENT BY INTEGRATION 591

12 FUNDAMENTAL PROBLEMS

F12–1. Determine the slope and deflection of end A of the cantilevered beam. E = 200 GPa and I = 65.0(106) mm4 .

3 m

30 kN!m

F12–1

F12–2. Determine the slope and deflection of end A of the cantilevered beam. E = 200 GPa and I = 65.0(10 6 ) mm 4 .

3 m 10 kN!m

10 kN

F12–2

F12–3. Determine the slope of end A of the cantilevered beam. E = 200 GPa and I = 65.0(106) mm4 .

3 m

10 kN 3 kN/m

F12–3

F12–4. Determine the maximum deflection of the simply supported beam. The beam is made of wood having a modulus of elasticity of Ew = 1.5(103) ksi and a rectangular cross section of width b = 3 in. and height h = 6 in.

A B

12 ft

100 lb/ft

F12–4

F12–5. Determine the maximum deflection of the simply supported beam. E = 200 GPa and I = 39.9(10-6) m4 .

6 m

40 kN!m 10 kN!m

A B

F12–5

F12–6. Determine the slope of the simply supported beam at A . E = 200 GPa and I = 39.9(10-6) m4 .

3 m

20 kN

10 kN!m 10 kN!m

3 m

A B

F12–6

592 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 PROBLEMS

12–1. An L2 steel strap having a thickness of 0.125 in. and a width of 2 in. is bent into a circular arc of radius 600 in. Determine the maximum bending stress in the strap.

12–2. The L2 steel blade of the band saw wraps around the pulley having a radius of 12 in. Determine the maximum normal stress in the blade. The blade has a width of 0.75 in. and a thickness of 0.0625 in.

12 in.

Prob. 12–2

12–3. A picture is taken of a man performing a pole vault, and the minimum radius of curvature of the pole is estimated by measurement to be 4.5 m. If the pole is 40 mm in diameter and it is made of a glass-reinforced plastic for which Eg = 131 GPa, determine the maximum bending stress in the pole.

r ! 4.5 m

Prob. 12–3

*12–4. Determine the equations of the elastic curve using the x1 and x2 coordinates. EI is constant.

x1 x2

Prob. 12–4

12–5. Determine the deflection of end C of the 100-mm- diameter solid circular shaft. The shaft is made of steel having a modulus elasticity of E = 200 GPa.

2 m

6 kN

1 m

C B

Prob. 12–5

12–6. Determine the equations of the elastic curve for the beam using the x1 and x3 coordinates. Specify the beam’s maximum deflection. EI is constant.

L 2

Prob. 12–6

12.2 SLOPE AND DISPLACEMENT BY INTEGRATION 593

12 12–10. Determine the equations of the elastic curve for the beam using the x coordinate. Specify the slope at A and maximum deflection. EI is constant.

12–11. Determine the deflection at the center of the beam and the slope at B . EI is constant.

Probs. 12–10/11

*12–12. Determine the equations of the elastic curve for the beam using the x1 and x2 coordinates. Specify the slope at A and the maximum displacement of the beam. EI is constant.

A B

x1 x2

a a

Prob. 12–12

12–13. Determine the maximum deflection of the beam and the slope at A . EI is constant.

a a a

M0 M0

Prob. 12–13

12–7. The beam is made of two rods and is subjected to the concentrated load P . Determine the maximum deflection of the beam if the moments of inertia of the rods are IAB and IBC, and the modulus of elasticity is E .

Prob. 12–7

*12–8. The shaft is supported at A by a journal bearing that exerts only vertical reactions on the shaft, and at C by a thrust bearing that exerts horizontal and vertical reactions on the shaft. Determine the equations of the elastic curve using the coordinates x 1 and x 2 . EI is constant.

A B

a b

x1 x2

Prob. 12–8

12–9. Determine the equations of the elastic curve for the beam using the x1 and x2 coordinates. EI is constant.

x1 x2

L 2

Prob. 12–9

594 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 *12–16. The pipe can be assumed roller supported at its ends and by a rigid saddle C at its center. The saddle rests on a cable that is connected to the supports. Determine the force that should be developed in the cable if the saddle keeps the pipe from sagging or deflecting at its center. The pipe and fluid within it have a combined weight of 125 lb>ft. EI is constant.

12.5 ft

1 ft A C

12.5 ft

Prob. 12–16

12–17. Determine the elastic curve in terms of the x 1 and x 2 coordinates. What is the deflection of end C of the shaft? EI is constant.

L 2

CBA

Prob. 12–17

12–18. The bar is supported by a roller constraint at B , which allows vertical displacement but resists axial load and moment. If the bar is subjected to the loading shown, determine the slope at A and the deflection at C . EI is constant.

12–19. Determine the deflection at B of the bar in Prob. 12–18 .

L— 2

A C

Probs. 12–18/19

12–14. The simply supported shaft has a moment of inertia of 2 I for region BC and a moment of inertia I for regions AB and CD . Determine the maximum deflection of the shaft due to the load P .

CA D

– 4 L –

4 L –

4 L

Prob. 12–14

12–15. The two wooden meter sticks are separated at their centers by a smooth rigid cylinder having a diameter of 50 mm. Determine the force F that must be applied at each end in order to just make their ends touch. Each stick has a width of 20 mm and a thickness of 5 mm. E w = 11 GPa.

F0.5 m 0.5 mF

Prob. 12–15

12.2 SLOPE AND DISPLACEMENT BY INTEGRATION 595

12 12–23. Determine the equations of the elastic curve using the coordinates x 1 and x 2 , and specify the deflection and slope at C . EI is constant.

B C

x1 x2

Probs. 12–23

*12–24. Determine the equations of the elastic curve using the coordinates x 1 and x 2 , and specify the slope at A . EI is constant.

B C

x1 x2

Probs. 12–24

12–25. The floor beam of the airplane is subjected to the loading shown. Assuming that the fuselage exerts only vertical reactions on the ends of the beam, determine the maximum deflection of the beam. EI is constant.

8 ft 2 ft 2 ft

80 lb/ft

Prob. 12–25

*12–20. Determine the equations of the elastic curve using the x1 and x2 coordinates, and specify the slope at A and the deflection at C. EI is constant.

A B C

x1 x2 20 kip!ft

8 kip

20 ft 10 ft

Prob. 12–20

12–21. Determine the maximum deflection of the solid circular shaft. The shaft is made of steel having E = 200 GPa. It has a diameter of 100 mm.

1.5 m 1.5 m

6 kN!m 6 kN!m

8 kN

BCA

Prob. 12–21

12–22. Determine the elastic curve for the cantilevered W14 * 30 beam using the x coordinate. Specify the maximum slope and maximum deflection. E = 29(103) ksi .

B A

3 kip/ ft

9 ft

Prob. 12–22

596 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 *12–28. Determine the slope at end B and the maximum deflection of the cantilever triangular plate of constant thickness t . The plate is made of material having a modulus of elasticity of E .

b 2

B x

Prob. 12–28

12–29. Determine the equation of the elastic curve using the coordinates x 1 and x 2 , and specify the slope and deflection at B . EI is constant.

12–30. Determine the equations of the elastic curve using the coordinates x 1 and x 3 , and specify the slope and deflection at point B . EI is constant.

x2 x3

Probs. 12–29/30

12–26. Determine the maximum deflection of the rectangular simply supported beam. The beam is made of wood having a modulus of elasticity of E = 1.5 (10 3 ) ksi.

8 ft 4 ft

3 in.

6 in.

600 lb

x1 x2

Prob. 12–26

12–27. Determine the elastic curve and the maximum deflection of the cantilever beam.

w 0

Prob. 12–27

12.3 DISCONTINUITY FUNCTIONS 597

1212121212 *12.3 Discontinuity Functions The method of integration, used to find the equation of the elastic curve for a beam or shaft, is convenient if the load or internal moment can be expressed as a continuous function throughout the beam’s entire length. If several different loadings act on the beam, however, the method becomes more tedious to apply, because separate loading or moment functions must be written for each region of the beam. Furthermore, integration of these functions requires the evaluation of integration constants using both boundary and continuity conditions. For example, the beam shown in Fig. 12–14 requires four moment functions to be written. They describe the moment in regions AB , BC , CD , and DE . When applying the moment-curvature relationship, EI d2v>dx2 = M , and integrating each moment equation twice, we must evaluate eight constants of integration. These involve two boundary conditions that require zero displacement at points A and E , and six continuity conditions for both slope and displacement at points B , C , and D .

In this section, we will discuss a method for finding the equation of the elastic curve for a multiply loaded beam using a single expression , either formulated from the loading on the beam, w = w(x), or from the beam’s internal moment, M = M(x). If the expression for w is substituted into EI d4v>dx4 = w(x) and integrated four times, or if the expression for M is substituted into EI d2v>dx2 = M(x) and integrated twice, the constants of integration will be determined only from the boundary conditions. Since the continuity equations will not be involved, the analysis will be greatly simplified.

Discontinuity Functions. In order to express the load on the beam or the internal moment within it using a single expression, we will use two types of mathematical operators known as discontinuity functions .

A E

P w

CB D

Fig. 12–14

For safety, the beams supporting these bags of cement must be designed for strength and a restricted amount of deflection.

598 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

1212121212

Macaulay Functions. For purposes of beam or shaft deflection, Macaulay functions, named after the mathematician W. H. Macaulay, can be used to describe distributed loadings . These functions can be written in general form as

8x - a9n = b 0 foff r x 6 a (x - a)a n foff r x Ú a (12–11) n Ú 0

Here x represents the coordinate position of a point along the beam, and a is the location on the beam where a “discontinuity” occurs, namely the point where a distributed loading begins . Note that the Macaulay function 8x - a9n is written with angle (or Macaulay) brackets to distinguish it from the ordinary function (x - a)n, written with parentheses. As stated by the equation, only when x Ú a is 8x - a9n = (x - a)n, otherwise it is zero. Furthermore, these functions are valid only for exponential values n Ú 0. Integration of Macaulay functions follows the same rules as for ordinary functions, i.e.,

L 8x - a9ndx = = 8x - a9n+1n + 1 + C (12–12) Note how the Macaulay functions describe both the uniform load

w0 (n = 0) and triangular load (n = 1), shown in Table 12–2 , items 3 and 4. This type of description can, of course, be extended to distributed loadings having other forms. Also, it is possible to use superposition with the

(1)

(2)

(3)

(4) slope ! m

Loading Loading Function w ! w(x)

w ! M0˚x"a¬"2

w ! P˚x"a¬"1

w ! w0˚x"a¬0

w ! m˚x"a¬1x a

x a

Shear V ! w(x)dx Moment M ! Vdx

V ! M0˚x"a¬"1

V ! P˚x"a¬0

V ! w0˚x"a¬1

V ! ˚x"a¬2 2 m

M ! M0˚x"a¬0

M ! P˚x"a¬1

M !

˚x"a¬2 2

M ! ˚x"a¬3 6 m

TABLE 12–2

12.3 DISCONTINUITY FUNCTIONS 599

1212121212 uniform and triangular loadings to create the Macaulay function for a trapezoidal loading. Using integration, the Macaulay functions for shear, V = 1w(x) dx, and moment, M = 1V dx, are also shown in the table. Singularity Functions. These functions are only used to describe the point location of concentrated forces or couple moments acting on a beam or shaft. Specifically, a concentrated force P can be considered as a special case of an infinitely large distributed loading, where the intensity of the loading is w = P>P such that its length is P, where P S 0, Fig. 12–15 . The area under this loading diagram is equivalent to P, positive upwards , and has this value only when x = a . We will use a symbolic representation w to express this result, namely

w = P8x - a9-1 = b 0 for x ! a P for x = a

(12–13)

This expression is referred to as a singularity function since it takes on the value P only at the point x = a where the load acts, otherwise it is zero. *

In a similar manner, a couple moment M0, considered positive clockwise , is a limit as P S 0 of two distributed loadings as shown in Fig. 12–16 . Here the following function describes its value.

w = M08x - a9-2 = b0 for x ! aM0 for x = a (12–14) The exponent n = -2, in order to ensure that the units of w, force per length, are maintained.

Integration of the above two singularity functions follow the rules of operational calculus and yields results that are different from those of Macaulay functions. Specifically,

L 8x - a9ndx = 8x - a9n+1, n = -1, -2 (12–15) Using this formula, notice how M0 and P , described in Table 12–2 , items 1 and 2, are integrated once, then twice, to obtain the internal shear and moment in the beam.

Application of Eqs. 12–11 through 12–15 provides a rather direct means for expressing the loading or the internal moment in a beam as a function of x . When doing so, close attention must be paid to the signs of the external loadings. As stated above, and as shown in Table 12–2 , concentrated forces and distributed loads are positive upward, and couple moments are positive clockwise . If this sign convention is followed, then the internal shear and moment are in accordance with the beam sign convention established in Sec. 6.1 .

w ! P P

Fig. 12–15

= a

w ! P P

M0 P2

w ! P P

Fig. 12–16

* This function 8 x − a 9 −1 is also referred to as a unit impulse function or the Dirac delta.

600 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

1212121212 As an example of how to apply discontinuity functions to describe

the loading or internal moment consider the beam loaded as shown in Fig. 12–17 a . Here the reactive 2.75-kN force created by the roller, Fig. 12–17 b , is positive since it acts upward, and the 1.5-kN · m couple moment is also positive since it acts clockwise. Finally, the trapezoidal loading is negative and has been separated into triangular and uniform loadings. From Table 12–2 , the loading at any point x on the beam is therefore

3 m

3 kN/m 1.5 kN!m

6 kN/m

(a)

3 m 3 m

3 kN/m

2.75 kN

3 kN/m

(b)

m " " 1 kN/m2 3 m

3 kN/m

1.5 kN!m

Fig. 12–17

w = 2.75 kN8x - 09-1 + 1.5 kN # m8x - 3 m9-2 - 3 kN>m8x - 3 m90 - 1 kN>m28x - 3 m91 The reactive force at B is not included here since x is never greater than 6 m, and furthermore, this value is of no consequence in calculating the slope or deflection. We can determine the moment expression directly from Table 12–2 , rather than integrating this expression twice. In either case,

M = 2.75 kN8x - 091 + 1.5 kN # m8x - 3 m90 - 3 kN>m 2

8x - 3 m92 - 1 kN>m2 6

8x - 3 m93 = 2.75x + 1.58x - 390 - 1.58x - 392 - 1

6 8x - 393

The deflection of the beam can now be determined after this equation is integrated two successive times and the constants of integration are evaluated using the boundary conditions of zero displacement at A and B .

12.3 DISCONTINUITY FUNCTIONS 601

1212121212 Procedure for Analysis

The following procedure provides a method for using discontinuity functions to determine a beam’s elastic curve. This method is particularly advantageous for solving problems involving beams or shafts subjected to several loadings , since the constants of integration can be evaluated by using only the boundary conditions, while the compatibility conditions are automatically satisfied.

Elastic Curve.

• Sketch the beam’s elastic curve and identify the boundary conditions at the supports.

• Zero displacement occurs at all pin and roller supports, and zero slope and zero displacement occur at fixed supports.

• Establish the x axis so that it extends to the right and has its origin at the beam’s left end.

Load or Moment Function.

• Calculate the support reactions and then use the discontinuity functions in Table 12–2 to express either the loading w or the internal moment M as a function of x . Make sure to follow the sign convention for each loading as it applies for this equation.

• Note that the distributed loadings must extend all the way to the beam’s right end to be valid. If this does not occur, use the method of superposition, which is illustrated in Example 12.6 .

Slope and Elastic Curve.

• Substitute w into EI d 4v>dx4 = w(x) , or M into the moment curvature relation EI d 2v>dx2 = M, and integrate to obtain the equations for the beam’s slope and deflection.

• Evaluate the constants of integration using the boundary conditions, and substitute these constants into the slope and deflection equations to obtain the final results.

• When the slope and deflection equations are evaluated at any point on the beam, a positive slope is counterclockwise , and a positive displacement is upward .

602 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.5

Determine the maximum deflection of the beam shown in Fig. 12–18 a . EI is constant.

10 ft

(a)

20 ft

8 kip

120 kip!ft

vC vD

10 ft

30 ft

8 kip

120 kip!ft

x 6 kip 2 kip

(b) Fig. 12–18

SOLUTION Elastic Curve. The beam deflects as shown in Fig. 12–18 a . The boundary conditions require zero displacement at A and B .

Loading Function. The reactions have been calculated and are shown on the free-body diagram in Fig. 12–18 b . The loading function for the beam can be written as

w = -8 kip 8x - 09-1 + 6 kip 8x - 10 ft9 -1 The couple moment and force at B are not included here, since they are located at the right end of the beam, and x cannot be greater than 30 ft. Integrating dV >dx = w(x), we get

V = -88x - 090 + 68x - 1090 In a similar manner, dM >dx = V yields

M = -88x - 091 + 68x - 1091 = (-8x + 68x - 1091) kip # ft

12.3 DISCONTINUITY FUNCTIONS 603

12 Notice how this equation can also be established directly using the results of Table 12–2 for moment.

Slope and Elastic Curve. Integrating twice yields

EI d2v dx2

= -8x + 68x - 1091 EI

dv dx

= -4x2 + 38x - 1092 + C1 EIv = - 4

3 x3 + 8x - 1093 + C1x + C2 (1)

From Eq. 1 , the boundary condition v = 0 at x = 10 ft and v = 0 at x = 30 ft gives

0 = -1333 + (10 - 10)3 + C1(10) + C2 0 = -36 000 + (30 - 10)3 + C1(30) + C2

Solving these equations simultaneously for C1 and C2, we get C1 = 1333 and C2 = -12 000. Thus,

EI dv dx

= -4x2 + 38x - 1092 + 1333 (2) EIv = - 4

3 x3 + 8x - 1093 + 1333x - 12 000 (3)

From Fig. 12–18 a , maximum displacement may occur either at C, or at D , where the slope dv>dx = 0. To obtain the displacement of C , set x = 0 in Eq. 3 . We get

vC = - 12 000 kip # ft3

EI Ans.

The negative sign indicates that the displacement is downward as shown in Fig. 12–18 a . To locate point D , use Eq. 2 with x 7 10 ft and dv>dx = 0. This gives

0 = -4xD 2 + 3(xD - 10)2 + 1333

xD 2 + 60xD - 1633 = 0

Solving for the positive root, xD = 20.3 ft

Hence, from Eq. 3 ,

EIvD = - 4 3

(20.3)3 + (20.3 - 10)3 + 1333(20.3) - 12 000

vD = 5006 kip # ft3

Comparing this value with vC, we see that vmax = vC .

604 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.6

Determine the equation of the elastic curve for the cantilevered beam shown in Fig. 12–19 a . EI is constant.

SOLUTION Elastic Curve. The loads cause the beam to deflect as shown in Fig. 12–19 a . The boundary conditions require zero slope and displacement at A .

Loading Function. The support reactions at A have been calculated and are shown on the free-body diagram in Fig. 12–19 b . Since the distributed loading in Fig. 12–19 a does not extend to C as required, we can use the superposition of loadings shown in Fig. 12–19 b to represent the same effect. By our sign convention, the beam’s loading is therefore

w = 52 kN8x - 09-1 - 258 kN # m8x - 09-2 - 8 kN>m8x - 090 + 50 kN # m8x - 5 m9-2 + 8 kN>m8x - 5 m90 The 12-kN load is not included here, since x cannot be greater than 9 m. Because dV >dx = w(x), then by integrating, neglecting the constant of integration since the reactions are included in the load function, we have

12 kN

5 m

(a)

4 m

8 kN/m 50 kN!m

A B

(b)

12 kN

4 m

8 kN/m

50 kN!m A B C

8 kN/m52 kN

258 kN!m

5 m

Fig. 12–19

V = 528x - 090 - 2588x - 09-1 - 88x - 091 + 508x - 59-1 + 88x - 591 Furthermore, dM >dx = V , so that integrating again yields

M = -2588x - 090 + 528x - 091 - 1 2

(8)8x - 092 + 508x - 590 + 1 2

(8)8x - 592 = (-258 + 52x - 4x2 + 508x - 590 + 48x - 592) kN # m

This same result can be obtained directly from Table 12–2 .

Slope and Elastic Curve. Applying Eq. 12–10 and integrating twice, we have

EI d2v

dx2 = -258 + 52x - 4x2 + 508x - 590 + 48x - 592

EI dv dx

= -258x + 26x2 - 4 3

x3 + 508x - 591 + 4 3

8x - 593 + C1 EIv = -129x2 + 26

3 x3 - 1

3 x4 + 258x - 592 + 1

3 8x - 594 + C1x + C2

Since dv>dx = 0 at x = 0, C1 = 0; and v = 0 at x = 0, so C2 = 0. Thus,

v = 1 EI

a -129x2 + 26 3

x3 - 1 3

x4 + 258x - 592 + 1 3

8x - 594b m Ans.

12.3 DISCONTINUITY FUNCTIONS 605

12–31. The shaft is supported at A by a journal bearing that exerts only vertical reactions on the shaft, and at C by a thrust bearing that exerts horizontal and vertical reactions on the shaft. Determine the equation of the elastic curve. EI is constant.

A B

a b

Prob. 12–31

*12–32. The shaft supports the two pulley loads shown. Determine the equation of the elastic curve. The bearings at A and B exert only vertical reactions on the shaft. EI is constant.

A B

40 lb

20 in.20 in.20 in.

60 lb

Prob. 12–32

12–33. Determine the equation of the elastic curve, the maximum deflection in region BC , and the deflection of end A of the shaft. EI is constant.

P a2aa

A B

C D

Prob. 12–33

12–34. Determine the equation of the elastic curve, the maximum deflection in region AB , and the deflection of end C of the shaft. EI is constant.

P P aaa

A CB

Prob. 12–34

12–35. The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant.

A x

8 ft4 ft 4 ft

5 kip!ft 5 kip!ft 3 kip/ft

Prob. 12–35

*12–36. Determine the equation of the elastic curve, the slope at A , and the deflection at B of the simply supported beam. EI is constant.

12–37. Determine the equation of the elastic curve and the maximum deflection of the simply supported beam. EI is constant.

A CB

L 3

M0 M0

Probs. 12–36/37

PROBLEMS

606 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 12–42. The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant.

A B

3 m1.5 m

6 kN/m 20 kN

1.5 m

Prob. 12–42

12–43. Determine the equation of the elastic curve. EI is constant.

B 6 kip!ft

8 ft

4 kip

8 ft8 ft

6 kip!ft

Prob. 12–43

*12–44. The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant.

A B

3 m1.5 m

20 kN

1.5 m

20 kN

Prob. 12–44

12–38. The beam is subjected to the loads shown. Determine the equation of the elastic curve. EI constant.

2 kip

8 ft

4 kip

4 kip!ft

8 ft8 ft

Prob. 12–38

12–39. Determine the maximum deflection of the cantilevered beam. The beam is made of material having an E = 200 GPa and I = 65.0(10 6 ) mm4.

30 kN/m

1.5 m 1.5 m

15 kN

Prob. 12–39

*12–40. Determine the slope at A and the deflection of end C of the overhang beam. E = 29(10 3 ) ksi and I = 204 in 4 .

12–41. Determine the maximum deflection in region AB of the overhang beam. E = 29(10 3 ) ksi and I = 204 in 4 .

6 ft6 ft 6 ft

2 kip/ft 3 kip

6 kip

C B

Probs. 12–40/41

12.3 DISCONTINUITY FUNCTIONS 607

12 12–45. Determine the deflection at each of the pulleys C, D , and E . The shaft is made of steel and has a diameter of 30 mm. The bearings at A and B exert only vertical reactions on the shaft. E st = 200 GPa.

150 N 60 N 150 N

250 mm 250 mm250 mm 250 mm

D EC

Prob. 12–45

12–46. Determine the slope of the shaft at the bearings at A and B. The shaft is made of steel and has a diameter of 30 mm. The bearings at A and B exert only vertical reactions on the shaft. E st = 200 GPa.

150 N 60 N 150 N

250 mm 250 mm250 mm 250 mm

D EC

Prob. 12–46

12–47. The shaft is made of steel and has a diameter of 15 mm. Determine its maximum deflection. The bearings at A and B exert only vertical reactions on the shaft. E st = 200 GPa.

15 mm

250 N 80 N

200 mm 200 mm300 mm

Prob. 12–47

*12–48. The wooden beam is subjected to the load shown. Determine the equation of the elastic curve. Specify the deflection at the end C . E w = 1.6(10 3 ) ksi.

6 in.

12 in.A C

9 ft

1.5 kip

0.8 kip/ft

9 ft

Prob. 12–48

12–49. Determine the displacement C and the slope at A of the beam. EI is constant.

A B

6 ft 9 ft

8 kip/ ft

Prob. 12–49

12–50. The beam is subjected to the load shown. Determine the equations of the slope and elastic curve. EI is constant.

A B

5 m 3 m

x C

3 kN/m

15 kN!m

Prob. 12–50

608 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

1212121212 *12.4 Slope and Displacement by the Moment-Area Method

The moment-area method provides a semigraphical technique for finding the slope and displacement at specific points on the elastic curve of a beam or shaft. Application of the method requires calculating areas associated with the beam’s moment diagram; and so if this diagram consists of simple shapes, the method is very convenient to use. Normally this is the case when the beam is loaded with concentrated forces and couple moments.

To develop the moment-area method we will make the same assumptions we used for the method of integration: The beam is initially straight, it is elastically deformed by the loads, such that the slope and deflection of the elastic curve are very small, and the deformations are only caused by bending. The moment-area method is based on two theorems, one used to determine the slope and the other to determine the displacement at a point on the elastic curve.

Theorem 1. Consider the simply supported beam with its associated elastic curve, shown in Fig. 12–20 a . A differential segment dx of the beam is isolated in Fig. 12–20 b . Here the beam’s internal moment M deforms the element such that the tangents to the elastic curve at each side of the element intersect at an angle du. This angle can be determined from Eq. 12–10 , written as

EI d 2v

dx2 = EI

d dx

a dv dx

b = M Since the slope is small , u = dv>dx, and therefore du =

M EI

dx (12–16)

If the moment diagram for the beam is constructed and divided by the flexural rigidity, EI, Fig. 12–20 c , then this equation indicates that du is equal to the area under the “ M / EI diagram” for the beam segment dx . Integrating from a selected point A on the elastic curve to another point B , we have

uB>A = L B

A M EI

dx (12–17)

This equation forms the basis for the first moment-area theorem.

Theorem 1: The angle between the tangents at any two points on the elastic curve equals the area under the M>EI diagram between these two points .

The notation uB>A is referred to as the angle of the tangent at B measured with respect to the tangent at A . From the proof it should be evident that this angle is measured counterclockwise , from tangent A to

A B

tan B tan A

(a)

Elastic curve

uB/A

(b)

M M

dx BA

(c)

—E I M

—E I M Diagram

Fig. 12–20

12.4 SLOPE AND DISPLACEMENT BY THE MOMENT-AREA METHOD 609

1212121212 tangent B , if the area under the M>EI diagram is positive . Conversely, if the area is negative , or lies below the x axis, the angle uB>A is measured clockwise from tangent A to tangent B . Furthermore, from the dimensions of Eq. 12–17 , uB>A will be in radians . Theorem 2. The second moment-area theorem is based on the relative deviation of tangents to the elastic curve. Shown in Fig. 12–21 a is a greatly exaggerated view of the vertical deviation dt of the tangents on each side of the differential element dx . This deviation is caused by the curvature of the element and has been measured along a vertical line passing through point A on the elastic curve. Since the slope of the elastic curve and its deflection are assumed to be very small, it is satisfactory to approximate the length of each tangent line by x and the arc ds! by dt . Using the circular-arc formula s = ur, where r is the length x and s is dt , we can write dt = x du. Substituting Eq. 12–16 into this equation and integrating from A to B , the vertical deviation of the tangent at A with respect to the tangent at B can then be determined; that is,

tA >B = L B

A x

M EI

dx (12–18)

Since the centroid of an area is found from x 1dA = 1x dA , and 1(M >EI) dx represents the area under the M > EI diagram, we can also write

tA >B = x L B

A M EI

dx (12–19)

Here x is the distance from A to the centroid of the area under the M/EI diagram between A and B , Fig. 12–21 b .

The second moment-area theorem can now be stated in reference to Fig. 12–21 a as follows:

Theorem 2: The vertical distance between the tangent at a point (A) on the elastic curve and the tangent extended from another point (B) equals the moment of the area under the M>EI diagram between these two points (A and B). This moment is calculated about the point (A) where the vertical distance (tA >B) is to be determined .

Note that tA >B is not equal to tB>A , which is shown in Fig. 12–21 c . Specifically, the moment of the area under the M > EI diagram between A and B is calculated about point A to determine tA >B, Fig. 12–21 b , and it is calculated about point B to determine tB>A , Fig. 12–21 c .

If the moment of a positive M > EI area between A and B is found for tA >B, it indicates that point A is above the tangent extended from point B , Fig. 12–21 a . Similarly, negative M > EI areas indicate that point A is below the tangent extended from point B . This same rule applies for tB>A.

A B

tan B

tan A

(a)

du ds¿

A dx

tA/B

(b)

—E I M

x _ x

tanB tanA

(c)

A B tB/AtA/B

M EI

¿ _ x

Fig. 12–21

610 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

1212121212 Procedure for Analysis

The following procedure provides a method that may be used to apply the two moment-area theorems.

M/EI Diagram. • Determine the support reactions and draw the beam’s M/EI

diagram. If the beam is loaded with concentrated forces, the M/EI diagram will consist of a series of straight line segments, and the areas and their moments required for the moment-area theorems will be relatively easy to calculate. If the loading consists of a series of distributed loads, the M/EI diagram will consist of parabolic or perhaps higher-order curves, and it is suggested that the table on the inside front cover be used to locate the area and centroid under each curve.

Elastic Curve. • Draw an exaggerated view of the beam’s elastic curve. Recall that

points of zero slope and zero displacement always occur at a fixed support, and zero displacement occurs at all pin and roller supports.

• If it becomes difficult to draw the general shape of the elastic curve, use the moment (or M/EI ) diagram. Realize that when the beam is subjected to a positive moment , the beam bends concave up , whereas negative moment bends the beam concave down . Furthermore, an inflection point or change in curvature occurs where the moment in the beam (or M / EI ) is zero.

• The unknown displacement and slope to be determined should be indicated on the curve.

• Since the moment-area theorems apply only between two tangents , attention should be given as to which tangents should be constructed so that the angles or vertical distance between them will lead to the solution of the problem. In this regard, the tangents at the supports should be considered , since the beam has zero displacement and/or zero slope at the supports.

Moment-Area Theorems. • Apply Theorem 1 to determine the angle between any two

tangents on the elastic curve and Theorem 2 to determine the vertical distance between the tangents.

• The algebraic sign of the answer can be checked from the angle or vertical distance indicated on the elastic curve.

• A positive uB>A represents a counterclockwise rotation of the tangent at B with respect to the tangent at A , and a positive tB>A indicates that point B on the elastic curve lies above the extended tangent from point A .

12.4 SLOPE AND DISPLACEMENT BY THE MOMENT-AREA METHOD 611

12 EXAMPLE 12.7

Determine the slope of the beam shown in Fig. 12–22 a at point B . EI is constant.

(a)

(b)

B L

A x

PL EI

M EI

tan A

A uB/A

Fig. 12–22

SOLUTION M/EI Diagram. See Fig. 12–22 b .

Elastic Curve. The force P causes the beam to deflect as shown in Fig. 12–22 c . (The elastic curve is concave downward, since M/EI is negative.) The tangent at B is indicated since we are required to find uB. Also, the tangent at the support ( A ) is shown. This tangent has a known zero slope. By the construction, the angle between tan A and tan B , that is, uB>A , is equivalent to uB, or

uB = uB>A Moment-Area Theorem. Applying Theorem 1 , uB>A is equal to the area under the M > EI diagram between points A and B ; that is,

uB = uB>A = 12 a - PLEI bL = - PL

2EI Ans .

The negative sign indicates that the angle measured from the tangent at A to the tangent at B is clockwise . This checks, since the beam slopes downward at B .

612 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.8

Determine the displacement of points B and C of the beam shown in Fig. 12–23 a . EI is constant.

(a)

A B C M0

L 2

(b)

BA x C

L 2

L 4

L 2

M EI

M0 EI

(c)

tan B

tan C

tan A

tB/A " #B tC/A " #C

Fig. 12–23

SOLUTION M/EI Diagram. See Fig. 12–23 b . Elastic Curve. The couple moment at C causes the beam to deflect as shown in Fig. 12–23 c . The tangents at B and C are indicated since we are required to find !B and !C . Also, the tangent at the support ( A ) is shown since it is horizontal. The required displacements can now be related directly to the vertical distance between the tangents at B and A and C and A . Specifically,

!B = tB>A !C = tC>A

Moment-Area Theorem. Applying Theorem 2 , tB>A is equal to the moment of the shaded area under the M > EI diagram between A and B calculated about point B (the point on the elastic curve), since this is the point where the vertical distance is to be determined. Hence, from Fig. 12–23 b ,

!B = tB>A = aL4 b J ¢ - M0EI ≤ aL2 b R = - M0L28EI Ans . Likewise, for tC>A we must determine the moment of the area under

the entire M > EI diagram from A to C about point C (the point on the elastic curve). We have

!C = tC>A = aL2 b J ¢ - M0EI ≤(L) R = - M0L22EI Ans . NOTE: Since both answers are negative , they indicate that points B and C lie below the tangent at A . This checks with Fig. 12–23 c .

12.4 SLOPE AND DISPLACEMENT BY THE MOMENT-AREA METHOD 613

12 EXAMPLE 12.9

Determine the slope at point C of the shaft in Fig. 12–24 a . EI is constant.

(a)

A B D C

L 2

L 4

(b)

x D C

L 4

PL 4 EI PL

8 EI

M EI

(c)

tan C

tan D (horizontal)

uC/D

Fig. 12–24

SOLUTION M/EI Diagram. See Fig. 12–24 b . Elastic Curve. Since the loading is applied symmetrically to the shaft, the elastic curve is symmetric, and the tangent at D is horizontal, Fig. 12–24 c . Also the tangent at C is drawn, since we must find the slope uC . By the construction, the angle uC>D between the tangents at tan D and C is equal to uC; that is,

uC = uC>D Moment-Area Theorem. Using Theorem 1 , uC>D is equal to the shaded area under the M > EI diagram between points D and C . We have uC = uC>D = a PL8EI b aL4 b + 12 a PL4EI - PL8EI b aL4 b = 3PL264EI Ans .

What does the positive result indicate?

614 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.10

(a)

A B C

2 m

16 kN

4 m 2 m

(b)

x C

2 m 4 m 2 m

A B

M EI

8 EI

24 EI

(c)

tan C

tan A

C tan B BA

uC/A

uC tB/A

Fig. 12–25

Determine the slope at point C for the steel beam in Fig. 12–25 a . Take Est = 200 GPa, I = 17(106) mm4.

SOLUTION M/EI Diagram. See Fig. 12–25 b . Elastic Curve. The elastic curve is shown in Fig. 12–25 c . The tangent at C is shown since we are required to find uC . Tangents at the supports, A and B , are also constructed as shown. Angle uC>A is the angle between the tangents at A and C . The slope at A , uA, in Fig. 12–25 c can be found using ! uA ! = ! tB>A ! >LAB . This equation is valid since tB>A is actually very small, so that tB>A in meters can be approximated by the length of a circular arc defined by a radius of LAB = 8 m and a sweep of uA in radians. (Recall that s = ur. ) From the geometry of Fig. 12–25 c , we have

! uC ! = ! uA ! - ! uC>A ! = 2 tB>A8 2 - ! uC>A ! (1) Note that Example 12.9 could also be solved using this method.

Moment-Area Theorems. Using Theorem 1 , uC>A is equivalent to the area under the M > EI diagram between points A and C ; that is,

uC>A = 12 (2 m)a 8 kN # mEI b = 8 kN # m2EI Applying Theorem 2 , tB>A is equivalent to the moment of the area

under the M > EI diagram between B and A about point B (the point on the elastic curve), since this is the point where the vertical distance is to be determined. We have

tB>A = a2 m + 13 (6 m)b c 12 (6 m)a 24 kN # mEI b d + a 2

3 (2 m)b c 1

2 (2 m)a 24 kN # m

EI b d

= 320 kN # m3

Substituting these results into Eq. 1 , we get

uC = 320 kN # m2

(8 m) EI - 8 kN

# m2 EI

= 32 kN # m2

EI b

We have calculated this result in units of kN and m, so converting EI into these units, we have

uC = 32 kN # m2

[200(106) kN>m2[17(10-6) m4] = 0.00941 rad b Ans .

12.4 SLOPE AND DISPLACEMENT BY THE MOMENT-AREA METHOD 615

12 EXAMPLE 12.11

Determine the displacement at C for the beam shown in Fig. 12–26 a . EI is constant.

(a)

A C

M0 B

L 2

(b)

x A C BL

2 L 2

M EI

M0 EI

M0 2EI

(c)

tan C C

B tan A

tan B

!¿

tC/B tA/B

L 2

Fig. 12–26

SOLUTION M/EI Diagram. See Fig. 12–26 b . Elastic Curve. The tangent at C is drawn on the elastic curve since we are required to find !C, Fig. 12– 26 c . (Note that C is not the location of the maximum deflection of the beam, because the loading and hence the elastic curve are not symmetric .) Also indicated in Fig. 12–26 c are the tangents at the supports A and B . It is seen that !C = !" - tC>B . If tA >B is determined, then !" can be found from proportional triangles, that is, !">(L >2) = tA >B>L or !" = tA >B>2. Hence, !C =

tA >B 2

- tC>B (1) Moment-Area Theorem. Applying Theorem 2 to determine tA >B and tC>B, we have

tA >B = a 13 (L)b J12 (L)¢M0EI ≤ R = M0L26EI tC>B = a 13 aL2 b b J12 aL2 b ¢ M02EI ≤ R = M0L248EI

Substituting these results into Eq. 1 gives

!C = 1 2

¢M0L2 6EI ≤ - ¢M0L2

48EI ≤

= M0L

16EI T Ans .

616 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.12

Determine the displacement at point C for the steel overhanging beam shown in Fig. 12–27 a . Take Est = 29(103) ksi, I = 125 in4.

(a)

A B

5 kip

10 kip5 kip

12 ft 12 ft

SOLUTION M/EI Diagram. See Fig. 12–27 b . Elastic Curve. The loading causes the beam to deflect as shown in Fig. 12–27 c . We are required to find !C . By constructing tangents at C and at the supports A and B , it is seen that !C = ! tC>A ! - !". However, !" can be related to tB>A by proportional triangles; that is, !">24 = ! tB>A ! >12 or !" = 2 ! tB>A ! . Hence !C = ! tC>A ! - 2 ! tB>A ! (1) Moment-Area Theorem. Applying Theorem 2 to determine tC>A and tB>A , we have

tC>A = (12 ft)a 12 (24 ft)a - 60 kip # ftEI b b = -

8640 kip # ft3 EI

tB>A = a 13 (12 ft)b c 12 (12 ft)a - 60 kip # ftEI b d = - 1440 kip # ft3EI Why are these terms negative? Substituting the results into Eq. 1 yields

!C = 8640 kip # ft3

EI - 2¢ 1440 kip # ft3

EI ≤ = 5760 kip # ft3

EI T

Realizing that the calculations were made in units of kip and ft, we have

!C = 5760 kip # ft3(1728 in3>ft3) [29(103) kip>in2](125 in4) = 2.75 in. T Ans .

tan A tan B

A tC/A tB/A

!¿

Fig. 12–27

(b)

12 ft 12 ft

A B C

M EI

"60 EI

12.4 SLOPE AND DISPLACEMENT BY THE MOMENT-AREA METHOD 617

F12–7. Determine the slope and deflection of end A of the cantilevered beam. E = 200 GPa and I = 65.0(10- 6) m4.

B A

6 kN

20 kN!m

3 m

F12–7

F12–8. Determine the slope and deflection of end A of the cantilevered beam. E = 200 GPa and I = 126(10-6) m4 .

B A

20 kN 10 kN

1 m 1 m

F12–8

F12–9. Determine the slope and deflection of end A of the cantilevered beam. E = 200 GPa and I = 121(10-6) m4 .

B A

60 kN

30 kN!m

1 m 1 m

F12–9

F12–10. Determine the slope and deflection at A of the cantilevered beam. E = 29(103) ksi, I = 245 in4.

3 kip

3 ft 3 ft

2 kip/ft

F12–10

F12–11. Determine the maximum deflection of the simply supported beam. E = 200 GPa and I = 42.8(10-6) m4.

3 m

20 kN

10 kN!m 10 kN!m

3 m

A B

F12–11

F12–12. Determine the maximum deflection of the simply supported beam. E = 200 GPa and I = 39.9(10-6) m4.

6 m

40 kN!m 10 kN!m

A B

F12–12

FUNDAMENTAL PROBLEMS

618 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12–51. Determine the slope and deflection at C . EI is constant.

15 kip

15 ft30 ft

Prob. 12–51

*12–52. Determine the slope and deflection at C . EI is constant.

10 kN

3 m6 m

Prob. 12–52

12–53. Determine the deflection of end B of the cantilever beam. EI is constant.

L 2

P P

Prob. 12–53

12–54. If the bearings at A and B exert only vertical reactions on the shaft, determine the slope at B and the deflection at C . EI is constant.

a a a

P P

Prob. 12–54

12–55. The composite simply supported steel shaft is subjected to a force of 10 kN at its center. Determine its maximum deflection. E st = 200 GPa.

5 kN

200 mm

5 kN

200 mm

200 mm A

B 20 mm

40 mm

Prob. 12–55

*12–56. If the bearings at A and B exert only vertical reactions on the shaft, determine the slope at A and the maximum deflection of the shaft. EI is constant.

50 lb!ft

2 ft 4 ft 2 ft

50 lb!ft

Prob. 12–56

PROBLEMS

12.4 SLOPE AND DISPLACEMENT BY THE MOMENT-AREA METHOD 619

12 12–61. Determine the position a of roller support B in terms of L so that the deflection at end C is the same as the maximum deflection of region AB of the overhang beam. EI is constant.

A B

Prob. 12–61

12–62. The flat spring is made of A-36 steel and has a rectangular cross section as shown. Determine the maximum elastic load P that can be applied. What is the deflection at B when P reaches its maximum value? Assume that the spring is fixed supported at A .

0.1 in.

1.5 in.3 in.

14 in.

Prob. 12–62

12–63. Determine the slope and the deflection of end B of the cantilever beam. EI is constant.

M0M0 B

2 L 2

Prob. 12–63

12–57. Determine the maximum deflection of the shaft. EI is constant. The bearings exert only vertical reactions on the shaft.

_ aa 2

a_ 2

A B

P P

Prob. 12–57

12–58. Determine the deflection at C and the slope of the beam at A , B , and C . EI is constant.

6 m

8 kN!m

3 m

Prob. 12–58

12–59. Determine the slope at A of the solid circular shaft of diameter 100 mm. The shaft is made of steel having a modulus elasticity of E = 200 GPa.

*12–60. Determine the deflection at C of the solid circular shaft of diameter 100 mm. The shaft is made of steel having a modulus elasticity of E = 200 GPa.

A B C

1.5 m1.5 m

6 kN 6 kN

1.5 m

Probs. 12–59/60

620 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 *12–68. Determine the slope at A and the maximum deflection of the simply supported beam. EI is constant.

L A B

M0 M0

Prob. 12–68

12–69. Determine the slope at C and the deflection at B . EI is constant.

A B

M0 Pa=

Prob. 12–69

12–70. Determine the slope at A and the maximum deflection in the beam. EI is constant.

A B

6 ft6 ft 12 ft

12 kip

24 kip!ft

Prob. 12–70

12–71. The beam is made of a ceramic material. In order to obtain its modulus of elasticity, it is subjected to the loading shown. If the moment of inertia is I and the beam has a measured maximum deflection ! , determine E . The supports at A and D exert only vertical reactions on the beam.

A D

a a

B C

P P

Prob. 12–71

*12–64. The beam is subjected to the loading shown. Determine the slope at B and deflection at C . EI is constant.

C a

Prob. 12–64

12–65. The beam is subjected to the loading shown. Determine the slope at A and the displacement at C . Assume the support at A is a pin and B is a roller. EI is constant.

A C B

PPP

a a a a

Prob. 12–65

12–66. The shaft supports the gear at its end C . Determine the deflection at C and the slopes at the bearings A and B . EI is constant. The bearings exert only vertical reactions on the shaft.

12–67. The shaft supports the gear at its end C . Determine its maximum deflection within region AB . EI is constant. The bearings exert only vertical reactions on the shaft.

A B C

P2 ––L

2 ––L

Probs. 12–66/67

12.4 SLOPE AND DISPLACEMENT BY THE MOMENT-AREA METHOD 621

12 12–75. Determine the slope at B and the deflection at C of the beam. E = 200 GPa and I = 65.0(10 6 ) mm 4 .

C B

3 m 3 m

18 kN!m

40 kN

Prob. 12–75

*12–76. Determine the slope at point A and the maximum deflection of the simply supported beam. The beam is made of material having a modulus of elasticity E . The moment of inertia of segments AB and CD of the beam is I , while the moment of inertia of segment BC is 2 I .

A B C

L 2

L 4

Prob. 12–76

12–77. Determine the position a of roller support B in terms of L so that deflection at end C is the same as the maximum deflection of region AB of the overhang beam. EI is constant.

B C

Prob. 12–77

*12–72. A beam having a constant EI is supported as shown. Attached to the beam at A is a pointer, free of load. Both the beam and pointer are originally horizontal when no load is applied to the beam. Determine the distance between the end of the beam and the pointer after each has been displaced by the loading shown.

15 kN 20 kN

1 m 1 m

2 m

A B

Prob. 12–72

12–73. Determine the slope at A and the maximum deflection of the shaft. EI is constant.

M0 M0

A B

L 3

Prob. 12–73

12–74. The rod is constructed from two shafts for which the moment of inertia of AB is I and of BC is 2 I . Determine the maximum slope and deflection of the rod due to the loading. The modulus of elasticity is E .

A B C

L 2

Prob. 12–74

622 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 12–81. The two bars are pin connected at D . Determine the slope at A and the deflection at D . EI is constant.

–2 L –

2 LL

A C DB

Prob. 12–81

12–82. Determine the maximum deflection of the beam. EI is constant.

B C D

w w

Prob. 12–82

12–78. The beam is subjected to the loading shown. Determine the slope at B and deflection at C . EI is constant.

a a_a 2

_a 2

A B C

Prob. 12–78

12–79. The cantilevered beam is subjected to the loading shown. Determine the slope and displacement at C . EI is constant.

A B

Prob. 12–79

*12–80. Determine the slope at C and deflection at B . EI is constant.

a a

Prob. 12–80

12.5 METHOD OF SUPERPOSITION 623

1212121212 12.5 Method of Superposition The differential equation EI d4v>dx4 = w(x) satisfies the two necessary requirements for applying the principle of superposition; i.e., the load w(x) is linearly related to the deflection v(x), and the load is assumed not to change significantly the original geometry of the beam or shaft. As a result, the deflections for a series of separate loadings acting on a beam may be superimposed. For example, if v1 is the deflection for one load and v2 is the deflection for another load, the total deflection for both loads acting together is the algebraic sum v1 + v2 . Using tabulated results for various beam loadings, such as the ones listed in Appendix C , or those found in various engineering handbooks, it is therefore possible to find the slope and displacement at a point on a beam subjected to several different loadings by algebraically adding the effects of its various component parts.

The following examples illustrate how to use the method of superposition to solve deflection problems, where the deflection is caused not only by beam deformations, but also by rigid-body displacements, such as those that occur when the beam is supported by springs.

The resultant deflection at any point on this beam can be determined from the superposition of the deflections caused by each of the separate loadings acting on the beam.

624 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.13

Determine the displacement at point C and the slope at the support A of the beam shown in Fig. 12–28 a . EI is constant.

(a)

8 kN

4 m

vC =

2 kN/m

4 m 4 m

A C

2 kN/m

(vC)1

(vC)2

(b)

4 m

A C

(c)

8 kN

4 m

uA (uA)1

(uA)2

Fig. 12–28

SOLUTION The loading can be separated into two component parts as shown in Figs. 12–28 b and 12–28 c . The displacement at C and slope at A are found using the table in Appendix C for each part.

For the distributed loading,

(uA)1 = 3wL3

128EI =

3(2 kN>m)(8 m)3 128EI

= 24 kN # m2

EI b

(vC)1 = 5wL4

768EI =

5(2 kN>m)(8 m)4 768EI

= 53.33 kN # m3

EI T

For the 8-kN concentrated force,

(uA)2 = PL2

16EI =

8 kN(8 m)2

16EI =

32 kN # m2 EI

(vC)2 = PL3

48EI =

8 kN(8 m)3

48EI =

85.33 kN # m3 EI

The displacement at C and the slope at A are the algebraic sums of these components. Hence,

(+b) uA = (uA)1 + (uA)2 = 56 kN # m2

EI b Ans .

(+ T ) vC = (vC)1 + (vC)2 = 139 kN # m3

EI T Ans .

12.5 METHOD OF SUPERPOSITION 625

B A

(a) =

4 m

5 kN/m

2 m

10 kN

B A

(b)

4 m

5 kN/m

2 m

B A

(c)

4 m 2 m

2 m

10 kN

(d) B

10 kN

20 kN!m

(uB)2

(uB)1

(vC)3

(vC)2

(vC)1

Fig. 12–29

EXAMPLE 12.14

Determine the displacement at the end C of the overhanging beam shown in Fig. 12–29 a . EI is constant.

SOLUTION Since the table in Appendix C does not include beams with overhangs, the beam will be separated into a simply supported and a cantilevered portion. First we will calculate the slope at B , as caused by the distributed load acting on the simply supported span, Fig. 12–29 b .

(uB)1 = wL3

24EI =

5 kN>m(4 m)3 24EI

= 13.33 kN # m2

EI d

Since this angle is small , (uB)1 ! tan(uB)1, and the vertical displacement at point C is

(vC)1 = (2 m)¢ 13.33 kN # m2EI ≤ = 26.67 kN # m3EI c Next, the 10-kN load on the overhang causes a statically equivalent

force of 10 kN and couple moment of 20 kN # m at the support B of the simply supported span, Fig. 12–29 c . The 10-kN force does not cause a displacement or slope at B ; however, the 20@kN # m couple moment does cause a slope. The slope at B due to this moment is

(uB)2 = M0L

3EI =

20 kN # m(4 m) 3EI

= 26.67 kN # m2

EI b

so that the extended point C is displaced

(vC)2 = (2 m)¢ 26.7 kN # m2EI ≤ = 53.33 kN # m3EI T Finally, the cantilevered portion BC is displaced by the 10-kN force,

Fig. 12–29 d . We have

(vC)3 = PL3

3EI =

10 kN(2 m)3

3EI =

26.67 kN # m3 EI

Summing these results algebraically, we obtain the displacement of point C ,

(+ T ) vC = - 26.7 EI

+ 53.3 EI

+ 26.7 EI

= 53.3 kN # m3

EI T Ans.

626 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.15

Determine the displacement at the end C of the cantilever beam shown in Fig. 12–30 . EI is constant.

6 m 2 m

A B C

4 kN/m

vC uB

Fig. 12–30

SOLUTION Using the table in Appendix C for the triangular loading, the slope and displacement at point B are

uB = w0L

24EI =

4 kN>m(6 m)3 24EI

= 36 kN # m2

vB = w0L

30EI =

4 kN>m(6 m)4 30EI

= 172.8 kN # m3

The unloaded region BC of the beam remains straight, as shown in Fig. 12–30 . Since uB is small, the displacement at C becomes

(+ T ) vC = vB + uB(LBC)

= 172.8 kN # m3

EI + 36 kN

# m2 EI

(2 m)

= 244.8 kN # m3

EI T Ans .

12.5 METHOD OF SUPERPOSITION 627

12 EXAMPLE 12.16

1 kip2 kip Rigid body displacement

(b)

6 ft

A B

3 ft

k ! 15 kip/ft k ! 15 kip/ft

3 kip

6 ft

A B

3 ft C

3 kip Original position

(a)

6 ft3 ft

3 kip

Deformable body displacement

(c)

(vC)2

(vA)1 (vC)1 (vB)1

Fig. 12–31

The steel bar shown in Fig. 12–31 a is supported by two springs at its ends A and B . Each spring has a stiffness of k = 15 kip>ft and is originally unstretched. If the bar is loaded with a force of 3 kip at point C , determine the vertical displacement of the force. Neglect the weight of the bar and take Est = 29(103) ksi, I = 12 in4.

SOLUTION The end reactions at A and B are calculated and shown in Fig. 12–31 b . Each spring deflects by an amount

(vA)1 = 2 kip

15 kip>ft = 0.1333 ft (vB)1 =

1 kip 15 kip>ft = 0.0667 ft

If the bar is considered to be rigid , these displacements cause it to move into the position shown in Fig. 12–31 b . For this case, the vertical displacement at C is

(vC)1 = (vB)1 + 6 ft 9 ft

[(vA)1 - (vB)1]

= 0.0667 ft + 2 3

[0.1333 ft - 0.0667 ft] = 0.1111 ft T

We can find the displacement at C caused by the deformation of the bar, Fig. 12–31 c , by using the table in Appendix C . We have

(vC)2 = Pab

6EIL (L2 - b2 - a2)

= 3 kip(3 ft)(6 ft)[(9 ft)2 - (6 ft)2 - (3 ft)2]

6[29(103)kip>in2](144 in2>1 ft2)(12 in4)(1 ft4>20 736 in4)(9 ft) = 0.0149 ft T

Adding the two displacement components, we get

(+ T ) vC = 0.1111 ft + 0.0149 ft = 0.126 ft = 1.51 in. T Ans .

628 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12–83. The W10 * 15 cantilevered beam is made of A-36 steel and is subjected to the loading shown. Determine the slope and displacement at its end B .

6 ft 6 ft

3 kip/ft

B A

Prob. 12–83

*12–84. The W10 * 15 cantilevered beam is made of A-36 steel and is subjected to the loading shown. Determine the displacement at B and the slope at B .

6 ft 6 ft

6 kip 4 kip

Prob. 12–84

12–85. Determine the slope and deflection at end C of the overhang beam. EI is constant.

12–86. Determine the slope at A and the deflection at point D of the overhang beam. EI is constant.

BD C

aaa

Probs. 12–85/86

12–87. The W12 * 45 simply supported beam is made of A-36 steel and is subjected to the loading shown. Determine the deflection at its center C .

12 kip

12 ft 12 ft

50 kip!ft

Prob. 12–87

*12–88. Determine the slope at A and the deflection at point C of the simply supported beam. The modulus of elasticity of the wood is E = 10 GPa.

A B

3 kN 3 kN

1.5 m 1.5 m 3 m

100 mm

200 mmC

Prob. 12–88

12–89. The W8 * 24 simply supported beam is made of A-36 steel and is subjected to the loading shown. Determine the deflection at its center C .

6 kip/ft

8 ft 8 ft

A B

5 kip!ft

Prob. 12–89

PROBLEMS

12.5 METHOD OF SUPERPOSITION 629

12 *12–92. The W10 * 30 cantilevered beam is made of A-36 steel and is subjected to unsymmetrical bending caused by the applied moment. Determine the deflection of the centroid at its end A due to the loading. Hint: Resolve the moment into components and use superposition

15 ft

30 A

M ! 4.5 kip"ft

Prob. 12–92

12–93. The rod is pinned at its end A and attached to a torsional spring having a stiffness k , which measures the torque per radian of rotation of the spring. If a force P is always applied perpendicular to the end of the rod, determine the displacement of the force. EI is constant.

Prob. 12–93

12–94. Determine the vertical deflection and slope at the end A of the bracket. Assume that the bracket is fixed supported at its base, and neglect the axial deformation of segment AB . EI is constant.

6 in.

3 in.

8 kip

Prob. 12–94

12–90. The simply supported beam carries a uniform load of 2 kip>ft . Code restrictions, due to a plaster ceiling, require the maximum deflection not to exceed 1>360 of the span length. Select the lightest-weight A992 steel wide-flange beam from Appendix B that will satisfy this requirement and safely support the load. The allowable bending stress is sallow = 24 ksi and the allowable shear stress is tallow = 14 ksi. Assume A is a pin and B a roller support.

4 ft

A B

8 ft

8 kip

4 ft

2 kip/ft

8 kip

Prob. 12–90

12–91. The simply supported beam is made of A-36 steel and is subjected to the loading shown. Determine the deflection at its center C . I = 0.1457(10-3) m4.

4 kN/m

B A

5 m

20 kN

5 m

Prob. 12–91

630 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 12–97. The wide-flange beam acts as a cantilever. Due to an error it is installed at an angle u with the vertical. Determine the ratio of its deflection in the x direction to its deflection in the y direction at A when a load P is applied at this point. The moments of inertia are Ix and Iy. For the solution, resolve P into components and use the method of superposition. Note: The result indicates that large lateral deflections ( x direction) can occur in narrow beams, Iy V Ix, when they are improperly installed in this manner. To show this numerically, compute the deflections in the x and y directions for an A992 steel W10 * 15, with P = 1.5 kip, u = 10!, and L = 12 ft.

Vertical

Prob. 12–97

12–98. Determine the vertical deflection at the end A of the bracket. Assume that the bracket is fixed supported at its base B and neglect axial deflection. EI is constant.

a P

Prob. 12–98

12–95. The pipe assembly consists of three equal-sized pipes with flexibility stiffness EI and torsional stiffness GJ . Determine the vertical deflection at point A .

L– 2

L– 2 B

Prob. 12–95

*12–96. The framework consists of two A992 steel cantilevered beams CD and BA and a simply supported beam CB. If each beam is made of steel and has a moment of inertia about its principal axis Ix = 118 in4, determine the deflection at the center G of beam CB.

16 ft

8 ft

8 ftC G

15 kip

Prob. 12–96

12.6 STATICALLY INDETERMINATE BEAMS AND SHAFTS 631

1212121212 12.6 Statically Indeterminate Beams and Shafts

The analysis of statically indeterminate axially loaded bars and torsionally loaded shafts has been discussed in Secs. 4.4 and 5.5 , respectively. In this section we will illustrate a general method for determining the reactions on statically indeterminate beams and shafts. Specifically, a member of any type is classified as statically indeterminate if the number of unknown reactions exceeds the available number of equilibrium equations.

The additional support reactions on the beam or shaft that are not needed to keep it in stable equilibrium are called redundants . The number of these redundants is referred to as the degree of indeterminacy . For example, consider the beam shown in Fig. 12–32 a . If the free-body diagram is drawn, Fig. 12–32 b , there will be four unknown support reactions, and since three equilibrium equations are available for solution, the beam is classified as being indeterminate to the first degree. Either Ay, By, or MA can be classified as the redundant, for if any one of these reactions is removed, the beam remains stable and in equilibrium ( Ax cannot be classified as the redundant, for if it were removed, !Fx = 0 would not be satisfied.) In a similar manner, the continuous beam in Fig. 12–33 a is indeterminate to the second degree, since there are five unknown reactions and only three available equilibrium equations, Fig. 12–33 b . Here the two redundant support reactions can be chosen among Ay, By, Cy, and Dy.

(a)

P1 P2 P3

A D

Fig. 12–33

(a)

A B

Fig. 12–32

(b)

P1 P2 P3

Ay By Cy Dy

632 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

1212121212 To determine the reactions on a beam (or shaft) that is statically

indeterminate, it is first necessary to specify the redundant reactions. We can determine these redundants from the load-displacement relationship and the conditions of geometry known as compatibility conditions . Once determined, the redundants are then applied to the beam, and the remaining reactions are determined from the equations of equilibrium.

In the following sections we will illustrate this procedure for solution using the method of integration, Sec. 12.7 ; the moment-area method, Sec. 12.8 ; and the method of superposition, Sec. 12.9 .

12.7 Statically Indeterminate Beams and Shafts—Method of Integration

The method of integration, discussed in Sec. 12.2 , requires applying the load-displacement relationship, represented by the differential equation d2v>dx2 = M >EI . If the beam is statically indeterminate, M will be expressed in terms of position x and in terms of the unknown redundants. After integrating this equation twice, there will be two constants of integration along with the redundants to be determined. Although this is the case, these unknowns can always be found from the equilibrium equations and the boundary and/or continuity conditions for the problem.

The following example problems illustrate specific applications of this method using the procedure for analysis outlined in Sec. 12.2 .

An example of a statically indeterminate beam used to support a bridge deck.

12.7 STATICALLY INDETERMINATE BEAMS AND SHAFTS—METHOD OF INTEGRATION 633

12 EXAMPLE 12.17

x L

(a)

Bx A

w0L

(b) Ay

L 2 3

1 3

1 2

w0 w0

A 2 3

1 3

1 2

x L

Fig. 12–34

The beam is subjected to the distributed loading shown in Fig. 12–34 a . Determine the reaction at A . EI is constant.

SOLUTION Elastic Curve. The beam deflects as shown in Fig. 12–34 a . Only one coordinate x is needed. For convenience we will take it directed to the right, since the internal moment is easy to formulate.

Moment Function. The beam is indeterminate to the first degree as indicated from the free-body diagram, Fig. 12–34 b . We can express the internal moment M in terms of the redundant force at A using the segment shown in Fig. 12–34 c . Here

M = A y x - 1 6

w0 x3

Slope and Elastic Curve. Applying Eq. 12–10 , we have

EI d2v dx2

= A y x - 1 6

w0 x3

EI dv dx

= 1 2

A y x 2 - 1

24 w0

L + C1

EIv = 1 6

Ay x 3 - 1

120 w0

L + C1 x + C2

The three unknowns A y , C1 , and C2 are determined from the boundary conditions x = 0, v = 0; x = L, dv>dx = 0; and x = L, v = 0. Applying these conditions yields

x = 0, v = 0; 0 = 0 - 0 + 0 + C2 x = L,

dv dx

= 0; 0 = 1 2

A y L 2 - 1

24 w0L

3 + C1

x = L, v = 0; 0 = 1 6

A y L 3 - 1

120 w0 L

4 + C1 L + C2

Solving,

A y = 1

10 w0 L Ans.

C1 = - 1

120 w0L

3 C2 = 0

NOTE: Using the result for A y, the reactions at B can be determined from the equations of equilibrium, Fig. 12–34 b . Show that Bx = 0, By = 2w0 L >5, and MB = w0 L2>15.

634 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.18

x L

(a)

The beam in Fig. 12–35 a is fixed supported at both ends and is subjected to the uniform loading shown. Determine the reactions at the supports. Neglect the effect of axial load.

SOLUTION

Elastic Curve. The beam deflects as shown in Fig. 12–35 a . As in the previous problem, only one x coordinate is necessary for the solution since the loading is continuous across the span.

Moment Function. From the free-body diagram, Fig. 12–35 b , the respective shear and moment reactions at A and B must be equal, since there is symmetry of both loading and geometry. Because of this, the equation of equilibrium, !Fy = 0, requires

V A = V B = wL 2

Ans .

The beam is indeterminate to the second degree, with M " the redundant at each end. Using the beam segment shown in Fig. 12–35 c , the internal moment M can be expressed in terms of M " as follows:

M = wL 2

x - w 2

x2 - M "

Slope and Elastic Curve. Applying Eq. 12–10 , we have

EI d2v dx

= wL 2

x - w 2

x2 - M "

EI dv dx

= wL 4

x2 - w 6

x3 - M "x + C1

EIv = wL 12

x3 - w 24

x4 - M " 2

x2 + C1 x + C2

The three unknowns, M ", C1, and C2, can be determined from the three boundary conditions v = 0 at x = 0, which yields C2 = 0; dv>dx = 0 at x = 0, which yields C1 = 0; and v = 0 at x = L, which yields

M " = wL2

12 Ans .

Using these results, notice that because of symmetry the remaining boundary condition dv>dx = 0 at x = L is automatically satisfied. NOTE: It should be realized that this method of solution is generally suitable when only one x coordinate is needed to describe the elastic curve. If several x coordinates are needed, equations of continuity must be written, thus complicating the solution process.

(b)

MB ! M¿ MA ! M¿

VB ! wL 2VA !

wL 2

L 2

(c)

M¿ x

wL 2

x 2

Fig. 12–35

12.7 STATICALLY INDETERMINATE BEAMS AND SHAFTS—METHOD OF INTEGRATION 635

12 PROBLEMS

12–102. Determine the reactions at the supports A and B , then draw the shear and moment diagrams. Use discontinuity functions. EI is constant.

A B

L– 2

Prob. 12–102

12–103. Determine the reactions at the supports A and B , then draw the shear and moment diagrams. EI is constant.

A B

Prob. 12–103

*12–104. The loading on a floor beam used in the airplane is shown. Use discontinuity functions and determine the rections at the supports A and B , and then draw the moment diagram for the beam.

120 in.

30 lb/in.

120 in.

Prob. 12–104

12–99. Determine the reactions at the supports A and B , then draw the shear and moment diagram. EI is constant. Neglect the effect of axial load.

L–3

A B

1 L–3 2

Prob. 12–99

*12–100. Determine the value of a for which the maximum positive moment has the same magnitude as the maximum negative moment. EI is constant.

Prob. 12–100

12–101. Determine the reactions at the supports A , B , and C ; then draw the shear and moment diagrams. EI is constant.

CA B

P P

L 2

Prob. 12–101

636 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 *12–108. Determine the moment reactions at the supports A and B , then draw the shear and moment diagrams. Solve by expressing the internal moment in the beam in terms of A y and M A . EI is constant.

A B

Prob. 12–108

12–109. The beam has a constant E1I1 and is supported by the fixed wall at B and the rod AC . If the rod has a cross- sectional area A 2 and the material has a modulus of elasticity E2 , determine the force in the rod.

A B

Prob. 12–109

12–110. The beam is supported by a pin at A , a roller at B , and a post having a diameter of 50 mm at C . Determine the support reactions at A , B , and C . The post and the beam are made of the same material having a modulus of elasticity E = 200 GPa, and the beam has a constant moment of inertia I = 255(106) mm4.

BC A

15 kN/m

6 m

1 m

6 m

Prob. 12–110

12–105. Use discontinuity functions and determine the reactions at the supports, then draw the shear and moment diagrams. EI is constant.

4 m 4 m

8 kN/m

CA B

Prob. 12–105

12–106. Determine the reactions at the support A and B . EI is constant.

Prob. 12–106

12–107. Determine the reactions at pin support A and roller supports B and C . EI is constant.

A C

L L

Prob. 12–107

12.8 STATICALLY INDETERMINATE BEAMS AND SHAFTS—MOMENT-AREA METHOD 637

1212121212 *12.8 Statically Indeterminate Beams and Shafts—Moment-Area Method

If the moment-area method is used to determine the unknown redundants of a statically indeterminate beam or shaft, then the M>EI diagram must be drawn such that the redundants are represented as unknowns on this diagram. Once the M>EI diagram is established, the two moment-area theorems can then be applied to obtain the proper relationships between the tangents on the elastic curve in order to meet the conditions of displacement and/or slope at the supports of the beam. In all cases the number of these compatibility conditions will be equivalent to the number of redundants, and so a solution for the redundants can be obtained.

Moment Diagrams Constructed by the Method of Superposition. Since application of the moment-area theorems requires calculation of both the area under the M>EI diagram and the centroidal location of this area, it is often convenient to use separate M>EI diagrams for each of the known loads and redundants rather than using the resultant diagram to calculate these geometric quantities. This is especially true if the resultant moment diagram has a complicated shape. The method for drawing the moment diagram in parts is based on the principle of superposition.

Most loadings on cantilevered beams or shafts will be a combination of the four loadings shown in Fig. 12–36 . Construction of the associated moment diagrams, also shown in this figure, has been discussed in the examples of Chapter 6 . Based on these results, we will now show how to use the method of superposition to represent the resultant moment diagram by a series of separate moment diagrams for the cantilevered beam shown in Fig. 12–37 a . To do this, we will first replace the loads by a system of statically equivalent loads. For example, the three cantilevered beams shown in Fig. 12–37 a are statically equivalent to the

!PL (a)

Sloping line

(b)

Zero sloping line

(c)

Parabolic curve

!wL2

(d)

Cubic curve

!w0L 2

Fig. 12–36

638 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

1212121212

resultant beam, since the load at each point on the resultant beam is equal to the superposition or addition of the loadings on the three separate beams. Thus, if the moment diagrams for each separate beam are drawn, Fig. 12–37 b , the superposition of these diagrams will yield the moment diagram for the resultant beam, shown at the top. For example, from each of the separate moment diagrams, the moment at end A is MA = -8 kN # m - 30 kN # m - 20 kN # m = -58 kN # m, as verified by the top moment diagram. This example demonstrates that it is sometimes easier to construct a series of separate statically equivalent moment diagrams for the beam, rather than constructing its more complicated resultant moment diagram. Obviously, the area and location of the centroid for each part are easier to establish than those of the centroid for the resultant diagram.

x (m)

!58

2 4

!40

!10

x (m)

!30

2 x (m)

!20

M (kN"m)

Superposition of moment diagrams

(b)

2 m 2 m

30 kN"m

5 kN4 kN/m 13 kN

58 kN"m

2 m

4 kN/m 8 kN

8 kN"m

30 kN"m

4 m

5 kN 5 kN

20 kN"m

Superposition of loadings

(a)

2 m

! !

Fig. 12–37

12.8 STATICALLY INDETERMINATE BEAMS AND SHAFTS—MOMENT-AREA METHOD 639

1212121212

In a similar manner, we can also represent the resultant moment diagram for a simply supported beam by using a superposition of moment diagrams for each loading acting on a series of simply supported beams. For example, the beam loading shown at the top of Fig. 12–38 a is equivalent to the sum of the beam loadings shown below it. Consequently, the sum of the moment diagrams for each of these three loadings can be used rather than the resultant moment diagram shown at the top of Fig. 12–38 b .

The examples that follow should also clarify some of these points and illustrate how to use the moment-area theorems to obtain the redundant reactions on statically indeterminate beams and shafts. The solutions follow the procedure for analysis outlined in Sec. 12.4 .

x (m)

Resultant moment diagram !20 !20

Superposition of moment diagrams

(b)

x (m)

x (m) 12

!20

12 m

20 kN"m 5 kN/m

20 kN"m

12 m

5 kN/m

12 m

Superposition of loadings

(a)

20 kN"m

" "

M (kN"m)

Fig. 12–38

640 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.19

The beam is subjected to the concentrated force shown in Fig. 12–39 a . Determine the reactions at the supports. EI is constant.

(a) L

B A

SOLUTION M/EI Diagram. The free-body diagram is shown in Fig. 12–39 b . Using the method of superposition, the separate M >EI diagrams for the redundant reaction By and the load P are shown in Fig. 12–39 c . Elastic Curve. The elastic curve for the beam is shown in Fig. 12–39 d . The tangents at the supports A and B have been constructed. Since !B = 0, then

tB>A = 0 Moment-Area Theorem. Applying Theorem 2 , we have

tB>A = a 2 3 L b J 1 2 ¢ByL EI ≤L R + a L 2 b c -PL EI (L) d + a 2

3 L b c 1

2 a -PL

EI b (L) d = 0

By = 2.5P Ans.

Equations of Equilibrium. Using this result, the reactions at A on the free-body diagram, Fig. 12–39 b , are

S + "Fx = 0; A x = 0 Ans. + c "Fy = 0; -A y + 2.5P - P = 0

A y = 1.5P Ans. a+ "MA = 0; -MA + 2.5P(L) - P(2L) = 0 MA = 0.5PL Ans.

