UNIVERSITY OF WINDSOR
Faculty of Engineering
Mechanics of Deformable Bodies (GENG-2180)
Laboratory #4 – Stress Distribution Under Eccentric Load
Submitted on March 29 2019
Lorenzo, Pernasilici
Student ID: 104 808 810
Objectives
The objective of this test is to determine the behavior of steel and aluminum tubular
specimens while subjected eccentric loading.
Testing Apparatus:
Figure 1: Cross sectional area of testing apparatus
Figure 2: Positions of strain gauges
Figure 3: Diagram of testing apparatus
Instrumentation Used:
● Universal Testing Machine
● Spherically Seated Compression Block
● 6 Strain Gauges
● Loading Block and Pin
● Datascan Analog Measurement Processor
● Dalite Datascan Software
Test Procedure:
Pre-Testing Procedure
1. The inner and outer diameters of each specimen was verified.
2. The six strain gauges were fixed to each specimen in the specified locations around its
outer diameter.
Testing Procedure
1. The loading block was placed on the specimen with the loading pin in one of the three
positions.
2. The specimen was placed in the universal testing machine such that the specimen was
centered under the compression tools.
3. The Dalite software was setup to record the six strain gauges.
4. A load of 12.5 kN was applied followed by a 25 kN load. Measurement were recorded at
each load.
5. The above procedure was repeated for the other two loading positions.
Results:
Sample Calculations:
Cross-sectional area: A = π4 (do2 − d2i ) = π4 (72.89o2 − 63.562i ) = 999.87mm2
Moment of Inertia: I =
4
π
64 (do
− d4i ) =
Young's Modulus (12.5 kN): E =
4
4
π
4
64 (77.89 − 63.56 ) = 584477.6697mm
P
12.5kN
Aεave = 999.87mm2 ×189.56 = 198.702 Error = 0.65%
Experimental Stress (steel, 12.5 kN, 20mm eccentricity): σ = E ε = 198.702 × 33.81 = 2.278 M P a
Theoretical Stress (steel, 12.5 kN, 20mm eccentricity) σ =
P
A
−
Average % error (steel, 12.5 kN, 20mm eccentricity): E rror =
My
−20×12500(−36.44)
12.5kN
I = 999.87 − 584477.6697mm4
σ
σ T he
Σ( Exp−
σ T he ) = 17.05%
Experimental Eccentricity (steel, 12.5 kN, 20mm eccentricity.):
σ max− σ min
4
dexp = ymax−
× PI = 2.565−−25.676
× 584477.6697mm
= 18.116 Error = 9.48%
y
72.89
12.5 kN
min
= 3.09
Table 1: Steel Results Summary
Eccentricity
Load
(KN)
Strain
Gauge
% error
1
Centre
20 mm
25 mm
2
3
4
5
6
12.5 Strain
-44.039
-70.752
-75.603
-43.980
-65.935
-77.189
σ(exp)
-8.751
-14.059
-15.022
-8.739
-13.101
-15.338
-12.502
-12.502
-12.502
-12.502
-12.502
-12.502
25 Strain
-87.654
-117.570
-142.396
-102.671
-137.530
-144.768
σ(exp)
-17.950
-24.076
-29.160
-21.025
-28.163
-29.645
σ(theor)
-25.003
-25.003
-25.003
-25.003
-25.003
-25.003
12.5 Strain
12.907
-34.709
-106.193
-129.217
-90.873
-20.071
σ(exp)
2.565
-6.897
-21.101
-25.676
-18.057
-3.988
σ(theor)
3.087
-4.707
-20.296
-28.090
-20.296
-4.707
25 Strain
24.975
-62.862
-201.122
-263.564
-182.580
-40.181
σ(exp)
5.114
-12.873
-41.186
-53.972
-37.389
-8.228
σ(theor)
6.174
-9.414
-40.592
-56.181
-40.592
-9.414
12.5 Strain
-133.509
-107.519
-32.105
30.077
-21.679
-121.436
σ(exp)
-26.529
-21.364
-6.379
5.976
-4.308
-24.129
σ(theor)
-31.987
-22.245
-2.759
6.984
-2.759
-22.245
25 Strain
-280.727
-209.688
-51.492
55.009
-35.354
-235.679
σ(exp)
-57.487
-42.940
-10.545
11.265
-7.240
-48.262
σ(theor)
-63.975
-44.489
-5.517
13.969
-5.517
-44.489
σ(theor)
20.03%
15.94%
17.05%
13.30%
38.56%
27.30%
Eccentricity
Load (KN)
Modulus of elasticity E
(Gpa)
Experimental
Centre
20 mm
25 mm
Average Stress (Mpa)
Theoretical
Experimental
Theoretical
Eccentricity d (mm)
Experiment
al
Theoretical
12.500
198.702
200.000
-12.502
-12.502
25.000
204.779
200.000
-25.003
-25.003
12.500
-12.192
-12.502
18.116
20.000
25.000
-24.756
-25.003
18.952
20.000
12.500
-12.789
-12.502
20.852
25.000
25.000
-25.868
-25.003
22.052
25.000
Table 2: Aluminum Results Summary
Eccentricity
Load
(KN)
Strain
Gauge
% error
1
Centre
20 mm
25 mm
2
3
4
5
6
12.5 Strain
-129.501
-240.521
-172.149
-139.956
-249.433
-181.797
σ(exp)
-8.725
-16.204
-11.598
-9.429
-16.805
-12.248
-12.502
-12.502
-12.502
-12.502
-12.502
-12.502
25 Strain
-284.763
-442.447
-347.521
