UNIFIED LAB MECHANICAL ENGINEERING REPORT READ MY POST FIRST !

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EGME 306A The Beam

Page 1 of 18 Group 2

EXPERIMENT 3:The Beam Group 2 Members: Ahmed Shehab

Marvin Penaranda

Edwin Estrada

Chris May

Bader Alrwili

Paola Barcenas

Deadline Date: 10/23/2015 Submission Date: 10/23/2015

EGME 306A – UNIFIED LABORATORY

EGME 306A The Beam

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Abstract (Bader): The main objective for this experiment was to determine the stress, deflection, and strain of a supported beam

under loading, and to experimentally verify the beam stress and flexure formulas. Additionally, maximum

bending stress and maximum deflection were determined. To accomplish this, a 1018 steel I-beam with a strain

gage bonded to the underside was utilized in conjunction with a dial indicator to monitor beam deflection. In

order to determine the values for strain and deflection, the beam underwent testing utilizing the MTS Tensile

Testing machine, which applied a controlled, incrementally increasing load to the beam. This data was then

utilized along with calculations for the beams neutral axis, moment of inertia, and section modulus to determine

the required objective values. Final values of 12,150 psi for the maximum actual stress (vs. 12,784.8 psi for

theoretical stress), and 0.0138 in for the maximum actual deflection (vs. .0130 in for theoretical deflection)

correlated closely with each other, and successfully verify established beam stress and flexure formulas.

EGME 306A The Beam

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Table of Contents: List of Symbols and Units 4

Theory 5

Procedure and Experimental Set-up 8

Results 9

Sample Calculations and Error Analysis 12

Discussion and Conclusion 15

Bibliography 16

Appendix 17

EGME 306A The Beam

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List of Symbols and Units (Chris):

List of Symbols and Units Name of variables (units) Units

𝜎 Stress psi

𝑃 Applied load lbf

𝐼 Moment of Inertia in.4

𝜀 Strain in/in

𝐿 Length of the bar in

Z Section Modulus of Beam in3

𝑐 Distance to Beam Neutral Axis in

𝐸 Modulus of Elasticity psi

EGME 306A The Beam

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Theory (Edwin): There are two main objectives for this experiment: to determine maximum bending stress values in

the beam and to determine the deflection in the beam. To help visualize this phenomena, imagine

cutting a section of a symmetrically loaded beam:

Now, examine diagrams of this section before (Fig. A) and after bending (Fig. B):

(Fig. A)

(Fig. B) The main points to take away from the above diagrams are as follows: When the moment, M is applied

as shown in Fig. A, forces will be in compression near the top (positive moment) and in tension near

the bottom (negative moment). The effects from this moment are seen in Fig. B.

For determining max stress values, one concept to note is that our bending moment M can help

calculate bending stress. First, we recall our basic definition of normal strain:

𝜀 = 𝑙′ − 𝑙

𝑙

EGME 306A The Beam

Page 6 of 18 Group 2

Since the beam is bending, however, we will need to alter the above formula by taking into account

the radius of curvature, r, and the differential angle, 𝜃. Since there are locations where forces are compressive and locations where forces are tensile, there must be a location where forces are neither

compressive nor tensile. This axis is called the Neutral Axis and is labeled in Fig. B as N-N. We can

now alter the above formula by measuring the strain at distance +𝜂 from the neutral axis:

𝜀 = 𝑙′ − 𝑙

𝑙 =

(𝑟 + 𝜂)∆𝜃 − 𝑟∆𝜃 𝑟∆𝜃

= 𝜂 𝑟

The strain equation from above can be converted to stress by using Hooke’s Law:

𝜎 = 𝐸 𝜂 𝑟

Now, if we allow c to be the max distance from the neutral axis, we can build upon the above

expression as follows:

𝜎 = 𝜎𝑚 𝜂 𝑐

To obtain the beam stress formula, we still need to define where the neutral axis is located. We can do

this by relating the radius of curvature and the bending moment, which will be determined by applying

a moment equilibrium equation about the neutral axis:

𝑑𝐹 = ∫ 𝜎 𝐴

𝑑𝐴 = 𝜎𝑚 𝑐

∫ 𝜂 𝐴

𝑑𝐴 = 0

Since moment due to all forces is summed up by the product of their forces and the moment arm about

the neutral axis, we deduce:

𝑀 = ∫ 𝜂𝑑𝐹 = ∫ 𝜂𝜎𝑑𝐴 = 𝜎𝑚 𝑐

∫ 𝜂2𝑑𝐴

We will then let I be defined as the second moment of area about the neutral axis, commonly called

the moment of inertia:

𝐼 = ∫ 𝜂2𝑑𝐴

Finally, we deduce the moment equation as:

𝜎𝑚 = 𝑀𝑐 𝐼

= 𝑀 𝑍

Our final objective involved determining beam deflection. Using similar methods from bending stress,

we will examine the relationship between the radius of curvature, r and the moment, M, at any given

point on a beam. We begin with the radius of curvature for any point of a function from calculus:

EGME 306A The Beam

Page 7 of 18 Group 2

𝐼 𝑟

= 𝑑2𝑦 𝑑𝑥2

(1 + 𝑑 2𝑦

𝑑𝑥2 )

3 2

We will note that for small deflections, the first derivative or slope is small and becomes even smaller

when it is squared and can therefore be neglected. This will greatly simplify the radius of curvature

to:

1 𝑟

= 𝑑2𝑦 𝑑𝑥2

By combing the above equation with the bending stress equation, we develop the standard moment

curvature equation:

𝐸𝐼 𝑑2𝑦 𝑑𝑥2

= 𝑀(𝑥)

The differential equation will only be useful if we apply it to the beam with specific boundaries, or

boundaries of integration. Integrating with M(x)=Px/2 for 0≤x≤a andM(x)=Pa/2 for a≤x≤a+b, we get:

𝑦(𝑥) = 𝑃

2𝐸𝐼 (

𝑥3

6 −

𝑎𝑥(𝑎 + 𝑏) 2

) , 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 𝑎

𝑦(𝑥) = 𝑃

2𝐸𝐼 (

𝑎3

6 −

𝑎𝑥(2𝑎 + 𝑏) 2

+ 𝑎𝑥2

2 ) , 𝑓𝑜𝑟 𝑎 ≤ 𝑥 ≤ 𝑎 + 𝑏

The max deflection at x = a + b/2 is:

−𝑦𝑚 = 𝑃𝑎 2𝐸𝐼

( 𝑎2

3 +

𝑎𝑏 2

+ 𝑏2

8 )

Lastly, by allowing a=b=L/3:

−𝑦𝑚 = 23𝑃𝑎3

48𝐸𝐼 =

23𝑃𝐿3

1296𝐸𝐼

EGME 306A The Beam

Page 8 of 18 Group 2

Procedure and Experimental Set-Up (Marvin): This experiment requires the use of MTS Insight Tensile Testing Machine, Beam Deflection Dial indicator,

digital caliper, a 1018 steel beam and two separate data acquisition software. The experiment begins with the

measurement of the length and the cross-sectional dimensions of the 1018 steel beam with the use of the digital

caliper. The beam is then carefully placed into the MTS machine while making sure that the beam's black marks

are aligned with the roller supports of the bending fixture. Once the beam is squarely nested on the roller

supports, TestWorks 4 software is accessed in order to zero the load reading. Now that the parameters have been

set up, the MTS' handset is used to position the upper bending fixture over the beam. While the crosshead

lowers, the digital load readout from the software is observed. When the load increases as the upper loading

component of the fixture makes contact with the beam, the fixture is raised until a preload of 0.2 lb is applied.

The Beam Deflection Dial indicator is positioned in the center of the beam on the bottom side. Do note, the dial

indicator needs to be set to zero. Now, the LabVIEW software is accessed. On the software, the strain indicator

is zeroed by pressing “zero strain”. The software also requires the inputs dialog for the thickness and width, a

value of 0.125 is used for thickness and a value of 0.500 is used for width. The MTS machine loads the beam

up to 1000 lb in increments of 100 lb. When the digital load readout reaches approximately 100 lb, the “Pause”

button is pressed since the data readings need to be taken for every 100 lbs. Pausing for every 100 lb increments

are continued up until 1000 lbs. The procedure is repeated again since the average of the 2 sets of readings will

be used for experimental calculations. Keep in mind, the dial indicator and the strain indicator needs to be re-

zeroed again for the next set of experiment.