(b)

P MA

Ay By

(c)

L 2L x

M EI

2PL EI

PL EI

ByL EI

(d)

tB/A " 0

tanA

tanB

Fig. 12–39

12.8 STATICALLY INDETERMINATE BEAMS AND SHAFTS—MOMENT-AREA METHOD 641

The beam is subjected to the couple moment at its end C as shown in Fig. 12–40 a . Determine the reaction at B . EI is constant.

SOLUTION M/EI Diagram. The free-body diagram is shown in Fig. 12–40 b . By inspection, the beam is indeterminate to the first degree. In order to obtain a direct solution, we will choose By as the redundant. Using superposition, the M > EI diagrams for By and M0, each applied to a simply supported beam, are shown in Fig. 12–40 c . (Note that for such a beam A x, A y, and Cy do not contribute to an M>EI diagram.) Elastic Curve. The elastic curve for the beam is shown in Fig. 12–40 d . The tangents at A , B , and C have been established. Since !A = !B = !C = 0, then the vertical distances shown must be proportional; i.e.,

tB>C = 1 2 tA >C (1) From Fig. 12–40 c , we have

tB>C = a 1 3 L b J 1 2 ¢By L 2EI ≤(L) R + a 2 3 L b J 1 2 ¢ -M0 2EI ≤(L) R + a L

2 b J ¢ -M0

2EI ≤(L) R

tA >C = (L)J 1 2 ¢By L 2EI ≤(2L) R + a 2 3 (2L)b J 1 2 ¢ -M0 EI ≤(2L) R Substituting into Eq. 1 and simplifying yields

By = 3M0 2L

Ans .

Equations of Equilibrium. The reactions at A and C can now be determined from the equations of equilibrium, Fig. 12–40 b . Show that A x = 0, Cy = 5M0 >4L, and A y = M0>4L.

Note from Fig. 12–40 e that this problem can also be worked in terms of the vertical distances,

tB>A = 1 2 tC>A

EXAMPLE 12.20

(a)

B A

M0 C

Fig. 12–40

(b)

Ay Cy

(c)

2L x

M EI

M0 2EI

! M0 2EI

! M0 EI

ByL 2EI

(d)

tanA tanC

L tanB

tB/C tA/C

(e)

BA tanB

tanC

tanA

L L

tC/A tB/A

642 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12–111. Determine the moment reactions at the supports A and B . EI is constant.

L– 2

A B

L– 2

Prob. 12–111

*12–112. Determine the moment reactions at the supports, and then draw the shear and moment diagrams. EI is constant.

L– 2

A B

L– 2

Prob. 12–112

12–113. Determine the reactions at the bearing support A and fixed support B , then draw the shear and moment diagrams for the beam. EI is constant.

3 ft 3 ft 3 ft

3 kip 2 kip

Prob. 12–113

12–114. Determine the reactions at the supports A and B , then draw the shear and moment diagrams. EI is constant.

A B

– 3 L –

3 L –

3 L

Prob. 12–114

12–115. Determine the vertical reactions at the bearing supports, then draw the shear and moment diagrams. EI is constant.

CA B

L 2

Prob. 12–115

*12–116. Determine the reactions at the journal bearing support A and fixed support B , then draw the shear and moment diagrams for the shaft. EI is constant.

3 kip!ft

3 ft 6 ft

Prob. 12–116

PROBLEMS

12.9 STATICALLY INDETERMINATE BEAMS AND SHAFTS—METHOD OF SUPERPOSITION 643

1212121212 12.9 Statically Indeterminate Beams and Shafts—Method of Superposition

The method of superposition has been used previously to solve for the redundant loading on axially loaded bars and torsionally loaded shafts. In order to apply this method to the solution of statically indeterminate beams (or shafts), it is first necessary to identify the redundant support reactions as explained in Sec. 12.6 . By removing them from the beam we obtain the so-called primary beam , which is statically determinate and stable, and is subjected only to the external load. If we add to this beam a succession of similarly supported beams, each loaded with a separate redundant, then by the principle of superposition, we obtain the actual loaded beam. Finally, in order to solve for the redundants, we must write the conditions of compatibility that exist at the supports where each of the redundants acts. Since the redundant forces are determined directly in this manner, this method of analysis is sometimes called the force method . Once the redundants are obtained, the other reactions on the beam can then be determined from the three equations of equilibrium.

To clarify these concepts, consider the beam shown in Fig. 12–41 a . If we choose the reaction By at the roller as the redundant, then the primary beam is shown in Fig. 12–41 b , and the beam with the redundant By acting on it is shown in Fig. 12–41 c . The displacement at the roller is to be zero, and since the displacement of point B on the primary beam is vB , and By causes point B to be displaced upward vB= , we can write the compatibility equation at B as

(+ c ) 0 = -vB + vB=

The displacements vB and vB= can be related to the loads using any one of the methods discussed in Secs. 12.2 through 12.5 . Here we will obtain them directly from the table in Appendix C . We have

vB = 5PL3

48EI and v=B =

ByL 3

3EI

Substituting into the compatibility equation, we get

0 = - 5PL 3

48EI +

By L 3

3EI

By = 5 16

Now that By is known, the reactions at the wall are determined from the three equations of equilibrium applied to the free-body diagram of the beam, Fig. 12–41 d . The results are

A x = 0 A y = 11 16

MA = 3 16

Actual beam

(a)

(b)

(c)

v¿B

Only redundant By applied

Redundant By removed

L 2

Ay P

(d)

MA L 2

L 2 5

Fig. 12–41

644 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

1212121212

As stated in Sec. 12.6 , choice of the redundant is arbitrary , provided the primary beam remains stable. For example, the moment at A for the beam in Fig. 12–42 a can also be chosen as the redundant. In this case the capacity of the beam to resist MA is removed, and so the primary beam is then pin supported at A , Fig. 12–42 b . To it we add the beam with the redundant at A acting on it, Fig. 12–42 c . Referring to the slope at A caused by the load P as uA , and the slope at A caused by the redundant MA as uA= , the compatibility equation for the slope at A requires

(c+) 0 = uA + uA=

Again using the table in Appendix C , to relate these rotations to the loads, we have

uA = PL2

16EI and uA

= = MAL 3EI

Thus,

0 = PL2

16EI +

MAL 3EI

MA = - 3

16 PL

This is the same result determined previously. Here the negative sign for MA simply means that MA acts in the opposite sense of direction of that shown in Fig. 12–42 c .

Redundant MA removed

(b)

Only redundant MA applied

BA(c)

MA !

Actual beam

(a)

L 2

u¿A

Fig. 12–42

12.9 STATICALLY INDETERMINATE BEAMS AND SHAFTS—METHOD OF SUPERPOSITION 645

1212121212

Another example that illustrates this method is given in Fig. 12–43 a . In this case the beam is indeterminate to the second degree and therefore two compatibility equations will be necessary for the solution. We will choose the forces at the roller supports B and C as redundants. The primary (statically determinate) beam deforms as shown in Fig. 12–43 b when the redundants are removed. Each redundant force deforms this beam as shown in Figs. 12–43 c and 12–43 d , respectively. By superposition, the compatibility equations for the displacements at B and C are

(+ T ) 0 = vB + v!B + vB > (12–20) (+ T ) 0 = vC + v!c + v"c

Here the displacement components vB= and vC= will be expressed in terms of the unknown By , and the components v B > and v"c will be expressed in terms of the unknown Cy. When these displacements have been determined and substituted into Eq. 12–20 , these equations may then be solved simultaneously for the two unknowns By and Cy.

A(a)

Actual beam

A(b)

Redundant By and Cy removed

vB vC

B A(c)

C D

Only redundant By applied

B A(d)

C D

v¿¿

Only redundant Cy applied

v¿B v¿C

v¿¿C

Fig. 12–43

646 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

1212121212

The following examples illustrate application of this procedure. For brevity, all displacements and slopes have been found using the table in Appendix C .

Procedure for Analysis

The following procedure provides a means for applying the method of superposition (or the force method) to determine the reactions on statically indeterminate beams or shafts.

Elastic Curve.

• Specify the unknown redundant forces or moments that must be removed from the beam in order to make it statically determinate and stable.

• Using the principle of superposition, draw the statically indeterminate beam and show it equal to a sequence of corresponding statically determinate beams .

• The first of these beams, the primary beam, supports the same external loads as the statically indeterminate beam, and each of the other beams “added” to the primary beam shows the beam loaded with a separate redundant force or moment.

• Sketch the deflection curve for each beam and indicate symbolically the displacement (slope) at the point of each redundant force (moment).

Compatibility Equations.

• Write a compatibility equation for the displacement (slope) at each point where there is a redundant force (moment).

Load-Displacement Equations. • Relate all the displacements or slopes to the forces or moments

using an appropriate method as explained in Secs. 12.2 through 12.5 . • Substitute the results into the compatibility equations and solve

for the unknown redundants. • If a numerical value for a redundant is positive , it has the same

sense of direction as originally assumed. Similarly, a negative numerical value indicates the redundant acts opposite to its assumed sense of direction .

Equilibrium Equations.

• Once the redundant forces and/or moments have been determined, the remaining unknown reactions can be found from the equations of equilibrium applied to the loadings shown on the beam’s free-body diagram.

12.9 STATICALLY INDETERMINATE BEAMS AND SHAFTS—METHOD OF SUPERPOSITION 647

12 EXAMPLE 12.21

Actual beam

B A

10 ft

(a)

v¿B

5 ft 8 kip

10 ft

2 kip/ ft

Redundant By removed

Only redundant By applied 10 ft

5 ft 8 kip

2 kip/ ft

(b)

(c)

5 ft

2 kip/ ft18 kip

40 kip!ft 0

8 kip

V (kip)

M (kip!ft)

x (ft)

"40

18 8

"10

(kip)

10 kip 5 ft

(d)

(e)

Fig. 12–44

Determine the reactions at the roller support B of the beam shown in Fig. 12–44 a , then draw the shear and moment diagrams. EI is constant.

SOLUTION Principle of Superposition. By inspection, the beam is statically indeterminate to the first degree. The roller support at B will be chosen as the redundant so that By will be determined directly . Figures 12–44 b and 12–44 c show application of the principle of superposition. Here we have assumed that By acts upward on the beam.

Compatibility Equation. Taking positive displacement as downward, the compatibility equation at B is

(+ T ) 0 = vB - v!B (1)

Load-Displacement Equations. These displacements can be obtained directly from the table in Appendix C .

vB = wL4

8EI + 5PL

48EI

= 2 kip>ft(10 ft)4

8EI +

5(8 kip)(10 ft)3

48EI =

3333 kip # ft3 EI

v!B = PL3

3EI =

By (10 ft) 3

3EI =

333.3 ft3 By EI

Substituting into Eq. 1 and solving yields

0 = 3333 EI

- 333.3By

By = 10 kip Ans.

Equilibrium Equations. Using this result and applying the three equations of equilibrium, we obtain the results shown on the beam’s free-body diagram in Fig. 12–44 d . The shear and moment diagrams are shown in Fig. 12–44 e .

648 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.22

5 ft A

5 ft v¿¿

Actual beam and rod

8 kip

(a)

8 ft

SOLUTION I Principle of Superposition. By inspection, this problem is indeterminate to the first degree. Here B will undergo an unknown displacement v!B, since the rod will stretch. The rod will be treated as the redundant and hence the force of the rod is removed from the beam at B , Fig. 12–45 b , and then reapplied, Fig. 12–45 c .

Compatibility Equation. At point B we require

(+ T ) v!B = vB - v"B (1) Load-Displacement Equations. The displacements vB and v"B are determined from the table in Appendix C . v!B is calculated from Eq. 4–2 . Working in kilopounds and inches, we have

v!B = PL AE

= FBC (8 ft)(12 in.>ft)

(p>4)1 1 2 in.22[29(103) kip>in2] = 0.01686FBC T vB =

5PL3

48EI =

5(8 kip)(10 ft)3 (12 in.>ft)3 48[29(103) kip>in2](475 in4) = 0.1045 in. T

v"B = PL3

3EI =

FBC (10 ft) 3 (12 in.>ft)3

3[29(10)3 kip>in2](475 in4) = 0.04181FBC c Thus, Eq. 1 becomes

(+ T ) 0.01686FBC = 0.1045 - 0.04181FBC FBC = 1.78 kip Ans .

Redundant FBC removed

8 kip

(b)

B vB A

Only redundant FBC applied (c)

B FBC

v¿B

The beam in Fig. 12–45 a is fixed supported to the wall at A and pin connected to a 1 2 @in.@diameter rod BC . If E = 29(103) ksi for both members, determine the force developed in the rod due to the loading. The moment of inertia of the beam about its neutral axis is I = 475 in4.

Fig. 12–45

12.9 STATICALLY INDETERMINATE BEAMS AND SHAFTS—METHOD OF SUPERPOSITION 649

SOLUTION II Principle of Superposition. We can also solve this problem by removing the pin support at C and keeping the rod attached to the beam. In this case the 8-kip load will cause points B and C to be displaced downward the same amount vC, Fig. 12–45 e , since no force exists in rod BC . When the redundant force FBC is applied at point C , it causes the end C of the rod to be displaced upward v!C and the end B of the beam to be displaced upward v!B, Fig. 12–45 f . The difference in these two displacements, vBC, represents the stretch of the rod due to FBC, so that v!C = vBC + v!B. Hence, from Figs. 12–45 d , 12–45 e , and 12–45 f , the compatibility of displacement at point C is

(+ T ) 0 = vC - (vBC + v!B) (2)

From Solution I, we have

vC = vB = 0.1045 in. T

vBC = v"B = 0.01686FBC c v!B = 0.04181FBCc

Therefore, Eq. 2 becomes

(+ T ) 0 = 0.1045 - (0.01686FBC + 0.04181FBC) FBC = 1.78 kip Ans .

Fig. 12–45 (cont.)

A 5 ft

Actual beam and rod

8 kip

(d)

B 5 ft

Redundant FBC removed

8 kip

(e)

C vC

Only redundant FBC applied (f)

v¿C

v¿B

vBC

FBC

650 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 EXAMPLE 12.23

Determine the moment at B for the beam shown in Fig. 12–46 a . EI is constant. Neglect the effects of axial load.

SOLUTION Principle of Superposition. Since the axial load on the beam is neglected, there will be a vertical force and moment at A and B . Here there are only two available equations of equilibrium (!M = 0, !Fy = 0) and so the problem is indeterminate to the second degree. We will assume that By and MB are redundant, so that by the principle of superposition, the beam is represented as a cantilever, loaded separately by the distributed load and reactions By and MB , Figs. 12–46 b , 12–46 c , and 12–46 d .

Redundants MB and By removed

A(b)

6 ft

3 kip/ ft

Actual beam

A(a)

6 ft

3 kip/ ft

6 ft B

Only redundant By applied

A(c)

12 ft B

Only redundant MB applied

A(d)

v¿¿B

v¿B

6 ft

12 ft u¿¿

u¿

Fig. 12–46

12.9 STATICALLY INDETERMINATE BEAMS AND SHAFTS—METHOD OF SUPERPOSITION 651

12 Compatibility Equations. Referring to the displacement and slope at B , we require

(c+) 0 = uB + u!B + u"B (1)

(+ T ) 0 = vB + v!B + v"B (2)

Load-Displacement Equations. Using the table in Appendix C to calculate the slopes and displacements, we have

uB = wL3

48EI =

3 kip>ft (12 ft)3 48EI

= 108 kip # ft2

EI b

vB = 7wL4

384EI =

7(3 kip>ft)(12 ft)4 384EI

= 1134 kip # ft3

EI T

u!B = PL2

2EI =

By(12 ft) 2

2EI =

72By EI

v!B = PL3

3EI =

By(12 ft) 3

3EI =

576By EI

u"B = ML EI

= MB(12 ft)

EI =

12MB EI

v"B = ML2

2EI =

MB(12 ft) 2

2EI =

72MB EI

Substituting these values into Eqs. 1 and 2 and canceling out the common factor EI , we get

(c+) 0 = 108 + 72By + 12MB

(+ T ) 0 = 1134 + 576By + 72MB

Solving these equations simultaneously gives

By = -3.375 kip

MB = 11.25 kip # ft Ans. NOTE: The reactions at A can now be determined from the equilibrium equations.

652 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

F12–16. Determine the reaction at the roller B . EI is constant.

A B C

L L

F12–16

F12–17. Determine the reaction at the roller B . EI is constant.

4 m 6 m2 m

50 kN

F12–17

F12–18. Determine the reaction at the roller support B if it settles 5 mm. E = 200 GPa and I = 65.0(10- 6) m4.

A B

6 m 6 m

10 kN/m

F12–18

F12–13. Determine the reactions at the fixed support A and the roller B . EI is constant.

A B

40 kN

4 m 2 m

F12–13

F12–14. Determine the reactions at the fixed support A and the roller B . EI is constant.

F12–14

F12–15. Determine the reactions at the fixed support A and the roller B . Support B settles 2 mm. E = 200 GPa, I = 65.0(10-6) m4.

A B

6 m

10 kN/m

F12–15

FUNDAMENTAL PROBLEMS

12.9 STATICALLY INDETERMINATE BEAMS AND SHAFTS—METHOD OF SUPERPOSITION 653

12 PROBLEMS

12–117. Determine the reactions at the bearing supports A , B , and C of the shaft, then draw the shear and moment diagrams. EI is constant. Each bearing exerts only vertical reactions on the shaft.

400 N

1 m 1 m

CA B

1 m 1 m

400 N

Prob. 12–117

12–118. Determine the reactions at the supports A and B . EI is constant.

A B L 2

Prob. 12–118

12–119. Determine the reactions at the supports A , B , and C, then draw the shear and moment diagrams. EI is constant.

6 ft 12 ft

3 kip/ ft

A B C

6 ft

12 kip

Prob. 12–119

*12–120. Determine the moment reactions at the supports A and B. EI is constant.

L– 2

A B

L– 2

Prob. 12–120

12–121. Determine the deflection at the end B of the clamped A-36 steel strip. The spring has a stiffness of k = 2 N>mm. The strip is 5 mm wide and 10 mm high. Also, draw the shear and moment diagrams for the strip.

50 N

200 mm

10 mmA

k= 2 N/mm

Prob. 12–121

12–122. Determine the reactions at the supports A and B . EI is constant.

Prob. 12–122

654 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 12–126. Determine the force in the spring. EI is constant.

A B

Prob. 12–126

12–127. The beam is supported by the bolted supports at its ends. When loaded these supports initially do not provide an actual fixed connection, but instead allow a slight rotation a before becoming fixed after the load is fully applied. Determine the moment at the connections and the maximum deflection of the beam.

L— 2

Prob. 12–127

*12–128. Each of the two members is made from 6061-T6 aluminum and has a square cross section 1 in. : 1 in. They are pin connected at their ends and a jack is placed between them and opened until the force it exerts on each member is 500 lb. Determine the greatest force P that can be applied to the center of the top member without causing either the two members to yield. For the analysis neglect the axial force in each member. Assume the jack is rigid.

6 ft 6 ft

Prob. 12–128

12–123. Determine the reactions at support C . EI is the same for both beams.

A C

L 2

Prob. 12–123

*12–124. Before the uniform distributed load is applied on the beam, there is a small gap of 0.2 mm between the beam and the post at B. Determine the support reactions at A, B, and C. The post at B has a diameter of 40 mm, and the moment of inertia of the beam is I = 875(106) mm4. The post and the beam are made of material having a modulus of elasticity of E = 200 GPa.

B C

6 m

1 m

6 m

0.2 mm

30 kN/m

Prob. 12–124

12–125. The fixed supported beam AB is strengthened using the simply supported beam CD and the roller at F which is set in place just before application of the load P . Determine the reactions at the supports if EI is constant.

C D

L— 4

Prob. 12–125

12.9 STATICALLY INDETERMINATE BEAMS AND SHAFTS—METHOD OF SUPERPOSITION 655

12 12–131. The 1-in.-diameter A-36 steel shaft is supported by bearings at A and C . The bearing at B rests on a simply supported A992 steel wide-flange beam having a moment of inertia of I = 500 in4 . If the belt loads on the pulley are 400 lb each, determine the vertical reactions at A , B , and C .

400 lb400

2 ft

3 ft

5 ft

Prob. 12–131

*12–132. The assembly consists of three simply supported beams for which the bottom of the top beam rests on the top of the bottom two. if a uniform load of 3 kN>m is applied to the beam, determine the vertical reactions at each of the supports. EI is constant.

3 m 2 m

2 m

3 kN/m

2 m

3 m

G A

Prob. 12–132

12–129. The beam is made from a soft linear elastic material having a constant EI . If it is originally a distance ! from the surface of its end support, determine the length a that rests on this support when it is subjected to the uniform load w0, which is great enough to cause this to happen.

L a

Prob. 12–129

12–130. The A992 steel beam and rod are used to support the load of 8 kip. If it is required that the allowable normal stress for the steel is s allow = 18 ksi, and the maximum deflection not exceed 0.05 in., determine the smallest diameter rod that should be used. The beam is rectangular, having a height of 5 in. and a thickness of 3 in.

4 ft

A B

5 ft

8 kip

Prob. 12–130

656 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12 CHAPTER REVIEW

The elastic curve represents the centerline deflection of a beam or shaft. Its shape can be determined using the moment diagram. Positive moments cause the elastic curve to be concave upwards and negative moments cause it to be concave downwards. The radius of curvature at any point is determined from

1 r

= M EI

Moment diagram

Inflection point

Elastic curve

The equation of the elastic curve and its slope can be obtained by first finding the internal moment in the member as a function of x . If several loadings act on the member, then separate moment functions must be determined between each of the loadings. Integrating these functions once using EI(d2v>dx2) = M(x) gives the equation for the slope of the elastic curve, and integrating again gives the equation for the deflection. The constants of integration are determined from the boundary conditions at the supports, or in cases where several moment functions are involved, continuity of slope and deflection at points where these functions join must be satisfied.

u ! 0 v ! 0

x1 x2

v1 ! v2

v ! 0

dv2 dx2

dv1 dx1

Boundary conditions

Continuity conditions

Discontinuity functions allow one to express the equation of the elastic curve as a continuous function, regardless of the number of loadings on the member. This method eliminates the need to use continuity conditions, since the two con stants of integration can be determined solely from the two boundary conditions.

CHAPTER REVIEW 657

12 The moment-area method is a semi-graphical technique for finding the slope of tangents or the vertical distance between tangents at specific points on the elastic curve. It requires finding area segments under the M/EI diagram, or the moment of these segments about points on the elastic curve. The method works well for M/EI diagrams composed of simple shapes, such as those produced by concentrated forces and couple moments.

A B

tan B tan AuB/A BA x

M EI uB/A ! Area

tanB

tanA

A B BA

xtB/A

M EI

_ x ¿

tB/A ! _ x ¿(Area)

The deflection or slope at a point on a member subjected to combinations of loadings can be determined using the method of superposition. The table in Appendix C is available for this purpose.

Statically indeterminate beams and shafts have more unknown support reactions than available equations of equilibrium. To solve, one first identifies the redundant reactions. The method of integration or the moment- area theorems can then be used to solve for the unknown redundants. It is also possible to determine the redundants by using the method of superposition, where one considers the conditions of continuity at the redundant. Here the displacement due to the external loading is determined with the redundant removed, and again with the redundant applied and the external loading removed. The tables in Appendix C can be used to determine these necessary displacements.

658 CHAPTER 12 DEFLECT ION OF BEAMS AND SHAFTS

12–133. The shaft supports the two pulley loads shown. Using discontinuity functions, determine the equation of the elastic curve.The bearings at A and B exert only vertical reactions on the shaft. EI is constant.

A B

12 in. 12 in. 36 in.

70 lb

180 lb

Prob. 12–133

12–134. The shaft is supported by a journal bearing at A , which exerts only vertical reactions on the shaft, and by a thrust bearing at B , which exerts both horizontal and vertical reactions on the shaft. Draw the bending-moment diagram for the shaft and then, from this diagram, sketch the deflection or elastic curve for the shaft’s centerline. Determine the equations of the elastic curve using the coordinates x1 and x2. EI is constant. Use the method of integration.

A B

12 in.

80 lb

4 in.

x2 4 in.

80 lb

12 in.

Prob. 12–134

12–135. Determine the moment reactions at the supports A and B. Use the method of integration. EI is constant.

Prob. 12–135

*12–136. Determine the equations of the elastic curve for the beam using the x1 and x2 coordinates. Specify the slope at A and the maximum deflection. EI is constant. Use the method of integration.

A B

L L

x1 x2

Prob. 12–136

12–137. Determine the maximum deflection between the supports A and B . EI is constant. Use the method of integration.

A B

L L

x1 x2

Prob. 12–137

12–138. If the bearings at A and B exert only vertical reactions on the shaft, determine the slope at B and the deflection at C . Use the moment-area theorems.

BA C

a a

Prob. 12–138

REVIEW PROBLEMS

REVIEW PROBLEMS 659

12 12–141. Using the method of superposition, determine the deflection at C of beam AB . The beams are made of wood having a modulus of elasticity of E = 1.5(103) ksi.

AB C

6 ft

100 lb/ ft

6 ft 4 ft4 ft

3 in.

6 in.

Section a – a Prob. 12–141

12–142. The rim on the flywheel has a thickness t , width b , and specific weight g. If the flywheel is rotating at a constant rate of v , determine the maximum moment developed in the rim. Assume that the spokes do not deform. Hint : Due to symmetry of the loading, the slope of the rim at each spoke is zero. Consider the radius to be sufficiently large so that the segment AB can be considered as a straight beam fixed at both ends and loaded with a uniform centrifugal force per unit length. Show that this force is w = btgv2r>g.

Prob. 12–142

12–139. The bearing supports A , B , and C exert only vertical reactions on the shaft. Determine these reactions, then draw the shear and moment diagrams. EI is constant. Use the moment-area theorems.

200 N

1 m 2 m1 m

CA B

Prob. 12–139

*12–140. Using the method of superposition, determine the magnitude of M 0 in terms of the distributed load w and dimension a so that the deflection at the center of the beam is zero. EI is constant.

a aa

0 M0

Prob. 12–140

The columns of this water tank are braced at their length in order to reduce their chance of buckling.

Chapter 13

661

13.1 Critical Load Whenever a member is designed, it is necessary that it satisfy specific strength, deflection, and stability requirements. In the preceding chapters, we have discussed some of the methods used to determine a member’s strength and deflection, while assuming that the member was always in stable equilibrium. Some members, however, may be subjected to compressive loadings, and if these members are long and slender the loading may be large enough to cause the member to deflect laterally or sidesway. To be specific, long slender members subjected to an axial compressive force are called columns , and the lateral deflection that occurs is called buckling . Quite often the buckling of a column can lead to a sudden and dramatic failure of a structure or mechanism, and as a result, special attention must be given to the design of columns so that they can safely support their intended loadings without buckling.

CHAPTER OBJECTIVES

■ In this chapter, we will discuss the behavior of columns and indicate some of the methods used for their design. The chapter begins with a general discussion of buckling, followed by a determination of the axial load needed to buckle a so-called ideal column. Afterwards, a more realistic analysis is considered, which accounts for any bending of the column. Also, inelastic buckling of a column is presented as a special topic. At the end of the chapter we will discuss some of the methods used to design both concentrically and eccentrically loaded columns made of common engineering materials.

Buckling of Columns

662 CHAPTER 13 BUCKL ING OF COLUMNS

The maximum axial load that a column can support when it is on the verge of buckling is called the critical load , Pcr, Fig. 13–1 a . Any additional loading will cause the column to buckle and therefore deflect laterally as shown in Fig. 13–1 b . In order to better understand the nature of this instability, consider a two-bar mechanism consisting of weightless bars that are rigid and pin connected as shown in Fig. 13–2 a . When the bars are in the vertical position, the spring, having a stiffness k , is unstretched, and a small vertical force P is applied at the top of one of the bars. We can upset this equilibrium position by displacing the pin at A by a small amount !, Fig. 13–2 b . As shown on the free-body diagram of the pin when the bars are displaced, Fig. 13–2 c , the spring will produce a restoring force F = k!, while the applied load P develops two horizontal components, Px = P tan u, which tend to push the pin (and the bars) further out of equilibrium. Since u is small, ! ! u(L >2) and tan u ! u. Thus the restoring spring force becomes F = kuL >2, and the disturbing force is 2Px = 2Pu.

If the restoring force is greater than the disturbing force, that is, kuL >2 7 2Pu, then, noticing that u cancels out, we can solve for P , which gives

P 6 kL 4 stable equilibrium

This is a condition for stable equilibrium since the force developed by the spring would be adequate to restore the bars back to their vertical position. However, if kLu>2 6 2Pu, or

P 7 kL 4 unstable equilibrium

then the mechanism would be in unstable equilibrium . In other words, if this load P is applied, and a slight displacement occurs at A , the mechanism will tend to move out of equilibrium and not be restored to its original position.

Pcr

Pcr (a)

P " Pcr

(b) Fig. 13–1

13.1 CRITICAL LOAD 663

The intermediate value of P , which requires kLu>2 = 2Pu, is the critical load . Here

Pcr = kL 4 neutral equilibrium

This loading represents a case of the mechanism being in neutral equilibrium . Since Pcr is independent of the (small) displacement u of the bars, any slight disturbance given to the mechanism will not cause it to move further out of equilibrium, nor will it be restored to its original position. Instead, the bars will remain in the deflected position.

These three different states of equilibrium are represented graphically in Fig. 13–3 . The transition point where the load is equal to the critical value P = Pcr is called the bifurcation point . At this point the mechanism will be in equilibrium for any small value of u, measured either to the right or to the left of the vertical. Physically, Pcr represents the load for which the mechanism is on the verge of buckling. It is quite reasonable to determine this value by assuming small displacements as done here; however, it should be understood that Pcr may not be the largest value of P that the mechanism can support. Indeed, if a larger load is placed on the bars, then the mechanism may have to undergo a further deflection before the spring is compressed or elongated enough to hold the mechanism in equilibrium.

Like the two-bar mechanism just discussed, the critical buckling loads on columns supported in various ways can be obtained, and the method used to do this will be explained in the next section. Although in engineering design the critical load may be considered to be the largest load the column can support, realize that, like the two-bar mechanism in the deflected or buckled position, a column may actually support an

(a)

L 2

(b)

L 2

! " u( )

P tan u

F A

Bifurcation point

Unstable equilibrium

Neutral equilibrium

Stable equilibrium

Pcr " kL 4

Fig. 13–3

664 CHAPTER 13 BUCKL ING OF COLUMNS

even greater load than Pcr. Unfortunately, however, this loading may require the column to undergo a large deflection, which is generally not tolerated in engineering structures or machines. For example, it may take only a few newtons of force to buckle a meterstick, but the additional load it may support can be applied only after the stick undergoes a relatively large lateral deflection.

13.2 Ideal Column with Pin Supports In this section we will determine the critical buckling load for a column that is pin supported as shown in Fig. 13–4 a . The column to be considered is an ideal column , meaning one that is perfectly straight before loading, is made of homogeneous material, and upon which the load is applied through the centroid of the cross section. It is further assumed that the material behaves in a linear-elastic manner and that the column buckles or bends in a single plane. In reality, the conditions of column straightness and load application are never accomplished; however, the analysis to be performed on an “ideal column” is similar to that used to analyze initially crooked columns or those having an eccentric load application. These more realistic cases will be discussed later in this chapter.

Since an ideal column is straight, theoretically the axial load P could be increased until failure occurs by either fracture or yielding of the material. However, when the critical load Pcr is reached, the column will be on the verge of becoming unstable, so that a small lateral force F , Fig. 13–4 b , will cause the column to remain in the deflected position when F is removed, Fig. 13–4 c . Any slight reduction in the axial load P from Pcr will allow the column to straighten out, and any slight increase in P , beyond Pcr, will cause further increases in lateral deflection.

The dramatic failure of this off-shore oil platform was caused by the horizontal forces of hurricane winds, which led to buckling of its supporting columns.

(a)

(b)

Pcr

(c)

Pcr

Fig. 13–4

13.2 IDEAL COLUMN WITH PIN SUPPORTS 665

Whether or not a column will remain stable or become unstable when subjected to an axial load will depend on its ability to restore itself, which is based on its resistance to bending. Hence, in order to determine the critical load and the buckled shape of the column, we will apply Eq. 12–10 , which relates the internal moment in the column to its deflected shape, i.e.,

EI d2v dx2

= M (13–1)

Recall that this equation assumes that the slope of the elastic curve is small and that deflections occur only by bending. When the column is in its deflected position, Fig. 13–5 a , the internal bending moment can be determined by using the method of sections. The free-body diagram of a segment in the deflected position is shown in Fig. 13–5 b . Here both the deflection v and the internal moment M are shown in the positive direction according to the sign convention used to establish Eq. 13–1 . Moment equilibrium requires M = -Pv. Thus Eq. 13–1 becomes

EI d2v dx2

= -Pv

d2v dx2

+ a P EI

bv = 0 (13–2) This is a homogeneous, second-order, linear differential equation

with constant coefficients. It can be shown by using the methods of differential equations, or by direct substitution into Eq. 13–2 , that the general solution is

v = C1 sinaA PEI x b + C2 cosaA PEI x b (13–3) The two constants of integration are determined from the boundary conditions at the ends of the column. Since v = 0 at x = 0, then C2 = 0. And since v = 0 at x = L, then

C1 sinaA PEI L b = 0 This equation is satisfied if C1 = 0; however, then v = 0, which is a trivial solution that requires the column to always remain straight, even though the load may cause the column to become unstable. The other possibility is for

sinaA PEI L b = 0 which is satisfied if

A PEI L = np

L v

(a)

(b)

Fig. 13–5

666 CHAPTER 13 BUCKL ING OF COLUMNS

* n represents the number of curves in the deflected shape of the column. For example, if n = 2, then two curves will appear, Fig. 13–5 c . Here the critical load is 4 Pcr just prior to buckling, which practically speaking will not exist.

P = n2p2EI

L2 n = 1, 2, 3, c (13–4)

The smallest value of P is obtained when n = 1, so the critical load for the column is therefore *

Pcr = p2EI

This load is sometimes referred to as the Euler load , named after the Swiss mathematician Leonhard Euler, who originally solved this problem in 1757. The corresponding buckled shape is defined by the equation

v = C1 sin px L

Here the constant C1 represents the maximum deflection, vmax, which occurs at the midpoint of the column, Fig. 13–5 c . Specific values for C1 cannot be obtained, since the exact deflected form for the column is unknown once it has buckled. It has been assumed, however, that this deflection is small.

Note that the critical load is independent of the strength of the material; rather it only depends on the column’s dimensions ( I and L ) and the material’s stiffness or modulus of elasticity E . For this reason, as far as elastic buckling is concerned, columns made, for example, of high- strength steel offer no advantage over those made of lower-strength steel, since the modulus of elasticity for both is approximately the same. Also note that the load-carrying capacity of a column will increase as the moment of inertia of the cross section increases. Thus, efficient columns are designed so that most of the column’s cross-sectional area is located as far away as possible from the principal centroidal axes for the section. This is why hollow sections such as tubes are more economical than solid sections. Furthermore, wide-flange sections, and columns that are “built up” from channels, angles, plates, etc., are better than sections that are solid and rectangular.

vmax

x n ! 1

L 2

P ! 4Pcr

L 2

Fig. 13–5 (cont.)

13.2 IDEAL COLUMN WITH PIN SUPPORTS 667

It is also important to realize that a column will buckle about the principal axis of the cross section having the least moment of inertia (the weakest axis) provided it is supported the same way about each axis. For example, a column having a rectangular cross section, like a meter stick, as shown in Fig. 13–6 , will buckle about the a – a axis, not the b – b axis. As a result, engineers usually try to achieve a balance, keeping the moments of inertia the same in all directions. Geometrically, then, circular tubes would make excellent columns. Also, square tubes or those shapes having Ix ! Iy are often selected for columns.

Summarizing the above discussion, the buckling equation for a pin-supported long slender column can be rewritten, and the terms defined as follows:

Pcr = p 2EI

L2 (13–5)

where Pcr = critical or maximum axial load on the column just before it

begins to buckle. This load must not cause the stress in the column to exceed the proportional limit

E = modulus of elasticity for the material I = least moment of inertia for the column’s cross-sectional area L = unsupported length of the column, whose ends are pinned

For purposes of design, the above equation can also be written in a more useful form by expressing I = Ar 2, where A is the cross-sectional area and r is the radius of gyration of the cross-sectional area. Thus,

Pcr = p2E(Ar 2)

a P A b

cr =

p 2E (L >r)2

scr = p 2E

(L >r)2 (13–6) Here

scr = critical stress, which is an average normal stress in the column just before the column buckles. This stress is an elastic stress and therefore scr … sY

E = modulus of elasticity for the material L = unsupported length of the column, whose ends are pinned r = smallest radius of gyration of the column, determined from

r = 2I>A , where I is the least moment of inertia of the column’s cross-sectional area A

The geometric ratio L >r in Eq. 13–6 is known as the slenderness ratio . It is a measure of the column’s flexibility, and as will be discussed later, it serves to classify columns as long, intermediate, or short.

Failure of this crane boom was caused by the localized buckling of one of its tubular struts.

Fig. 13–6

668 CHAPTER 13 BUCKL ING OF COLUMNS

It is possible to graph Eq. 13–6 using axes that represent the critical stress versus the slenderness ratio. Examples of this graph for columns made of a typical structural steel and aluminum alloy are shown in Fig. 13–7 . Note that the curves are hyperbolic and are valid only for critical stresses below the material’s yield point (proportional limit), since the material must behave elastically. For the steel the yield stress is (sY)st = 36 ksi [Est = 29(103) ksi], and for the aluminum it is (sY)al = 27 ksi [Eal = 10(103) ksi]. Substituting scr = sY into Eq. 13–6 , the smallest allowable slenderness ratios for the steel and aluminum columns are therefore (L >r)st = 89 and (L >r)al = 60.5, respectively. Thus, for a steel column, if (L >r)st Ú 89, Euler’s formula can be used to determine the critical load since the stress in the column remains elastic. On the other hand, if (L >r)st 6 89, the column’s stress will exceed the yield point before buckling can occur, and therefore the Euler formula is not valid in this case.

L r

50 100 150 200

Aluminum alloy

Structural steel

60.5 89

scr (103) ksi

(sY ! 36 ksi)

(sY ! 27 ksi)

Fig. 13–7

Important Points

• Columns are long slender members that are subjected to axial compressive loads.

• The critical load is the maximum axial load that a column can support when it is on the verge of buckling. This loading represents a case of neutral equilibrium .

• An ideal column is initially perfectly straight, made of homogeneous material, and the load is applied through the centroid of the cross section.

• A pin-connected column will buckle about the principal axis of the cross section having the least moment of inertia.

• The slenderness ratio is L >r , where r is the smallest radius of gyration of the cross section. Buckling will occur about the axis where this ratio gives the greatest value.

13.2 IDEAL COLUMN WITH PIN SUPPORTS 669

EXAMPLE 13.1

The A992 steel W8 * 31 member shown in Fig. 13–8 is to be used as a pin-connected column. Determine the largest axial load it can support before it either begins to buckle or the steel yields.

12 ft

y y

Fig. 13–8

SOLUTION From the table in Appendix B , the column’s cross-sectional area and moments of inertia are A = 9.13 in2, Ix = 110 in4, and Iy = 37.1 in4. By inspection, buckling will occur about the y – y axis. Why? Applying Eq. 13–5 , we have

Pcr = p2EI

L2 =

p2[29(103) kip>in2](37.1 in4) [12 ft(12 in.>ft)]2 = 512 kip

When fully loaded, the average compressive stress in the column is

scr = Pcr A

= 512 kip

9.13 in2 = 56.1 ksi

Since this stress exceeds the yield stress (50 ksi), the load P is determined from simple compression:

50 ksi = P

9.13 in2 ; P = 456 kip Ans.

In actual practice, a factor of safety would be placed on this loading.

670 CHAPTER 13 BUCKL ING OF COLUMNS

13.3 Columns Having Various Types of Supports

The Euler load was derived for a column that is pin connected or free to rotate at its ends. Oftentimes, however, columns may be supported in some other way. For example, consider the case of a column fixed at its base and free at the top, Fig. 13–9 a . As the column buckles the load displaces d and at x the displacement is v . From the free-body diagram in Fig. 13–9 b , the internal moment at the arbitrary section is M = P(d - v). Consequently, the differential equation for the deflection curve is

EI d2v dx2

= P(d - v)

d2v dx2

+ P EI

v = P EI

d (13–7)

Unlike Eq. 13–2 , this equation is nonhomogeneous because of the nonzero term on the right side. The solution consists of both a complementary and a particular solution, namely,

v = C1 sinaA PEI x b + C2 cosaA PEI x b + d The constants are determined from the boundary conditions. At x = 0, v = 0, so that C2 = -d. Also,

dv dx

= C1A PEI cosaA PEI x b - C2A PEI sinaA PEI x b At x = 0, dv>dx = 0, so that C1 = 0. The deflection curve is therefore v = d c 1 - cosaA PEI x b d (13–8)

Since the deflection at the top of the column is d, that is, at x = L, v = d, we require

d cosaA PEI L b = 0 The trivial solution d = 0 indicates that no buckling occurs, regardless of the load P . Instead,

cosaA PEI L b = 0 or A PEI L = np2 , n = 1, 3, 5c The smallest critical load occurs when n = 1, so that

Pcr = p2EI 4L2

(13–9)

By comparison with Eq. 13–5 , it is seen that a column fixed supported at its base and free at its top will support only one-fourth the critical load that can be applied to a column pin supported at both ends.

L v

x P

(a)

x P

(b)

Fig. 13–9

The tubular columns used to support this water tank have been braced at three locations along their length to prevent them from buckling.

13.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS 671

Other types of supported columns are analyzed in much the same way and will not be covered in detail here. * Instead, we will tabulate the results for the most common types of column support and show how to apply these results by writing Euler’s formula in a general form.

Effective Length. As stated previously, the Euler formula, Eq. 13–5 , was developed for the case of a column having ends that are pinned or free to rotate. In other words, L in the equation represents the unsupported distance between the points of zero moment. This formula can be used to determine the critical load on columns having other types of support provided “ L ” represents the distance between the zero- moment points. This distance is called the column’s effective length , Le. Obviously, for a pin-ended column Le = L, Fig. 13–10 a . For the fixed- and free-ended column, the deflection curve, Eq. 13–8 , was found to be one-half that of a column that is pin connected and has a length of 2L, Fig. 13–10 b . Thus the effective length between the points of zero moment is Le = 2L. Examples for two other columns with different end supports are also shown in Fig. 13–10 . The column fixed at its ends, Fig. 13–10 c , has inflection points or points of zero moment L >4 from each support. The effective length is therefore represented by the middle half of its length, that is, Le = 0.5L. Lastly, the pin- and fixed-ended column, Fig. 13–10 d , has an inflection point at approximately 0.7L from its pinned end, so that Le = 0.7L.

Rather than specifying the column’s effective length, many design codes provide column formulas that employ a dimensionless coefficient K called the effective-length factor . This factor is defined from

Le = KL (13–10)

Specific values of K are also given in Fig. 13–10 . Based on this generality, we can therefore write Euler’s formula as

Pcr = p2EI

(KL)2 (13–11)

scr = p2E

(KL >r)2 (13–12) Here (KL >r) is the column’s effective-slenderness ratio . For example, if the column is fixed at its base and free at its end, we have K = 2, and therefore Eq. 13–11 gives the same result as Eq. 13–9 .

* See Problems 13–43 , 13–44 , and 13–45 .

Pinned ends

(a) K ! 1

L ! Le

Fixed and free ends

(b) K ! 2

Le ! 2L

Fixed ends

L Le ! 0.5L

Pinned and fixed ends

(d) K ! 0.7

Le ! 0.7L

Fig. 13–10

672 CHAPTER 13 BUCKL ING OF COLUMNS

EXAMPLE 13.2

A W6 * 15 steel column is 24 ft long and is fixed at its ends as shown in Fig. 13–11 a . Its load-carrying capacity is increased by bracing it about the y – y (weak) axis using struts that are assumed to be pin connected to its midheight. Determine the load it can support so that the column does not buckle nor the material exceed the yield stress. Take Est = 29(103) ksi and sY = 60 ksi.

SOLUTION The buckling behavior of the column will be different about the x – x and y – y axes due to the bracing. The buckled shape for each of these cases is shown in Figs. 13–11 b and 13–11 c . From Fig. 13–11 b , the effective length for buckling about the x – x axis is (KL)x = 0.5(24 ft) = 12 ft = 144 in., and from Fig. 13–11 c , for buckling about the y – y axis, (KL)y = 0.7(24 ft>2) = 8.40 ft = 100.8 in. The moments of inertia for a W6 * 15 are found from the table in Appendix B . We have Ix = 29.1 in4, Iy = 9.32 in4.

Applying Eq. 13–11 ,

(Pcr)x = p2EIx (KL)x

2 = p2[29(103) ksi]29.1 in4

(144 in.)2 = 401.7 kip (1)

(Pcr)y = p2EIy

(KL)y 2 =

p2[29(103) ksi]9.32 in4

(100.8 in.)2 = 262.5 kip (2)

By comparison, buckling will occur about the y – y axis. The area of the cross section is 4.43 in2, so the average compressive

stress in the column is

scr = Pcr A

= 262.5 kip

4.43 in2 = 59.3 ksi

Since this stress is less than the yield stress, buckling will occur before the material yields. Thus, Pcr = 263 kip Ans.

NOTE: From Eq. 13–12 it can be seen that buckling will always occur about the column axis having the largest slenderness ratio, since a large slenderness ratio will give a small critical stress. Thus, using the data for the radius of gyration from the table in Appendix B , we have

aKL r b

x =

144 in. 2.56 in.

= 56.2

aKL r b

y =

100.8 in. 1.46 in.

= 69.0

Hence, y – y axis buckling will occur, which is the same conclusion reached by comparing Eqs. 1 and 2 .

(a)

12 ft

(b)

x–x axis buckling

(c)

y–y axis buckling

8.40 ft

Fig. 13–11

13.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS 673

EXAMPLE 13.3

The aluminum column is braced at its top by cables so as to prevent movement at the top along the x axis, Fig. 13–12 a . If it is assumed to be fixed at its base, determine the largest allowable load P that can be applied. Use a factor of safety for buckling of F.S. = 3.0. Take Eal = 70 GPa, sY = 215 MPa, A = 7.5(10-3) m2, Ix = 61.3(10-6) m4, Iy = 23.2(10-6) m4.

SOLUTION Buckling about the x and y axes is shown in Figs. 13–12 b and 13–12 c , respectively. Using Fig. 13–10 a , for x – x axis buckling, K = 2, so (KL)x = 2(5 m) = 10 m. Also, for y – y axis buckling, K = 0.7, so (KL)y = 0.7(5 m) = 3.5 m.

Applying Eq. 13–11 , the critical loads for each case are

(Pcr)x = p2EIx (KL)x

2 = p2[70(109) N>m2](61.3(10-6) m4)

(10 m)2

= 424 kN

(Pcr)y = p2EIy

(KL)y 2 =

p2[70(109) N>m2](23.2(10-6) m4) (3.5 m)2

= 1.31 MN

By comparison, as P is increased the column will buckle about the x – x axis. The allowable load is therefore

Pallow = Pcr

F.S. =

424 kN 3.0

= 141 kN Ans.

Since

scr = Pcr A

= 424 kN

7.5(10-3) m2 = 56.5 MPa 6 215 MPa

Euler’s equation can be applied.

(a)

5 m

x–x axis buckling

(b)

Le ! 10 m

y–y axis buckling

(c)

Le ! 3.5 m

Fig. 13–12

674 CHAPTER 13 BUCKL ING OF COLUMNS

FUNDAMENTAL PROBLEMS

F13–4. A steel pipe is fixed supported at its ends. If it is 5 m long and has an outer diameter of 50 mm and a thickness of 10 mm, determine the maximum axial load P that it can carry without buckling. Est = 200 GPa, sY = 250 MPa.

F13–5. Determine the maximum force P that can be supported by the assembly without risk of causing member AC to buckle. The member is made of A992 steel and has a diameter of 2 in. Take F.S. = 2 against buckling.

3 ft

4 ft

C A

F13–5

F13–6. The A992 steel rod BC has a diameter of 50 mm and is used as a strut to support the beam. Determine the maximum intensity w of the uniform distributed load that can be applied to the beam without risk of causing the strut to buckle. Take F.S. = 2 against buckling.

6 m 3 m

F13–6

F13–1. A 50-in.-long rod is made from a 1-in.-diameter steel rod. Determine the critical buckling load if the ends are fixed supported. E = 29(103) ksi, sY = 36 ksi.

F13–2. A 12-ft wooden rectangular column has the dimensions shown. Determine the critical load if the ends are assumed to be pin connected. E = 1.6(103) ksi. Yielding does not occur.

12 ft 4 in.

2 in.

F13–2

F13–3. The A992 steel column can be considered pinned at its top and bottom and braced against its weak axis at the mid-height. Determine the maximum allowable force P that the column can support without buckling. Apply a F.S. = 2 against buckling. Take A = 7.4(10-3) m2, Ix = 87.3(10-6) m4, and Iy = 18.8(10-6) m4.

6 m

x y

F13–3

13.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS 675

PROBLEMS

13–3. The aircraft link is made from an A992 steel rod. Determine the smallest diameter of the rod, to the nearest 116 in., that will support the load of 4 kip without buckling. The ends are pin connected.

18 in.

4 kip4 kip

Prob. 13–3

*13–4. Rigid bars AB and BC are pin connected at B . If the spring at D has a stiffness k , determine the critical load Pcr for the system.

Prob. 13–4

13–1. Determine the critical buckling load for the column. The material can be assumed rigid.

L 2

Prob. 13–1

13–2. The column consists of a rigid membr that is pinned at its bottom and attached to a spring at its top. If the spring is unstretched when the column is in the vertical position, determine the critical load that can be placed on the column.

B k

Prob. 13–2

676 CHAPTER 13 BUCKL ING OF COLUMNS

13–9. A column is constructed using four A992 steel angles that are laced together as shown. The length of the column is to be 25 ft and the ends are assumed to be pin connected. Each angle shown below has an area of A = 2.75 in 2 and moments of inertia of I x = I y = 2.22 in 4 . Determine the distance d between the centroids C of the angles so that the column can support an axial load of P = 350 kip without buckling. Neglect the effect of the lacing.

13–10. A column is constructed using four A992 steel angles that are laced together as shown. The length of the column is to be 40 ft and the ends are assumed to be fixed connected. Each angle shown below has an area of A = 2.75 in 2 and moments of inertia of I x = I y = 2.22 in 4 . Determine the distance d between the centroids C of the angles so that the column can support an axial load of P = 350 kip without buckling. Neglect the effect of the lacing.

0.932 in.

C C

0.932 in. x

2.068 in.

Probs. 13–9/10

13–11. The A992 steel angle has a cross-sectional area of A = 2.48 in2 and a radius of gyration about the x axis of rx = 1.26 in. and about the y axis of ry = 0.879 in. The smallest radius of gyration occurs about the a – a axis and is ra = 0.644 in. If the angle is to be used as a pin-connected 10-ft-long column, determine the largest axial load that can be applied through its centroid C without causing it to buckle.

x x

Prob. 13–11

13–5. A rod made from polyurethane has a stress-strain diagram in compression as shown. If the rod is pinned at its ends and is 37 in. long, determine its smallest diameter so it does not fail from elastic buckling.

13–6. A rod made from polyurethane has a stress-strain diagram in compression as shown. If the rod is pinned at its top and fixed at its base, and is 37 in. long, determine its smallest diameter so it does not fail from elastic buckling.

0.003

s (ksi)

P (in./in.)

Probs. 13–5/6

13–7. A 2014-T6 aluminum alloy hollow circular tube has an outer diameter of 150 mm and inner diameter of 100 mm. If it is pinned at both ends, determine the largest axial load that can be applied to the tube without causing it to buckle. The tube is 6 m long.

*13–8. A 2014-T6 aluminum alloy hollow circular tube has an outer diameter of 150 mm and inner diameter of 100 mm. If it is pinned at one end and fixed at the other end, determine the largest axial load that can be applied to the tube without causing it to buckle. The tube is 6 m long.

13.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS 677

13–14. The two steel channels are to be laced together to form a 30-ft-long bridge column assumed to be pin connected at its ends. Each channel has a cross-sectional area of A = 3.10 in2 and moments of inertia Ix = 55.4 in4, Iy = 0.382 in4. The centroid C of its area is located in the figure. Determine the proper distance d between the centroids of the channels so that buckling occurs about the x–x and y !9y ! axes due to the same load. What is the value of this critical load? Neglect the effect of the lacing. Est = 29(103) ksi, sy = 50 ksi.

1.231 in.0.269 in.

y y¿

C C

y y¿

x x

Prob. 13–14

13–15. An A992 steel W200 * 46 column of length 9 m is pinned at both of its ends. Determine the allowable axial load the column can support if F.S. = 2 is to be used against buckling.

*13–16. An A992 steel W200 * 46 column of length 9 m is fixed at one end and free at its other end. Determine the allowable axial load the column can support if F.S. = 2 is to be used against buckling.

13–17. The 10-ft wooden rectangular column has the dimensions shown. Determine the critical load if the ends are assumed to be pin connected. Ew = 1.6(103) ksi, sY = 5 ksi.

13–18. The 10-ft wooden column has the dimensions shown. Determine the critical load if the bottom is fixed and the top is pinned. Ew = 1.6(103) ksi, sY = 5 ksi.

10 ft 4 in.

2 in.

Probs. 13–17/18

*13–12. The control linkage for a machine consists of two L2 steel rods BE and FG , each with a diameter of 1 in. If a device at G causes the end G to freeze up and become pin connected, determine the maximum horizontal force P that could be applied to the handle without causing either of the two rods to buckle. The members are pin connected at A , B , D , E , and F .

B E

F G

4 in. 2 in.

15 in. 20 in.

12 in.

4 in.

Prob. 13–12

13–13. An A992 steel column has a length of 5 m and is fixed at both ends. If the cross-sectional area has the dimensions shown, determine the critical load.

10 mm50 mm

10 mm

100 mm

Prob. 13–13

678 CHAPTER 13 BUCKL ING OF COLUMNS

13–22. The linkage is made using two A992 steel rods, each having a circular cross section. Determine the diameter of each rod to the nearest 18 in. that will support a load of P = 6 kip. Assume that the rods are pin connected at their ends. Use a factor of safety with respect to buckling of 1.8.

13–23. The linkage is made using two A992 steel rods, each having a circular cross section. If each rod has a diameter of 34 in., determine the largest load it can support without causing any rod to buckle. Assume that the rods are pin connected at their ends.

45! 30!

12 ft

A C

Probs. 13–22/23

*13–24. An L-2 tool steel link in a forging machine is pin connected to the forks at its ends as shown. Determine the maximum load P it can carry without buckling. Use a factor of safety with respect to buckling of F.S. = 1.75. Note from the figure on the left that the ends are pinned for buckling, whereas from the figure on the right the ends are fixed.

P P

24 in.

1.5 in. 0.5 in.

Prob. 13–24

13–19. Determine the maximum force P that can be applied to the handle so that the A992 steel control rod AB does not buckle. The rod has a diameter of 1.25 in. It is pin connected at its ends.

3 ft 2 ft

3 ft

Prob. 13–19

*13–20. The A992 steel tube has the cross-sectional area shown. If it has a length of 15 ft and is pinned at both ends, determine the maximum axial load that the tube can support without causing it to buckle.

13–21. The A992 steel tube has the cross-sectional area shown. If it has a length of 15 ft and is fixed at one end and free at the other end, determine the maximum axial load that the tube can support without causing it to buckle.

3 in.

0.5 in. 0.5 in.

0.5 in.

Probs. 13–20/21

13.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS 679

13–27. The strongback BC is made of an A992 steel hollow circular section with d o = 60 mm and d i = 40 mm. Determine the allowable maximum lifting force P without causing the strong back to buckle. F.S. = 2 against buckling is desired. *13–28. The strongback is made of an A992 steel hollow circular section with the outer diameter of d o = 60 mm. If it is designed to withstand the lifting force of P = 60 kN, determine the minimum required wall thickness of the strong back so that it will not buckle. Use F.S. = 2 against buckling.

45! 45! a

4 m Section a – a

D E

Probs. 13–27/28 13–29. The beam supports the load of P = 6 kip. As a result, the A992 steel member BC is subjected to a compressive load. Due to the forked ends on the member, consider the supports at B and C to act as pins for x–x axis buckling and as fixed supports for y–y axis buckling. Determine the factor of safety with respect to buckling about each of these axes. 13–30. Determine the greatest load P the beam will support without causing the A992 steel member BC to buckle. Due to the forked ends on the member, consider the supports at B and C to act as pins for x–x axis buckling and as fixed supports for y–y axis buckling.

P 4 ft

A B

4 ft

3 ft 3 in.

1 in.

Probs. 13–29/30

13–25. The W14 * 30 is used as a structural A992 steel column that can be assumed pinned at both of its ends. Determine the largest axial force P that can be applied without causing it to buckle.

25 ft

Prob. 13–25

13–26. The A992 steel bar AB has a square cross section. If it is pin connected at its ends, determine the maximum allowable load P that can be applied to the frame. Use a factor of safety with respect to buckling of 2.

10 ft 1.5 in.

BA 1.5 in.

1.5 in. 30!

Prob. 13–26

680 CHAPTER 13 BUCKL ING OF COLUMNS

13–34. A 6061-T6 aluminum alloy solid circular rod of length 4 m is pinned at both of its ends. If it is subjected to an axial load of 15 kN and F.S. = 2 is required against buckling, determine the minimum required diameter of the rod to the nearest mm.

13–35. A 6061-T6 aluminum alloy solid circular rod of length 4 m is pinned at one end while fixed at the other end. If it is subjected to an axial load of 15 kN and F.S. = 2 is required against buckling, determine the minimum required diameter of the rod to the nearest mm.

*13–36. The members of the truss are assumed to be pin connected. If member BD is an A992 steel rod of radius 2 in., determine the maximum load P that can be supported by the truss without causing the member to buckle.

13–37. Solve Prob. 13–36 in the case of member AB , which has a radius of 2 in.

B D

C F

16 ft 16 ft

12 ft

16 ft

Probs. 13–36/37

13–38. The truss is made from A992 steel bars, each of which has a circular cross section with a diameter of 1.5 in. Determine the maximum force P that can be applied without causing any of the members to buckle. The members are pin connected at their ends.

13–39. The truss is made from A992 steel bars, each of which has a circular cross section. If the applied load P = 10 kip, determine the diameter of member AB to the nearest 18 in. that will prevent this member from buckling. The members are pin connected at their ends.

D C

3 ft

4 ft

4 ft4 ft

Probs. 13–38/39

13–31. The steel bar AB has a rectangular cross section. If it is pin connected at its ends, determine the maximum allowable intensity w of the distributed load that can be applied to BC without causing bar AB to buckle. Use a factor of safety with respect to buckling of 1.5. E st = 200 GPa, s Y = 360 MPa.

5 m

20 mm

3 m

B C

x x 30 mm

20 mm

Prob. 13–31

*13–32. The frame supports the load of P = 4 kN. As a result, the A992 steel member BC is subjected to a compressive load. Due to the forked ends on this member, consider the supports at B and C to act as pins for x – x axis buckling and as fixed supports for y – y axis buckling. Determine the factor of safety with respect to buckling about each of these axes.

13–33. Determine the greatest load P the frame will support without causing the A992 steel member BC to buckle. Due to the forked ends on the member, consider the supports at B and C to act as pins for x – x axis buckling and as fixed supports for y – y axis buckling.

4 m

2 m

1 m

35 mm

25 mm

Probs. 13–32/33

13.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS 681

13–43. The column with constant EI has the end constraints shown. Determine the critical load for the column.

Prob. 13–43

*13–44. Consider an ideal column as in Fig. 13–10 c , having both ends fixed. Show that the critical load on the column is given by Pcr = 4p2EI>L2. Hint : Due to the vertical deflection of the top of the column, a constant moment M! will be developed at the supports. Show that d2v>dx2 + (P>EI)v = M !>EI. The solution is of the form v = C1 sin(1P>EIx) + C2 cos(1P>EIx) + M !>P. 13–45. Consider an ideal column as in Fig. 13–10 d , having one end fixed and the other pinned. Show that the critical load on the column is given by Pcr = 20.19EI>L2. Hint : Due to the vertical deflection at the top of the column, a constant moment M! will be developed at the fixed support and horizontal reactive forces R! will be developed at both supports. Show that d2v>dx2 + (P>EI)v = (R !>EI)(L - x). The solution is of the form v = C1sin (1P>EIx) + C2 cos (1P>EIx) + (R !>P)(L - x). After application of the boundary conditions show that tan (1P>EIL) = 1P>EI L. Solve by trial and error for the smallest nonzero root.

*13–40. The steel bar AB of the frame is assumed to be pin con nected at its ends for y–y axis buckling. If P = 18 kN, determine the factor of safety with respect to buckling about the y–y axis due to the applied loading. Est = 200 GPa, sY = 360 MPa.

6 m

P 3 m

4 m

50 mm

x 50 mm

Prob. 13–40

13–41. The ideal column has a weight w (force> length) and rests in the horizontal position when it is subjected to the axial load P . Determine the maximum moment in the column at midspan. EI is constant. Hint: Establish the differential equation for deflection, Eq. 13–1 , with the origin at the mid span. The general solution is v = C1 sin kx + C2 cos kx + (w >(2P))x2 - (wL >(2P))x - (wEI>P 2 ) where k2 = P>EI.

Prob. 13–41

13–42. The ideal column is subjected to the force F at its midpoint and the axial load P . Determine the maximum moment in the column at midspan. EI is constant. Hint: Establish the differential equation for deflection, Eq. 13–1 . The general solution is v = C1 sin kx + C2 cos kx - c2x >k2, where c2 = F>2EI, k2 = P>EI.

L 2

Prob. 13–42

682 CHAPTER 13 BUCKL ING OF COLUMNS

*13.4 The Secant Formula The Euler formula was derived assuming the load P is always applied through the centroid of the column’s cross-sectional area and that the column is perfectly straight. This is actually quite unrealistic, since manufactured columns are never perfectly straight, nor is the application of the load known with great accuracy. In reality, then, columns never suddenly buckle; instead they begin to bend, although ever so slightly, immediately upon application of the load. As a result, the actual criterion for load application should be limited either to a specified deflection of the column or by not allowing the maximum stress in the column to exceed an allowable stress.

To study this effect, we will apply the load P to the column at a short eccentric distance e from its centroid, Fig. 13–13 a . This loading on the column is statically equivalent to the axial load P and bending moment M ! = Pe shown in Fig. 13–13 b . As shown, in both cases, the ends A and B are supported so that they are free to rotate (pin supported). As before, we will only consider small slopes and deflections and linear-elastic material behavior. Furthermore, the x – v plane is a plane of symmetry for the cross-sectional area.

From the free-body diagram of the arbitrary section, Fig. 13–13 c , the internal moment in the column is

M = -P(e + v) (13–13)

The differential equation for the deflection curve is therefore

EI d 2v dx2

= -P(e + v) The column supporting this crane is unusually long. It will be subjected not only to uniaxial load, but also a bending moment. To ensure it will not buckle, it should be braced at the roof as a pin connection.

P x

e A

(a)

M¿ ! Pe

M¿ ! Pe P x

(b)

P M

13.4 THE SECANT FORMULA 683

d 2v

dx2 + P

EI v = - P

EI e

This equation is similar to Eq. 13–7 and has a general solution consisting of the complementary and particular solutions, namely,

v = C1 sin A PEI x + C2 cos A PEI x - e (13–14) To evaluate the constants we must apply the boundary conditions. At x = 0, v = 0, so C2 = e. And at x = L, v = 0, which gives

C1 = e[1 - cos(2P>EI L)]

sin(2P>EI L) Since 1 - cos(2P>EI L) = 2 sin2(2P>EI L >2) and sin(2P>EI L) = 2 sin(2P>EI L >2) cos (2P>EI L >2), we have

C1 = e tanaA PEI L2 b Hence, the deflection curve, Eq. 13–14 , can be written as

v = e c tanaA PEI L2 b sinaA PEI x b + cosaA PEI x b - 1 d (13–15) Maximum Deflection. Due to symmetry of loading, both the maximum deflection and maximum stress occur at the column’s mid point. Therefore, when x = L >2, v = vmax, so vmax = e c secaA PEI L2 b - 1 d (13–16) Notice that if e approaches zero, then vmax approaches zero. However, if the terms in the brackets approach infinity as e approaches zero, then vmax will have a nonzero value. Mathematically, this would represent the behavior of an axially loaded column at failure when subjected to the critical load Pcr. Therefore, to find Pcr we require

sec¢APcrEI L2 b = ! APcrEI L2 = p2 Pcr =

p2EI

L2 (13–17)

which is the same result found from the Euler formula, Eq. 13–5 . If Eq. 13–16 is plotted as load P versus deflection ymax for various

values of eccentricity e , the family of colored curves shown in Fig. 13–14 results. Here the critical load becomes an asymptote to the curves, and of

684 CHAPTER 13 BUCKL ING OF COLUMNS

course represents the unrealistic case of an ideal column (e = 0). As stated earlier, e is never zero due to imperfections in initial column straightness and load application; however, as e S 0, the curves tend to approach the ideal case. Furthermore, these curves are appropriate only for small deflections , since the curvature was approximated by d2v>dx2 when Eq. 13–16 was developed. Had a more exact analysis been performed, all these curves would tend to turn upward, intersecting and then rising above the line P = Pcr. This, of course, indicates that a larger load P is needed to create larger column deflections. We have not considered this analysis here, however, since most often engineering design restricts the deflection of columns to small values.

It should also be noted that the colored curves in Fig. 13–14 apply only for linear-elastic material behavior. Such is the case if the column is long and slender. However, if a short or intermediate-length stocky column is considered, then the applied load, as it is increased, may eventually cause the material to yield, and the column will begin to behave in an inelastic manner . This occurs at point A for the black curve in Fig. 13–14 . As the load is further increased, the curve never reaches the critical load, and instead the load reaches a maximum value at B . Afterwards, a sudden decrease in load-carrying capacity occurs as the column continues to yield and deflect by larger amounts.

Lastly, the colored curves in Fig. 13–14 also illustrate that a nonlinear relationship occurs between the load P and the deflection v. As a result, the principle of superposition cannot be used to determine the total deflection of a column caused by applying successive loads to the column. Instead, the loads must first be added, and then the corresponding deflection due to their resultant can be determined. Physically, the reason that successive loads and deflections cannot be superimposed is that the column’s internal moment depends on both the load P and the deflection v, that is, M = -P(e + v), Eq. 13–13 .

Pcr Ideal column (small deflections)

Inelastic behavior

eas 0

spl is reached

vmax

e ! 0

Fig. 13–14

13.4 THE SECANT FORMULA 685

The Secant Formula. The maximum stress in the column can be determined by realizing that it is caused by both the axial load and the moment, Fig. 13–15 a . Maximum moment occurs at the column’s midpoint, and using Eqs. 13–13 and 13–16 , it has a magnitude of

M = ! P(e + vmax) ! M = Pe secaA PEI L2 b (13–18) As shown in Fig. 13–15 b , the maximum stress in the column is compressive, and it has a value of

smax = P A

+ Mc I

; smax = P A

+ Pec I

secaA PEI L2 b Since the radius of gyration is defined as r2 = I>A , the above equation can be written in a form called the secant formula :

smax = P A

J1 + ec r2

seca L 2r

A PEA b R (13–19) Here

smax = maximum elastic stress in the column, which occurs at the inner concave side at the column’s midpoint. This stress is compressive.

P = vertical load applied to the column. P 6 Pcr unless e = 0; then P = Pcr ( Eq. 13–5 ).

e = eccentricity of the load P , measured from the centroidal axis of the column’s cross-sectional area to the line of action of P

c = distance from the centroidal axis to the outer fiber of the column where the maximum compressive stress smax occurs

A = cross-sectional area of the column L = unsupported length of the column in the plane of bending .

Application is restricted to members that are pin connected, Le = L or have one end free and the other end fixed, Le = 2L.

E = modulus of elasticity for the material r = radius of gyration, r = 2I>A , where I is calculated about the

centroidal or bending axis

Like Eq. 13–16 , Eq. 13–19 indicates that there is a nonlinear relationship between the load and the stress. Hence, the principle of superposition does not apply, and therefore the loads have to be added before the stress is determined. Furthermore, due to this nonlinear relationship, any factor of safety used for design purposes applies to the load and not to the stress.

For a given value of smax, graphs of Eq. 13–19 can be plotted as the slenderness ratio KL >r versus the average stress P>A for various values of the eccentricity ratio ec>r 2. A specific set of graphs for a structural-grade A-36 steel having a yield point of smax = sY = 36 ksi

P e

P M

e e

(a)

Axial stress

(b)

Bending stress

Resultant stress

smax

Fig. 13–15

686 CHAPTER 13 BUCKL ING OF COLUMNS

and a modulus of elasticity of Est = 29(103) ksi is shown in Fig. 13–16 . Note that when e S 0, or when ec>r2 S 0, Eq. 13–19 gives smax = P>A , where P is the critical load on the column, defined by Euler’s formula. This results in Eq. 13–6 , which has been plotted in Fig. 13–7 and repeated in Fig. 13–16 . Since both Eqs. 13–6 and 13–19 are valid only for elastic loadings, the stresses shown in Fig. 13–16 cannot exceed sY = 36 ksi, represented here by the horizontal line.

The curves in Fig. 13–16 indicate that differences in the eccentricity ratio have a marked effect on the load-carrying capacity of columns that have small slenderness ratios. However, columns that have large slenderness ratios tend to fail at or near the Euler critical load regardless of the eccentricity ratio. When using Eq. 13–19 for design purposes, it is therefore important to have a somewhat accurate value for the eccentricity ratio for shorter-length columns.

Design. Once the eccentricity ratio has been determined, the column data can be substituted into Eq. 13–19 . If a value of smax = sY is chosen, then the corresponding load PY can be determined from a trial-and-error procedure, since the equation is transcendental and cannot be solved explicitly for PY . As a design aid, computer software, or graphs such as those in Fig. 13–16 , can also be used to determine PY directly.

Realize that PY is the load that will cause the column to develop a maximum compressive stress of sY at its inner concave fibers. Due to the eccentric application of PY , this load will always be smaller than the critical load Pcr, which is determined from the Euler formula that assumes (unrealistically) that the column is axially loaded.

50 100 A-36 structural steel

150 200

Euler’s formula Eq. 13–60.5

1.0

1.5

(ksi)P A

KL r

r2 ! 0

Est ! 29 (103) ksi, sY ! 36 ksi

= 0.1—ec r2

Fig. 13–16

Important Points

• Due to imperfections in manufacturing or specific application of the load, a column will never suddenly buckle; instead, it begins to bend.

• The load applied to a column is related to its deflection in a nonlinear manner, and so the principle of superposition does not apply.

• As the slenderness ratio increases, eccentrically loaded columns tend to fail at or near the Euler buckling load.

13.4 THE SECANT FORMULA 687

EXAMPLE 13.4

The W8 * 40 A992 steel column shown in Fig. 13–17 a is fixed at its base and braced at the top so that it is fixed from displacement, yet free to rotate about the y – y axis. Also, it can sway to the side in the y–z plane. Determine the maximum eccentric load the column can support before it either begins to buckle or the steel yields.

