1

Mechanical Laboratory 2

Lab 3: Bending Test

Ante Kulas

MMAE-419-L01

Professor: Dr. Murat Vural

TA: George Kim

09/12/2018

2

EXPERIMENTAL SETUP AND PROCEDURE:

For this experiment students were able to find elastic modulus and Poisson’s ratio for five different

materials, that include: steel, aluminum, glass fiber reinforced plastic (GFRP), clear plastic and wood.

Each cantilever beam specimen was in form shown in Fig 1. Initial measurements were taken for each

specimen before experiment (Tables 6, 8, 10, 12 and 14), where L is distance from loading point to the

wall, Li is distance from loading point to center of the gages, h is height and b is width, c is distance

from neutral plane or h/2. Beam for all materials was already set in device shown in Fig 2. Load was

applied by pulling cantilever beam up. Displacement was recorded by dial gauge and four strains (top

longitudinal, top transversal, bottom longitudinal and bottom transversal) were recorded with strain

indicator (Figures 7, 9, 11, 13, 15). Around 18 measurements were recorded, and first half were while

increasing load and other half while decreasing load in similar increments. Increments were adjusted

depending what material was tested since all materials had different maximum load.

Figure 2. Experiment Setup

Figure 1. Diagram of the Specimen

3

ANALYSIS OF DATA:

After we took measurements of cross-section of the beam we can calculate moment of inertia:

𝐼 = 𝑏 ∙ ℎ3

12 (1)

Stress (longitudinal) at the top and bottom is used to plot stress-strain plot and is calculated by:

𝜎 = 𝑃 ∙ 𝐿𝑖 ∙𝑐

𝐼=

𝑃 ∙ 𝐿𝑖 ∙ℎ

2 ∙ 𝐼 (2)

Now since we measured strain and calculated stress by using MATLAB stress-strain plots (Figures 3, 6,

9, 12 and 15) were made and regression line fitted in order to determine elastic modulus. Slope of fitted

line is elastic modulus (Tables 1-5).

Displacement method to find elastic modulus is simpler but less accurate. Using formula for

displacement method,

𝛿 ∙ 𝐸 = 𝑃 ∙ 𝐿3

3 ∙𝐼 (3)

we plotted (Figures 4, 7, 10, 13 and 16) 𝑃 ∙ 𝐿3

3 ∙𝐼 vs. 𝛿 , fit regression line through and slope of that

line is our elastic modulus (Tables 1-5)

In addition to elastic modulus, we found Poisson’s ratio of each material tested. Using measured values

of strain and knowing Poisson ratio relation:

𝜐 = −𝜀𝑡𝑟𝑎𝑛𝑠𝑣𝑒𝑠𝑎𝑙

𝜀𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 (4)

we plotted (Figures 5, 8, 11, 14 and 17) −휀𝑡𝑟𝑎𝑛𝑠𝑣𝑒𝑠𝑎𝑙 vs. 휀𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 , fitted regression line

trough and from slope of that line found Poisson’s ratio (Tables 1-5)

