Lab 1
Determination of Material Properties
Ting Zhang – First Author, Abstract, Experimental Theory, Results, Error Analysis
Kanchan Bhattacharyya – Discussion, Conclusion
Matthew Steven – Results, Error Analysis
Xie Zheng – Introduction, Experimental Theory, Experimental Procedures,
Specimens and Instrumentation
Abstract
The purpose of this experiment is to let students to learn to determine material properties and
to observe material responses at different stages of loading. In this experiment, students will familiarize
with strain gage, digital strain indicator, Universal Digital Testing Machine, Torsion Testing Machine and
LVDT system. This experiment consists of three parts. The first part is determination of Young’s Modulus
and Poisson’s ratio for both steel and aluminum specimens. The second part of the experiment is to do a
tensile test in order to obtain a stress- strain diagram of an aluminum specimen. The last part of the
experiment is determination of shear modulus for specimens of carbon steel and aluminum by using
Torsion Testing machine and Torsiometer. We use linearly regression method in order to get the value
of Young’s Modulus, Shear Modulus and Poisson’s ratio. From the results, values calculated for the
linear regression slope constants can be represented as Young's Modulus, Poisson’s ratio or Shear,
depending on which experiment students are working on. From the result, the values we get from the
experiment are acceptable when compared to the corresponding values in theory.
Introduction
This lab consists of three parts which are dealing with Young’s modulus, shear modulus
and strain-stress graph respectively. The purpose of the experiments is looking for material’s
properties. By adding load to specimens and finding out the relationship between the strain
and stress, we are able to determine material’s properties such as its Young’s modulus and
shear modulus. Also, after doing a time consuming experiment, we are getting the plot of
stress-strain diagram which would presents the relation between stress and strain of the
specimen. Young’s modulus is named after
To determine the modulus of the specimens, we are using Hooke’s law as basic
equation. It is named after the 19th-century British scientist Thomas Young. However, the
concept was developed in 1727 by Leonhard Euler, and the first experiments that used the
concept of Young's modulus in its current form were performed by the Italian
scientist Giordano Riccati in 1782. The importance of determination of material properties
could be found from our daily life. For example, when we are building up a building for
residence or public purpose, we need to consider the safety of the building parts such as pillar
and beam. Once we can determine the young’s modulus and shear modulus of the certain
material (we may use concrete in general), we can easily design the size of the beam or pillar
and consider the structure of the building. All the design should base on the material properties
and then engineers can come up with a proper plan. To come up with a proper and safe plan or
even an idea of tool for daily usage, we have to know more enough about the material
properties.
Specimens and Instrumentation
Testing specimens
Aluminum testing specimen with LVDTs for doing the tensile test for the stress-strain diagram.
Transducer Indicator
Tinius Olsen 1000 Universal Digital
Testing Machine
E101 Digital Measuring System
P3 Digital Strain Indicator
Torsion Testing Machine
Theory
Determination of Young’s Modulus
Young's modulus is also called as the tensile modulus or elastic modulus, it indicates
that every certain material holds a linear relationship between the tensile load which is applied
to it and the elongation. It is a measure of the stiffness of an elastic isotropic material and is a
quantity used to characterize materials. To determine the Young’s modulus, Hoode’s law is
used as the basic equation to solve for E which stands for Young’s modulus.
𝛿=
𝑃𝐿
𝐴𝐸
(Eq. 1)
Where
P = force producing extension of bar,
L = length of bar,
A = cross-section area of bar,
δ = total elongation of bar,
E = elastic constant of the material, called Modulus of Elasticity or Young’s Modulus.
According to the Eq.2 and Eq.3 , the stress and strain can be determined:
𝜎=
𝑃
𝐴
(Eq.2)
𝜀=
𝛿
𝐿
(Eq.3)
Through the observation, the axial elongation is always accompanied by lateral contraction of
the bar within the elastic region where the ratio is called Poisson’s ratio. According to Fig. 1.1,
we can see that in the elastic region (from O to A), the relation between strain and stress is
linear which means the specimen is in an elastic deformation. And point A is the proportional
limit. After point A, between A and D, the relation between strain and stress becomes
complicated, the curve goes down a little bit and then presents a curve which looks like a graph
of quadratic equation. And the point C indicates the maximum load applied to the specimen.
Finally at point D, the specimen would be broken and the fracture of the specimen occurs.
Determination of Shear Modulus
Assuming a material obeys Hook’s law, the shearing stress of the material, which contributes to its
distortion, is proportional to the shearing strain . The relationship between shearing stress and shearing
strain in this material can be expressed as
𝜏 = 𝐺𝛾
where G is called shear modulus and it is a constant, 𝜏 is shear stress and 𝛾 is shear strain
We can observe from Fig that for small values of 𝛾, we can express the arc length AA’ as AA’ =L𝛾. But on
the other hand, we have 𝐴𝐴′ = 𝜌Φ
So,
𝛾=
𝜌Φ
𝐿
where 𝛾 and Φ both expressed in radians.
The equation obtained show that the shearing stress at a given point of a shaft is proportional to the
angle of twisted. It also shows that 𝛾 is proportional to the distance from the axis of the shaft to the
point under consideration.
The angle of twisted angle per unit length of the shaft varies directly as the applied torque and inversely
as the Shear Modulus G. The equation yields to
Φ=
𝑀𝐿
𝐺𝐼
Where M represents torque, I represent Polar Moment of Inertial
The equation above can be transformed to G =
𝑀𝐿
Φ𝐼
in order to calculate shear modulus of Aluminum
and Steel in this experiment.
Procedures
PART 1– Determination of Young’s Modulus E and Poisson’s ν.
1. To avoid the error from flatness of the specimen, four strain gages are attached to each of
the testing specimen. So there would be four channels to receive the signal from gages.
2. Measure the size of the specimen (steel and aluminum testing specimens).
3. The experiment is already setup and check to see if the gages are connected to the channel
on P3soft. The first specimen is made of steel, so one of the wires for each gage is
connected to D 120.