Note: for a detailed procedure, see Appendix A.1

EGME 306A The Beam

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Results (Chris):

-TEST 1 / TEST 2 - - AVERAGE OF TEST 1 & 2 -

LOAD ( lbf )

STRAIN GAGE

ELONGATIO N ( ε )

INDICATOR DEFLECTIO

N (in.)

LOA D

( lbf )

STRAIN GAGE

ELONGATIO N ( ε )

INDICATOR DEFLECTIO

N (in.)

100/99 47/46 .0011/.0016 99.5 46.5 .00135

202/19

91/88 .0023/.0029 200 89.5 .0026

316/29

144/131 .004/.0041 305.5 137.5 .00405

396/39

179/174 .005/.0059 395.5 176.5 .00545

497/50

222/224 .0064/.0071 498.5 223 .00675

596/59

268/268 .008/.0087 597 268 .00835

695/69

312/310 .0093/.010 695.5 311 .00965

797/80

354/355 .0109/.0114 798.5 354.5 .01115

897/89

318/400 .0122/.0129 897.5 359 .01255

996/99

365/445 .0134/.0142 996 405 .0138

TABLE 1. Strain gauge, indicator, and load data (reference Appendix for calculations)

EGME 306A The Beam

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LOAD ( lbf )

ACTUAL VALUES / DEFLECTION ( in. )

THEORETICAL VALUES / DEFLECTION ( in. )

99.5 0.00135 0.0013

200 0.0026 0.0026

305.5 0.0041 0.0039

395.5 0.0055 0.0051

498.5 0.00675 0.0065

597 0.00835 0.0077

695.5 0.00965 0.0091

798.5 0.01115 0.0104

897.5 0.01255 0.0117

996 0.0138 0.013 TABLE 2. Actual and Theoretical Values for Beam Deflection (reference Appendix for calculations)

Figure 2-1. Actual vs. Theoretical Deflection using dial indicator and calculated values. The Load vs. Deflection curve consists of the actual experimental values obtained during the

experiment compared with theoretical calculated values to measure deflection in relation to

applied load.

200

400

600

800

1000

1200

0 0.005 0.01 0.015

Lo ad

( lb

f )

Deflection ( in. )

Actual vs. Theoretical Deflection

Actual Deflection

Theoretical Deflection

Linear (Actual Deflection)

Linear (Theoretical Deflection)

EGME 306A The Beam

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LOAD ( lbf )

ACTUAL STRESS ( psi )

THEORETICAL STRESS ( psi )

99.5 1,410 1,277.20

200 2,685 2,567.23

305.5 4,125 3,921.44

395.5 5,295 5,076.70

498.5 6,690 6,398.82

597 8,040 7,663.18

695.5 9,330 8,927.54

798.5 10,635 10,249.66

897.5 10,770 11,520.44

996 12,150 12,784.80 TABLE 3. Actual and Theoretical Values for Beam Stress (reference Appendix for calculations)

Figure 2-1. Actual vs. Theoretical Stress curve using experimental and calculated values. The Load vs. Stress curve consists of the actual experimental values obtained during the

experiment compared with theoretical calculated values to measure stress in relation to applied

load.

200

400

600

800

1000

1200

0 5,000 10,000 15,000

Lo ad

( lb

f )

Stress ( psi )

Actual vs. Theoretical Stress

Actual Stress

Theoretical Stress

Linear (Actual Stress)

Linear (Theoretical Stress)

EGME 306A The Beam

Page 12 of 18 Group 2

Sample Calculations and Analysis (Paola):

E=30x106 psi

Load =P= 996 lb

Dimensions: a = b= 4”

Bending Moment:

𝑀 = 𝑃 × 𝑎

𝑀 = 996 𝑙𝑏 × 4" 𝑀 = 3984 lb-in

Shear force:

𝑉 = 𝑃 2

𝑉 = 996 𝑙𝑏

𝑉 = 498 𝑙𝑏 Height of the beam: 0.278+0.438+0.290= 1.006”

Neutral axis (c):

𝑐 = ℎ 2

0.278” 0.438”

0.290”

0.992”

0.191”

12”

4” 4”

4”

498 lb

3984 lb-in

996 𝑙𝑏 2

498 𝑙𝑏

996 𝑙𝑏 2

498 𝑙𝑏

EGME 306A The Beam

Page 13 of 18 Group 2

𝑐 = 1.006

2 = 0.503"