SOLUTION From the support conditions it is seen that about the y – y axis the column behaves as if it were pinned at its top and fixed at the bottom and subjected to an axial load P , Fig. 13–17 b . About the x – x axis the column is free at the top and fixed at the bottom, and it is subjected to both an axial load P and moment M = P(9 in.), Fig. 13–17 c .

y–y Axis Buckling. From Fig. 13–10 d the effective length factor is Ky = 0.7, so (KL)y = 0.7(12) ft = 8.40 ft = 100.8 in. Using the table in Appendix B to determine Iy for the W8 * 40 section and applying Eq. 13–11 , we have

(Pcr)y = p2EIy

(KL)y 2 =

p2[29(103) ksi](49.1 in4)

(100.8 in.)2 = 1383 kip

x–x Axis Yielding. From Fig. 13–10 b , Kx = 2, so (KL)x = 2(12) ft = 24 ft = 288 in. Again using the table in Appendix B to determine A = 11.7 in2, c = 8.25 in.>2 = 4.125 in., and rx = 3.53 in., and applying the secant formula, we have

sY = Px A

J1 + ec rx

2 sec¢ (KL)x2rx A PxEA ≤ R Substituting the data and simplifying yields

585 = Px[1 + 2.979 sec(0.07002Px)] Solving for Px by trial and error, noting that the argument for the secant is in radians, we get

Px = 115 kip Ans.

Since this value is less than (Pcr)y = 1383 kip, failure will occur about the x – x axis.

(a)

9 in.

12 ft

P z

(b) y–y axis buckling

8.40 ft 12 ft

12 ft

(c)

x–x axis yielding

M ! P(9 in.)

Fig. 13–17

688 CHAPTER 13 BUCKL ING OF COLUMNS

*13.5 Inelastic Buckling In engineering practice, columns are generally classified according to the type of stresses developed within the column at the time of failure. Long slender columns will become unstable when the compressive stress remains elastic. The failure that occurs is referred to as elastic instability . Intermediate columns fail due to inelastic instability , meaning that the compressive stress at failure is greater than the material’s proportional limit. And short columns , sometimes called posts , do not become unstable; rather the material simply yields or fractures.

Application of the Euler equation requires that the stress in the column remain below the material’s yield point (actually the proportional limit) when the column buckles, and so this equation applies only to long columns. In practice, however, most columns are selected to have intermediate lengths. The behavior of these columns can be studied by modifying the Euler equation so that it applies for inelastic buckling. To show how this can be done, consider the material to have a stress–strain diagram as shown in Fig. 13–18 a . Here the proportional limit is spl , and the modulus of elasticity, or slope of the line AB , is E .

If the column has a slenderness ratio that is less than (KL >r)pl , then the critical stress in the column must be greater than spl . For example, suppose a column has a slenderness ratio of (KL >r)1 6 (KL >r)pl , with corresponding critical stress sD 7 spl needed to cause instability. When the column is about to buckle , the change in stress and strain that occurs in the column is within a small range !s and !P, so that the modulus of elasticity or stiffness for the material can be taken as the tangent modulus Et = !s>!P defined as the slope of the s-P diagram at point D , Fig. 13–18 a . In other words, at the time of failure, the column behaves as if it were made from a material that has a lower stiffness than when it behaves elastically, Et 6 E.

(a)

spl

Fig. 13–18

This crane boom failed by buckling caused by an overload. Note the region of localized collapse.

13.5 INELASTIC BUCKLING 689

In general, therefore, as the slenderness ratio ( KL>r ) decreases, the critical stress for a column continues to rise; and from the s-P diagram, the tangent modulus for the material decreases . Using this idea, we can modify Euler’s equation to include these cases of inelastic buckling by substituting the material’s tangent modulus Et for E , so that

scr = p2Et

(KL >r)2 (13–20) This is the so-called tangent modulus or Engesser equation , proposed

by F. Engesser in 1889. A plot of this equation for intermediate and short- length columns of a material defined by the s-P diagram in Fig. 13–18 a is shown in Fig. 13–18 b .

No actual column can be considered to be either perfectly straight or loaded along its centroidal axis, as assumed here, and therefore it is indeed very difficult to develop an expression that will provide a complete analysis of this phenomenon. As a result, other methods of describing the inelastic buckling of columns have been considered. One of these methods was developed by the aeronautical engineer F. R. Shanley and is called the Shanley theory of inelastic buckling. Although it provides a better description of the phenomenon than the tangent modulus theory, as explained here, experimental testing of a large number of columns, each of which approximates the ideal column, has shown that Eq. 13–20 is reasonably accurate in predicting the column’s critical stress. Furthermore, the tangent modulus approach to modeling inelastic column behavior is relatively easy to apply.

Fig. 13–18 (cont.)

Long columns

(b)

Short and intermediate length columns

ElasticInelastic

scr

spl

scr ! p2Et

(KL/r)2

scr ! p2E

(KL/r)2

KL rKL

r 1 KL

r pl

690 CHAPTER 13 BUCKL ING OF COLUMNS

A solid rod has a diameter of 30 mm and is 600 mm long. It is made of a material that can be modeled by the stress–strain diagram shown in Fig. 13–19 . If it is used as a pin-supported column, determine the critical load.

SOLUTION The radius of gyration is

r = A I A = B (p>4)(15 mm)4 p(15 mm)2 = 7.5 mm and therefore the slenderness ratio is

KL r

= 1(600 mm)

7.5 mm = 80

Applying Eq. 13–20 we have,

scr = p 2E t

(KL >r) 2 = p 2Et (80) 2 = 1.542(10-3)Et (1) First we will assume that the critical stress is elastic. From Fig. 13–19 ,

E = 150 MPa

0.001 = 150 GPa

Thus, Eq. 1 becomes

s cr = 1.542(10-3)[150(103)] MPa = 231.3 MPa

Since s cr 7 s pl = 150 MPa, inelastic buckling occurs. From the second line segment of the s-P diagram, Fig. 13–19 , we

have

E t = !s !P =

270 MPa - 150 MPa 0.002 - 0.001 = 120 GPa

Applying Eq. 1 yields

scr = 1.542(10-3)[120(103)] MPa = 185.1 MPa

Since this value falls within the limits of 150 MPa and 270 MPa, it is indeed the critical stress.

The critical load on the rod is therefore

Pcr = scr A = 185.1(106) Pa[p(0.015 m)2 ] = 131 kN Ans.

EXAMPLE 13.5

270

0.001 0.002

s (MPa)

spl ! 150

Fig. 13–19

13.5 INELASTIC BUCKLING 691

PROBLEMS

13–46. The wood column has a square cross section with dimensions 100 mm by 100 mm. It is fixed at its base and free at its top. Determine the load P that can be applied to the edge of the column without causing the column to fail either by buckling or by yielding. E w = 12 GPa, s Y = 55 MPa.

120 mm100 mm

100 mm

2 m

Prob. 13–46 13–47. The hollow red brass C83400 copper alloy shaft is fixed at one end but free at the other end. Determine the maximum eccentric force P the shaft can support without causing it to buckle or yield. Also, find the corresponding maximum deflection of the shaft. *13–48. The hollow red brass C83400 copper alloy shaft is fixed at one end but free at the other end. If the eccentric force P = 5 kN is applied to the shaft as shown, determine the maximum normal stress and the maximum deflection.

150 mm

2 m

30 mm

20 mm

Section a – a

Probs. 13–47/48

13–49. The tube is made of copper and has an outer diameter of 35 mm and a wall thickness of 7 mm. Determine the eccentric load P that it can support without failure. The tube is pin supported at its ends. Ecu = 120 GPa, sY = 750 MPa.

13–50. Solve Prob. 13–49 if instead the left end is free and the right end is fixed-supported.

2 m

14 mm

P P

Probs. 13–49/50

13–51. Assume that the wood column is pin connected at its base and top. Determine the maximum eccentric load P that can be applied without causing the column to buckle or yield. Ew = 1.8(103) ksi, sY = 8 ksi.

*13–52. Assume that the wood column is pinned top and bottom for movement about the x–x axis, and fixed at the bottom and free at the top for movement about the y–y axis. Determine the maximum eccentric load P that can be applied without causing the column to buckle or yield. Ew = 1.8(103) ksi, sY = 8 ksi.

10 ft

10 in.

4 in. x

P x

Probs. 13–51/52

692 CHAPTER 13 BUCKL ING OF COLUMNS

*13–56. The wood column is pinned at its base and top. Determine the maximum eccentric force P the column can support without causing it to either buckle or yield. Take E = 10 GPa and s Y = 15 MPa.

3.5 m

150 mm x

75 mm 75 mm

25 mm

25 mm xy

Probs. 13–55/56

13–57. The 6061-T6 aluminum alloy solid shaft is fixed at one end but free at the other end. If the shaft has a diameter of 100 mm, determine its maximum allowable length L if it is subjected to the eccentric force P = 80 kN.

13–58. The 6061-T6 aluminum alloy solid shaft is fixed at one end but free at the other end. If the length is L = 3 m, determine its minimum required diameter if it is subjected to the eccentric force P = 60 kN.

100 mm

Probs. 13–57/58

13–53. A W12 * 26 structural A992 steel column is pin connected at its ends and has a length L = 11.5 ft. Determine the maximum eccentric load P that can be applied so the column does not buckle or yield. Compare this value with an axial critical load P ! applied through the centroid of the column.

6 in. P

Prob. 13–53

13–54. A W14 * 30 structural A-36 steel column is pin connected at its ends and has a length L = 10 ft. Determine the maximum eccentric load P that can be applied so the column does not buckle or yield. Compare this value with an axial critical load P ! applied through the centroid of the column.

6 in. P

Prob. 13–54

13–55. The wood column is pinned at its base and top. If the eccentric force P = 10 kN is applied to the column, investigate whether the column is adequate to support this loading without buckling or yielding. Take E = 10 GPa and s Y = 15 MPa.

13.5 INELASTIC BUCKLING 693

13–61. The A992 steel rectangular hollow section column is pinned at both ends. If it has a length of L = 14 ft, determine the maximum allowable eccentric force P it can support without causing it to either buckle or yield.

13–62. The A992 steel rectangular hollow section column is pinned at both ends. If it is subjected to the eccentric force P = 45 kip, determine its maximum allowable length L without causing it to either buckle or yield.

6 in.

6 in. P a

5 in.

2 in. 0.5 in.

0.5 in.

Section a – a

Probs. 13–61/62

13–63. The W10 * 30 structural A992 steel columm is pinned at its top and bottom. Determine the maximum load P it can support.

*13–64. The W10 * 30 structural A992 steel columm is fixed at its bottom and free at its top. If it is subjected to the eccentric load of P = 85 kip, determine if the column fails by yielding. The column is braced so that it does not buckle about the y – y axis.

15 ft

8 in. y

y x

680 kip!in.

Probs. 13–63/64

13–59. The wood column is pinned at its base and top. If L = 7 ft, determine the maximum eccentric load P that can be applied without causing the column to buckle or yield. E w = 1.8(10 3 ) ksi, s Y = 8 ksi.

10 in.

4 in. x

P x

Prob. 13–59

*13–60. The wood column is pinned at its base and top. If L = 5 ft, determine the maximum eccentric load P that can be applied without causing the column to buckle or yield. E w = 1.8(10 3 ) ksi, s Y = 8 ksi.

10 in.

4 in. x

P x

Prob. 13–60

694 CHAPTER 13 BUCKL ING OF COLUMNS

13–69. The W14 * 53 column is fixed at its base and free at its top. Determine the maximum eccentric load P that it can support without causing it to buckle or yield. E st = 29(10 3 ) ksi, s Y = 50 ksi.

18 ft

10 in. P

Probs. 13–68/69 13–70. A column of intermediate length buckles when the compressive stress is 40 ksi. If the slenderness ratio is 60, determine the tangent modulus.

13–71. The aluminum rod is fixed at its base free at its top. If the eccentric load P = 200 kN is applied, determine the greatest allowable length L of the rod so that it does not buckle or yield. E al = 72 GPa, s Y = 410 MPa.

*13–72. The aluminum rod is fixed at its base free at its top. If the length of the rod is L = 2 m, Determine the greatest allowable load P that can be applied so that the rod does not buckle or yield. Also, determine the largest sidesway deflection of the rod due to the loading. E al = 72 GPa, s Y = 410 MPa.

5 mm

200 mm

Probs. 13–71/72

13–65. Determine the maximum eccentric load P the 2014-T6-aluminum-alloy strut can support without causing it either to buckle or yield. The ends of the strut are pin connected.

3 m

100 mm

150 mm

100 mm

50 mm

150 mm a

P P

Section a – a

Prob. 13–65

13–66. The W10 * 45 structural A992 steel column is assumed to be pinned at its top and bottom. If the 12-kip load is applied at an eccentric distance of 8 in., determine the maximum stress in the column. Take L = 12.6 ft.

13–67. The W10 * 45 structural A992 steel column is assumed to be pinned at its top and bottom. If the 12-kip load is applied at an eccentric distance of 8 in., determine the maximum stress in the column. Take L = 9 ft.

12 kip 8 in.

Probs. 13–66/67

*13–68. The W14 * 53 structural A992 steel column is fixed at its base and free at its top. If P = 75 kip, determine the sidesway deflection at its top and the maximum stress in the column.

13.5 INELASTIC BUCKLING 695

13–75. The stress–strain diagram of the material can be approximated by the two line segments shown. If a bar having a diameter of 80 mm and a length of 1.5 m is made from this material, determine the critical load provided the ends are pinned. Assume that the load acts through the axis of the bar. Use Engesser’s equation.

550

100

0.001 0.007 P (mm/mm)

s (MPa)

Prob. 13–75 *13–76. The stress–strain diagram of the material can be approximated by the two line segments shown. If a bar having a diameter of 80 mm and a length of 1.5 m is made from this material, determine the critical load provided the ends are fixed. Assume that the load acts through the axis of the bar. Use Engesser’s equation.

550

100

0.001 0.007 P (mm/mm)

s (MPa)

Prob. 13–76 13–77. The stress–strain diagram of the material can be approximated by the two line segments shown. If a bar having a diameter of 80 mm and a length of 1.5 m is made from this material, determine the critical load provided one end is pinned and the other is fixed. Assume that the load acts through the axis of the bar. Use Engesser’s equation.

550

100

0.001 0.007 P (mm/mm)

s (MPa)

Prob. 13–77

13–73. The stress-strain diagram of the material of a column can be approximated as shown. Plot P>A vs. KL>r for the column.

P (in./in.)

200

350

0.0010 0.004

s (MPa)

Prob. 13–73

13–74. Construct the buckling curve, P>A versus L>r , for a column that has a bilinear stress–strain curve in compression as shown.

0.002 0.005 P (in./in.)

s (ksi)

Prob. 13–74

696 CHAPTER 13 BUCKL ING OF COLUMNS

*13.6 Design of Columns for Concentric Loading

The theory presented thus far applies to columns that are perfectly straight, made of homogeneous material, and originally stress free. Practically speaking, though, as stated previously, columns are not perfectly straight, and most have residual stresses in them, primarily due to nonuniform cooling during manufacture. Also, the supports for columns are less than exact, and the points of application and directions of loads are not known with absolute certainty. In order to compensate for these effects, which actually vary from one column to the next, many design codes specify the use of column formulas that are empirical. By performing experimental tests on a large number of axially loaded columns, the results may be plotted and a design formula developed by curve-fitting the mean of the data.

An example of such tests for wide-flange steel columns is shown in Fig. 13–20 . Notice the similarity between these results and those of the family of curves determined from the secant formula, Fig. 13–16 . The reason for this similarity has to do with the influence of an “accidental” eccentricity ratio on the column’s strength. As stated in Sec. 13.4 , this ratio has more of an effect on the strength of short and intermediate- length columns than on those that are long. Tests have indicated that ec>r 2 can range from 0.1 to 0.6 for most axially loaded columns.

In order to account for the behavior of different-length columns, design codes usually specify several formulas that will best fit the data within the short, intermediate, and long column range. Hence, each formula will apply only for a specific range of slenderness ratios, and so it is important that the engineer carefully observe the KL > r limits for which a particular formula is valid. Examples of design formulas for steel, aluminum, and wood columns that are currently in use will now be discussed. The purpose is to give some idea as to how columns are designed in practice. These formulas should not, however, be used for the design of actual columns, unless the code from which they are referenced is consulted.

These long unbraced timber columns are used to support the roof of this building.

Euler formula Eq. 13–6

Short column

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

Intermediate column Long column ——KL r

scr

Fig. 13–20

13.6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING 697

Steel Columns. Columns made of structural steel can be designed on the basis of formulas proposed by the Structural Stability Research Council (SSRC). Factors of safety have been applied to these formulas and adopted as specifications for building construction by the American Institute of Steel Construction (AISC). Basically these specifications provide two formulas for column design, each of which gives the maximum allowable stress in the column for a specific range of slenderness ratios. *

For long columns the Euler formula is proposed, i.e., smax = p2 E>(KL >r)2.

Application of this formula requires that a factor of safety F.S. = 23 12 ! 1.92 be applied. Thus, for design,

sallow = 12p2E

23(KL >r)2 aKLr b c … KLr … 200 (13–21) As stated, this equation is applicable for a slenderness ratio bounded by 200 and (KL >r)c . A specific value of (KL >r)c is obtained by requiring the Euler formula to be used only for elastic material behavior. Through experiments it has been determined that compressive residual stresses can exist in rolled-formed steel sections that may be as much as one-half the yield stress. Consequently, if the stress in the Euler formula is greater than 1 2 sY , the equation will not apply. Therefore the value of (KL >r)c is determined as follows:

1 2

sY = p2E

(KL >r)c 2 or aKLr b c = B2p2EsY (13–22) Columns having slenderness ratios less than (KL >r)c are designed on

the basis of an empirical formula that is parabolic and has the form

smax = J1 - (KL >r)2 2(KL >r)c2 RsY Since there is more uncertainty in the use of this formula for longer columns, it is divided by a factor of safety defined as follows:

F.S. = 5 3

+ 3 8

(KL >r)

(KL >r)c - (KL >r)3 8(KL >r)c3 Here it is seen that F.S. = 5 3 ! 1.67 at KL >r = 0 and increases to F.S. = 2312 ! 1.92 at (KL >r)c . Hence, for design purposes, sallow =

J1 - (KL >r)2 2(KL >r)c2 RsY

(5>3) + [(3>8)(KL >r)>(KL >r)c] - 3(KL >r)3>8(KL >r)c 34 (13–23) Equations 13–21 and 13–23 are plotted in Fig. 13–21 . When applying any of these equations, either FPS or SI units can be used for the calculations.

* The current AISC code enables engineers to use one of two methods for design, namely, Load and Resistance Factor Design, and Allowable Stress Design. The latter is explained here.

0.6

0.261

Eq. 13–23

Eq. 13–21

——KLr c

——KLr0

———— sallow sY

Fig. 13–21

698 CHAPTER 13 BUCKL ING OF COLUMNS

Aluminum Columns. Column design for structural aluminum is specified by the Aluminum Association using three equations, each applicable for a specific range of slenderness ratios. Since several types of aluminum alloy exist, there is a unique set of formulas for each type. For a common alloy (2014-T6) used in building construction, the formulas are

sallow = 28 ksi 0 … KL r

… 12 (13–24)

sallow = c 30.7 - 0.23aKL r b d ksi 12 6 KL r 6 55 (13–25) sallow =

54 000 ksi (KL >r)2 55 … KL r (13–26)

These equations are plotted in Fig. 13–22 . As shown, the first two represent straight lines and are used to model the effects of columns in the short and intermediate range. The third formula has the same form as the Euler formula and is used for long columns.

Timber Columns. Columns used in timber construction are designed on the basis of formulas published by the National Forest Products Association (NFPA) or the American Institute of Timber Construction (AITC). For example, the NFPA formulas for the allowable stress in short, intermediate, and long columns having a rectangular cross section of dimensions b and d , where d is the smallest dimension of the cross section, are

sallow = 1.20 ksi 0 … KL d

… 11 (13–27)

sallow = 1.20 c 1 - 1 3 aKL >d 26.0 b2 d ksi 11 6 KL d … 26 (13–28) sallow =

540 ksi (KL >d)2 26 6 KL d … 50 (13–29)

Here wood has a modulus of elasticity of Ew = 1.8(103) ksi and an allowable compressive stress of 1.2 ksi parallel to the grain. In particular, Eq. 13–29 is simply Euler’s equation having a factor of safety of 3. These three equations are plotted in Fig. 13–23 .

12 55 ——KL r

Eq. 13–24

Eq. 13–25

Eq. 13–26

sallow(ksi)

Fig. 13–22

1.2

——KL d

sallow(ksi)

Eq. 13–28

0.8

0.216

11 26 50

Eq. 13–27

Eq. 13–29

Fig. 13–23

13.6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING 699

These timber columns can be considered pinned at their bottom and fixed connected to the beams at their tops.

Procedure for Analysis

Column Analysis.

• When using any formula to analyze a column, that is, to find its allowable load, it is first necessary to calculate the slenderness ratio in order to determine which column formula applies.

• Once the average allowable stress has been calculated, the allowable load on the column is determined from P = sallow A.

Column Design.

• If a formula is used to design a column, that is, to determine the column’s cross-sectional area for a given loading and effective length, then a trial-and-check procedure generally must be followed when the column has a composite shape, such as a wide-flange section.

• One possible way to apply a trial-and-check procedure would be to assume the column’s cross-sectional area, A!, and calculate the corresponding stress s! = P>A!. Also, use an appropriate design formula to determine the allowable stress sallow. From this, calculate the required column area A req>d = P>sallow.

• If A! 7 A req>d , the design is safe. When making the comparison, it is practical to require A! to be close to but greater than A req>d , usually within 2–3%. A redesign is necessary if A! 6 A req>d .

• Whenever a trial-and-check procedure is repeated, the choice of an area is determined by the previously calculated required area. In engineering practice this method for design is usually shortened through the use of computer software or published tables and graphs.

700 CHAPTER 13 BUCKL ING OF COLUMNS

An A992 steel W10 * 100 member is used as a pin-supported column, Fig. 13–24 . Using the AISC column design formulas, determine the largest load that it can safely support.

SOLUTION The following data for a W10 * 100 is taken from the table in Appendix B .

A = 29.4 in2 rx = 4.60 in. ry = 2.65 in.

Since K = 1 for both x and y axis buckling, the slenderness ratio is largest if ry is used. Thus,

KL r

= 1(16 ft)(12 in.>ft)

2.65 in. = 72.45

From Eq. 13–22 , we have

aKL r

b c = B2p2 E sY = B2p2[29(103) ksi] 50 ksi = 107

Here 0 6 KL >r 6 (KL >r)c, so Eq. 13–23 applies. sallow =

J1 - (KL >r)2 2(KL >r)c2 RsY

(5>3) + [(3>8)(KL >r)>(KL >r)c] - 3(KL >r)3>8(KL >rc )34 =

[1 - (72.45)2>2(107)2] 50 ksi (5>3) + [(3>8)(72.45>107)] - [(72.45)3>8(107)3]

= 23.93 ksi

The allowable load P on the column is therefore

sallow = P A

; 23.93 kip>in2 = P 29.4 in2

P = 704 kip Ans.

EXAMPLE 13.6

16 ft

Fig. 13–24

13.6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING 701

EXAMPLE 13.7

The steel rod in Fig. 13–25 is to be used to support an axial load of 18 kip. If Est = 29(103 ) ksi and sY = 50 ksi, determine the smallest diameter of the rod as allowed by the AISC specification. The rod is fixed at both ends.

18 kip18 kip

15 ft

Fig. 13–25

SOLUTION For a circular cross section the radius of gyration becomes

r = A I A = B (1>4)p(d>2)4 (1>4)pd2 = d 4 Applying Eq. 13–22 , we have

aKL r

b c = B2p2 E sY = B2p2[29(103 ) ksi] 50 ksi = 107.0

Since the rod’s radius of gyration is unknown, KL > r is unknown, and therefore a choice must be made as to whether Eq. 13–21 or Eq. 13–23 applies. We will consider Eq. 13–21 . For a fixed-end column K = 0.5, so

sallow = 12p2E

23(KL >r)2

18 kip

(1>4)pd2 = 12p2[29(103) kip>in2] 23[0.5(15 ft)(12 in.>ft)>(d>4)]2 22.92

d2 = 1.152d2

d = 2.11 in.

Use d = 2.25 in. = 2 14 in. Ans.

For this design, we must check the slenderness-ratio limits; i.e.,

KL r

= 0.5(15 ft)(12 in.>ft)

(2.25 in.>4) = 160 Since 107.0 6 160 6 200, use of Eq. 13–21 is appropriate.

702 CHAPTER 13 BUCKL ING OF COLUMNS

A bar having a length of 30 in. is used to support an axial compressive load of 12 kip, Fig. 13–26 . It is pin supported at its ends and made of a 2014-T6 aluminum alloy. Determine the dimensions of its cross- sectional area if its width is to be twice its thickness.

SOLUTION Since KL = 30 in. is the same for both x and y axis buckling, the larger slenderness ratio is determined using the smaller radius of gyration, i.e., using Imin = Iy:

KL ry

= KL

2Iy >A = 1(30) 2(1>12)2b(b3)>[2b(b)] = 103.9 b (1) Here we must apply Eq. 13–24 , 13–25 , or 13–26 . Since we do not as yet know the slenderness ratio, we will begin by using Eq. 13–24 .

P A

= 28 ksi

12 kip 2b(b)

= 28 kip>in2 b = 0.463 in.

Checking the slenderness ratio, we have

KL r

= 103.9 0.463

= 224.5 7 12

Try Eq. 13–26 , which is valid for KL >r Ú 55,

P A

= 54 000 ksi (KL >r)2

2b(b) =

54 000 (103.9>b)2

b = 1.05 in. Ans.

From Eq. 1 ,

KL r

= 103.9 1.05

= 99.3 7 55 OK

NOTE: It would be satisfactory to choose the cross section with dimensions 1 in. by 2 in.

EXAMPLE 13.8

12 kip

30 in.

12 kip

2bb

Fig. 13–26

13.6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING 703

A board having cross-sectional dimensions of 5.5 in. by 1.5 in. is used to support an axial load of 5 kip, Fig. 13–27 . If the board is assumed to be pin supported at its top and bottom, determine its greatest allowable length L as specified by the NFPA.

EXAMPLE 13.9

5 kip

5.5 in.1.5 in.

y x

5 kip

Fig. 13–27

SOLUTION By inspection, the board will buckle about the y axis. In the NFPA equations, d = 1.5 in. Assuming that Eq. 13–29 applies, we have

P A

= 540 ksi

(KL >d)2

5 kip (5.5 in.)(1.5 in.)

= 540 ksi

(1 L >1.5 in.)2 L = 44.8 in. Ans.

Here

KL d

= 1(44.8 in.)

1.5 in. = 29.8

Since 26 6 KL >d … 50, the solution is valid.

704 CHAPTER 13 BUCKL ING OF COLUMNS

13–87. A 5-ft-long rod is used in a machine to transmit an axial compressive load of 3 kip. Determine its smallest diameter if it is pin connected at its ends and is made of a 2014-T6 aluminum alloy.

*13–88. Determine the largest length of a W8 * 31 structural A992 steel column if it is to support an axial load of 10 kip. The ends are pinned.

13–89. Using the AISC equations, check if a column having the cross section shown can support an axial force of 1500 kN. The column has a length of 4 m, is made from A992 steel, and its ends are pinned.

350 mm

10 mm

300 mm

20 mm20 mm

Prob. 13–89

13–90. The A992-steel tube is pinned at both ends. If it is subjected to an axial force of 150 kN, determine the maximum length of the tube using the AISC column design formulas.

100 mm

80 mm

Prob. 13–90

13–78. Determine the largest length of a W10 * 12 structural A992 steel section if it is pin supported and is subjected to an axial load of 28 kip. Use the AISC equations.

13–79. Using the AISC equations, select from Appendix B the lightest-weight structural A992 steel column that is 14 ft long and supports an axial load of 40 kip. The ends are fixed.

*13–80. Using the AISC equations, select from Appendix B the lightest-weight structural A992 steel column that is 14 ft long and supports an axial load of 40 kip. The ends are pinned. Take sY = 50 ksi.

13–81. Determine the largest length of a W8 * 31 structural A992 steel section if it is pin supported and is subjected to an axial load of 130 kip. Use the AISC equations.

13–82. Using the AISC equations, select from Appendix B the lightest-weight structural A992 steel column that is 12 ft long and supports an axial load of 20 kip. The ends are pinned.

13–83. Determine the largest length of a W10 * 12 structural A992 steel section if it is fixed supported and is subjected to an axial load of 28 kip. Use the AISC equations.

*13–84. Using the AISC equations, select from Appendix B the lightest-weight structural A992 steel column that is 30 ft long and supports an axial load of 200 kip. The ends are fixed.

13–85. Determine the largest length of a W8 * 31 structural A992 steel section if it is pin supported and is subjected to an axial load of 18 kip. Use the AISC equations.

13–86. Using the AISC equations, select from Appendix B the lightest-weight structural A992 steel column that is 12 ft long and supports an axial load of 40 kip. The ends are fixed.

PROBLEMS

13.6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING 705

13–95. The tube is 0.5 in. thick, is made of aluminum alloy 2014-T6, and is fixed connected at its ends. Determine the largest axial load that it can support.

*13–96. The tube is 0.5 in. thick, is made from aluminum alloy 2014-T6, and is fixed at its bottom and pinned at its top. Determine the largest axial load that it can support.

8 in.

y x 8 in.

12 ft

Probs. 13–95/96

13–97. The tube is 0.25 in. thick, is made of a 2014-T6 aluminum alloy, and is fixed at its bottom and pinned at its top. Determine the largest axial load that it can support.

13–98. The tube is 0.25 in. thick, is made of a 2014-T6 aluminum alloy, and is fixed connected at its ends. Determine the largest axial load that it can support.

13–99. The tube is 0.25 in. thick, is made of 2014-T6 aluminum alloy and is pin connected at its ends. Determine the largest axial load it can support.

6 in. yx

y x 6 in.

10 ft

Probs. 13–97/98/99

13–91. The bar is made of a 2014-T6 aluminum alloy. Determine its smallest thickness b if its width is 5b. Assume that it is pin connected at its ends.

*13–92. The bar is made of a 2014-T6 aluminum alloy. Determine its smallest thickness b if its width is 5b. Assume that it is fixed connected at its ends.

5b b

8 ft

600 lb

Probs. 13–91/92

13–93. The 2014-T6 aluminum column of 3-m length has the cross section shown. If the column is pinned at both ends and braced against the weak axis at its mid-height, determine the allowable axial force P that can be safely supported by the column.

13–94. The 2014-T6 aluminum column has the cross section shown. If the column is pinned at both ends and subjected to an axial force P = 100 kN, determine the maximum length the column can have to safely support the loading.

100 mm

15 mm170 mm

15 mm

Probs. 13–93/94

706 CHAPTER 13 BUCKL ING OF COLUMNS

*13–104. The bar is made of aluminum alloy 2014-T6. Determine its thickness b if its width is 1.5 b . Assume that it is fixed connected at its ends.

1.5b b

5 ft

800 lb

Prob. 13–104

13–105. The column is made of wood. It is fixed at its bottom and free at its top. Use the NFPA formulas to determine its greatest allowable length if it supports an axial load of P = 6 kip.

13–106. The column is made of wood. It is fixed at its bottom and free at its top. Use the NFPA formulas to determine the largest allowable axial load P that it can support if it has a length L = 6 ft.

6 in. 3 in.

x x

Probs. 13–105/106

*13–100. A rectangular wooden column has the cross section shown. If the column is 6 ft long and subjected to an axial force of P = 15 kip, determine the required minimum dimension a of its cross-sectional area to the nearest 1 16 in. so that the column can safely support the loading. The column is pinned at both ends.

13–101. A rectangular wooden column has the cross section shown. If a = 3 in. and the column is 12 ft long, determine the allowable axial force P that can be safely supported by the column if it is pinned at its top and fixed at its base.

13–102. A rectangular wooden column has the cross section shown. If a = 3 in. and the column is subjected to an axial force of P = 15 kip, determine the maximum length the column can have to safely support the load. The column is pinned at its top and fixed at its base.

Probs. 13–100/101/102

13–103. The timber column has a square cross section and is assumed to be pin connected at its top and bottom. If it supports an axial load of 50 kip, determine its smallest side dimension a to the nearest 1 2 in. Use the NFPA formulas.

14 ft

Prob. 13–103

13.7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING 707

*13.7 Design of Columns for Eccentric Loading

Occasionally a column may be required to support a load acting either at its edge or on an angle bracket attached to its side, such as shown in Fig. 13–28 a . The bending moment M = Pe, which is caused by the eccentric loading, must be accounted for when the column is designed. There are several acceptable ways in which this is done in engineering practice. We will discuss two of the most common methods.

Use of Available Column Formulas. The stress distribution acting over the cross-sectional area of the column shown in Fig. 13–28 a is determined from a superposition of both the axial force P and the bending moment M = Pe. In particular, the maximum compressive stress is

smax = P A

+ Mc I

(13–30)

A typical stress profile is shown in Fig. 13–28 b . If we conservatively assume that the entire cross section is subjected to the uniform stress smax as determined from Eq. 13–30 , then we can compare smax with sallow, which is determined using the formulas given in Sec. 13.6 . Calculation of sallow is usually done using the largest slenderness ratio for the column, regardless of the axis about which the column experiences bending. This requirement is normally specified in design codes and will in most cases lead to a conservative design. If

smax … sallow

then the column can carry the specified loading. If this inequality does not hold, then the column’s area A must be increased, and a new smax and sallow must be calculated. This method of design is rather simple to apply and works well for columns that are short or of intermediate length.

Interaction Formula. When designing an eccentrically loaded column it is desirable to see how the bending and axial loads interact , so that a balance between these two effects can be achieved. To do this, we will consider the separate contributions made to the total column area by the axial force and moment. If the allowable stress for the axial load is (sa )allow, then the required area for the column needed to support the load P is

A a = P

(sa )allow

Similarly, if the allowable bending stress is (sb)allow, then since I = Ar2, the required area of the column needed to support the eccentric moment is determined from the flexure formula, that is,

(a)

M ! Pe

(b)

smax

Fig. 13–28

708 CHAPTER 13 BUCKL ING OF COLUMNS

A b = Mc

(sb )allow r 2

The total area A for the column needed to resist both the axial load and moment requires that

A a + A b = P

(sa )allow + Mc

(sb)allow r 2 … A

P>A

(sa )allow +

Mc>Ar2 (sb)allow

… 1

(sa )allow +

(sb)allow … 1 (13–31)

Here

sa = axial stress caused by the force P and determined from sa = P>A , where A is the cross-sectional area of the column

sb = bending stress caused by an eccentric load or applied moment M ; sb is found from sb = Mc>I, where I is the moment of inertia of the cross-sectional area calculated about the bending or centroidal axis

(sa )allow = allowable axial stress as defined by formulas given in Sec. 13.6 or by other design code specifications. For this purpose, always use the largest slenderness ratio for the column, regardless of the axis about which the column experiences bending

(sb)allow = allowable bending stress as defined by code specifications

Notice that, if the column is subjected only to an axial load, then the bending-stress ratio in Eq. 13–31 would be equal to zero and the design will be based only on the allowable axial stress. Likewise, when no axial load is present, the axial-stress ratio is zero and the stress requirement will be based on the allowable bending stress. Hence, each stress ratio indicates the contribution of axial load or bending moment. Since Eq. 13–31 shows how these loadings interact, this equation is sometimes referred to as the interaction formula . This design approach requires a trial-and-check procedure, where it is required that the designer pick an available column and then check to see if the inequality is satisfied. If it is not, a larger section is then picked and the process repeated. An economical choice is made when the left side is close to but less than 1.

The interaction method is often specified in codes for the design of columns made of steel, aluminum, or timber. In particular, for allowable stress design, the American Institute of Steel Construction specifies the use of this equation only when the axial-stress ratio sa >(sa )allow … 0.15. For other values of this ratio, a modified form of Eq. 13–31 is used.

Typical example of a column used to support an eccentric roof loading.

13.7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING 709

The column in Fig. 13–29 is made of aluminum alloy 2014-T6 and is used to support an eccentric load P . Determine the maximum magnitude of P that can be supported if the column is fixed at its base and free at its top. Use Eq. 13–30 .

EXAMPLE 13.10

80 in.

2 in. 1 in.

2 in. 2 in.

Fig. 13–29

SOLUTION From Fig. 13–10 b , K = 2. The largest slenderness ratio for the column is therefore

KL r

= 2(80 in.)

2[(1>12)(4 in.)(2 in.)3][(2 in.) 4 in.] = 277.1 By inspection, Eq. 13–26 must be used (277.1 7 55). Thus,

sallow = 54 000 ksi (KL >r)2 = 54 000 ksi (277.1)2 = 0.7031 ksi

The maximum compressive stress in the column is determined from the combination of axial load and bending. We have

smax = P A

+ (Pe)c

= P

2 in.(4 in.) +

P(1 in.)(2 in.)

(1>12)(2 in.)(4 in.)3 = 0.3125P

Assuming that this stress is uniform over the cross section, we require

sallow = smax; 0.7031 = 0.3125P P = 2.25 kip Ans.

710 CHAPTER 13 BUCKL ING OF COLUMNS

The A-36 steel W6 * 20 column in Fig. 13–30 is pin connected at its ends and is subjected to the eccentric load P . Determine the maximum allowable value of P using the interaction method if the allowable bending stress is (sb)allow = 22 ksi.

SOLUTION Here K = 1. The necessary geometric properties for the W6 * 20 are taken from the table in Appendix B .

A = 5.87 in2 Ix = 41.4 in4 ry = 1.50 in. d = 6.20 in.

We will consider ry because this will lead to the largest value of the slenderness ratio. Also, Ix is needed since bending occurs about the x axis (c = 6.20 in.>2 = 3.10 in.). To determine the allowable compressive stress, we have

KL r

= 1[15 ft(12 in.>ft)]

1.50 in. = 120

Since

aKL r

b c = B2p2E sY = B2p2 [29(103) ksi] 36 ksi = 126.1

then KL >r 6 (KL >r)c and so Eq. 13–23 must be used. sallow =

[1 - (KL >r)2>2(KL >r)c ]2sY (5>3) + [(3>8)(KL >r)>(KL >r)c] - 3(KL >r)3>8(KL >r)c 34

= [1 - (120)2>2(126.1)2] 36 ksi

(5>3) + [(3>8)(120)>(126.1)] - [(120)3>8(126.1)3] = 10.28 ksi

Applying the interaction Eq. 13–31 yields

(sa)allow +

(sb)allow … 1

P>5.87 in2 10.28 ksi

+ P(30 in.)(3.10 in.)>(41.4 in4)

22 ksi = 1

P = 8.43 kip Ans.

Checking the application of the interaction method for the steel section, we require

(sa)allow =

8.43 kip>(5.87 in.) 10.28 kip>in2 = 0.140 6 0.15 OK

EXAMPLE 13.11

M ! P(30 in.)

15 ft

30 in.

Fig. 13–30

13.7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING 711

The timber column in Fig. 13–31 is made from two boards nailed together so that the cross section has the dimensions shown. If the column is fixed at its base and free at its top, use Eq. 13–30 to determine the eccentric load P that can be supported.

EXAMPLE 13.12

60 in.

x y

3 in.

3 in. 1 in.

3 in.

Fig. 13–31

SOLUTION From Fig. 13–10 b , K = 2. Here we must calculate KL >d to determine which equation from Eqs. 13–27 through 13–29 should be used. Since sallow is determined using the largest slenderness ratio, we choose d = 3 in. This is done to make this ratio as large as possible, and thereby yields the lowest possible allowable axial stress. We have

KL d

= 2(60 in.)

3 in. = 40

Since 26 6 KL >d 6 50 the allowable axial stress is determined using Eq. 13–29 . Thus,

sallow = 540 ksi

(KL >d)2 = 540 ksi (40)2 = 0.3375 ksi Applying Eq. 13–30 with sallow = smax, we have

sallow = P A

+ Mc I

0.3375 ksi = P

(3 in.)(6 in.) +

P(4 in.)(3 in.)

(1>12)(3 in.)(6 in.)3 P = 1.22 kip Ans.

712 CHAPTER 13 BUCKL ING OF COLUMNS

13–107. The W8 * 15 structural A992 steel column is assumed to be pinned at its top and bottom. Determine the largest eccentric load P that can be applied using Eq. 13–30 and the AISC equations of Sec. 13.6 . The load at the top consists of a force P and a moment M = P(8 in.).

*13–108. Solve Prob. 13–107 if the column is fixed at its top and bottom.

13–109. Solve Prob. 13–107 if the column is fixed at its bottom and pinned at its top.

10 ft

x M

x y

Probs. 13–107/108/109

13–110. The W10 * 19 structural A992 steel column is assumed to be pinned at its top and bottom. Determine the largest eccentric load P that can be applied using Eq. 13–30 and the AISC equations of Sec. 13.6 .

12 ft

P 20 kip

6 in. x

x y

Prob. 13–110

13–111. The W8 * 15 structural A992 steel column is fixed at its top and bottom. If it supports end moments of M = 5 kip·ft, determine the axial force P that can be applied. Bending is about the x – x axis. Use the AISC equations of Sec. 13.6 and Eq. 13–30 .

16 ft

y x

P M

yyx M

Prob. 13–111

*13–112. The W8 * 15 structural A992 steel column is fixed at its top and bottom. If it supports end moments of M = 23 kip·ft, determine the axial force P that can be applied. Bending is about the x – x axis. Use the interaction formula with (s b ) = 24 ksi.

16 ft

y x

P M

yyx M

Prob. 13–112

PROBLEMS

13.7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING 713

13–113. The A992-steel W10 * 45 column is fixed at its base. Its top is constrained so that it cannot move along the x–x axis but it is free to rotate about and move along the y–y axis. Determine the maximum eccentric force P that can be safely supported by the column using the allowable stress method.

13–114. The A992-steel W10 * 45 column is fixed at its base. Its top is constrained so that it cannot move along the x–x axis but it is free to rotate about and move along the y–y axis. Determine the maximum eccentric force P that can be safely supported by the column using an interaction formula. The allowable bending stress is (sb)allow = 15 ksi.

13–115. The A-36-steel W12 * 50 column is fixed at its base. Its top is constrained so that it cannot move along the x–x axis but it is free to rotate about and move along the y–y axis. If the eccentric force P = 15 kip is applied to the column, investigate if the column is adequate to support the loading. Use the allowable stress method.

*13–116. The A-36-steel W12 * 50 column is fixed at its base. Its top is constrained so that it cannot move along the x–x axis but it is free to rotate about and move along the y–y axis. If the eccentric force P = 15 kip is applied to the column, investigate if the column is adequate to support the loading. Use the interaction formula. The allowable bending stress is (sb)allow = 15 ksi.

x y

24 ft

12 in.

Probs. 13–113/114/115/116

13–117. A 16-ft-long column is made of aluminum alloy 2014-T6. If it is fixed at its top and bottom, and a concentric load P and a moment M = P(4.5 in.) are applied at point A , determine the maximum allowable magnitude of P using the equations of Sec. 13.6 and Eq. 13–30 .

13–118. A 16-ft-long column is made of aluminum alloy 2014-T6. If it is fixed at its top and bottom, and a concentric load P and a moment M = P(4.5 in.) are applied at point A , determine the maximum allowable magnitude of P using the equations of Sec. 13.6 and the interaction formula with (sb)allow = 20 ksi.

8 in.

0.5 in. 0.5 in.

y M

x x

8 in. y

A 0.5 in.

Probs. 13–117/118 13–119. The 2014-T6 aluminum hollow column is fixed at its base and free at its top. Determine the maximum eccentric force P that can be safely supported by the column. Use the allowable stress method. The thickness of the wall for the section is t = 0.5 in.

*13–120. The 2014-T6 aluminum hollow column is fixed at its base and free at its top. Determine the maximum eccentric force P that can be safely supported by the column. Use the interaction formula. The allowable bending stress is (sb)allow = 30 ksi. The thickness of the wall for the section is t = 0.5 in.

P6 in.

6 in.

3 in.

8 ft

Probs. 13–119/120

714 CHAPTER 13 BUCKL ING OF COLUMNS

13–121. The 10-ft-long bar is made of aluminum alloy 2014-T6. If it is fixed at its bottom and pinned at the top, determine the maximum allowable eccentric load P that can be applied using the formulas in Sec. 13.6 and Eq. 13–30 .

13–122. The 10-ft-long bar is made of aluminum alloy 2014-T6. If it is fixed at its bottom and pinned at the top, determine the maximum allowable eccentric load P that can be applied using the equations of Sec. 13.6 and the interaction formula with (sb)allow = 18 ksi.

3 in.1.5 in.

1.5 in.

x y y

2 in.

Probs. 13–121/122

13–123. Determine if the column can support the eccentric compressive load of 1.5 kip. Assume that the ends are pin connected. Use the NFPA equations in Sec. 13.6 and Eq. 13–30 .

*13–124. Determine if the column can support the eccentric compressive load of 1.5 kip. Assume that the bottom is fixed and the top is pinned. Use the NFPA equations in Sec. 13.6 and Eq. 13–30 .

1.5 kip

6 ft

3 in. 12 in.

1.5 in.

1.5 kip

Probs. 13–123/124

13–125. The 10-in.-diameter utility pole supports the transformer that has a weight of 600 lb and center of gravity at G . If the pole is fixed to the ground and free at its top, determine if it is adequate according to the NFPA equations of Sec. 13.6 and Eq. 13–30 .

18 ft

15 in.

Prob. 13–125

13–126. Using the NFPA equations of Sec. 13–6 and Eq. 13–30 , determine the maximum allowable eccentric load P that can be applied to the wood column. Assume that the column is pinned at both its top and bottom.

13–127. Using the NFPA equations of Sec. 13.6 and Eq. 13–30 , determine the maximum allowable eccentric load P that can be applied to the wood column. Assume that the column is pinned at the top and fixed at the bottom.

6 in.

12 ft

P 0.75 in.

3 in.

Probs. 13–126/127

CHAPTER REVIEW 715

CHAPTER REVIEW

Buckling is the sudden instability that occurs in columns or members that support an axial compressive load. The maximum axial load that a member can support just before buckling is called the critical load Pcr.

The critical load for an ideal column is determined from Euler’s formula, where K = 1 for pin supports, K = 0.5 for fixed supports, K = 0.7 for a pin and a fixed support, and K = 2 for a fixed support and a free end.

If the axial loading is applied eccentrically to the column, then the secant formula can be used to determine the maximum stress in the column.

When the axial load causes yielding of the material, then the tangent modulus should be used with Euler’s formula to determine the critical load for the column. This is referred to as Engesser’s equation.

smax = P A

c 1 + ec r2

seca L 2rA PEA b d

scr = p2 Et

(KL >r)2

Pcr = p2 EI

(KL)2

Pcr

Empirical formulas based on experimental data have been developed for use in the design of steel, aluminum, and timber columns.

716 CHAPTER 13 BUCKL ING OF COLUMNS

*13–128. The wood column has a thickness of 4 in. and a width of 6 in. Using the NFPA equations of Sec. 13.6 and Eq. 13–30 , determine the maximum allowable eccentric load P that can be applied. Assume that the column is pinned at both its top and bottom.

6 in.

10 ft

P 1 in.

4 in.

Prob. 13–128

13–129. The wood column has a thickness of 4 in. and a width of 6 in. Using the NFPA equations of Sec. 13.6 and Eq. 13–30 , determine the maximum allowable eccentric load P that can be applied. Assume that the column is pinned at the top and fixed at the bottom.

6 in.

10 ft

P 1 in.

4 in.

Prob. 13–129

13–130. A steel column has a length of 5 m and is free at one end and fixed at the other end. If the cross-sectional area has the dimensions shown, determine the critical load. E st = 200 GPa, s Y = 360 MPa.

10 mm60 mm

10 mm

80 mm

Prob. 13–130

13–131. The square structural A992 steel tubing has outer dimensions of 8 in. by 8 in. Its cross-sectional area is 14.40 in 2 and its moments of inertia are I x = I y = 131 in 4 . Determine the maximum load P it can support. The column can be assumed fixed at its base and free at its top.

x y

12 ft

P3 in.

Prob. 13–131

REVIEW PROBLEMS

REVIEW PROBLEMS 717

*13–132. If the A-36 steel solid circular rod BD has a diameter of 2 in., determine the allowable maximum force P that can be supported by the frame without causing the rod to buckle. Use F.S. = 2 against buckling.

13–133. If P = 15 kip, determine the required minimum diameter of the A992 steel solid circular rod BD to the nearest 116 in. Use F.S. = 2 against buckling.

3 ft 3 ft

4.5 in.

4 ft

A B C

Probs. 13–132/133

13–134. The steel pipe is fixed supported at its ends. If it is 4 m long and has an outer diameter of 50 mm, determine its required thickness so that it can support an axial load of P = 100 kN without buckling. E st = 200 GPa, s Y = 250 MPa.

4 m

Prob. 13–134

13–135. The W200 * 46 A992-steel column can be considered pinned at its top and fixed at its base. Also, the column is braced at its mid-height against the weak axis. Determine the maximum axial load the column can support without causing it to buckle.

6 m

Prob. 13–135

*13–136. The structural A992 steel column has the cross section shown. If it is fixed at the bottom and free at the top, determine the maximum force P that can be applied at A without causing it to buckle or yield. Use a factor of safety of 3 with respect to buckling and yielding.

13–137. The structural A992 steel column has the cross section shown. If it is fixed at the bottom and free at the top, determine if the column will buckle or yield when the load P = 10 kN is applied. Use a factor of safety of 3 with respect to buckling and yielding.

20 mm

4 m

10 mm

100 mm

10 mm 150 mm

A 100 mm

10 mm

Probs. 13–136/137

These diving boards must be made from a material that can store a high value of elastic strain energy due to bending. This allows it to have a large

flexure, and thereby transfer this energy to the diver, as the board begins to straighten out.

Chapter 14

719

14.1 External Work and Strain Energy The deflection of joints on a truss or points on a beam or shaft can be determined using energy methods. Before developing any of these methods, however, we will first define the work caused by an external force and couple moment and show how to express this work in terms of a body’s strain energy. The formulations to be presented here and in the next section will provide the basis for applying the work and energy methods that follow throughout the chapter.

CHAPTER OBJECTIVES

■ In this chapter, we will show how to apply energy methods to solve problems involving deflection. The chapter begins with a discussion of work and strain energy, followed by a development of the principle of conservation of energy. Using this principle, the stress and deflection of a member are determined when the member is subjected to impact. The method of virtual work and Castigliano’s theorem are then developed, and these methods are used to determine the displacement and slope at points on structural members and mechanical elements.

Energy Methods

720 CHAPTER 14 ENERGY METHODS

Work of a Force. In mechanics, a force does work when it undergoes a displacement dx that is in the same direction as the force. The work done is a scalar, defined as dUe = F dx. If the total displacement is ! , the work becomes

Ue = L !

0 F dx (14–1)

To show how to apply this equation, we will calculate the work done by an axial force applied to the end of the bar shown in Fig. 14–1 a . As the magnitude of the force is gradually increased from zero to some limiting value F = P, the final displacement of the end of the bar becomes !. If the material behaves in a linear-elastic manner, then the force will be directly proportional to the displacement; that is, F = (P>!)x. Substituting into Eq. 14–1 and integrating from 0 to !, we get

Ue = 1 2

P! (14–2)

Therefore, as the force is gradually applied to the bar, its magnitude builds from zero to some value P , and consequently, the work done is equal to the average force magnitude , P>2 , times the total displacement !. We can represent this graphically as the light-blue shaded area of the triangle in Fig. 14–1 c .

Suppose, however, that P is already applied to the bar and that another force P" is now applied, so that the end of the bar is displaced further by an amount !", Fig. 14–1 b . The work done by P" is equal to the gray shaded triangular area, but now the work done by P when the bar undergoes this further displacement is

Ue= = P!" (14–3)

Here the work represents the dark-blue shaded rectangular area in Fig. 14–1 c . In this case P does not change its magnitude, since the bar’s displacement !" is caused only by P". Therefore, work here is simply the force magnitude P times the displacement !".

(a)

P¿

(b)

!¿ !

P¿

(c)

x ! !¿

Fig. 14–1

14.1 EXTERNAL WORK AND STRAIN ENERGY 721

Work of a Couple Moment. A couple moment M does work when it undergoes an angular displacement du along its line of action. The work is defined as dUe = M du, Fig. 14–2 . If the total angular displacement is u rad, the work becomes

Ue = L u

0 M du (14–4)

As in the case of force, if the couple moment is applied to a body having linear elastic material behavior, such that its magnitude is increased gradually from zero at u = 0 to M at u, then the work is

Ue = 1 2

Mu (14–5)

However, if the couple moment is already applied to the body and other loadings further rotate the body by an amount u!, then the work is

U=e = Mu!

Strain Energy. When loads are applied to a body, they will deform the material. Provided no energy is lost in the form of heat, the external work done by the loads will be converted into internal work called strain energy . This energy, which is always positive , is stored in the body and is caused by the action of either normal or shear stress.

Normal Stress. If the volume element shown in Fig. 14–3 is subjected to the normal stress sz, then the force created on the element’s top and bottom faces is dFz = sz dA = sz dx dy. If this force is applied gradually to the element, like the force P discussed previously, its magnitude is increased from zero to dFz, while the element undergoes an elongation d"z = Pz dz. The work done by dFz is therefore dUi = 12 dFz d"z =

1 2 [sz dx dy]Pz dz. Since the volume of the

element is dV = dx dy dz, we have

dUi = 1 2

szPz dV (14–6)

Notice that dUi is always positive , even if sz is compressive, since sz and Pz will always be in the same direction.

In general then, if the body is subjected only to a uniaxial normal stress s, the strain energy in the body is then

Ui = LV sP 2

dV (14–7)

Fig. 14–2

Fig. 14–3

722 CHAPTER 14 ENERGY METHODS

Also, if the material behaves in a linear-elastic manner, then Hooke’s law applies, and we can express the strain energy in terms of the normal stress as

Ui = LV s2

2E dV (14–8)

Shear Stress. A strain-energy expression similar to that for normal stress can also be established for the material when it is subjected to shear stress. Consider the volume element shown in Fig. 14–4 . Here the shear stress causes the element to deform such that only the shear force dF = t(dx dy), acting on the top face of the element, is displaced g dz relative to the bottom face. The vertical faces only rotate, and therefore the shear forces on these faces do no work. Hence, the strain energy stored in the element is

dUi = 1 2

[t(dx dy)]g dz

or since dV = dx dy dz

dUi = 1 2

tg dV (14–9)

The strain energy stored in the body is therefore

Ui = LV tg

2 dV (14–10)

Like the case for normal strain energy, shear strain energy is always positive since t and g are always in the same direction. If the material is linear elastic, then, applying Hooke’s law, g = t>G, we can express the strain energy in terms of the shear stress as

Ui = LV t2

2G dV (14–11)

dz t

gdz

Fig. 14–4

14.1 EXTERNAL WORK AND STRAIN ENERGY 723

In the next section, we will use Eqs. 14–8 and 14–11 to obtain formal expressions for the strain energy stored in members subjected to several types of loads. Once this is done we will then be able to develop the energy methods necessary to determine the displacement and slope at points on a body.

Multiaxial Stress. The previous development may be expanded to determine the strain energy in a body when it is subjected to a general state of stress, Fig. 14–5 a . The strain energies associated with each of the normal and shear stress components can be obtained from Eqs. 14–6 and 14–9 . Since energy is a scalar, the total strain energy in the body is therefore

Ui = LV c 12 sxPx + 12 syPy + 12 szPz + 1

2 txygxy +

1 2

tyzgyz + 1 2

txzgxz d dV (14–12) The strains can be eliminated by using the generalized form of Hooke’s law given by Eqs. 10–18 and 10–19 . After substituting and combining terms, we have

Ui = LV c 12E 1sx2 + sy2 + sz22 - nE (sxsy + sysz + sxsz) + 1

2G 1txy2 + tyz2 + txz2 2 d dV (14–13)

If only the principal stresses s1, s2, s3 act on the element, Fig. 14–5 b , this equation reduces to a simpler form, namely,

Ui = LV c 12E 1s12 + s22 + s32 2 - nE (s1s2 + s2s3 + s1s3) d dV (14–14) This equation was used in Sec. 10.7 as a basis for developing the maximum-distortion-energy theory.

(a)

sy sx

txz tyz

txy

(b)

Fig. 14–5

724 CHAPTER 14 ENERGY METHODS

14.2 Elastic Strain Energy for Various Types of Loading

Using the equations for elastic strain energy developed in the previous section, we will now formulate the strain energy stored in a member when it is subjected to an axial load, bending moment, transverse shear, and torsional moment. Examples will be given to show how to calculate the strain energy in members subjected to each of these loadings.

Axial Load. Consider a bar of variable yet slightly tapered cross section, Fig. 14–6 . The internal axial force at a section located a distance x from one end is N . If the cross-sectional area at this section is A , then the normal stress on the section is s = N >A . Applying Eq. 14–8 , we have

Ui = LV sx

2E dV = LV

2EA 2 dV

If we choose an element or differential slice having a volume dV = A dx, the general formula for the strain energy in the bar is therefore

Ui = L L

2AE dx (14–15)

For the more common case of a prismatic bar of constant cross-sectional area A , length L , and constant axial load N , Fig. 14–7 , Eq. 14–15 , when integrated, gives

Ui = N2L 2AE

(14–16)

Notice that the bar’s elastic strain energy will increase if the length of the bar is increased, or if the modulus of elasticity or cross-sectional area is decreased. For example, an aluminum rod [Eal = 10(103) ksi] will store approximately three times as much energy as a steel rod [Est = 29(103) ksi] having the same size and subjected to the same load. However, doubling the cross-sectional area of a rod will decrease its ability to store energy by one-half. The following example illustrates this point numerically.

Fig. 14–6

Fig. 14–7

14.2 ELASTIC STRAIN ENERGY FOR VARIOUS TYPES OF LOADING 725

EXAMPLE 14.1

One of the two high-strength steel bolts A and B shown in Fig. 14–8 is to be chosen to support a sudden tensile loading. For the choice it is necessary to determine the greatest amount of elastic strain energy that each bolt can absorb. Bolt A has a diameter of 0.875 in. for 2 in. of its length and a root (or smallest) diameter of 0.731 in. within the 0.25-in. threaded region. Bolt B has “upset” threads, such that the diameter throughout its 2.25-in. length can be taken as 0.731 in. In both cases, neglect the extra material that makes up the threads. Take Est = 29(103) ksi, sY = 44 ksi.

2 in.

0.25 in.

0.875 in.

0.731 in.

2.25 in. 0.731 in.

Fig. 14–8

SOLUTION Bolt A. If the bolt is subjected to its maximum tension, the maximum stress of sY = 44 ksi will occur within the 0.25-in. region. This tension force is

Pmax = sY A = 44 ksi Jpa 0.731 in.2 b2R = 18.47 kip Applying Eq. 14–16 to each region of the bolt, we have

Ui = a N 2L

2AE

= (18.47 kip)2(2 in.)

2[p(0.875 in.>2)2][29(103) ksi] + (18.47 kip)2(0.25 in.)2[p(0.731 in.>2)2][29(103) ksi] = 0.0231 in. # kip Ans.

Bolt B. Here the bolt is assumed to have a uniform diameter of 0.731 in. throughout its 2.25-in. length. Also, from the calculation above, it can support a maximum tension force of Pmax = 18.47 kip. Thus,

Ui = N2L 2AE

= (18.47 kip)2(2.25 in.)

2[p(0.731 in.>2)2][29(103) ksi] = 0.0315 in. # kip Ans. NOTE: By comparison, bolt B can absorb 36% more elastic energy than bolt A , because it has a smaller cross section along its shank.

726 CHAPTER 14 ENERGY METHODS

Bending Moment. Since a bending moment applied to a straight prismatic member develops normal stress in the member, we can use Eq. 14–8 to determine the strain energy stored in the member due to bending. For example, consider the axisymmetric beam shown in Fig. 14–9 . Here the internal moment is M , and the normal stress acting on the arbitrary element a distance y from the neutral axis is s = My >I. If the volume of the element is dV = dA dx, where dA is the area of its exposed face and dx is its length, the elastic strain energy in the beam is

Ui = LV s2

2E dV = LV

1 2E

aMy I

b2 dA dx or

Ui = L L

0 M2

2EI2 ¢LAy2 dA ≤ dx

Realizing that the area integral represents the moment of inertia of the area about the neutral axis, the final result can be written as

Ui = L L

0 M2 dx

2EI (14–17)

To evaluate the strain energy, therefore, we must first express the internal moment as a function of its position x along the beam, and then perform the integration over the beam’s entire length. * The following examples illustrate this procedure.

x z

y dA

Fig. 14–9

* Recall that the flexure formula, as used here, can also be used with justifiable accuracy to determine the stress in slightly tapered beams. (See Sec. 6.4 .) So in the general sense, I in Eq. 14–17 may also have to be expressed as a function of x.

14.2 ELASTIC STRAIN ENERGY FOR VARIOUS TYPES OF LOADING 727

EXAMPLE 14.2

Determine the elastic strain energy due to bending of the cantilevered beam in Fig. 14–10 a . EI is constant.

(a)

SOLUTION The internal moment in the beam is determined by establishing the x coordinate with origin at the left side. The left segment of the beam is shown in Fig. 14–10 b . We have

a+ !MNA = 0; M + wx a x2 b = 0 M = -w ¢ x2

2 ≤

Applying Eq. 14–17 yields

Ui = L L

M2 dx 2EI

= L L

[-w(x2>2)]2 dx 2EI

= w2

8EI L L

0 x4 dx

Ui = w2L5

40EI Ans.

We can also obtain the strain energy using an x coordinate having its origin at the right side of the beam and extending positive to the left, Fig. 14–10 c . In this case,

a+ !MNA = 0; -M - wx a x2 b + wL(x) - wL22 = 0 M = - wL

2 + wLx - w ¢ x2

2 ≤

Applying Eq. 14–17 , we obtain the same result as before; however, more calculations are involved in this case.

V x

(b)

x 2

(c)

x 2

wL2

Fig. 14–10

728 CHAPTER 14 ENERGY METHODS

EXAMPLE 14.3

Determine the bending strain energy in region AB of the beam shown in Fig. 14–11 a . EI is constant.

SOLUTION A free-body diagram of the beam is shown in Fig. 14–11 b . To obtain the answer we can express the internal moment in terms of any one of the indicated three “ x ” coordinates and then apply Eq. 14–17 . Each of these solutions will now be considered.

0 … x1 … L. From the free-body diagram of the section in Fig. 14–11 c , we have

a+ !MNA = 0; M1 + Px1 = 0

M1 = -Px1

Ui = L M2 dx

2EI = L

0 (-Px1)2 dx1

2EI =

P 2L3

6EI Ans.

0 … x2 … L . Using the free-body diagram of the section in Fig. 14–11 d gives

a+ !MNA = 0; -M2 + 2P(x2) - P(x2 + L) = 0

M2 = P(x2 - L)

Ui = L M2 dx

2EI = L

0 [P(x2 - L)]2 dx2

2EI =

P 2L3

6EI Ans.

L … x3 … 2L. From the free-body diagram in Fig. 14–11 e , we have

a+ !MNA = 0; -M3 + 2P(x3 - L) - P(x3) = 0

M3 = P(x3 - 2L)

Ui = L M2 dx

2EI = L

L [P(x3 - 2L)]2 dx3

2EI =

P 2L3

6EI Ans.

NOTE: This and the previous example indicate that the strain energy for the beam can be found using any suitable x coordinate. It is only necessary to integrate over the range of the coordinate where the internal energy is to be determined. Here the choice of x1 provides the simplest solution.

A B

(a)

A B

(b)

x1 x2

2 P

(c)

(d)

M2 x2 L

2 P

(e)

(x3 ! L)

2 P

Fig. 14–11

14.2 ELASTIC STRAIN ENERGY FOR VARIOUS TYPES OF LOADING 729

Transverse Shear. The strain energy due to shear stress in a beam element can be determined by applying Eq. 14–11 . Here we will consider the beam to be prismatic and to have an axis of symmetry about the y axis as shown in Fig. 14–12 . If the internal shear at the section x is V , then the shear stress acting on the volume element of material, having an area dA and length dx , is t = VQ>It. Substituting into Eq. 14–11 , the strain energy for shear becomes

Ui = LV t 2

2G dV = LV

1 2G

aVQ It

b2 dA dx Ui = L

V 2

2GI 2 ¢LA Q2t 2 dA ≤ dx

The integral in parentheses can be simplified if we define the form factor for shear as

fs = A I 2

LA Q2

t 2 dA (14–18)

Substituting into the above equation, we get

Ui = L L

0 fs V

2 dx 2GA

(14–19)

The form factor defined by Eq. 14–18 is a dimensionless number that is unique for each specific cross-sectional area. For example, if the beam has a rectangular cross section of width b and height h , Fig. 14–13 , then

t = b

dA = b dy

I = 1

12 bh3

Q = y !A ! = ay + (h>2) - y 2

b ba h 2

- y b = b 2

¢ h2 4

- y2≤ Substituting these terms into Eq. 14–18 , we get

fs = bh1 112 bh322 Lh>2-h>2 b24b2 ¢ h24 - y2≤2 b dy = 65 (14–20)

The form factor for other sections can be determined in a similar manner. Once obtained, this factor is substituted into Eq. 14–19 and the strain energy for transverse shear can then be evaluated.

x z

y dA

Fig. 14–12

A¿

N A

h 2

! y)(

Fig. 14–13

730 CHAPTER 14 ENERGY METHODS

EXAMPLE 14.4

Determine the strain energy in the cantilevered beam due to shear if the beam has a square cross section and is subjected to a uniform distributed load w, Fig. 14–14 a . EI and G are constant.

(a)

a a

(b)

x 2

Fig. 14–14

SOLUTION From the free-body diagram of an arbitrary section, Fig. 14–14 b , we have

+ c !Fy = 0; -V - wx = 0 V = -wx

Since the cross section is square, the form factor fs = 65 ( Eq. 14–20 ) and therefore Eq. 14–19 becomes

(Ui)s = L L

0 6 5 (-wx)2 dx

2GA =

3w2

5GA L

0 x2 dx

(Ui)s = w2L3

5GA Ans.

NOTE: Using the results of Example 14.2 , with A = a2, I = 112 a4, the ratio of shear to bending strain energy is

(Ui)s (Ui)b

= w2L3>5Ga2

w2L5>40E1 112 a42 = 23 a aL b2 EG Since G = E>2(1 + v) and v … 12 ( Sec. 10.6 ) , then as an upper

bound, E = 3G, so that

(Ui)s (Ui)b

= 2a a L b2

It can be seen that this ratio will increase as L decreases. However, even for very short beams, where, say, L = 5a, the contribution due to shear strain energy is only 8% of the bending strain energy. For this reason, the shear strain energy stored in beams is usually neglected in engineering analysis.

14.2 ELASTIC STRAIN ENERGY FOR VARIOUS TYPES OF LOADING 731

Torsional Moment. To determine the internal strain energy in a circular shaft or tube due to an applied torsional moment, we must apply Eq. 14–11 . Consider the slightly tapered shaft in Fig. 14–15 . A section of the shaft taken a distance x from one end is subjected to an internal torque T . The shear stress distribution that causes this torque varies linearly from the center of the shaft. On the arbitrary element of area dA and length dx , the stress is t = Tr>J. The strain energy stored in the shaft is thus

Ui = LV t2

2G dV = LV

1 2G

aTr J b2 dA dx = LL0 T22GJ 2 ¢LA r2 dA ≤ dx

Since the area integral represents the polar moment of inertia J for the shaft at the section, the final result can be written as

Ui = L L

0 T 2

2GJ dx (14–21)

The most common case occurs when the shaft (or tube) has a constant cross-sectional area and the applied torque is constant, Fig. 14–16 . Integration of Eq. 14–21 then gives

Ui = T 2L 2GJ

(14–22)

From this equation we may conclude that, like an axially loaded member, the energy-absorbing capacity of a torsionally loaded shaft is decreased by increasing the diameter of the shaft, since this increases J .

Fig. 14–15

Fig. 14–16

Important Points

• A force does work when it moves through a displacement . When a force is applied to a body and its magnitude is increased gradually from zero to F , the work is U = (F>2)!, whereas if the force is constant when the displacement occurs then U = F!.

• A couple moment does work when it displaces through a rotation . • Strain energy is caused by the internal work of the normal and

shear stresses. It is always a positive quantity.

• The strain energy can be related to the resultant internal loadings N, V, M , and T .

• As the beam becomes longer, the strain energy due to bending becomes much larger than the strain energy due to shear. For this reason, the shear strain energy in beams can generally be neglected .

The following example illustrates how to determine the strain energy in a circular shaft due to a torsional loading.

732 CHAPTER 14 ENERGY METHODS

EXAMPLE 14.5

The tubular shaft in Fig. 14–17 a is fixed at the wall and subjected to two torques as shown. Determine the strain energy stored in the shaft due to this loading. G = 75 GPa.

SOLUTION Using the method of sections, the internal torque is first determined within the two regions of the shaft where it is constant, Fig. 14–17 b . Although these torques ( 40 N # m and 15 N # m ) are in opposite directions, this will be of no consequence in determining the strain energy, since the torque is squared in Eq. 14–22 . In other words, the strain energy is always positive. The polar moment of inertia for the shaft is

J = p

2 [(0.08 m)4 - (0.065 m)4] = 36.30(10-6) m4

Applying Eq. 14–22 , we have

Ui = a T 2L

2GJ

= (40 N # m)2(0.750 m)

2[75(109) N>m2]36.30(10-6) m4 + (15 N # m)2(0.300 m)2[75(109) N>m2]36.30(10-6) m4 = 233 mJ Ans.

40 N!m

55 N!m

750 mm

300 mm

15 mm

80 mm

(a)

40 N!m 40 N!m 55 N!m

(b)

T " 40 N!m T " 15 N!m

Fig. 14–17

14.2 ELASTIC STRAIN ENERGY FOR VARIOUS TYPES OF LOADING 733

14–5. Using bolts of the same material and cross-sectional area, two possible attachments for a cylinder head are shown. Compare the strain energy developed in each case, and then explain which design is better for resisting an axial shock or impact load.

(a)

(b)

Prob. 14–5

14–6. If P = 60 kN, determine the total strain energy stored in the truss. Each member has a cross-sectional area of 2.5(103) mm2 and is made of A-36 steel.

14–7. Determine the maximum force P and the corresponding maximum total strain energy stored in the truss without causing any of the members to have permanent deformation. Each member has the cross-sectional area of 2.5(103) mm2 and is made of A-36 steel.

1.5 m

2 m

Probs. 14–6/7

PROBLEMS

14–1. A material is subjected to a general state of plane stress. Express the strain energy density in terms of the elastic constants E, G , and n and the stress components sx, sy, and txy.

txy

Prob. 14–1

14–2. The strain-energy density must be the same whether the state of stress is represented by sx, sy, and txy, or by the principal stresses s1 and s2. This being the case, equate the strain–energy expressions for each of these two cases and show that G = E>[2(1 + n)]. 14–3. Determine the strain energy in the stepped rod assembly. Portion AB is steel and BC is brass. Ebr = 101 GPa, Est = 200 GPa, (sY )br = 410 MPa, (sY )st = 250 MPa.