4. Press “Force Zero” button on Tinius Olsen 1000 Digital Testing Machine to adjust the force
on the machine is 0. And then press ext zero.
5. Run the P3software on computer and then select “BAL” to set all the strain gages initially
zero.
6. Start to add the initial load to specimen to around 20 lbf to tight the specimen.
7. Click “RECORD” and select all the four channels and click “on this computer”.
8. Hit record to record the reading for initial condition which is used as “zero” for later
calculation.
9. Start to add the load to specimen with increment of 50 lbf, save the data in program and
record the reading on Tinius Olsen 1000 Digital Testing Machine for every load till the
maximum load which is around 520 lbf.
10. Save the data on computer for first trial and unload the force on specimen, then repeat
steps 4 to 9 for another two trials.
11. After getting three trials, change the specimen to aluminum one, repeat steps 4 to 10 to get
the data for aluminum specimen.
Part 2—Tensile test for the stress-strain diagram.
1. Load a thin aluminum specimen on Tinius Olsen 1000 Digital Testing Machine for this
experiment.
2. Measure the thickness of the block three times and then take the average of them.
3. Measure the size of specimen.
4. Adjust the height of the LVDT’s to make sure the maximum distance of deflection will be in
its detecting range.
5. Press “Force Zero” button on Tinius Olsen 1000 Digital Testing Machine to adjust the force
on the machine is 0. And then press ext zero.
6. Start to add the initial load to specimen to around 20 lbf to tight the specimen.
7. Start to add the load to specimen with increment of 50 lbf, record the reading on Tinius
Olsen 1000 Digital Testing Machine and RDP Transducer Indicator E309 for every load.
8. When the load is added up to 570 lbf, we set the speed of load applying to 9 since it is
getting slower to add load on the specimen.
9. When the load is much harder to add to specimen, we take increment of 0.5 of the reading
on RDP Transducer Indicator E309 since the load is much harder to add (it is in the range of
plastic deformation).
10. Keep adding load to the specimen, the aluminum specimen is finally broken with the
fracture on the upper position.
Part 3-- Determination of Shear Modulus.
1. Measure the diameter of two specimens (steel on and aluminum one).
2. The lab is already setup, check the connections between Torsion Testing Machine and E101
Digital Measuring System.
3. Set deflection arm approximately by adjusting the hand wheel which is located at the right
hand side of the Torsion Testing Machine.
4. Set the dial gauge on the Torsiometer to zero by rotating the outer bezel.
5. Zero the E101 Digital Measuring System by pressing the “zero” button which is located at
the back.
6. Carefully rotating two hand wheels at the same time till the reading on the E101 Digital
Measuring System is about 2.5. Then, record the reading on E101 Digital Measuring System
and the dial gauge on the Torsiometer.
7. Unload the specimen by turning the hand wheels until the reading on E101 Digital
Measuring System is back to 0 and then zero it.
8. Repeat steps 3 to 7 twice to get another two trials for steel specimen.
9. Change the specimen to aluminum one and then repeat steps 3 to 8 get the record.
Results
Determination of Young’s Modulus and Poisson’s Ratio of Aluminum
a)
Material Aluminum
Width (in)
0.5
b)
Material
Steel
Width (in) 0.5055
Thickness (in)
0.121
Thickness (in)
0.121
Area (in2)
0.0605
Area (in2)
0.0612
Table 1: Testing Material Property Data
The specified materials and measurements for the width and thickness of the cross-sectional area in
testing, and the calculated cross-sectional area for each specimen.
𝐴𝑟𝑒𝑎
= 𝑆𝑝𝑒𝑐𝑖𝑚𝑒𝑛 𝑊𝑖𝑑𝑡ℎ 𝑥 𝑆𝑝𝑒𝑐𝑖𝑚𝑒𝑛 𝑇ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠
𝐴𝐴𝑙 = 0.5𝑖𝑛 𝑥 0.1210𝑖𝑛 = 0.0605𝑖𝑛2
𝐴𝑆𝑡𝑒𝑒𝑙 = 0.5055𝑖𝑛 𝑥 0.1210𝑖𝑛
= 0.0612𝑖𝑛2
Fig. : Calculation of Cross-Sectional Areas in Tensile Testing
The calculations for the cross-sectional areas of both the aluminum and steel specimens used in tensile
Testing. These calculations were needed to compute the Normal Stresses induced with each load.
P (lb)
εxx (in/in)-6 εyy (in/in)-6
σxx (lb/in2)
2.80E-05
-1.50E-05
20
330.5785124
1.11E-04
-4.45E-05
70.3
1161.983471
1.95E-04
-7.20E-05
120
1983.471074
2.81E-04
-1.00E-04
171
2826.446281
3.67E-04
-1.28E-04
220.6
3646.280992
4.53E-04
-1.56E-04
270
4462.809917
5.39E-04
-1.84E-04
320.8
5302.479339
Table 2: Part One: First Trial Aluminum Tensile Testing Results
The applied loads, average normal and transverse strains as measured by the strain indicator, and
corresponding normal stresses measured during the first aluminum tensile test.
𝜎𝑥𝑥𝑖 =
𝑃𝑖
𝐴𝐴𝑙
𝐸𝑥. 𝜎
𝑥𝑥1 =
20 𝑙𝑏.
𝑙𝑏
= 330.5785 2
0.0605 𝑖𝑛2
𝑖𝑛
Fig. : Calculation of the Axial Stress σxx associated with the Induced Loads during Aluminum Tensile
Test Number One The formula used to compute the axial stress induced using the measured values for
the applied loads and the cross sectional area of the testing material. The example given using the first
recorded load measurement demonstrates how the each experimental load was used to calculate the
associated axial stress
Stress vs. Strain
7000
6000
Stress (psi)
5000
y = 1E+07x + 82.442
4000
3000
2000
1000
0
0.00E+00
-1000
1.00E-04
2.00E-04
3.00E-04
4.00E-04
Strain εxx (in/in)
Stress vs. Strain
5.00E-04
6.00E-04
Linear (Stress vs. Strain)
Fig. : Graph plotting the Axial Stress against recorded Axial Strain during Aluminum Tensile Test
Number One
Graph plotting the stresses computed against the associated axial strain as recorded by the strain
𝑙𝑏
indicator. The linear approximation function 𝑦 = (10 𝑥 106 )𝑥 + 82.442 𝑖𝑛2 is represented by the
dashed line connecting the initial and final measurements.
