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Lab 3 Flexural Stresses in Beams
Mechanics of Deformable Bodies (University of Windsor)
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UNIVERSITY OF WINDSOR
Faculty of Engineering
Mechanics of Deformable Bodies (GENG 2180)
Laboratory #3 – Flexural Stress In Beams
Submitted on Wednesday March 30, 2022
Ratanjot Pabla
Student ID: 110038085
Section 53-B
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Table of Contents:
Objectives ………………………………………………………………………………………. 2
Procedures …………………………………………………………………………………...... 2-4
Results …………………………………………………………………………..…………..… 5-9
Discussions …………………………………………………….………………….…………. 9-10
Conclusion ……………………………………………………………………………………... 11
References ……………………………………………………………………………………… 12
List of Figures:
Figure 1 ………………………………………………………………………………………….. 3
Figures 2-5 …………………………………………………………...……………………….. 3-4
Figure 6-7 ……………………………………….……………………………………………..… 7
Figure 8.a-9b ……..…………………………………………………………………..…..…… 8-9
List of Tables:
Table 1 …………………………………………………………………………………………... 5
Table 2 …………………………………………………………………………..…….………… 5
Table 3 …………………………………………………………………………..…….………… 6
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Objectives
The motive of this lab is to conduct and study the behaviour of flexural stress distribution on a
cross-section of aluminum and steel beams in the elastic range. The flexural stresses and normal
strains will be experimented on the two beams with the use of electrical resistance strain gauges,
Hooke’s law and flexure formula. After calculating the stresses and strains, the average percent
error will be computed between the theoretical and experimental values for each level of load.
Finally, 4 different graphs will showcase the distribution of flexural stresses at various distances
from the bottom of the beam for both the aluminum and steel theoretical and experimental
values. The formula for Hooke’s law is given in equation 1 below where the experimental
flexural (longitudinal) stress (σ) is equal to the modulus of elasticity (E) multiplied by the normal
strain (ε). Using the flexure formula as discussed before will be used to compare the
experimental flexural stresses and the theoretical flexural stresses. This is given in equation 2
below where the theoretical flexural stress at distance ‘y’ from the neutral axis is equal to the
negative bending moment (M) multiplied by the distance from the neutral axis (y) which all be
divided by the moment of inertia of the cross-section about the axis of bending that passes
through the centroid of the cross-section (I).
Experimental flexural stress: σ = E* ε
(1)
Theoretical flexural stress: σ = -(M*y)/I
(2)
Procedures
The instrumentation required for this lab are as follows:
1.
2.
3.
4.
5.
6.
MTS Universal Testing Machine
Datascan Analog Measurement Processor
Dalite Datascan Configurator
Strain Gauges
Scales
Weights
The specimens for the flexural stress lab will include non-ferrous material aluminum and ferrous
material steel. A ferrous material is defined as a material that contains iron. While steel is
considered to be an alloy created by adding iron to carbon in order to harden the iron
furthermore, other materials can be diffused such as chromium and nickel to toughen the
structure of steel. An alloy is to be considered as steel if it is composed of less than 2.1 wt.%
carbon. This category can be split into two sub-categories including low alloy such as low carbon
steel (<0.25 wt.% carbon) and high alloy such as stainless steel [2]. Aluminum is considered to
be a non-ferrous metal as it doesn’t contain a percentage of iron and is used in the industry for
being a lightweight metal (3 times lighter than steel).
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The setup of this lab is divided into three different sections. The students will be gathering
around the desk station to record and watch the live experiment of the flexure test on the two
beams. The second station is for the machine operator(s) who will be at the MTS Universal
Testing Machine who will safely apply loads on the beams and run the Dalite Datascan
Configurator. The final station will be where the strain values for all six strain gauges will be
outputted on to a monitor at each load (from 5-25 kN for the steel I-beam and 10-50 lbs for the
aluminum box beam). This will be recorded and used later on for further calculations to see any
discrepancies between the theoretical stresses and strains using the flexure formula compared to
the experimental stresses and strains using Hooke’s law. Below is a drawing of each setup in
order during the flexure test lab in figure 1.
Figure 1: Lab Setup
The experimental testing procedure is as follows:
1. Before beginning the lab, take each beam and verify the measurements of the span length,
locations of the strain gauge line, and the distance of each gauge on the gauge line from
the bottom of the beam. The positioning and distances required for measurements carried
out later on can be seen through figures 2-5 below for steel and aluminum beams [3].