A B C

75 mm 20 kN

30 kN

100 mm

0.5 m1.5 m

Prob. 14–3

*14–4. Determine the torsional strain energy in the A-36 steel shaft. The shaft has a radius of 30 mm.

0.5 m

0.5 m 3 kN!m

4 kN!m

Prob. 14–4

734 CHAPTER 14 ENERGY METHODS

14–11. Determine the bending strain energy in the A-36 structural steel W10 * 12 beam. Obtain the answer using the coordinates (a) x1 and x4 , and (b) x2 and x3 .

12 ft 6 ft x2 x3

x4x1

6 kip

Prob. 14–11

*14–12. If P = 10 kip, determine the total strain energy stored in the truss. Each member has a diameter of 2 in. and is made of A992 steel.

14–13. Determine the maximum force P and the corresponding maximum total strain energy that can be stored in the truss without causing any of the members to have permanent deformation. Each member of the truss has a diameter of 2 in. and is made of A-36 steel.

4 ft

3 ft3 ft P

Probs. 14–12/13

14–14. Consider the thin-walled tube of Fig. 5–28. Use the formula for shear stress, tavg = T >2t A m, Eq. 5–18, and the general equation of shear strain energy, Eq. 14–11, to show that the twist of the tube is given by Eq. 5–20. Hint: Equate the work done by the torque T to the strain energy in the tube, determined from integrating the strain energy for a differential element, Fig. 14–4, over the volume of material.

*14–8. Determine the torsional strain energy in the A-36 steel shaft. The shaft has a diameter of 40 mm.

0.5 m

900 N!m

200 N!m

300 N!m

Prob. 14–8

14–9. Determine the torsional strain energy in the A-36 steel shaft. The shaft has a radius of 40 mm.

0.6 m

0.4 m

0.5 m6 kN!m

12 kN!m

8 kN!m

Prob. 14–9

14–10. The shaft assembly is fixed at C . The hollow segment BC has an inner radius of 20 mm and outer radius of 40 mm, while the solid segment AB has a radius of 20 mm. Determine the torsional strain energy stored in the shaft. The shaft is made of 2014-T6 aluminum alloy. The coupling at B is rigid.

600 mm

60 N!m

30 N!m 20 mm

20 mm

40 mm

Prob. 14–10

14.2 ELASTIC STRAIN ENERGY FOR VARIOUS TYPES OF LOADING 735

14–18. Determine the bending strain energy stored in the simply supported beam subjected to the triangular distributed load. EI is constant.

Prob. 14–18

14–19. Determine the strain energy in the horizontal curved bar due to torsion. There is a vertical force P acting at its end. JG is constant.

90!

Prob. 14–19

*14–20. The concrete column contains six 1-in.-diameter steel reinforcing rods. If the column supports a load of 300 kip, determine the strain energy in the column. E st = 29(10 3 ) ksi, E c = 3.6(10 3 ) ksi.

5 ft

300 kip

12 in.

Prob. 14–20

14–15. Determine the bending strain energy stored in the simply supported beam subjected to the uniform distributed load. EI is constant.

L 2

Prob. 14–15

*14–16. Determine the bending strain energy in the A992 steel beam due to the loading shown. Obtain the answer using the coordinates (a) x1 and x4 , and (b) x2 and x3 . I = 53.4 in 4 .

8 kip/ft

10 ft 5 ft

x2 x3

x4x1

Prob. 14–16

14–17. Determine the bending strain energy stored in the simply supported beam subjected to the triangular distributed load. EI is constant.

L 2

Prob. 14–17

736 CHAPTER 14 ENERGY METHODS

14–23. Determine the bending strain energy in the cantilevered beam due to a uniform load w . Solve the problem two ways. ( a ) Apply Eq. 14–17 . ( b ) The load w dx acting on a segment dx of the beam is displaced a distance y , where y = w(-x4 + 4L3 x - 3L4)>(24EI) , the equation of the elastic curve. Hence the internal strain energy in the differential segment dx of the beam is equal to the external work, i.e., dUi = 12 (w dx)(-y) . Integrate this equation to obtain the total strain energy in the beam. EI is constant.

L dx x

w dx

Prob. 14–23

*14–24. Determine the bending strain energy in the simply supported beam due to a uniform load w . Solve the problem two ways. ( a ) Apply Eq. 14–17 . ( b ) The load w dx acting on the segment dx of the beam is displaced a distance y , where y = w(-x4 + 2Lx3 - L3x)>(24EI) , the equation of the elastic curve. Hence the internal strain energy in the differential segment dx of the beam is equal to the external work, i.e., dUi = 12 (w dx)(-y) . Integrate this equation to obtain the total strain energy in the beam. EI is constant.

L dxx

w dx

Prob. 14–24

14–21. A load of 5 kN is applied to the center of the A992 steel beam, for which I = 4.5(10 6 ) mm 4 . If the beam is supported on two springs, each having a stiffness of k = 8 MN>m, determine the strain energy in each of the springs and the bending strain energy in the beam.

5 kN

3 m 3 m

k k

Prob. 14–21

14–22. Determine the bending strain energy in the beam and the axial strain energy in each of the two posts. All members are made of aluminum and have a square cross section 50 mm by 50 mm. Assume the posts only support an axial load. E al = 70 GPa.

1.5 m

8 kN/m

3 m

Prob. 14–22

14.3 CONSERVATION OF ENERGY 737

14.3 Conservation of Energy All energy methods used in mechanics are based on a balance of energy, often referred to as the conservation of energy. In this chapter, only mechanical energy will be considered in the energy balance; that is, the energy developed by heat, chemical reactions, and electromagnetic effects will be neglected. As a result, if a loading is applied slowly to a body, then physically the external loads tend to deform the body so that the loads do external work Ue as they are displaced. This external work on the body is transformed into internal work or strain energy Ui, which is stored in the body. Furthermore, when the loads are removed, the strain energy restores the body back to its original undeformed position, provided the material’s elastic limit is not exceeded. The conservation of energy for the body can therefore be stated mathematically as

Ue = Ui (14–23)

We will now show three examples of how this equation can be applied to determine the displacement of a point on a deformable member or structure. As the first example, consider the truss in Fig. 14–18 subjected to the load P . Provided P is applied gradually, the external work done by P is determined from Eq. 14–2 , that is, Ue = 12 P!, where ! is the vertical displacement of the truss at the joint where P is applied. Assuming that P develops an axial force N in a particular member, the strain energy stored in this member is determined from Eq. 14–16 , that is, Ui = N2L >2AE. Summing the strain energies for all the members of the truss, we can write Eq. 14–23 as

1 2

P! = a N 2L

2AE (14–24)

Once the internal forces ( N ) in all the members of the truss are determined and the terms on the right calculated, it is then possible to determine the unknown displacement !.

Fig. 14–18

738 CHAPTER 14 ENERGY METHODS

As a second example, consider finding the vertical displacement ! under the load P acting on the beam in Fig. 14–19 . Again, the external work is Ue = 12 P!. In this case the strain energy is the result of internal shear and moment loadings caused by P . In particular, the contribution of strain energy due to shear is generally neglected in most beam deflection problems unless the beam is short and supports a very large load. (See Example 14.4 .) Consequently, the beam’s strain energy will be determined only by the internal bending moment M , and therefore, using Eq. 14–17 , Eq. 14–23 can be written symbolically as

1 2

P! = L L

0 M2

2EI dx (14–25)

Once M is expressed as a function of position x and the integral is evaluated, ! can then be determined.

As a last example, we will consider a beam loaded by a couple moment M0 as shown in Fig. 14–20 . This moment causes the rotational displacement u at the point of application of the couple moment. Since the couple moment only does work when it rotates , using Eq. 14–5 , the external work is Ue = 12 M0 u. Therefore Eq. 14–23 becomes

1 2

M0 u = L L

0 M 2

2EI dx (14–26)

Here the strain energy is the result of the internal bending moment M caused by application of the couple moment M0. Once M has been expressed as a function of x and the strain energy evaluated, then u which measures the slope of the elastic curve can be determined.

In each of the above examples, it should be noted that application of Eq. 14–23 is quite limited , because only a single external force or couple moment must act on the member or structure. Also, the displacement can only be calculated at the point and in the direction of the external force or couple moment. If more than one external force or couple moment were applied, then the external work of each loading would involve its associated unknown displacement. As a result, all these unknown displacements could not be determined, since only the single Eq. 14–23 is available for the solution. Although application of the conservation of energy as described here has these restrictions, it does serve as an introduction to more general energy methods, which we will consider throughout the rest of this chapter.

Fig. 14–19

Fig. 14–20

14.3 CONSERVATION OF ENERGY 739

EXAMPLE 14.6

The three-bar truss in Fig. 14–21 a is subjected to a horizontal force of 5 kip. If the cross-sectional area of each member is 0.20 in2, determine the horizontal displacement at point B . E = 29(103) ksi.

SOLUTION We can apply the conservation of energy to solve this problem because only a single external force acts on the truss and the required displacement happens to be in the same direction as the force. Furthermore, the reactive forces on the truss do no work since they are not displaced.

Using the method of joints, the force in each member is determined as shown on the free-body diagrams of the pins at B and C , Fig. 14–21 b .

Applying Eq. 14–24 , we have

1 2

P! = a N 2L

2AE

1 2

(5 kip)(!B)h = (2.89 kip)2(2 ft)

2AE +

(-5.77 kip)2(4 ft) 2AE

+ (5 kip)2(3.46 ft)

2AE

(!B)h = 47.32 kip # ft

Notice that since N is squared, it does not matter if a particular member is in tension or compression. Substituting in the numerical data for A and E and solving, we get

(!B)h = 47.32 kip # ft(12 in.>ft)

(0.2 in2)[29(103) kip>in2] = 0.0979 in. S Ans.

2 ft 4 ft

5 kip

30!

(a)

60!

5 kip

30! 60! C

NAB " 2.89 kip

NBC " 5.77 kip NAC " 5 kip

5.77 kip

(b)

Fig. 14–21

740 CHAPTER 14 ENERGY METHODS

EXAMPLE 14.7

The cantilevered beam in Fig. 14–22 a has a rectangular cross section and is subjected to a load P at its end. Determine the displacement of the load. EI is constant.

SOLUTION The internal shear and moment in the beam as a function of x are determined using the method of sections, Fig. 14–22 b .

When applying Eq. 14–23 we will consider the strain energy due to both shear and bending. Using Eqs. 14–19 and 14–17 , we have

1 2

P! = L L

0 fsV

2 dx 2GA

+ L L

0 M 2 dx

2EI

= L L

0 1652(-P)2dx

2GA + L

0 (-Px)2dx

2EI =

3P 2L 5GA

+ P 2L3

6EI (1)

The first term on the right side of this equation represents the strain energy due to shear, while the second is the strain energy due to bending. As stated in Example 14.4 , for most beams the shear strain energy is much smaller than the bending strain energy. To show when this is the case for the beam in Fig. 14–22 a , we require

3 5

P 2L GA

V P 2L3

6EI

3 5

P 2L

G(bh) V P

2L3

6E3 112 (bh3)4 3

5G V 2L

Eh2

Since E … 3G (see Example 14.4 ), then

0.9 V aL h b2

Hence if L is relatively long compared with h , the beam becomes slender and the shear strain energy can be neglected. In other words, the shear strain energy becomes important only for short, deep beams . For example, beams for which L = 5h have approximately 28 times more bending strain energy than shear strain energy, so neglecting the shear strain energy represents an error of about 3.6%. With this in mind, Eq. 1 can be simplified to

1 2

P! = P2L3

6EI

so that

! = PL3

3EI Ans.

L (a)

b h

x M ! "Px

V ! "P

(b)

Fig. 14–22

14.3 CONSERVATION OF ENERGY 741

PROBLEMS

14–27. Determine the horizontal displacement of joint A . The members of the truss are A992 steel rods, each having a diameter of 2 in.

15 kip

4 ft

3 ft 3 ft

4 ft

D E

Prob. 14–27

*14–28. Determine the vertical displacement of joint C . The members of the truss are 2014-T6 aluminum, 40 mm diameter rods.

30 kN

C BA

2 m2 m

1.5 m

Prob. 14–28

14–25. Determine the horizontal displacement of joint B . The members of the truss are A992 steel bars, each with a cross-sectional area of 2500 mm 2 .

2 m

1.5 m

15 kN

Prob. 14–25

14–26. Determine the vertical displacement of joint C . AE is constant.

Prob. 14–26

742 CHAPTER 14 ENERGY METHODS

14–33. The A992 steel bars are pin connected at C and D . If they each have the same rectangular cross section, with a height of 200 mm and a width of 100 mm, determine the vertical displacement at B . Neglect the axial load in the bars.

500 N

2 m 2 m 3 m 3 m

DCB

Prob. 14–33

14–34. The A992 steel bars are pin connected at C . If they each have a diameter 2 in., determine the displacement at E .

A D

6 ft 6 ft 10 ft 8 ft

2 kip

Prob. 14–34

14–35. Determine the slope of the beam at the pin support A . Consider only bending strain energy. EI is constant.

Prob. 14–35

*14–36. The curved rod has a diameter d . Determine the vertical displacement of end B of the rod. The rod is made of material having a modulus of elasticity of E . Consider only bending strain energy.

Prob. 14–36

14–29. Determine the displacement of point B on the A992 steel beam. I = 250 in 4 .

15 ft B

10 ft

8 kip

Prob. 14–29

14–30. Determine the vertical displacement of end B of the cantilevered 6061-T6 aluminum alloy rectangular beam. Consider both shearing and bending strain energy.

100 mm

300 mm

Section a – a

150 kN

1 m

Prob. 14–30

14–31. Determine the displacement of point B on the A992 steel beam. I = 80(10 6 ) mm 4 .

3 m 5 m

A C

20 kN

Prob. 14–31

*14–32. Determine the slope at point A of the beam. EI is constant.

a a a

CA B

Prob. 14–32

14.3 CONSERVATION OF ENERGY 743

*14–40. The rod has a circular cross section with a moment of inertia I . If a vertical force P is applied at A , determine the vertical displacement at this point. Only consider the strain energy due to bending. The modulus of elasticity is E .

Prob. 14–40

14–41. Determine the vertical displacement of end B of the frame. Consider only bending strain energy. The frame is made using two A-36 steel W460 * 68 wide-flange sections.

20 kN

4 m

3 m

Prob. 14–41

14–37. The load P causes the open coils of the spring to make an angle u with the horizontal when the spring is stretched. Show that for this position this causes a torque T = PR cos u and a bending moment M = PR sin u at the cross section. Use these results to determine the maximum normal stress in the material.

14–38. The coiled spring has n coils and is made from a material having a shear modulus G . Determine the stretch of the spring when it is subjected to the load P . Assume that the coils are close to each other so that u ! 0! and the deflection is caused entirely by the torsional stress in the coil.

R d

Probs. 14–37/38

14–39. The pipe assembly is fixed at A . Determine the vertical displacement of end C of the assembly. The pipe has an inner diameter of 40 mm and outer diameter of 60 mm and is made of A-36 steel. Neglect the shearing strain energy.

800 mm

400 mm

B600 N

Prob. 14–39

744 CHAPTER 14 ENERGY METHODS

14.4 Impact Loading Throughout this text we have considered all loadings to be applied to a body in a gradual manner, such that when they reach a maximum value the body remains static. Some loadings, however, are dynamic; that is, they vary with time. A typical example would be caused by the collision of objects. This is called an impact loading. Specifically, impact occurs when one object strikes another, such that large forces are developed between the objects during a very short period of time.

If we assume no energy is lost during impact, due to heat, sound or localized plastic deformations, then we can study the mechanics of impact using the conservation of energy. To show how this is done, we will first analyze the motion of a simple block-and-spring system as shown in Fig. 14–23 . When the block is released from rest, it falls a distance h, striking the spring and compressing it a distance !max before momentarily coming to rest. If we neglect the mass of the spring and assume that the spring responds elastically, then the conservation of energy requires that the energy of the falling block be transformed into stored (strain) energy in the spring; or in other words, the work done by the block’s weight, falling h + !max, is equal to the work needed to displace the end of the spring by an amount !max. Since the force in a spring is related to !max by the equation F = k!max, where k is the spring stiffness, then applying the conservation of energy and Eq. 14–2 , we have

Ue = Ui

W (h + !max) = 1 2

(k!max) !max

W (h + !max) = 1 2

k!2max (14–27)

!2max - 2W k

!max - 2aWk bh = 0 This quadratic equation may be solved for !max. The maximum root is

!max = W k

+ CaWk b2 + 2aWk bh If the weight W is supported statically by the spring, then the top displacement of the spring is !st = W >k. Using this simplification, the above equation becomes

!max = !st + 2(!st)2 + 2!sth or

!max = !st J1 + C1 + 2¢ h!st ≤ R (14–28)

!max

Fig. 14–23

This crash barrier is designed to absorb the impact energy of moving vehicles.

14.4 IMPACT LOADING 745

Once !max is calculated, the maximum force applied to the spring can be determined from

Fmax = k!max (14–29)

It should be realized, however, that this force and associated displacement occur only at an instant. Provided the block does not rebound off the spring, it will continue to vibrate until the motion dampens out and the block assumes the static position, !st. Note also that if the block is held just above the spring, h = 0, and released, then, from Eq. 14–28 , the maximum displacement of the block is

!max = 2!st

In other words, when the block is released from the top of the spring (a dynamic load), the displacement is twice what it would be if it were set on the spring (a static load).

Using a similar analysis, it is also possible to determine the maximum displacement of the end of the spring if the block is sliding on a smooth horizontal surface with a known velocity v just before it collides with the spring, Fig. 14–24 . Here the block’s kinetic energy,* 12 (W>g)v2, will be transformed into stored energy in the spring. Hence,

Ue = Ui

1 2

aW g bv2 = 1

2 k!2max

!max = BWv2gk (14–30) Since the static displacement at the top of the spring caused by the weight W resting on it is !st = W >k, then !max = B!stv2g (14–31)

The results of this simplified analysis can be used to determine both the approximate deflection and the stress developed in a deformable member when it is subjected to impact. To do this we must make the necessary assumptions regarding the collision, so that the behavior of the colliding bodies is similar to the response of the block-and-spring models discussed above. Hence we will consider the moving body to be rigid like the block and the stationary body to be deformable like the spring. Also, it is assumed that the material behaves in a linear-elastic manner. When collision occurs, the bodies remain in contact until the elastic body reaches its maximum deformation, and during the motion the inertia or mass of the elastic body is neglected. Realize that each of these assumptions will lead to a conservative estimate of both the maximum stress and deflection of the elastic body. In other words, their values will be larger than those that actually occur.

k v

!max

Fig. 14–24

*Recall from physics that kinetic energy is “energy of motion.” For the translation of a body it is determined from 12 mv

2, where m is the body’s mass, m = W >g.

746 CHAPTER 14 ENERGY METHODS

A few examples of when this theory can be applied are shown in Fig. 14–25 . Here a block of known weight is dropped onto a post and a beam, causing them to deform a maximum amount !max. The energy of the falling block is transformed momentarily into axial strain energy in the post and bending strain energy in the beam.* In order to determine the deformation!max, we could use the same approach as for the block– spring system, and that is to write the conservation-of-energy equation for the block and post or block and beam, and then solve for !max. However, we can also solve these problems in a more direct manner by modeling the post and beam by an equivalent spring. For example, if a force P displaces the top of the post ! = PL >AE, then a spring having a stiffness k = AE>L would be displaced the same amount by P, that is, ! = P>k . In a similar manner, from Appendix C, a force P applied to the center of a simply supported beam displaces the center ! = PL3>48EI, and therefore an equivalent spring would have a stiffness of k = 48EI>L3. It is not necessary, however, to actually find the equivalent spring stiffness to apply E q. 14–28 or 14–30 . All that is needed to determine the dynamic displacement, !max, is to calculate the static displacement, !st, due to the weight Pst = W of the block resting on the member.

Once !max is determined, the maximum dynamic force can then be calculated from Pmax = k!max. If we consider Pmax to be an equivalent static load then the maximum stress in the member can be determined using statics and the theory of mechanics of materials. Recall that this stress acts only for an instant. In reality, vibrational waves pass through the material, and the stress in the post or the beam, for example, does not remain constant.

The ratio of the equivalent static load Pmax to the static load Pst = W is called the impact factor, n. Since Pmax = k!max and Pst = k!st , then from Eq. 14–28 , we can express it as

n = 1 + C1 + 2¢ h!st ≤ (14–32) This factor represents the magnification of a statically applied load so that it can be treated dynamically. Using Eq. 14–32 , n can be calculated for any member that has a linear relationship between load and deflection. For a complicated system of connected members, however, impact factors are determined from experience and experimental testing. Once n is determined, the dynamic stress and deflection at the point of impact are easily found from the static stress sst and static deflection !st caused by the load W, that is, smax = nsst and !max = n!st.

!max

Fig. 14–25

The members of this crash guard must be designed to resist a prescribed impact loading in order to arrest the motion of a rail car.

*Strain energy due to shear is neglected for reasons discussed in Example 14.4 .

14.4 IMPACT LOADING 747

EXAMPLE 14.8

The aluminum pipe shown in Fig. 14–26 is used to support a load of 150 kip. Determine the maximum displacement at the top of the pipe if the load is (a) applied gradually, and (b) applied by suddenly releasing it from the top of the pipe when h = 0. Take Eal = 10(103) ksi and assume that the aluminum behaves elastically.

SOLUTION Part (a). When the load is applied gradually, the work done by the weight is transformed into elastic strain energy in the pipe. Applying the conservation of energy, we have

Ue = Ui

1 2

W !st = W 2L 2AE

!st = WL AE

= 150 kip(12 in.)

p[(3 in.)2 - (2.5 in.)2] 1011032kip>in2 = 0.02083 in. = 0.0208 in. Ans.

Part (b). Here Eq. 14–28 can be applied, with h = 0. Hence,

!max = !st J1 + C1 + 2¢ h!st ≤ R = 2!st = 2(0.02083 in.)

= 0.0417 in. Ans.

Hence, the displacement of the weight when applied dynamically is twice as great as when the load is applied statically. In other words, the impact factor is n = 2, Eq. 14–32 .

Important Points

• Impact occurs when a large force is developed between two objects which strike one another during a short period of time.

• We can analyze the effects of impact by assuming the moving body is rigid, the material of the stationary body is linear elastic, no energy is lost during collision, the bodies remain in contact during collision, and the inertia of the elastic body is neglected.

• The dynamic load on a body can be determined by multiplying the static load by an impact factor.

3 in. h t ! 0.5 in.

12 in.

150 kip

Fig. 14–26

748 CHAPTER 14 ENERGY METHODS

The A992 steel beam shown in Fig. 14–27a is a W10 * 39. Determine the maximum bending stress in the beam and the beam’s maximum deflection if the weight W = 1.50 kip is dropped from a height h = 2 in. onto the beam. Est = 29(103) ksi.

SOLUTION I We will apply Eq. 14–28 . First, however, we must calculate !st. Using the table in Appendix C , and the data in Appendix B for the properties of a W10 * 39, we have

!st = WL3

48EI =

(1.50 kip)(16 ft)3(12 in.>ft)3 48[29(103) ksi](209 in4)

= 0.03649 in.

!max = !st J1 + C1 + 2¢ h!st ≤ R = 0.03649 in.J1 + B1 + 2a 2 in.0.03649 in. b R = 0.420 in. Ans.

The equivalent static load that causes this displacement is therefore

Pmax = 48EI

L3 !max =

48(29(103) ksi)(209 in4)

(16 ft)3(12 in.>ft)3 (0.420 in.) = 17.3 kip The internal moment caused by this load is maximum at the center

of the beam, such that by the method of sections, Fig. 14–27b , Mmax = Pmax L >4. Applying the flexure formula to determine the bending stress, we have

smax = Mmax c

I =

Pmax Lc 4I

= 12E !max c

= 12[29(103) kip>in2](0.420 in.) (9.92 in.>2)

(16 ft)2(12 in.>ft)2 = 19.7 ksi Ans. SOLUTION II It is also possible to obtain the dynamic or maximum deflection!max from first principles. The external work of the falling weight W is Ue = W (h + !max) . Since the beam deflects !max, and Pmax = 48EI!max>L3, then

Ue = Ui

W (h + !max) = 1 2

¢ 48EI!max L3

≤ !max (1.50 kip)(2 in. + !max) =

1 2

J48[29(103)kip>in2]209 in4 (16 ft)3(12 in.>ft)3 R !2max

20.55!2max - 1.50!max - 3.00 = 0 Solving and choosing the positive root yields

!max = 0.420 in. Ans.

EXAMPLE 14.9

8 ft

h ! 2 in. W

(a) 8 ft

Mmax

(b)

— 2 L

2 Pmax

Fig. 14–27

14.4 IMPACT LOADING 749

A railroad car that is assumed to be rigid and has a mass of 80 Mg is moving forward at a speed of v = 0.2 m>s when it strikes a steel 200-mm by 200-mm post at A, Fig. 14–28a . If the post is fixed to the ground at C, determine the maximum horizontal displacement of its top B due to the impact. Take Est = 200 GPa.

SOLUTION Here the kinetic energy of the railroad car is transformed into internal bending strain energy only for region AC of the post. (Region BA is not subjected to an internal loading.) Assuming that point A is displaced (!A)max, then the force Pmax that causes this displacement can be determined from the table in Appendix C . We have

Pmax = 3EI(!A)max

L3AC (1)

Ue = Ui ; 1 2

mv2 = 1 2

Pmax(!A)max

1 2

mv2 = 1 2

3EI L3AC

(!A) 2 max; (!A)max = Bmv2L3AC3EI

Substituting in the numerical data yields

(!A)max = B 80(103) kg(0.2 m>s)2(1.5 m)33[200(109) N>m23 112 (0.2 m)4 4 = 0.01162 m = 11.62 mm Using Eq. 1, the force Pmax is therefore

Pmax = 3[200(109) N>m2]3 112 (0.2 m)44(0.01162 m)

(1.5 m)3 = 275.4 kN

With reference to Fig. 14–28b , segment AB of the post remains straight. To determine the maximum displacement at B, we must first determine the slope at A. Using the appropriate formula from the table in Appendix C to determine uA , we have

uA = PmaxLAC

2EI =

275.4(103) N (1.5 m)2

2[200(109) N>m2]312 (0.2 m)44 = 0.01162 rad The maximum displacement at B is thus

(!B)max = (!A)max + uALAB = 11.62 mm + (0.01162 rad) 1(103) mm = 23.2 mm Ans.

EXAMPLE 14.10

1 m 1.5 m

200 mm

(a)

v ! 0.2 m/s

(b)

1 m

1.5 m

(!A)max

(!B)max

Pmax

Fig. 14–28

750 CHAPTER 14 ENERGY METHODS

14–42. A bar is 4 m long and has a diameter of 30 mm. If it is to be used to absorb energy in tension from an impact loading, determine the total amount of elastic energy that it can absorb if (a) it is made of steel for which Est = 200 GPa, sY = 800 MPa, and (b) it is made from an aluminum alloy for which Eal = 70 GPa, sY = 405 MPa.

14–43. Determine the diameter of a red brass C83400 bar that is 8 ft long if it is to be used to absorb 800 ft # lb of energy in tension from an impact loading. No yielding occurs.

*14–44. Determine the speed v of the 50-Mg mass when it is just over the top of the steel post, if after impact, the maximun stress developed in the post is 550 MPa. The post has a length of L = 1 m and a cross-sectional area of 0.01 m2. Est = 200 GPa, sY = 600 MPa.

Prob. 14–44

14–45. The collar has a weight of 50 lb and falls down the titanium bar. If the bar has a diameter of 0.5 in., determine the maximum stress developed in the bar if the weight is (a) dropped from a height of h = 1 ft, (b) released from a height h ≈ 0, and (c) placed slowly on the flange at A. Eti = 16(103) ksi, sY = 60 ksi.

h 4 ft

0.5 in.

Prob. 14–45

14–46. The collar has a weight of 50 lb and falls down the titanium bar. If the bar has a diameter of 0.5 in., determine the largest height h at which the weight can be released and not permanently damage the bar after striking the flange at A. Eti = 16(103) ksi, sY = 60 ksi.

h 4 ft

0.5 in.

Prob. 14–46

14–47. A steel cable having a diameter of 0.4 in. wraps over a drum and is used to lower an elevator having a weight of 800 lb. The elevator is 150 ft below the drum and is descending at the constant rate of 2 ft>s when the drum suddenly stops. Determine the maximum stress developed in the cable when this occurs. Est = 29(103) ksi, sY = 50 ksi.

150 ft

Prob. 14–47

PROBLEMS

14.4 IMPACT LOADING 751

*14–48. A steel cable having a diameter of 0.4 in. wraps over a drum and is used to lower an elevator having a weight of 800 lb. The elevator is 150 ft below the drum and is descending at the constant rate of 3 ft>s. when the drum suddenly stops. Determine the maximum stress developed in the cable when this occurs. Est = 29(103) ksi, sY = 50 ksi.

150 ft

Prob. 14–48

14–49. The steel beam AB acts to stop the oncoming railroad car, which has a mass of 10 Mg and is coasting towards it at v = 0.5 m>s . Determine the maximum stress developed in the beam if it is struck at its center by the car. The beam is simply supported and only horizontal forces occur at A and B. Assume that the railroad car and the supporting framework for the beam remain rigid. Also, compute the maximum deflection of the beam. Est = 200 GPa, sY = 250 MPa.

1 m

200 mm 200 mm

v ! 0.5 m/s

Prob. 14–49

14–50. The aluminum bar assembly is made from two segments having diameters of 40 mm and 20 mm. Determine the maximum axial stress developed in the bar if the 10-kg collar is dropped from a height of h = 150 mm. Take Eal = 70 GPa, sY = 410 MPa.

1.2 m

40 mm

20 mm

0.6 m

Probs. 14–50

14–51. Rods AB and AC have a diameter of 20 mm and are made of 6061-T6 aluminum alloy. They are connected to the rigid collar A which slides freely along the vertical guide rod. If the 50-kg block D is dropped from height h = 200 mm above the collar, determine the maximum normal stress developed in the rods.

*14–52. Rods AB and AC have a diameter of 20 mm and are made of 6061-T6 aluminum alloy. They are connected to the rigid collar which slides freely along the vertical guide rod. Determine the maximum height h from which the 50-kg block D can be dropped without causing yielding in the rods when the block strikes the collar.

400 mm 400 mm30" 30"

Probs. 14–51/52

752 CHAPTER 14 ENERGY METHODS

14–53. The composite aluminum 2014-T6 bar is made from two segments having diameters of 7.5 mm and 15 mm. Determine the maximum axial stress developed in the bar if the 10-kg collar is dropped from a height of h = 100 mm.

h 15 mm

7.5 mm

200 mm

300 mm

Prob. 14–53

14–54. The composite aluminum 2014-T6 bar is made from two segments having diameters of 7.5 mm and 15 mm. Determine the maximum height h from which the 10-kg collar should be dropped so that it produces a maximum axial sterss in the bar of smax = 300 MPa.

h 15 mm

7.5 mm

200 mm

300 mm

Prob. 14–54

14–55. When the 100-lb block is at h = 3 ft above the cylindrical post and spring assembly, it has a speed of v = 20 ft>s. If the post is made of 2014-T6 aluminum and the spring has the stiffness of k = 250 kip> in., determine the required minimum diameter d of the post to the nearest 1 8 in. so that it will not yield when it is struck by the block.

1.5 ft

Prob. 14–55

*14–56. When the 100-lb block is at h = 3 ft above the cylindrical post and spring assembly, it has a speed of v = 20 ft>s. If the post is made of 2014-T6 aluminum and has a diameter of d = 2 in., determine the required minimum stiffness k of the spring so that the post will not yield when it is struck by the block.

14–57. When the 100-lb block is at h = 3 ft above the post and spring assembly, it has a speed of v = 20 ft>s. If the post has a diameter of d = 2 in., and is made of 2014-T6 aluminum, and the spring has a stiffness of k = 500 kip/in., determine the maximum normal stress developed in the post.

1.5 ft

Probs. 14–56/57

14.4 IMPACT LOADING 753

14–58. The tugboat has a weight of 120 000 lb and is traveling forward at 2 ft>s when it strikes the 12-in.-diameter fender post AB used to protect a bridge pier. If the post is made from treated white spruce and is assumed fixed at the river bed, determine the maximum horizontal distance the top of the post will move due to the impact. Assume the tugboat is rigid and neglect the effect of the water.

12 ft

3 ft

Prob. 14–58

14–59. The wide-flange beam has a length of 2L, a depth 2c, and a constant EI. Determine the maximum height h at which a weight W can be dropped on its end without exceeding a maximum elastic stress smax in the beam.

Prob. 14–59

*14–60. The weight of 175 lb is dropped from a height of 4 ft from the top of the A992 steel beam. Determine the maximum deflection and maximum stress in the beam if the supporting springs at A and B each have a stiffness of k = 500 lb> in . The beam is 3 in. thick and 4 in. wide. 14–61. The weight of 175 lb, is dropped from a height of 4 ft from the top of the A992 steel beam. Determine the load factor n if the supporting springs at A and B each have a stiffness of k = 500 lb> in . The beam is 3 in. thick and 4 in. wide.

8 ft

4 ft

3 in.

4 in.kk

Probs. 14–60/61

14–62. The 200-lb block has a downward velocity of 4 ft>s when it is 3 ft from the top of the wooden beam. Determine the maximum stress in the beam due to the impact and com pute the maximum deflection of its end C.Ew = 1.9(103) ksi, sY = 6 ksi.

14–63. The 100-lb block has a downward velocity of 4 ft>s when it is 3 ft from the top of the wooden beam. Determine the maximum stress in the beam due to the impact and compute the maximum deflection of point B.Ew = 1.9(103) ksi, sY = 8 ksi.

5 ft5 ft

4 ft/s

18 in.

12 in.

3 ft

Probs. 14–62/63

754 CHAPTER 14 ENERGY METHODS

*14–64. The 50 kg block is dropped from h = 0.9 m onto the cantilever beam. If the beam is made from A992 steel and is a W200 * 46 wide-flange section, determine the maximum bending stress developed in the beam.

14–65. Determine the maximum height h from which the 50-kg block can be dropped without causing yielding in the cantilever beam. The beam is made from A992 steel and is a W200 * 46 wide-flange section.

A B

3 m

Probs. 14–64/65

14–66. The overhang beam is made of 2014-T6 aluminum. If the 75-kg block has a speed of v = 3 m>s at h = 0.75 m, determine the maximum bending stress developed in the beam.

14–67. The overhang beam is made of 2014-T6 aluminum. Determine the maximum height h from which the 100-kg block can be dropped from rest (v = 0), without causing the beam to yield.

A B C

2 m4 m

75 mm

150 mm

Probs. 14–66/67

*14–68. A 40-lb weight is dropped from a height of h = 2 ft onto the center of the cantilevered A992 steel beam. If the beam is a W10 * 15, determine the maximum bending stress developed in the beam.

14–69. If the maximum allowable bending stress for the W10 * 15 structural A992 steel beam is sallow = 20 ksi, determine the maximum height h from which a 50-lb weight can be released from rest and strike the center of the beam.

14–70. A 40-lb weight is dropped from a height of h = 2 ft onto the center of the cantilevered A992 steel beam. If the beam is a W10 * 15, determine the vertical displacement of its end B due to the impact.

5 ft

B A

5 ft

Probs. 14–68/69/70

14–71. The car bumper is made of polycarbonate- polybutylene terephthalate. If E = 2.0 GPa, determine the maximum deflection and maximum stress in the bumper if it strikes the rigid post when the car is coasting at v = 0.75 m>s . The car has a mass of 1.80 Mg, and the bumper can be considered simply supported on two spring supports connected to the rigid frame of the car. For the bumper take I = 300(106) mm4, c = 75 mm, sY = 30 MPa and k = 1.5 MN>m.

k k

0.9 m 0.9 m

v ! 0.75 m/s

Prob. 14–71

14.5 PRINCIPLE OF VIRTUAL WORK 755

*14.5 Principle of Virtual Work The principle of virtual work was developed by John Bernoulli in 1717, and like other energy methods of analysis, it is based on the conservation of energy. Although the principle of virtual work has many applications in mechanics, in this text we will use it to obtain the displacement and slope at a point on a deformable body.

To do this, we will consider the body to be of arbitrary shape as shown in Fig. 14–29b , and to be subjected to the “real loads” P1, P2, and P3. It is assumed that these loads cause no movement of the supports; however, in general they can strain the material beyond the elastic limit. Suppose that it is necessary to determine the displacement ! of point A on the body. Since there is no force acting at A, then ! will not be included as an external “work term” in the equation when the conservation of energy principle is applied to the body. In order to get around this limitation, we will place an imaginary or “virtual” force P" on the body at point A, such that P" acts in the same direction as !. Furthermore, this load is applied to the body before the real loads are applied, Fig. 14–29a . For convenience, which will be made clear later, we will choose P" to have a “unit” magnitude; that is, P" = 1. It is to be emphasized that the term “virtual ” is used to describe this load because it is imaginary and does not actually exist as part of the real loading.

This external virtual load, however, does create an internal virtual load u in a representative element or fiber of the body, as shown in Fig. 14–29a . As expected, P" and u can be related by the equations of equilibrium. Also, because of P" and u, the body and the element will each undergo a virtual (imaginary) displacement, although we will not be

L u

Application of virtual unit load (a)

P¿!1

Application of real loads (b)

Fig. 14–29

756 CHAPTER 14 ENERGY METHODS

concerned with their magnitudes. Once the virtual load is applied and then the body is subjected to the real loads P1, P2, and P3, point A will be displaced a real amount ! , which causes the element to be displaced dL, Fig. 14–29b . As a result, the external virtual force P" and internal virtual load u “ride along” or are displaced by ! and dL, respectively. Consequently these loads perform external virtual work 1 # ! on the body and internal virtual work u # dL on the element. Considering only the conservation of virtual energy, the external virtual work is then equal to the internal virtual work done on all the elements of the body. Therefore, we can write the virtual-work equation as

virtual loadings

1 # ! = #u # dL (14–33) real displacements

Here P" = 1 = external virtual unit load acting in the direction of !

u = internal virtual load acting on the element ! = external displacement caused by the real loads

dL = external displacement of the element in the direction of u, caused by the real loads

By choosing P" = 1, it can be seen that the solution for ! follows directly, since ! = #u dL.

L uu

Application of virtual unit couple moment

(a)

M¿!1

Application of real loads (b)

Fig. 14–30

14.5 PRINCIPLE OF VIRTUAL WORK 757

In a similar manner, if the angular displacement or slope of the tangent at a point on the body is to be determined at A, Fig. 14–30b , then a virtual couple moment M!, having a“unit” magnitude, is applied at the point, Fig. 14–30a. As a result, this couple moment causes a virtual load uu in one of the elements of the body. Assuming the real loads P1, P2, P3 deform the element an amount dL, the angular displacement u can be found from the virtual-work equation

virtual loadings

1 # u = "uu dL (14–34) real displacements

Here

M ! = 1 = external virtual unit couple moment acting in the direction of u

uu = internal virtual load acting on an element u = external angular displacement in radians caused by the real loads

dL = internal displacement of the element in the direction of uu, caused by the real loads

This method for applying the principle of virtual work is often referred to as the method of virtual forces, since a virtual force is applied, resulting in a determination of an external real displacement. The equation of virtual work in this case represents a statement of compatibility requirements for the body. Although it is not important here, realize that we can also apply the principle of virtual work as a method of virtual displacements . In this case, virtual displacements are imposed on the body when the body is subjected to real loadings. This method can be used to determine the external reactive force on the body or an unknown internal loading. When it is used in this manner, the equation of virtual work is a statement of the equilibrium requirements for the body.*

Internal Virtual Work. The terms on the right side of Eqs. 14–34 and 14–35 represent the internal virtual work developed in the body. The real internal displacements dL in these terms can be produced in several different ways. For example, these displacements may result from geometric fabrication errors, from a change in temperature, or more commonly from stress. In particular, no restriction has been placed on the magnitude of the external loading, so the stress may be large enough to cause yielding or even strain hardening of the material.

*See Engineering Mechanics: Statics , 13th edition, R.C. Hibbeler, Prentice Hall, Inc., 2012.

758 CHAPTER 14 ENERGY METHODS

If we assume that the material behavior is linear elastic and the stress does not exceed the proportional limit, we can then formulate the expressions for internal virtual work caused by stress using the equations of elastic strain energy developed in Sec. 14.2. They are listed in the center column of Table 14–1 . Recall that each of these expressions assumes that the internal loading N, V, M, or T was applied gradually from zero to its full value. As a result, the work done by these resultants is shown in these expressions as one-half the product of the internal loading and its displacement. In the case of the virtual-force method, however, the “full” virtual internal loading is applied before the real loads cause displacements, and therefore the work of the virtual loading is simply the product of the virtual load and its real displacement. Referring to these internal virtual loadings (u) by the corresponding lowercase symbols n, v, m, and t, the virtual work due to axial load, shear, bending moment, and torsional moment is listed in the right-hand column of Table 14–1 . Using these results, the virtual-work equation for a body subjected to a general loading can therefore be written as

1 # ! = L nN AE

dx + L mM EI

dx + L fsvV

GA dx + L

tT GJ

dx (14–35)

In the following sections we will apply the above equation to problems involving the displacement of joints on trusses, and points on beams and mechanical elements. We will also include a discussion of how to handle the effects of fabrication errors and differential temperature. For application it is important that a consistent set of units be used for all the terms. For example, if the real loads are expressed in kilonewtons and the body’s dimensions are in meters, a 1-kN virtual force or 1@kN # m virtual couple should be applied to the body. By doing so a calculated displacement ! will be in meters, and a calculated slope will be in radians.

TABLE 14–1

Axial load N

Shear V

Bending moment M

Torsional moment T

Deformation caused by

Strain energy

Internal virtual work

mM EI

—— dx L

fsvV GA

—— dx L

nN EA —— dx

tT GJ

—— dx L

2EI —— dx L

fsV 2

2GA —— dx L

2EA —— dx L

2GJ —— dx L

14.6 METHOD OF VIRTUAL FORCES APPLIED TO TRUSSES 759

*14.6 Method of Virtual Forces Applied to Trusses

In this section, we will apply the method of virtual forces to determine the displacement of a truss joint. To illustrate the principles, the vertical displacement of joint A of the truss shown in Fig. 14–31b will be determined. To do this, we must place a virtual unit force at this joint, Fig. 14–31a , so that when the real loads P1 and P2 are applied to the truss, they cause the external virtual work 1 # !. The internal virtual work in each member is n!L. Since each member has a constant cross-sectional area A, and n and N are constant throughout the member’s length, then from Table 14–1 , the internal virtual work for each member is

L L

0 nN AE

dx = nNL AE

(14–36)

Therefore, the virtual-work equation for the entire truss is

1 # ! = a nNLAE (14–37) Here

1 = external virtual unit load acting on the truss joint in the direction of ! ! = joint displacement caused by the real loads on the truss n = internal virtual force in a truss member caused by the external virtual unit load N = internal force in a truss member caused by the real loads L = length of a member A = cross@sectional area of a member E = modulus of elasticity of a member

Application of virtual unit load

(a)

NN L

Application of real loads

(b)

!L " NL AE

Fig. 14–31

760 CHAPTER 14 ENERGY METHODS

Temperature Change. Truss members can change their length due to a change in temperature. If a is the coefficient of thermal expansion for a member and !T is the change in temperature, the change in length of a member is !L = a !TL ( Eq. 4–4 ). Hence, we can determine the displacement of a selected truss joint due to this temperature change from Eq. 14–34 , written as

1 # ! = "na !TL (14–38)

Here 1 = external virtual unit load acting on the truss joint in the

direction of ! ! = joint displacement caused by the temperature change n = internal virtual force in a truss member caused by the external

virtual unit load a = coefficient of thermal expansion of material

!T = change in temperature of member L = length of member

Fabrication Errors. Occasionally errors in fabricating the lengths of the members of a truss may occur. If this happens, the displacement ! in a particular direction of a truss joint from its expected position can be determined from direct application of Eq. 14–34 written as

1 # ! = "n !L (14–39)

Here 1 = external virtual unit load acting on the truss joint in the

direction of ! ! = joint displacement caused by the fabrication errors n = internal virtual force in a truss member caused by the external virtual unit load

!L = difference in length of the member from its intended length caused by a fabrication error

A combination of the right-hand sides of Eqs. 14–37 through 14–39 will be necessary if external loads act on the truss and some of the members undergo a temperature change or have been fabricated with the wrong dimensions.

14.6 METHOD OF VIRTUAL FORCES APPLIED TO TRUSSES 761

Procedure for Analysis

The following procedure provides a method that may be used to determine the displacement of any joint on a truss using the method of virtual forces.

Virtual Forces n.

• Place the virtual unit load on the truss at the joint where the displacement is to be determined. The load should be directed along the line of action of the displacement.

• With the unit load so placed and all the real loads removed from the truss, calculate the internal n force in each truss member. Assume that tensile forces are positive and compressive forces are negative.

Real Forces N.

• Determine the N forces in each member. These forces are caused only by the real loads acting on the truss. Again, assume that tensile forces are positive and compressive forces are negative.

Virtual-Work Equation.

• Apply the equation of virtual work to determine the desired displacement. It is important to retain the algebraic sign for each of the corresponding n and N forces when substituting these terms into the equation.

• If the resultant sum !nNL >AE is positive, the displacement " is in the same direction as the virtual unit load. If a negative value results, " is opposite to the virtual unit load.

• When applying 1 # " = !na "TL, an increase in temperature, "T, will be positive; whereas a decrease in temperature will be negative.

• For 1 # " = !n "L, when a fabrication error causes an increase in the length of a member, "L is positive, whereas a decrease in length is negative.

• When applying this method, attention should be paid to the units of each numerical quantity. Notice, however, that the virtual unit load can be assigned any arbitrary unit: pounds, kips, newtons, etc., since the n forces will have these same units, and as a result, the units for both the virtual unit load and the n forces will cancel from both sides of the equation.

762 CHAPTER 14 ENERGY METHODS

Determine the vertical displacement of joint C of the steel truss shown in Fig. 14–32a . The cross-sectional area of each member is A = 400 mm2 and Est = 200 GPa.

EXAMPLE 14.11

D C

B A

2 m 2 m

2 m

(a)

100 kN

1 kN

(b)

! 1 4

14 kN

Virtual forces

1 kN D C

B A

SOLUTION Virtual Forces n. Since the vertical displacement at joint C is to be determined, only a vertical 1-kN virtual load is placed at joint C; and the force in each member is calculated using the method of joints. The results of this analysis are shown in Fig. 14–32b . Using our sign convention, positive numbers indicate tensile forces and negative numbers indicate compressive forces.

Real Forces N. The applied load of 100 kN causes forces in the members that are also calculated using the method of joints. The results of this analysis are shown in Fig. 14–32c .

Virtual-Work Equation. Arranging the data in tabular form, we have

200 kN

(c)

!100 kN 100 kN

Real forces

! 14

1.4 kN

141.4 kN

D C

B A

Fig. 14–32

Member n N L nNL AB 0 -100 4 0 BC 0 141.4 2.828 0 AC -1.414 -141.4 2.828 565.7 CD 1 200 2 400 ! 965.7 kN2 # m

Thus,

1 kN # "C v

= a nNLAE = 965.7 kN 2 # m

Substituting the numerical values for A and E, we have

1 kN # "Cv = 965.7 kN 2 # m

[400(10-6) m2][200(106) kN>m2] "Cv = 0.01207 m = 12.1 mm Ans.

14.6 METHOD OF VIRTUAL FORCES APPLIED TO TRUSSES 763

EXAMPLE 14.12

30!

4 m

30!

(a)

6 kN

SOLUTION Virtual Forces n. A horizontal 1-kN virtual load is applied to the truss at joint B, and the forces in each member are calculated, Fig. 14–33b .

Real Forces N. Since the n forces in members AC and BC are zero, the N forces in these members do not have to be determined. Why? For completeness, though, the entire“real” force analysis is shown in Fig. 14–33c .

Virtual-Work Equation. The loads, temperature, and the fabrication error all affect the displacement of point B; therefore, Eqs. 14–37 , 14–38 , and 14–39 must be combined, which gives

1 kN # !Bh = a nNLAE + "na !TL + "n!L = 0 + 0 +

(-1.155 kN)(-12 kN)(4 m) [250(10-6) m2][200(106) kN>m2]

+ 0 + 0 + (-1.155 kN)[12(10-6)>#C](60#C)(4 m) + (-1.155 kN)(-0.003 m)

!Bh = 0.00125 m

= 1.25 mm d Ans.

!1.155 kN

1 kN

A 0

(b)

Virtual forces

!12 kN

6 kN

A 10.39 kN

!1 2 k

(c)

Real forces

Fig. 14–33

Determine the horizontal displacement of the roller at B of the truss shown in Fig. 14–33a . Due to radiant heating, member AB is subjected to an increase in temperature of !T = +60#C, and this member has been fabricated 3 mm too short. The members are made of steel, for which ast = 12(10-6)>#C and Est = 200 GPa. The cross-sectional area of each member is 250 mm2.

764 CHAPTER 14 ENERGY METHODS

*14–72. Determine the vertical displacement of joint A. Each A992 steel member has a cross-sectional area of 400 mm2.

2 m

3 m1.5 m E

40 kN 60 kN

Prob. 14–72

14–73. Determine the horizontal displacement of joint B. Each A992 steel member has a cross-sectional area of 2 in2.

14–74. Determine the vertical displacement of joint B. Each A992 steel member has a cross-sectional area of 2 in2.

C B

3 ft

600 lb

5 ft

Probs. 14–73/74

14–75. Determine the vertical displacement of joint B. For each A992 steel member A = 1.5 in2.

*14–76. Determine the vertical displacement of joint E. For each A992 steel member A = 1.5 in2.

6 ft

DEF

8 ft

2 kip 2 kip 2 kip

Probs. 14–75/76

14–77. Determine the vertical displacement of point B. Each A992 steel member has a cross-sectional area of 4.5 in2.

14–78. Determine the vertical displacement of point E. Each A992 steel member has a cross-sectional area of 4.5 in2.

6 ft

DEF

5 kip 8 ft

8 ft

Probs. 14–77/78

PROBLEMS

14.6 METHOD OF VIRTUAL FORCES APPLIED TO TRUSSES 765

14–79. Determine the horizontal displacement of joint B of the truss. Each A992 steel member has a cross-sectional area of 400 mm2.

*14–80. Determine the vertical displacement of joint C of the truss. Each A992 steel member has a cross-sectional area of 400 mm2.

1.5 m

C B

2 m 4 kN

A D

5 kN

Probs. 14–79/80

14–81. Determine the horizontal displacement of joint C on the truss. Each A992 steel member has a cross-sectional area of 3 in2.

14–82. Determine the horizontal displacement of joint B on the truss. Each A992 steel member has a cross-sectional area of 3 in2.

4 ft

3 ft

15 kipC

2 kip

Probs. 14–81/82

14–83. Determine the vertical displacement of joint A. The truss is made from A992 steel rods having a diameter of 30 mm.

*14–84. Determine the vertical displacement of joint D. The truss is made from A992 steel rods having a diameter of 30 mm.

D C

20 kN

2 m

1.5 m 1.5 m

Probs. 14–83/84

14–85. Determine the vertical displacement of joint C on the truss. Each A992 steel member has a cross-sectional area of A = 300 mm2.

14–86. Determine the vertical displacement of joint D on the truss. Each A992 steel member has a cross-sectional area of A = 300 mm2.

4 m

20 kN

3 m3 m B

30 kN

Probs. 14–85/86

766 CHAPTER 14 ENERGY METHODS

*14.7 Method of Virtual Forces Applied to Beams

In this section we will apply the method of virtual forces to determine the displacement and slope at a point on a beam. To illustrate the principles, the vertical displacement ! of point A on the beam shown in Fig. 14–34b will be determined. To do this we must place a vertical unit load at this point, Fig. 14–34a , so that when the “real” distributed load w is applied to the beam it will cause the internal virtual work 1 # !. Because the load causes both a shear V and moment M within the beam, we must actually consider the internal virtual work due to both of these loadings. In Example 14.7 , however, it was shown that beam deflections due to shear are negligible compared with those caused by bending, particularly if the beam is long and slender. Since this type of beam is most often used in practice, we will only consider the virtual strain energy due to bending, Table 14–1 . Hence, the real load causes the element dx to deform so its sides rotate by an angle du = (M >EI)dx , which causes internal virtual work m du. Applying Eq. 14–34 , the virtual-work equation for the entire beam, we have

1 # ! = L L

0 mM EI

dx (14–40)

Here 1 = external virtual unit load acting on the beam in the direction of ! ! = displacement caused by the real loads acting on the beam m = internal virtual moment in the beam, expressed as a function of

x and caused by the external virtual unit load M = internal moment in the beam, expressed as a function of x and

caused by the real loads E = modulus of elasticity of the material I = moment of inertia of the cross-sectional area about the

neutral axis

In a similar manner, if the slope u of the tangent at a point on the beam’s elastic curve is to be determined, a virtual unit couple moment must be applied at the point, and the corresponding internal virtual moment mu has to be determined. If we apply Eq. 14–35 for this case and neglect the effect of shear deformations, we have

1 # u = L L

0 muM

EI dx (14–41)

14.7 METHOD OF VIRTUAL FORCES APPLIED TO BEAMS 767

When applying these equations, keep in mind that the integrals on the right side represent the amount of virtual bending strain energy that is stored in the beam. If concentrated forces or couple moments act on the beam or the distributed load is discontinuous, a single integration cannot be performed across the beam’s entire length. Instead, separate x coordinates must be chosen within regions that have no discontinuity of loading. Also, it is not necessary that each x have the same origin; however, the x selected for determining the real moment M in a particular region must be the same x as that selected for determining the virtual moment m or mu within the same region. For example, consider the beam shown in Fig. 14–35 . In order to determine the displacement at D, we can use x1 to determine the strain energy in region AB, x2 for region BC, x3 for region DE, and x4 for region DC. In any case, each x coordinate should be selected so that both M and m (or mu) can easily be formulated.

Unlike beams, as discussed here, some members may also be subjected to significant virtual strain energy caused by axial load, shear, and torsional moment. When this is the case, we must include in the above equations the energy terms for these loadings as formulated in Eq. 14–36 .

x dx

Virtual loads

(a)

1 A

x !

(b)

Real loads

Virtual load

E DCB

(a)

x2 x4

x3x1

Real loads

E DCB

P w

(b)

x2 x4

x3x1

Fig. 14–34

Fig. 14–35

768 CHAPTER 14 ENERGY METHODS

Procedure for Analysis

The following procedure provides a method that may be used to determine the displacement and slope at a point on the elastic curve of a beam using the method of virtual forces.

Virtual Moments m or mu. • Place a virtual unit load on the beam at the point and directed

along the line of action of the desired displacement. • If the slope is to be determined, place a virtual unit couple

moment at the point. • Establish appropriate x coordinates that are valid within regions

of the beam where there is no discontinuity of both real and virtual load.

• With the virtual load in place, and all the real loads removed from the beam, calculate the internal moment m or mu as a function of each x coordinate.

• Assume that m or mu acts in the positive direction according to the established beam sign convention for positive moment, Fig. 6–3 .

Real Moments. • Using the same x coordinates as those established for m or mu,

determine the internal moments M caused by the real loads. • Since positive m or mu was assumed to act in the conventional

“positive direction,” it is important that positive M acts in this same direction. This is necessary since positive or negative internal virtual work depends on the directional sense of both the virtual load, defined by {m or {mu, and displacement, caused by {M.

Virtual-Work Equation. • Apply the equation of virtual work to determine the desired

displacement ! or slope u. It is important to retain the algebraic sign of each integral calculated within its specified region.

• If the algebraic sum of all the integrals for the entire beam is positive, ! or u is in the same direction as the virtual unit load or virtual unit couple moment. If a negative value results, ! or u is opposite to the virtual unit load or couple moment.

14.7 METHOD OF VIRTUAL FORCES APPLIED TO BEAMS 769

EXAMPLE 14.13

Determine the displacement of point B on the beam shown in Fig. 14–36a . EI is constant.

(a)

B A

Fig. 14–36

x v

m ! "1 x

(b) Virtual loads

M!"wx x2

–x 2

SOLUTION Virtual Moment m. The vertical displacement of point B is obtained by placing a virtual unit load at B, Fig. 14–36b. By inspection, there are no discontinuities of loading on the beam for both the real and virtual loads. Thus, a single x coordinate can be used to determine the virtual strain energy. This coordinate will be selected with its origin at B, so that the reactions at A do not have to be determined in order to find the internal moments m and M. Using the method of sections, the internal moment m is shown in Fig. 14–36b .

Real Moment M. Using the same x coordinate, the internal moment M is shown in Fig. 14–36c .

Virtual-Work Equation. The vertical displacement at B is thus

1 # !B = L mM EI

dx = L L

0 (-1x)(-wx2>2) dx

!B = wL4

8EI Ans.

770 CHAPTER 14 ENERGY METHODS

Determine the slope at point B of the beam shown in Fig. 14–37a . EI is constant.

EXAMPLE 14.14

(a)

1v2

Virtual loads

mu2 ! 1

mu1 ! 0

(b) Real load

(c)

— 2 L—

2 L

x2 x2 x1

x2 x2

M1 ! "Px1

M2 ! "P( L 2

#x2) — 2 L

— 2 L

Fig. 14–37 SOLUTION Virtual Moments mU. The slope at B is determined by placing a virtual unit couple moment at B, Fig. 14–37b . Two x coordinates must be selected in order to determine the total virtual strain energy in the beam. Coordinate x1 accounts for the strain energy within segment AB, and coordinate x2 accounts for the strain energy in segment BC. Using the method of sections the internal moments mu within each of these segments are shown in Fig. 14–37b .

Real Moments M. Using the same coordinates x1 and x2 (Why?), the internal moments M are shown in Fig. 14–37c .

Virtual-Work Equation. The slope at B is thus

1 # uB = L muM

EI dx

= L L>2

0 0(-Px1) dx1

EI + L

L>2 0

1{-P[(L >2) + x2]} dx2

uB = - 3PL2

8EI Ans.

The negative sign indicates that uB is opposite to the direction of the virtual couple moment shown in Fig. 14–37b .

14.7 METHOD OF VIRTUAL FORCES APPLIED TO BEAMS 771

PROBLEMS

*14–92. Determine the displacement of point C of the beam made from A992 steel and having a moment of inertia of I = 53.8 in4.

14–93. Determine the slope at B of the beam made from A992 steel and having a moment of inertia of I = 53.8 in4.

5 ft

8 kip

5 ft 10 ft

A B

Probs. 14–92/93

14–94. The beam is made of Douglas fir. Determine the slope at C.

8 kN

1.5 m

B C

120 mm

180 mm

1.5 m

Prob. 14–94

14–95. Determine the displacement at pulley B. The A992 steel shaft has a diameter of 30 mm.

A 4 kN

1 kN 1 kN

0.4 m B

0.4 m

0.3 m

3 kN

Prob. 14–95

14–87. Determine the displacement at point C. EI is constant.

a a–2 a –

2 a

Prob. 14–87

*14–88. The beam is made of southern pine for which Ep = 13 GPa. Determine the displacement at A.

3 m

1.5 m

4 kN/m

B C

120 mm

180 mm

15 kN

Prob. 14–88

14–89. Determine the displacement at point C. EI is constant.

14–90. Determine the slope at point C. EI is constant.

14–91. Determine the slope at point A. EI is constant.

A C B

Probs. 14–89/90/91

772 CHAPTER 14 ENERGY METHODS

14–103. Determine the slope at A of the 2014-T6 aluminum shaft having a diameter of 100 mm.

*14–104. Determine the displacement at C of the 2014-T6 aluminum shaft having a diameter of 100 mm.

A BC

1 m 1 m 0.5 m

8 kN 8 kN

0.5 m

Probs. 14–103/104

14–105. Determine the displacement of point B. The moment of inertia of the center portion DG of the shaft is 2I, whereas the end segments AD and GC have a moment of inertia I. The modulus of elasticity for the material is E.

aa a a

B G

Prob. 14–105

14–106. Determine the displacement of point C of the W14 * 26 beam made from A992 steel.

14–107. Determine the slope at A of the W14 * 26 beam made from A992 steel.

5 ft

8 kip

5 ft

A B

8 kip

5 ft 5 ft C

Probs. 14–106/107

*14–96. The A992 steel beam has a moment of inertia of I = 125(106) mm4. Determine the displacement at D.

14–97. The A992 steel beam has a moment of inertia of I = 125(106) mm4. Determine the slope at A.

14–98. The A992 structural steel beam has a moment of inertia of I = 125(106) mm4. Determine the slope of the beam at B.

4 m

18 kN!m

3 m 3 m

D B C

18 kN!m

Probs. 14–96/97/98

14–99. Determine the displacement at C of the shaft. EI is constant.

*14–100. Determine the slope at A of the shaft. EI is constant.

A B C

L L2

Probs. 14–99/100

14–101. Determine the slope of end C of the overhang beam. EI is constant.

14–102. Determine the displacement of point D of the overhang beam. EI is constant.

A C B

D L 2

L 2

Probs. 14–101/102

14.7 METHOD OF VIRTUAL FORCES APPLIED TO BEAMS 773

*14–112. Determine the displacement of the shaft C. EI is constant.

L 2–

A C

Prob. 14–112

14–113. Determine the slope of the shaft at the bearing support A. EI is constant.

L 2–

A C

Prob. 14–113

14–114. Determine the vertical displacement of point A on the angle bracket due to the concentrated force P. The bracket is fixed connected to its support. EI is constant. Consider only the effect of bending.

Prob. 14–114

*14–108. Determine the slope at A. EI is constant.

B C

Prob. 14–108

14–109. Determine the slope at end C of the overhang white spruce beam.

14–110. Determine the displacement at point D of the overhang white spruce beam.

CDA B

4 ft4 ft4 ft

300 lb

3 in.

6 in.

Section a – a

150 lb/ft

Probs. 14–109/110

14–111. The simply supported beam having a square cross section is subjected to a uniform load w. Determine the maximum deflection of the beam caused only by bending, and caused by bending and shear. Take E = 3G.

L a

Prob. 14–111

774 CHAPTER 14 ENERGY METHODS

14–119. The L-shaped frame is made from two segments, each of length L and flexural stiffness EI. If it is subjected to the uniform distributed load, determine the horizontal displacement of the end C.

*14–120. The L-shaped frame is made from two segments, each of length L and flexural stiffness EI. If it is subjected to the uniform distributed load, determine the vertical displacement of point B.

L A

Probs. 14–119/120

14–121. Determine the displacement at C. EI is constant.

B C

Prob. 14–121

14–122. Determine the slope at B. EI is constant.

B C

Prob. 14–122

14–115. Beam AB has a square cross section of 100 mm by 100 mm. Bar CD has a diameter of 10 mm. If both members are made of A992 steel, determine the vertical displacement of point B due to the loading of 10 kN.

*14–116. Beam AB has a square cross section of 100 mm by 100 mm. Bar CD has a diameter of 10 mm. If both members are made of A992 steel, determine the slope at A due to the loading of 10 kN.

3 m 2 m

10 kN

A D B

2 m

Probs. 14–115/116

14–117. Bar ABC has a rectangular cross section of 300 mm by 100 mm. Attached rod DB has a diameter of 20 mm. If both members are made of A-36 steel, determine the vertical displacement of point C due to the loading. Consider only the effect of bending in ABC and axial force in DB.

14–118. Bar ABC has a rectangular cross section of 300 mm by 100 mm. Attached rod DB has a diameter of 20 mm. If both members are made of A-36 steel, determine the slope at A due to the loading. Consider only the effect of bending in ABC and axial force in DB.

3 m

20 kN

A B

4 m

100 mm

300 mm

3 m

Probs. 14–117/118

14.8 CASTIGLIANO’S THEOREM 775

*14.8 Castigliano’s Theorem In 1879, Alberto Castigliano, an Italian railroad engineer, published a book in which he outlined a method for determining the displacement and slope at a point in a body. This method, which is referred to as Castigliano’s second theorem, applies only to bodies that have constant temperature and material with linear-elastic behavior. If the displacement at a point is to be determined, the theorem states that the displacement is equal to the first partial derivative of the strain energy in the body with respect to a force acting at the point and in the direction of displacement. In a similar manner, the slope of the tangent at a point in a body is equal to the first partial derivative of the strain energy in the body with respect to a couple moment acting at the point and in the direction of the slope angle.

To derive Castigliano’s second theorem, consider a body of any arbitrary shape, which is subjected to a series of n forces P1, P2, . . . , Pn, Fig. 14–38 . According to the conservation of energy, the external work done by these forces is equal to the internal strain energy stored in the body. However, the external work is a function of the external loads, Ue = ! 1P dx, Eq. 14–1 , so the internal work is also a function of the external loads. Thus,

Ui = Ue = f(P1 , P2 , . . . , Pn) (14–42)

Now, if any one of the external forces, say Pj, is increased by a differential amount dPj, the internal work will also be increased, such that the strain energy becomes

Ui + dUi = Ui + 0Ui 0Pj

dPj (14–43)

Fig. 14–38

776 CHAPTER 14 ENERGY METHODS

This value, however, will not depend on the sequence in which the n forces are applied to the body. For example, we could apply dPj to the body first, then apply the loads P1, P2, . . . , Pn. In this case, dPj would cause the body to displace a differential amount d!j in the direction of dPj. By Eq. 14–2 (Ue = 12 Pj !j), the increment of strain energy would be 12 dPj d!j. This quantity, however, is a second-order differential and may be neglected. Further application of the loads P1, P2, . . . , Pn causes dPj to move through the displacement !j so that now the strain energy becomes

Ui + dUi = Ui + dPj !j (14–44)

Here, as above, Ui is the internal strain energy in the body, caused by the loads P1, P2, . . . , Pn, and dPj !j is the additional strain energy caused by dPj.

In summary, Eq. 14–43 represents the strain energy in the body determined by first applying the loads P1, P2, . . . , Pn, then dPj ; Eq. 14–44 represents the strain energy determined by first applying dPj and then the loads P1, P2, . . . , Pn. Since these two equations must be equal, we require

!j = 0Ui 0Pj

(14–45)

which proves the theorem; i.e., the displacement !j in the direction of Pj is equal to the first partial derivative of the strain energy with respect to Pj.

Castigliano’s second theorem, Eq. 14–45, is a statement regarding the body’s compatibility requirements, since it is a condition related to displacement. Also, the above derivation requires that only conservative forces be considered for the analysis. These forces can be applied in any order, and furthermore, they do work that is independent of the path and therefore create no energy loss. As long as the material has linear- elastic behavior, the applied forces will be conservative and the theorem is valid. Castigliano’s first theorem is similar to his second theorem; however, it relates the load Pj to the partial derivative of the strain energy with respect to the corresponding displacement, that is, Pj = 0Ui>0!j. The proof is similar to that given above. This theorem is another way of expressing the equilibrium requirements for the body; however, it has limited application and therefore it will not be discussed here.

14.9 CASTIGLIANO’S THEOREM APPLIED TO TRUSSES 777

*14.9 Castigliano’s Theorem Applied to Trusses

Since a truss member is only subjected to an axial load, the strain energy for the member is given by Eq. 14–16 , Ui = N2L >2AE. Substituting this equation into Eq. 14–45 and omitting the subscript i, we have

! = 0

0P a N 2L

2AE

It is generally easier to perform the differentiation prior to summation. Also, L, A, and E are constant for a given member, and therefore we can write

! = aN a 0N0P b LAE (14–46) Here

! = displacement of the truss joint P = an external force of variable magnitude applied to the truss joint

in the direction of ! N = internal axial force in a member caused by both force P and the

actual loads on the truss L = length of a member A = cross@sectional area of a member E = modulus of elasticity of the material

By comparison, Eq. 14–46 is similar to that used for the method of virtual forces, Eq. 14–37 (1 # ! = "nNL >AE), except that n is replaced by 0N >0P. These terms, n and 0N >0P, are the same, since they represent the change of the member’s axial force with respect to the load P or, in other words, the axial force per unit load.

778 CHAPTER 14 ENERGY METHODS

Procedure for Analysis

The following procedure provides a method that may be used to determine the displacement of any joint on a truss using Castigliano’s second theorem.

External Force P.

• Place a force P on the truss at the joint where the displacement is to be determined. This force is assumed to have a variable magnitude and should be directed along the line of action of the displacement.

Internal Forces N.

• Determine the force N in each member in terms of both the actual (numerical) loads and the (variable) force P. Assume that tensile forces are positive and compressive forces are negative.

• Find the respective partial derivative 0N >0P for each member. • After N and 0N >0P have been determined, assign P its numerical value if it has actually replaced a real

force on the truss. Otherwise, set P equal to zero.

Castigliano’s Second Theorem.

• Apply Castigliano’s second theorem to determine the desired displacement !. It is important to retain the algebraic signs for corresponding values of N and 0N >0P when substituting these terms into the equation.

• If the resultant sum "N(0N >0P)L >AE is positive, ! is in the same direction as P. If a negative value results, ! is opposite to P.

EXAMPLE 14.15

Determine the vertical displacement of joint C of the steel truss shown in Fig. 14–39a . The cross-sectional area of each member is A = 400 mm2, and Est = 200 GPa.

SOLUTION

External Force P. A vertical force P is applied to the truss at joint C, since this is where the vertical displacement is to be determined, Fig. 14–39b .

D C

2 m 2 m

2 m

(a) 100 kN

Fig. 14–39

Internal Forces N. The reactions at the truss supports A and D are calculated and the results are shown in Fig. 14–39b. Using the method of joints, the N forces in each member are determined, Fig. 14–39c .* For convenience, these results along with their partial derivatives 0N >0P are listed in tabular form. Note that since P does not actually exist as a real load on the truss, we require P = 0.

14.9 CASTIGLIANO’S THEOREM APPLIED TO TRUSSES 779

(b)

D C

B A

200 kN ! P

100 kN ! P 100 kN

200 kN ! P

100 kN ! P

141.4 kN ! 1.414 P

100 kN A

(c)

45"

100 kN

100 kN B

141.4 kN

45"

*It may be more convenient to analyze the truss with just the 100-kN load on it, then analyze the truss with the P load on it. The results can then be summed algebraically to give the N forces.

Fig. 14–39 (cont.)

Member N 0N 0P N(P = 0) L Na 0N0P bL

AB -100 0 -100 4 0 BC 141.4 0 141.4 2.828 0 AC - (141.4 + 1.414P) -1.414 -141.4 2.828 565.7 CD 200 + P 1 200 2 400 ! 965.7 kN # m

Castigliano’s Second Theorem. Applying Eq. 14–46 , we have

"Cv = !N a 0N0P b LAE = 965.7 kN # mAE Substituting the numerical values for A’ and E, we get

"Cv = 965.7 kN # m

[400(10-6) m2] 200(106) kN>m2 = 0.01207 m = 12.1 mm Ans.

This solution should be compared with that of Example 14.11 , using the virtual-work method.

780 CHAPTER 14 ENERGY METHODS

14–123. Solve Prob. 14–73 using Castigliano’s theorem.

*14–124. Solve Prob. 14–74 using Castigliano’s theorem.

14–125. Solve Prob. 14–75 using Castigliano’s theorem.

14–126. Solve Prob. 14–76 using Castigliano’s theorem.

14–127. Solve Prob. 14–77 using Castigliano’s theorem.

*14–128. Solve Prob. 14–78 using Castigliano’s theorem.