Lateral Strain εxx (in/in)-6
Axial Strain vs. Lateral Strain
0.00E+00
5.00E-05
0.00E+00
-5.00E-05
-1.00E-04
-1.50E-04
-2.00E-04
-2.50E-04
1.00E-04
2.00E-04
3.00E-04
4.00E-04
5.00E-04
y = -0.3282x - 7E-06
Axial Strain vs. Lateral Strain
Axial Strain εxx (in/in)-6
Linear (Axial Strain vs. Lateral Strain)
6.00E-04
Fig. : Axial Strain vs. Transverse Train - Aluminum Tensile Test Number One
Graph plotting the axial strain induced from each load against the lateral strain as recorded by the strain
indicator. The linear approximation function 𝑦 = (−0.3282)𝑥 − (7𝑥10−6 ) is represented by the dashed
line connecting the initial and final measurements.
εxx (in/in)-6
3.65E-05
1.14E-04
2.01E-04
2.88E-04
3.74E-04
4.62E-04
5.47E-04
P (lb)
21.8
68.2
119
169
220.1
271
321.5
εyy (in/in)-6
-1.15E-05
-3.65E-05
-6.45E-05
-9.30E-05
-1.21E-04
-1.49E-04
-1.77E-04
σxx (lb/in2)
360.3305785
1127.272727
1966.942149
2793.38843
3638.016529
4479.338843
5314.049587
Table 3: Part One: Second Trial Aluminum Tensile Testing Results
The applied loads, average normal and transverse strains as measured by the strain indicator, and
corresponding normal stresses measured during the second aluminum tensile test.
𝜎𝑥𝑥𝑖 =
𝑃𝑖
𝐴𝐴𝑙
𝐸𝑥. 𝜎
𝑥𝑥1 =
21.8 𝑙𝑏.
𝑙𝑏
= 360.3306 2
0.0605 𝑖𝑛2
𝑖𝑛
Fig. : Calculation of the Axial Stress σxx associated with the Induced Loads during Aluminum Tensile
Test Number Two The formula used to compute the axial stress induced using the measured values for
the applied loads and the cross sectional area of the testing material. The example given using the first
recorded load measurement demonstrates how the each experimental load was used to calculate the
associated axial stress
Stress vs. Strain
7000
6000
Stress (psi)
5000
4000
y = 1E+07x + 17.009
3000
2000
1000
0
0.00E+00
-1000
1.00E-04
2.00E-04
3.00E-04
4.00E-04
5.00E-04
6.00E-04
Strain εxx (in/in)
Stress vs. Strain
Linear (Stress vs. Strain)
Fig. : Graph plotting the Axial Stress against recorded Axial Strain during Aluminum Tensile Test
Number Two
Graph plotting the stresses computed against the associated axial strain as recorded by the strain
𝑙𝑏
indicator. The linear approximation function 𝑦 = (10 𝑥 106 )𝑥 + 17.009 𝑖𝑛2 is represented by the
dashed line connecting the initial and final measurements.
Lateral Strain εxx (in/in)-6
Axial Strain vs. Lateral Strain
0.00E+00
5.00E-05
0.00E+00
-5.00E-05
-1.00E-04
-1.50E-04
-2.00E-04
-2.50E-04
1.00E-04
2.00E-04
3.00E-04
4.00E-04
5.00E-04
6.00E-04
y = -0.3234x + 3E-07
Axial Strain vs. Lateral Strain
Axial Strain εxx (in/in)-6
Linear (Axial Strain vs. Lateral Strain)
Fig. : Axial Strain vs. Transverse Train - Aluminum Tensile Test Number Two
Graph plotting the axial strain induced from each load against the lateral strain as recorded by the strain
indicator. The linear approximation function 𝑦 = (−0.3234)𝑥 + (3𝑥10−7 ) is represented by the dashed
line connecting the initial and final measurements.
P (lb)
εxx (in/in)-6 εyy (in/in)-6 σxx (lb/in2)
3.35E-05
-1.10E-05
19.9
328.92562
1.20E-04
-3.85E-05 1158.67769
70.1
2.00E-04
-6.45E-05 1971.90083
119.3
2.70E-04
-8.70E-05 2634.71074
159.4
3.58E-04
-1.15E-04 3469.42149
209.9
4.42E-04
-1.43E-04 4302.47934
260.3
5.31E-04
-1.71E-04 5148.76033
311.5
Table 4: Part One: Third Trial Aluminum Tensile Testing Results
The applied loads, average normal and transverse strains as measured by the strain indicator, and
corresponding normal stresses measured during the third aluminum tensile test.
𝜎𝑥𝑥𝑖 =
𝑃𝑖
𝐴𝐴𝑙
𝐸𝑥. 𝜎
𝑥𝑥1 =
19.9𝑙𝑏.
𝑙𝑏
= 328.9256 2
0.0605 𝑖𝑛2
𝑖𝑛
Fig. : Calculation of the Axial Stress σxx associated with the Induced Loads during Aluminum Tensile
Test Number Three The formula used to compute the axial stress induced using the measured values for
the applied loads and the cross sectional area of the testing material. The example given using the first
recorded load measurement demonstrates how the each experimental load was used to calculate the
associated axial stress
Stress vs. Strain
7000
6000
Stress (psi)
5000
4000
y = 1E+07x + 12.357
3000
2000
1000
0
0.00E+00
-1000
1.00E-04
2.00E-04
3.00E-04
4.00E-04
5.00E-04
6.00E-04
Strain εxx (in/in)
Stress vs. Strain
Linear (Stress vs. Strain)
Fig. : Graph plotting the Axial Stress against recorded Axial Strain during Aluminum Tensile Test
Number Three
Graph plotting the stresses computed against the associated axial strain as recorded by the strain
𝑙𝑏
indicator. The linear approximation function 𝑦 = (10 𝑥 106 )𝑥 + 12.357 𝑖𝑛2 is represented by the
dashed line connecting the initial and final measurements.
