a 1,000-word essay. In the essay, you must summarize what you consider to be the key concepts from the chapters covered, and you must include at least three photographs that you've taken to help explain the concepts.
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6186695_67669615_engr213e_week3_chapter_05.pdf
AXIAL DEFORMATION
Chapter 5
Perspec've This is a very important chapter. The remainder of the course is, for the most part, a more complicated version of this material. The chapter begins by introducing St. Venant’s Principle, and limi'ng our subsequent focus to “slender structural elements”. With this restric'on in place, equa'ons are derived for the deflec'on of a structure under axial load. This is the key to sa'sfying deflec'on-‐based design requirements, and to evalua'on of sta'cally-‐indeterminate structures (something that sta'cs alone does not accomplish). If you learn this material, you are in good shape for the remainder of the class.
INTRODUCTION 5.1
Real Applica'ons • Axially loaded structures may seem like an insignificant niche
in the mechanics world, but look around – they are real • Here is what I found walking around campus at Oregon State
These axially loaded structural elements support the overhang that keeps Beaver fans out of the rain as they wait to get into the stadium.
SAINT-‐VENANT’S PRINCIPLE
5.2
Sta'cally-‐Equivalent • Even though it is just a “principle”, and you will not directly
solve “St. Venant’s problems”, the concept is very important
• In a nutshell: At a “sufficient distance” away from the loca'on where a load is applied, how the load is applied is unimportant for the overall behavior of the component
• Two important implica'ons: 1. We can make meaningful calcula'ons based simply on the magnitude
of the applied loads, not on the applica'on details 2. We need to limit our analyses to rela'vely long and slender structural
elements if we want results of reasonable accuracy
Slender Structural Elements • Think about it: If the load applica'on regions are influenced
by the manner of load applica'on, a rela'vely short and thick structure is mostly a collec'on of these interac'ng regions
• It takes more advanced techniques, some of them computer-‐ based, to properly analyze these structures
• If we instead limit ourselves to structures where the load applica'ons regions influence a small por'on of the overall structure, then we can proceed with confidence
• Thus “slender structural elements”
DEFORMATIONS IN AXIALLY LOADED BARS
5.3
A Key Equa'on … • This equa'on is simple, but very important:
• Focus first on the individual terms:
δ = FL AE
δ = overall change in length of the bar F = internal reaction force throughout the bar L = original length of the bar A = cross-sectional area of the bar E = modulus of elasticity of the bar material
When Does This Equa'on Apply? • A key engineering skill is selec'ng the correct equa'on for a
given situa'on
• The basic axial deforma'on equa'on applies when the terms F, A, and E are uniform (constant magnitude) over the en're length (L) of the bar
• Think of it this way: – If F, A, and/or E varied over the length (L) of the bar, what values
would you plug into the equa'on? – We must modify the equa'on to account for terms that vary
Internal Reac'on Force • I highlighted “internal reac'on force” in the original equa'on
presenta'on for a reason – it is the single most significant source of confusion in the en're class
Equa'ons for stress and deflec'on contain terms that represent internal reac'on forces. These terms are dis'nct from applied loads, although in some situa'ons they may have equivalent magnitudes. Carefully dis'nguish between applied loads and internal reac'ons!
What is an Internal Reac'on? • In Sta'cs you frequently isolated a structure from a constraint
or from another object to create free body diagrams • The loads that developed at these severed connec'ons were
referred to as external reac'ons • An internal reac'on is simply the load that exists when you
create a free body diagram by cuang a structural component
It doesn’t maber where along bar (2) I cut to create a free body diagram, the internal reac'on on the cut face (F) must be equal in magnitude and opposite in direc'on to the applied load (P) for sta'c equilibrium.
F
A Bar With Step-‐Changes • This structure is analyzed by considering the behavior of the
individual components – note that both (1) and (2) are uniform along their length, but different from each other
Internal reac'on force (F) uniform and equal in magnitude to (P).
Internal reac'on force (F) uniform and equal in magnitude to (P).