14–129. Solve Prob. 14–81 using Castigliano’s theorem.

14–130. Solve Prob. 14–82 using Castigliano’s theorem.

14–131. Solve Prob. 14–85 using Castigliano’s theorem.

*14–132. Solve Prob. 14–86 using Castigliano’s theorem.

PROBLEMS

*14.10 Castigliano’s Theorem Applied to Beams

The internal strain energy for a beam is caused by both bending and shear. However, as pointed out in Example 14.7 , if the beam is long and slender, the strain energy due to shear can be neglected compared with that of bending. Assuming this to be the case, the internal strain energy for a beam is given by Ui = 1M2 dx >2EI, Eq. 14–17 . Omitting the subscript i, Castigliano’s second theorem, !i = 0Ui >0Pi, becomes

! = 0

0P L L

0 M2 dx

2EI

Rather than squaring the expression for internal moment, integrating, and then taking the partial derivative, it is generally easier to differentiate prior to integration. Then we have

! = L L

0 M a 0M0P b dxEI (14–47)

14.10 CASTIGLIANO’S THEOREM APPLIED TO BEAMS 781

Here

! = displacement of the point caused by the real loads acting on the beam

P = an external force of variable magnitude applied to the beam at the point and in the direction of !

M = internal moment in the beam, expressed as a function of x and caused by both the force P and the actual loads on the beam

E = modulus of elasticity of the material I = moment of inertia of cross-sectional area about the neutral axis

If the slope of the tangent u at a point on the elastic curve is to be determined, the partial derivative of the internal moment M with respect to an external couple moment M " acting at the point must be found. For this case,

u = L L

0 M a 0M0M " b dxEI (14–48)

The above equations are similar to those used for the method of virtual forces, Eqs. 14–40 and 14–41 , except m and mu replace 0M >0P and 0M >0M ", respectively.

In addition, if axial load, shear, and torsion cause significant strain energy within the member, then the effects of all these loadings should be included when applying Castigliano’s theorem. To do this we must use the strain-energy functions developed in Sec. 14.2 , along with their associated partial derivatives. The result is

! = #N a 0N0P b LAE + LL0 fs V a 0V0P b dxGA + LL0 M a 0M0P b dxEI + LL0 T a 0T0P b dxGJ (14–49) The method of applying this general formulation is similar to that used to apply Eqs. 14–47 and 14–48 .

782 CHAPTER 14 ENERGY METHODS

Procedure for Analysis

The following procedure provides a method that may be used to apply Castigliano’s second theorem.

External Force P or Couple Moment M!.

• Place a force P on the beam at the point and directed along the line of action of the desired displacement.

• If the slope of the tangent is to be determined at the point, place a couple moment M! at the point.

• Assume that both P and M! have a variable magnitude.

Internal Moments M.

• Establish appropriate x coordinates that are valid within regions of the beam where there is no discontinuity of force, distributed load, or couple moment.

• Determine the internal moments M as a function of x, the actual (numerical) loads, and P or M !, and then find the partial derivatives 0M >0P or 0M >0M ! for each coordinate x.

• After M and 0M >0P or 0M >0M ! have been determined, assign P or M ! its numerical value if it has actually replaced a real force or couple moment. Otherwise, set P or M ! equal to zero.

Castigliano’s Second Theorem.

• Apply Eq. 14–47 or 14–48 to determine the desired displacement " or u. It is important to retain the algebraic signs for corresponding values of M and 0M >0P or 0M >0M !.

• If the resultant sum of all the definite integrals is positive, " or u is in the same direction as P or M!. If a negative value results, " or u is opposite to P or M!.

Determine the displacement of point B on the beam shown in Fig. 14–40a . EI is constant.

EXAMPLE 14.16

(a)

Fig. 14–40

SOLUTION External Force P. A vertical force P is placed on the beam at B as shown in Fig. 14–40b .

Internal Moments M. A single x coordinate is needed for the solution, since there are no discontinuities of loading between A and B. Using the method of sections, Fig. 14–40c , the internal moment and its partial derivative are determined as follows:

a+ !MNA = 0; M + wx a x2 b + P(x) = 0 M = - wx

2 - Px

0M 0P = -x

Setting P = 0 gives

M = -wx2

2 and

0M 0P = -x

Castigliano’s Second Theorem. Applying Eq. 14–47 , we have

"B = L L

0 M a 0M0P b dxEI = LL0 (-wx2>2)(-x) dxEI

= wL4

8EI Ans.

The similarity between this solution and that of the virtual-work method, Example 14.13 , should be noted.

(b)

B A

(c)

P x 2

14.10 CASTIGLIANO’S THEOREM APPLIED TO BEAMS 783

784 CHAPTER 14 ENERGY METHODS

EXAMPLE 14.17

Determine the slope at point B of the beam shown in Fig. 14–41a . EI is constant.

SOLUTION External Couple Moment M’. Since the slope at point B is to be determined, an external couple moment M! is placed on the beam at this point, Fig. 14–41b .

Internal Moments M. Two coordinates, x1 and x2, must be used to completely describe the internal moments within the beam since there is a discontinuity, M!, at B. As shown in Fig. 14–41b , x1 ranges from A to B and x2 ranges from B to C. Using the method of sections, Fig. 14–41c , the internal moments and the partial derivatives for x1 and x2 are determined as follows:

a+ "MNA = 0; M1 = -Px1, 0M1 0M ! = 0

a+ "MNA = 0; M2 = M ! - PaL2 + x2b , 0M20M ! = 1 Castigliano’s Second Theorem. Setting M ! = 0 and applying Eq. 14–48, we have

AC x1B

(b)

M¿

Fig. 14–41

uB = L L

0 M a 0M0M ! b dxEI = LL/20 (-Px1)(0) dx1EI + LL/20 -P[(L >2) + x2](1) dx2EI = - 3PL28EI Ans.

Note the similarity between this solution and that of Example 14.14 .

PROBLEMS

14–133. Solve Prob. 14–90 using Castigliano’s theorem.

14–134. Solve Prob. 14–91 using Castigliano’s theorem.

14–135. Solve Prob. 14–92 using Castigliano’s theorem.

*14–136. Solve Prob. 14–93 using Castigliano’s theorem.

14–137. Solve Prob. 14–95 using Castigliano’s theorem.

14–138. Solve Prob. 14–96 using Castigliano’s theorem.

14–139. Solve Prob. 14–97 using Castigliano’s theorem.

*14–140. Solve Prob. 14–98 using Castigliano’s theorem.

14–141. Solve Prob. 14–108 using Castigliano’s theorem.

14–142. Solve Prob. 14–119 using Castigliano’s theorem.

14–143. Solve Prob. 14–120 using Castigliano’s theorem.

*14–144. Solve Prob. 14–121 using Castigliano’s theorem.

14–145. Solve Prob. 14–122 using Castigliano’s theorem.

A C

(a)

BL 2

L 2

(c)

M¿

x1 M1

L 2

CHAPTER REVIEW 785

CHAPTER REVIEW

When a force (couple moment) acts on a deformable body it will do external work when it displaces (rotates). The internal stresses produced in the body also undergo displacement, thereby creating elastic strain energy that is stored in the material. The conservation of energy states that the external work done by the loading is equal to the internal elastic strain energy produced by the stresses in the body.

Ue = Ui

The conservation of energy can be used to solve problems involving elastic impact, which assumes the moving body is rigid and all the strain energy is stored in the stationary body. This leads to use of an impact factor n, which is a ratio of the dynamic load to the static load. It is used to determine the maximum stress and displacement of the body at the point of impact.

n = 1 + C1 + 2a h!stb smax = nsst

!max = n!st

The principle of virtual work can be used to determine the displacement of a joint on a truss or the slope and the displacement of points on a beam. It requires placing an external virtual unit force (virtual unit couple moment) at the point where the displacement (rotation) is to be determined. The external virtual work that is produced by the external loading is then equated to the internal virtual strain energy in the structure.

1 # ! = a nNLAE 1 # ! = L

0 mM EI

1 # u = L L

0 muM

EI dx

Castigliano’s second theorem can also be used to determine the displacement of a joint on a truss or the slope and the displacement at a point on a beam. Here a variable force P (couple moment M") is placed at the point where the displacement (slope) is to be determined. The internal loading is then determined as a function of P (M") and its partial derivative with respect to P (M") is determined. Castigliano’s second theorem is then applied to obtain the desired displacement (rotation).

! = aN a 0N0P b LAE ! = L

0 M a 0M0P b dxEI

u = L L

0 M a 0M0M " b dxEI

786 CHAPTER 14 ENERGY METHODS

REVIEW PROBLEMS

14–149. The A992 steel bars are pin connected at B and C. If they each have a diameter of 30 mm, determine the slope at E.

300 N!m

A D

3 m 2 m 2 m 3 m

Prob. 14–149

14–150. The steel chisel has a diameter of 0.5 in. and a length of 10 in. It is struck by a hammer that weighs 3 lb, and at the instant of impact it is moving at 12 ft>s. Determine the maximum compressive stress in the chisel, assuming that 80% of the impacting energy goes into the chisel. Est = 29(103) ksi, sY = 100 ksi.

10 in.

Prob. 14–150

14–146. Determine the total axial and bending strain energy in the A992 steel beam. A = 2300 mm2, I = 9.5(106) mm4.

10 m

1.5 kN/m

15 kN

Prob. 14–146

14–147. The 200-kg block D is dropped from a height h = 1 m onto end C of the A992 steel W200 * 36 overhang beam. If the spring at B has a stiffness k = 200 kN>m, determine the maximum bending stress developed in the beam.

*14–148. Determine the maximum height h from which the 200-kg block D can be dropped without causing the A992 steel W200 * 36 overhang beam to yield. The spring at B has a stiffness k = 200 kN>m.

A B

2 m4 m

Probs. 14–147/148

14–154. The cantilevered beam is subjected to a couple moment M0 applied at its end. Determine the slope of the beam at B. EI is constant. Use the method of virtual work.

14–155. Solve Prob. 14–154 using Castigliano’s theorem.

L B

A M0

Probs. 14–154/155

*14–156. Determine the slope and displacement at point C. EI is constant.

a 2

B C

Prob. 14–156

14–157. Determine the displacement at B. EI is constant.

L— 2

Prob. 14–157

14–151. Determine the total axial and bending strain energy in the A992 structural steel W8 * 58 beam.

5 kip

10 ft 3 kip10 ft

30!

Prob. 14–151

*14–152. Determine the vertical displacement of joint C. The truss is made from A992 steel rods each having a diameter of 1 in.

14–153. Determine the horizontal displacement of joint B. The truss is made from A992 steel rods each having a diameter of 1 in.

6 ft

60! 60!

12 kip

Probs. 14–152/153

REVIEW PROBLEMS 787

A.1 Centroid of an Area The centroid of an area refers to the point that defines the geometric center for the area. If the area has an arbitrary shape, as shown in Fig. A–1 a , the x and y coordinates defining the location of the centroid C are determined using the formulas

x = LA x dA

LA dA y = LA

y dA

LA dA (A–1)

The numerators in these equations are formulations of the “moment” of the area element dA about the y and the x axis, respectively, Fig. A–1 b ; the denominators represent the total area A of the shape.

(a)

_ y

_ x

Fig. A–1

(b)

x x

788

Geometric Properties of an Area A

APPENDIX

A.1 CENTROID OF AN AREA 789

The location of the centroid for some areas may be partially or completely specified by using symmetry conditions. In cases where the area has an axis of symmetry, the centroid for the area will lie along this axis. For example, the centroid C for the area shown in Fig. A–2 must lie along the y axis, since for every elemental area dA a distance +x to the right of the y axis, there is an identical element a distance -x to the left. The total moment for all the elements about the axis of symmetry will therefore cancel; that is, 1x dA = 0 ( Eq. A–1 ), so that x = 0. In cases where a shape has two axes of symmetry, it follows that the centroid lies at the intersection of these axes, Fig. A–3 . Based on the principle of symmetry, or using Eq. A–1 , the locations of the centroid for common area shapes are listed on the inside front cover.

Composite Areas. Often an area can be sectioned or divided into several parts having simpler shapes. Provided the area and location of the centroid of each of these “composite shapes” are known, one can eliminate the need for integration to determine the centroid for the entire area. In this case, equations analogous to Eq. A–1 must be used, except that finite summation signs replace the integrals; i.e.,

x = !x~A !A

y = !y~A

!A (A–2)

Here x~ and y~ represent the algebraic distances or x, y coordinates for the centroid of each composite part, and !A represents the sum of the areas of the composite parts or simply the total area . In particular, if a hole, or a geometric region having no material, is located within a composite part, the hole is considered as an additional composite part having a negative area. Also, as discussed above, if the total area is symmetrical about an axis, the centroid of the area lies on the axis.

The following example illustrates application of Eq. A–2 .

"x #x

Fig. A–2

Fig. A–3

790 APPENDIX A GEOMETR IC PROPERT IES OF AN AREA

Locate the centroid C of the cross-sectional area for the T-beam shown in Fig. A–4 a .

SOLUTION I The y axis is placed along the axis of symmetry so that x = 0, Fig. A–4 a . To obtain y we will establish the x axis (reference axis) through the base of the area. The area is segmented into two rectangles as shown, and the centroidal location y! for each is established. Applying Eq. A–2 , we have

y = !y~A !A

= [5 in.](10 in.)(2 in.) + [11.5 in.](3 in.)(8 in.)

(10 in.)(2 in.) + (3 in.)(8 in.)

= 8.55 in. Ans.

SOLUTION II Using the same two segments, the x axis can be located at the top of the area, Fig. A–4 b . Here

y = !y~A !A

= [-1.5 in.](3 in.)(8 in.) + [-8 in.](10 in.)(2 in.)

(3 in.)(8 in.) + (10 in.)(2 in.)

= -4.45 in. Ans.

The negative sign indicates that C is located below the x axis, which is to be expected. Also note that from the two answers 8.55 in. + 4.45 in. = 13.0 in.,which is the depth of the beam.

SOLUTION III It is also possible to consider the cross-sectional area to be one large rectangle less two small rectangles shown shaded in Fig. A–4 c . Here we have

y = !y~A !A

= [6.5 in.](13 in.)(8 in.) - 2[5 in.](10 in.)(3 in.)

(13 in.)(8 in.) - 2(10 in.)(3 in.)

= 8.55 in. Ans.

EXAMPLE A.1

2 in.

5 in.

11.5 in.10 in.

3 in.

8 in.

(a)

_ y

2 in.

"1.5 in.

"8 in.

10 in.

3 in.

(b)

_ y

3 in.

5 in. 6.5 in.

10 in. 13 in.

8 in.

(c)

3 in.

_ y

Fig. A–4

A.2 MOMENT OF INERTIA FOR AN AREA 791

A.2 Moment of Inertia for an Area The moment of inertia of an area often appears in formulas used in mechanics of materials. It is a geometric property that is calculated about an axis, and for the x and y axes shown in Fig. A–5 , it is defined as

Ix = LAy2 dA

Iy = LAx2 dA (A–3)

These integrals have no physical meaning, but they are so named because they are similar to the formulation of the moment of inertia of a mass, which is a dynamical property of matter.

We can also calculate the moment of inertia of an area about the pole O or z axis, Fig. A–5 . This is referred to as the polar moment of inertia ,

JO = LAr2 dA = Ix + Iy (A–4)

Here r is the perpendicular distance from the pole ( z axis) to the element dA . The relationship between JO and Ix , Iy is possible since r2 = x2 + y2, Fig. A–5 .

From the above formulations it is seen that Ix , Iy , and JO will always be positive , since they involve the product of distance squared and area. Furthermore, the units for moment of inertia involve length raised to the fourth power, e.g., m4, mm4, or ft4, in4.

Using the above equations, the moments of inertia for some common area shapes have been calculated about their centroidal axes and are listed on the inside front cover.

Parallel-Axis Theorem for an Area. If the moment of inertia for an area is known about a centroidal axis, we can determine the moment of inertia of the area about a corresponding parallel axis using the parallel-axis theorem . To derive this theorem, consider finding the moment of inertia of the shaded area shown in Fig. A–6 about the x axis. In this case, a differential element dA is located at the arbitrary distance y ! from the centroidal x ! axis, whereas the fixed distance between the

Fig. A–5

y¿dA

x¿

dx C

y¿

Fig. A–6

792 APPENDIX A GEOMETR IC PROPERT IES OF AN AREA

parallel x and x ! axes is dy . Since the moment of inertia of dA about the x axis is dIx = (y ! + dy)2 dA, then for the entire area,

Ix = LA (y ! + dy)2 dA = LAy !2 dA + 2dy LAy ! dA + dy 2LA dA The first term on the right represents the moment of inertia of the area about the x ! axis, I x! . The second term is zero since the x ! axis passes through the area’s centroid C , that is, 1y ! dA = y !A = 0 since y ! = 0. The final result is therefore

Ix = I x! + Ady 2 (A–5)

A similar expression can be written for Iy, that is,

Iy = I y! + Adx 2 (A–6)

And finally, for the polar moment of inertia about an axis perpendicular to the x – y plane and passing through the pole O ( z axis), Fig. A–6 , we have

JO = JC + Ad 2 (A–7)

The form of each of the above equations states that the moment of inertia of an area about an axis is equal to the area’s moment of inertia about a parallel axis passing through the “centroid” plus the product of the area and the square of the perpendicular distance between the axes.

Composite Areas. Many cross-sectional areas consist of a series of connected simpler shapes, such as rectangles, triangles, and semicircles. In order to properly determine the moment of inertia of such an area about a specified axis, it is first necessary to divide the area into its composite parts and indicate the perpendicular distance from the axis to the parallel centroidal axis for each part. Using the table on the inside front cover of the book, the moment of inertia of each part is determined about the centroidal axis. If this axis does not coincide with the specified axis, the parallel-axis theorem, I = I + Ad 2, should be used to determine the moment of inertia of the part about the specified axis. The moment of inertia of the entire area about this axis is then determined by summing the results of its composite parts. In particular, if a composite part has a “hole,” the moment of inertia for the composite is found by “subtracting” the moment of inertia for the hole from the moment of inertia of the entire area including the hole.

A.2 MOMENT OF INERTIA FOR AN AREA 793

EXAMPLE A.2

Determine the moment of inertia of the cross-sectional area of the T-beam shown in Fig. A–7 a about the centroidal x ! axis.

3 in.

13 in.

3 in.

(b)

x¿

5 in. 8.55 in.10 in.

2 in.

4.45 in.

6.5 in.

x¿ C

5 in. 8.55 in.10 in.

8 in.

(a)

2 in.

4.45 in. 1.5 in. 1.5 in.

Fig. A–7

SOLUTION I The area is segmented into two rectangles as shown in Fig. A–7 a , and the distance from the x ! axis and each centroidal axis is determined. Using the table on the inside front cover, the moment of inertia of a rectangle about its centroidal axis is I = 112 bh3. Applying the parallel- axis theorem, Eq. A–5 , to each rectangle and adding the results, we have

I = "(I x! + Ady 2)

= c 1 12

(2 in.)(10 in.)3 + (2 in.)(10 in.)(8.55 in. - 5 in.)2 d + c 1

12 (8 in.)(3 in.)3 + (8 in.)(3 in.)(4.45 in. - 1.5 in.)2 d

I = 646 in4 Ans.

SOLUTION II The area can be considered as one large rectangle less two small rectangles, shown shaded in Fig. A–7 b . We have

I = "(I x! + Ady 2)

= c 1 12

(8 in.)(13 in.)3 + (8 in.)(13 in.)(8.55 in. - 6.5 in.)2 d - 2 c 1

12 (3 in.)(10 in.)3 + (3 in.)(10 in.)(8.55 in. - 5 in.)2 d

I = 646 in4 Ans.

794 APPENDIX A GEOMETR IC PROPERT IES OF AN AREA

EXAMPLE A.3

Determine the moments of inertia of the beam’s cross-sectional area shown in Fig. A–8 a about the x and y centroidal axes.

SOLUTION The cross section can be considered as three composite rectangular areas A, B , and D shown in Fig. A–8 b . For the calculation, the centroid of each of these rectangles is located in the figure. From the table on the inside front cover, the moment of inertia of a rectangle about its centroidal axis is I = 112 bh3. Hence, using the parallel-axis theorem for rectangles A and D , the calculations are as follows:

Rectangle A :

400 mm

100 mm

600 mm

100 mm

(a)

300 mm

100 mm

(b)

D200 mm

250 mm

200 mm 250 mm300 mm

Fig. A–8

Ix = Ix! + Ady 2 = 1

12 (100 mm)(300 mm)3 + (100 mm)(300 mm)(200 mm)2

= 1.425(109) mm4

Iy = Iy! + Adx 2 = 1

12 (300 mm)(100 mm)3 + (100 mm)(300 mm)(250 mm)2

= 1.90(109) mm4

Rectangle B :

Ix = 1

12 (600 mm)(100 mm)3 = 0.05(109) mm4

Iy = 1

12 (100 mm)(600 mm)3 = 1.80(109) mm4

Rectangle D :

Ix = Ix! + Ady 2 = 1

12 (100 mm)(300 mm)3 + (100 mm)(300 mm)(200 mm)2

= 1.425(109) mm4

Iy = Iy! + Adx 2 = 1

12 (300 mm)(100 mm)3 + (100 mm)(300 mm)(250 mm)2

= 1.90(109) mm4

The moments of inertia for the entire cross section are thus

Ix = 1.425(109) + 0.05(109) + 1.425(109)

= 2.90(109) mm4 Ans.

Iy = 1.90(109) + 1.80(109) + 1.90(109)

= 5.60(109) mm4 Ans.

A.3 PRODUCT OF INERTIA FOR AN AREA 795

A.3 Product of Inertia for an Area In general, the moment of inertia for an area is different for every axis about which it is computed. In some applications of mechanical or structural design it is necessary to know the orientation of those axes that give, respectively, the maximum and minimum moments of inertia for the area. The method for determining this is discussed in Sec. A.4. To use this method, however, one must first determine the product of inertia for the area as well as its moments of inertia for given x, y axes.

The product of inertia for the area A shown in Fig. A–9 is defined as

Ixy = LAxy dA (A–8) Like the moment of inertia, the product of inertia has units of length

raised to the fourth power, e.g., m4, mm4 or ft4, in4. However, since x or y may be a negative quantity, while dA is always positive. The product of inertia may be positive, negative, or zero, depending on the location and orientation of the coordinate axes. For example, the product of inertia Ixy for an area will be zero if either the x or y axis is an axis of symmetry for the area. To show this, consider the shaded area in Fig. A–10 , where for every element dA located at point ( x, y ) there is a corresponding element dA located at (x, -y), Since the products of inertia for these elements are, respectively, xy dA and -xy dA , their algebraic sum or the integration of all the elements of area chosen in this way will cancel each other. Consequently, the product of inertia for the total area becomes zero.

x y

Fig. A–10

Fig. A–9

796 APPENDIX A GEOMETR IC PROPERT IES OF AN AREA

Parallel-Axis Theorem. Consider the shaded area shown in Fig. A–11 , where x ! and y ! represent a set of centroidal axes, and x and y represent a corresponding set of parallel axes. Since the product of inertia of dA with respect to the x and y axes is dIxy = (x ! + dx)(y ! + dy) dA , then for the entire area,

Ixy = LA (x ! + dx)(y ! + dy) dA = LAx !y ! dA + dx LAy ! dA + dy LAx ! dA + dxdy LA dA

The first term on the right represents the product of inertia of the area with respect to the centroidal axis, I x!y! . The second and third terms are zero since the moments of the area are taken about the centroidal axis. Realizing that the fourth integral represents the total area A , we therefore have

Ixy = I x!y! + Adxdy (A–9)

The similarity between this equation and the parallel-axis theorem for moments of inertia should be noted. In particular, it is important that here the algebraic signs for dx and dy be maintained when applying Eq. A–9 .

y¿

x¿

y¿

Fig. A–11

A.3 PRODUCT OF INERTIA FOR AN AREA 797

EXAMPLE A.4

Determine the product of inertia of the beam’s cross-sectional area, shown in Fig. A–12 a , about the x and y centroidal axes.

400 mm

100 mm

600 mm

100 mm

(a)

300 mm

100 mm

(b)

D200 mm

250 mm

200 mm 250 mm300 mm

Fig. A–12

SOLUTION As in Example A.3 , the cross section can be considered as three composite rectangular areas A, B , and D , Fig. A–12 b . The coordinates for the centroid of each of these rectangles are shown in the figure. Due to symmetry, the product of inertia of each rectangle is zero about a set of x !, y ! axes that pass through the rectangle’s centroid. Hence, application of the parallel-axis theorem to each of the rectangles yields

Rectangle A : Ixy = I x!y! + Adxdy = 0 + (300 mm)(100 mm)(-250 mm)(200 mm) = -1.50(109) mm4

Rectangle B : Ixy = I x!y! + Adxdy = 0 + 0 = 0

Rectangle D : Ixy = I x!y! + Adxdy = 0 + (300 mm)(100 mm)(250 mm)(-200 mm) = -1.50(109) mm4

The product of inertia for the entire cross section is thus

Ixy = [-1.50(109) mm4] + 0 + [-1.50(109) mm4] = -3.00(109) mm4 Ans.

798 APPENDIX A GEOMETR IC PROPERT IES OF AN AREA

A.4 Moments of Inertia for an Area about Inclined Axes

In mechanical or structural design, it is sometimes necessary to calculate the moments and product of inertia Ix!, Iy! and Ix!y! for an area with respect to a set of inclined x ! and y ! axes when the values for u, Ix, Iy, and Ixy are known . As shown in Fig. A–13 , the coordinates to the area element dA from each of the two coordinate systems are related by the transformation equations

x ! = x cos u + y sin u

y ! = y cos u - x sin u

Using these equations, the moments and product of inertia of dA about the x ! and y ! axes become

dIx! = y !2 dA = (y cos u - x sin u)2 dA

dIy! = x !2 dA = (x cos u + y sin u)2 dA

dIx!y! = x !y ! dA = (x cos u + y sin u)(y cos u - x sin u) dA

Expanding each expression and integrating, realizing that Ix = 1y2 dA , Iy = 1x2 dA , and Ixy = 1xy dA , we obtain

Ix! = Ix cos2 u + Iy sin2 u - 2Ixy sin u cos u

Iy! = Ix sin2 u + Iy cos2 u + 2Ixy sin u cos u

Ix!y! = Ix sin u cos u - Iy sin u cos u + Ixy(cos2 u - sin2 u)

These equations may be simplified by using the trigonometric identities sin 2u = 2 sin u cos u and cos 2u = cos2 u - sin2 u, in which case

Ix! = Ix + Iy

2 +

Ix - Iy 2

cos 2u - Ixy sin 2u

Iy! = Ix + Iy

2 -

Ix - Iy 2

cos 2u + Ixy sin 2u (A–10)

Ix!y! = Ix - Iy

2 sin 2u + Ixy cos 2u

x¿

y¿

x¿ y cos u

x sin u

y sin u x cos u

Fig. A–13

A.4 MOMENTS OF INERTIA FOR AN AREA ABOUT INCLINED AXES 799

Principal Moments of Inertia. Note that Ix!, Iy!, and Ix!y! depend on the angle of inclination, u, of the x!, y ! axes. We will now determine the orientation of these axes about which the moments of inertia for the area, Ix! and Iy!, are maximum and minimum. This particular set of axes is called the principal axes of inertia for the area, and the corresponding moments of inertia with respect to these axes are called the principal moments of inertia . In general, there is a set of principal axes for every chosen origin O ; however, in mechanics of materials the area’s centroid is the most important location for O .

The angle u = up, which defines the orientation of the principal axes for the area, can be found by differentiating the first of Eq. A–10 with respect to u and setting the result equal to zero. Thus,

dIx! du

= -2¢ Ix - Iy 2 ≤ sin 2u - 2Ixy cos 2u = 0

Therefore, at u = up,

tan 2up = -Ixy

(Ix - Iy)>2 (A–11) This equation has two roots, up1 and up2, which are 90" apart and so specify the inclination of each principal axis.

The sine and cosine of 2up1 and 2up2 can be obtained from the triangles shown in Fig. A–14 , which are based on Eq. A–11 . If these trigonometric relations are substituted into the first or second of Eq. A–10 and simplified, the result is

Imax min

= Ix + Iy

2 { C¢ Ix - Iy2 ≤2 + Ixy 2 (A–12)

Depending on the sign chosen, this result gives the maximum or minimum moment of inertia for the area. Furthermore, if the above trigonometric relations for up1 and up2 are substituted into the third of Eq. A–10 , it will be seen that Ix!y! = 0; that is, the product of inertia with respect to the principal axes is zero . Since it was indicated in Sec. A.3 that the product of inertia is zero with respect to any symmetrical axis, it therefore follows that any symmetrical axis and the one perpendicular to it represent principal axes of inertia for the area . Also, notice that the equations derived in this section are similar to those for stress and strain transformation developed in Chapters 9 and 10 , respectively.

Ixy

2up1

2up2

!Ixy

Ixy

Ix ! Iy 2

" Ixy 2

Ix ! Iy 2

Fig. A–14

800 APPENDIX A GEOMETR IC PROPERT IES OF AN AREA

EXAMPLE A.5

Determine the principal moments of inertia for the beam’s cross- sectional area shown in Fig. A–15 with respect to an axis passing through the centroid C .

SOLUTION The moments and product of inertia of the cross section with respect to the x, y axes have been determined in Examples A.3 and A.4 . The results are

Ix = 2.90(109) mm4 Iy = 5.60(109) mm4 Ixy = -3.00(109) mm4

Using Eq. A–11 , the angles of inclination of the principal axes x! and y! are

tan 2up = -Ixy

(Ix - Iy)>2 = 3.00(109)[2.90(109) - 5.60(109)]>2 = -2.22 2up1 = 114.2" and 2up2 = -65.8"

Thus, as shown in Fig. A–15 ,

up1 = 57.1" and up2 = -32.9"

The principal moments of inertia with respect to the x ! and y ! axes are determined by using Eq. A–12 .

400 mm

100 mm

600 mm

100 mm C

y¿

x¿

up2 # $32.9"

up1 # 57.1"

Fig. A–15

Imax min

= Ix + Iy

2 { C¢ Ix - Iy2 ≤2 + Ixy 2

= 2.90(109) + 5.60(109)

2 { CJ2.90(109) - 5.60(109)2 R 2 + [-3.00(109)]2

= 4.25(109) { 3.29(109)

Imax = 7.54(109) mm4 Imin = 0.960(109) mm4 Ans.

Specifically, the maximum moment of inertia, Imax = 7.54(109) mm4, occurs with respect to the x ! axis (major axis), since by inspection most of the cross-sectional area is farthest away from this axis. To show this, substitute the data with u = 57.1" into the first of Eq. A–10 .

A.5 MOHR’S CIRCLE FOR MOMENTS OF INERTIA 801

A.5 Mohr’s Circle for Moments of Inertia Equations A–10 through A–12 have a semi-graphical solution that is convenient to use and generally easy to remember. Squaring the first and third of Eq. A–10 and adding, it is found that

¢ Ix! - Ix + Iy2 ≤2 + Ix!y! 2 = ¢ Ix - Iy2 ≤2 + Ixy 2 (A–13) In any given problem, Ix! and Ix!y! are variables , and Ix, Iy, and Ixy are known constants . Thus, the above equation may be written in compact form as

(Ix! - a)2 + Ix!y! 2 = R2

When this equation is plotted, the resulting graph represents a circle of radius

R = C ¢ Ix - Iy2 ≤2 + Ixy 2 and having its center located at point ( a , 0), where a = (Ix + Iy)>2. The circle so constructed is called Mohr’s circle . Its application is similar to that used for stress and strain transformation developed in Chapters 9 and 10 , respectively.

Procedure for Analysis

The main purpose for using Mohr’s circle here is to have a convenient means of transforming Ix, Iy, and Ixy into the principal moments of inertia for the area. The following procedure provides a method for doing this.

Calculate Ix , Iy , Ixy . Establish the x, y axes for the area, with the origin located at the point P of interest, usually the centroid, and determine Ix, Iy, and Ixy, Fig. A–16 a .

x¿

y¿ Minor axis for principal moment of inertia, Imin

Major axis for principal moment of inertia, Imax

(a)

up1

Fig. A–16

802 APPENDIX A GEOMETR IC PROPERT IES OF AN AREA

Procedure for Analysis (continued)

Construct the Circle. Establish a rectangular coordinate system such that the horizontal axis represents the moment of inertia I , and the vertical axis represents the product of inertia Ixy, Fig. A–16 b . Determine the center of the circle, C , which is located a distance (Ix + Iy)>2 from the origin, and plot the “reference point” A having coordinates (Ix , Ixy). By definition, Ix is always positive, whereas Ixy can either be positive or negative. Connect the reference point A with the center of the circle, and determine the distance CA by trigonometry. This distance represents the radius of the circle, Fig. A–16 b . Finally, draw the circle.

Principal Moments of Inertia. The points where the circle intersects the I axis give the values of the principal moments of inertia Imin and Imax. Here the product of inertia will be zero at these points , Fig. A–16 b .

To find the orientation of the major principal axis, determine by trigonometry the angle 2up1, measured from the radius CA to the positive I axis , Fig. A–16 b . This angle represents twice the angle from the x axis to the axis of maximum moment of inertia Imax, Fig. A–16 a . Both the angle on the circle, 2up1, and the angle on the area, up1, must be measured in the same sense , as shown in Fig. A–16 . The minor axis is for minimum moment of inertia Imin , which is always perpendicular to the major axis defining Imax.

Fig. A–16 (cont.)

(b)

2up1

Imin

Imax

Ix " Iy 2

Ix # Iy 2

# Ixy 2

Ix " Iy 2

2Ixy

Ixy

A.5 MOHR’S CIRCLE FOR MOMENTS OF INERTIA 803

Use Mohr’s circle to determine the principal moments of inertia for the beam’s cross-sectional area, shown in Fig. A–17 a , with respect to principal axes passing through the centroid C .

SOLUTION Compute Ix , Iy , Ixy . The moments of inertia and the product of inertia have been determined in Examples A.3 and A.4 with respect to the x, y axes shown in Fig. A–17 a . The results are Ix = 2.90(109) mm4, Iy = 5.60(109) mm4, and Ixy = -3.00(109) mm4.

Construct the Circle. The I and Ixy axes are shown in Fig. A–17 b . The center of the circle, C , lies at a distance (Ix + Iy)>2 = (2.90 + 5.60)>2 = 4.25 from the origin. When the reference point A (2.90, -3.00) is connected to point C , the radius CA is determined from the shaded triangle CBA using the Pythagorean theorem:

CA = 2(1.35)2 + (-3.00)2 = 3.29 The circle is constructed in Fig. A–17 c .

Principal Moments of Inertia. The circle intersects the I axis at points (7.54, 0) and (0.960, 0). Hence

Imax = 7.54(109) mm4 Ans.

Imin = 0.960(109) mm4 Ans.

As shown in Fig. A–17 c , the angle 2up1 is determined from the circle by measuring counterclockwise from CA to the positive I axis. Hence,

2up1 = 180! - tan -1a ! BA !

! BC ! b = 180! - tan-1a 3.00

1.35 b = 114.2!

up1 = 57.1!

The major principal axis (for Imax = 7.54(109) mm4 ) is therefore oriented at an angle up1 = 57.1!, measured counterclockwise , from the positive x axis. The minor axis is perpendicular to this axis. The results are shown in Fig. A–17 a .

EXAMPLE A.6

400 mm

100 mm

600 mm

100 mm

(a)

y¿

x¿

up1 " 57.1!

Fig. A–17

(b)

4.25

2.90 C

1.35

3.00 B

A(2.90, #3.00)

Ixy(109) mm4

I(109) mm4

(c)

A(2.90, #3.00)

3.29

Imin " 0.960

Imax " 7.54

Ixy(109) mm4

I(109) mm4

2up1

Geometric Properties of Structural Shapes

APPENDIX

Designation

Web

thickness tw

Flange

Area A

Depth d

width bf

thickness tf

x – x axis y – y axis

I S r I S r

in. : lb>ft in2 in. in. in. in. in4 in3 in. in4 in3 in. W24 * 104 30.6 24.06 0.500 12.750 0.750 3100 258 10.1 259 40.7 2.91 W24 * 94 27.7 24.31 0.515 9.065 0.875 2700 222 9.87 109 24.0 1.98 W24 * 84 24.7 24.10 0.470 9.020 0.770 2370 196 9.79 94.4 20.9 1.95 W24 * 76 22.4 23.92 0.440 8.990 0.680 2100 176 9.69 82.5 18.4 1.92 W24 * 68 20.1 23.73 0.415 8.965 0.585 1830 154 9.55 70.4 15.7 1.87 W24 * 62 18.2 23.74 0.430 7.040 0.590 1550 131 9.23 34.5 9.80 1.38 W24 * 55 16.2 23.57 0.395 7.005 0.505 1350 114 9.11 29.1 8.30 1.34 W18 * 65 19.1 18.35 0.450 7.590 0.750 1070 117 7.49 54.8 14.4 1.69 W18 * 60 17.6 18.24 0.415 7.555 0.695 984 108 7.47 50.1 13.3 1.69 W18 * 55 16.2 18.11 0.390 7.530 0.630 890 98.3 7.41 44.9 11.9 1.67 W18 * 50 14.7 17.99 0.355 7.495 0.570 800 88.9 7.38 40.1 10.7 1.65 W18 * 46 13.5 18.06 0.360 6.060 0.605 712 78.8 7.25 22.5 7.43 1.29 W18 * 40 11.8 17.90 0.315 6.015 0.525 612 68.4 7.21 19.1 6.35 1.27 W18 * 35 10.3 17.70 0.300 6.000 0.425 510 57.6 7.04 15.3 5.12 1.22 W16 * 57 16.8 16.43 0.430 7.120 0.715 758 92.2 6.72 43.1 12.1 1.60 W16 * 50 14.7 16.26 0.380 7.070 0.630 659 81.0 6.68 37.2 10.5 1.59 W16 * 45 13.3 16.13 0.345 7.035 0.565 586 72.7 6.65 32.8 9.34 1.57 W16 * 36 10.6 15.86 0.295 6.985 0.430 448 56.5 6.51 24.5 7.00 1.52 W16 * 31 9.12 15.88 0.275 5.525 0.440 375 47.2 6.41 12.4 4.49 1.17 W16 * 26 7.68 15.69 0.250 5.500 0.345 301 38.4 6.26 9.59 3.49 1.12 W14 * 53 15.6 13.92 0.370 8.060 0.660 541 77.8 5.89 57.7 14.3 1.92 W14 * 43 12.6 13.66 0.305 7.995 0.530 428 62.7 5.82 45.2 11.3 1.89 W14 * 38 11.2 14.10 0.310 6.770 0.515 385 54.6 5.87 26.7 7.88 1.55 W14 * 34 10.0 13.98 0.285 6.745 0.455 340 48.6 5.83 23.3 6.91 1.53 W14 * 30 8.85 13.84 0.270 6.730 0.385 291 42.0 5.73 19.6 5.82 1.49 W14 * 26 7.69 13.91 0.255 5.025 0.420 245 35.3 5.65 8.91 3.54 1.08 W14 * 22 6.49 13.74 0.230 5.000 0.335 199 29.0 5.54 7.00 2.80 1.04

Wide-Flange Sections or W Shapes FPS Units

Web thickness

Flange

Area

A Depth

d width

bf thickness

x – x axis y – y axis

Designation I S r I S r

in. : lb>ft in2 in. in. in. in. in4 in3 in. in4 in3 in. W12 * 87 25.6 12.53 0.515 12.125 0.810 740 118 5.38 241 39.7 3.07 W12 * 50 14.7 12.19 0.370 8.080 0.640 394 64.7 5.18 56.3 13.9 1.96 W12 * 45 13.2 12.06 0.335 8.045 0.575 350 58.1 5.15 50.0 12.4 1.94 W12 * 26 7.65 12.22 0.230 6.490 0.380 204 33.4 5.17 17.3 5.34 1.51 W12 * 22 6.48 12.31 0.260 4.030 0.425 156 25.4 4.91 4.66 2.31 0.847 W12 * 16 4.71 11.99 0.220 3.990 0.265 103 17.1 4.67 2.82 1.41 0.773 W12 * 14 4.16 11.91 0.200 3.970 0.225 88.6 14.9 4.62 2.36 1.19 0.753 W10 * 100 29.4 11.10 0.680 10.340 1.120 623 112 4.60 207 40.0 2.65 W10 * 54 15.8 10.09 0.370 10.030 0.615 303 60.0 4.37 103 20.6 2.56 W10 * 45 13.3 10.10 0.350 8.020 0.620 248 49.1 4.32 53.4 13.3 2.01 W10 * 39 11.5 9.92 0.315 7.985 0.530 209 42.1 4.27 45.0 11.3 1.98 W10 * 30 8.84 10.47 0.300 5.810 0.510 170 32.4 4.38 16.7 5.75 1.37 W10 * 19 5.62 10.24 0.250 4.020 0.395 96.3 18.8 4.14 4.29 2.14 0.874 W10 * 15 4.41 9.99 0.230 4.000 0.270 68.9 13.8 3.95 2.89 1.45 0.810 W10 * 12 3.54 9.87 0.190 3.960 0.210 53.8 10.9 3.90 2.18 1.10 0.785 W8 * 67 19.7 9.00 0.570 8.280 0.935 272 60.4 3.72 88.6 21.4 2.12 W8 * 58 17.1 8.75 0.510 8.220 0.810 228 52.0 3.65 75.1 18.3 2.10 W8 * 48 14.1 8.50 0.400 8.110 0.685 184 43.3 3.61 60.9 15.0 2.08 W8 * 40 11.7 8.25 0.360 8.070 0.560 146 35.5 3.53 49.1 12.2 2.04 W8 * 31 9.13 8.00 0.285 7.995 0.435 110 27.5 3.47 37.1 9.27 2.02 W8 * 24 7.08 7.93 0.245 6.495 0.400 82.8 20.9 3.42 18.3 5.63 1.61 W8 * 15 4.44 8.11 0.245 4.015 0.315 48.0 11.8 3.29 3.41 1.70 0.876 W6 * 25 7.34 6.38 0.320 6.080 0.455 53.4 16.7 2.70 17.1 5.61 1.52 W6 * 20 5.87 6.20 0.260 6.020 0.365 41.4 13.4 2.66 13.3 4.41 1.50 W6 * 16 4.74 6.28 0.260 4.030 0.405 32.1 10.2 2.60 4.43 2.20 0.966 W6 * 15 4.43 5.99 0.230 5.990 0.260 29.1 9.72 2.56 9.32 3.11 1.46 W6 * 12 3.55 6.03 0.230 4.000 0.280 22.1 7.31 2.49 2.99 1.50 0.918 W6 * 9 2.68 5.90 0.170 3.940 0.215 16.4 5.56 2.47 2.19 1.11 0.905

WIDE-FLANGE SECTIONS OR W SHAPES FPS UNITS 805

806 APPENDIX B GEOMETR IC PROPERT IES OF STRUCTURAL SHAPES

xx d

American Standard Channels or C Shapes FPS Units

Web

thickness tw

Flange

Area A

Depth d

width bf

thickness tf

x – x axis y – y axis

Designation I S r I S r

in. : lb>ft in2 in. in. in. in. C15 * 50 14.7 15.00 0.716 11/16 3.716 C15 * 40 11.8 15.00 0.520 1/2 3.520 312 C15 * 33.9 9.96 15.00 0.400 3/8 3.400 C12 * 30 8.82 12.00 0.510 1/2 3.170 318 C12 * 25 7.35 12.00 0.387 3/8 C12 * 20.7 6.09 12.00 0.282 C10 * 30 8.82 10.00 0.673 11/16 C10 * 25 7.35 10.00 0.526 1/2 2.886 278 C10 * 20 5.88 10.00 0.379 3/8 2.739 234 C10 * 15.3 4.49 10.00 0.240 1/4 2.600 C9 * 20 5.88 9.00 0.448 7/16 2.648 C9 * 15 4.41 9.00 0.285 5/16 2.485 212 C9 * 13.4 3.94 9.00 0.233 1/4 2.433 238 C8 * 18.75 5.51 8.00 0.487 1/2 2.527 C8 * 13.75 4.04 8.00 0.303 5/16 2.343 C8 * 11.5 3.38 8.00 0.220 1/4 2.260 214 C7 * 14.75 4.33 7.00 0.419 7/16 2.299 C7 * 12.25 3.60 7.00 0.314 5/16 2.194 C7 * 9.8 2.87 7.00 0.210 3/16 2.090 218 C6 * 13 3.83 6.00 0.437 7/16 2.157 218 C6 * 10.5 3.09 6.00 0.314 5/16 C6 * 8.2 2.40 6.00 0.200 3/16 1.920 178 C5 * 9 2.64 5.00 0.325 5/16 1.885 178 C5 * 6.7 1.97 5.00 0.190 3/16 1.750 134 C4 * 7.25 2.13 4.00 0.321 5/16 1.721 C4 * 5.4 1.59 4.00 0.184 3/16 1.584 C3 * 6 1.76 3.00 0.356 3/8 1.596 C3 * 5 1.47 3.00 0.258 1/4 1.498 C3 * 4.1 1.21 3.00 0.170 3/16 1.410 in4 in3 in. in4 in3 in. 3 34 0.650 5/8 404 53.8 5.24 11.0 3.78 0.867 0.650 5/8 349 46.5 5.44 9.23 3.37 0.886 3 38 0.650 5/8 315 42.0 5.62 8.13 3.11 0.904 0.501 1/2 162 27.0 4.29 5.14 2.06 0.763 3.047 3 0.501 1/2 144 24.1 4.43 4.47 1.88 0.780 5/16 2.942 3 0.501 1/2 129 21.5 4.61 3.88 1.73 0.799 3.033 3 0.436 7/16 103 20.7 3.42 3.94 1.65 0.669 0.436 7/16 91.2 18.2 3.52 3.36 1.48 0.676 0.436 7/16 78.9 15.8 3.66 2.81 1.32 0.692 258 0.436 7/16 67.4 13.5 3.87 2.28 1.16 0.713 258 0.413 7/16 60.9 13.5 3.22 2.42 1.17 0.642 0.413 7/16 51.0 11.3 3.40 1.93 1.01 0.661 0.413 7/16 47.9 10.6 3.48 1.76 0.962 0.669 212 0.390 3/8 44.0 11.0 2.82 1.98 1.01 0.599 238 0.390 3/8 36.1 9.03 2.99 1.53 0.854 0.615 0.390 3/8 32.6 8.14 3.11 1.32 0.781 0.625 214 0.366 3/8 27.2 7.78 2.51 1.38 0.779 0.564 214 0.366 3/8 24.2 6.93 2.60 1.17 0.703 0.571 0.366 3/8 21.3 6.08 2.72 0.968 0.625 0.581 0.343 5/16 17.4 5.80 2.13 1.05 0.642 0.525 2.034 2 0.343 5/16 15.2 5.06 2.22 0.866 0.564 0.529 0.343 5/16 13.1 4.38 2.34 0.693 0.492 0.537 0.320 5/16 8.90 3.56 1.83 0.632 0.450 0.489 0.320 5/16 7.49 3.00 1.95 0.479 0.378 0.493 134 0.296 5/16 4.59 2.29 1.47 0.433 0.343 0.450 158 0.296 5/16 3.85 1.93 1.56 0.319 0.283 0.449 158 0.273 1/4 2.07 1.38 1.08 0.305 0.268 0.416 112 0.273 1/4 1.85 1.24 1.12 0.247 0.233 0.410 138 0.273 1/4 1.66 1.10 1.17 0.197 0.202 0.404

Angles Having Equal Legs FPS Units

Size and thickness

Weight per foot Area A

x – x axis y – y axis z – z axis

I S r y I S r x r

in. lb in2 in4 in3 in. in. in4 in3 in. in. in.

L8 * 8 * 1 51.0 15.0 89.0 15.8 2.44 2.37 89.0 15.8 2.44 2.37 1.56 L8 * 8 * 34 38.9 11.4 69.7 12.2 2.47 2.28 69.7 12.2 2.47 2.28 1.58 L8 * 8 * 12 26.4 7.75 48.6 8.36 2.50 2.19 48.6 8.36 2.50 2.19 1.59 L6 * 6 * 1 37.4 11.0 35.5 8.57 1.80 1.86 35.5 8.57 1.80 1.86 1.17 L6 * 6 * 34 28.7 8.44 28.2 6.66 1.83 1.78 28.2 6.66 1.83 1.78 1.17 L6 * 6 * 12 19.6 5.75 19.9 4.61 1.86 1.68 19.9 4.61 1.86 1.68 1.18 L6 * 6 * 38 14.9 4.36 15.4 3.53 1.88 1.64 15.4 3.53 1.88 1.64 1.19 L5 * 5 * 34 23.6 6.94 15.7 4.53 1.51 1.52 15.7 4.53 1.51 1.52 0.975 L5 * 5 * 12 16.2 4.75 11.3 3.16 1.54 1.43 11.3 3.16 1.54 1.43 0.983 L5 * 5 * 38 12.3 3.61 8.74 2.42 1.56 1.39 8.74 2.42 1.56 1.39 0.990 L4 * 4 * 34 18.5 5.44 7.67 2.81 1.19 1.27 7.67 2.81 1.19 1.27 0.778 L4 * 4 * 12 12.8 3.75 5.56 1.97 1.22 1.18 5.56 1.97 1.22 1.18 0.782 L4 * 4 * 38 9.8 2.86 4.36 1.52 1.23 1.14 4.36 1.52 1.23 1.14 0.788 L4 * 4 * 14 6.6 1.94 3.04 1.05 1.25 1.09 3.04 1.05 1.25 1.09 0.795 L312 * 3

1 2 *

1 2 11.1 3.25 3.64 1.49 1.06 1.06 3.64 1.49 1.06 1.06 0.683

L312 * 3 1 2 *

1 2 8.5 2.48 2.87 1.15 1.07 1.01 2.87 1.15 1.07 1.01 0.687

L312 * 3 1 2 *

1 4 5.8 1.69 2.01 0.794 1.09 0.968 2.01 0.794 1.09 0.968 0.694

L3 * 3 * 12 9.4 2.75 2.22 1.07 0.898 0.932 2.22 1.07 0.898 0.932 0.584 L3 * 3 * 38 7.2 2.11 1.76 0.833 0.913 0.888 1.76 0.833 0.913 0.888 0.587 L3 * 3 * 14 4.9 1.44 1.24 0.577 0.930 0.842 1.24 0.577 0.930 0.842 0.592 L212 * 2

1 2 *

1 2 7.7 2.25 1.23 0.724 0.739 0.806 1.23 0.724 0.739 0.806 0.487

L212 * 2 1 2 * 38 5.9 1.73 0.984 0.566 0.753 0.762 0.984 0.566 0.753 0.762 0.487

L212 * 2 1 2 *

1 4 4.1 1.19 0.703 0.394 0.769 0.717 0.703 0.394 0.769 0.717 0.491

L2 * 2 * 38 4.7 1.36 0.479 0.351 0.594 0.636 0.479 0.351 0.594 0.636 0.389 L2 * 2 * 14 3.19 0.938 0.348 0.247 0.609 0.592 0.348 0.247 0.609 0.592 0.391 L2 * 2 * 18 1.65 0.484 0.190 0.131 0.626 0.546 0.190 0.131 0.626 0.546 0.398

y x

ANGLES HAVING EQUAL LEGS FPS UNITS 807

808 APPENDIX B GEOMETR IC PROPERT IES OF STRUCTURAL SHAPES

Wide-Flange Sections or W Shapes FPS Units

Web

thickness tw

Flange

Area

A Depth

d width

bf thickness

x – x axis y – y axis

Designation I S r I S r

mm : kg>m mm2 mm mm mm mm 106 mm4 103 mm3 mm 106 mm4 103 mm3 mm W610 * 155 19 800 611 12.70 324.0 19.0 1 290 4 220 255 108 667 73.9 W610 * 140 17 900 617 13.10 230.0 22.2 1 120 3 630 250 45.1 392 50.2 W610 * 125 15 900 612 11.90 229.0 19.6 985 3 220 249 39.3 343 49.7 W610 * 113 14 400 608 11.20 228.0 17.3 875 2 880 247 34.3 301 48.8 W610 * 101 12 900 603 10.50 228.0 14.9 764 2 530 243 29.5 259 47.8 W610 * 92 11 800 603 10.90 179.0 15.0 646 2 140 234 14.4 161 34.9 W610 * 82 10 500 599 10.00 178.0 12.8 560 1 870 231 12.1 136 33.9 W460 * 97 12 300 466 11.40 193.0 19.0 445 1 910 190 22.8 236 43.1 W460 * 89 11 400 463 10.50 192.0 17.7 410 1 770 190 20.9 218 42.8 W460 * 82 10 400 460 9.91 191.0 16.0 370 1 610 189 18.6 195 42.3 W460 * 74 9 460 457 9.02 190.0 14.5 333 1 460 188 16.6 175 41.9 W460 * 68 8 730 459 9.14 154.0 15.4 297 1 290 184 9.41 122 32.8 W460 * 60 7 590 455 8.00 153.0 13.3 255 1 120 183 7.96 104 32.4 W460 * 52 6 640 450 7.62 152.0 10.8 212 942 179 6.34 83.4 30.9 W410 * 85 10 800 417 10.90 181.0 18.2 315 1 510 171 18.0 199 40.8 W410 * 74 9 510 413 9.65 180.0 16.0 275 1 330 170 15.6 173 40.5 W410 * 67 8 560 410 8.76 179.0 14.4 245 1 200 169 13.8 154 40.2 W410 * 53 6 820 403 7.49 177.0 10.9 186 923 165 10.1 114 38.5 W410 * 46 5 890 403 6.99 140.0 11.2 156 774 163 5.14 73.4 29.5 W410 * 39 4 960 399 6.35 140.0 8.8 126 632 159 4.02 57.4 28.5 W360 * 79 10 100 354 9.40 205.0 16.8 227 1 280 150 24.2 236 48.9 W360 * 64 8 150 347 7.75 203.0 13.5 179 1 030 148 18.8 185 48.0 W360 * 57 7 200 358 7.87 172.0 13.1 160 894 149 11.1 129 39.3 W360 * 51 6 450 355 7.24 171.0 11.6 141 794 148 9.68 113 38.7 W360 * 45 5 710 352 6.86 171.0 9.8 121 688 146 8.16 95.4 37.8 W360 * 39 4 960 353 6.48 128.0 10.7 102 578 143 3.75 58.6 27.5 W360 * 33 4 190 349 5.84 127.0 8.5 82.9 475 141 2.91 45.8 26.4

WIDE-FLANGE SECTIONS OR W SHAPES SI UNITS 809

Wide-Flange Sections or W Shapes SI Units

Web

thickness tw

Flange

Area

A Depth

d width

bf thickness

x – x axis y – y axis

Designation I S r I S r

mm : kg>m mm2 mm mm mm mm 106 mm4 103 mm3 mm 106 mm4 103 mm3 mm W310 * 129 16 500 318 13.10 308.0 20.6 308 1940 137 100 649 77.8 W310 * 74 9 480 310 9.40 205.0 16.3 165 1060 132 23.4 228 49.7 W310 * 67 8 530 306 8.51 204.0 14.6 145 948 130 20.7 203 49.3 W310 * 39 4 930 310 5.84 165.0 9.7 84.8 547 131 7.23 87.6 38.3 W310 * 33 4 180 313 6.60 102.0 10.8 65.0 415 125 1.92 37.6 21.4 W310 * 24 3 040 305 5.59 101.0 6.7 42.8 281 119 1.16 23.0 19.5 W310 * 21 2 680 303 5.08 101.0 5.7 37.0 244 117 0.986 19.5 19.2 W250 * 149 19 000 282 17.30 263.0 28.4 259 1840 117 86.2 656 67.4 W250 * 80 10 200 256 9.40 255.0 15.6 126 984 111 43.1 338 65.0 W250 * 67 8 560 257 8.89 204.0 15.7 104 809 110 22.2 218 50.9 W250 * 58 7 400 252 8.00 203.0 13.5 87.3 693 109 18.8 185 50.4 W250 * 45 5 700 266 7.62 148.0 13.0 71.1 535 112 7.03 95 35.1 W250 * 28 3 620 260 6.35 102.0 10.0 39.9 307 105 1.78 34.9 22.2 W250 * 22 2 850 254 5.84 102.0 6.9 28.8 227 101 1.22 23.9 20.7 W250 * 18 2 280 251 4.83 101.0 5.3 22.5 179 99.3 0.919 18.2 20.1 W200 * 100 12 700 229 14.50 210.0 23.7 113 987 94.3 36.6 349 53.7 W200 * 86 11 000 222 13.00 209.0 20.6 94.7 853 92.8 31.4 300 53.4 W200 * 71 9 100 216 10.20 206.0 17.4 76.6 709 91.7 25.4 247 52.8 W200 * 59 7 580 210 9.14 205.0 14.2 61.2 583 89.9 20.4 199 51.9 W200 * 46 5 890 203 7.24 203.0 11.0 45.5 448 87.9 15.3 151 51.0 W200 * 36 4 570 201 6.22 165.0 10.2 34.4 342 86.8 7.64 92.6 40.9 W200 * 22 2 860 206 6.22 102.0 8.0 20.0 194 83.6 1.42 27.8 22.3 W150 * 37 4 730 162 8.13 154.0 11.6 22.2 274 68.5 7.07 91.8 38.7 W150 * 30 3 790 157 6.60 153.0 9.3 17.1 218 67.2 5.54 72.4 38.2 W150 * 22 2 860 152 5.84 152.0 6.6 12.1 159 65.0 3.87 50.9 36.8 W150 * 24 3 060 160 6.60 102.0 10.3 13.4 168 66.2 1.83 35.9 24.5 W150 * 18 2 290 153 5.84 102.0 7.1 9.19 120 63.3 1.26 24.7 23.5 W150 * 14 1 730 150 4.32 100.0 5.5 6.84 91.2 62.9 0.912 18.2 23.0

810 APPENDIX B GEOMETR IC PROPERT IES OF STRUCTURAL SHAPES

xx d

American Standard Channels or C Shapes SI Units

Web

thickness tw

Flange

x – x axis y – y axis Area

A Depth

d width

bf thickness

tf Designation I S r I S r

mm : kg>m mm2 mm mm mm mm 106 mm4 103 mm3 mm 106 mm4 103 mm3 mm C380 * 74 9 480 381.0 18.20 94.4 16.50 168 882 133 4.58 61.8 22.0 C380 * 60 7 610 381.0 13.20 89.4 16.50 145 761 138 3.84 55.1 22.5 C380 * 50 6 430 381.0 10.20 86.4 16.50 131 688 143 3.38 50.9 22.9 C310 * 45 5 690 305.0 13.00 80.5 12.70 67.4 442 109 2.14 33.8 19.4 C310 * 37 4 740 305.0 9.83 77.4 12.70 59.9 393 112 1.86 30.9 19.8 C310 * 31 3 930 305.0 7.16 74.7 12.70 53.7 352 117 1.61 28.3 20.2 C250 * 45 5 690 254.0 17.10 77.0 11.10 42.9 338 86.8 1.61 27.1 17.0 C250 * 37 4 740 254.0 13.40 73.3 11.10 38.0 299 89.5 1.40 24.3 17.2 C250 * 30 3 790 254.0 9.63 69.6 11.10 32.8 258 93.0 1.17 21.6 17.6 C250 * 23 2 900 254.0 6.10 66.0 11.10 28.1 221 98.4 0.949 19.0 18.1 C230 * 30 3 790 229.0 11.40 67.3 10.50 25.3 221 81.7 1.01 19.2 16.3 C230 * 22 2 850 229.0 7.24 63.1 10.50 21.2 185 86.2 0.803 16.7 16.8 C230 * 20 2 540 229.0 5.92 61.8 10.50 19.9 174 88.5 0.733 15.8 17.0 C200 * 28 3 550 203.0 12.40 64.2 9.90 18.3 180 71.8 0.824 16.5 15.2 C200 * 20 2 610 203.0 7.70 59.5 9.90 15.0 148 75.8 0.637 14.0 15.6 C200 * 17 2 180 203.0 5.59 57.4 9.90 13.6 134 79.0 0.549 12.8 15.9 C180 * 22 2 790 178.0 10.60 58.4 9.30 11.3 127 63.6 0.574 12.8 14.3 C180 * 18 2 320 178.0 7.98 55.7 9.30 10.1 113 66.0 0.487 11.5 14.5 C180 * 15 1 850 178.0 5.33 53.1 9.30 8.87 99.7 69.2 0.403 10.2 14.8 C150 * 19 2 470 152.0 11.10 54.8 8.70 7.24 95.3 54.1 0.437 10.5 13.3 C150 * 16 1 990 152.0 7.98 51.7 8.70 6.33 83.3 56.4 0.360 9.22 13.5 C150 * 12 1 550 152.0 5.08 48.8 8.70 5.45 71.7 59.3 0.288 8.04 13.6 C130 * 13 1 700 127.0 8.25 47.9 8.10 3.70 58.3 46.7 0.263 7.35 12.4 C130 * 10 1 270 127.0 4.83 44.5 8.10 3.12 49.1 49.6 0.199 6.18 12.5 C100 * 11 1 370 102.0 8.15 43.7 7.50 1.91 37.5 37.3 0.180 5.62 11.5 C100 * 8 1 030 102.0 4.67 40.2 7.50 1.60 31.4 39.4 0.133 4.65 11.4 C75 * 9 1 140 76.2 9.04 40.5 6.90 0.862 22.6 27.5 0.127 4.39 10.6 C75 * 7 948 76.2 6.55 38.0 6.90 0.770 20.2 28.5 0.103 3.83 10.4 C75 * 6 781 76.2 4.32 35.8 6.90 0.691 18.1 29.8 0.082 3.32 10.2

ANGLES HAVING EQUAL LEGS SI UNITS 811

y x

Angles Having Equal Legs SI Units

Size and thickness

Mass per

Meter Area

x – x axis y – y axis z–z axis

I S r y I S r x r

mm kg mm2 106 mm4 106 mm3 mm mm 106 mm4 106 mm3 mm mm mm

L203 * 203 * 25.4 75.9 9 680 36.9 258 61.7 60.1 36.9 258 61.7 60.1 39.6 L203 * 203 * 19.0 57.9 7 380 28.9 199 62.6 57.8 28.9 199 62.6 57.8 40.1 L203 * 203 * 12.7 39.3 5 000 20.2 137 63.6 55.5 20.2 137 63.6 55.5 40.4 L152 * 152 * 25.4 55.7 7 100 14.6 139 45.3 47.2 14.6 139 45.3 47.2 29.7 L152 * 152 * 19.0 42.7 5 440 11.6 108 46.2 45.0 11.6 108 46.2 45.0 29.7 L152 * 152 * 12.7 29.2 3 710 8.22 75.1 47.1 42.7 8.22 75.1 47.1 42.7 30.0 L152 * 152 * 9.5 22.2 2 810 6.35 57.4 47.5 41.5 6.35 57.4 47.5 41.5 30.2 L127 * 127 * 19.0 35.1 4 480 6.54 73.9 38.2 38.7 6.54 73.9 38.2 38.7 24.8 L127 * 127 * 12.7 24.1 3 060 4.68 51.7 39.1 36.4 4.68 51.7 39.1 36.4 25.0 L127 * 127 * 9.5 18.3 2 330 3.64 39.7 39.5 35.3 3.64 39.7 39.5 35.3 25.1 L102 * 102 * 19.0 27.5 3 510 3.23 46.4 30.3 32.4 3.23 46.4 30.3 32.4 19.8 L102 * 102 * 12.7 19.0 2 420 2.34 32.6 31.1 30.2 2.34 32.6 31.1 30.2 19.9 L102 * 102 * 9.5 14.6 1 840 1.84 25.3 31.6 29.0 1.84 25.3 31.6 29.0 20.0 L102 * 102 * 6.4 9.8 1 250 1.28 17.3 32.0 27.9 1.28 17.3 32.0 27.9 20.2 L89 * 89 * 12.7 16.5 2 100 1.52 24.5 26.9 26.9 1.52 24.5 26.9 26.9 17.3 L89 * 89 * 9.5 12.6 1 600 1.20 19.0 27.4 25.8 1.20 19.0 27.4 25.8 17.4 L89 * 89 * 6.4 8.6 1 090 0.840 13.0 27.8 24.6 0.840 13.0 27.8 24.6 17.6 L76 * 76 * 12.7 14.0 1 770 0.915 17.5 22.7 23.6 0.915 17.5 22.7 23.6 14.8 L76 * 76 * 9.5 10.7 1 360 0.726 13.6 23.1 22.5 0.726 13.6 23.1 22.5 14.9 L76 * 76 * 6.4 7.3 927 0.514 9.39 23.5 21.3 0.514 9.39 23.5 21.3 15.0 L64 * 64 * 12.7 11.5 1 450 0.524 12.1 19.0 20.6 0.524 12.1 19.0 20.6 12.4 L64 * 64 * 9.5 8.8 1 120 0.420 9.46 19.4 19.5 0.420 9.46 19.4 19.5 12.4 L64 * 64 * 6.4 6.1 766 0.300 6.59 19.8 18.2 0.300 6.59 19.8 18.2 12.5 L51 * 51 * 9.5 7.0 877 0.202 5.82 15.2 16.2 0.202 5.82 15.2 16.2 9.88 L51 * 51 * 6.4 4.7 605 0.146 4.09 15.6 15.1 0.146 4.09 15.6 15.1 9.93 L51 * 51 * 3.2 2.5 312 0.080 2.16 16.0 13.9 0.080 2.16 16.0 13.9 10.1

Simply Supported Beam Slopes and Deflections

Slopes and Deflections of Beams C

APPENDIX

Beam Slope Deflection Elastic Curve

vmaxumax

L 2

umax = -PL2 16EI

vmax = -PL3 48EI

v =

-Px 48EI

(3L2 - 4x2)

0 … x … L >2

a b

u2u1

u1 = -Pab(L + b)

6EIL

u2 = Pab(L + a)

6EIL

v 2 x = a

= -Pba 6EIL

(L2 - b2 - a2) v =

-Pbx 6EIL

(L2 - b2 - x2)

0 … x … a

L u2u1

x M0

u1 = -M0L

6EI

u2 = M0L

3EI

vmax = -M0L2 923EI

at x = 0.5774L v =

-M0x 6EIL

(L2 - x2)

L w

vmaxumax

umax = -wL3 24EI

vmax = -5wL4 384EI

v = -wx 24EI

(x3 - 2Lx2 + L3)

L 2

u1 = -3wL3 128EI

u2 = 7wL3 384EI

v 2 x = L>2 = -5wL

4 768EI

vmax = -0.006563 wL4 EI

at x = 0.4598L

v = -wx

384EI (16x3 - 24Lx2 + 9L3)

0 … x … L >2 v =

-wL 384EI

(8x3 - 24Lx2

+ 17L2 x - L3) L >2 … x 6 L

u1 u2

u1 = -7w0 L3

360EI

u2 = w0 L

45EI

vmax = -0.00652 w0 L

at x = 0.5193L v =

-w0x 360EIL

(3x4 - 10L2x2 + 7L4)

812

22C

Cantilevered Beam Slopes and Deflections

Beam Slope Deflection Elastic Curve

vmax

umax

umax = -PL2 2EI

vmax = -PL3 3EI

v = -Px2 6EI

(3L - x)

vmax

L 2

umax

umax = -PL2 8EI

vmax = -5PL3 48EI

v =

-Px2 12EI

(3L - 2x) 0 … x … L >2 v =

-PL2 48EI

(6x - L) L >2 … x … L

vmax

umax

umax = -wL3

6EI vmax =

-wL4 8EI

v = -wx2 24EI

(x2 - 4Lx + 6L2)

x M0 vmax

v umax

umax = M0L

EI vmax =

M0L 2

2EI v =

M0 x 2

2EI

w v

vmax

L 2

umax

umax = -wL3 48EI

vmax = -7wL4 384EI

v = -wx2 24EI

1x2 - 2Lx + 32 L22 0 … x … L >2

v = -wL3 384EI

(8x - L)

L >2 … x … L

w0 vmax

umax

umax = -w0 L3

24EI vmax =

-w0L4 30EI

v = -w0 x2 120EIL

(10L3 - 10L2x + 5Lx2 - x3)

CANTILEVERED BEAM SLOPES AND DEFLECTIONS 813

814

Solutions and Answers for Preliminary Problems P1–1a.

2a 2a C Cx

CyTB

a+ !MC = 0; get TB

2a a A NA

+S !Fx = 0; get NA + c !Fy = 0; get V A a+ !MA = 0; get MA

P1–1b.

a aa

w(2a)

a+ !MB = 0; get Cy

VA Cy

w(a)

a/2 a/2

+S !Fx = 0; get NA + c !Fy = 0; get V A a+ !MA = 0; get MA

P1–1c.

aaa

By Cy

a+ !MB = 0; get Cy

a/2

+S !Fx = 0; get NA + c !Fy = 0; get V A a+ !MA = 0; get MA

P1–1d.

y x

NA MA

+ a !Fy = 0; get NA + Q !Fx = 0; get V A

a+!MO = 0 or !MA = 0; get MA

SOLUTIONS AND ANSWERS FOR PRELIMINARY PROBLEMS 815

P1–1e.

ByBx

a+!MB = 0; get NC

P Nc

VA NA

+ Q !Fx = 0; get NA + a !Fy = 0; get VA a+!MA = 0; get MA

P1–1f.

2a 3a

+ c !Fy = 0; get Cy (= P) a+ !MB = 0; get Dx

NA MA

+ c !Fy = 0; get NA (= 0) +S !Fx = 0; get V A a+ !MA = 0; get MA

P1–2a.

6 kN

3 kN

2 kN

5 kNFA " 2 kN

FB " 5 kN

V max = 3 kN

P1–2b.

4 kN

5 kN 4 kN

5 kN

6 kN

5 kN

5 kN 10 kN

4 kN

4 kN 8 kN

4 kN

20 kN

FB " 3 kN

FA " 11 kN

FC " 16 kN

V max = 5 kN

816 SOLUTIONS AND ANSWERS FOR PRELIMINARY PROBLEMS

P1–3.

10 kN

6 kN

6 kN2 kN

NAB ! 10 kN

NBC ! 4 kN

NCD ! 6 kN

Fx ! 1 kN NDF ! 1 kN

N max = 10 kN

P1–4.

(8 kN/m)(3 m) ! 24 kN

NA ! 24 kN

P1–5.

20 N 20 N

0.4 m

0.1 mV

"M = 0; 20 N (0.4 m) - V(0.1 m) = 0 V = 80 N

P1–6.

30# 5 kN

5 kN

N = (5 kN) cos 30# = 4.33 kN V = (5 kN) sin 30# = 2.5 kN

P2–1.

L 3L

! ! ¿

L =

3L , $% =

PAB = $ >3 L>2 = 2$3L

PCD = $

P2–2.

2 L !

! ¿

$% 2 L

= $

L $% = 2$

PCD = $L PAB =

2$ L>2 = 4$L

P2–3.

PAB = LA%B - LAB

LAB

P2–4.

PAB = LAB% - LAB

LAB , PAC =

LAC% - LAC LAC

PBC = LB%C% - LBC

LBC , (gA)xy = ap2 - ub rad

P2–5.

(gA)xy = p

2 - ap

2 + u1b

= (-u1) rad

(gB)xy = p

2 - (p - u2)

= a-p 2

+ u2b rad

P4–1a.