Lateral Strain εxx (in/in)-6
Axial Strain vs. Lateral Strain
0.00E+00
0.00E+00
1.00E-04
2.00E-04
3.00E-04
4.00E-04
5.00E-04
6.00E-04
-5.00E-05
-1.00E-04
y = -0.3211x - 3E-07
-1.50E-04
-2.00E-04
Axial Strain εxx (in/in)-6
Axial Strain vs. Lateral Strain
Linear (Axial Strain vs. Lateral Strain)
Fig. : Axial Strain vs. Transverse Train - Aluminum Tensile Test Number Three
Graph plotting the axial strain induced from each load against the lateral strain as recorded by the strain
indicator. The linear approximation function 𝑦 = (−0.3211)𝑥 − (3𝑥10−7 ) is represented by the dashed
line connecting the initial and final measurements.
(a)
Trial
1
2
3
Eexp
(lb/in2)
9.70E+06
9.68E+06
9.69E+06
EAl(lb/in2)
1.00E+07
1.00E+07
1.00E+07
ε (lb/in2)
2.96E+05
3.17E+05
3.08E+05
εr (%)
2.96%
3.17%
3.08%
A (%)
97.04%
96.83%
96.92%
Table 5: Comparison of Experimentally determined values for Young’s modulus and associated errors
and accuracies with known value of Young’s Modulus for Aluminum
Table giving the values calculated from linear regression for the Young’s Modulus of Aluminum, or the
slope of the Stress vs. Strain plots and their absolute (ε) and relative (εr) errors and accuracy (A) with
𝑙𝑏
respect to the known value for Young’s Modulus of Aluminum of 𝐸𝐴𝑙 = 10𝑥106 𝑖𝑛2 . Detailed
calculations for Young’s Modulus determined from each trial set and the uncertainties in the results can
be found in Appendices A1-A6.
Trial
1
2
3
ѵexp
0.32820763
0.32339856
0.32111065
ѵAl
0.333
0.333
0.333
ε
4.79E-03
9.60E-03
1.19E-02
εr
1.44%
2.88%
3.57%
A
98.56%
97.12%
96.43%
Table 6: Comparison of Experimentally determined values for Poisson’s Ratio and associated errors
and accuracies with known value of Poisson’s Ratio for Aluminum
Table giving the values calculated from linear regression for the Young’s Modulus of Aluminum, or the
slope of the Transverse Strain vs. Axial Strain plots and their absolute (ε) and relative (εr) errors and
accuracy (A) with respect to the known value for Young’s Modulus of Aluminum of 𝐸𝐴𝑙 = 0.333.
Detailed calculations for Poisson’s Ratio determined from each trial set and the uncertainties in the
results can be found in Appendices B1-B6.
Determination of Young’s Modulus and Poisson’s Ratio of Steel
P (lb) εxx (in/in)-6 εyy (in/in)-6 σxx (lb/in2)
1.05E-05
-1.50E-06 286.108999
17.5
3.80E-05
-8.00E-06 1067.59529
65.3
6.65E-05
-1.60E-05 1922.65248
117.6
9.65E-05
-2.40E-05 2735.20203
167.3
1.27E-04
-3.20E-05 3570.64031
218.4
1.53E-04
-4.00E-05 4365.20588
267.0
1.83E-04
-4.80E-05 5182.66016
317.0
2.15E-04
-5.60E-05 6006.65408
367.4
2.42E-04
-6.40E-05 6832.2829
417.9
2.69E-04
-7.10E-05 7499.3256
458.7
2.98E-04
-8.00E-05 8305.33552
508.0
Table 7: Part One: Second Trial Steel Tensile Testing Results
The applied loads, average normal and transverse strains as measured by the strain indicator, and
corresponding normal stresses measured during the first steel tensile test.
𝜎𝑥𝑥𝑖 =
𝑃𝑖
𝐴𝐴𝑙
𝐸𝑥. 𝜎
𝑥𝑥1 =
17.5𝑙𝑏.
𝑙𝑏
= 286.1090 2
0.0612 𝑖𝑛2
𝑖𝑛
Fig. : Calculation of the Axial Stress σxx associated with the Induced Loads during Steel Tensile Test
Number
One
The formula used to compute the axial stress induced using the measured values for the applied loads
and the cross sectional area of the testing material. The example given using the first recorded load
measurement demonstrates how the each experimental load was used to calculate the associated axial
stress.
Stress vs. Strain
10000
8000
y = 3E+07x + 37.105
Stress (psi)
6000
4000
2000
0
0.00E+00
5.00E-05
1.00E-04
1.50E-04
-2000
2.00E-04
2.50E-04
3.00E-04
3.50E-04
Strain εxx (in/in)
Stress vs. Strain
Linear (Stress vs. Strain)
Fig. : Axial Stress vs. Axial Strain - Steel Tensile Test Number One
Graph plotting the stresses computed against the associated axial strain as recorded by the strain
𝑙𝑏
indicator. The linear approximation function 𝑦 = (30 𝑥 106 )𝑥 + 37.105 𝑖𝑛2 is represented by the
dashed line connecting the initial and final measurements.
Axial Strain vs. Lateral Strain
Lateral Strain εxx (in/in)-6
0.00E+00 5.00E-05
2.00E-05
1.00E-04
1.50E-04
2.00E-04
2.50E-04
3.00E-04
3.50E-04
0.00E+00
-2.00E-05
-4.00E-05
y = -0.2732x + 2E-06
-6.00E-05
-8.00E-05
-1.00E-04
Axial Strain εxx (in/in)-6
Axial Strain vs. Lateral Strain
Linear (Axial Strain vs. Lateral Strain)
Fig. : Axial Strain vs. Transverse Train - Steel Tensile Test Number One
Graph plotting the axial strain induced from each load against the lateral strain as recorded by the strain
indicator. The linear approximation function 𝑦 = (−0.2732)𝑥 + (2𝑥10−6 ) is represented by the dashed
line connecting the initial and final measurements.