Figure 2: Simply supported steel I-beam
Figure 3: Cantilever aluminum box beam
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Figure 4: Cross-section of aluminum box-beam,
and location of strain gauges (not in scale)
Figure 5: Cross-section of steel I-beam, and
location of strain gauges (not in scale)
2.
a. Starting with the steel I-beam, the moment of inertia of the cross section is given as I =
22.4 in4 (9.3236 x 106 mm4).
b. Open the Dalite Datascan Configurator at the DB2L1S.OVL file and click on the
monitor every second window. Initially, check that the load is zero and record the
initial strains values for all six strain gauges. It will be noticed that although the strain
values should be equal to zero for all six gauges, the values seen will be different due to
external forces.
c. Now, load the steel I-beam steadily up to 5 kN. Using a table as shown in table 1 record
the strain for all six strain gauges.
d. Finally, increase the load increments by 5 kN until 25 kN has been reached. Record the
strains for all six strain gauges at each level of load.
3.
a. For the aluminum box beam, open the Dalite Datascan Configurator once again with
the DB1L1A.OVL file and click on every monitor every second window as was done
earlier for the steel beam. As before, check to see that the load is at zero and record the
initial strain values for all six strain gauges. This will be equal to zero throughout all
gauges.
b. Add 10 lb (44.48 N) of weights to the end of the aluminum cantilever beam safely and
steadily using a hook that is attached to the end of the beam. Record the strain values
for all six strain gauges in the table as shown in table 2.
c. Increase the load in increments of 10 lb or 44.48N up to a total of a maximum load of
50 lb and record the experimental strain values for all six strain gauges at each level of
load.
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Results
After conducting the lab, results were collected and inputted into a Microsoft Excel sheet. The
results in the Excel sheet were the experimental strains for all six strain gauges. As mentioned
earlier, the loads applied to each beam would increase in increments. Below in tables 1 and 2
show the live readings of the strains for each specimen at each gauge and load while conducting
the flexure test.
Table 1: Steel I-Beam Strain Gauges for Loads 5-25 kN
Load (kN)
5
10
15
20
25
Steel I Beam
Strain Gauge
1
2
3
-22.9984
-23.1675
-10.8474
-42.3981
-40.145
-19.7361
-61.7977
-56.3079
-31.0544
-82.0047
-73.2852
-41.5654
-101.4043 -90.2627
-52.0691
4
3.687
7.7314
10.9609
15.0015
19.0422
5
12.5486
26.2891
40.8445
52.9702
66.7104
6
25.0044
47.6409
71.8923
94.5215
117.9656
Table 2: Aluminum Box Beam Strain Gauges for Loads 10-50 lbs
Al Box beam
Strain Gauge
Load (lb)
10
20
30
40
50
1
25.1064
52.5872
78.4534
104.3196
130.1858
2
14.434
33.8335
49.1964
63.7441
77.4846
3
8.0728
16.1541
26.6572
36.3611
47.6793
4
-7.182
-17.6854
-28.1964
-35.7416
-42.7443
5
-18.5616
-31.4948
-46.05
-63.835
-82.4227
6
-32.3078
-57.3661
-83.2323
-107.4836
-131.735
From the live experimental results above, the values for experimental stresses using the Hooke’s law
were calculated. Since the values for the modulus of elasticity were provided to the lab students where
the modulus of elasticity for steel is 200000 MPa and for aluminum is 72000 MPa, the experimental
stresses can be calculated using equation 1 from above. As steel is more elastic than aluminum, it will be
able to handle more stress per unit of strain elongation. Following the experimental values, lab students
would be required to compute the values for the theoretical stresses and strains. As mentioned earlier,
the flexure formula would be used to calculate the theoretical stresses as given in equation 2 above. In
order to do so, the values for bending moment (M), moment of inertia of the cross-section (I), and the
distance from the neutral axis (y) would need to be required. The moment can be calculated by
multiplying each load by the distance between the applied force and the location of the strain gauges
along the length of the beam. This can be clearly seen in figures 2 and 3 for steel and aluminum,
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respectively. The distance for the steel I-beam is given as 0.1m and for the aluminum box-beam is given
as 0.65m as the total length of the beam is 0.75 m and the location of the strain gauges is offset 0.13m
from the fixed surface. It must be kept in mind that the loads for aluminum are given as lbs which must
be converted to newtons. This can be completed by multiplying each load in pounds by 0.00444822
kilonewtons. As for the moment of inertia of the cross-section (I), the value for steel was given as I =
22.4 in4 (9.3236 x 106 mm4). For the aluminum box-beam, this will need to be calculated using the
dimensions of the inner and outer lengths of given in figure 5. To calculate this, formula 3 below will be
utilized [1].