100 N 200 N 400 N

200 N 400 N

400 N

NBA ! 400 N

NCB ! 500 N

NDC ! 300 N

NED ! 700 N

700 N

x u1

SOLUTIONS AND ANSWERS FOR PRELIMINARY PROBLEMS 817

P4–1b.

NAB ! 600 N

NBC ! 200 N

NCD ! 500 N

600 N

600 N 400 N 300 N

400 N

P4–2.

900 N NCB ! 900 N

400 N NED ! 400 N

400 N 500 N NDB ! 900 N

NAB ! 1800 N

900 N

P4–3.

N ! 8x

(8x) N

P4–4.

N ! (800 " 100x) N

800 N

100x

P4–5.

2 m 4 m

60 kN

40 kN 20 kN

#B = PL AE

= 20(103) N (3 m)

2(10- 3) m2 (60(109) N>m2) = 0.5(10- 3) m = 0.5 mm

P5–1.

300 N$m

Equilibrium

DC 500 N$m

500 N$m

800 N$m

300 N$m

Equilibrium

818 SOLUTIONS AND ANSWERS FOR PRELIMINARY PROBLEMS

P5–2.

400 N!m

C 200 N!m

P5–3.

tmax

P5–4. P = Tv

(10 hp)a550 ft # lb>s 1 hp

b = Ta1200 rev min

b a1 min 60 s

b2p rad 1 rev

T = 43.8 lb # ft

P6–1a.

P6–1b.

SOLUTIONS AND ANSWERS FOR PRELIMINARY PROBLEMS 819

P6–1c.

P6–1d.

P6–1e.

P6–1f.

820 SOLUTIONS AND ANSWERS FOR PRELIMINARY PROBLEMS

P6–1g.

P6–1h.

P6–2. I = c 1 12

(0.2 m)(0.4 m)3 d - c 1 12

(0.1 m)(0.2 m)3 d = 1.0 (10-3) m4

P6–3.

y = !y!A

!A =

(0.05 m)(0.2 m)(0.1 m) + (0.25 m)(0.1 m)(0.3 m) (0.2 m)(0.1 m) + (0.1 m)(0.3 m)

= 0.17 m

I = c 1 12

(0.2 m)(0.1 m)3 + (0.2 m)(0.1 m)(0.17 m - 0.05 m)2 d + c 1

12 (0.1 m)(0.3 m)3 + (0.1 m)(0.3 m)(0.25 m - 0.17 m)2 d

= 0.722 (10-3) m4

P6–4a.

P6–4b.

MM B

M M

SOLUTIONS AND ANSWERS FOR PRELIMINARY PROBLEMS 821

P6–5a.

P6–5b.

P7–1a.

0.1 m

0.05 m 0.15 m

0.4 m tA

Q = y!A! = (0.1 m) (0.1 m) (0.4 m) = 4(10- 3) m3

t = 0.4 m

P7–1b.

0.3 m

0.1 m

0.15 m 0.2 m

Q = y!A! = (0.15 m) (0.3 m) (0.1 m) = 4.5(10- 3) m3

t = 0.3 m

P7–1c.

0.1 m

0.5 m

0.1 m

0.2 m 0.1 m

0.1 m A

tAV

Q = y!A! = (0.2 m) (0.1 m) (0.5 m) = 0.01 m3

t = 3 (0.1 m) = 0.3 m

P7–1d.

0.35 m

0.6 m0.1 m

Q = y!A! = (0.35 m) (0.6 m) (0.1 m) = 0.021 m3

t = 0.6 m

822 SOLUTIONS AND ANSWERS FOR PRELIMINARY PROBLEMS

P7–1e.

0.2 m

0.25 m

0.1 m

Q = y!A! = (0.25 m) (0.2 m) (0.1 m) = 5(10- 3) m3

t = 0.2 m

P7–1f.

0.1 m

0.25 m

0.35 m

0.1 m 0.2 m

0.5 m

tAV

Q = "y !A ! = (0.25 m) (0.1 m) (0.1 m) + (0.35 m) (0.1 m) (0.5 m) = 0.02 m3 t = 0.1 m

P8–1a.

140 N#m

300 N

P8–1b.

600 N#m

200 N

300 N

100 N

400 N#m P8–1c.

50 N#m

100 N#m

100 N

P8–1d.

300 N!m

1200 N

500 N

300 N

200 N

600 N!m

P8–2a.

P Element A

M V T

! ! !

P M V T

! ! !Element B

SOLUTIONS AND ANSWERS FOR PRELIMINARY PROBLEMS 823

P8–2b.

V M T ! !Element A

V M T

! !Element B

P9–1.

u " 20#

8 kPa

5 kPa

y¿

x ¿

u " $160#

8 kPa

5 kPa

x ¿

y¿

P9–1b.

u " 120#5 kPa

8 kPa

y¿

x ¿

5 kPa

8 kPa

y¿

x ¿

u " $60#

P9–1c.

u " 50#

5 kPa 8 kPa

y¿ x ¿

u " $130#5 kPa

8 kPa y¿

x ¿

P9–2.

tmax = A asx - sy2 b2 + t xy2 = A a4 - (-4)2 b2 + (0)2 = 4 MPa

savg = sx + sy

2 =

4 - 4 2

= 0

tan 2us = - (sx - sy)>2

txy =

[4 - (-4)]>2 0

= - %

us = -45#

tx&y& = - sx - sy

2 sin 2u + txy cos 2u

= - 4 - (-4)

2 sin 2(-45#) + 0 = 4 MPa

4 MPa

us " $45# x

x¿

y¿

P12–1a.

x 32 N'm

8 kN

M = (8x - 32) kN # m x = 0,

dx = 0

x = 0, y = 0

P12–1b.

5 kN

M = (5x) kN # m x = 0, y = 0 x = 2 m, y = 0

824 SOLUTIONS AND ANSWERS FOR PRELIMINARY PROBLEMS

1 m1 m

x2 2 kN 10 kN

4 kN

M2 = 10(x2 - 2) - 4(x2 - 1) - 2x2 M2 = (4x2 - 16) kN # m x1 = 0, y1 = 0 x1 = 2 m, y1 = 0 x2 = 2 m, y2 = 0

x1 = x2 = 2 m, dy1 dx1

= dy2 dx2

P12–1f.

V21.5 kN

M1 = (1.5x1) kN # m

2 m

V2 x2

1.5 kN

3 (x2 ! 2) kN

(x2 ! 2) 1 2

M2 = 1.5x2 - 3(x2 - 2) a12 b (x2 - 2) M2 = -1.5x22 + 7.5x2 - 6

x1 = 0, y1 = 0

x2 = 4 m, y2 = 0

x1 = x2 = 2 m, dy1 dx1

= dy2 dx2

x1 = x2 = 2 m, y1 = y2

P12–1c.

4 kN

(2 x) kN

x1 2

M = 4 x - (2x)112 x2 M = (4 x - x2) kN # m x = 0, y = 0 x = 4 m, y = 0

P12–1d.

V12 kN

M1 = (-2x1) kN # m

V22 kN

8 kN"m

M2 = (-2 x + 8) kN # m x1 = 0, y1 = 0 x2 = 4 m, y2 = 0

x1 = x2 = 2 m, dy1 dx1

= dy2 dx2

x1 = x2 = 2 m, y1 = y2

P12–1e.

5 kN

(2 x1) kN

x1 1 2

M1 = -2x1 - (2x1)112 x12 M1 = (-2x1 - x21) kN # m

825

Fundamental Problems Partial Solutions and Answers Chapter 1

F1–1 Entire beam: a+ !MB = 0; 60 - 10(2) - A y (2) = 0 A y = 20 kN

Left segment: S + !Fx = 0; N C = 0 Ans. + c !Fy = 0; 20 - V C = 0 V C = 20 kN Ans. a+ !MC = 0; M C + 60 - 20(1) = 0 M C = -40 kN · m Ans.

F1–2 Entire beam: a+ !MA = 0; B y (3) - 100(1.5)(0.75) - 200(1.5)(2.25) = 0 B y = 262.5 N

Right segment: S + !Fx = 0; N C = 0 Ans. + c !Fy = 0; V C + 262.5 - 200(1.5) = 0 V C = 37.5 N Ans. a+ !MC = 0; 262.5(1.5) - 200(1.5)(0.75) - M C = 0 M C = 169 N · m Ans.

F1–3 Entire beam: S + !Fx = 0; B x = 0 a+ !MA = 0; 20(2)(1) - B y (4) = 0 B y = 10 kN

Right segment: S + !Fx = 0; N C = 0 Ans. + c !Fy = 0; V C - 10 = 0 V C = 10 kN Ans. a+ !MC = 0; - M C - 10(2) = 0 M C = -20 kN · m Ans.

F1–4 Entire beam:

a+ !MB = 0; 1 2

(10)(3)(2) + 10(3)(4.5) - A y(6) = 0 A y = 27.5 kN

Left segment: S + !Fx = 0; N C = 0 Ans. + c !Fy = 0; 27.5 - 10(3) - V C = 0 V C = -2.5 kN Ans. a+ !MC = 0; M C + 10(3)(1.5) - 27.5(3) = 0 M C = 37.5 kN · m Ans.

F1–5 Entire beam: S + !Fx = 0; A x = 0

a+ !MB = 0; 300(6)(3) - 1 2

(300)(3)(1) - A y (6) = 0 A y = 825 lb

Left segment: S + !Fx = 0; N C = 0 Ans. + c !Fy = 0; 825 - 300(3) - V C = 0 V C = -75 lb Ans. a+ !MC = 0; M C + 300(3)(1.5) - 825(3) = 0 M C = 1125 lb · ft Ans.

826 FUNDAMENTAL PROBLEMS PART IAL SOLUT IONS AND ANSWERS

F1–6. Entire beam:

a+ !MA = 0; FBD a35 b(4) - 5(6)(3) = 0 F BD = 37.5 kN S + !Fx = 0; 37.5a45 b - A x = 0 A x = 30 kN + c !Fy = 0; A y + 37.5a35 b - 5(6) = 0 A y = 7.5 kN Left segment:

S + !Fx = 0; N C - 30 = 0 N C = 30 kN Ans. + c !Fy = 0; 7.5 - 5(2) - V C = 0 V C = -2.5 kN Ans. a+ !MC = 0; M C + 5(2)(1) - 7.5(2) = 0 M C = 5 kN · m Ans.

F1–7 Beam:

!MA = 0; T CD = 2 w !Fy = 0; T AB = w

Rod AB :

s = P A

; 300(103) = w 10

;

w = 3 N>m Rod CD :

s = P A

; 300(103) = 2w 15

;

w = 2.25 N>m Ans. F1–8 A = p(0.1 2 - 0.08 2 ) = 3.6(10 -3 )p m 2

savg = P A

= 300(103)

3.6(10- 3)p = 26.5 MPa Ans.

F1–9 A = 3[4(1)] = 12 in 2

savg = P A

= 15 12

= 1.25 ksi Ans.

F1–10 Consider the cross section to be a rectangle and two triangles.

y = ! ~yA

!A =

0.15[(0.3)(0.12)] + (0.1) c 1 2

(0.16)(0.3) d 0.3(0.12) + 1

2 (0.16)(0.3)

= 0.13 m = 130 mm Ans.

savg = P A

= 600(103)

0.06 = 10 MPa Ans.

F1–11

A A = A C = p

4 (0.52) = 0.0625p in2, AB =

4 (12) = 0.25p in2

sA = NA A A

= 3

0.0625p = 15.3 ksi (T) Ans.

sB = NB A B

= -6

0.25p = -7.64 ksi = 7.64 ksi (C) Ans.

sC = NC A C

= 2

0.0625p = 10.2 ksi (T) Ans.

F1–12

F AD = 50(9.81) N = 490.5 N

+ c !Fy = 0; FAC a35 b - 490.5 = 0 F AC = 817.5 N S + !Fx = 0; 817.5a45 b - FAB = 0 F AB = 654 N A AB =

4 (0.0082) = 16(10- 6)p m2

(sAB)avg = FAB A AB

= 654

16(10- 6)p = 13.0 MPa Ans.

F1–13 Ring C :

+ c !Fy = 0; 2F cos 60° - 200(9.81) = 0 F = 1962 N

(sallow)avg = F A

; 150(106) = 1962 p

4 d2

d = 0.00408 m = 4.08 mm Use d = 5 mm Ans.

FUNDAMENTAL PROBLEMS PARTIAL SOLUTIONS AND ANSWERS 827

F1–14 Entire frame: !Fy = 0; A y = 600 lb !MB = 0; A x = 800 lb FA = 2(600)2 + (800)2 = 1000 lb (tA)avg =

FA >2 A

= 1000>2 p 4(0.25)

2 = 10.2 ksi Ans.

F1–15 Double shear: !Fx = 0; 4 V - 10 = 0 V = 2.5 kip

A = p

4 a3

4 b2 = 0.140625p in2

tavg = V A

= 2.5

0.140625p = 5.66 ksi Ans.

F1–16 Single shear:

!Fx = 0; P - 3 V = 0 V = P 3

A = p

4 (0.0042) = 4(10- 6)p m2

(tavg)allow = V A

; 60(106) =

P 3

4(10- 6)p

P = 2.262(10 3 ) N = 2.26 kN Ans.

F1–17 S + !Fx = 0; V - P cos 60° = 0 V = 0.5 P

A = a 0.05 sin 60"

b (0.025) = 1.4434(10-3) m2 (tavg)allow =

V A

; 600(103) = 0.5P

1.4434(10- 3)

P = 1.732(10 3 ) N = 1.73 kN Ans.

F1–18 The resultant force on the pin is F = 2302 + 402 = 50 kN. Here we have double shear:

V = F 2

= 50 2

= 25 kN

A = p

4 (0.032) = 0.225(10- 3)p m2

tavg = V A

= 25(103)

0.225(10- 3)p = 35.4 MPa Ans.

F1–19 S + !Fx = 0; 30 - N = 0 N = 30 kN

sallow = sY

F.S. =

250 1.5

= 166.67 MPa

sallow = N A

; 166.67(106) = 30(103) p

4 d2

d = 15.14 mm Use d = 16 mm Ans.

F1–20. S + !Fx = 0; N AB - 30 = 0 N AB = 30 kip

S + !Fx = 0; N BC - 15 - 15 - 30 = 0 N BC = 60 kip

sallow = sY

F.S. =

50 1.5

= 33.33 ksi

Segment AB :

sallow = NAB A AB

; 33.33 = 30

h1(0.5)

h 1 = 1.8 in. Segment BC :

sallow = NBC A BC

; 33.33 = 60

h2(0.5)

h 2 = 3.6 in.

Use h1 = 1 7 8

in. and h2 = 3 5 8

in. Ans.

F1–21 N = P

sallow = sY

F.S. =

250 2

= 125 MPa

A r = p

4 (0.042) = 1.2566(10- 3) m2

A a – a = 2(0.06 - 0.03)(0.05) = 3(10 -3 ) m 2

The rod will fail first.

sallow = N A r

; 125(106) = P

1.2566(10- 3)

P = 157.08(10 3 ) N = 157 kN Ans.

F1–22 S + !Fx = 0; 80 - 2V = 0 V = 40 kN

tallow = tfail

F.S. =

100 2.5

= 40 MPa

tallow = V A

; 40(106) = 40(103) p

4 d2

d = 0.03568 m = 35.68 mm Use d = 36 mm Ans.

F1–23 V = P

tallow = tfail

F.S. =

120 2.5

= 48 MPa

Area of shear plane for bolt head and plate: A b = pdt = p(0.04)(0.075) = 0.003p m2 A p = pdt = p(0.08)(0.03) = 0.0024p m2

828 FUNDAMENTAL PROBLEMS PART IAL SOLUT IONS AND ANSWERS

Since the area of shear plane for the plate is smaller,

tallow = V A p

; 48(10 6 ) = P

0.0024p

P = 361.91(10 3 ) N = 362 kN Ans.

F1–24

a+ !MB = 0; 1 2

(300)(9)(6) - 6V (9) = 0 V = 150 lb

tallow = tfail

F.S. =

16 2

= 8 ksi

tallow = V A

; 8(10 3 ) = 150 p

4 d2

d = 0.1545 in.

Use d = 3

16 in. Ans.

Chapter 2

F2–1 dC

600 =

0.2 400

; dC = 0.3 mm

PCD = dC

LCD =

0.3 300

= 0.001 mm>mm Ans. F2–2

u = a0.02" 180"

bp rad = 0.3491(10- 3) rad dB = uLAB = 0.3491(10- 3)(600) = 0.2094 mm

dC = uLAC = 0.3491(10- 3)(1200) = 0.4189 mm

PBD = dB

LBD =

0.2094 400

= 0.524(10- 3) mm>mm Ans. PCE =

LCE =

0.4189 600

= 0.698(10- 3) mm>mm Ans. F2–3

a = 2

400 = 0.005 rad b =

4 300

= 0.01333 rad

(gA)xy = 1 2

- u

= p

2 - ap

2 - a + bb

= a - b = 0.005 - 0.01333 = -0.00833 rad Ans.

F2–4 L BC = 23002 + 4002 = 500 mm L B'C = 2(300 - 3)2 + (400 + 5)2 = 502.2290 mm a =

3 405

= 0.007407 rad

(PBC)avg = LB#C - LBC

LBC =

502.2290 - 500 500

= 0.00446 mm>mm Ans. (gA)xy =

2 -u = p

2 - ap

2 + ab = -a = -0.00741 rad Ans.

F2–5 L AC = 2LCD2 + LAD2 = 23002 + 3002 = 424.2641 mm L AC = 2LC#D#2 + LA #D#2 = 23062 + 2962 = 425.7370 mm u

2 = tan-1aLC#D#

LA #D# b ; u = 2 tan-1a 306

296 b = 1.6040 rad

(PAC)avg = LA #C# - LAC

LAC =

425.7370 - 424.2641 424.2641

= 0.00347 mm>mm Ans. (gE)xy =

2 - u = p

2 - 1.6040 = - 0.0332 rad Ans.

Chapter 3 F3–1 Material has uniform properties throughout. Ans. F3–2 Proportional limit is A . Ans. Ultimate stress is D . Ans. F3–3 The initial slope of the s - P diagram. Ans. F3–4 True. Ans. F3–5 False. Use the original cross-sectional area

and length. Ans. F3–6 False. It will normally decrease. Ans.

F3–7 e = s

E =

P AE

d = PL = PL AE

= 100(103)(0.100)

4 (0.015)2 200(109)

= 0.283mm Ans.

F3–8 P = s E

= P

d = PL = PL AE

;

0.003 = (10 000)(8)

12E

E = 2.22(106) psi Ans.

FUNDAMENTAL PROBLEMS PARTIAL SOLUTIONS AND ANSWERS 829

F3–9 P = s E

= P

d = PL = PL AE

= 6(103)4

4 (0.01)2100(109)

= 3.06 mm Ans.

F3–10 s = P A

= 100(103)

4 (0.02)2

= 318.31 MPa

Since s 6 sY = 450 MPa, Hooke’s Law is applicable.

E = sY

PY =

450(106) 0.00225

= 200 GPa

P = s E

= 318.31(106)

200(109) = 0.001592 mm>mm

d = PL = 0.001592(50) = 0.0796 mm Ans.

F3–11 s = P A

= 150(103) p

4 (0.022)

= 477.46 MPa

Since s 7 sY = 450 MPa, Hooke’s Law is not applicable. From the geometry of the stress-strain diagram,

P - 0.00225

0.03 - 0.00225 = 477.46 - 450

500 - 450

P = 0.017493 When the load is removed, the strain recovers

along a line parallel to the original elastic line.

Here E = sY

PY =

450(106) 0.00225

= 200 GPa.

The elastic recovery is

Pr = s

E =

477.46(106)

200(109) = 0.002387 mm>mm

Pp = P - Pr = 0.017493 - 0.002387 = 0.01511 mm>mm dp = PpL = 0.01511(50) = 0.755 mm Ans.

F3–12 PBC = dBC

LBC =

0.2 300

= 0.6667(10- 3) mm>mm sBC = EPBC = 200(109)[0.6667(10- 3)] = 133.33 MPa Since sBC 6 sY = 250 MPa, Hooke’s Law is valid.

sBC = FBC A BC

; 133.33(106) = FBC

4 (0.0032)

FBC = 942.48 N a+ !MA = 0; 942.48(0.4) - P(0.6) = 0 P = 628.31 N = 628 N Ans.

F3–13 s = P A

= 10(103)

4 (0.015)2

= 56.59 MPa

Plong = s

E =

56.59(106)

70(109) = 0.808(10- 3)

Plat = -nPlong = -0.35(0.808(10- 3)) = -0.283(10- 3) dd = (-0.283(10- 3))(15 mm) = -4.24(10- 3) mm Ans.

F3–14 s = P A

= 50(103) p

4 (0.022)

= 159.15 MPa

Plong = d

L =

1.40 600

= 0.002333 mm>mm E =

Plong =

159.15(106) 0.002333

= 68.2 GPa Ans.

Plat = d" - d

d =

19.9837 - 20 20

= -0.815(10- 3) mm>mm n = -

Plat Plong

= - -0.815(10- 3)

0.002333 = 0.3493

G = E

2(1 + n) = 68.21

2(1 + 0.3493) = 25.3 GPa Ans.

F3–15 a = 0.5 150

= 0.003333 rad

g = p

2 - u = p

2 - ap

2 - ab

= a = 0.003333 rad t = Gg = [26(109)](0.003333) = 86.67 MPa

t = V A

; 86.67(106) = P

0.15(0.02)

P = 260 kN Ans.

F3–16 a = 3

150 = 0.02 rad

g = p

2 - u = p

2 - ap

2 - ab = a = 0.02 rad

When P is removed, the shear strain recovers along a line parallel to the original elastic line.

gr = gY = 0.005 rad gp = g - gr = 0.02 - 0.005 = 0.015 rad Ans.

830 FUNDAMENTAL PROBLEMS PART IAL SOLUT IONS AND ANSWERS

Chapter 4 F4–1 A =

4 (0.022) = 0.1(10- 3)p m2

dC = 1

AE {40(103)(400) + [-60(103)(600)]}

= -20(106) N # mm

= -0.318 mm Ans.

F4–2 A AB = A CD = p

4 (0.022) = 0.1(10- 3)p m2

A BC = p

4 (0.042 - 0.032) = 0.175(10- 3)p m2

dD>A = [-10(103)](400)[0.1(10- 3)p][68.9(109)] +

[10(103)](400)

[0.175(10- 3)p][68.9(109)]

+ [-20(103)](400)

[0.1(10- 3)p][68.9(109)]

= -0.449 mm Ans.

F4–3 A = p

4 (0.032) = 0.225(10- 3)p m2

dC = 1

0.225(10- 3)p[200(109)] e c -90(103)

+ 2a4 5 b30(103) d (0.4) + [-90(103)(0.6)] f

= -0.772(10- 3) m = -0.772 mm Ans.

F4–4 dA>B = PLAE = [60(103)](0.8)[0.1(10- 3)p][200(109)] = 0.7639(10- 3)m T

dB = Fsp k

= 60(103)

50(106) = 1.2(10- 3)m T

+ T dA = dB + dA>B dA = 1.2(10- 3) + 0.7639(10- 3) = 1.9639(10- 3) m = 1.96 mm T Ans.

F4–5 A = p

4 (0.022) = 0.1(10- 3)p m2

Internal load P ( x ) = 30(10 3 ) x

dA = L P(x)dx

= 1

[0.1(10- 3)p][73.1(109)] L 0.9 m

0 30(103)x dx

= 0.529(10- 3) m = 0.529 mm Ans.

F4–6 Distributed load P ( x ) = 45(103)

0.9 x = 50(10 3 ) x N>m

Internal load P ( x ) = 1 2

(50(103))x(x) = 25(10 3 ) x 2

dA = L L

0 P(x)dx

= 1

[0.1(10- 3)p][73.1(109)] L

0.9 m

0 [25(103)x2]dx

= 0.265 mm Ans.

Chapter 5

F5–1 J = p

2 (0.044) = 1.28(10 -6 ) p m 4

tA = tmax = Tc J

= 5(103)(0.04)

1.28(10- 6)p = 49.7 MPa

Ans.

tB = TrB

J =

5(103)(0.03)

1.28(10- 6)p = 37.3 MPa Ans.

F5–2 J = p

2 (0.064 - 0.044) = 5.2(10 -6 )p m 4

tB = tmax = Tc J

= 10(103)(0.06)

5.2(10- 6)p = 36.7 MPa

Ans.

tA = TrA

J =

10(103)(0.04)

5.2(10- 6)p = 24.5 MPa Ans.

F5–3 J AB = p

2 (0.044 - 0.034) = 0.875(10 -6 )p m 4

J BC = p

2 (0.044) = 1.28(10 -6 )p m 4

( tAB) max = TAB cAB

J AB =

[2(103)](0.04)

0.875(10- 6)p = 29.1 MPa

( tBC ) max = TBC cBC

J BC =

[6(103)](0.04)

1.28(10- 6)p

= 59.7 MPa Ans.

F5–4 T AB = 0, T BC = 600 N · m, T CD = 0

J = p

2 (0.024) = 80(10 -9 )p m 4

tmax = Tc J

= 600(0.02)

80(10- 9)p = 47.7 MPa Ans.

FUNDAMENTAL PROBLEMS PARTIAL SOLUTIONS AND ANSWERS 831

F5–5 J BC = p

2 (0.044 - 0.034) = 0.875(10 -6 )p m 4

( tBC ) max = TBC cBC

J BC =

2100(0.04)

0.875(10- 6)p

= 30.6 MPa Ans.

F5–6 t = 5(10 3 ) N · m>m Internal torque is T = 5(10 3 )(0.8) = 4000 N · m

J = p

2 (0.044) = 1.28(10 -6 )p m 4

tAB = TAc

J =

4000(0.04)

1.28(10- 6)p = 39.8 MPa Ans.

F5–7 Maximum internal torque is in region AB.

TAB = 250 N # m t

max abs =

TAB c

J =

250 (0.025) p

2 (0.025)4

= 10.2 MPa Ans.

F5–8 P = Tv; 3(550) ft # lb>s = T c 150 a2p 60

b rad>s d T = 105.04 ft # lb

tallow = Tc

J ; 12(103) =

105.04 (12)(d>2) p

2 (d>2)4

d = 0.812 in.

Use d = 7 8

in. Ans.

F5–9 J = p

2 (0.034) = 0.405(10 -6 )p m 4

fA>C = -2 (103)(0.6) + (103)(0.4)[0.405(10- 6)p][75(109)] = - 0.00838 rad = -0.480° Ans.

F5–10 J = p

2 (0.024) = 80(10 -9 )p m 4

fB>A = 600(0.45)[80(10- 9)p][75(109)] = 0.01432 rad = 0.821° Ans.

F5–11 J = p

2 (0.044 - 0.034) = 0.875(10 -6 )p m 4

fA>B = TAB LABJG = 3(103)(0.9)[0.875(10- 6)p][26(109)] = 0.03778 rad

fB = TB k

= 3(103)

90(103) = 0.03333 rad

fA = fB + fA>B = 0.03333 + 0.03778 = 0.07111 rad = 4.07° Ans.

F5–12 J = p

2 (0.024) = 80(10 -9 )p m 4

fB>A = 0.02[600 + (-300) + 200 + 500][80(10- 9)p][75(109)] = 0.01061 rad = 0.608° Ans.

F5–13 J = p

2 (0.044) = 1.28(10 -6 )p m 4

t = 5(10 3 ) N · m>m Internal torque is 5(10 3 ) x N · m

fA>B = L L

0 T(x)dx

= 1

[1.28(10- 6)p][75(109)] L 0.8 m

0 5(103)xdx

= 0.00531 rad = 0.304° Ans.

F5–14 J = p

2 (0.044) = 1.28(10 -6 )p m 4

Distributed torque is t = 15(103)

0.6 (x)

= 25(10 3 ) x N · m>m Internal torque is T ( x ) =

1 2

(25x)(103)(x)

= 12.5(10 3 ) x 2 N · m

fA>C = L L

T(x)dx JG

+ TBC LBC

= 1

[1.28(10- 6)p][75(109)] cL0.6 m0 12.5(103)x2 dx+4500(0.4)d

= 0.008952 rad = 0.513° Ans.

832 FUNDAMENTAL PROBLEMS PART IAL SOLUT IONS AND ANSWERS

Chapter 6 F6–1 a+ !MB = 0; A y (6) - 30 = 0 A y = 5 kN +c gFy = 0; - V - 5 = 0 V = -5 kN Ans. a+ !M0 = 0; M + 5x = 0 M = {-5x} kN # m

Ans.

"30

x (m)

V (kN)

M (kN#m)

F6–2 + c gFy = 0; - V - 9 = 0 V = -9 kN Ans. a+ !MO = 0; M + 9 x = 0 M = {-9 x } kN · m Ans.

"27

x (m)

V (kN)

x (m)

M (kN#m)

F6–3 + c gFy = 0; - V - 2 x = 0 V = {-2 x } kip Ans. a+ !MO = 0; M + 2x a x2 b - 18 = 0 M = {18 - x 2 } kip · ft Ans.

94.2418

"63

"18

x (ft)

V (kip)

x (ft)

M (kip#ft)

F6–4 + c gFy = 0; -V - 12 (4x)(x) = 0 V = {-2 x 2 } kN Ans.

a+gMO = 0; M + c 12 (4x)(x) d a x3 b = 0 M = e - 2

3 x3f kN # m Ans.

M (kN#m)

"18

x (m)

V (kN)

"18

x (m)

FUNDAMENTAL PROBLEMS PARTIAL SOLUTIONS AND ANSWERS 833

F6–5

M (kN!m)

64.5

1.5

x (m)

V (kN)

4.5 61.5

"4.5

x (m)

F6–6

M (kN!m)

"15

x (m)

V (kN)

x (m)

F6–7

M (kip!ft)

5.25 6 9 12

1050

"750

2756 2700 2250

"150 x (ft)

V (kip)

x (ft)

F6–8

M (kN!m)

20 4

1.5 30

"50

x (m)

V (kN)

3 4 6

1.5

22.5

"40

x (m)

F6–9 I = 2 c 1 12

(0.02)(0.23) d + 1 12

(0.26)(0.023)

= 26.84(10- 6) m4

smax = Mc

I =

20(103)(0.1)

26.84(10- 6) = 74.5 MPa Ans.

F6–10 y = 0.3 3

= 0.1 m

I = 1

36 (0.3)(0.33) = 0.225 (10- 3) m 4

( smax ) c = Mc

I =

50(103)(0.3 - 0.1) 0.225(10- 3)

= 44.4 MPa (C) Ans.

( smax ) t = My

I =

50(103)(0.1)

0.225(10- 3) = 22.2 MPa (T) Ans.

F6–11 I = 1

12 (0.2)(0.33) - 1

12 (0.18)(0.263)

= 0.18636(10 -3 ) m 4

smax = Mc

I =

50(103)(0.15)

0.18636(10- 3) = 40.2 MPa Ans.

F6–12

I = 2 c 1 12

(0.03)(0.43) d + 2 c 1 12

(0.14)(0.033) + 0.14(0.03)(0.152) d = 0.50963 (10- 3) m 4

smax = Mc

I =

10(103)(0.2)

0.50963(10- 3) = 3.92 MPa Ans.

sA = 3.92 MPa (C) sB = 3.92 MPa (T)

834 FUNDAMENTAL PROBLEMS PART IAL SOLUT IONS AND ANSWERS

F6–13 I = 1

12 (0.05)(0.4)3 + 2 c 1

12 (0.025)(0.3)3 d

= 0.37917(10 -3 ) m 4

sA = MyA

I =

5(103)(-0.15) 0.37917(10- 3)

= 1.98 MPa (T) Ans.

F6–14 M y = 50a45 b = 40 kN # m M z = 50a35 b = 30 kN # m I y =

1 12

(0.3)(0.23) = 0.2 (10- 3) m 4

I z = 1

12 (0.2)(0.33) = 0.45 (10- 3) m 4

s = - Mzy

Iz +

My z

sA = - [30(103)](-0.15)

0.45(10- 3) +

[40(103)](0.1)

0.2(10- 3)

= 30 MPa (T) Ans.

sB = - [30(103)](0.15)

0.45(10- 3) +

[40(103)](0.1)

0.2(10- 3)

= 10 MPa (T) Ans.

tan a = Iz Iy

tan u

tan a = c 0.45(10- 3) 0.2(10- 3)

d a4 3 b

a = 71.6° Ans.

F6–15 Maximum stress occurs at D or A .

( smax ) D = (50 cos 30!)12(3)

1 12(4)(6)

3 +

(50 sin 30!)12(2) 1 12(6)(4)

= 40.4 psi Ans.

Chapter 7

F7–1 I = 2 c 1 12

(0.02)(0.23) d + 1 12

(0.26)(0.023)

= 26.84 (10- 6) m 4 Q A = 2[0.055(0.09)(0.02)] = 198(10 -6 ) m 3

tA = VQA

It =

100(103)[198(10- 6)]

[26.84(10- 6)]2(0.02)

= 18.4 MPa Ans.

F7–2 I = 1

12 (0.1)(0.33) + 1

12 (0.2)(0.13) = 0.24167(10 -3 ) m 4

Q A = y ¿ 1 A ¿ 1 + y ¿ 2 A ¿ 2

= c 1 2

(0.05) d (0.05)(0.3) + 0.1(0.1)(0.1) = 1.375(10 -3 ) m 3

Q B = y ¿ 2 A ¿ 2 = 0.1(0.1)(0.1) = 1(10 -3 ) m 3

t A = VQ It

= 600(103)[1.375(10- 3)]

[0.24167(10- 3)](0.3) = 11.4 MPa Ans.

t B = VQ It

= 600(103)[1(10- 3)]

[0.24167(10- 3)](0.1) = 24.8 MPa Ans.

F7–3 V max = 4.5 kip

I = 1

12 (3)(63) = 54 in 4

Q max = y " A " = 1.5(3)(3) = 13.5 in 3

(t max ) abs = Vmax Qmax

It =

4.5(103)(13.5) 54(3)

= 375 psi

Ans.

F7–4 I = 2 c 1 12

(0.03)(0.4)3 d + 2 c 1 12

(0.14)(0.03)3

+ 0.14(0.03)(0.152)d = 0.50963(10- 3) m4 Qmax = 2y "1 A"1 + y "2 A"2 = 2(0.1)(0.2)(0.03) + (0.15)(0.14)(0.03) = 1.83(10- 3) m3

tmax = VQmax

It =

20(103)[1.83(10- 3)]

0.50963(10- 3)[2(0.03)] = 1.20 MPa

Ans.

F7–5 I = 1

12 (0.05)(0.4)3+ 2 c 1

12 (0.025)(0.3)3 d

= 0.37917(10- 3) m4 Qmax = 2y "1 A "1 + y "2 A "2 = 2(0.075)(0.025)(0.15)

+ (0.1)(0.05)(0.2) = 1.5625(10- 3) m3

tmax = VQmax

It =

20(103)[1.5625(10- 3)]

[0.37917(10- 3)][2(0.025)]

= 1.65 MPa Ans.

F7–6 I = 1

12 (0.3)(0.23) = 0.2 (10- 3) m 4

Q = y " A " = 0.05(0.1)(0.3) = 1.5 (10- 3) m 3

FUNDAMENTAL PROBLEMS PARTIAL SOLUTIONS AND ANSWERS 835

q allow = 2aFs b = 2[15(103)]s = 30(103)s q allow =

VQ I

; 30(103)

s =

50(103)[1.5(10- 3)]

0.2(10- 3)

s = 0.08 m = 80 mm Ans.

F7–7 I = 1

12 (0.2)(0.343) - 1

12 (0.19)(0.283)

= 0.3075 (10- 3) m 4 Q = y ! A ! = 0.16(0.02)(0.2) = 0.64 (10- 3) m 3

q allow = 2aFs b = 2[30(103)]s = 60(103)s q allow =

VQ I

; 60(103)

s =

300(103)[0.64(10- 3)]

0.3075(10- 3)

s = 0.09609 m = 96.1 mm Use s = 96 mm Ans.

F7–8

I = 2 c 1 12

(0.025)(0.33)d + 2c 1 12

(0.05)(0.23) + 0.05(0.2)(0.152)d = 0.62917 (10- 3) m 4 Q = y ! A ! = 0.15(0.2)(0.05) = 1.5 (10- 3) m 3

q allow = 2aFs b = 2[8(103)]s = 16(103)s q allow =

VQ I

; 16(103)

s =

20(103)[1.5(10- 3)]

0.62917(10- 3)

s = 0.3356 m = 335.56 mm Use s = 335 mm Ans.

F7–9 I = 1

12 (1)(63) + 4 c 1

12 (0.5)(43) + 0.5(4)(32) d

= 100.67 in 4 Q = y ! A ! = 3(4)(0.5) = 6 in 3

q allow = F s

= 6 s

q allow = VQ

I ;

6 s

= 15(6)

100.67

s = 6.711 in.

Use s = 6 5 8

in. Ans.

Chapter 8

F8–1 + c gFz = ( F R ) z ; -500 - 300 = P P = -800 kN

g Mx = 0; 300(0.05) - 500(0.1) = M x M x = -35 kN · m g My = 0; 300(0.1) - 500(0.1) = M y M y = -20 kN · m A = 0.3(0.3) = 0.09 m 2

I x = I y = 1

12 (0.3)(0.33) = 0.675 (10- 3) m 4

s A = -800(103)

0.09 +

[20(103)](0.15)

0.675(10- 3) +

[35(103)](0.15)

0.675(10- 3)

= 3.3333 MPa = 3.33 MPa (T) Ans.

s B = -800(103)

0.09 +

[20(103)](0.15)

0.675(10- 3) -

[35(103)](0.15)

0.675(10- 3)

= -12.22 MPa = 12.2 MPa (C) Ans.

F8–2 + c g Fy = 0; V - 400 = 0 V = 400 kN a+ "MA = 0; - M - 400(0.5) = 0 M = -200 kN · m

I = 1

12 (0.1)(0.33) = 0.225 (10- 3) m 4

QA = y !A ! = 0.1(0.1)(0.1) = 1(10- 3) m3

s A = My

I =

[200(103)](-0.05) 0.225(10- 3)

= -44.44 MPa = 44.4 MPa (C) Ans.

t A = VQ It

= 400(103)[1(10- 3)]

0.225(10- 3)(0.1) = 17.8 MPa Ans.

F8–3 Left reaction is 20 kN. Left segment:

+ c gFy = 0; 20 - V = 0 V = 20 kN a+ "Ms = 0; M - 20(0.5) = 0 M = 10 kN · m

I = 1

12 (0.1)(0.23) - 1

12 (0.09)(0.183)

= 22.9267 (10- 6) m 4

QA = y !1A!1 + y !2 A!2 = 0.07(0.04)(0.01) + 0.095(0.1)(0.01) = 0.123(10- 3) m3

s A = - MyA

I = -

[10(103)](0.05)

22.9267(10- 6)

= -21.81 MPa = 21.8 MPa (C) Ans.

t A = VQA

It =

20(103)[0.123(10- 3)]

[22.9267(10- 6)](0.01)

= 10.7 MPa Ans.

F8–4 At the section through centroidal axis: N = P V = 0

836 FUNDAMENTAL PROBLEMS PART IAL SOLUT IONS AND ANSWERS

M = (2 + 1) P = 3 P

s = P A

+ Mc I

30 = P

2(0.5) +

(3P)(1) 1

12 (0.5)(2)3

P = 3 kip Ans.

F8–5 At section through B : N = 500 lb, V = 400 lb M = 400(10) = 4000 lb · in. Axial load:

s x = P A

= 500 4(3)

= 41.667 psi (T)

Shear load:

t xy = VQ It

= 400[(1.5)(3)(1)]

[ 1

12 (3)(4)3]3

= 37.5 psi

Bending moment:

s x = My

I =

4000(1) 1

12 (3)(4)3

= 250 psi (C)

Thus s x = 41.667 - 250 = 208 psi (C) Ans. s y = 0 Ans. t xy = 37.5 psi Ans.

F8–6 Top segment: gFy = 0; V y + 1000 = 0 V y = -1000 N gFx = 0; V x - 1500 = 0 V x = 1500 N gMz = 0; T z - 1500(0.4) = 0 T z = 600 N · m gMy = 0; M y - 1500(0.2) = 0 M y = 300 N · m gMx = 0; M x - 1000(0.2) = 0 M x = 200 N · m I y = I x =

4 (0.024) = 40 (10- 9) p m 4

J = p

2 (0.024) = 80 (10- 9) p m 4

( Q y ) A = 4(0.02)

3p cp

2 (0.022) d = 5.3333 (10- 6) m 3

s A = Mxy

Ix =

Myx

Iy =

200(0)

40(10- 9)p +

300(0.02)

40(10- 9)p

= 47.7 MPa (T) Ans.

[(t zy ) T ] A = Tzc

J =

600(0.02)

80(10- 9)p = 47.746 MPa

[(t zy ) V ] A = Vy(Qy)A

Ixt =

1000[5.3333(10- 6)]

[40(10- 9)p](0.04)

= 1.061 MPa

Combining these two shear stress components,

(t zy ) A = 47.746 + 1.061 = 48.8 MPa Ans.

F8–7 Right Segment: !Fz = 0; V z - 6 = 0 V z = 6 kN !My = 0; T y - 6(0.3) = 0 T y = 1.8 kN · m !Mx = 0; M x - 6(0.3) = 0 M x = 1.8 kN · m

Ix = p

4 (0.054 - 0.044) = 0.9225(10- 6)p m4

J = p

2 (0.054 - 0.044) = 1.845(10- 6)p m4

(Qz)A = y2 "A 2 " - y1 "A 1 "

= 4(0.05)

3p cp

2 (0.052) d - 4(0.04)

3p cp

2 (0.042) d

= 40.6667(10- 6) m3

sA = Mxz

Ix =

1.8(103)(0)

0.9225(10- 6)p = 0 Ans.

[(tyz)T]A = Ty c

J =

[1.8(103)](0.05)

1.845(10- 6)p = 15.53 MPa

[(tyz)V]A = V z(Qz)A

Ixt =

6(103)[40.6667(10- 6)]

[0.9225(10- 6)p](0.02)

= 4.210 MPa Combining these two shear stress components,

(t yz ) A = 15.53 - 4.210 = 11.3 MPa Ans.

F8–8 Left Segment: !Fz = 0; V z - 900 - 300 = 0 V z = 1200 N

!My = 0; T y + 300(0.1) - 900(0.1) = 0 T y = 60 N · m

!Mx = 0; M x + (900 + 300)0.3 = 0 M x = -360 N · m

Ix = p

4 (0.0254 - 0.024) = 57.65625(10- 9)p m4

J = p

2 (0.0254 - 0.024) = 0.1153125(10- 6)p m4

( Q y ) A = 0

sA = Mxy

Ix =

(360)(0.025)

57.65625(10- 9)p = 49.7 MPa Ans.

[(txy)T]A = TyrA

J =

60(0.025)

0.1153125(10- 6 )p = 4.14 MPa Ans.

[(tyz)V]A = Vz (Qz)A

Ixt = 0 Ans.

Chapter 9 F9–1 u = 120° s x = 500 kPa s y = 0 t xy = 0 Apply Eqs. 9–1 , 9–2 .

FUNDAMENTAL PROBLEMS PARTIAL SOLUTIONS AND ANSWERS 837

sx ! = 125 kPa Ans. tx!y! = 217 kPa Ans.

F9–2 u = -45° s x = 0 s y = -400 kPa t xy = -300 kPa Apply Eqs. 9–1 , 9–3 , 9–2 . sx! = 100 kPa Ans. sy! = -500 kPa Ans. tx!y! = 200 kPa Ans.

F9–3 s x = 80 MPa s y = 0 t xy = 30 MPa Apply Eqs. 9–5 , 9–4 . s 1 = 90 MPa s 2 = -10 MPa Ans. u p = 18.43° and 108.43° From Eq. 9–1 ,

sx! = 80 + 0

2 + 80 - 0

2 cos 2 (18.43")

+ 30 sin 2(18.43") = 90 MPa = s1 Thus, (u p ) 1 = 18.4° and (u p ) 2 = 108° Ans.

F9–4 s x = 100 kPa s y = 700 kPa t xy = -400 kPa Apply Eqs. 9–7 , 9–8 . t max in-plane = 500 kPa Ans. s avg = 400 kPa Ans.

F9–5 At the cross section through B : N = 4 kN V = 2 kN M = 2(2) = 4 kN # m sB =

P A

+ Mc I

= 4(103)

0.03(0.06) +

4(103)(0.03) 1 12(0.03)(0.06)

= 224 MPa (T) Note tB = 0 since Q = 0. Thus s1 = 224 MPa Ans. s2 = 0

F9–6 A y = By = 12 kN Segment AC :

V C = 0 MC = 24 kN # m tC = 0 (since V C = 0 ) sC = 0 (since C is on neutral axis) s1 = s2 = 0 Ans.

F9–7 savg = sx + sy

2 =

500 + 0 2

= 250 kPa

The coordinates of the center C of the circle and the reference point A are

A (500, 0) C (250, 0)

R = CA = 500 - 250 = 250 kPa u = 120° (counterclockwise). Rotate the radial

line CA counterclockwise 2u = 240° to the coordinates of point P(sx!, tx!y!).

a = 240° - 180° = 60° sx! = 250 - 250 cos 60° = 125 kPa Ans. tx!y! = 250 sin 60° = 217 kPa Ans.

F9–8 savg = sx + sy

2 =

80 + 0 2

= 40 kPa

The coordinates of the center C of the circle and the reference point A are

A (80, 30) C (40, 0)

R = CA = 2(80 - 40)2 + 302 = 50 MPa s1 = 40 + 50 = 90 MPa Ans. s2 = 40 - 50 = -10 MPa Ans.

tan 2(u p ) 1 = 30

80 - 40 = 0.75

(u p ) 1 = 18.4° (counterclockwise) Ans.

F9–9 The coordinates of the reference point A and the center C of the circle are

A(30, 40) C(0, 0) R = CA = 50 MPa s1 = 50 MPa s2 = -50 MPa

F9–10 J = p

2 (0.044 - 0.034) = 0.875(10 -6 )p m 4

t = Tc J

= 4(103)(0.04)

0.875(10- 6)p = 58.21 MPa

s x = s y = 0 and t xy = -58.21 MPa

savg = sx + sy

2 = 0

The coordinates of the reference point A and the center C of the circle are

A (0, -58.21) C (0, 0) R = CA = 58.21 MPa s 1 = 0 + 58.21 = 58.2 MPa Ans. s 2 = 0 - 58.21 = -58.2 MPa Ans.

F9–11

+ c #Fy = 0; V - 30 = 0 V = 30 kN

a+#MO = 0; -M - 30(0.3) = 0 M = -9 kN · m

838 FUNDAMENTAL PROBLEMS PART IAL SOLUT IONS AND ANSWERS

I = 1

12 (0.05)(0.153) = 14.0625(10- 6) m4

QA = y !A ! = 0.05(0.05)(0.05) = 0.125(10- 3) m3

sA = - MyA

I =

[-9(103)] (0.025) 14.0625(10- 6)

= 16 MPa (T)

tA = VQA

It =

30(103)[0.125(10- 3)]

14.0625(10- 6)(0.05) = 5.333 MPa

s x = 16 MPa, s y = 0, and t xy = -5.333 MPa

savg = sx + sy

2 =

16 + 0 2

= 8 MPa

The coordinates of the reference point A and the center C of the circle are

A (16, -5.333) C (8, 0)

R = CA = 2(16 - 8)2 + (-5.333)2 = 9.615 MPa s 1 = 8 + 9.615 = 17.6 MPa Ans.

s 2 = 8 - 9.615 = –1.61 MPa Ans.

F9–12 a+"MB = 0; 60(1) - A y (1.5) = 0 A y = 40 kN + c "Fy = 0; 40 - V = 0 V = 40 kN a+"MO = 0; M - 40(0.5) = 0 M = 20 kN · m

I = 1

12 (0.1)(0.23) - 1

12 (0.09)(0.183) = 22.9267(10- 6) m4

QA = y !A ! = 0.095(0.01)(0.1) = 95(10- 6) m3

sA = - MyA

I = -

[20(103)](0.09)

22.9267(10- 6) = -78.51 MPa

= 78.51 MPa (C)

tA = VQA

It =

40(103)[95(10- 6)]

[22.9267(10- 6)](0.01) = 16.57 MPa

s x = -78.51 MPa, s y = 0, and t xy = -16.57 MPa

savg = sx + sy

2 =

-78.51 + 0 2

= -39.26 MPa

The coordinates of the reference point A and the center C of the circle are

A (-78.51, -16.57) C (-39.26, 0)

R = CA = 2[-78.51 - (-39.26)]2 + (-16.57)2 = 42.61 MPa t max in-plane = |R| = 42.6 MPa Ans.

Chapter 11 F11–1 V max = 12 kN M max = 18 kN · m

sallow = Mmax c

I ; 10(10 6 ) =

18(103)(a) 2 3

a = 0.1392 m = 139.2 mm

Use a = 140 mm Ans.

I = 2 3

(0.144) = 0.2561(10 -3 ) m 4

Q max = 0.14

2 (0.14)(0.14) = 1.372(10 -3 ) m 3

t max = V max Q max

It =

12(103)[1.372(10- 3)]

[0.2561(10- 3)](0.14)

= 0.459 MPa < t allow = 1 MPa (OK)

F11–2

V max = 3 kip M max = 12 kip · ft

I = p

4 a d

2 b4 = pd4

sallow = Mmax c

I ; 20 =

12(12)a d 2 b

pd4

d = 4.19 in.

Use d = 4 1 4

in. Ans.

I = p

64 (4.254) = 16.015 in 4

Qmax = 4(4.25>2)

3p c 1

2 ap

4 b(4.252) d = 6.397 in3

tmax = V max Q max

It =

3(6.397) 16.015(4.25)

= 0.282 ksi < t allow = 10 ksi (OK)

F11–3

V max = 10 kN M max = 5 kN · m

I = 1

12 (a)(2a)3 =

2 3

sallow = Mmaxc

I ; 12(106) =

5(103)(a) 2 3

a = 0.0855 m = 85.5 mm Use a = 86 mm Ans.

FUNDAMENTAL PROBLEMS PARTIAL SOLUTIONS AND ANSWERS 839

I = 2 3

(0.0864) = 36.4672(10 -6 ) m 4

Q max = 0.086

2 (0.086)(0.086)

= 0.318028(10 -3 ) m 3

tmax = V max Q max

It =

10(103)[0.318028(10- 3)]

[36.4672(10- 6)](0.086)

= 1.01 MPa 6 tallow = 1.5 MPa (OK)

F11–4

V max = 4.5 kip M max = 6.75 kip · ft

I = 1

12 (4)(h3) =

sallow = Mmax c

I ; 2 =

6.75(12)a h 2 b

h = 7.794 in.

Qmax = y !A ! = h 4

a h 2 b(4) = h2

tmax = V max Q max

It ; 0.2 =

4.5ah2 2 b

3 (4)

h = 8.4375 in. (controls)

Use h = 8 1 2

in. Ans.

F11–5

V max = 25 kN M max = 20 kN · m

I = 1

12 (b)(3b)3 = 2.25b4

sallow = Mmaxc

I ; 12(106) =

20(103)(1.5b)

2.25b4

b = 0.1036 m = 103.6 mm Use b = 104 mm Ans. I = 2.25(0.104 4 ) = 0.2632(10 -3 ) m 4 Q max = 0.75(0.104)[1.5(0.104)(0.104)] = 1.2655(10 -3 ) m 3

tmax = Vmax Q max

It =

25(103)[1.2655(10- 3)]

[0.2632(10- 3)](0.104)

= 1.156 MPa 6 tallow = 1.5 MPa (OK) .

F11–6

V max = 150 kN M max = 150 kN · m

S req’d = Mmax sallow

= 150(103)

150(106) = 0.001 m 3 = 1000(10 3 ) mm 3

Select W410 × 67 [ S x = 1200(10 3 ) mm 3 , d = 410 mm, and t w = 8.76 mm]. Ans.

tmax = V

twd =

150(103) 0.00876(0.41)

= 41.76 MPa 6 tallow = 75 MPa (OK)

Chapter 12 F12–1

a+ "MO = 0; M ( x ) = 30 kN # m EI

d2v

dx2 = 30

EI dv dx

= 30 x + C 1

EIv = 15 x 2 + C 1 x + C 2

At x = 3 m, dv dx

= 0.

C 1 = -90 kN # m 2 At x = 3 m, v = 0 . C 2 = 135 kN # m 3 dv dx

= 1 EI

(30x - 90)

v = 1 EI

(15x2 - 90x + 135)

For end A , x = 0

uA = dv dx

` x = 0

= - 90(103)

200(109)[65.0(10- 6)] = -0.00692 rad

Ans.

vA = v ! x = 0 = 135(103)

200(109)[65.0(10- 6)] = 0.01038 m = 10.4 mm

Ans.

F12–2

a+ "MO = 0; M ( x ) = (-10 x - 10) kN # m EI

d2v

dx2 = -10 x - 10

EI dv dx

= -5 x 2 - 10 x + C 1

EIv = - 5 3

x3 - 5x2 + C1 x + C2

At x = 3 m, dv dx

= 0.

EI (0) = -5(3 2 ) - 10(3) + C 1 C 1 = 75 kN # m 2 At x = 3 m, v = 0.

EI(0) = - 5 3

(33) - 5(32) + 75(3) + C2 C 2 = -135 kN # m 3

840 FUNDAMENTAL PROBLEMS PART IAL SOLUT IONS AND ANSWERS

dv dx

= 1 EI

(-5x2 - 10x + 75)

v = 1 EI

a - 5 3

x3 - 5x2 + 75x - 135b For end A , x = 0

uA = dv dx

` x = 0

= 1 EI

[-5(0) - 10(0) + 75]

= 75(103)

200(109)[65.0(10- 6)] = 0.00577 rad Ans.

vA = v ! x = 0 = 1 EI

c - 5 3

(03) - 5(02) + 75(0) - 135 d = -

135(103)

200(109)[65.0(10- 6)] = -0.01038 m = -10.4 mm

Ans.

F12–3

a+ !MO = 0; M ( x ) = a - 32 x2 - 10x b kN # m EI

d2v

dx2 = -3

2 x2 - 10x

EI dv dx

= - 1 2

x3 - 5x2 + C1

At x = 3 m, dv dx

= 0.

EI (0) = - 1 2

(33) - 5(32) + C1 C 1 = 58.5 kN # m 2

dv dx

= 1 EI

a - 1 2

x3 - 5x2 + 58.5b For end A , x = 0

uA = dv dx

! x = 0 = 58.5(103)

200(109)[65.0(10- 6)] = 0.0045 rad Ans.

F12–4

A y = 600 lb a+ !MO = 0; M ( x ) = (600 x - 50 x 2 ) lb # ft EI

d2v

dx2 = 600 x - 50 x 2

EI dv dx

= 300 x 2 - 16.667 x 3 + C 1

EIv = 100 x 3 - 4.1667 x 4 + C 1 x + C 2 At x = 0, v = 0. EI (0) = 100(0 3 ) - 4.1667(0 4 ) + C 1 (0) + C 2 C 2 = 0 At x = 12 ft, v = 0. EI (0) = 100(12 3 ) - 4.1667(12 4 ) + C 1 (12)

C 1 = -7200 lb · ft 2

dv dx

= 1 EI

(300x2 - 16.667x3 - 7200)

v = 1 EI

(100x3 - 4.1667x4 - 7200x)

vmax occurs where dv dx

= 0.

300 x 2 - 16.667 x 3 - 7200 = 0 x = 6 ft Ans.

v = 1 EI

[100(63) - 4.1667(64) - 7200(6)]

= -27 000(12 in.>ft)3

1.5(106) c 1 12

(3)(63) d = -0.576 in. Ans.

F12–5

a+!MO = 0; M ( x ) = (40 - 5 x ) kN · m

EI d2v

dx2 = 40 - 5 x

EI dv dx

= 40 x - 2.5 x 2 + C 1

EIv = 20 x 2 - 0.8333 x 3 + C 1 x + C 2 At x = 0, v = 0. EI (0) = 20(0 2 ) - 0.8333(0 3 ) + C 1 (0) + C 2 C 2 = 0 At x = 6 m, v = 0. EI (0) = 20(6 2 ) - 0.8333(6 3 ) + C 1 (6) + 0

C 1 = -90 kN · m 2

dv dx

= 1 EI

(40x - 2.5x2 - 90)

v = 1 EI

(20x2 - 0.8333x3 - 90x)

vmax occurs where dv dx

= 0.

40 x - 2.5 x 2 - 90 = 0 x = 2.7085 m

v = 1 EI

[20(2.70852) - 0.83333(2.70853) - 90(2.7085)]

= - 113.60(103)

200(109)[39.9(10- 6)] = -0.01424 m = -14.2 mm

Ans. F12–6

a+!MO = 0; M ( x ) = (10 x + 10) kN · m

EI d2v

dx2 = 10 x + 10

EI dv dx

= 5 x 2 + 10 x + C 1

Due to symmetry, dv dx

= 0 at x = 3 m.

FUNDAMENTAL PROBLEMS PARTIAL SOLUTIONS AND ANSWERS 841

EI (0) = 5(3 2 ) + 10(3) + C 1 C 1 = -75 kN · m 2

dv dx

= 1 EI

[5x2 +10x - 75]

At x = 0,

dv dx

= -75(103)

200(109)(39.9(10- 6)) = -9.40(10- 3) rad Ans.

F12–7

Since B is a fixed support, uB = 0.

uA = | uA>B | = 12 a38EI + 20EI b(3) = 87 kN # m2 EI =

87(103)

200(109)[65(10- 6)] = 0.00669 radb Ans.

! A = | t A/B | = (1.5) c 20EI (3) d + 2 c 12 a18EI b(3) d =

144(103)

200(109)[65(10- 6)] = 0.01108 m = 11.1 mm T

Ans.

F12–8 Since B is a fixed support, uB = 0.

uA = | uA>B | = 12 a50EI + 20EI b(1) + 12 a20EI b(1) = 45 kN # m2 EI =

45(103)

200(109)[126(10- 6)] = 0.00179 radb Ans.

! A = | t A/B | =

(1.6667) c 1 2

a30 EI

b(1) d + 1.5 c 20 EI

(1) d + 0.6667 c 1 2

a20 EI

b(1) d =

61.667 kN # m3 EI

= 61.667(103)

200(109)[126(10- 6)]

= 0.002447 m = 2.48 mm T Ans.

F12–9

Since B is a fixed support, uB = 0.

uA = | uA>B | = 12 c 60EI (1) d + 30EI (2) = 90 kN # m2EI =

90(103)

200(109)[121(10- 6)] = 0.00372 radb Ans.

! A = | t A/B | = 1.6667c12 a60EIb(1)d + (1)c30EI (2)d =

110 kN # m3 EI

= 110(103)

200(109)[121(10- 6)] = 0.004545 m = 4.55 mm T Ans.

F12–10

Since B is a fixed support, uB = 0.

uA = | uA>B | = 12 a18EI b(6) + 13 a 9EI b(3) = 63 kip # ft2EI =

63(122)

29(103)(245) = 0.00128 radb Ans.

! A = | t A/B | = 4 c 12 a18EI b(6) d + (3 + 2.25) c 13 a 9EI b(3) d =

263.25 kip # ft3 EI

= 263.25(123)

29(103)(245) = 0.0640 in. T Ans.

F12–11

Due to symmetry, the slope at the midspan of the beam (point C ) is zero, i.e., uC = 0.

! max = ! C = |t A/C | = (2) c 12 a30EI b(3) d + 1.5 c 10EI (3) d =

135 kN # m3 EI

= 135(103)

200(109)[42.8(10- 6)] = 0.0158 m = 15.8 mm T Ans.

F12–12

t A>B = 2 c 12 a30EI b(6) d + 3 c 10EI (6) d = 360EI uB =

! tA>B ! L

360 EI 6

= 60 EI

The maximum deflection occurs at point C where the slope of the elastic curve is zero.

uB = uB>C 60 EI

= a10 EI

bx + 1 2

a5x EI

bx 2.5 x 2 + 10 x - 60 = 0

x = 3.2915 m ! max = |t B/C | =

2 3

(3.2915)e 1 2

c5(3.2915) EI

d (3.2915)f + 1 2

(3.2915)c10 EI

(3.2915)d =

113.60 kN # m3 EI

= 113.60(103)

200(109)[39.9(10- 6)] = 0.01424 m = 14.2 mm T Ans.

842 FUNDAMENTAL PROBLEMS PART IAL SOLUT IONS AND ANSWERS

F12–13

(vB)1 = Px2

6EI (3L - x) =

40(42) 6EI

[3(6) - 4] = 1493.33 EI

(vB)2 = PL3

3EI =

By(4 3)

3EI =

21.33By EI

(+ c) vB = 0 = (vB)1 + (vB)2

0 = - 1493.33 EI

+ 21.33By

B y = 70 kN Ans. S + !Fx = 0; A x = 0 Ans. + c !Fy = 0; 70 - 40 - A y = 0 A y = 30 kN Ans. a+ !MA = 0; 70(4) - 40(6) - M A = 0 M A = 40 kN # m Ans. F12–14

To use the deflection tables, consider loading as a superposition of uniform distributed load minus a triangular load.

(vB)1 = w0 L4

8EI T (vB)2 =

w0 L4

30EI c (vB)3 =

By L 3

3EI c

(+ c ) vB = 0 = (vB)1 + (vB)2 + (vB)3

0 = - w0 L4

8EI +

w0 L4

30EI +

By L 3

3EI

By = 11w0 L

40 Ans.

S + !Fx = 0; A x = 0 Ans.

+ c !Fy = 0; A y + 11w0 L

40 - 1

2 w0 L = 0

A y = 9w0 L

40 Ans.

a+ !MA = 0; MA + 11w0 L

40 (L) - 1

2 w0 L a23 L b = 0

M A = 7w0 L2

120 Ans.

F12–15

(vB)1 = wL4

8EI =

[10(103)](64)

8[200(109)][65.0(10- 6)] = 0.12461 m T

(vB)2 = By L

3EI =

By (6 3)

3[200(109)][65.0(10- 6)] = 5.5385(10 -6 ) B y c

(+ T ) vB = (vB)1 + (vB)2 0.002 = 0.12461 - 5.5385(10 -6 ) B y B y = 22.14(103) N = 22.1 kN Ans.

S + !Fx = 0; A x = 0 Ans.

+ c !Fy = 0; A y + 22.14 - 10(6) = 0 A y = 37.9 kN Ans. a+ !MA = 0; M A + 22.14(6) - 10(6)(3) = 0 M A = 47.2 kN # m Ans. F12–16

(vB)1 = MOL

6EI(2L) [(2L)2 - L2] =

MO L 2

4EI T

(vB)2 = By (2L)

48EI =

By L 3

6EI c

(+ c ) vB = 0 = (vB)1 + (vB)2

0 = - MO L

4EI +

By L 3

6EI

B y = 3MO 2L

Ans.

F12–17

(vB)1 = Pbx

6EIL (L2 - b2 - x2) =

50(4)(6) 6EI(12)

(122 - 42 - 62)

= 1533.3 kN # m3

EI T

(vB)2 = By L

48EI =

By (12 3)

48EI =

36By EI

(+ c ) vB = 0 = (vB)1 + (vB)2

0 = - 1533.3 kN # m3

EI +

36By EI

B y = 42.6 kN Ans.

F12–18

(vB)1 = 5wL4

384EI =

5[10(103)](124)

384[200(109)][65.0(10- 6)] = 0.20769 T

(vB)2 = By L

48EI =

By (12 3)

48[200(109)][65.0(10- 6)] = 2.7692(10 -6 ) B y c

(+ c ) vB = (vB)1 + (vB)2 - 0.005 = - 0.20769 + 2.7692(10 -6 ) B y B y = 73.19(10 3 ) N = 73.2 kN Ans.

Chapter 13 F13–1

P = p2 EI

(KL)2 =

p2 c29(103 )d cp 4

(0.5)4 d [0.5(50)]2

= 22.5 kip Ans.

s = P A

= 22.5

p(0.5)2 = 28.6 ksi < s Y OK

FUNDAMENTAL PROBLEMS PARTIAL SOLUTIONS AND ANSWERS 843

F13–2

P = p2 EI

(KL)2 =

p2 c1.6(1032d c 1 12

(4)(2)3d [1(12)(12)]2

= 2.03 kip Ans.

F13–3

For buckling about the x axis, K x = 1 and L x = 12 m.

P cr = p2 EIx

(Kx Lx) 2 =

p2 [200(109)][87.3(10- 6)]

[1(12)]2 = 1.197(10 6 ) N

For buckling about the y axis, L = 6 m and K y = 1.

P cr = p2 EIy

(Ky Ly) 2 =

p2 [200(109)][18.8(10- 6)]

[1(6)]2

= 1.031(10 6 ) N (controls)

P allow = Pcr

F.S. =

1.031(106) 2

= 515 kN Ans.

s cr = Pcr A

= 1.031(106)

7.4(10- 3) = 139.30 MPa < s Y = 345 MPa

(OK)

F13–4

A = p[(0.025) 2 - (0.015) 2 ] = 1.257(10 -3 ) m 2

I = 1 4

p3(0.025)4 - (0.015)44 = 267.04(10 -9 ) m 4 P =

p2 EI

(KL)2 =

p2 3200(109)43267.04(10- 9)4 [0.5(5)]2

= 84.3 kN Ans.

s = P A

= 84.3(103)

1.257(10- 3) = 67.1 MPa < 250 MPa (OK)

F13–5

+ c !Fy = 0; FAB a35 b - P = 0 F AB = 1.6667 P (T) S + !Fx = 0; 1.6667Pa45 b - FAC = 0

F AC = 1.3333 P (C)

A = p

4 (22) = p in 2 I =

4 (14) =

4 in4

P cr = F (F.S.) = 1.3333 P (2) = 2.6667 P

P cr = p2 EI

(KL)2

2.6667 P = p2 [29(103)]ap

4 b

[1(4)(12)]2

P = 36.59 kip = 36.6 kip Ans.

s cr = Pcr A

= 2.6667(36.59)

p = 31.06 ksi < s Y = 50 ksi

(OK)

F13–6

a+ !MA = 0; w (6)(3) - F BC (6) = 0 F BC = 3 w

A = p

4 (0.052) = 0.625(10 -3 )p m 2 I =

4 (0.0254)

= 97.65625(10 -9 )p m 4 P cr = F BC (F.S.) = 3 w (2) = 6 w

P cr = p2 EI

(KL)2

6 w = p2 [200(109)][97.65625(10- 9)p]

[1(3)]2

w = 11.215(10 3 ) N>m = 11.2 kN>m Ans. s cr =

Pcr A

= 6[11.215(103)]

0.625(10- 3)p = 34.27 MPa < s Y = 345 MPa

(OK)

Chapter 1 1–1. NE = 0, VE = -200 lb, ME = -2.40 kip # ft 1–2. Na = 500 lb, V a = 0, Nb = 433 lb, V b = 250 lb 1–3. NC = 0, V C = 3.50 kip, MC = -47.5 kip # ft,

ND = 0, V D = 0.240 kip, MD = -0.360 kip # ft 1–5. ND = 0, V D = 0.750 kip, MD = 13.5 kip # ft,

NE = 0, V E = -9.00 kip, ME = -24.0 kip # ft 1–6. NC = -30.0 kN, V C = -8.00 kN,

MC = 6.00 kN # m 1–7. P = 0.533 kN, NC = -2.00 kN, V C = -0.533 kN,

MC = 0.400 kN # m 1–9. ND = 0, V D = -1.875 kN,

MD = 3.94 kN # m 1–10. NA = 0, V A = 450 lb, MA = -1.125 kip # ft,

NB = 0, V B = 850 lb, MB = -6.325 kip # ft, VC = 0, NC = -1.20 kip, MC = -8.125 kip # ft 1–11. NE = -22.5 N, V E = -64.5 N, ME = -2.26 N # m 1–13. Na- a = -100 N, Va - a = 0, Ma - a = -15 N # m 1–14. Nb - b = -86.6 N, V b - b = 50 N,

Mb - b = -15 N # m 1–15. ND = 300 lb, V D = -150 lb, MD = -150 lb # ft 1–17. Na- a = -3.75 kN, V a- a = 1.25 kN,

Ma- a = 3.75 kN # m, Nb - b = -1.77 kN, V b - b = 3.54 kN # m, Mb - b = 3.75 kN # m

1–18. NC = -80 lb, VC = 0, MC = -480 lb # in. 1–19. NC = 0, V C = 4.50 kip, MC = 31.5 kip # ft 1–21. Na- a = 779 N, V a- a = 450 N, Ma- a = 180 N # m: 1–22. FBC = 1.39 kN, FA = 1.49 kN, ND = 120 N,

VD = 0, MD = 36.0 N # m 1–23. NE = 0, V E = 120 N, ME = 48.0 N # m,

Short link: V = 0, N = 1.39 kN, M = 0 1–25. 1V B2x = 105 lb, 1VB2y = 0, 1NB2z = 0,1MB2x = 0, 1MB2y = 788 lb # ft,1TB2z = 52.5 lb # ft 1–26. 1V C2x = -250 N, 1NC2y = 0, 1V C2z = -240 N,1MC2x = -108 N # m, 1TC2y = 0,1MC2z = -138 N # m 1–27. 1NC2x = 150 N, 1V C2y = -260 N,1V C2z = -520 N, 1TC2x = -77.9 N # m,1MC2y = 153 N # m, 1MC2z = -99.0 N # m 1–29. NB = -wru cos u, VB = -wru sin u,

MB = wr21u cos u - sin u2 1–31. tavg = 119 MPa

1–33. savg = P A

sin2 u, tavg = P

2A sin 2u

1–34. sD = 13.3 MPa 1C2, sE = 70.7 MPa 1T2

Selected Answers 1–35. 1sa- a2avg = 1.80 ksi, sb = 4.58 ksi 1–37. P = 40 MN, d = 2.40 m 1–38. s = 66.7 psi, t = 115 psi 1–39. savg = 5 MPa 1–41. P = 37.7 kN 1–42. 1tavg2A = 50.9 MPa 1–43. 1savg2BD = 23.6 MPa, 1savg2CF = 69.4 MPa 1–45. x = 4 in., y = 4 in., s = 9.26 psi 1–46. u = 60.8! 1–47. m = 148 kg 1–49. sAB = 2.17 ksi, sBC = 0.819 ksi 1–50. tavg = 16 psi 1–51. P = 4 kip, 1ta- a2avg = 250 psi 1–53. P = 9.05 kN 1–54. s = 121 psi 1–55. sAB = 127 MPa, sAC = 129 MPa

1–57. savg = g

3 c 1z + h23 - h31z + h22 d

1–58. P = 68.3 kN 1–59. s = 8.15 ksi, t = 5.87 ksi 1–61. tavg = 1.77 MPa 1–62. 1tA2avg = 3.71 ksi 1–63. 1tB2avg = 1.59 ksi 1–65. P = 1.81 kN 1–66. P = 62.5 kN 1–67. s = 5.86 psi

1–69. Use h = 2 3 4

in.

1–70. d = 5.71 mm 1–71. d = 13.5 mm 1–73. F = 3.09 kip

1–74. Use a = 6 1 2

in.