Moment of inertia:

Ai xi Ai xi Ici di Ix

1 0.275776 0.867 0.2390977 0.00177608 0.367397 0.0389556913

2 0.083658 0.509 0.04258192 0.001337 0.009694 0.001345

3 0.28768 0.145 0.0417136 0.002016 0.354603 0.038189

∑ 0.647114 0.32339322 0.0784896913

𝑦 = ∑ 𝑦𝑖 𝐴𝑖 ∑ 𝐴𝑖

= 0.32339322

0.647114 = .499746 𝑖𝑛.

Moment of inertia:

𝐼𝑐 = ∑(𝐼𝑐 + 𝐴𝑖 𝑑𝑖 2) = ∑ 𝐼𝑥

Ic= 0.07837 in4

Section modulus of the beam cross-section:

𝑍 = 𝐼 𝑐

= 0.07837 𝑖𝑛4

0.503" = 0.15581 𝑖𝑛4

Maximum Bending Stress:

𝜎𝑚𝑎𝑥 = 𝑀𝑐

𝐼 =

(3984 𝑙𝑏 − 𝑖𝑛)(0.503 𝑖𝑛) 0.07837 𝑖𝑛4

= 25570.4 𝑝𝑠𝑖

Maximum Deflection:

−𝑦𝑚 = 23 𝑃𝐿3

1296 𝐸𝐼 =

23(996𝑙𝑏)(123) 1296(30 × 106𝑝𝑠𝑖)(0.07837𝑖𝑛4)

= 0.01299 𝑖𝑛

Error Analysis:

A) Determining the maximum bending stress values in the beam experimentally and theoretically

B) Determining the deflection in the beam by experimental and theoretical methods.

A-1) Experimental Method:

𝜎𝑚𝑎𝑥 = 𝑀𝑐

𝐼 =

(3984 𝑙𝑏 − 𝑖𝑛)(0.503 𝑖𝑛) 0.07837 𝑖𝑛4

= 25570.4 𝑝𝑠𝑖

𝜎𝑚𝑎𝑥 = 𝐸𝜀𝑚𝑎𝑥 = (30 × 106𝑝𝑠𝑖)𝜀𝑚𝑎𝑥 = 25570.4 𝑝𝑠𝑖

𝜀𝑚𝑎𝑥 = 8.523 × 10−4 𝑖𝑛 𝑖𝑛

Bending stress uncertainty:

Δ𝜎 𝜎

= Δ𝜀 𝜀

= 10−6

8.523 × 10−4 𝑖𝑛 𝑖𝑛

= 0.0011732

EGME 306A The Beam

Page 14 of 18 Group 2

Bending stress uncertainty: 0.0011732

A-2) Theoretical Method:

Bending stress uncertainty:

Δ𝜎 𝜎

= Δ𝑝 𝑝

+ Δℎ ℎ

+ ℎ3∆𝑏 + 3𝑏ℎ2∆ℎ + 2𝑑3∆𝑠 + 6𝑠𝑑2∆𝑑

𝑏ℎ3 − 6𝑠𝑑3

Δ𝜎 𝜎

= 0.0001

996 +

0.001 in 0.438

+ 0.4383 (0.001) + 3(0.191)(0.4382 )0.001 + 2(0.2783 )0.001 + 6(.992)(0.2782 )0.001

(0.191)(0.4383) − 6(0.992)(0.278)3

Δ𝜎 𝜎

= -0.023679

B-1) Experimental Method for finding the deflection:

The deflection value, which is directly the uncertainty of the dial indicator, is 10-4in.

B-2) Theoretical Method:

Δ𝑦 𝑦

= Δ𝑝 𝑝

+ 3Δ𝑙

𝑙 +

ℎ3∆𝑏 + 3𝑏ℎ2∆ℎ + 2𝑑3∆𝑠 + 6𝑠𝑑2∆𝑑 𝑏ℎ3 − 6𝑠𝑑3

Δ𝑦 𝑦

= 0.0001

996 +

3(0.001) 12

+ 0.4383 (0.001) + 3(0.191)(0.4382 )0.001 + 2(0.2783 )0.001 + 6(.992)(0.2782)0.001

(0.191)(0.4383) − 6(0.992)(0.278)3

Δ𝑦 𝑦

= −0.0054985

EGME 306A The Beam

Page 15 of 18 Group 2

Discussion and Conclusion (Ahmed): In this experiment we examined stress, deflection and strain of the load applied to a 1018 steel I-beam. The

beam was placed horizontally and a force P was applied to it by the MTS Insight Tensile Testing Machine. A

load was applied and readings were taken incrementally every 100 lbf until the test was concluded at 1000 lbf.