4

Steel

Figure 3. Stress-Strain Plot for Steel

Figure 5. Poisson’s Ratio Plot for Steel

Figure 4. Displacement Plot for Steel

Table 1. Elastic Modulus and Poisson’s Ratio

for Steel

E by Stress/Strain [GPa] 199.45

E by Displacement [GPa] 178.70

Poisson's Ratio 0.2313

5

Aluminum

Figure 6. Stress-Strain Plot for Aluminum

Figure 8. Poisson’s Ratio Plot for Aluminum

Figure 7. Displacement Plot for Aluminum

Table 2. Elastic Modulus and Poisson’s Ratio

for Aluminum

E by Stress/Strain [GPa] 65.536

E by Displacement [GPa] 58.773

Poisson's Ratio 0.3072

6

GFRP

Figure 9. Stress-Strain Plot for GFRP

Figure 11. Poisson’s Ratio Plot for GFRP

Figure 10. Displacement Plot for GFRP

Table 3. Elastic Modulus and Poisson’s Ratio

for GFRP

E by Stress/Strain [GPa) 24.385

E by Displacement [GPa] 23.480

Poisson's Ratio 0.1235

7

Clear Plastic

Figure 12. Stress-Strain Plot for Clear Plastic

Figure 14. Poisson’s Ratio Plot for Clear Plastic

Figure 13. Displacement Plot for Clear Plastic

Table 4. Elastic Modulus and Poisson’s Ratio

for Clear Plastic

E by Stress/Strain [GPa) 2.8305

E by Displacement [GPa] 2.5509

Poisson's Ratio 0.3640

8

Wood

Figure 15. Stress-Strain Plot for Wood

Figure 17. Poisson’s Ratio Plot for Wood

Figure 16. Displacement Plot for Wood

Table 5. Elastic Modulus and Poisson’s Ratio

for Wood

E by Stress/Strain [GPa) 2.4764

E by Displacement [GPa] 2.5078

Poisson's Ratio 0.3590

9

Table 6. Steel Specimen Measurements

L = 311.00 mm

Li = 298.00 mm

b = 25.46 mm

h = 4.91 mm

c = 2.45 mm

I = 251.125 mm4

Table 7. Steel Data

P d PL3/3I

eLTOP

eTTOP

eLBOTTOM

eTBOTTOM

[N] [mm] [N/mm] sLTOP

sLBOTTOM

(-)eTTOP

(-)eTBOTTOM

0.000000 0.000000 0.000000 0.000000 0 0 0 0 0 0 0

-0.000127 0.000028 0.000127 -0.000030 8.811 1.8034 351799.4562 -25.6686 25.66865 -0.000028 0.00003

-0.000245 0.000053 0.000245 -0.000058 17.622 3.5306 703598.9125 -51.3373 51.3373 -0.000053 0.000058

-0.000375 0.000085 0.000376 -0.000089 26.433 5.5372 1055398.369 -77.0059 77.00595 -0.000085 0.000089

-0.000517 0.000118 0.000519 -0.000123 35.823 7.747 1430295.769 -104.36 104.3599 -0.000118 0.000123

-0.000654 0.000150 0.000657 -0.000155 45.301 9.8806 1808746.699 -131.973 131.9732 -0.00015 0.000155

-0.000768 0.000176 0.000771 -0.000181 53.222 11.6332 2125010.857 -155.049 155.049 -0.000176 0.000181

-0.000946 0.000216 0.000950 -0.000221 65.727 14.4272 2624281.802 -191.478 191.4778 -0.000216 0.000221

-0.001007 0.000229 0.001012 -0.000234 69.776 15.3924 2785967.411 -203.275 203.275 -0.000229 0.000234

-0.000892 0.000204 0.000896 -0.000210 60.476 13.6652 2414623.541 -176.18 176.1803 -0.000204 0.00021

-0.000802 0.000185 0.000805 -0.000190 54.023 12.2682 2156992.626 -157.383 157.3825 -0.000185 0.00019

-0.000660 0.000153 0.000662 -0.000157 44.456 10.0584 1774988.166 -129.51 129.51 -0.000153 0.000157

-0.000530 0.000123 0.000531 -0.000126 35.689 8.001 1424965.474 -103.971 103.971 -0.000123 0.000126

-0.000371 0.000086 0.000371 -0.000088 24.965 5.5626 996765.126 -72.7278 72.72784 -0.000086 0.000088

-0.000260 0.000060 0.000259 -0.000062 17.444 3.8862 696491.8527 -50.8187 50.81874 -0.00006 0.000062

-0.000007 0.000001 0.000005 -0.000002 0.267 0.1016 10660.58958 -0.77784 0.777838 -0.000001 0.000002

0.000000 0.000000 0.000000 0.000000 0 0 0 0 0 0 0

StrainStrain Stress [MPa]

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Table 8. Aluminum Specimen Measurements

L = 311.00 mm

Li = 298.00 mm

b = 25.48 mm

h = 12.73 mm

c = 6.36 mm

I = 4374.896 mm4

Table 9. Aluminum Data

P d PL3/3I

eLTOP

eTTOP

eLBOTTOM

eTBOTTOM

[N] [mm] [N/mm] s LTOP

s LBOTTOM

(-)eTTOP

(-)eTBOTTOM

0.000000 0.000000 0.000000 0.000000 0.0445 0 1776.76493 -0.12964 0.12964 0 0

-0.000118 0.000036 0.000116 -0.000036 18.2005 0.6858 726696.8566 -53.0226 53.02261 -0.000036 0.000036