1–75. For nails at A: tavg = 1.53 ksi For nails at B: tavg = 1.68 ksi 1–77. P = 3.26 kip 1–78. dAB = 1.81 mm, dBC = 2.00 mm 1–79. 1F.S.2AB = 1.72, 1F.S.2BC = 2.50 1–81. t = 3.47 mm, r = 126 mm 1–82. 1F.S.2st = 2.14, 1F.S.2con = 3.53 1–83. P = 90 kN, A = 6.19110-32 m2, P max = 155 kN 1–85. Use lA =

1 2

in., lB = 3 4

in.

1–86. dAB = 15.5 mm, dAC = 13.0 mm 1–87. P = 7.54 kN 1–89. t = 1 in., b = 3.46 in. 1–90. P = 9.09 kip

844

1–91. aA! = 130 mm, aB! = 300 mm 1–93. dAB = 6.90 mm, dCD = 6.20 mm 1–94. h = 1.74 in.

1–95. Use aD = 19 5 8

in., aC = 21 1

16 in.

1–97. ND = -2.16 kip, VD = 0, MD = 2.16 kip # ft, V E = 0.540 kip, NE = 4.32 kip, ME = 2.16 kip # ft

1–98. ss = 208 MPa, 1tavg2a = 4.72 MPa,1tavg2b = 45.5 MPa 1–99. Use t =

1 4

in., dA = 1 1 8

in., dB = 13 16

in.

1–101. tavg = 25.5 MPa, sb = 4.72 MPa 1–102. sa- a = 200 kPa, ta- a = 115 kPa 1–103. s40 = 3.98 MPa, s30 = 7.07 MPa,

tavg = 5.09 MPa

Chapter 2 2–1. P = 0.167 in.> in. 2–2. P = 0.0472 in.> in. 2–3. PCE = 0.00250 mm>mm, PBD = 0.00107 mm>mm 2–5. PAC = PAB = 0.00578 mm>mm 2–6. Pavg =

h 1z + h2 - 1

2–7. 1Pavg2AC = 6.04110-32 mm>mm 2–9. "D = 4.38 mm 2–10. 1gA2nt = 0.0502 rad, 1gB2nt = -0.0502 rad 2–11. PAB = -0.00469 in.> in., PAC = 0.0200 in.> in.,

PDB = -0.0300 in.> in. 2–13. PDB = -0.00680 mm>mm,

PAD = 0.0281110-32 mm>mm 2–14. 1Pavg2AC = 0.0258 mm>mm 2–15. 1Pavg2AE = 0.0207 mm>mm 2–17. 1gG2x!y! = 4.50(10- 3) rad 2–18. 1gB2xy = 11.6110-32 rad, 1gA2xy = 11.6110-32 rad 2–19. 1gC2xy = 11.6110-32 rad, 1gD2xy = 11.6110-32 rad 2–21. 1gA2xy = 0.0292 rad, 1gB2xy = 0.0292 rad 2–22. gxy = 0.00880 rad 2–23. Px = 0.00443 mm>mm 2–25. 1Pavg2BC = 6.00110-32 mm>mm 2–26. 1Pavg2CA = -5.59110-32 mm>mm 2–27. 1gE2x!y! = 0.996110-32 rad 2–29. 1Pavg2AC = 0.0168 mm>mm, 1gA2xy = 0.0116 rad 2–30. 1Pavg2BD = 1.60110-32 mm>mm,1gB2xy = 0.0148 rad 2–31. 1"x2B = kL33 , 1Px2avg = kL23 2–33. PAB =

vB sin u L

- uA cos u

Chapter 3 3–1. 1sult2approx = 110 ksi, 1sR2approx = 93.1 ksi,1sY 2approx = 55 ksi, Eapprox = 32.01103) ksi 3–2. E = 55.311032 ksi, ur = 9.96 in # lbin3 3–3. 1ut2approx = 85.0 in # lbin3 3–5. ut = 16.3

in. # kip in3

3–6. E = 8.8311032 ksi 3–7. A = 0.209 in2, P = 1.62 kip 3–9. E = 5.5 psi, ut = 19.25 psi, ur = 11 psi 3–10. E = 30.011032 ksi, PY = 11.8 kip, Pult = 19.6 kip 3–11. Elastic Recovery = 0.003 in.> in., "L = 0.094 in. 3–13. E = 28.611032 ksi 3–14. dBD = 0.0632 in. 3–15. P = 570 lb 3–17. Eapprox = 6.50(103) ksi, sY S = 25.9 ksi 3–18. dP = 0.00637 in. 3–19. sY S = 2.03 MPa 3–21. P = 15.0 kip 3–22. A BC = 0.8 in2, A BA = 0.2 in2 3–23. n = 2.73, k = 4.23110-62 3–25. d = 0.126 mm, "d = -0.00377 mm 3–26. E = 67.9 GPa, n = 0.344, G = 25.3 GPa 3–27. P = 157 kN 3–29. P = 0.08660 mm>mm, g = 0.140 rad 3–30. g = 3.06110-32 rad 3–31. gP = 0.0189 rad 3–33. n = 0.330, h! = 2.000176 in.

3–34. d = Pa

2bhG 3–35. Gal = 4.3111032 ksi 3–37. x = 1.53 m, d!A = 30.008 mm 3–38. u = 0.0139# 3–39. P = 6.48 kip 3–41. L = 10.17 in.

3–42. dV = PL E

11 - 2n2 3–43. Pb = 0.00227 mm>mm, Ps = 0.000884 mm>mm

Chapter 4 4–1. dB = 2.31 mm, dA = 2.64 mm 4–2. dA>D = 0.111 in. away from end D. 4–3. sAB = 22.2 ksi 1T2, sBC = 41.7 ksi 1C2,

sCD = 25.0 ksi 1C2, dA>D = 0.00157 in. towards end D

4–5. dB = 1.59 mm, dA = 6.14 mm 4–6. dA = 0.0128 in. 4–7. dA = -0.194 in.

SELECTED ANSWERS 845

846 SELECTED ANSWERS

4–63. Use d = 1 5 8

in.

4–65. ss = 45.9 MPa, sb = 180 MPa 4–66. db = 59.4 mm, Fs = Fb = 502 kN 4–67. d = 4.90 in. 4–69. sal = 2.46 ksi, sbr = 5.52 ksi, sst = 22.1 ksi 4–70. F = 0.510 kip

4–71. sAB = 2 5 a1T2 - T12E, sBC = 85a1T2 - T12E

4–73. s = 19.1 ksi 4–74. F = 7.60 kip 4–75. d = 0.348 in., F = 19.5 kip

4–77. F = aAE

2 1TB - TA2

4–78. s = 180 MPa 4–79. s = 105 MPa 4–81. F = 904 N 4–82. T2 = 244! 4–83. FAC = FAB = 10.0 lb, FAD = 136 lb 4–85. FAB = FEF = 1.85 kN

4–86. d = c 2E2 + E1 31E2 + E12 dw

4–87. smax = 168 MPa 4–89. P = 49.1 kN 4–90. P = 5.47 kip 4–91. P = 1.34 kip 4–93. s max = 81.7 MPa 4–94. P = 15 kip, K = 1.60 4–95. P = 173 kN 4–97. (a) Fst = 444 N, Fal = 156 N, (b) Fst = 480 N, Fal = 240 N 4–98. dTot = 0.432 in. 4–99. w = 21.9 kN>m, dG = 4.24 mm 4–101. FAB = 3.14 kN, FCD = 2.72 kN,

dCD = 0.324 mm, dAB = 0.649 mm 4–102. (a) P = 2.62 kN, (b) P = 3.14 kN 4–103. FAC = 62 N, FBC = 42 N, P = 125 N 4–105. 1sCF2r = 17.7 MPa 1C2, 1sBE2r = 53.2 MPa 1T2,1sAD2r = 35.5 MPa 1C2 4–106. d =

g2L3

3c2

4–107. d = 3 5 ag

c b 23L53

4–109. dB = 17.8 mm 4–110. (a) dD = 0.375 in., (b) dD = 6.40 in. 4–111. P = 126 kip, "d = 0.00720 in.

4–113. P = sYA 12 cos u + 12, dA = sgLE cos u

4–9. dF = 0.453 mm 4–10. P = 4.97 kN 4–11. dl = 0.0260 in. 4–13. dD = 17.3 mm 4–14. F = 12.0 kN, dA>B = -0.864 mm 4–15. F = 17.0 kN, dA>B = -1.03 mm 4–17. P = 59.5 kN 4–18. 1dA2v = 0.0379 in. 4–19. P = 9.24 kip 4–21. d = 33.9 mm 4–22. W = 9.69 kN 4–25. d = 0.360 mm 4–26. d = 0.00257 in.

4–27. d = P

2apr0 2E

11 - e-2aL2 4–29. p0 = 250 kN>m, d = 2.93 mm 4–30. F = 420 kN, d = 4.57 mm 4–31. sst = 65.9 MPa, scon = 8.24 MPa 4–33. sst = 48.8 MPa, scon = 5.85 MPa 4–34. scon = 1.64 ksi, sst = 11.3 ksi 4–35. P = 114 kip

4–37. FC = c 918ka + pd2E2136ka + 18pd2E dP, FA = a 64ka + 9pd2E136ka + 18pd2E bP 4–38. TAB = 1.12 kip, TAC = 1.68 kip 4–39. A AB = 0.03 in2 4–41. FD = 71.4 kN, FC = 329 kN 4–42. FD = 219 kN, FC = 181 kN 4–43. sAB = sCD = 26.5 MPa, sEF = 33.8 MPa 4–45. sb = 32.4 MPa, st = 34.5 MPa 4–46. FD = 20.4 kN, FA = 180 kN 4–47. P = 198 kN 4–49. FA = 4.09 kip, FB = 2.91 kip 4–50. x = 28.9 in., P = 60.4 kip 4–51. TCD = 27.2 kip, TCB = 9.06 kip 4–53. FBD = 969 N, FAB = FBC = 620 N 4–54. sBD = 45.5 MPa, sAB = 69.1 MPa 4–55. sBE = 96.3 MPa, sAD = 79.6 MPa,

sCF = 113 MPa 4–57. u = 0.0633!

4–58. sAB = sCD = 6P pd2

4–59. u = 6PL pd2Ea

4–61. sAB = 7P

12A , sCD =

P 3A

, sEF = P

12A 4–62. sAB = sAC = 10.4 ksi, sAD = 13.8 ksi

SELECTED ANSWERS 847

5–47. tmax = 44.3 MPa, f = 11.9! 5–49. fA>D = 0.879! 5–50. t max = 2.83 ksi, f = 4.43! 5–51. T = 5.09 kN # m, fA>C = 3.53! 5–53. fB = 5.74! 5–54. Use d = 22 mm, fA>D = 2.54! 5–55. Use d = 25 mm 5–57. tmax = 9.12 MPa, fE>B = 0.585! 5–58. tmax = 14.6 MPa, fB>E = 1.11! 5–59. fB>D = 1.15! 5–61. fB = 1.53! 5–62. fA = 1.78! 5–63. T1 = 18.5 kN # m, T2 = 50.0 kN # m, fA>C = 0.155! 5–65. F = 6.03 N, s = 0.720 mm 5–66. t = 7.53 mm 5–67. v = 131 rad>s 5–69. k = 1.2011062 N>m2, f = 3.56! 5–70. k = 12.2811032 N>m2>3, f = 2.97! 5–73. f =

T 2apG

11 - e-4aL2 5–74. t0 =

4Pd L

, f = 4PLd

3pc4G 5–75. fA>B = 5.62! 5–77. 1tAC2max = 14.3 MPa, 1tCB2max = 9.55 MPa 5–78. t

max abs = 9.77 MPa

5–79. tmax = 29.3 ksi 5–81. TA = 12.3 kip # ft, TD = 7.72 kip # ft 5–82. fC = 0.116!, 1tst2max = 395 psi, 1gst2max = 34.3110-62 rad, 1tbr2max = 96.1 psi, 1gbr2max = 17.2110-62 rad 5–83. 1tBC2max = 1.47 ksi, 1tBD2max = 1.96 ksi,

fB>C = fB>D = 0.338! 5–85. 1tst2max = 86.5 MPa, 1tmg2max = 41.5 MPa, 1tmg2 ! r= 0.02 m = 20.8 MPa 5–86. TB = 222 N # m, TA = 55.6 N # m 5–87. fE = 1.66! 5–89. t

max abs = 64.1 MPa

5–90. fC>D = 6.22! 5–91. t

max abs = 5.50 ksi

5–93. TB = 37

189 T, TA =

152 189

5–94. TB = 7t0 L 12

, TA = 3t0 L

4 5–95. T = 7.74 N # m 5–97. Factor of increase in max. shear stress =

5–98. 1tBC2max = 0.955 MPa, 1tAC2max = 1.59 MPa, fB>A = 0.207!

4–114. sb = 33.5 MPa, sr = 16.8 MPa 4–115. T = 507! C 4–117. sAB = sBC = 145 ksi 4–119. FB = 2.13 kip, FA = 2.14 kip 4–121. FB = 86.6 lb, FC = 195 lb 4–122. dA>B = 0.491 mm

Chapter 5 5–1. r" = 0.841r 5–2. r" = 0.707r 5–3. tB = 6.04 MPa, tA = 6.04 MPa 5–5. tA = 3.45 ksi, tB = 2.76 ksi 5–6. 1tBC2max = 5.07 ksi, 1tDE2max = 3.62 ksi 5–7. 1tEF2max = 0, 1tCD2max = 2.17 ksi 5–9. tB = 6.79 MPa, tA = 7.42 MPa

5–10. n = 2r 3

Rd 2

5–11. tAB = 7.82 ksi, tBC = 2.36 ksi 5–13. Use t = 25 mm

5–14. Use d = 1 3 4

in.

5–15. Use d = 33 mm 5–17. tmax = 4.89 ksi 5–18. tmax = 7.33 ksi 5–19. T = 7.54 kN # m 5–21. tC = 35.8 MPa 5–22. d = 39.4 mm 5–23. t0 = 1.57 kN # m>m 5–25. t

max abs = 3.59 ksi

5–26. tmax = T

2pr i 2h

5–27. T = 1.57 kip # ft, P = 5.11 kip 5–29. tavg = 1.44 MPa 5–30. T0 = 670 N # m, tmax abs = 6.66 MPa 5–31. 1tAB2max = 1.04 MPa, 1tBC2max = 3.11 MPa 5–33. do = 1.09 in., di = 0.819 in. 5–34. v = 82.0 rad>s 5–35. v = 21.7 rad>s 5–37. tmax = 6.02 ksi 5–38. dA = 12.4 mm, dB = 16.8 mm 5–39. 1tmax2CF = 12.5 MPa, 1tmax2BC = 7.26 MPa 5–41. t = 2.3 mm 5–42. v = 17.7 rad>s 5–43. t max =

2TL3

p3rA 1L - x2 + rBx 43 5–45. 1tB2max = 18.3 ksi 5–46. Use di = 1

5 8

in.

848 SELECTED ANSWERS

5–99. 1tBC2max = 0.955 MPa, 1tAC2max = 1.59 MPa, fB>C = 0.0643! 5–101. T = 1.35 kip # ft, f = 3.42! 5–102. TB = 32 lb # ft, TA = 48 lb # ft, fC = 0.0925! 5–103. tavg = 3.35 ksi 5–105. tmax = 61.1 MPa, fB = 0.700! 5–106. Use a = 47 mm, fB = 0.897! 5–107. T = 6.65 kip # ft 5–109. Factor of increase = 1.66 5–110. Factor of increase = 2.85 5–111. qsq =

4 qcr

5–113. t = 0.104 in. 5–114. T = 2 kip # ft 5–115. b = 0.773 in. 5–117. T = 4.73 MN # m, f = 0.428!>m 5–118. tavg = 119 MPa, f = 0.407!>m 5–119. 1tavg2A = (tavg)B = 357 kPa 5–121. No, it is not possible. 5–122. P = 101 kW 5–123. tmax = 50.6 MPa 5–125. r = 0.075 in. 5–126. TY = 1.26 kN # m, f = 3.58!, f" = 4.86! 5–127. Tp = 0.105 N # m 5–129. T = 20.8 kN # m, f = 34.4!, fr = 12.2! 5–130. T = 14.4 kip # ft 5–131. TY = 12.6 kN # m, Tp = 16.8 kN # m 5–133. T = 110 lb # ft 5–134. T = 19.2 kN # m, f = 24.9!, fr = 6.72! 5–135. rY = 0.542 in., f = 6.34! 5–137. T = 331 lb # ft 5–138. T = 41.2 kip # ft, 1tr2r= co = 2.44 ksi, 1tr2r= ci = -3.78 ksi 5–139. T = 3.27 kN # m, f = 68.8! 5–141. Tt = 7.39 kN # m, Tc = 7.61 kN # m 5–142. Tp = 13.6 kN # m, fr = 12.3! 5–143. Use d = 26 mm, fA>C = 2.11! 5–145. t = 88.3 MPa, f = 4.50! 5–146. T = 381 kN # m, f = 0.0310! 5–147. The circular shaft will resist the largest torque. For the square shaft: 73.7%, For the triangular shaft: 62.2% 5–149. P = 2.80 kip 5–150. fA = 1.59! 5–151. P = 1.10 kW, tmax = 825 kPa

Chapter 6 6–1. T1 = 250 lb, T2 = 200 lb

M (lb#in)

V (lb)

$600

$50

200

6–2.

M (lb#in)

V (lb)

82.2

2.24

1151 1196

$108

$420

6–3.

M (lb#in)

V (lb)

$6000

$2000

1200

SELECTED ANSWERS 849

6–5.

M (kN!m)

V (kN)

"39 "15

"75

6–6. For 0 … x 6 L 2

: V = w0 L 24

, M = w0 L 24

For L 2

6 x … L: V = w0 24L

3L2 - 612x - L224 , M =

24L 3L2x - 12x - L234

0 0.5 L

0.5 L 0.704 L L

0.704 L L

5 24

w0L

0.0208 w0L2 0.0265 w0L 2

w0L 24

6–7.

M (kip"ft)

V (kip)

!6 !4

6 ft 4 ft

!24

6–9. For 0 … x 6 4 m: V = 59 - 6x6 kN, M = 59x - 3x26 kN # m,

For 4 m 6 x … 6 m: V = 5616 - x26 kN # m, M = - 5316 - x226 kN # m

M (kN!m)

x (m)

V (kN)

1.5 4

x (m) 1.5

4 6.75

"12

"15

6–10.

M (lb!ft)

x (ft)

V (lb)

150

"225

1.5 3

x (ft) 1.5

225

6–11.

M (kN!m)

V (kN)

"0.4

850 SELECTED ANSWERS

6–13.

M (kN!m)

V (kN)

"30

6–14.

M (lb!in)

V (lb)

9 40

"3875

"393.8"378.8

115

"12

6–17. V = 5-300 - 16.67x26 lb, M = 5-300x - 5.556x36 lb # ft

M (lb!ft)

V (lb)

"300

"900

"3000

6–18. For 0 … x 6 6 ft: V = 530.0 - 2x6 kip, M = 5-x2 + 30.0x - 2166 kip # ft,

For 6 ft 6 x … 10 ft: V = 8.00 kip, M = 58.00x - 1206 kip # ft

x (ft)

V (kip)

30.0

"216

6 10

18.0 8.00

x (ft)

M (kip!ft)

6 10

"72.0 "40.0

6–19.

M (kip!ft)

V (kip)

"10

"0.5

"27.5"25

2.5

6–21. Mmax = 281 lb # ft

M (lb!ft)

V (lb)

281

"75

SELECTED ANSWERS 851

6–22.

M (kN!m)

V (kN)

"10

"18

x (m)

3 6

6–23.

M (kip!ft)

V (kip)

21 21

6–25. V = 1050 - 150x M = -75x2 + 1050x - 3200

M (lb!ft)

V (lb)

"200 "200

"450

475

450

6–27.

7w0L 36

"w0L 18

0.0345w0L2

0.707 L

"0.00617w0L2

"w0L 4

6–29.

M (kN!m)

x (m)

V (kN)

x (m)

"12"12

"9 "9

12 12

1.5 3

1.5 3 4.5 6

4.5

6–30.

M (kip!ft)

V (kip)

852 SELECTED ANSWERS

6–35. For 0 … x 6 3 m: V = 200 N, M = 1200 x2 N # m, For 3 m 6 x … 6 m: V = e - 100

3 x2 + 500 f N,

M = e - 100 9

x3 + 500x - 600 f N # m

x (m)

V (N)

0 200

3.87

3 3.87

691600

x (m)

M (N!m)

"700

6–37.

M (kN!m)

V (kN)

"112.5

169

112.5

6–38.

M (kN!m)

V (kN)

"10.5 "11.4

"11

4 3.252.95

2.75

3.25

2.75 2.95

x (m)

6–31.

"wL

wL2

6–33.

M (N!m)

x (m)

V (N)

"250

400 650

1 2 3

1 0

650

2 3

6–34.

M (kN!m)

x (m)

V (kN)

31.5

SELECTED ANSWERS 853

6–43.

M (kN!m)

x (m)

V (kN)

"4.5

4.5

6 4.5

x (m) 4.5 6

3.375

3 0

"19.5

6.5

6–45.

M (kN!m)

x (m)

V (kN)

"7.5

x (m) 6 7.5

6 7.5

11.25

4.5

7.5

"54

14.25

6–46. b = 0.207 L

x L

"0.293ga2L "0.207ga2L

"0.0214ga2L2"0.0214ga2L2

0.0214ga2L2

0.293ga2L 0.207ga2L

0.207L

0.5L 0.793L

0.207L

0.5L

0.793L

6–39.

M (lb!ft)

x (ft)

V (lb)

"400

x (ft)

"600

"1200 "1200

"300

600 400

3 6

3 6 9 12

6–41.

M (N!m)

x (m)

V (N)

"800

"400

400

200

2 0

"2800

1000

2 4

5 6

6–42.

x (m)

V (kN)

0 5.00

"10.0 "5.00

10.0

5.00

431

x (m)

M (kN!m)

"7.50

2.50

1 2

3 4

854 SELECTED ANSWERS

6–47. r = 909 mm, M = 61.9 N # m 6–49. 1st2max = 3.72 ksi, 1sc2max = 1.78 ksi 6–50. M = 2.50 kip # ft 6–51. 1smax2t = 2.40 ksi, 1smax2c = 4.80 ksi 6–53. FR = 200 kN 6–54. 1s max 2c = 78.1 MPa, 1s max 2t = 165 MPa 6–55. M = 50.3 kN # m 6–57. s max = 49.4 MPa 6–58. sB = 3.61 MPa, sC = 1.55 MPa 6–59. 1smax2t = 6.71 MPa, 1smax2c = 3.61 MPa 6–61. % of moment carried by web = 22.6% 6–62. sA = 6.21 MPa 1C2, sB = 5.17 MPa 1T2 6–63. sA = 214 psi (C), sB = 33.0 psi (T),

sC = 115 psi (T) 6–65. (a) smax = 497 kPa, (b) smax = 497 kPa 6–66. (a) smax = 249 kPa, (b) smax = 249 kPa 6–67. smax = 158 MPa 6–69. smin = 74.7 MPa 6–70. smax = 22.1 ksi 6–71. smax = 21.1 ksi 6–73. d = 1.28 in. 6–74. smax = 45.1 ksi 6–75. smax = 52.8 MPa 6–77. 1smax2c = 1.87 ksi, 1smax2t = 1.37 ksi 6–78. M = 2.92 kip # ft 6–79. FR = 4.23 kip 6–81. M = 123 kN # m 6–82. smax = 24.4 ksi

6–83. Use d = 3 1

16 in.

6–85. w = 18.75 kN>m 6–86. s max = 19.1 ksi 6–87. d = 2 in. 6–89. a = 66.9 mm

6–90. s max = 23w0 L2

36 bh2 6–91. s max = 119 MPa 6–93. s max = 25.8 ksi 6–94. P = 10.4 kN 6–95. s max = 11.5 MPa 6–97. P = 114 kip 6–98. s max = 7.59 ksi 6–99. s max = 5.60 ksi 6–101. d = 410 mm 6–102. 1smax2t = 2.77 MPa, 1smax2c = 3.84 MPa 6–103. P = 2.60 kN 6–105. sA = -119 kPa, sB = 446 kPa, sD = -446 kPa,

sE = 119 kPa

6–106. a = 0; b = - aMzIy + MyIyz IyIz - I2yz

b ; c = MyIz + MzIyz IyIz - I2yz

6–107. smax = 5.66 MPa 1C2, a = 79.8! 6–109. s max = 4.70 MPa, a = -76.0! 6–110. M = 5.11 kN # m 6–111. smax = 131 MPa 1C2, a = -66.5! 6–113. When u = 3!: sA = 925 psi, a = -25.3!,

When u = 0!: sA = 800 psi 6–114. smax = 3.33 ksi (T), a = -63.1! 6–115. smax = 2.90 MPa, a = -66.6! 6–117. sA = 2.60 MPa 6–118. smax = 151 MPa, a = 72.5! 6–119. w = 4.37 kN>m 6–121. M = 128 kN # m 6–122. 1smax2st = 22.6 ksi, 1s max2al = 13.3 ksi 6–123. w = 0.875 kip>ft 6–125. M = 35.0 kN # m 6–126. 1s max2st = 123 MPa, 1smax2w = 5.14 MPa 6–127. smax = 20.1 MPa 6–129. 1sst2 max = 3.70 MPa, 1sw2 max = 0.179 MPa 6–130. 1s pvc2max = 1.53 ksi 6–131. d = 531 mm, M = 98.6 kN # m 6–133. M = 103 kN # m 6–134. 1smax2st = 154 MPa, 1smax2al = 171 MPa 6–135. 1smax2st = 8.51 ksi, 1smax2w = 0.558 ksi 6–137. % of error = 22.3% 6–138. M = 51.8 kN # m 6–139. smax = 842 psi 1T2 6–141. sA = 43.7 MPa 1T2, sB = 7.77 MPa 1T2,

sC = 65.1 MPa 1C2 6–142. P = 6.91 kN 6–143. 1s max2t = 204 psi, 1s max2c = 120 psi 6–145. 1smax2c = 792 kPa, 1smax2t = 1.02 MPa 6–146. 1smax2t = 1.16 ksi, 1smax2c = 0.994 ksi 6–147. sA = 10.1 ksi 1C2, sB = 0.955 ksi 1T2,

sC = 5.94 ksi 1T2 6–149. smax = {1.39 MPa 6–150. r = 8.0 mm 6–151. s max = 14.6 MPa 6–153. s max = 54.4 MPa 6–154. P = 122 lb 6–155. s max = 29.5 ksi 6–157. M = 41.7 N # m 6–158. k = 1.17 6–159. stop = sbottom = 43.5 Mpa 6–161. stop = sbottom = 142 MPa 6–162. s top = s bottom = 67.1 MPa 6–163. Mp = 172 kip # ft 6–165. MY = 143 kip # ft, Mp = 243 kip # ft 6–166. k = 1.16

SELECTED ANSWERS 855

7–33. F = 675 lb 7–34. V = 16.7 kN 7–35. V = 28.2 kN 7–37. V = 2571 lb, tb = 14.0 ksi 7–38. t = 35.2 MPa 7–39. F = 12.5 kN 7–41. s = 71.3 mm

7–42. V = 8.82 kip, use s = 1 1 8

in.

7–43. V = 1.24 kip 7–45. P = 6.60 kN 7–46. tavg = 97.2 MPa 7–47. s = 8.66 in., s! = 1.21 in. 7–50. qA = 228 kN>m, qB = 462 kN>m 7–51 qC = 0, qD = 601 kN>m 7–53. qC = 38.6 kN>m 7–54. qA = 1.39 kN>m, qB = 1.25 kN>m 7–55. q max = 1.63 kN>m 7–57. q max = 414 lb> in. 7–58. qA = 215 kN>m 7–59. q max = 232 kN>m 7–61. qA = 196 lb> in., qB = 452 lb> in.,

q max = 641 lb> in. 7–62. t =

V pR2t

2R2 - y2 7–63. e = 1.07 in. 7–65. VAB = 1.47 kip

7–66. e = 3

14 a

7–67. e = 33h2b2 - 1h - 2h122b214

h3 + 6bh2 + 6b11h - 2h122 7–69. e = c 31p + 42

4 + 3p d r 7–70. e =

4r1sin a - a cos a2 2a - sin 2a

7–71. FC = 197 lb, FD = 1.38 kip 7–73. qA = 0, qB = 1.21 kN>m, qC = 3.78 kN>m 7–74. tB = 795 psi, tC = 596 psi 7–75. t max = 928 psi

Chapter 8 8–1. t = 18.8 mm 8–2. ro = 75.5 in. 8–3. (a) s1 = 1.04 ksi, s2 = 0, (b) s1 = 1.04 ksi, s2 = 520 psi 8–5. s1 = 600 psi, s2 = 0 8–6. s1 = 600 psi, s2 = 300 psi 8–7. (a) s1 = 127 MPa, (b) s1! = 79.1 MPa, 1tavg2b = 322 MPa 8–9. shoop = 7.20 ksi, slong = 3.60 ksi

6–167. Elastic: P = 66.7 kN, Plastic: P = 100 kN 6–169. MY = 130.5 kip # ft, Mp = 223 kip # ft 6–170. Elastic: P = 37.3 kip, Plastic: P = 45.5 kip

6–171. k = 16ro1r3o - r3i 2 3p1r4o - r4i 2

6–173. MY = 18 kip # ft, Mp = 36 kip # ft

6–174. Elastic: w = 4.27 kip>ft, Plastic: w = 6.40 kip>ft

6–175. Elastic: w0 = 18.0 kip>ft, Plastic: w0 = 22.8 kip>ft

6–177. P = 100 lb 6–178. M = 94.7 N # m 6–179. Maximum elastic moment: M = 35.0 kip # ft,

Ultimate moment: M = 59.8 kip # ft 6–181. M = 73.5 kip # ft 6–182. M = 81.7 kip # ft 6–183. k = 1.22 6–186. M = 14.9 kN # m 6–187. M = 26.4 kN # m 6–189. s max = 8.41 ksi 6–190. sA = 225 kPa 1C2, sB = 265 kPa 1T2 6–191. V = 20 - 2x, M = -x2 + 20x - 166

6–194. smax = 6M

a3 1cos u + sin u2,

u = 45", a = 45"

Chapter 7 7–1. tA = 2.56 MPa 7–2. t max = 3.46 MPa 7–3. Vw = 19.0 kN 7–5. Vf = 3.82 kip 7–6. V max = 100 kN 7–7. t max = 17.9 MPa 7–9. V = 32.1 kip 7–10. t max = 4.48 ksi 7–11. t max = 45.0 MPa 7–13. t max = 4.22 MPa 7–14. V = 190 kN 7–15. V AB = 50.3 kN 7–17. tA = 1.99 MPa, tB = 1.65 MPa 7–18. t max = 4.62 MPa 7–19. Vw = 27.1 kN 7–21. P = 1.28 kip 7–22. tB = 4.41 MPa 7–23. tmax = 4.85 MPa 7–25. t max = 3.67 MPa 7–26. a = 1.27 in. 7–27. t max = 22.0 MPa, 1t max 2s = 66.0 MPa

856 SELECTED ANSWERS

8–58. sC = 107 MPa (C), tC = 15.3 MPa, sD = 0, tD = 15.8 MPa

8–59. smax = 71.0 MPa (C) 8–61. (smax)t = 106 MPa, (smax)c = -159 MPa 8–62. P = 9.08 kN 8–63. sC = 15.6 ksi (T), sD = 124 ksi (T),

tD = 62.4 ksi, tC = 52.4 ksi 8–65. sA = 605 psi (T), tA = 327 psi 8–66. sB = 466 psi (C), tB = 422 psi 8–67. a = 61.9 mm. Remove material equally from

both sides, s = 15.5 MPa. 8–69. sB = -21.7 MPa, tB = 0 8–70. tA = 0, sA = 30.2 ksi (C) 8–71. sB = 0, tB = 0.377 ksi 8–73. s = 1.48 psi (T), t = 384 psi 8–74. 1st2max = 15.8 ksi, 1sc2max = -10.5 ksi 8–75. sE = 802 kPa, tE = 69.8 kPa 8–77. u = 0.286! 8–78. u = 0.215! 8–79. 1smax2AB = 667 psi, 1smax2CD = 40.7 ksi 8–81. P = 94.2 kN 8–82. smax = 236 psi (C) 8–83. s1 = 7.07 MPa, s2 = 0 8–85. sC = 11.6 ksi, tC = 0, sD = -23.2 ksi, tD = 0 8–86. sC = 10.4 ksi, tC = 0, sD = -20.8 ksi, tD = 0

Chapter 9 9–2. sx" = -4.05 ksi, tx"y" = -0.404 ksi 9–3. sx" = -388 psi, tx"y" = 455 psi 9–5. sx" = 1.45 ksi, tx"y" = 3.50 ksi 9–6. sx" = -5 MPa, tx"y" = 40 MPa 9–7. sx" = -5 MPa, tx"y" = 40 MPa 9–9. sx" = 56.25 MPa, sy" = -31.25 MPa, tx"y" = -75.8 MPa 9–10. sx" = 47.5 MPa, sy" = 202 MPa, tx"y" = -15.8 MPa 9–11. sx" = 177 MPa, sy" = 72.5 MPa,

tx"y" = -59.2 MPa 9–13. sx" = -898 psi, tx"y" = 605 psi, sy" = 598 psi 9–14. s1 = 4.21 ksi, s2 = -34.2 ksi,

up2 = 19.3! and up1 = -70.7!, tmax

in@plane = 19.2 ksi, savg = -15 ksi, us = -25.7!

and 64.3! 9–15. s1 = 53.0 MPa, s2 = -68.0 MPa, up1 = 14.9! and up2 = -75.1!, savg = -7.50 MPa, tmaxin@plane = 60.5 MPa, us = -30.1! and 59.9!

8–10. s1 = 1.60 ksi, p = 25 psi, d = 0.00140 in. 8–11. (a) T = 18.2 kip # ft,

(b) P = 18.1 kip, (c) F = 9.05 kip

8–13. T1 = 128!, s1 = 12.1 ksi, p = 252 psi

8–14. dri = pr2i

E1ro - ri2 8–15. p =

E1r2 - r32 r 22

r2 - r1 +

r 23 r4 - r3

8–17. sfil = pr

t + t"w >L + Twt", sw =

t + t"w >L - TLt 8–18. d = 66.7 mm 8–19. sL = 66.7 MPa (C), sR = 33.3 MPa (T) 8–21. smax = sL = 13.9 ksi (T), sR = 13.6 ksi (C) 8–22. smax = 1.07 MPa 8–23. sconst = 1.07 MPa 8–25. sB = 5.35 ksi, tB = 0 8–26. smax = 2.34 MPa (C) 8–27. P = 128 kN 8–29. P = 11.8 kN 8–30. sA = 25 MPa (C), sB = 0,

tA = 0, tB = 5 MPa 8–31. d = 66.7 mm 8–33. sB = 8.89 ksi (C), tB = 0,

sA = 720 psi (T), tA = 0 8–34. sE = 8.89 ksi (T), tE = 0, sF = 0, tF = 240 psi 8–35. sA = sB = 306 psi (C),

tA = 8.46 ksi, tB = 5.64 ksi 8–37. sB = 1.53 MPa (C), tB = 100 MPa 8–38. sD = -88.0 MPa, tD = 0 8–39. sE = 57.8 MPa, tE = 864 kPa 8–41. sB = 3.26 MPa (T), tB = 0.209 MPa 8–42. sA = 3.31 ksi (T), tA = 0.581 ksi 8–43. sB = 1.99 ksi (C), tB = 0.510 ksi 8–45. sA = 1.00 ksi (C), sB = 3.00 ksi (C)

8–46. smax = 1.33P

a2 (C), smin =

3a2 (T)

8–47. sD = 0, tD = 80.8 psi, sE = -501 psi, tE = 93.9 psi

8–49. sD = -126 psi, tD = 57.2 psi, sE = -347 psi, tE = 66.4 psi

8–51. 6ey + 18ez 6 5a 8–53. sA = 6.61 ksi (T), tA = 1.39 ksi 8–54. sB = 5.76 ksi (C), tB = 1.36 ksi 8–55. sA = 7.20 MPa (T), tA = 0.6 MPa 8–57. sA = 107 MPa (T), tA = 15.3 MPa,

sB = 0, tB = 14.8 MPa

SELECTED ANSWERS 857

9–54. s1 = 87.7 MPa, s2 = -128 MPa, 1up21 = 10.9!1clockwise2, savg = -20 MPa,

tmax in@plane

= -108 MPa, us = 34.1! 1counterclockwise2 9–55. s1 = 3.51 ksi, s2 = -28.5 ksi, 1up21 = 19.3!(clockwise), tmaxin@plane = 16.0 ksi, savg = -12.5 ksi, us = 25.7! 1counterclockwise2 9–57. s1 = 64.1 MPa, s2 = -14.1 MPa, up2 = 25.1! savg = 25.0 MPa, tmaxin@plane = 39.1 MPa, us = -19.9! 9–58. sx" = -421 MPa, tx"y" = -354 MPa, sy" = 421 MPa 9–59. (a) s1 = -5.53 ksi, s2 = -14.5 ksi, up = -31.7! (b) tmax

in@plane = 4.47 ksi, savg = -10 ksi,

us = 13.3! 9–62. sx" = 19.5 kPa, tx"y" = -53.6 kPa 9–63. tmax

in@plane = 41.0 psi, s1 = 0.976 psi,

s2 = -81.0 psi 9–65. s1 = 29.4 ksi, s2 = -17.0 ksi 9–66. s1 = 0.361 MPa, s2 = -5.36 MPa,

tmax in@plane

= 2.86 MPa 9–67. s1 = 0.942 MPa, s2 = -3.30 MPa,

tmax in@plane

= 2.12 MPa 9–69. s1 = 4.71 ksi, s2 = -0.0262 ksi 9–70. Regardless of the orientation of the element,

the shear stress is zero and the state of stress is represented by the same two normal stress components.

9–71. sx" = 500 MPa, tx"y" = -167 MPa 9–73. s1 = 1.15 MPa, s2 = -0.0428 MPa 9–74. s1 = 2.97 ksi, s2 = -2.97 ksi, up1 = 45.0!,

up2 = -45.0!, tmaxin@plane = 2.97 ksi, us = 0! 9–75. s1 = 2.59 ksi, s2 = -3.61 ksi, up1 = -40.3!,

up2 = 49.7, tmaxin@plane = 3.10 ksi, us = 4.73! 9–79. s1 = 100 MPa, s2 = 50 MPa, s3 = 0, tabs

max = 50 MPa

9–81. s2 = -0.807 ksi, s3 = -6.19 ksi, s1 = 0, tabs

max = -3.10 ksi

9–82. s1 = 222 MPa, s2 = 0 MPa, s3 = -102 MPa, tabs

max = 162 MPa

9–83. s1 = 6.73 ksi, s2 = 0, s3 = -4.23 ksi, tabs

max = 5.48 ksi

9–85. The stress in every direction is s1 = s2 = s3 = -p

9–86. s1 = 2.50 ksi, s2 = s3 = 0, tabsmax = 1.25 ksi 9–87. s1 = 583 psi, s2 = 0, s3 = -926 psi, tabs

max = 755 psi

9–17. s1 = 137 MPa, s2 = -86.8 MPa, up1 = -13.3!, up2 = 76.7!, tmaxin@plane = 112 MPa, us = 31.7! and 122!, savg = 25 MPa 9–18. sx = 33.0 MPa, sy = 137 MPa, txy = -30 MPa 9–19. s1 = 5.90 MPa, s2 = -106 MPa,

up1 = 76.7! and up2 = -13.3!, tmax

in@plane = 55.9 MPa, savg = -50 MPa,

us = 31.7! and 122! 9–21. ta = -1.96 ksi, s1 = 80.1 ksi, s2 = 19.9 ksi 9–22. sx" = 19.5 kPa, tx"y" = -53.6 kPa 9–23. sx" = 0.507 MPa, tx"y" = 0.958 MPa 9–25. sy = -824 psi 9–26. s1 = 29.8 ksi, s2 = 0, tmaxin@plane = 14.9 ksi, us = -45! and 45! 9–27. s1 = 0, s2 = -22.90 ksi, tmaxin@plane = 11.5 ksi, us = 45! and 135! 9–29. s1 = 0.939 ksi, s2 = -1.36 ksi, tmaxin@plane = 1.15 ksi 9–30. sx" = -63.3 MPa, tx"y" = 35.7 MPa 9–31. s1 = 6.38 MPa, s2 = -0.360 MPa,

up1 = 13.4! and up2 = -76.6! 9–33. Point A: s1 = 0, s2 = -192 MPa, up1 = 0, up2 = 90!, Point B: s1 = 24.0 MPa, s2 = -24.0 MPa, up1 = -45.0!, up2 = 45.0! 9–34. s1 = 0, s2 = -1.34 ksi, tmaxin@plane = 668 psi, us = 45! and -45! 9–35. tmax

in@plane = 5 kPa, savg = 0

9–37. s1 = 4

pd2 a2PL

d - Fb , s2 = 0,

tmax in@plane

= 2

pd2 a2PL

d - Fb

9–38. tx"y" = -47.5 kPa 9–39. sx" = 82.3 kPa 9–41. s1 = 1.37 MPa, s2 = -198 MPa 9–42. s1 = 2.97 ksi, s2 = -4.12 ksi, tmaxin@plane = 3.55 ksi

9–43. s1 = 4.33 MPa, s2 = -13.0 MPa 9–45. sx" = -5 MPa, tx"y" = 40 MPa 9–46. s1 = 4.21 ksi, s2 = -34.2 ksi,

up2 = 19.3! and up1 = -70.7!, tmax

in@plane = 19.2 ksi, savg = -15 ksi, us = 64.3!

9–47. sx" = 47.5 MPa, tx"y" = -15.8 MPa, sy" = 202 MPa 9–49. s1 = 54.2 MPa, s2 = -4.15 MPa, up = -15.5! savg = 25 MPa, tmaxin@plane = 29.2 MPa, us = 29.5!

9–51. sx" = 10 ksi, tx"y" = -5 ksi, sy" = 0 9–53. sx" = 736 MPa, sy" = -156 MPa, tx"y" = -188 MPa

858 SELECTED ANSWERS

10–18. gmax in@plane

= 335110- 62, Pavg = -30110- 62, us = -31.7! 10–19. P1 = 1039110- 62, P2 = 291110- 62, up = 30.2!, gmax

in@plane = 748110- 92, Pavg = 665110- 62,

us = -14.8! 10–21. Px" = 466110- 62, gx"y" = -393110- 62, Py" = -116110- 62 10–22. (a) P1 = 773110- 62, P2 = 76.8110- 62, (b) gmax

in@plane = 696110- 62,

= 773110- 62 10–23. (a) P1 = 870110- 62, P2 = 405110- 62, (b) gmax

in@plane = 465110- 62,

= 870110- 62 10–25. P1 = 1434110- 62, P2 = -304110- 62, gmax

in@plane = 1738110- 62, Pavg = 565110- 62

10–26. P1 = 336110- 62, P2 = -536110- 62, up1 = 11.7!1counterclockwise2, gmax

in@plane = 872110- 62, Pavg = -100110- 62,

us = 33.3!1clockwise2 10–27. P1 = 301110- 62, P2 = -401110- 62,

up2 = 27.6!1clockwise2, gmax

in@plane = 702110- 62, Pavg = -50110- 62,

us = 17.4!1counterclockwise2 10–33. E = 17.4 GPa, #d = -12.6110- 62 mm 10–34. #u = -0.0103! 10–35. npvc = 0.164 10–37. (a) kr = 3.33 ksi,

(b) kg = 5.1311032 ksi 10–38. p = 0.967 ksi, gmax

in@plane = 1.30110- 32

10–39. P = 13.0 kip, gxy = -13.7110- 62 10–41. Px = 23.0110- 62, gxy = -3.16110- 62 10–42. Px = 2.35110- 32, Py = -0.972110- 32,

Pz = -2.44110- 32 10–43. p = 4 MPa

10–45. u = tan - 1a 12n b 10–46. s1 = 8.37 ksi, s2 = 6.26 ksi 10–47. P1 = 833110- 62, P2 = 168110- 62, P3 = -763110- 62 10–49. sx = 15.5 ksi1C2, sy = 16.8 ksi1C2 10–50. Px = Py = 0, gxy = -160110- 62, T = 65.2 N # m 10–51. Px" = -2.52110- 32, Py" = 2.52110- 32 10–53. #d = 0.800 mm, sAB = 315 MPa 10–54. #d = 0.680 mm 10–57. #V = 0.0168 m3 10–58. k = 1.35 10–59. s2x + s2y - sxsy + 3t2xy = s2y 10–61. s1 = 10.2 ksi

9–89. s1 = 10.7 MPa, s2 = -35.8 MPa 9–90. tmax

in@plane = 23.2 MPa

9–91. s1 = 119 psi, s2 = -119 psi 9–93. sx" = -0.611 ksi, tx"y" = 7.88 ksi, sy" = -3.39 ksi 9–94. Point A: s1 = 0, s2 = -1.20 ksi,

Point B: s1 = 9.88 psi, s2 = -43.1 psi 9–95. s1 = 3.03 ksi, s2 = -33.0 ksi, up1 = -16.8! and up2 = 73.2!, tmax

in@plane = 18.0 ksi, savg = -15 ksi, us = 28.2!

9–97. Point A: s1 = 61.7 psi, s2 = 0 Point B: s1 = 0, s2 = -46.3 psi

9–98. s1 = 53.0 MPa, s2 = -68.0 MPa, up1 = 14.9!, up2 = -75.1!, tmaxin@plane = 60.5 MPa, savg = -7.50 MPa, us = -30.1! and 59.9! 9–99. sx" = -16.5 ksi, tx"y" = 2.95 ksi

Chapter 10 10–2. Px" = 248110- 62, gx"y" = -233110- 62, Py" = -348110- 62 10–3. P1 = 138110- 62, P2 = -198110- 62, up1 = 13.3!, up2 = -76.7!, gmax

in@plane = 335110- 62, Pavg = -30.0110- 62,

us = -31.7! and 58.3! 10–5. P1 = 622110- 62, P2 = -862110-62, up1 = -15.2! and up2 = 74.8!, gmax

in@plane = -1484110- 62, Pavg = -120110- 62,

us = 29.8! and -60.2! 10–6. Px" = -116110- 62, Py" = 466110- 62, gx"y" = 393110- 62 10–7. Px" = 466110- 62, Py" = -116110- 62, gx"y" = -393110- 62 10–9. P1 = 188110- 62, P2 = -128110- 62, up1 = -9.22!, up2 = 80.8!, gmax

in@plane = 316110- 62, us = 35.8! and -54.2!,

Pavg = 30110- 62 10–10. (a) P1 = 713110-6), P2 = 36.6110- 62, up1 = 133!

(b) gmax in@plane

= 677110- 62, Pavg = 375110- 62, us = -2.12! 10–11. Px" = 86.6110- 62, gx"y" = 620110- 62, Py" = 213110- 62 10–13. P1 = 17.7110- 62, P2 = -318110- 62, up1 = 76.7! and up2 = -13.3!, gmax

in@plane = 335110- 62, us = 31.7! and 122!,

Pavg = -150110- 62 10–14. P1 = 368110- 62, P2 = 182110- 62, up1 = -52.8! and up2 = 37.2!, gmax

in@plane = 187110- 62, us = -7.76! and 82.2!,

Pavg = 275 110- 62 10–17. P1 = 138110- 62, P2 = -198110- 62, up = 13.3!

SELECTED ANSWERS 859

Chapter 11 11–1. b = 211 mm, h = 264 mm 11–2. Use b = 4 in. 11–3. Use b = 5 in. 11–5. Use W12 * 16 11–6. Yes. 11–7. No. 11–9. Use W360 * 45 11–10. Yes. 11–11. P = 2.49 kN 11–13. Use W12 * 26 11–14. Use W24 * 62 11–15. P = 12.5 kip, treq=d = 466 psi 11–17. P = 103 kN

11–18. Use a = 3 1 8

in.

11–19. P = 750 lb 11–21. P = 6.24 kN

11–22. Use h = 9 1 8

in.

11–23. s = 0.710 in., s! = 0.568 in., s" = 2.84 in. 11–25. P = 13.7 kN 11–26. Use W14 * 22 11–27. P = 2.90 kN 11–29. b = 5.86 in. 11–30. a = 106 mm, s = 44.3 mm

11–31. w = w0 L

11–33. x = L 3

, 2L 3

, s absmax =

18PL

25bh0 2

11–34. h = h0

L3>2 (3L2x - 4x3)1>2 11–35. d = hA xL 11–37. h =

h0 L

11–38. b = b0 L2

11–39. Use d = 21 mm 11–41. smax = 13.4 MPa 11–42. Use d = 23 mm

11–43. Use d = 1 5 8

in.

11–45. Use d = 36 mm 11–46. Use d = 35 mm

11–47. y = c 4P psallow

x d 12 11–49. Use d = 44 mm 11–50. Use d = 41 mm 11–51. Use W18 * 50

10–62. s1 = 11.6 ksi 10–63. d = 1.88 in. 10–65. t = 25 ksi

10–66. Te = A43M 2 + T 2 10–67. Me = AM 2 + 34T 2 10–69. No. 10–70. Me = 2M2 + T 2 10–71. sx = 105 ksi 10–73. Yes. 10–74. No. 10–75. No. 10–77. F.S. = 1.43 10–78. F.S. = 1.64 10–79. s2 = 38.9 ksi 10–81. s1 = 160 MPa 10–82. sY = 94.3 ksi 10–83. sY = 91.0 ksi 10–85. sY = 19.7 ksi

10–86. (a) p = t r

sy,

(b) p = 2t23r sy

10–89. (a) F.S. = 1.67, (b) F.S. = 1.92

10–90. (a) t = 22.5 mm, (b) t = 19.5 mm

10–91. d = 1.50 in. 10–93. T = 9.67 kN # m 10–95. da = 0.367 mm, db = -0.255 mm,

dt = -0.00167 mm 10–97. No. 10–98. Pavg = 83.3110- 62, P1 = 880110- 62, P2 = -713110- 62, up = 54.8# 1clockwise2, gmax

in@plane = -1593110- 62,

us = 9.78#1clockwise2 10–99. Px! = -380110- 62, Py! = -130110- 62,

gx!y! = 1.21110- 32 10–101. P1 = 996110- 62, P2 = 374110- 62, up1 = -15.8, up2 = 74.2, gmaxin@plane = 622110- 62, Pavg = 685110- 62, us = 29.2# and 119# 10–102. P1 = 283110- 62, P2 = -133110- 62, up1 = 84.8# , up2 = -5.18#, gmaxin@plane = 417110- 62, Pavg = 75.0110- 62, us = 39.8# and 130# 10–103. P1 = 480110- 62, P2 = 120110- 62, up1 = -28.2# 1clockwise2, gmax

in@plane = -361110- 62,

us = 16.8# 1counterclockwise2, Pavg = 300110- 62

860 SELECTED ANSWERS

11–53. h = 0.643 in. Yes, the joist will support the load. 11–54. P = 178 lb, s = 12.0 in.

Chapter 12 12–1. s = 3.02 ksi 12–2. s = 75.5 ksi 12–3. s = 582 MPa 12–5. vC = 6.11 mmT

12–6. v1 = Px1

12EI 1-x12 + L22,

v3 = P

12EI 12x33 - 9Lx32 + 10L2x3 - 3L32,

vmax = PL3

8EI T

12–7. vmax = P

3EIAB e a1 - IAB

IBC b l3 - L3 f

12–9. v1 = P

12EI 12x13 - 3x212, v2 = PL248EI (-6x2 + L)

12–10. uA = - M0L

3 EI , v =

M0 6 EIL

13Lx2 - x3 - 2L2x2, v max =

-0.0642M0L2

12–11. uB = M0L

6EI , v ! x = L>2 = -M0L216EI

12–13. uA = - M0a

2EI , vmax = -

5M0a 2

8EI

12–14. vmax = 3PL3

256EI T

12–15. F = 1.375 N

12–17. v1 = M0

6EIL 1-x31 + L2x12,

v2 = M0

24EI 1-12x22 + 20Lx2 - 7L22,

vC = 7M0L

24EI T

12–18. uA = - 3PL2

8EI , vC =

-PL3

6EI

12–19. v0 = - 11PL3

48EI 12–21. vmax = 11.5 mmT

12–22. v = 1 EI

(2.25 x3 - 0.002778 x5 - 40.5 x22 kip # ft3, u = 0.00466 rad (clockwise), vmax = 0.369 in.

12–23. uC = - 4M0L

3EI , v1 =

M0 6EIL

3-x31 + L2x14 , v2 =

M0 6EIL

3-3Lx22 + 8L2x2 - 5L34 , vC = -

5M0L 2

6EI

12–25. vmax = -18.8 kip # ft3

EI 12–26. vmax = 0.396 in.T

12–27. v = w0x

120EIL 110L2x - x3 - 20L32,

vmax = 11w0L

120 EI T

12–29. uB = - wa3

6EI , v1 =

w 24EI

3-x41 + 4ax31 - 6a2x214 , v2 =

wa3

24EI 3-4x2 + a4 ,

vB = wa3

24EI 1-4L + a2

12–30. uB = - wa3

6EI , v1 =

wx21 24EI

3-x21 + 4ax1 - 6a24 . v2 =

wa3

24EI 34x3 + a - 4L4 , vB = wa324EI 1a - 4L2

12–31. v = 1 EI

c - Pb 6a

x3 + P1a + b2

6a 6 x - a 7 3 + Pab

6 x d

12–33. v = P

6EI 3-x3 + 6x - a73 + 6x - 3a7 3

+ 9a2x - 8a34 , 1vmax2BC = Pa32EI c , vA = 4Pa33EI T 12–34. v =

P 12EI

3-26x - a73 + 46x - 2a73 + a2x 4 , 1vmax2AB = 0.106Pa3EI c , vC = 3Pa34EI T 12–35. v =

1 EI

c -2.5x2 + 2 6x - 473 - 1 8

6x - 47 4

+ 2 6x - 1273 + 1 8

6x - 1274

- 24x + 1364 kip # ft3 12–37. v =

M0 6EI

c 3h x - L 3 i2 - 3h x - 2

3 L i2 - Lx d ,

vmax = 5M0L

72EI T

12–38. v = 1 EI

30.417x3 - 0.333 6x - 87 3 - 0.667 6x - 1673 - 169x 4 kip # ft3 12–39. vmax = 12.9 mmT

12–41. 1vmax2AB = 0.0867 in.c 12–42. v =

1 EI

3-0.25x4 + 0.208 6x - 1.57 3 + 0.25 6x - 1.57 4 + 4.625 6x - 4.57 3 + 25.1x - 36.44 kN # m3

SELECTED ANSWERS 861

12–43. v = 1 EI

3-0.0833x3 + 3 6x - 872 + 3 6x - 1672 + 8.00x 4 12–45. vC = -0.501 mm, vD = -0.698 mm, vE = -0.501 mm 12–46. uA = -0.128!, uB = 0.128! 12–47. vmax = -3.64 mm

12–49. uA = 302 EI

kip # ft2, vC = - 3110EI kip # ft 3

12–50. dv dx

= 1 EI

32.25x2 - 0.5x3 + 5.25 6x - 57 2 + 0.5 6x - 573 - 3.1254 kN # m2, v =

1 EI

30.75x3 - 0.125x4 + 1.75 6x - 57 3 + 0.125 6x - 57 4 - 3.125x 4 kN # m3 12–51. uC = -

3937.5 EI

, "C = 50 625

EI T

12–53. uB = - 5PL2

8 EI , "B =

7PL3

16EI T

12–54. uB = - Pa2

12EI , "C =

Pa3

12EI 12–55. " max = 12.2 mm

12–57. " max = 11Pa3

48EI

12–58. "C = - 84 EI

, uA = 8 EI

, uB = - 16 EI

, uC = - 40 EI

12–59. uA = 0.00115 rad 12–61. a = 0.858 L 12–62. P = 6.43 lb, "B = 2.14 in.T

12–63. uB = - 3M0L

2EI , "B =

7M0L 2

8EI T

12–65. uA = - 5Pa2

2EI , "C =

19Pa3

6EI T

12–66. "C = - PL3

12EI , uA =

PL2

24EI , uB = -

PL2

12EI

12–67. " max = 0.00802PL3

12–69. uC = - 5Pa2

2EI , "B =

25Pa3

6EI T

12–70. uA = - 336 kip # ft2

EI , vmax =

3048 kip # ft3 EI

12–71. E = Pa

24I" 13L2 - 4a22

12–73. uA = - M0L

6EI , "max =

5M0L 2

72EI T

12–74. umax = 5PL2

16EI , "max =

3PL3

16EI T

12–75. uB = 0.00658 rad, "C = 13.8 mmT

12–77. a = 0.865 L

12–78. uB = 7wa3

12EI , "C =

25wa4

48EI T

12–79. uC = - a 2

6EI 112P + wa2,

"C = a3

24EI 164P + 7wa2 T

12–81. uA = PL2

12EI , "D =

PL3

8EI T

12–82. " max = 3wa4

8EI 12–83. uB = -0.00778 rad, vB = 0.981 in.T

12–85. uC = - wa3

6EI , "C =

wa4

8EI T

12–86. uA = wa3

6EI , "D =

wa4

12EI T

12–87. "C = 0.895 in.T 12–89. "C = 1.90 in.T 12–90. Use W14 * 34 12–91. "C = 23.2 mmT

12–93. " = PL2a1 k

+ L 3EI

b 12–94. "A =

72 EI

T , uA = 36 EI

12–95. "A = PL3a 112EI + 18GJb T 12–97.

xmax ymax

= Ix Iy

tan u,

ymax = 0.736 in., xmax = 3.09 in.

12–98. "A = Pa2 (3b + a)

3EI T

12–99. A y = 20 27

P, MA = 4 27

PL, By = 7

27 P, MB =

2 27

12–101. A x = 0, Cy = 5

16 P, By =

11 8

P, A y = 5

16 P

12–102. By = 5

16 P, A y =

11 16

P, MA = 3PL 16

12–103. A x = 0, By = 3wL

8 , A y =

5wL 8

, MA = wL2

8 12–105. Cx = 0, A y = 12.0 kN, By = 40.0 kN,

Cy = 12.0 kN

12–106. A x = 0, By = w0L

10 , A y =

2w0L

5 , MA =

w0L 2

12–107. A x = 0, A y = 7wL 16

, Cy = - wL 16

, By = 5wL

12–109. TAC = 3A 2E2wL

4 1

81A 2E2L31 + 3E1I1L22 12–110. A x = 0, FC = 112 kN , A y = 34.0 kN,

By = 34.0 kN

862 SELECTED ANSWERS

12–111. MA = 5wL2

192 , MB =

11wL2

192

12–113. A y = 1.48 kip, Bx = 0, By = 3.52 kip, MB = 7.67 kip # ft

12–114. By = 2 3

P, MA = PL 3

, Ay = 4 3

P, A x = 0

12–115. By = 11P 16

, Cy = 13P 32

, A y = 3P 32

12–117. By = 550 N, A y = 125 N, Cy = 125 N

12–118. A x = 0, By = 7P 4

, Ay = 3P 4

, MA = PL 4

12–119. Cx = 0, By = 30.75 kip, A y = 2.625 kip, Cy = 14.6 kip

12–121. !B = 1.50 mmT

12–122. A x = 0, By = 3M0 2L

, Ay = 3M0 2L

, MA = M0 2

12–123. Cx = 0, Cy = P 3

12–125. MA = MB = 1 24

PL, A y = By = 1 6

Cy = Dy = 1 3

P, Dx = 0

12–126. Fsp = 3kwL4

24EI + 8kL3

12–127. M = PL 8

- 2EI L

a, !max = PL3

192EI + aL

12–129. a = L - a72!EI wo

b 14 12–130. d = 0.708 in. 12–131. By = 634 lb , A y = 243 lb , Cy = 76.8 lb

12–133. v = 1 EI

( -30x3 + 46.25 6x - 127 3

- 11.7 6x - 2473 + 38,700x - 412,560) lb # in3 12–134. v1 =

1 EI

14.44x31 - 640x12 lb # in3, v2 =

1 EI

1-4.44x32 + 640x22 lb # in3 12–135. MB =

w0L 2

30 , MA =

w0L 2

12–137. 1v22max = wL4 1823EI

12–138. uB = Pa2

4EI , !C =

Pa3

4EI c

12–139. By = 138 N, A y = 81.3 N, Cy = 18.8 N 12–141. !C = 0.644 inT

12–142. Mmax = p2brgv2r3

108g

Chapter 13 13–1. Pcr =

kL 4

13–2. Pcr = kL 13–3. Use d =

9 16

in.

13–5. d = 2.58 in. 13–6. d = 1.81 in. 13–7. Pcr = 400 kN 13–9. d = 6.07 in. 13–10. d = 4.73 in. 13–11. Pcr = 20.4 kip 13–13. Pcr = 272 kN 13–14. d = 8.43 in., Pcr = 245 kip 13–15. Pallow = 186 kN 13–17. Pcr = 2.92 kip 13–18. Pcr = 5.97 kip 13–19. P = 17.6 kip 13–21. Pcr = 32.2 kip

13–22. Use dAB = 2 1 8

in., dBC = 2 in.

13–23. P = 207 lb 13–25. P = 62.3 kip 13–26. P = 2.42 kip 13–27. P = 63.0 kN 13–29. x -x axis buckling: F.S. = 8.94 y -y axis buckling: F.S. = 3.98 13–30. P = 23.9 kip 13–31. w = 1.17 kN>m 13–33. P = 8.46 kN 13–34. Use d = 62 mm 13–35. Use d = 52 mm 13–37. P = 37.5 kip 13–38. P = 5.79 kip

13–39. Use d = 1 3 4

in.

13–41. M max = - wEI

P c sec aL

2 A PEI b - 1 d

13–42. M max = - F 2AEIP tan aL2 A PEI b

13–43. Pcr = p2EI

4L2

13–46. P = 31.4 kN 13–47. P = 5.87 kN, vmax = 42.1 mm 13–49. P max = 16.9 kN 13–50. P max = 4.60 kN 13–51. s max = 65.8 kip 13–53. P = 156 kip 13–54. P" = 390 kip, P = 139 kip 13–55. Yes. 13–57. L = 2.53 m

SELECTED ANSWERS 863

13–125. Yes. 13–126. P = 1.69 kip 13–127. P = 3.44 kip 13–129. P = 9.01 kip 13–130. Pcr = 12.1 kN 13–131. P = 161 kip

13–133. Use d = 2 1 8

in.

13–134. t = 5.92 mm 13–135. Pcr = 839 kN 13–137. No, it does not buckle or yield.

Chapter 14

14–1. Ui V

= 1

2E (s2x + s2y - 2nsxsy) +

t2xy

14–3. Ui = 3.28 J

14–5. UA = N2L1 2AE

, UB = N2L2 2AE

Since UB 7 UA, i.e., L2 7 L1, the design for case (b) is better able to absorb energy.

14–6. Ui = 43.2 J 14–7. P = 375 kN, Ui = 1.69 kJ 14–9. Ui = 149 J 14–10. Ui = 0.0638 J 14–11. (Ub)i = 4.31 in. # kip 14–13. P = 113 kip, Ui = 7.37 in # kip 14–15. (Ub)i =

17w2L5

15 360 EI

14–17. (Ub)i = 17w20L

10 080 EI

14–18. (Ub)i = w20 L

945 EI

14–19. (Ut)i = P 2r 3

JG a3p

8 - 1b

14–21. (Ub)i = 62.5 J, (Ui)sp = 0.391 J

14–22. (Ub)i = 1.78 kJ, (Ua)i = 0.617 J

14–23. (Ub)i = w2L5

40 EI

14–25. (!B)h = 0.2025 mm

14–26. (!C)v = 2PL AE

14–27. (!h)A = 0.0830 in. 14–29. !B = 2.15 in. 14–30. !B = 3.46 mm

13–58. d = 98.3 mm 13–59. P = 73.5 kip 13–61. P max = 61.2 kip 13–62. L = 21.2 ft 13–63. P = 129 kip 13–65. Pcr = 83.5 kN 13–66. s max = 2.86 ksi 13–67. s max = 2.86 ksi 13–69. P = 88.5 kip 13–70. Et = 14.6(103) ksi 13–71. L = 8.34 m 13–75. Pcr = 661 kN 13–77. Pcr = 1350 kN 13–78. L = 8.99 ft 13–79. Use W6 * 9 13–81. L = 15.1 ft 13–82. Use W6 * 12 13–83. L = 18.0 ft 13–85. L = 33.7 ft 13–86. Use W6 * 9 13–87. d = 1.42 in. 13–89. Yes. 13–90. L = 4.46 m 13–91. b = 0.704 in. 13–93. Pallow = 422 kN 13–94. L = 3.08 m 13–95. Pallow = 380 kip 13–97. Pallow = 129 kip 13–98. Pallow = 143 kip 13–99. Pallow = 109 kip 13–101. Pallow = 8.61 kip 13–102. L = 8.89 ft

13–103. Use a = 7 1 2

in.

13–105. L = 5.03 ft 13–106. Pallow = 4.22 kip 13–107. P = 8.83 kip 13–109. P = 17.4 kip 13–110. P = 3.91 kip 13–111. P = 29.6 kip 13–113. P = 8.60 kip 13–114. P = 14.6 kip 13–115. No. 13–117. P = 95.7 kip 13–118. P = 98.6 kip 13–119. P = 2.79 kip 13–121. P = 98.0 kip 13–122. P = 132 kip 13–123. Yes.

864 SELECTED ANSWERS

14–31. !B = 11.7 mm 14–33. !B = 0.100 mm 14–34. !g = 5.46 in.

14–35. uA = - M0L

3EI

14–37. s max = 16 PR

pd3 (sin u + 1)

14–38. ! = 64nPR3

d 4G 14–39. (!C)v = 2.13 mm 14–41. (!B)v = 15.2 mm 14–42. (a) Ui = 4.52 kJ (b) Ui = 3.31 kJ 14–43. d = 5.35 in. 14–45. (a) smax = 45.4 ksi (b) smax = 509 psi (c) smax = 254 psi 14–46. h = 1.75 ft 14–47. smax = 20.3 ksi 14–49. !max = 3.95 mm, smax = 237 MPa 14–50. s max = 85.7 MPa 14–51. (s max)AB = (smax)AC = 233 MPa 14–53. smax = 414 MPa 14–54. h = 95.6 mm

14–55. Use d = 2 1 8

in.

14–57. smax = 57.1 ksi 14–58. (!A)max = 15.4 in.

14–59. h = smaxL

3Ec c smaxI

WLc - 2 d

14–61. n = 16.7 14–62. smax = 4.55 ksi, !C = 0.799 in. 14–63. !B = 0.226 in., smax = 3.22 ksi 14–65. h = 1.37 m 14–66. smax = 137 MPa 14–67. h = 8.66 m 14–69. h = 7.45 in. 14–70. (!B)max = 0.661 in. 14–71. !max = 23.3 mm, smax = 4.89 MPa 14–73. (!B)h = 0.223(10-3) in. d 14–74. (!B)v = 0.00112 in.T 14–75. (!B)v = 0.0132 in.T 14–77. (!B)v = 0.0124 in.T

14–78. (!E)v = 0.00966 in.T 14–79. (!B)h = 0.367 mm 14–81. (!C)h = 0.0482 in. 14–82. (!B)h = 0.0420 in. 14–83. (!A)v = 3.18 mm 14–85. (!C)v = 20.4 mm 14–86. (!D)v = 4.88 mm

14–87. !C = 23Pa 3

24EI

14–89. !C = 2Pa 3

3EI

14–90. uC = - 5Pa 2

6EI

14–91. uA = Pa 2

6EI

14–93. uB = -0.353" 14–94. uC = 5.89 (10-3) rad 14–95. !B = 47.8 mm 14–97. uA = 0.289" 14–98. uB = 0.124"

14–99. !C = PL3

8EI

14–101. uC = - 13wL3

576 EI

14–102. !D = wL4

96 EI T

14–103. uA = -0.0223 rad

14–105. !B = 65wa4

48EI

14–106. !C = 0.122 in. 14–107. uA = 4.05 (10-3) rad 14–109. uC = -0.0198 rad 14–110. !D = 0.219 in.