This process was repeated twice and the average of the two tests was taken. This data was then utilized along with calculations for the beams neutral axis, moment of inertia, and section modulus to determine the required

objective values. Final values of 12,150 psi for the maximum actual stress (vs. 12,784.8 psi for theoretical

stress), and 0.0138 in for the maximum actual deflection (vs. .0130 in for theoretical deflection) correlated

closely with each other, and successfully verified established beam stress and flexure formulas.

EGME 306A The Beam

Page 16 of 18 Group 2

BIBLIOGRAPHY (Edwin): 1) Beckwith, T. G., Buck, N. L. and Marangoni, R. D., Mechanical Measurements, Addison-Wesley.

2) Popov, E.P. Mechanics of Materials, Prentice-Hall Inc.

3) Sharma, P. The Beam (EGME 306A lab manual). CSU-Fullerton.

4) Thomas, G.B., and Finney R.L., Calculus and Analytical Geometry, Addison-Wesley,

EGME 306A The Beam

Page 17 of 18 Group 2

Appendix (All): A.1 Procedures

1. Measure the cross-sectional dimensions of the beam by using the caliper and measure the location of

the loading and support pins.

2. Place the beam into the MTS machine while making sure that beam's black marks aligns with the

roller supports of the lower bending fixture.

3. Access the TestWorks 4 software by double clicking its icon on the desktop

4. If prompted, make sure that the name field under the User Login says "306A_lab" then click OK to

5. Under the Open Method dialog, select "exp-3 4 Point Flex Mod X'

6. Select the Motor Reset button in the bottom right corner by clicking on it

7. Zero the "load" readout by right clicking on the "Load cell" icon and selecting "zero channel"

8. Do not close the TestWorks 4 software as it will be used again in the later parts of the experiment.

Leave the software running.

9. Next, use the MTS' handset to position the upper bending fixture over the beam

10. Enable the handset by pressing "unlock" button at the top right of the handset

11. Slowly lower the crosshead using the down arrow until fixture is nearly touching the beam. Make

sure not to pinch the strain gauge lead wires between the fixture and the beam while lowering.

12. Once the crosshead is near the beam, use the thumb wheel of the handset to lower/raise the fixture

onto the beam.

13. Observe the digital load readout from the screen while lowering/raising the fixture and aim for a pre-

load value of 0.2 lb

14. Once finished, return control to the computer software by locking the handset by pressing the

"unlock" button.

15. Take the magnetic base holding the dial indicator and position it with the dial indicator in the center

of the beam on the bottom side. Make sure the dial indicator is not touching the strain gauge.

16. Once the dial indicator is in position, lock it to the MTS frame by activating the magnetic base.

17. Zero the dial indicator by turning the dial of the indicator.

18. Start the LabVIEW software by double clicking its icon on the desktop. NOTE: LabVIEW and TestWorks 4 should be open side-by-side.

19. In LabVIEW, select "Open" and double click "exp2&3-Strain Mod 9-15 LV7.1"

20. Press the white arrow near the upper left corner of the screen to start the strain gauge acquisition

EGME 306A The Beam

Page 18 of 18 Group 2

21. Press the "Zero Strain" button to zero-out the strain indicator

22. Press the Green Arrow on the TestWorks 4 window

23. Name a sample ID for the test

24. Under the required inputs dialog, input 0.125 for thickness and 0.500 for width.

25. Press OK when ready to start

26. When the digital load readout reaches approximately 100 lb, press the "Pause" button to pause the

test.

27. Record the actual load from the digital readout, also record the strain reading from the LabVIEW

window, and record the deflection of the beam from the dial indicator.

28. Continue taking readings in increments of 100 lb until reaching approximately 1000 lb (steps 26-27

will be repeated 10 times)

29. Steps 18-29 will be repeated again for the next set of experiment.