I = [(Bo*Ho3)/12] - [(Bi*Hi3)/12]
(3)
For the box beam, the moment of inertia using formula 3 above can be calculated to be the
following:
I = [(Bo*Ho3)/12] - [(Bi*Hi3)/12] = [(50*753)/12] – [(44*693)/12] = 553279.5 mm4.
Hence the moment of inertia of the cross-section about the axis of bending that passes through
the centroid of the cross-section is given the value of 553279.5 mm4.
Finally, the next step would be to calculate the values for the distances from the neutral axis to
each strain gauge in each specimen. The distances (y) for each material are given below in table
3.
Table 3: Distances(y) from the neutral axis to each strain gauge in each specimen
Steel
Strain Gauge 1 = 74 mm
Strain Gauge 2 = 49 mm
Strain Gauge 3 = 23.5 mm
Strain Gauge 4 = -23.5 mm
Strain Gauge 5 = -49 mm
Strain Gauge 6 = -74 mm
Aluminum
Strain Gauge 1 = 37.5 mm
Strain Gauge 2 = 25 mm
Strain Gauge 3 = 12.5 mm
Strain Gauge 4 = -12.5 mm
Strain Gauge 5 = -25mm
Strain Gauge 6 = -37.5 mm
After these values have been calculated, the theoretical stress can be computed using equation 2
above. As for the theoretical strain, this will be similar to calculating the experimental stress
where Hooke’s law will be used. In this case, for the theoretical strain, the formula will be strain
is equal to the theoretical stress divided by the modulus of elasticity. All throughout these
calculations, the units of MPa will be used to cancel out the units for stress and modulus of
elasticity in order to have a unitless strain value. Finally, as for the percentage error in stresses
and strains when comparing between the theoretical and experimental, it will be seen that for
each load, the percentage errors for the stress and strain are equal due to being equally
proportional during the calculations in figures 6 and 7. In order to calculate the percentage error,
this will be equal to the absolute value summation of the difference between the experimental
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and theoretical which all gets divided by the theoretical value for each load. The summations will
then be divided by the number of strain gauges which is equal to 6 for each specimen. To get the
value as a percentage, multiply the final quotient by 100%. For further ease, formula 4 can be
shown as the following:
%Error = 1/n*(Σ ((Experimental -Theoretical)/Experimental) * 100%
(4)
The formula above will be used for both the average percentage error in strains and stresses.
Figures 6 and 7 below showcases the experimental and theoretical strains and strains, along with
the percentage errors for each specimen.
Figure 6: Steel I-Beam Experimental, Theoretical and Percentage Errors of Flexure Stresses and Strains
Figure 7: Aluminum Box Beam Experimental, Theoretical and Percentage Errors of Flexure Stresses and
Strains
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Following the calculations, plotting graphs were an excellent way to show the distribution of the
flexural stresses at various distances (from the bottom of the beam for the two beams). Those
stresses with a positive value would be under tension, while those with a negative value would
be under compressions. The following four graphs plotted were the following:
o
o
o
o
Steel I-beam: distance from bottom of the beam vs. experimental flexural stresses
Steel I-beam: distance from bottom of the beam vs. theoretical flexural stresses
Aluminum box-beam: distance from bottom of the beam vs. experimental flexural stresses
Aluminum box-beam: distance from bottom of the beam vs. theoretical flexural stresses
These graphs can be shown in figures 8.a-9.b below:
Figure 8.a: Distance from bottom of Steel I-Beam vs. Experimental Flexural Stresses
Figure 8.b: Distance from bottom of Steel I-Beam vs. Theoretical Flexural Stresses
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Figure 9.a: Distance from bottom of Aluminum Box-Beam vs. Experimental Flexural Stresses
Figure 9.b: Distance from bottom of Aluminum Box-Beam vs. Theoretical Flexural Stresses
As it can be seen, the intersection point of all trendlines is the location of the neutral axis from
the bottom of the beam. For this lab, the neutral axis was 74 mm above the bottom of the steel Ibeam and 37.5 mm above the bottom of the aluminum box-beam.