P (lb) εxx (in/in)-6 εyy (in/in)-6 σxx (lb/in2)
1.20E-05
-2.00E-06 333.521348
20.4
4.10E-05
-9.00E-06 1154.24545
70.6
7.00E-05
-1.65E-05 1978.23937
121
9.90E-05
-2.45E-05 2787.51911
170.5
1.28E-04
-3.20E-05 3611.51303
220.9
1.59E-04
-4.05E-05 4443.68149
271.8
1.88E-04
-4.90E-05 5259.50086
321.7
2.17E-04
-5.65E-05 6076.95515
371.7
2.46E-04
-6.45E-05 6876.42544
420.6
2.77E-04
-7.20E-05
471.5
7708.5939
3.07E-04
-8.05E-05
521.7
8529.318
Table 8: Part One: Second Trial Steel Tensile Testing Results
The applied loads, average normal and transverse strains as measured by the strain indicator, and
corresponding normal stresses measured during the second steel tensile test.
Stress vs. Strain
10000
Stress (psi)
8000
6000
y = 3E+07x + 28.104
4000
2000
0
0.00E+00 5.00E-05
1.00E-04
1.50E-04
-2000
2.00E-04
2.50E-04
3.00E-04
3.50E-04
Strain εxx (in/in)
Stress vs. Strain
Linear (Stress vs. Strain)
Fig. : Axial Stress vs. Axial Strain - Steel Tensile Test Number Two
Graph plotting the stresses computed against the associated axial strain as recorded by the strain
indicator. The linear approximation function 𝑦 = (30 𝑥 106 )𝑥 + 28.104
𝑙𝑏
𝑖𝑛2
is represented by the
dashed line connecting the initial and final measurements.
Axial Strain vs. Lateral Strain
Lateral Strain εxx (in/in)-6
0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 3.50E-04
0.00E+00
-2.00E-05
-4.00E-05
y = -0.2684x + 2E-06
-6.00E-05
-8.00E-05
-1.00E-04
Axial Strain εxx (in/in)-6
Axial Strain vs. Lateral Strain
Linear (Axial Strain vs. Lateral Strain)
Fig. : Axial Strain vs. Transverse Train - Steel Tensile Test Number Two
Graph plotting the axial strain induced from each load against the lateral strain as recorded by the strain
indicator. The linear approximation function 𝑦 = (−0.2684)𝑥 + (2𝑥10−6 ) is represented by the dashed
line connecting the initial and final measurements.
P (lb) εxx (in/in)-6 εyy (in/in)-6 σxx (lb/in2)
1.10E-05
-2.50E-06 328.616622
20.1
3.95E-05
-1.00E-05 1159.15017
70.9
6.90E-05
-1.75E-05
120.2
1965.1601
9.80E-05
-2.55E-05
171.6
2805.5031
1.28E-04
-3.30E-05 3611.51303
220.9
1.56E-04
-4.10E-05 4443.68149
271.8
1.86E-04
-4.95E-05 5259.50086
321.7
2.18E-04
-5.85E-05 6135.81185
375.3
2.48E-04
-6.65E-05
426.0
6964.7105
2.77E-04
-7.40E-05 7759.27606
474.6
3.05E-04
-8.15E-05 8534.22272
522.0
Table 9: Part One: Third Trial Steel Tensile Testing Results
The applied loads, average normal and transverse strains as measured by the strain indicator, and
corresponding normal stresses measured during the third steel tensile test.
Stress (psi)
Stress vs. Strain
9000
8000
7000
6000
y = 3E+07x + 52.701
5000
4000
3000
2000
1000
0
0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 3.50E-04
Strain εxx (in/in)
Stress vs. Strain
Linear (Stress vs. Strain)
Fig. : Axial Stress vs. Axial Strain - Steel Tensile Test Number Three
Graph plotting the stresses computed against the associated axial strain as recorded by the strain
𝑙𝑏
indicator. The linear approximation function 𝑦 = (30 𝑥 106 )𝑥 + 52.701 𝑖𝑛2 is represented by the
dashed line connecting the initial and final measurements.
Lateral Strain εxx (in/in)-6
Axial Strain vs. Lateral Strain
0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04
0.00E+00
-1.00E-05
-2.00E-05
-3.00E-05
-4.00E-05
-5.00E-05
y = -0.2709x + 9E-07
-6.00E-05
-7.00E-05
-8.00E-05
-9.00E-05
Axial Strain εxx (in/in)-6
Axial Strain vs. Lateral Strain
3.00E-04
3.50E-04
Linear (Axial Strain vs. Lateral Strain)
Fig. Axial Strain vs. Transverse Train - Steel Tensile Test Number Three
Graph plotting the axial strain induced from each load against the lateral strain as recorded by the strain
indicator. The linear approximation function 𝑦 = (−0.2709)𝑥 + (9𝑥10−7 ) is represented by the dashed
line connecting the initial and final measurements.
Trial Eexp (lb/in2) EAl(lb/in2) ε (lb/in2)
εr (%)
A (%)
1
2.79E+07 3.00E+07 2.10E+06 6.99E-02 0.930
2
2.78E+07 3.00E+07 2.18E+06 7.26E-02 0.927
3
2.79E+07 3.00E+07 2.09E+06 6.96E-02 0.930
Table 10: Comparison of Experimentally determined values for Young’s modulus and associated errors
and accuracies with known value of Young’s Modulus for Steel
Table giving the values calculated from linear regression for the Young’s Modulus of Steel, or the slope
of the Stress vs. Strain plots and their absolute (ε) and relative (εr) errors and accuracy (A) with respect
𝑙𝑏
to the known value for Young’s Modulus of Steel of 𝐸𝐴𝑙 = 30𝑥106 𝑖𝑛2. Detailed calculations for Young’s
Modulus determined from each trial set and the uncertainties in the results can be found in Appendices
C1-C6.