14–111. !tot = awG b aLa b2 c a 596 b aLa b2 + 320 d , !b =

5w 96G

aL a b4

14–113. uA = - 5w0L

192 EI

14–114. (!A)v = 4PL3

3EI

14–115. !B = 43.5 mm 14–117. !C = 17.9 mm

SELECTED ANSWERS 865

14–141. uA = wL3

24EI

14–142. !C = 5wL4

8EI

14–143. !B = wL4

4EI

14–145. uB = wL3

8EI

14–146. Ui = 496 J 14–147. smax = 116 MPa 14–149. uE = -3.15" 14–150. smax = 43.6 ksi 14–151. Ui = 45.5 ft # lb 14–153. (!B)h = 0.0876 in. d

14–154. uB = M0 L

14–155. uB = M0 L

14–157. !B = wL4

128 EI

14–118. uA = 0.991(10-3) rad

14–119. !C = 5wL4

8EI

14–121. !C = wL4

4EI

14–122. uB = wL3

8EI

14–123. (!B)h = 0.223(10-3) in. d 14–125. (!B)v = 0.0132 in.T 14–126. (!E)v = 0.0149 in. 14–127. (!B)v = 0.0124 in.T 14–129. (!C)v = 0.0482 in. 14–130. (!B)h = 0.0420 in. 14–131. (!C)v = 20.4 mm

14–133. uC = - 5Pa 2

6EI

14–134. uA = Pa 2

6EI

14–135. !C = 0.369 in. 14–137. !B = 47.8 mm 14–138. !D = 3.24 mm 14–139. uA = 0.289"

866

A Absolute maximum shear strain, 506 – 507 , 536 Absolute maximum shear stress ( t max ), 184 – 185 , 187 , 477 – 481 , 485

stress concentration, 239 , 255 stress transformations, 477 – 481 , 485 torsional loads, 184 – 185 , 187 , 226 , 239 , 255

Allowable stress design (ASD), 46 – 47 , 62 Aluminum column specifications, 698 Angle of twist ( f ), 182 – 183 , 204 – 211 , 226 , 230 , 254

circular shafts, 182 – 183 , 204 – 211 , 254 constant torque and, 205 – 206 cross-sectional area for, 204 – 208 multiple torques and, 206 – 207 noncircular shafts, 226 procedure for analysis of, 208 sign convention for, 206 – 207 thin-walled tubes, 230 torsional deformation and, 182 – 183

Anisotropic material, 24 Annulus (differential ring), 186 , 241 – 242 Area ( A ), 788 – 803

centroid, 788 – 790 composite, 789 , 792 geometric properties of, 788 – 803 inclined axes, 798 – 800 Mohr’s circle for, 801 – 803 moment of inertia for, 791 – 794 , 798 – 803 parallel-axis theorem, 791 – 792 , 796 principle moments of inertia, 799 procedure for analysis of, 801 – 802 product of inertia for, 795 – 797 , 799 transformation equations, 798

Average normal strain ( P avg ), 68 Average normal stress, 24 – 31 , 62 . See also Normal stress ( s ) Average shear stress ( t avg ), 32 – 37 , 62 , 229 – 230 , 255 Axial loads, 24 – 31 , 62 , 109 – 111 , 117 , 120 – 179 , 724 – 725

average normal stress distribution, 24 – 31 , 62 compatibility (kinematic) conditions, 139 – 140 , 145 – 146 constant quantities, 24 – 25 , 124 – 125 cross-sectional areas, 24 – 25 , 124 – 125 , 160 – 163 , 175 deformation and, 120 – 179 displacement ( d ), 124 – 130 , 138 – 146 , 153 – 156 , 175 – 176 elastic deformation of, 124 – 130 , 153 – 156 , 160 – 163 , 175 – 176 elastic strain energy ( U i ), 724 – 725 equilibrium and, 25 – 26 , 164 – 165 failure and, 109 – 111 , 117 fatigue, 110 – 111 , 117 force (flexibility) method of analysis, 145 – 146 inelastic deformation, 164 – 170 , 176 internal axial force, 724 load-displacement relationship, 139 – 146 , 175 material properties of, 24 normal stress (s) in, 24 – 31 prismatic bars, 24 – 31 procedures for analysis of, 27 , 127 , 140 , 146 relative displacement (d) of, 124 – 130 , 175

residual stresses ( t r ), 166 – 170 , 176 Saint Venant’s principle, 121 – 123 , 175 sign convention for, 26 , 126 , 175 statically indeterminate, 139 – 146 , 153 , 166 , 175 stress concentrations in, 160 – 163 , 176 stress–cycle ( S – N ) diagrams for, 110 – 111 superposition, principle of, 138 , 175 thermal (stress) ( d T ), 153 – 156 , 176 uniaxial stress, 25 – 26 uniform deformation, 24 – 25

Axial (longitudinal) stress, 326 , 410 – 411 Axis of symmetry, 306 , 323 , 357 – 358 , 396 – 397

B Beams, 259 – 361 , 362 – 407 , 541 – 551 , 558 – 561 , 569 , 572 – 659 , 766 – 770 ,

780 – 785 , 804 – 811 . See also Deflection angles with equal legs, 807 , 811 bearing plates for, 542 bending and, 259 – 361 built-up members, 382 – 386 , 405 , 546 , 569 cantilevered, 259 Castigliano’s theorem applied to, 780 – 785 channels (C shape), 806 , 810 composite, 316 – 318 , 357 cross sections of, 285 – 296 , 306 – 312 , 357 curved, 323 – 329 , 358 deflection of, 542 , 572 – 659 deformation of by bending, 285 – 288 design of, 541 – 551 , 558 – 561 , 569 discontinuity functions, 597 – 604 , 656 displacement, 573 – 574 , 577 – 589 , 608 – 616 , 632 – 634 , 656 – 657 elastic curve, 573 – 574 , 608 – 609 , 656 fabricated, 545 – 546 flexural rigidity ( EI ), 576 flexure formula for, 289 – 296 , 357 force (flexibility) method of analysis, 145 – 146 , 643 – 651 fully stressed, 558 – 561 , 569 geometric properties for shapes of, 804 – 811 hyperbolic stress variations, 324 – 325 inelastic bending of, 339 – 349 , 358 inflection point, 574 integration method for, 577 – 589 , 632 – 634 , 656 – 657 linear stress variations, 286 – 289 moment-area method for, 608 – 616 , 637 – 641 , 657 moment-curvature relationship, 575 – 576 moment diagrams for, 637 – 641 nonprismatic, 558 – 561 , 569 overhanging, 259 plane cross-sections of, 286 , 316 , 323 prismatic, 544 – 551 , 569 procedures for analysis of, 261 , 269 , 292 , 327 , 370 , 398 , 547 , 581 ,

601 , 610 , 646 , 768 , 782 radius of curvature, 576 reinforced concrete, 319 – 322 residual stress of, 342 – 343 , 358 section modulus ( S ), 544 , 558

Index

INDEX 867

shear and moment diagrams for, 259 – 275 , 356 shear center ( O ), 396 – 401 , 406 shear flow ( q ), 382 – 386 , 391 – 395 , 405 – 406 shear formula for, 365 – 375 , 405 shear stresses (t) in, 362 – 407 sign conventions for, 260 , 267 – 268 , 291 , 309 simply supported, 259 slope, 573 – 574 , 577 – 589 , 608 – 616 , 632 – 634 , 656 – 657 statically indeterminate, 631 – 657 straight members, 285 – 322 , 356 – 357 , 363 – 365 stress concentrations in, 330 – 332 , 358 stress trajectories, 542 – 543 superposition, method of, 623 – 627 , 643 – 651 , 657 thin-walled members, 391 – 401 , 406 transformation factor ( n ), 317 – 318 transformed section method, 316 – 318 , 357 transverse shear in, 362 – 407 twisting, 396 – 397 unsymmetric bending of, 306 – 312 , 357 virtual work analysis of, 766 – 770 warping, 364 – 365 wide-flange (W shape) sections, 804 – 805 , 808 – 809

Bearing plates, 542 Bending, 259 – 361 . See also Moments ( M )

composite beams, 316 – 318 , 357 curved beams, 323 – 329 , 358 deformation, 285 – 288 elastic behavior, 330 – 331 , 357 – 358 flexure formula for, 289 – 296 , 357 inelastic, 339 – 349 , 358 plastic moment, 340 – 341 procedures for analysis of, 261 , 269 , 292 , 327 reinforced concrete beams, 319 – 322 residual stress by, 342 – 343 , 358 resultant forces ( F R ), 289 , 339 resultant moment ( M R ), 339 shear and moment diagrams for, 259 – 275 , 356 sign conventions for, 260 , 267 – 268 , 291 , 309 straight members, 285 – 322 , 356 – 357 stress concentrations and, 330 – 332 , 358 transformed section method, 316 – 318 , 357 ultimate moment, 344 unsymmetric, 306 – 312 , 357

Bending moment ( M ), 8 , 260 , 268 , 285 , 289 – 290 , 357 , 726 – 728 change in, 268 concentrated, 268 deformation of beams, 260 , 285 , 357 elastic strain energy (U i ), 726 – 728 equilibrium and internal loadings as, 8 flexure formula and, 289 – 290 shear and moment diagrams and, 260 , 268 sign convention, 260 , 268 , 290

Biaxial stress, 411 Bifurcation point, 663 Body forces, 5 Boundary conditions, 580

Brittle materials, 91 , 110 , 116 , 239 , 331 , 528 – 529 , 537 bending and, 331 fatigue failure, 110 , 239 fracture, 528 – 529 fracture stress ( s f ), 91 material failure, 91 , 116 , 239 maximum-normal-stress theory, 528 , 537 Mohr’s failure criterion, 528 – 529 , 537 multiaxial stress and, 528 – 529 , 537 plane strain transformation and, 528 – 529 , 537 stress concentrations and, 160 , 239 , 331 torsional loadings and, 239

Buckling, 660 – 717 bifurcation point, 663 columns, 660 – 717 concentric loading, 696 – 703 critical load ( P cr ), 661 – 664 , 715 eccentric loading, 707 – 711 Engesser’s equation for, 689 , 715 Euler load, 666 – 667 , 715 ideal columns, 664 – 669 , 715 inelastic, 688 – 690 , 715 lateral deflection as, 661 – 664 least moment of inertia and, 667 maximum deflection ( v max ), 683 – 686 , 715 secant formula for, 652 – 587 , 715 tangent modulus ( E t ), 688 – 689

Built-up members, 382 – 386 , 405 , 546 , 569 design of, 546 , 569 shear flow ( q ) in, 382 – 386 , 405

Bulging, 225 Bulk modulus ( k ), 514 – 515 , 537

C Cantilevered beam, 259 Cartesian components of strain, 70 Castigliano’s theorem, 775 – 785

beams, applied to, 780 – 785 compatibility requirements, 776 energy displacement ( d ) from, 775 – 785 procedures for analysis using, 778 , 782 trusses, applied to, 777 – 779

Centroid, 7 , 9 , 323 , 396 , 788 – 790 Centroidal axis, beams, 290 Channel (C shape) properties, 806 , 810 Circular shafts, 181 – 224 , 154 Circumferential (hoop) stress, 326 , 410 – 411 Cohesive material, 22 Columns, 660 – 717

aluminum specifications, 698 buckling, 660 – 717 classification of, 688 – 689 concentric loading, 696 – 703 critical load ( P cr ), 661 – 664 , 715 cross sections of, 667 , 707 deflection, maximum ( v max ), 683 – 686 , 715

868 INDEX

Columns (continued) design of, 686 , 696 – 703 , 707 – 711 eccentric distance ( e ), 682 – 684 eccentric loading, 707 – 711 eccentricity ratio ( ec / r 2 ), 685 – 686 effective length ( L e ), 671 Engesser’s equation for, 689 , 715 equilibrium of, 662 – 663 Euler load, 666 – 667 , 715 ideal, 664 – 669 , 715 inelastic buckling, 688 – 690 , 715 interaction formula for, 707 – 708 least moment of inertia in, 667 pin-supported, 664 – 669 , 715 procedure for analysis of, 699 radius of gyration ( r ), 667 secant formula for, 652 – 587 , 715 slenderness ratio ( L / r ), 667 – 668 , 696 – 703 , 671 steel specifications, 697 stress distribution, 707 – 711 supports for, 664 – 669 , 670 – 673 tangent modulus ( E t ), 688 – 689 wood (timber) specifications, 698

Combined loadings, 408 – 439 biaxial stress, 411 circumferential (hoop) stress direction, 410 cylindrical vessels, 410 – 411 , 436 longitudinal (axial) stress direction, 410 procedure for analysis of, 416 – 417 radial stress, 411 spherical vessels, 411 , 436 state of stress caused by, 416 – 424 , 436 thin-walled pressure vessels, 409 – 412 , 436

Compatibility conditions, 139 – 140 , 145 – 146 , 632 , 643 – 645 , 776 axial loads, 139 – 140 , 145 – 146 Castigliano’s theorem, 776 deflections of shafts and beams, 632 , 643 – 645 displacement ( d ), 139 – 140 , 145 – 146 , 632 , 643 – 645 , 776 force (flexibility) method, 145 – 146 , 643 – 645 load-displacement relationship, 139 – 146 , 175 static indeterminacy, 139 – 140 , 632 superposition method, 139 – 140 , 643 – 645

Composite area, 789 , 792 Composite beams, 316 – 318 , 357 Compression (tension) test, 83 – 84 , 115 Compressive stress, 23 Concentrated forces, 4 , 238 – 240 , 255 , 268 Concentric loading, 696 – 703 Concrete (reinforced) beams, 319 – 322 Conservation of energy, 737 – 740 , 785 Constant loads, 24 – 25 , 125 , 175 , 205 – 206

axial, 125 , 175 stress ( s ), 24 – 25 torque ( T ), 205 – 206

Continuity conditions, 580 Continuous material, 22 Contraction, 104 , 126

Coplanar forces (loadings), 6 , 9 Couple moment, work of, 721 Couplings, torsional stress and, 238 Creep, 109 – 110 , 117 Critical load ( P cr ), 661 – 664 , 715 Cross sections, 7 , 9 – 10 , 24 – 25 , 124 – 125 , 160 – 165 , 175 – 176 , 182 – 191 ,

204 – 208 , 225 – 233 , 238 – 243 , 254 – 255 , 285 – 296 , 306 – 312 , 313 – 314 , 323 – 329 , 335 – 340 , 357 – 358 , 363 – 375 , 382 – 386 , 391 – 401 , 405 – 406 , 667 , 707

angle of twist ( f ) and, 204 – 208 annulus (differential ring), 186 , 241 – 242 average normal stress, for determination of, 24 – 25 axial loads, 24 – 25 , 124 – 125 , 160 – 165 , 176 axis of symmetry for, 306 , 323 , 357 – 358 beams, 285 – 296 , 306 – 312 , 313 – 314 , 323 – 329 , 335 – 340 , 357 – 358 ,

363 – 375 , 382 – 386 , 391 – 401 , 405 – 406 bending deformation and, 285 – 288 , 357 built-up members, 382 – 386 , 405 bulging, 225 centroid, 7 , 9 , 323 , 396 centroidal axis, 290 changes in, 238 – 240 , 255 circular shafts, 182 – 191 , 204 – 208 , 254 closed, 228 – 233 columns, 667 , 707 constant load and, 125 , 175 constant torque and, 205 – 206 coplanar loadings, 9 deformation and, 24 – 25 , 124 – 125 , 160 – 163 , 176 , 182 – 183 , 225 –

233 , 285 – 288 , 357 eccentric loads, 707 elastic deformation, 124 – 125 , 160 – 163 , 175 hyperbolic stress variation, 324 inelastic behavior, 164 – 165 , 176 , 241 – 243 , 255 , 339 – 349 , 358 internal resultant loadings, 7 least moment of inertia, 667 linear stress/strain variation, 183 – 184 , 286 – 289 longitudinal axis, 286 – 288 method of sections and, 7 , 9 – 10 neutral axis for beams, 286 , 290 – 291 , 309 , 323 – 324 noncircular shafts, 225 – 233 , 255 parabolic shear stress distribution, 372 plane, 286 , 316 , 323 polar moment of inertia ( J ), 185 – 186 radius of gyration ( r ), 667 shafts, 182 – 191 , 204 – 208 , 225 – 233 , 238 – 243 , 254 – 255 shape variations, 226 , 324 shear stress ( t ), 184 – 187 solid shafts, 186 , 225 – 227 stress concentration factor ( K ), 160 – 163 , 176 , 238 , 330 – 331 thin-walled members, 228 – 233 , 391 – 401 , 406 torsion formula and, 184 – 191 torsional loads, 182 – 191 , 204 – 208 , 225 – 233 , 238 – 243 , 254 – 255 transformed section method, 316 – 318 , 357 transverse shear, 363 – 375 , 382 – 386 , 391 – 401 , 405 – 406 tubular shafts, 187 , 228 – 233

INDEX 869

uniform deformation and, 24 – 25 unsymmetrical, 306 – 312 warping, 225 , 255 , 364 – 365

Curvature, moment relationship to, 575 – 576 Curved beams, 323 – 329 , 358 Cylindrical pressure vessels, 410 – 411 , 436

D Dead load, 48 Deflection, 542 , 572 – 659 , 660 – 771 . See also Buckling

beams, 542 , 572 – 659 columns, 660 – 717 coordinates, 579 discontinuity functions, 597 – 604 , 656 displacement, 573 – 574 , 577 – 589 , 608 – 616 , 632 – 634 , 656 – 657 elastic curve, 573 – 574 , 608 – 609 , 656 flexural rigidity ( EI ), 576 force (flexibility) method of analysis, 145 – 146 , 643 – 651 inflection point, 574 integration method for, 577 – 589 , 632 – 634 , 656 – 657 lateral (buckling), 661 – 664 M / EI diagrams for, 608 – 616 maximum ( v max ), 683 – 686 , 715 moment-area method for, 608 – 616 , 637 – 641 , 657 moment-curvature relationship, 575 – 576 moment diagrams for, 637 – 641 procedures for analysis of, 581 , 601 , 610 , 646 radius of curvature, 576 shafts, 572 – 659 sign conventions for, 579 , 608 – 609 slope, 573 – 574 , 577 – 589 , 608 – 616 , 632 – 634 , 656 – 657 statically determinate members, 573 – 630 , 656 – 657 statically indeterminate shafts and beams, 631 – 657 superposition, method of, 623 – 627 , 643 – 651 , 657

Deformable bodies, 4 – 14 cross section of, 7 equations of equilibrium, 6 equilibrium of, 4 – 14 external loads, 4 – 5 internal resultant loads, 7 – 9 procedure for analysis of, 10 right-hand rule for, 8 support reactions, 5

Deformation, 24 – 25 , 66 – 81 , 86 , 88 – 91 , 104 – 105 , 115 – 117 , 120 – 179 , 181 – 191 , 225 – 233 , 255 , 260 , 285 – 288 , 357 , 449 – 455 , 484 , 494 , 536 . See also Displacement ( d ); Strain ( t )

angle of twist ( f ), 182 – 183 , 226 , 230 axially loaded members, 24 – 25 , 120 – 179 beams (bending), 285 – 288 , 357 bending moment ( M ), 260 , 285 , 357 bulging, 225 changes in a body, 67 circular shafts, 181 – 191 cross-sections, 24 – 25 , 124 – 125 , 160 – 163 , 176 , 182 – 183 , 225 – 233 ,

285 – 288 , 257 displacement ( d ), 124 – 130 , 138 – 146 , 153 – 156 , 175 – 176 elastic, 86 , 88 , 115 – 116 , 124 – 130 , 153 – 156 , 160 – 163 , 175 – 176

inelastic, 164 – 165 , 176 localized, 122 – 123 mechanical material properties and, 86 , 88 – 91 , 104 – 105 ,

115 – 117 noncircular shafts, 225 – 233 , 255 plane transformation and, 449 – 455 , 484 , 494 , 536 plastic behavior, 86 , 93 , 115 – 116 Poisson’s ratio (n), 104 – 105 , 117 principal stresses (in-plane), 449 – 455 , 484 principal strains, 494 , 536 procedure for analysis of, 127 , 140 , 146 relative displacement (d), 124 – 130 , 175 Saint Venant’s principle, 121 – 123 , 175 shear strain ( g ) and, 181 – 183 shear stress ( t ) and, 184 – 191 , 225 – 233 strain and, 66 – 81 strain energy, 94 – 98 , 116 stress concentrations, 160 – 163 , 176 superposition, principle of, 138 , 175 thermal (stress) ( d T ), 153 – 156 , 176 torsional, 181 – 191 , 225 – 233 uniform, 24 – 25 warping, 225 , 255 yielding, 86 , 89 – 91 , 115

Degree of indeterminacy, 631 Design, 46 – 53 , 62 , 192 – 193 , 254 , 540 – 571 , 686 , 696 – 703 , 707 – 711

allowable stress design (ASD), 46 – 47 , 62 aluminum column specifications, 698 beams, 541 – 551 , 558 – 561 , 569 columns, 686 , 696 – 703 , 707 – 711 concentric loading, 696 – 703 eccentric loading, 707 – 711 factor of safety (F.S.), 46 – 47 , 62 , 697 interaction formula for, 707 – 708 limit state design (LSD), 48 load and resistance factor design (LRFD), 48 – 53 , 62 power ( P ) transmission and, 192 – 193 , 254 procedures of analysis for, 49 , 547 , 699 secant formula for, 686 shafts, 192 – 193 , 562 – 565 , 569 simple connections, 46 – 47 , 62 slenderness ratio ( L / r ) for, 696 – 703 steel column specifications, 697 torque diagrams for, 562 wood (timber) column specifications, 698

Differential ring (annulus), 186 , 241 – 242 Dilatation ( e ), 514 – 515 , 537 Direct (simple) shear, 32 Discontinuity functions, 597 – 604 , 656 Discontinuous functions, 260 Displacement ( d ), 124 – 130 , 138 – 146 , 153 – 156 , 175 – 176 , 573 – 574 ,

577 – 589 , 608 – 616 , 632 – 634 , 656 – 657 , 775 – 785 axially loaded members, 124 – 130 , 138 – 146 , 153 – 156 , 175 – 176 bending (shear), 260 , 267 , 356 Castigliano’s theorem for, 775 – 785 compatibility (kinematic) conditions, 139 – 140 , 145 – 146 ,

643 – 645

870 INDEX

Displacement (continued) deflection, 573 – 574 , 577 – 589 , 608 – 616 , 632 – 634 , 656 – 657 elastic curve, 573 – 574 , 608 – 609 , 656 elastic deformation, 124 – 130 , 175 energy, 775 – 785 fixed point, relative to, 126 force (flexibility) method of analysis, 145 – 146 integration method for, 577 – 589 , 632 – 634 , 656 – 657 load-displacement relationship, 139 – 146 , 175 moment-area method, 608 – 616 , 637 – 641 , 657 procedure for analysis of, 127 , 140 , 146 relative, 124 – 130 , 175 shear and moment diagrams, 260 , 267 , 356 sign convention, 126 , 579 , 608 – 609 statically indeterminate members, 139 – 146 , 175 , 608 – 616 ,

656 – 657 superposition, principle of, 138 , 175 thermal stress ( d T ), 153 – 156 , 176

Distributed loads, 260 , 266 – 268 , 356 , 598 – 600 Disturbing force, 662 Ductile materials, 89 – 90 , 115 , 239 , 331 , 524 – 527 , 537

bending, 331 failure of, 239 , 331 , 524 – 527 , 537 Lüder lines, 524 – 525 maximum-distortion-energy theory, 526 – 527 , 537 maximum-shear-stress theory, 524 – 525 , 537 multiaxial stress, 524 – 527 , 537 percent elongation, 89 , 115 percent reduction in area, 89 , 115 plane strain transformation and, 524 – 527 , 537 slipping, 524 – 525 strain-energy density, 526 stress concentration, 239 , 331 stress–strain diagrams for, 89 – 90 , 115 torsional loadings, 239 Tresca yield criterion, 525 yield strength, 89 – 90 yielding, 524 – 527

E Eccentric distance ( e ), 682 – 684 Eccentric loading, 707 – 711 Eccentricity ratio ( ec / r 2 ), 685 – 686 Effective length ( L e ), 671 Elastic behavior, 86 , 88 , 90 , 92 – 93 , 106 , 115 – 116 , 124 – 130 , 153 – 156 ,

160 – 165 , 175 – 176 , 238 – 240 , 255 , 330 – 331 , 357 – 358 axially loaded members, 124 – 130 , 153 – 156 , 160 – 163 , 175 – 176 bending (beams), 330 – 331 , 357 – 358 cross-sectional area of, 124 – 125 , 160 – 163 , 175 – 176 deformation, 86 , 115 – 116 , 124 – 130 , 153 – 156 , 175 – 176 displacement ( d ) and, 124 – 130 , 153 – 156 , 175 – 176 elastic limit, 86 , 88 , 115 nonlinear, 90 perfectly plastic (elastoplastic) materials, 164 – 165 , 176 procedure for analysis of, 127 proportional limit ( s pl ), 86 , 88 , 106 relative displacement (d) of, 124 – 130 , 175

shear modulus ( G ), 106 – 108 , 117 sign convention for, 126 stress concentrations, 160 – 163 , 176 , 238 – 240 , 255 , 330 – 331 ,

357 – 358 thermal stress ( d T ), 153 – 156 , 176 torsional loads, 238 – 240 , 255 Young’s modulus ( E ), 92 – 93 , 115

Elastic curve, 573 – 574 , 608 – 609 , 656 Elastic-plastic torque, 241 – 242 , 244 , 255 Elastic strain energy ( U i ), 94 , 724 – 732

axial loads, 724 – 725 bending moments, 726 – 728 development of, 94 internal work, 724 – 732 transverse shear, 729 – 730 torsional moments, 731 – 732

Elastoplastic behavior, 164 – 165 , 176 Electrical-resistance strain gauge, 84 , 508 Endurance (fatigue) limit, 110 – 111 , 117 Energy methods, 718 – 787

Castigliano’s theorem, 775 – 785 conservation of energy, 737 – 740 , 785 couple moment, work of, 721 displacement ( d ), 775 – 785 elastic strain energy ( U i ), 724 – 732 external work, 719 – 723 , 785 force, work of, 720 impact loading, 744 – 749 internal work, 724 – 732 , 737 – 740 method of virtual forces, 757 , 766 – 770 procedures for analysis of, 761 , 768 , 778 , 782 strain energy, 719 – 732 , 785 stress and, 721 – 723 virtual work, 755 – 763 , 757 , 766 – 770 , 785

Engesser’s equation, 689 , 715 Engineering (nominal) stress or strain, 85 Equilibrium, 4 – 14 , 25 – 26 , 33 , 62 , 164 – 165 , 365 – 366 , 662 – 663

axial loads, 25 – 26 , 164 – 165 balance of forces and moments, 6 – 9 , 62 column buckling and, 662 – 663 coplanar loads, 6 , 9 deformable bodies, 4 – 14 disturbing force, 662 equations of, 6 , 62 external loads, 4 – 5 free-body diagrams, 6 – 9 horizontal force, 365 – 366 inelastic axial deformation, 164 – 165 internal resultant loads, 7 – 8 neutral, 663 normal stress ( s ), 25 – 26 procedure for analysis of, 10 shear formula, 365 – 366 shear stress ( t ), 33 spring force and, 662 – 663 stable, 662 – 663 stress and, 4 – 14 , 25 – 26 , 33 , 62

INDEX 871

support reactions, 5 unstable, 662 – 663

Euler load, 666 – 667 , 715 Extensometer, 84 External loads, 4 – 5 , 542

F Fabricated beams, 545 – 546 , 569 Fabrication errors, trusses, 760 Factor of safety (F.S.), 46 – 47 , 62 , 697 Failure, 109 – 111 , 117 , 225 , 238 – 240 , 255 , 364 – 365 , 524 – 531 , 537 ,

664 – 669 axial loadings, 109 – 111 , 117 bending, 330 – 332 brittle materials, 110 , 239 , 331 , 528 – 529 , 537 creep, 109 – 110 , 117 ductile materials, 239 , 331 , 524 – 527 , 537 endurance (fatigue) limit ( S el ), 110 – 111 , 117 fatigue, 110 – 111 , 117 , 239 fracture, 528 – 529 ideal columns, 664 – 559 maximum-distortion-energy theory, 526 – 527 , 537 maximum-normal-stress theory, 528 , 537 maximum-shear-stress theory, 524 – 525 , 537 Mohr’s failure criterion, 528 – 529 , 537 multiaxial stress and, 524 – 531 , 537 slipping, 524 – 525 strain transformation and, 524 – 531 , 537 stress concentrations and, 109 – 111 , 117 , 238 – 240 , 330 – 332 stress–cycle ( S – N ) diagrams for, 110 – 111 torsional loadings, 238 – 240 , 225 , 255 transverse shear, 364 – 365 Tresca yield criterion, 525 warping, 225 , 255 , 364 – 365 yielding, 524 – 527 , 664 – 669

Fatigue, 110 – 111 , 117 , 239 Flexibility (force) method, 145 – 146 , 643 – 645 Flexural rigidity ( EI ), 576 Flexure formula, 289 – 296 , 357 Force ( F ), 4 – 10 , 24 – 26 , 62 , 94 , 260 , 268 , 289 , 339 , 365 – 366 , 662 – 663 ,

720 , 766 – 770 axially loaded bars, 24 – 26 balance of, 6 body, 5 buckling from, 662 – 663 compression (internal), 260 concentrated, 4 , 268 coplanar, 6 , 9 disturbing, 662 equilibrium and, 6 – 9 , 365 – 366 , 662 – 663 external loads, 4 – 5 internal resultant loads, 7 – 8 , 10 , 24 – 26 horizontal, 365 – 366 loading and distribution of, 4 – 10 , 62 normal ( N ), 8 restoring, 662 resultant ( F R ), 4 , 7 – 8 , 289 , 339

shear ( V ), 8 , 32 shear and moment diagrams for, 260 , 268 stress and distribution of, 22 – 23 , 62 spring, 662 – 663 support reactions, 5 surface, 4 virtual, method of, 766 – 770 weight, 5 work of, 94 , 720

Force (flexibility) method of analysis, 145 – 146 , 643 – 651 Fracture, 528 – 529 . See also Brittle materials Fracture stress ( s f ), 91 Frequency of rotation ( f ), 192 Fully stressed beams, 558 – 561 , 569

G Gauge-length distance, 84 Gauge pressure, 409 Glulam beams, 546

H Homogeneous material, 24 Hooke’s law, 92 – 93 , 115 , 512 – 513 , 537 Hoop (circumferential) stress, 326 , 410 – 411 Hyperbolic stress/strain variation, 324

I Ideal columns, 664 – 669 , 715

critical load ( P cr ), 664 Euler load, 666 – 667 , 715 failure of, 664 least moment of inertia in, 667 pin-supported, 664 – 669 , 715 radius of gyration ( r ), 667 slenderness ratio ( L / r ), 667 – 668 trivial solution for, 665 – 666

Impact factor ( n ), 746 Impact loading, 744 – 749 Inclined axes, 798 – 800 Inelastic behavior, 164 – 170 , 176 , 241 – 248 , 255 , 339 – 349 , 358

annulus (differential ring), 241 – 242 axial loads, 164 – 170 , 176 bending (beams), 339 – 349 , 358 cross section of, 164 – 165 , 176 , 241 – 243 , 255 , 339 – 349 , 358 deformation, 164 – 165 , 176 elastic-plastic torque, 241 – 242 , 244 , 255 linear normal-strain distribution, 339 perfectly plastic (elastoplastic) materials, 164 – 165 , 176 plastic load ( P p ), 165 plastic moment ( M Y ), 340 – 341 , 358 plastic torque ( T p ), 243 , 255 residual stress ( t r ), 166 – 170 , 176 , 243 – 248 , 255 , 342 – 343 , 358 resultant force ( F R ), 339 resultant moment ( M R ), 339 torsional loads, 241 – 248 , 255 ultimate moment, 343 – 344 , 358

872 INDEX

Inelastic buckling, 688 – 690 , 715 Inelastic torsion, 241 – 243 , 255 Inertia ( I ), 185 – 186 , 291 , 306 – 309 , 667 , 791 – 794 , 798 – 803

area ( A ) moments of, 791 – 794 , 798 – 803 bending, 291 , 306 – 309 column buckling, 667 inclined axes, 798 – 800 least moment of, 667 Mohr’s circle for moments of, 801 – 803 moments of, 291 , 306 – 309 , 791 – 794 , 798 – 803 parallel-axis theorem for, 791 – 792 , 796 polar moment of ( J ), 185 – 186 principal axes of, 306 – 307 , 799 – 800 product of, 307 , 795 – 797 , 799 torsional loading, 185 – 186 unsymmetric bending, 306 – 309

Inflection point, 574 In-plane shear strain, 494 , 536 In-plane shear stress, 449 – 455 , 484 Integration method, 577 – 589 , 632 – 634 , 656 – 657

boundary conditions, 580 continuity conditions, 580 deflection and, 573 – 585 , 628 – 630 displacement by, 577 – 589 , 632 – 634 , 656 – 657 elastic curve for, 573 – 574 , 656 procedure for analysis, 581 sign convention, 579 slope by, 577 – 589 , 632 – 634 , 656 – 657 statically determinate shafts and beams, 577 – 589 , 656 statically indeterminate shafts and beams, 632 – 634 , 657

Interaction formula, 707 – 708 Internal loadings, 7 – 10 , 22 , 24 – 26 , 32 , 62

bending moment ( M ) and, 8 – 9 coplanar forces and, 9 force ( F ) distribution and, 7 – 10 , 62 method of sections for, 7 – 8 , 10 normal force ( N ) and, 8 procedure for analysis of, 10 resultant force ( P ), 24 – 26 shear force ( V ) and, 8 , 32 stress and, 22 , 24 – 25 , 62 three-dimensional resultant, 8 torque ( T ) and, 8

Isotropic material, 24

K Keyways, torsional stress and, 238 Kinematic (compatibility) conditions, 139 – 140 , 145 – 146

L Lateral contraction, 104 Lateral deflection, 661 – 664 . See also Buckling Least moment of inertia, 667 Limit state design (LSD), 48 . See also Load and resistance factor

design (LRFD) Linear coefficient of thermal expansion, 153 Linear distributed loads w ( s ), 4

Linear normal-strain distribution, 339 Linear stress/strain variations, 183 – 184 , 286 – 289 Live load, 48 Load ( P ), 4 – 19 , 24 – 32 , 48 , 62 , 120 – 179 , 205 – 206 , 260 , 266 – 268 , 356 ,

408 – 439 , 661 – 664 , 666 – 667 , 696 – 703 , 707 – 711 , 715 , 724 – 732 , 744 – 749 . See also Force ( F ); Torsion

axial, 24 – 31 , 120 – 179 , 724 – 725 bifurcation point, 663 column bucking, 662 – 664 , 666 – 667 , 696 – 703 , 707 – 711 combined, 408 – 439 concentric, 696 – 703 constant, 24 – 25 , 125 , 175 , 205 – 206 coplanar, 9 critical ( P cr ), 661 – 664 , 715 cross section, 7 , 9 dead, 48 deflection and, 597 – 604 deformable bodies, 4 – 9 discontinuity functions for, 597 – 604 direct (simple) shear, 32 distributed, 260 , 266 – 268 , 356 , 598 – 600 eccentric, 707 – 711 elastic strain energy for, 724 – 732 equilibrium and, 4 – 9 , 662 – 663 Euler formula for, 666 – 667 , 715 external, 4 – 5 force ( F ) distribution and, 4 – 10 impact, 744 – 749 internal, 7 – 8 , 10 , 22 , 24 – 26 , 62 linear distributed w ( s ), 4 live, 48 Macauley functions for, 598 – 599 method of sections for, 7 , 9 – 10 moments ( M ) and, 6 – 9 procedure for analysis of, 10 , 416 – 417 shear and moment diagram regions, 260 , 266 – 268 , 356 singularity functions for, 599 – 600 stress ( s ), 24 – 25 three-dimensional resultant, 8 torque ( T ), 205 – 206

Load and resistance factor design (LRFD), 48 – 53 , 62 Load-displacement relationship, 139 – 146 , 175 Load factor ( g ), 48 Localized deformation, 122 – 123 Longitudinal axis, beams, 286 – 288 Longitudinal elongation, 104 Longitudinal (axial) stress, 326 , 410 – 411 Lüder lines, 524 – 525

M M / EI diagrams, 608 – 616 Macauley functions, 598 – 600 Magnitude, 26 , 33 , 382 Material properties, 22 , 24 – 25 , 82 – 119 , 512 – 519 , 537

anisotropic, 24 brittleness, 91 , 110 , 116 bulk modulus ( k ), 514 – 515 , 537

INDEX 873

cohesive material, 22 continuous material, 22 creep, 109 – 110 , 117 dilatation ( e ), 514 – 515 , 537 ductility, 89 – 90 , 115 elastic behavior, 86 , 88 , 90 , 92 – 93 , 115 – 116 failure, 109 – 111 , 117 fatigue, 110 – 111 , 117 homogeneous material, 24 Hooke’s law, 92 – 93 , 115 , 512 – 513 , 537 isotropic material, 24 mechanical, 82 – 119 modulus of elasticity ( E ), 92 – 93 , 115 modulus of resilience ( u r ), 94 , 116 modulus of rigidity ( G ), 106 – 108 , 117 modulus of toughness ( u t ), 95 , 116 multiaxial stress and strain, 512 – 519 , 537 necking, 87 , 115 permanent set, 93 plastic behavior, 86 , 93 , 115 – 116 Poisson’s ratio (n), 104 – 105 , 117 shear modulus (G), 106 – 108 , 117 , 514 , 537 strain energy, 94 – 98 , 116 strain hardening, 87 , 93 , 115 – 116 material-property relationships, 512 – 519 , 537 strain transformation relationships and, 512 – 519 , 537 stress and, 22 , 25 stress–cycle ( S – N ) diagrams for, 110 – 111 stress–strain ( s – P ) diagrams for, 85 – 98 , 106 – 108 , 115 – 117 tension (compression) test for, 83 – 84 , 115 uniform deformation, 24 – 25 yielding, 86 , 115

Maximum deflection ( v max ), 683 – 686 , 715 Maximum-distortion-energy theory, 526 – 527 , 537 Maximum in-plane shear stress, 451 – 455 , 484 Maximum-normal-stress theory, 528 , 537 Maximum-shear-stress theory, 524 – 525 , 537 Mechanics of materials, 3 – 4 Method of sections, 7 , 9 – 10 Method of virtual forces, 757 , 766 – 770 . See also Virtual work Modulus of elasticity ( E ), 92 – 93 , 115 Modulus of resilience ( u r ), 94 , 116 Modulus of rigidity ( G ), 106 – 108 , 117 Modulus of rupture ( t r or s r ), 244 , 342 Modulus of toughness ( u t ), 95 , 116 Mohr’s circle, 465 – 472 , 485 , 498 – 502 , 506 , 524 – 525 , 536 , 801 – 803

absolute maximum shear strain, 506 area ( A ) moments of inertia, 801 – 803 failure probability using, 524 – 525 plane-strain transformation, 498 – 502 , 536 plane-stress transformation, 465 – 472 , 485 procedures for analysis of, 467 – 468 , 498 – 499

Mohr’s failure criterion, 528 – 529 , 537 Moment-area method, 608 – 616 , 637 – 641 , 657 Moment diagrams, 637 – 641

Moments ( M ), 6 – 9 , 62 , 181 , 185 – 186 , 260 , 268 , 285 , 291 , 306 – 312 , 339 – 349 , 356 – 358 , 563 , 573 – 576 , 599 – 600 , 608 – 609 , 656 , 721 , 726 – 728 , 731 – 732 , 791 – 794 , 798 – 803

area ( A ), 791 – 794 , 798 – 803 arbitrarily applied, 308 – 309 balance of, 6 bending (beams), 8 – 9 , 285 , 291 , 306 – 312 , 339 – 349 , 356 – 358 ,

726 – 728 concentrated force and, 268 coplanar loads, 9 couple, work of, 721 curvature relationship, 575 – 576 deflection, 573 – 576 , 599 – 600 , 608 – 609 , 656 elastic curve, 573 – 574 , 608 – 609 , 656 elastic strain energy ( U i ), 726 – 728 , 731 – 732 energy and, 721 , 726 – 728 , 731 – 732 equilibrium and, 6 – 9 , 62 inelastic bending, 339 – 349 , 358 inertia ( I ), 291 , 306 – 309 , 791 – 794 , 798 – 803 internal, 7 – 9 , 260 , 573 – 574 neutral axis orientation and, 309 plastic ( M Y ), 340 – 341 , 358 polar inertia ( J ), 185 – 186 principal axis, applied along, 306 – 307 , 799 resultant ( M R ), 7 – 8 , 339 , 563 shear and moment diagram regions, 260 , 268 singularity functions, 599 – 600 torsional ( T ), 8 , 181 , 731 – 732 ultimate, 343 – 344 , 358 unsymmetric bending, 306 – 312 , 357

Multiaxial stress, 512 – 519 , 723 . See also Failure

N Necking, 87 , 115 Neutral axis, beams, 286 , 290 – 291 , 309 , 323 – 324 Neutral equilibrium, 663 Nominal (engineering) stress or strain, 85 Nominal strength ( P n ), 48 Noncircular shafts, 225 – 233 , 255

closed cross sections, 228 – 233 solid, 225 – 227 , 255 thin-walled tubes, 228 – 233 torsion in, 225 – 233 , 255

Nonlinear elastic behavior, 90 Nonprismatic beams, 558 – 561 , 569 Normal force ( N ), 8 Normal strain (P), 68 – 69 , 71 , 339 , 489 – 493 , 536

determination of, 68 – 69 hyperbolic variation of, 324 linear distribution, 339 linear variation of, 284 – 287 plane-strain transformation orientation, 489 – 493 , 536 principal strains,494, 536 small strain analysis, 71

Normal stress ( s ), 23 – 31 , 46 – 47 , 62 , 183 – 184 , 286 – 289 , 324 , 446 – 447 , 449 – 455 , 484 , 721 – 722

allowable ( s allow ), 46 – 47 , 62

874 INDEX

Normal stress (continued) average, 24 – 31 axially loaded bars, 24 – 31 compressive, 23 constant, 24 – 25 cross-sectional area for, 24 determination of, 23 , 62 distribution of average, 24 – 25 equilibrium and, 25 – 26 hyperbolic variation of, 324 in-plane principal stresses, 449 – 455 , 484 internal loading ( P ), 24 – 26 linear variation of, 183 – 184 , 286 – 289 magnitude and, 26 maximum average, 26 plane-stress transformation orientation, 446 – 447 , 484 prismatic bars and, 24 – 31 procedure for analysis of, 27 strain energy and, 721 – 722 stress transformation, 446 – 447 , 449 – 455 , 484 tensile, 23

O Offset method, 89 – 90 Overhanging beam, 259

P Parabolic shear stress distribution, 372 Parallel-axis theorem, 791 – 792 , 796 Percent elongation, 89 , 115 Percent reduction in area, 89 , 115 Perfectly plastic materials, 86 , 164 Permanent set of material, 93 , 116 Pin-supported columns, 664 – 669 , 715 Plane strain, 489 – 502 , 536

maximum in-plane shear, 494 , 536 Mohr’s circle for, 498 – 502 , 536 normal and shear component orientation, 489 – 493 , 536 principal strains, 494 , 536 procedure for analysis of, 498 – 499 sign convention for, 490 , 493 transformation equations for, 490 – 497 , 536

Plane stress, 441 – 448 , 465 – 472 , 484 – 485 component orientation, 441 – 445 , 484 in-plane principal stresses, 449 – 455 , 484 maximum in-plane shear, 451 – 455 , 484 Mohr’s circle for, 465 – 472 , 485 normal stress ( N ), 446 – 448 , 484 shear stress ( t ), 441 – 448 , 484 procedures for analysis of, 443 , 447 , 467 – 468 sign convention for, 446 – 447 state of, 442 – 445 transformation equations for, 446 – 448 , 484

Plastic behavior, 86 , 93 , 115 – 116 , 164 – 165 , 176 , 241 – 243 , 255 . See also Inelastic behavior

axial loads, 164 – 165 , 176

cross sections for, 164 – 165 , 176 , 241 – 243 , 255 deformation, 86 , 164 – 165 , 176 , 241 – 243 , 255 differential ring (annulus), 241 – 242 elastic-plastic torque, 241 – 242 , 244 , 255 elastoplastic, 164 – 165 , 176 perfectly, 86 , 164 , 243 permanent set, 93 plastic torque ( T p ), 243 , 255 strain hardening, 93 , 116 torsional loading, 241 – 243 , 255 yielding, 86 , 115

Plastic load ( P p ), 165 Plastic moment ( M Y ), 340 – 341 , 358 Plastic torque ( T p ), 243 , 255 Poisson’s ratio (n), 104 – 105 , 117 Polar moment of inertia ( J ), 185 – 186 Power ( P ) transmission, 192 – 193 , 254 Pressure vessels, see Thin-walled elements Principal axis, moments applied along, 306 – 307 , 799 Principal stresses, 449 – 455 , 484 Principle of superposition, 138 , 175 Prismatic bars, 24 – 31 Prismatic beams, 544 – 551 , 569 Product of inertia, 307 , 795 – 797 , 799 Proportional limit ( s pl ), 86 , 88 , 106 Pure shear, 33 , 106

R Radial distance ( r ), 182 – 183 Radial stress, 326 , 411 Radius of curvature, 576 Radius of gyration ( r ), 667 Reactions, 5 Redundants, 631 – 632 , 657 Reinforced concrete beams, 319 – 322 Relative displacement (d), 124 – 130 , 175 Relative rotation ( df ), 204 – 205 Residual stress, 166 – 170 , 176 , 243 – 248 , 255 , 342 – 343 , 358

axial loadings, 166 – 170 , 176 bending (beams), 342 – 343 , 358 modulus of rupture ( t r or s r ), 244 , 342 torsional loadings, 243 – 248 , 255

Resistance factor ( f ), 48 Resultant forces ( F R ), 4 , 7 – 8 , 289 , 339 Resultant moment ( M R ), 7 – 8 , 339 , 563 Right-hand rule, 8 , 184 , 206 – 207 Rotation of shafts, 182 – 183 , 192 , 204 – 205

S Saint Venant’s principle, 121 – 123 , 175 Secant formula, 652 – 587 , 715 Section modulus ( S ), 544 , 558 Shafts, 180 – 257 , 562 – 565 , 569 , 572 – 659

angle of twist ( f ), 182 – 183 , 204 – 211 , 226 , 230 , 254 average shear stress ( t avg ), 229 – 230 , 255 bulging, 225

INDEX 875

circular, 181 – 224 , 154 constant torque and, 205 – 206 cross sections of, 182 – 191 , 204 – 208 , 225 – 227 , 254 – 255 deflection of, 572 – 659 design, 192 – 193 , 562 – 565 , 569 discontinuity functions, 597 – 604 , 656 displacement, 573 – 574 , 577 – 589 , 608 – 616 , 632 – 634 , 656 – 657 elastic curve, 573 – 574 , 608 – 609 , 656 flexural rigidity ( EI ), 576 inflection point, 574 force (flexibility) method of analysis, 145 – 146 , 643 – 651 frequency of rotation ( f ), 192 inelastic torsion, 241 – 243 , 255 integration method for, 577 – 589 , 632 – 634 , 656 – 657 maximum torsional (shear) stress ( t max ), 184 , 187 , 227 , 239 , 255 moment-area method for, 608 – 616 , 637 – 641 , 657 moment-curvature relationship, 575 – 576 moment diagrams for, 637 – 641 multiple torques along, 206 – 207 noncircular, 225 – 233 , 255 polar moment of inertia ( J ), 185 – 186 power ( P ) transmission by, 192 – 193 , 254 procedures for analysis of, 188 , 208 , 219 , 581 , 601 , 610 , 646 radial distance ( r ), 182 – 183 resultant moment for, 563 rotation of, 182 – 183 , 192 , 204 – 205 shape variations, 226 shear-stress ( t ) distribution, 184 – 185 , 187 solid, 186 , 192 slope, 573 – 574 , 577 – 589 , 608 – 616 , 656 – 657 statically indeterminate, 218 – 221 , 254 , 631 – 657 stress concentration factor ( K ), 238 – 240 , 255 superposition method for, 623 – 627 , 643 – 651 , 657 torque diagrams for, 187 , 207 , 562 torsion formula for, 184 – 191 torsion loads on, 180 – 257 torsional deformation and, 181 – 191 tubular, 187 – 188 , 192 , 228 – 233 , 254 – 255 warping, 225 , 255

Shanley theory of inelastic buckling, 689 Shape factor, 341 Shear and moment diagrams, 259 – 275 , 356

bending (beams), 259 – 275 , 356 concentrated force and moment regions, 268 discontinuous functions of, 260 distributed load regions, 260 , 266 – 268 , 356 functions of, 260 graphical method for construction of, 266 – 275 , 356 internal moments (compression), 260 procedures for analysis of, 261 , 269 regions of, 260 , 266 – 275 slope (shear) of, 260 , 267 , 356 sign convention for, 260 , 267 – 268

Shear center ( O ), 396 – 401 , 406 Shear flow ( q ), 228 – 229 , 382 – 386 , 391 – 395 , 405 – 406

built-up members, 382 – 386 , 405 directional sense of, 394

linearity of, 394 magnitude of, 382 parallel, 391 – 392 , 394 thin-walled members, 391 – 395 , 406 thin-walled tubes, 228 – 229 torsional loading and, 228 – 229 transverse shear and, 382 – 386 , 391 – 395 , 405 – 406 vertical, 382 – 383 web, 393

Shear force ( V ), 8 , 32 , 106 – 108 , 117 , 260 , 268 , 363 – 365 average shear stress from, 32 bending moments ( M ) and, 260 , 268 development of, 8 modulus of elasticity ( G ), 106 – 108 , 117 straight members (beams), 363 – 365

Shear formula, 365 – 375 , 405 horizontal force equilibrium, 365 limitations on use of, 368 – 369 procedure for analysis using, 370 shear stress ( t ) for, 366 – 367

Shear strain ( g ), 69 , 182 – 183 , 489 – 494 , 506 – 507 , 536 component orientation, 489 – 493 , 536 determination of, 69 linear variation in, 183 maximum in-plane, 494 , 506 – 507 , 536 plain-strain transformation 489– 494 , 506 – 507 , 536 torsional deformation and, 182 – 183

Shear stress ( t ), 23 , 32 – 37 , 46 – 47 , 62 , 106 – 108 , 117 , 184 – 191 , 225 – 227 , 229 – 230 , 239 , 255 , 362 – 407 , 440 – 487 , 722 – 723

absolute maximum ( t max ), 184 – 185 , 187 , 227 , 239 , 255 , 477 – 481 , 485

allowable ( t allow ), 46 – 47 , 62 average ( t avg ), 32 – 37 , 62 , 229 – 230 , 255 beams, 362 – 407 complementary property of, 33 component orientation, 442 – 443 , 484 determination of, 23 , 62 direct (simple) loads, 32 equilibrium and, 33 in-plane transformations, 449 – 455 , 484 linear variation in, 184 modulus of elasticity/rigidity ( G ), 106 – 108 , 117 parabolic distribution, 372 plane-stress transformation, 446 – 448 , 477 – 481 , 484 – 485 procedure for analysis of, 34 , 443 , 447 , 467 – 468 proportional limit ( t pl ), 106 pure, 33 , 106 shafts, distribution in, 184 – 185 , 187 , 225 – 227 , 229 – 230 , 239 , 255 simple (direct) loads, 32 strain energy and, 722 – 723 thin-walled tubes, 225 – 227 , 255 torsional loads and, 184 – 191 , 225 – 227 , 229 – 230 , 239 , 255 transformations, 440 – 487 transverse, 362 – 407 ultimate ( t u ), 106

Shear stress—strain diagrams, 106 – 108 , 117

876 INDEX

Sign convention, 8 , 26 , 126 , 184 , 206 – 207 , 260 , 267 – 268 , 291 , 309 , 446 , 490 , 493 , 506 , 579 , 608 – 609

absolute maximum shear strain, 506 angle of twist ( f ), 206 – 207 axial loads, 26 , 126 bending, 260 , 267 – 268 , 291 , 309 coordinates and, 579 deflection, 579 , 608 – 609 displacement ( d ), 126 , 579 , 608 – 609 Poisson’s ratio ( n ), 104 right-hand rule for, 8 , 184 , 206 – 207 slope of elastic curve, 579 , 608 – 609 strain transformation, 490 , 493 , 506 stress transformation, 446 torque ( T ), 8 , 184 , 206 – 207

Simple connections, ASD for, 46 – 47 , 62 Simple (direct) shear, 32 Simply supported beam, 259 Singularity functions, 599 – 600 Slenderness ratio ( L / r ), 667 – 668 , 696 – 703 , 671 Slipping, 524 – 525 Slope, 260 , 267 , 356 , 573 – 574 , 577 – 589 , 608 – 616 , 632 – 634 , 656 – 657

bending (shear), 260 , 267 , 356 deflection, 573 – 574 , 577 – 589 , 608 – 616 , 632 – 634 , 656 – 657 elastic curve, 573 – 574 , 608 – 609 , 656 integration method for, 577 – 589 , 632 – 634 , 656 – 657 moment-area method for, 608 – 616 , 637 – 641 , 657 shear and moment diagrams, 260 , 267 , 356 sign convention, 579 , 608 – 609 statically indeterminate members, 608 – 616 , 656 – 657

Small strain analysis, 71 Spherical pressure vessels, 411 , 436 Spring force, 662 – 663 Stable equilibrium, 662 – 663 State of stress, 23 , 416 – 424 , 436 , 442 – 445

combined loadings and, 416 – 424 , 436 determination of, 23 plane stress transformation, 442 – 445 procedures for analysis of, 416 – 417 , 443

Statically indeterminate members, 139 – 146 , 153 , 166 , 175 , 218 – 221 , 254 , 631 – 657

axially-loaded, 139 – 146 , 153 , 166 , 175 beams, 631 – 657 compatibility conditions for, 139 – 140 , 145 – 146 , 632 deflection of, 631 – 657 degree of indeterminacy, 631 displacement ( d ), 139 – 146 , 175 force (flexibility) method of analysis, 145 – 146 , 643 – 651 integration method for, 632 – 634 , 657 load-displacement relationship, 139 – 146 , 175 moment-area method for, 637 – 641 , 657 moment diagrams for, 637 – 641 procedures for analysis of, 140 , 146 , 219 , 646 redundants, 631 – 632 , 657 residual stresses ( t r ), 166 shafts, 218 – 221 , 254 , 631 – 657

superposition method for, 643 – 651 , 657 thermal stress ( d T ), 153 torque-loaded, 218 – 221 , 254

Steel beams, 545 Steel column specifications, 697 Straight members, see Beams Strain, 66 – 81 , 85 , 104 – 105 , 117 , 182 – 183 , 339 , 488 – 539 . See also

Normal strain ( P ); Shear strain ( g ) Cartesian components of, 70 component orientation, 489 – 493 , 536 deformation and, 66 – 81 engineering, 85 inelastic behavior, 339 linear distribution, 339 multiaxial stress and, 512 – 519 nominal, 85 normal ( P ), 68 – 69 , 71 , 489 – 493 , 536 plane, 489 – 502 , 536 principals, 494 , 536 Poisson’s ratio (n), 104 – 105 , 117 shear ( g ), 69 , 182 – 183 , 489 – 493 , 536 small strain analysis, 71 state of, 70 transformation, 488 – 539 units of, 68 – 69

Strain energy ( u ), 94 – 98 , 116 , 526 , 719 – 732 , 785 deformation and, 94 – 98 , 116 density, 94 , 526 elastic, 94 , 724 – 732 external work and, 719 – 732 , 785 modulus of resilience ( u r ), 94 , 116 modulus of toughness ( u t ), 95 , 116 multiaxial stress, 723 normal stress ( s ), 721 – 722 shear stress ( t ), 722 – 723

Strain hardening, 87 , 93 , 115 – 116 Hooke’s law and, 93 , 115 permanent set of materials, 93 , 116 ultimate stress ( s u ), 87

Strain rosettes, 508 – 509 Strain transformation, 488 – 539

absolute maximum shear strain, 506 – 507 , 536 bulk modulus ( k ), 514 – 515 , 537 dilatation ( e ), 514 – 515 , 537 electrical-resistance strain gauge for, 508 equations for, 490 – 497 , 536 failure, theories of, 524 – 531 , 537 Hooke’s law for, 512 – 513 , 537 in-plane shear strain, 494 , 536 material-property relationships, 512 – 519 , 537 Mohr’s circle, 498 – 502 , 506 , 524 – 525 , 536 multiaxial stress, 512 – 519 normal and shear component orientation, 489 – 493 , 536 plane strain, 489 – 502 , 536 principal strains, 494 , 536 procedure for analysis of, 498 – 499

INDEX 877

shear modulus ( G ) and, 514 , 537 sign convention for, 490 , 493 , 506 strain rosettes, 508 – 509

Stress, 2 – 65 , 85 – 87 , 106 – 111 , 115 – 117 , 153 – 156 , 160 – 163 , 166 – 170 , 176 , 183 – 191 , 225 – 227 , 238 – 248 , 254 – 255 , 255 , 286 – 289 , 324 – 326 , 342 – 343 , 408 – 439 , 440 – 487 , 512 – 519 , 537 , 542 – 543 , 707 – 711 . See also Normal stress ( s ); Shear stress ( t ); Torque ( T ); Transverse shear

allowable stress design (ASD), 46 – 47 , 62 axial (longitudinal), 326 , 410 – 411 axially loaded members, 24 – 31 , 62 , 153 – 156 , 160 – 163 ,

166 – 170 , 176 bending (beams), 324 – 326 , 330 – 332 , 342 – 343 , 358 biaxial, 411 circumferential (hoop), 326 , 410 – 411 columns, distribution in, 707 – 711 combined loadings, 408 – 439 component orientation, 442 – 443 , 484 compressive, 23 , 707 concentration,160– 163 , 176 , 238 – 240 , 255 , 330 – 332 , 358 constant, 24 – 25 curved beams, 324 – 326 deformable bodies, 4 – 14 directional sense of, 394 , 410 elastic behavior, 160 – 163 , 176 endurance (fatigue) limit, 110 – 111 , 117 energy and, 721 – 723 engineering, 85 equilibrium and, 4 – 14 , 25 – 26 , 33 , 60 factor of safety (F.S.), 46 – 47 , 62 fatigue failure and, 109 – 111 , 117 force distribution and, 22 – 23 , 62 fracture ( s f ), 87 , 91 hoop (circumferential), 326 , 410 – 411 hyperbolic variation, 324 – 325 in-plane shear, 451 – 455 , 484 internal force ( P ) and, 22 , 24 – 26 , 62 linear variations, 183 – 184 , 286 – 289 load and resistance factor (LRFD), 48 – 53 , 62 longitudinal (axial), 326 , 410 – 411 material properties and, 22 , 24 – 25 mechanics of materials and, 3 – 4 multiaxial, 512 – 519 , 537 , 723 necking, 87 , 115 nominal, 85 normal ( s ), 23 – 31 , 62 , 442 – 443 , 484 , 721 – 722 plane, 441 – 448 , 465 – 472 , 484 – 485 principal, 449 – 455 , 484 prismatic bars, 24 – 31 prismatic beams, 542 – 543 procedures for analysis of, 27 , 34 , 49 radial, 326 , 411 residual ( t r ), 166 – 170 , 176 , 243 – 248 , 255 , 342 – 343 , 358 shear ( t ), 23 , 32 – 37 , 62 , 106 – 108 , 117 , 442 – 443 , 484 , 722 – 723 simple connections, 47 , 62 state of, 23 , 416 – 424 , 436 , 442 – 445 strain energy and, 721 – 723

tensile, 23 thermal ( d T ), 153 – 156 , 176 torsional, 184 – 191 , 225 – 227 , 238 – 248 , 254 – 255 trajectories, 542 – 543 transformation, 440 – 487 triaxial, 477 – 481 ultimate ( s u ), 87 , 106 uniaxial, 25 – 26 units of, 23 yield point ( s Y ), 86 , 115 – 116

Stress-concentration factor ( K ),160– 163 , 176 , 238 – 240 , 255 , 330 – 332 , 358

axial loads, 160 – 163 , 176 bending, 330 – 332 , 358 torsional loads, 238 – 240 , 255

Stress–cycle ( S – N ) diagrams, 110 – 111 Stress–strain ( s – P ) diagrams, 85 – 98 , 106 – 111 , 115 – 117

brittle materials, 91 , 116 conventional, 85 – 87 ductile materials, 89 – 90 , 115 elastic behavior, 86 , 92 – 93 , 106 , 115 – 116 endurance (fatigue) limit ( S el ), 110 – 111 fracture stress ( s f ), 87 Hooke’s law, 92 – 93 , 115 necking, 87 , 115 nominal (engineering) stress or strain, 85 offset method, 89 – 90 plastic behavior, 86 , 93 , 115 – 116 Poisson’s ratio (n), 104 – 105 , 117 proportional limit ( s pl ), 86 , 88 , 106 rigidity of materials, 106 shear, 106 – 108 , 117 strain energy ( u ), 94 – 98 , 116 strain hardening, 87 , 93 , 115 – 116 true, 87 – 88 ultimate stress ( s u ), 87 , 106 yield point ( s Y ), 86 , 115 – 116 yielding, 86 , 115

Stress transformation, 440 – 487 absolute maximum shear stress ( t max ), 477 – 481 , 485 equations for, 446 – 448 , 484 in-plane shear stress, 449 – 455 , 484 Mohr’s circle for, 465 – 472 , 485 normal and shear component orientation, 446 – 447 , 484 plane stress, 441 – 448 , 465 – 472 , 484 – 485 principal stresses, 449 – 455 , 484 procedures for analysis of, 443 , 447 , 467 – 468 sign convention for, 446 – 447 triaxial stress, 477 – 481

Structural design, LRFD for, 48 – 53 , 62 Superposition, 138 , 175 , 623 – 627 , 637 – 641 , 643 – 651 , 657

axial loaded members, 138 , 175 compatibility equations, 643 – 645 deflection solutions by, 623 – 627 , 643 – 651 , 657 moment diagrams constructed by, 643 – 651 principle of, 138 , 175

878 INDEX

Superposition (continued) procedure for analysis by, 646 statically indeterminate shafts and beams, 637 – 641 , 657

Support reactions, 5 , 580 Supports for columns, 664 – 669 , 670 – 673 Surface forces, 4

T Tangent modulus ( E t ), 688 – 689 Tensile stress, 23 Tension (compression) test, 83 – 84 , 115 Thermal (stress) ( d T ), 153 – 156 , 176 Thin-walled elements, 228 – 233 , 255 , 391 – 401 , 406 , 409 – 412 , 436

angle of twist ( f ), 230 average shear stress ( t avg ), 229 – 230 , 255 axis of symmetry, 396 – 397 beams, 391 – 401 , 406 biaxial stress, 411 circumferential (hoop) stress direction, 410 closed cross sections, 228 – 233 combined loadings, 409 – 412 , 436 cylindrical vessels, 410 – 411 , 436 longitudinal (axial) stress direction, 410 pressure vessels, 409 – 412 , 436 procedure for analysis of, 398 radial stress, 411 shear center ( O ), 396 – 401 , 406 shear flow ( q ), 228 – 229 , 391 – 395 , 406 spherical vessels, 411 , 436 transverse shear in, 391 – 395 , 406 tubes, 228 – 233 , 255 twisting, 396 – 397 web, 393

Three-dimensional resultant, 8 Torque ( T ), 8 , 181 – 191 , 204 – 211 , 241 – 243 , 254 – 255

angle of twist ( f ) and, 182 – 183 , 204 – 211 , 254 constant, 205 – 206 deformation from, 181 – 183 elastic-plastic, 241 – 242 , 244 , 255 external, 181 – 183 inelastic torsion and, 241 – 243 , 255 internal, 8 , 184 – 191 , 204 – 211 , 254 maximum elastic ( T Y ), 242 multiple, 206 – 207 plastic ( T p ), 243 , 255 residual stress and, 244 – 248 , 255 right-hand rule for, 184 , 206 – 207 sign convention for, 184 , 206 – 207 torsion formula for, 184 – 191 torsional moment, as, 8 , 181 ultimate ( T u ), 245

Torque diagrams, 187 , 207 , 562 Torsion, 180 – 257 , 731 – 732 . See also Torque ( T )

angle of twist ( f ), 182 – 183 , 204 – 211 , 206 , 226 , 230 , 254 couplings, 238 cross sections for, 182 – 191 , 204 – 208 , 225 – 233 , 238 – 240 ,

254 – 255

deformation, 181 – 183 elastic strain energy ( U i ), 731 – 732 formula for, 184 – 191 inelastic, 241 – 248 , 255 keyways, 238 linear stress/strain variations, 183 – 184 modulus of rupture ( t r ), 244 power transmission and, 192 – 193 , 254 procedures for analysis of, 188 , 208 , 219 radial distance ( r ), 182 – 183 residual stress ( t r ), 243 – 248 , 255 shafts, 180 – 257 shear strain ( g ) and, 182 – 183 shear stress ( t ) distribution, 184 – 191 , 225 – 227 sign conventions for, 184 , 206 – 207 static loadings, 239 statically indeterminate members, 218 – 221 , 254 stress concentration factor ( K ), 238 – 240 , 255 stress distribution, 184 – 191 , 225 – 227 , 238 – 248 , 254 – 255 tubes, 187 – 188 , 192 , 228 – 233 , 254 – 255 warping from, 225 , 255

Transformation equations, 798 Transformation factor ( n ), 317 – 318 Transformed section method, 316 – 318 , 357 Transverse shear, 362 – 407 , 729 – 730

beams and, 362 – 407 built-up members, 382 – 386 , 405 elastic strain energy ( U i ), 729 – 730 procedures for analysis of, 370 , 398 shear ( V ) in, 363 – 365 shear center ( O ), 396 – 401 , 406 shear flow ( q ), 382 – 386 , 391 – 395 , 405 – 406 shear formula for, 365 – 375 , 405 straight beams, 363 – 365 thin-walled members, 391 – 401 , 406 warping cross-sections, 364 – 365

Tresca yield criterion, 525 Triaxial stress, 477 – 481 True stress—strain diagrams, 87 – 88 Trusses, 759 – 763 , 777 – 779

Castigliano’s theorem, 777 – 779 fabrication errors, 760 procedures for analysis of, 761 , 778 temperature changes and, 760 virtual work analysis of, 759 – 763

Tubes, 187 – 188 , 192 , 208 , 228 – 233 , 255 angle of twist ( f ), 230 average shear stress ( t avg ), 229 – 230 , 255 closed cross sections, 228 – 233 cross section of, 187 , 228 – 233 , 255 procedure for analysis of, 188 , 208 power transmission by, 192 shear flow ( q ) in, 228 – 229 thin-walled, 288 – 233 , 255 torsion formula for, 187 – 188

Twisting, 396 – 397

INDEX 879

U Ultimate (bending) moment, 343 – 344 , 358 Ultimate stress ( s u ), 87 , 106 Ultimate torque ( T u ), 245 Uniaxial stress, 25 – 26 Uniform deformation, 24 – 25 Units, 23 , 68 – 69 , 153 , 192

power, 192 strain, 68 – 69 stress, 23 thermal stress, 153

Unstable equilibrium, 662 – 663 Unsymmetric bending, 306 – 312 , 357

V Virtual work, 755 – 763 , 757 , 766 – 770 , 785

application of, 755 – 757 beams, 766 – 770 equations for, 758 , 785 internal, 757 – 758 method of virtual forces, 757 , 766 – 770 procedures for analysis of, 761 , 768 temperature change and, 760 trusses, 759 – 763

W Warping, 225 , 255 , 364 – 365 Web, shear flow ( q ) in, 393

Weight, 5 Wide-flange (W shape) section properties, 804 – 805 , 808 – 809 Wood beams, 545 Wood (timber) column specifications, 698 Work, 94 , 192 , 719 – 732 , 737 – 740 , 755 – 763 , 785

conservation of energy, 737 – 740 , 785 couple moment, 721 elastic strain energy ( U i ), 724 – 732 external, 719 – 723 , 785 force ( F ) as, 94 , 720 internal, 724 – 732 , 737 – 740 , 757 – 758 power ( P ) as, 192 strain energy, 715 – 728 virtual, 755 – 763 , 785

Y Yield point ( s Y ), 86 , 115 – 116 Yield strength, 89 – 90 Yielding, 86 , 89 – 91 , 115 , 524 – 527 , 664 – 669 . See also Ductile

materials deformation from, 86 , 89 – 91 , 115 ideal columns (buckling), 664 – 669 failure from, 524 – 527 , 664 – 669

Young’s modulus ( E ), 92 – 93 , 115

Front Cover
Title Page
Copyright Page
Dedication Page
Preface
CONTENTS
Credits
1. Stress

1.1 Introduction
1.2 Equilibrium of a Deformable Body
1.3 Stress
1.4 Average Normal Stress in an Axially Loaded Bar
1.5 Average Shear Stress
1.6 Allowable Stress Design
1.7 Limit State Design
Chapter Review

2. Strain

2.1 Deformation
2.2 Strain

3. Mechanical Properties of Materials

3.1 The Tension and Compression Test
3.2 The Stress–Strain Diagram
3.3 Stress–Strain Behavior of Ductile and Brittle Materials
3.4 Hooke’s Law
3.5 Strain Energy
3.6 Poisson’s Ratio
3.7 The Shear Stress–Strain Diagram
3.8 Failure of Materials Due to Creep and Fatigue
Chapter Review

4. Axial Load

4.1 Saint-Venant’s Principle
4.2 Elastic Deformation of an Axially Loaded Member
4.3 Principle of Superposition
4.4 Statically Indeterminate Axially Loaded Member
4.5 The Force Method of Analysis for Axially Loaded Members
4.6 Thermal Stress
4.7 Stress Concentrations
4.8 Inelastic Axial Deformation
4.9 Residual Stress
Chapter Review

5. Torsion

5.1 Torsional Deformation of a Circular Shaft
5.2 The Torsion Formula
5.3 Power Transmission
5.4 Angle of Twist
5.5 Statically Indeterminate Torque-Loaded Members
5.6 Solid Noncircular Shafts
5.7 Thin-Walled Tubes Having Closed Cross Sections
5.8 Stress Concentration
5.9 Inelastic Torsion
5.10 Residual Stress
Chapter Review

6. Bending

6.1 Shear and Moment Diagrams
6.2 Graphical Method for Constructing Shear and Moment Diagrams
6.3 Bending Deformation of a Straight Member
6.4 The Flexure Formula
6.5 Unsymmetric Bending
6.6 Composite Beams
6.7 Reinforced Concrete Beams
6.8 Curved Beams
6.9 Stress Concentrations
6.10 Inelastic Bending
Chapter Review

7. Transverse Shear

7.1 Shear in Straight Members
7.2 The Shear Formula
7.3 Shear Flow in Built-Up Members
7.4 Shear Flow in Thin-Walled Members
7.5 Shear Center for Open Thin-Walled Members
Chapter Review

8. Combined Loadings

8.1 Thin-Walled Pressure Vessels
8.2 State of Stress Caused by Combined Loadings
Chapter Review

9. Stress Transformation

9.1 Plane-Stress Transformation
9.2 General Equations of Plane-Stress Transformation
9.3 Principal Stresses and Maximum In-Plane Shear Stress
9.4 Mohr’s Circle—Plane Stress
9.5 Absolute Maximum Shear Stress
Chapter Review

10. Strain Transformation

10.1 Plane Strain
10.2 General Equations of Plane-Strain Transformation
10.3 Mohr’s Circle—Plane Strain
10.4 Absolute Maximum Shear Strain
10.5 Strain Rosettes
10.6 Material-Property Relationships
10.7 Theories of Failure
Chapter Review

11. Design of Beams and Shafts

11.1 Basis for Beam Design
11.2 Prismatic Beam Design
11.3 Fully Stressed Beams
11.4 Shaft Design
Chapter Review

12. Deflection of Beams and Shafts

12.1 The Elastic Curve
12.2 Slope and Displacement by Integration
12.3 Discontinuity Functions
12.4 S lope and Displacement by the Moment-Area Method
12.5 Method of Superposition
12.6 Statically Indeterminate Beams and Shafts
12.7 Statically Indeterminate Beams and Shafts—Method of Integration
12.8 Statically Indeterminate Beams and Shafts — Moment-Area Method
12.9 Statically Indeterminate Beams and Shafts — Method of Superposition
Chapter Review

13. Buckling of Columns

13.1 Critical Load
13.2 Ideal Column with Pin Supports
13.3 Columns Having Various Types of Supports
13.4 The Secant Formula
13.5 Inelastic Buckling
13.6 Design of Columns for Concentric Loading
13.7 Design of Columns for Eccentric Loading
Chapter Review

14. Energy Methods

14.1 External Work and Strain Energy
14.2 Elastic Strain Energy for Various Types of Loading
14.3 Conservation of Energy
14.4 Impact Loading
14.5 Principle of Virtual Work
14.6 Method of Virtual Forces Applied to Trusses
14.7 Method of Virtual Forces Applied to Beams
14.8 Castigliano’s Theorem
14.9 Castigliano’s Theorem Applied to Trusses
14.10 Castigliano’s Theorem Applied to Beams
Chapter Review

APPENDICES

A. Geometric Properties of an Area

A.1 Centroid of an Area
A.2 Moment of Inertia for an Area
A.3 Product of Inertia for an Area
A.4 Moments of Inertia for an Area about Inclined Axes
A.5 Mohr’s Circle for Moments of Inertia

B. Geometric Properties of Structural Shapes

W Shapes Wide (FPS Units)
C Shapes (FPS Units)
Angle Shapes Equal (FPS Units)
W Shapes Wide (SI Units)
C Shapes (SI Units)
Angle Shapes Equal (SI Units)

C. Slopes and Deflections of Beams

Solutions and Answers for Preliminary Problems

CH01
CH02, CH04
CH05
CH06
CH07
CH08
CH09, CH12

Fundamental Problems Partial Solutions and Answers

CH01
CH02, CH03
CH04, CH05
CH06
CH07
CH08
CH09
CH11
CH12
CH13

Answers to Selected Problems

CH01
CH02
CH03
CH04
CH05
CH06
CH07, CH08
CH09
CH10
CH11
CH12
CH13
CH14

INDEX

A, B
C
D
E
F, G, H, I
K, L, M
N
O, P, R, S
T
U, V, W, Y

Fundamental Equations of Mechanics of Materials
Geometric Properties of Area Elements
Average Mechanical Properties of Typical Engineering Materials (SI Units)
Average Mechanical Properties of Typical Engineering Materials (U.S. Customary Units)