-0.000229 0.000068 0.000222 -0.000068 34.888 1.2954 1392983.705 -101.637 101.6375 -0.000068 0.000068

-0.000350 0.000106 0.000344 -0.000106 53.4445 2.032 2133894.682 -155.697 155.6972 -0.000106 0.000106

-0.000466 0.000142 0.000458 -0.000141 71.0665 2.6924 2837493.594 -207.035 207.0345 -0.000142 0.000141

-0.000589 0.000179 0.000579 -0.000178 89.7565 3.429 3583734.865 -261.483 261.4832 -0.000179 0.000178

-0.000697 0.000213 0.000685 -0.000211 105.8655 4.0894 4226923.77 -308.413 308.4127 -0.000213 0.000211

-0.000826 0.000253 0.000813 -0.000250 124.6445 4.826 4976718.57 -363.121 363.1206 -0.000253 0.00025

-0.000949 0.000291 0.000934 -0.000288 142.578 5.5372 5692754.837 -415.365 415.3654 -0.000291 0.000288

-0.001105 0.000339 0.001090 -0.000335 165.6735 6.4516 6614895.836 -482.648 482.6484 -0.000339 0.000335

-0.000922 0.000284 0.000910 -0.000281 140.264 5.4102 5600363.061 -408.624 408.6242 -0.000284 0.000281

-0.000785 0.000242 0.000775 -0.000240 119.171 4.5974 4758176.484 -347.175 347.175 -0.000242 0.00024

-0.000598 0.000185 0.000591 -0.000184 92.1595 3.5306 3679680.171 -268.484 268.4837 -0.000185 0.000184

-0.000463 0.000144 0.000458 -0.000143 71.7785 2.7432 2865921.833 -209.109 209.1087 -0.000144 0.000143

-0.000317 0.000099 0.000314 -0.000099 49.6175 1.8796 1981092.897 -144.548 144.5482 -0.000099 0.000099

-0.000194 0.000061 0.000193 -0.000061 30.616 1.143 1222414.272 -89.1921 89.19207 -0.000061 0.000061

-0.000109 0.000035 0.000109 -0.000034 17.622 0.635 703598.9125 -51.3373 51.3373 -0.000035 0.000034

0.000000 0.000001 0.000001 0.000000 0.4005 0 15990.88437 -1.16676 1.166757 -0.000001 0

Strain Stress [MPa] Strain

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Table 10. GFRP Specimen Measurements

L = 313.00 mm

Li = 300.00 mm

b = 25.43 mm

h = 6.30 mm

c = 3.15 mm

I = 529.5936 mm4

Table 11. GFRP Data

P d PL3/3I

eLTOP

eTTOP

eLBOTTOM

eTBOTTOM

[N] [mm] [N/mm] s LTOP

s LBOTTOM

(-)eTTOP

(-)eTBOTTOM

0.000000 0.000000 0.000000 0.000000 0 0 0 0 0 0 0

-0.000132 0.000016 0.000131 -0.000016 2.0915 1.397 83507.95173 -6.09306 6.093063 -0.000016 0.000016

-0.000296 0.000036 0.000293 -0.000037 4.183 3.175 167015.9035 -12.1861 12.18613 -0.000036 0.000037

-0.000464 0.000056 0.000459 -0.000058 6.319 5.08 252300.6201 -18.4088 18.40883 -0.000056 0.000058

-0.000662 0.000081 0.000656 -0.000082 9.0335 7.2644 360683.2809 -26.3168 26.31685 -0.000081 0.000082

-0.000818 0.000100 0.000811 -0.000102 11.214 9.0424 447744.7625 -32.6692 32.66919 -0.0001 0.000102

-0.000957 0.000116 0.000947 -0.000119 13.2165 10.6172 527699.1844 -38.503 38.50297 -0.000116 0.000119