Discussions
The purpose of flexural testing is to measure force required to bend a beam of a specific material
in order to determine the resistance to flexing or stiffness. As shown from the finalized table
above of the experimental and theoretical stresses and strains for both steel and aluminum
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specimens, steel is able to endure much more stress (MPa) than aluminum at each strain gauge
even after the load applied onto aluminum is much less than the loads applied for steel. For an
example, the first load applied on steel was 5 kilo-Newtons, whereas for aluminum it was
approximately 0.0444 kilo-Newtons; more than 100 times difference. As the stresses for steel at
each load and strain gauge is higher than for aluminum, this relates back to the modulus of
elasticity values given since the higher the modulus of elasticity, the greater the rigidity. In other
words, a higher force must be applied per unit area to produce a given deformation or strain.
From the graphs, it is quite evident at what locations of strain gauges where the beam was
compressed and under tension. For the aluminum specimen, the first three strain gauges had a
negative stress value which indicate compression. The last three strain gauges had a positive
stress value to indicate a tensile force at those specific locations. In other words, this produced a
downward linear trend. As for the steel specimen, this was the exact opposite where the first
three strain gauges had negative flexure stress values to indicate compression and the last three
strain gauges had positive stress values to indicate tension. The neutral axis was located in
similar locations in terms of where the intersection points of all 6 lines met up together along the
y-axis of the graphs. It was seem that the difference in neutral axis locations between the
experimental and theoretical were approximately 4 mm for steel and 0.5 mm for aluminum, In
the end, it can be said that the experimental values compare to the theoretical values for stresses
and strains are similar overall with a maximum percentage error of 9.14% in aluminum and 7.33
% in steel.
Although the average percentage errors did not exceed 10% throughout the duration of this lab
when comparing the experimental and theoretical values, there are reasonings for why an error
still had occurred. An example of an error is wedging stress where the distribution quantifies
axial wedging stresses on the bottom side of the contact points. It will be compressive under the
loading point but shift to tensile on both sides. Any unbalances can cause some deviations to the
experimental results of the strains and stresses. Another source of error is any dislocations. For
example, when applying the loads on the hanger off the beam, there could be a chance the load
was not set properly in place. Another dislocation could be that the location of the strain gauges
along the beams were not exactly at the locations as shown in the diagrams in figures 4 and 5. In
the end, any dislocations could cause an effect to the total load actually applied than what was
given prior to the loading such as loads of increments of 5 kN or 10 lbs. Another source of error
could be that the modulus of elasticity given prior to the lab of 200 GPa for steel and 72 GPa for
aluminum is not entirely exact. The modulus of elasticity was used while calculating for the
experimental stresses and theoretical strains. It is possible that the modulus may be incorrect to
the actual steel and aluminum modulus values due to the effect of temperature and effect of
impurities. When it comes to temperatures, as temperature increases, the modulus of elasticity of
the materials decreases. As well, any addition of impurities to the metal and alloy may increase
or decrease the elasticity. Adding impurities with a lower modulus compared to steel and
aluminum will only decrease the overall modulus of elasticity, and vice versa for an impurity
with a higher modulus.
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Conclusion
In the end, it is up to engineers to understand the aspects of the material’s behaviour, not only by
uniaxial tension or compression, but also bending. When the materials were subjected to
bending, it consisted of a combination of tensions, compression, and shear forces. Flexural tests
are useful for everyday objects that are to be used to support structure and return to its original
shape after any bending occurs. From the flexure testing within this lab, it can be said that steel
would be more predominant compared to aluminum when it comes to a material that is unlikely
to bend, warp or deform overall as it is approximately 3 times denser than aluminum [4]. It was
seen that for a higher load applied to steel, the experimental flexure stress was still higher to
cause a smaller elongation difference (strain) than aluminum. The experimental values for the
materials’ flexure stresses and strains were close to the theoretical value and any errors may be
subject to misalignment of the loads and/or strain gauges, impurities to cause the modulus of
elasticity for the materials to be different from the provided values, and also any unbalances of
added tension or compression from the wedging stresses.
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References
[1] Afsar, J 2013, Moment of Inertia of Hollow Rectangular Section. Engineering Intro, viewed
March 24, 2022. Available at: https://www.engineeringintro.com/mechanics-ofstructures/moment-of-inertia/moment-of-inertia-of-hollow-section/
[2] Carbon Steel: Properties, Production, Examples and Applications. Matmatch. [Online].
Available at: https://matmatch.com/learn/material/carbon-steel
[3] Dr. Gherib. (2022). Laboratory -3 Flexural Stresses in Beams. University of Windsor,
Windsor.
[4] Steel vs. Aluminum: How to Make the Best Choice for Your Product Design. Gabrian
International. [Online]. Available at: https://www.gabrian.com/steel-vs-aluminum/
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