Trial
1
2
3
Table
ѵexp
ѵSteel
ε
0.27316959 0.291 1.78E-02
0.26844712 0.291 2.26E-02
0.27090438 0.291 2.01E-02
11: Comparison of Experimentally
εr
A
6.13E-02 0.939
7.75E-02 0.922
6.91E-02 0.931
determined values for Poisson’s Ratio and associated errors
and accuracies with known value of Poisson’s Ratio for Steel
Table giving the values calculated from linear regression for the Young’s Modulus of Steel, or the slope
of the Transverse Strain vs. Axial Strain plots and their absolute (ε) and relative (εr) errors and accuracy
(A) with respect to the known value for Young’s Modulus of Steel of 𝐸𝐴𝑙 = 0.291. Detailed calculations
for Poisson’s Ratio determined from each trial set and the uncertainties in the results can be found in
Appendices D1-D6.
Determination Proportional Limit, Yield Point, and Ultimate Strength via the Stress Strain Curve
Thickness (in)
0.0325
Width (in)
Active Length
(in)
0.54
4.5
Table 12: Part Two Testing Material Property Data
The specified materials and measurements for the width and thickness of the cross-sectional area in
testing, and the calculated cross-sectional area for the aluminum specimen used during tensile testing.
𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
=
𝑈𝑛𝑖𝑡 𝑅𝑒𝑎𝑑𝑖𝑛𝑔 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑉𝐷𝑇 𝑅𝑒𝑎𝑑𝑖𝑛𝑔
∑ 𝐴𝑐𝑡𝑢𝑎𝑙 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡𝑠
𝑁
=
⁄∑ 𝐿𝑉𝐷𝑇 𝑅𝑒𝑎𝑑𝑖𝑛𝑔𝑠
(𝑁
𝑁
= 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡𝑠)
Fig. : LVDT Calibration Technique. The actual deflections of the aluminum specimen in thousands of an
inch had to be determined via calculation of the actual displacement in thousands of an inch per LVDT
deflection reading. The average displacement and LVDT readings were used to calculate the correction
factor used in determination of deflections associated with each induced load.
LVDT
Reading
(a)
Actual Displacement (1x10-3
in.)
-18.156
-6.065
(b)
Avg. LVDT Reading
Avg. Displacement (1x10-6 in.)
Displacement/Unit Reading
(in.)
-18.075
-6.283
-18.057
-5.999
Tables 12a, 12b: Calibration of LVDT Displacement Readings
-18.096
6.115666667
0.000337957
Table 12a: The LVDT displacement measurements taken and the corresponding actual displacements in
thousandths of an inch. The average of each of these measurements was taken to determine the actual
deflection per LVDT displacement reading.
Table 12b: The average values calculated from the respective LVDT displacement readings and the
actual displacements during calibration, and the calculated Displacement/Unit Reading used to calculate
the true deflection from each LVDT displacement reading during tensile testing.
𝐴𝑟𝑒𝑎 = 𝑇ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑥 𝑊𝑖𝑑𝑡ℎ = 0.0325 𝑥 0.54 = 0.01755 𝑖𝑛2
𝜎𝑥𝑥 𝑖 =
𝐿𝑜𝑎𝑑
𝐴𝑟𝑒𝑎
𝐸𝑥.
𝜎𝑥𝑥 1 =
𝜀𝑥𝑥𝑖 =
𝐷𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛
𝐴𝑐𝑡𝑖𝑣𝑒 𝐿𝑒𝑛𝑔𝑡ℎ
𝐸𝑥.
𝜀𝑥𝑥2 =
19.8 𝑙𝑏.
𝑙𝑏.
=
1128.2
0.01755 𝑖𝑛2
𝑖𝑛2
2.70 𝑥 10−6 𝑖𝑛.
𝑖𝑛
= 6.01 𝑥 10−7
4.5 𝑖𝑛.
𝑖𝑛
Fig. : Calculation Techniques used in determination of Axial Stresses and Axial Strains
The steps used to calculate the cross-sectional area pertinent in axial stress calculations, the axial stress
for each induced load, and the corresponding axial strain for each induced load considering actual
deflections and the active testing length of the aluminum specimen. Examples include data given in
Table 13.
Load (lbs)
σxx (lb/in2)
19.8
69.4
119.9
170.4
220
270.7
320.3
371.2
419.8
470.3
519.8
570.5
1128.2
3954.4
6831.9
9709.4
12535.6
15424.5
18250.7
21151
23920.2
26797.7
29618.2
32507.1
Deflection
(in)
0
2.70E-06
1.69E-05
3.18E-05
4.80E-05
6.56E-05
8.42E-05
0.000104429
0.000125044
0.000148701
0.000173034
0.000211899
εxx (in/in)
Load (lbs)
σxx (lb/in2)
0
6.01E-07
3.76E-06
7.06E-06
1.07E-05
1.46E-05
1.87E-05
2.32E-05
2.78E-05
3.30E-05
3.85E-05
4.71E-05
721.4
719.5
719.3
724.3
722.5
732.5
737.2
741
746.3
715.3
758
742
41105.4
40997.2
40985.8
41270.7
41168.1
41737.9
42005.7
42222.2
42524.2
40757.8
43190.9
42279.2
Deflection
(in)
0.00175163
0.001821249
0.001955756
0.002087897
0.002217673
0.002366712
0.002495811
0.002652285
0.002823967
0.002999029
0.003157193
0.003335634
εxx (in/in)
0.000389
0.000405
0.000435
0.000464
0.000493
0.000526
0.000555
0.000589
0.000628
0.000666
0.000702
0.000741
620.2
670.5
683.4
693.3
689.4
688
680.8
706
704
710
703
711
710.3
709.7
711.5
713.5
712.9
718.6
719
35339
38205.1
38940.2
39504.3
39282.1
39202.3
38792
40227.9
40114
40455.8
40057
40512.8
40472.9
40438.7
40541.3
40655.3
40621.1
40945.9
40968.7
0.000281518
0.000674562
0.001001028
0.001037527
0.001074027
0.001107147
0.001143646
0.001193326
0.001230163
0.00128018
0.001309245
0.001347434
0.001383933
0.001417053
0.00145727
0.001498501
0.001608674
0.001647202
0.001672886
6.26E-05
0.00015
0.000222
0.000231
0.000239
0.000246
0.000254
0.000265
0.000273
0.000284
0.000291
0.000299
0.000308
0.000315
0.000324
0.000333
0.000357
0.000366
0.000372
751.1
756
766
762.4
766
779
780
782
786.1
785.1
793
795
794
791
790
800
801.2
804
804.3
801
42797.7
43076.9
43646.7
43441.6
43646.7
44387.5
44444.4
44558.4
44792
44735
45185.2
45299.1
45242.2
45071.2
45014.2
45584
45652.4
45812
45829.1
45641
0.00350326
0.003677308
0.003836486
0.004089616
0.004255214
0.004448188
0.00461818
0.004793242
0.004992974
0.005195748
0.005364051
0.005531001
0.005705387
0.005861523
0.006039964
0.006208943
0.006370824
0.006542844
0.006677689
0.006825038
0.000779
0.000817
0.000853
0.000909
0.000946
0.000988
0.001026
0.001065
0.00111
0.001155
0.001192
0.001229
0.001268
0.001303
0.001342
0.00138
0.001416
0.001454
0.001484
0.001517
Table 13: Aluminum Tensile Testing Data
The recorded loads and deflections using the LVDT readings and correction factor, and the
corresponding calculated axial stresses and axial strains induced from each loading.