-0.001075 0.000131 0.001064 -0.000134 14.9075 11.9126 595216.2517 -43.4293 43.42928 -0.000131 0.000134

-0.001275 0.000155 0.001263 -0.000158 17.5775 14.1986 701822.1475 -51.2077 51.20766 -0.000155 0.000158

-0.001462 0.000178 0.001449 -0.000181 19.8915 16.3068 794213.9239 -57.9489 57.94892 -0.000178 0.000181

-0.001215 0.000149 0.001204 -0.000151 16.465 13.5636 657403.0243 -47.9667 47.96667 -0.000149 0.000151

-0.000946 0.000116 0.000936 -0.000119 12.7715 10.541 509931.535 -37.2066 37.20658 -0.000116 0.000119

-0.000728 0.000090 0.000720 -0.000091 10.057 8.1026 401548.8743 -29.2986 29.29856 -0.00009 0.000091

-0.000572 0.000071 0.000565 -0.000072 7.921 6.35 316264.1576 -23.0759 23.07586 -0.000071 0.000072

-0.000320 0.000040 0.000316 -0.000040 4.361 3.556 174122.9632 -12.7047 12.70468 -0.00004 0.00004

-0.000117 0.000015 0.000115 -0.000014 1.8245 1.2446 72847.36215 -5.31523 5.315225 -0.000015 0.000014

-0.000010 0.000002 0.000008 -0.000001 0 0.0508 0 0 0 -0.000002 0.000001

Strain Stress [MPa] Strain

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Table 12. Clear Plastic Specimen Measurements

L = 311.00 mm

Li = 298.00 mm

b = 25.02 mm

h = 12.17 mm

c = 6.08 mm

I = 3754.883 mm4

Table 13. Clear Plastic Data

P d PL3/3I

eLTOP

eTTOP

eLBOTTOM

eTBOTTOM [N] [mm] [N/mm] s L

TOPs L

BOTTOM (-)eTTOP (-)eT

BOTTOM

0.000000 0.000000 0.000000 0.000000 0 0 0 0 0 0 0

-0.000117 0.000047 0.000115 -0.000044 0.7565 0.762 30205.0038 -2.20387 2.203874 -4.7E-05 0.000044

-0.000304 0.000118 0.000300 -0.000110 1.8245 1.905 72847.3621 -5.31523 5.315225 -0.00012 0.00011

-0.000471 0.000180 0.000465 -0.000170 2.8035 2.921 111936.191 -8.1673 8.167297 -0.00018 0.00017

-0.000611 0.000233 0.000604 -0.000219 3.738 3.7846 149248.254 -10.8897 10.88973 -0.00023 0.000219

-0.000774 0.000294 0.000765 -0.000276 4.5835 4.7752 183006.788 -13.3529 13.35288 -0.00029 0.000276

-0.000971 0.000369 0.000968 -0.000348 5.7405 6.0198 229202.676 -16.7235 16.72351 -0.00037 0.000348

-0.001073 0.000406 0.001064 -0.000382 6.319 6.604 252300.62 -18.4088 18.40883 -0.00041 0.000382

-0.001237 0.000466 0.001226 -0.000438 7.209 7.5946 287835.919 -21.0016 21.00162 -0.00047 0.000438

-0.001455 0.000548 0.001443 -0.000516 8.4995 8.9408 339362.102 -24.7612 24.76117 -0.00055 0.000516

-0.001216 0.000459 0.001207 -0.000431 7.031 7.5184 280728.859 -20.4831 20.48306 -0.00046 0.000431

-0.001019 0.000387 0.001011 -0.000363 6.0965 6.2992 243416.795 -17.7606 17.76063 -0.00039 0.000363

-0.000901 0.000343 0.000894 -0.000322 5.251 5.588 209658.262 -15.2975 15.29748 -0.00034 0.000322

-0.000739 0.000284 0.000734 -0.000266 4.45 4.5974 177676.493 -12.964 12.96396 -0.00028 0.000266

-0.000627 0.000243 0.000624 -0.000226 3.6935 3.9116 147471.489 -10.7601 10.76009 -0.00024 0.000226