-322.568
-466.648
-349.937
σ(exp)
-19.296
-29.981
-23.549
-21.858
-31.621
-23.713
σ(theor)
-25.003
-25.003
-25.003
-25.003
-25.003
-25.003
12.5 Strain
33.811
-99.745
-284.778
-384.514
-308.158
-77.218
σ(exp)
2.278
-6.720
-19.186
-25.906
-20.761
-5.202
σ(theor)
3.087
-4.707
-20.296
-28.090
-20.296
-4.707
25 Strain
59.554
-181.803
-571.157
-790.777
-607.431
-143.993
σ(exp)
4.036
-12.319
-38.703
-53.585
-41.161
-9.757
σ(theor)
6.174
-9.414
-40.592
-56.181
-40.592
-9.414
12.5 Strain
-406.238
-370.850
-51.477
62.774
-82.907
-251.787
σ(exp)
-27.369
-24.985
-3.468
4.229
-5.586
-16.963
σ(theor)
-31.987
-22.245
-2.759
6.984
-2.759
-22.245
25 Strain
-832.612
-710.338
-91.706
124.721
-151.289
-530.131
σ(exp)
-56.420
-48.134
-6.214
8.451
-10.252
-35.923
σ(theor)
-63.975
-44.489
-5.517
13.969
-5.517
-44.489
σ(theor)
21.35%
15.46%
15.84%
13.30%
0.3635667
941
0.2953300
849
Eccentricity
Load (KN)
Modulus of elasticity E
(Gpa)
Experimental
Centre
20 mm
25 mm
Average Stress (Mpa)
Theoretical
Experimental
Theoretical
Eccentricity d (mm)
Experimental
Theoretical
12.500
67.372
200.000
-12.502
-12.502
25.000
67.763
200.000
-25.003
-25.003
12.500
-12.583
-12.502
18.079
20.000
25.000
-25.249
-25.003
18.482
20.000
12.500
-12.357
-12.502
20.270
25.000
25.000
-24.749
-25.003
20.807
25.000
Steel Graphs
Figure 4: Steel specimen 20mm eccentricity, 12.5 kN load
Figure 5: Steel specimen 20mm eccentricity, 25 kN load
Figure 6: Steel specimen 25mm eccentricity, 12.5 kN load
Figure 7: Steel specimen 25 mm eccentricity, 25 kN load
Aluminum Graphs
Figure 8: Aluminum specimen 20 mm eccentricity, 12.5 kN load
Figure 9: Aluminum specimen 20 mm eccentricity, 25 kN load
Figure 10: Aluminum specimen 25 mm eccentricity, 12.5 kN load
Figure 11: Aluminum specimen 25 mm eccentricity, 25 kN load
Discussion
Both the steel and aluminum tubular specimens were successfully tested using
experimental procedure. The test results produced strain readings for both specimens from all six
strain gauges affixed around the outer diameter of the specimens. Using the given parameters
experimental and theoretical values were determined for stress values, Young’s modulus and
eccentricity. Each of specimens behaved as expected while subjected to centric and eccentric
loads with the aluminum specimen experiencing more deformation than the steel specimen in
general. The calculated values for Young’s modulus and the eccentricity were of particular
interest as the percentage error from the theoretical values were quite low.
The stress-distribution graphs were also quite close to the theoretical relationship as there
were relatively low levels of error in those calculations, with peak average percentage error
being roughly 38% and most errors being around the 15-25% range. The main source of error in
this experiment is likely the positioning of specimen inside the universal testing machine. The
positioning of the specimen was done by hand which leaves the possibility for the load to no be
applied centrally to the specimen along either of its transverse axes, which may skew the values
one way or the other. Overall the level of error is acceptable. The overall shape of the
stress-distribution graphs agrees with the theoretical behavior whereby all of the gauges are
under some level of stress with the end nearest to the force having the greatest stress and the far
end having the least stress.
Conclusion
In conclusion, the tests were able to produce valid strain readings resulting from the
centric and eccentric loads. The values from the strain gauges were successfully used to produce
an experimentally calculated values for Young’s modulus and eccentricities along with an
experimental stress-distribution for each specimen. The experimental values and theoretically
calculated values for the experiment are mostly within a reasonable range of percentage error
with a few exceptions. The sources of error in this investigation were mostly mitigated through
the use of a loading block and pin and only small sources of systematic errors were present in the
experiment.