Axial Stress σxx
Stress vs. Strain
50000.0
45000.0
40000.0
35000.0
30000.0
25000.0
20000.0
15000.0
10000.0
5000.0
0.0
0.00E+00 2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-03 1.20E-03 1.40E-03 1.60E-03
Axial Strain εxx
Stress vs. Strain
Fig. : Axial Stress vs. Axial Strain Plot
A graph plotting the calculated axial stresses from each induced load against the calculated axial strains
determined from each deflection during aluminum tensile testing.
Determination of Young’s Modulus and Poisson’s Ratio of Aluminum
a)
b)
Material
Diameter D(in)
Length
L (in)
Moment of initial I (in^4)
Aluminum
0.25
2
0.000383495
Material
Diameter D(in)
Length
L (in)
Moment of initial I (in^4)
Steel
0.25
2
0.000383495
Table 1: Testing Material Property Data
The specified materials and measurements for the diameters and lengths and the calculated Moment of
Inertial for each specimen.
𝑃𝑜𝑙𝑎𝑟 𝑀𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝐼𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝑓𝑜𝑟 𝑠𝑡𝑒𝑒𝑙 𝑎𝑛𝑑 𝐴𝑙𝑢𝑚𝑖𝑛𝑢𝑚
= 𝑆𝑝𝑒𝑐𝑖𝑚𝑒𝑛 𝑊𝑖𝑑𝑡ℎ 𝑥 𝑆𝑝𝑒𝑐𝑖𝑚𝑒𝑛 𝑇ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠
𝐼𝐴𝑙 = 𝜋 ∗
𝐼𝑆𝑡𝑒𝑒𝑙
0.254
32
= 0.000383495 in4
0.254
=𝜋∗
= 0.000383495 in4
32
Fig. : Calculation of Polar Moment of Inertial in Torsion Testing
The calculations for polar moment of inertial for both the aluminum and steel specimens used in Torsion
Testing. These calculations were needed to compute the Shear modulus.
Mt (lb*ft)
0.6
1.1
2.6
ϕ(radians)
0.0095
0.0144
0.035
Mt*L(lb*in^2)
14.4
26.4
62.4
φ*I(in^4)
3.6432E-06
5.52233E-06
1.34223E-05
Table 2: Part One: First Trial Aluminum Torsion Testing Results
The applied torques and twisted angles of a specimen was measured by the torsion testing machine and
Torsiometer, respectively. Multiply Torque Mt with the length of the specimen L and twisted angle φ
with polar moment of inertial I in order to get the value of shear modulus by using equation as shown
below.
𝐺=
𝑀𝑡 𝐿
φ 𝐼𝑃
Fig. : Calculation of the shear modulus in Aluminum Torsion Testing Number One. The formula used to
compute shear modulus by using the measured values torque, twisted angle, length and calculated
value polar moment of inertial.
Fig. : Graph plotting Mt*L against φ*I during Aluminum Torsion Test Number One
Graph plotting the value of Mt*L against the value of φ*I. The linear approximation function 𝑦 =
(5 𝑥 106 )𝑥 − 1.7779 𝑙𝑏 ∗ 𝑖𝑛2 is represented by the dashed line connecting the initial and final
measurements.
Mt (lb*ft)
0.5
1
2.6
ϕ(radians)
0.0065
0.0138
0.035
Mt*L(lb*in^2)
12
24
62.4
φ*I(in^4)
2.49272E-06
5.29223E-06
1.34223E-05
Table 3: Part One: Second Trial Aluminum Torsion Testing Results
The applied torques and twisted angles of a specimen was measured by the torsion testing machine and
Torsiometer, respectively. Multiply Torque Mt with the length of the specimen L and twisted angle φ
with polar moment of inertial I in order to get the value of shear modulus.
Fig. : Graph plotting Mt*L against φ*I during Aluminum Torsion Test Number Two
Graph plotting the value of Mt*L against the value of φ*I. The linear approximation function 𝑦 =
(5 𝑥 106 )𝑥 + 0.0249 𝑙𝑏 ∗ 𝑖𝑛2 is represented by the dashed line connecting the initial and final
measurements.
Mt (lb*ft)
Mt*L(lb*in^2)
ϕ(radians)
0.5
0.006
12
1.7
0.027
40.8
2.4
0.038
57.6
Table 4: Part Two: Third Trial Aluminum Torsion Testing Results
φ*I(in^4)
2.30097E-06
1.03544E-05
1.45728E-05
The applied torques and twisted angles of a specimen was measured by the torsion testing machine and
Torsiometer, respectively. Multiply Torque Mt with the length of the specimen L and twisted angle φ
with polar moment of inertial I in order to get the value of shear modulus.