δ1 = PL1 A1E1
δ2 = PL2 A2E2
δ total = δ1 +δ2
Addi'onal Applied Loads • This structure is different, because in addi'onal to PD applied
at the free end, PC and PB are applied along the way – Internal reac'on force F1 in (1) has a magnitude of PB+PD-‐PC – Internal reac'on force F2 in (2) has a magnitude of PD-‐PC – Internal reac'on force F3 in (3) has a magnitude of PD
• This is just sta'cs of free body diagrams with the bars cut
Common Mistakes • I see this mistake a lot in abempts at problem solving:
– Internal reac'on force in sec'on (1) is PB (it is not) – Internal reac'on force in sec'on (2) is PC (it is not) – It is correct that the force in sec'on (3) is PD
• Learn what you can do quickly by inspec'on, and what takes a bit more thought – this is a key engineering skill
What Always Works?
If in doubt, construct a proper Free Body Diagram and calculate internal resultants by the methods of sta'cs. This works for everything we cover in this class: axial loading, torsion, beam bending, combined loads. If you cannot do sta'cs, you cannot complete this class.
DEFORMATIONS IN A SYSTEM OF AXIALLY
LOADED BARS
5.4
Sta'cs and Geometry • This sec'on applies the axial deforma'on equa'on
• Simply find the internal resultant force in each deformable bar (note that problems will ohen describe some structural elements as rigid) and use the deflec'on equa'on
• The rest is geometry. Similar triangles and finding the angle of 'lt from an arc-‐tangent (inverse tangent) rela'onship will generally solve the problems
• Tip: make careful sketches with deforma'ons exaggerated
STATICALLY INDETERMINATE AXIALLY LOADED
MEMBERS
5.5
What Is Sta'cally Indeterminate? • There are several ways to iden'fy a sta'cally indeterminate
structure – Try to solve with the method of sta'cs alone, and fail – Count the number of unknown loads and the number of sta'cs
equa'ons. If unknowns > equa'ons, it is indeterminate – Look for redundant supports. If any structural element can be
removed and the system remains stable, it is indeterminate – Get used to typical indeterminate configura'ons. Bars with both ends
constrained, nested tubes, mul'ple axial elements suppor'ng a rota'ng component, etc.
• The table towards the end of sec'on 5.5 is helpful
Solu'on Process • Cleary iden'fy the individual components of the system. Give
each a unique iden'fier (number or leber designa'on) • Write three different equa'on(s) for the system:
1. Equilibrium – these are the equa'ons you would write from a purely sta'cs analysis – they cannot be solved on their own
2. Force-‐Deforma'on – write this equa'on for each component:
3. Compa'bility – examine the problem geometry, and express how the deflec'ons of the different components relate to each other
δ = FL AE
Algebra • Now it is just algebra:
– Write an equa'on for the quan'ty you are asked to find from among the system equa'ons. You may have to re-‐arrange an equa'on to start.
– Clearly iden'fy the known and unknown quan''es. Subs'tute for any unknowns by rearranging any of the other system equa'ons.
– Simply as much as you can, and con'nue rearrangement/subs'tu'on un'l you have all know quan''es in the equa'on.
– Plug in values, and calculate the result.
THERMAL EFFECTS ON AXIAL DEFORMATION
5.6
Induced Deforma'on • This brief sec'on shows how temperature changes affect a
slender structural element (temperature alone does not cause deforma'on, a change from one temp to another does)
• The key ideas are: – The dominate effect is a length change – Thermally length change simply adds to loading length change – No stresses are directly induced, but can develop indirectly (for
example, if deforma'on is restrained) – The effect is most significant in really long structures
This is the expansion joint in a bridge. Now you know why it is there.
STRESS CONCENTRATIONS
5.7
Max Greater than Nominal • Axial loading stress calcula'on is based on the assump'on
that stresses are uniform over the cross-‐sec'on • That is reasonable – un'l you introduce a geometric feature
like a hole or step change in dimensions • Now the maximum stress is significantly greater than what
you predict from the uniform stress assump'on
A stress concentra'on factor, K, is the amount that the maximum stress at the site of the geometric feature is greater than the nominal (average, uniform) stress. A factor of 2 simply means the maximum is twice the nominal. Maximum is what mabers from a design standpoint. Stress concentra'on factors are a simple and useful way to es'mate the maximum from a very simple average stress calcula'on.
Typical Problem • Most problems follow this pabern:
– Find the nominal stress (average stress at the smallest cross-‐sec'on) – From the geometry, calculate the geometric factor (0.3 in this case) – Move up to the the curve in the stress concentra'on graph – Move across to the ver'cal axis and read the result (2.35 here) – Calculate the maximum stress by mul'plying nominal by factor