-0.000504 0.000198 0.000502 -0.000182 2.8925 3.2004 115489.72 -8.42658 8.426577 -0.0002 0.000182

-0.000334 0.000135 0.000335 -0.000123 2.0025 2.1336 79954.4219 -5.83378 5.833784 -0.00014 0.000123

-0.000158 0.000070 0.000161 -0.000060 0.9345 1.0668 37312.0635 -2.72243 2.722432 -0.00007 0.00006

-0.000029 0.000022 0.000032 -0.000015 0.1335 0.2794 5330.29479 -0.38892 0.388919 -2.2E-05 0.000015

Strain Stress MPa] Strain

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Table 14. Wood Specimen Measurements

L = 320.68 mm

Li = 283.36 mm

b = 25.04 mm

h = 12.24 mm

c = 6.12 mm

I = 3829.761 mm4

Table 15. Wood Data

P d PL3/3I

eLTOP

eTTOP

eLBOTTOM

eTBOTTOM

[N] [mm] [N/mm] s LTOP

s LBOTTOM

(-)eTTOP

(-)eTBOTTOM

-0.000204 0.000076 0.000204 -0.000071 1.068 0.9652 42642.35833 -3.11135 3.111351 -7.6E-05 0.000071

-0.000355 0.000133 0.000355 -0.000124 1.9135 2.1336 76400.89201 -5.5745 5.574505 -0.00013 0.000124

-0.000527 0.000196 0.000524 -0.000183 2.8035 3.2004 111936.1906 -8.1673 8.167297 -0.0002 0.000183

-0.000665 0.000247 0.000660 -0.000231 3.56 4.0386 142141.1944 -10.3712 10.37117 -0.00025 0.000231

-0.000832 0.000309 0.000826 -0.000289 4.4945 5.08 179453.258 -13.0936 13.0936 -0.00031 0.000289

-0.000998 0.000370 0.000992 -0.000346 5.3845 6.096 214988.5566 -15.6864 15.6864 -0.00037 0.000346

-0.001174 0.000435 0.001166 -0.000408 6.319 7.1628 252300.6201 -18.4088 18.40883 -0.00044 0.000408

-0.001405 0.000519 0.001395 -0.000487 7.565 8.6106 302050.0382 -22.0387 22.03874 -0.00052 0.000487

-0.001527 0.000565 0.001516 -0.000530 8.277 9.3472 330478.2771 -24.113 24.11297 -0.00057 0.00053

-0.001584 0.000586 0.001573 -0.000550 8.5885 9.7028 342915.6316 -25.0205 25.02045 -0.00059 0.00055

-0.001471 0.000544 0.001461 -0.000510 8.01 9.017 319817.6875 -23.3351 23.33514 -0.00054 0.00051

-0.001340 0.000496 0.001332 -0.000465 7.2535 8.255 289612.6837 -21.1313 21.13126 -0.0005 0.000465

-0.001154 0.000430 0.001148 -0.000401 6.2745 7.112 250523.8552 -18.2792 18.27919 -0.00043 0.000401

-0.001038 0.000386 0.001035 -0.000360 5.518 6.4008 220318.8514 -16.0753 16.07532 -0.00039 0.00036

-0.000844 0.000315 0.000842 -0.000293 4.539 4.9784 181230.0229 -13.2232 13.22324 -0.00032 0.000293

-0.000675 0.000255 0.000676 -0.000234 3.6045 4.1656 143917.9594 -10.5008 10.50081 -0.00026 0.000234

-0.000525 0.000200 0.000528 -0.000181 2.848 3.2512 113712.9556 -8.29694 8.296937 -0.0002 0.000181

Strain Stress [MPa] Strain

14

CONCLUSION:

While working on this experiment we get more insight on how to find mechanical properties for

materials with different strengths. As it was expected steel showed that it has highest elastic modulus

with both methods. To extend this experiment even further it would be interesting to test same material

with different cross sections to prove even more validation of our tests and experimental methods.

REFERENCES:

Mechanical Behavior of Materials 4th Ed. by Norman E. Dowling (Pearson, 2013 ISBN 0-13-139506-8)

Dr. Murat Vural, MMAE 419 Bending Test Handout, IIT, Chicago