Fig. : Graph plotting Mt*L against φ*I during Aluminum Torsion Test Number Three
Graph plotting the value of Mt*L against the value of φ*I. The linear approximation function 𝑦 =
(4 𝑥 106 )𝑥 + 3.2429 𝑙𝑏 ∗ 𝑖𝑛2 is represented by the dashed line connecting the initial and final
measurements.
2
2
Trial
Gexp (lb/in )
GAl(lb/in )
ε
εr
1
4.80E+06
4.00E+06
-8.05E+05 -2.01E-01
2
4.64E+06
4.00E+06
-6.36E+05 -1.59E-01
3
3.70E+06
4.00E+06
3.03E+05
7.57E-02
Table 5: Comparison of Experimentally determined values for Shear modulus and associated errors
and accuracies with known value of Shear Modulus for Aluminum
Table giving the values calculated from linear regression for the Shear Modulus of Aluminum, or the
slope of the Mt*L vs. φ*I plots and their absolute (ε) and relative (εr) errors with respect to the known
value for Shear Modulus of Aluminum of 𝐺𝐴𝑙 = 4𝑥106 𝑙𝑏/𝑖𝑛2 . Detailed calculations for Shear Modulus
determined from each trial set and the uncertainties in the results can be found in Appendices F1-F6.
Determination of Shear Modulus of Steel
Mt (lb*ft)
Mt*L(lb*in^2)
ϕ(radians)
2.6
0.0154
62.4
3.6
0.0217
86.4
4.4
0.026
105.6
Table 2: Part Two: First Trial Steel Torsion Testing Results
φ*I(in^4)
5.90583E-06
8.32185E-06
9.97088E-06
The applied torques and twisted angles of a specimen was measured by the torsion testing machine and
Torsiometer, respectively. Multiply Torque Mt with the length of the specimen L and twisted angle φ
with polar moment of inertial I in order to get the value of shear modulus by using equation as shown
below.
𝐺=
𝑀𝑡 𝐿
φ 𝐼𝑃
Fig. : Calculation of the shear modulus in Steel Torsion Testing Number One. The formula used to
compute shear modulus by using the measured values torque, twisted angle, length and calculated
value polar moment of inertial.
Fig. : Graph plotting Mt*L against φ*I during Steel Torsion Test Number One
Graph plotting the value of Mt*L against the value of φ*I. The linear approximation function 𝑦 =
(10 𝑥 106 )𝑥 − 0.5075 𝑙𝑏 ∗ 𝑖𝑛2 is represented by the dashed line connecting the initial and final
measurements.
Mt (lb*ft)
2.5
ϕ(radians)
0.015
Mt*L(lb*in^2)
60
φ*I(in^4)
5.75243E-06
3.6
0.0215
86.4
4.5
0.026
108
Table 3: Part One: Second Trial Steel Torsion Testing Results
8.24515E-06
9.97088E-06
The applied torques and twisted angles of a specimen was measured by the torsion testing machine and
Torsiometer, respectively. Multiply Torque Mt with the length of the specimen L and twisted angle φ
with polar moment of inertial I in order to get the value of shear modulus.
Fig. : Graph plotting Mt*L against φ*I during Steel Torsion Test Number Two
Graph plotting the value of Mt*L against the value of φ*I. The linear approximation function 𝑦 =
(10𝑥 106 )𝑥 − 5.6632 𝑙𝑏 ∗ 𝑖𝑛2 is represented by the dashed line connecting the initial and final
measurements.
Mt (lb*ft)
Mt*L(lb*in^2)
ϕ(radians)
2.4
0.0147
57.6
3.6
0.0213
86.4
4.4
0.0259
105.6
Table 4: Part One: Third Trial Aluminum Torsion Testing Results
φ*I(in^4)
5.63738E-06
8.16845E-06
9.93253E-06
The applied torques and twisted angles of a specimen was measured by the torsion testing machine and
Torsiometer, respectively. Multiply Torque Mt with the length of the specimen L and twisted angle φ
with polar moment of inertial I in order to get the value of shear modulus.
Fig. : Graph plotting Mt*L against φ*I during Aluminum Torsion Test Number Three
Graph plotting the value of Mt*L against the value of φ*I. The linear approximation function 𝑦 =
(10𝑥 106 )𝑥 − 5.3402 𝑙𝑏 ∗ 𝑖𝑛2 is represented by the dashed line connecting the initial and final
measurements.
Trial
Gexp (lb/in2)
GSteel(lb/in2)
1
1.06E+07
1.20E+07
2
1.13E+07
1.20E+07
3
1.12E+07
1.20E+07
Table 5: Comparison of Experimentally determined
ε
εr
1.42E+06
1.19E-01
6.77E+05
5.64E-02
8.10E+05
6.75E-02
values for Shear modulus and associated errors
and accuracies with known value of Shear Modulus for Steel
Table giving the values calculated from linear regression for the Shear Modulus of Aluminum, or the
slope of Mt*L vs. φ*I plots and their absolute (ε) and relative (εr) errors with respect to the known value
for Young’s Modulus of Aluminum of 𝐺𝑆𝑡𝑒𝑒𝑙 = 10𝑥106 𝑙𝑏/𝑖𝑛2 . Detailed calculations for Shear Modulus
determined from each trial set and the uncertainties in the results can be found in Appendices G1-A6.
Discussion
Determining Young’s Modulus “E”:
Young’s modulus “E” is defined as the constant of proportionality between the elongation of the
bar and the tensile force applied. This is best expressed as the case of the spring, F = kx. However, an
alternative form which incorporates the geometry of the beam is given in Eqn 1-4 (in the manual). This
takes into account using the quantity of stress, the fact that a given force over a small cross section will
produce a very large strain in response compared to a given force over a very large cross section, where
the strain response is far less. It also involves the length of the specimen – for two beams of the same
material such as a 1 m beam and a 10 cm beam – the 1 m beam may elongate much more than the 10
cm beam, however relative to the original size, they have the same percent elongation. This concept is
called “strain”.
“E” can be calculated from Eqn 1-6 (in the manual) and by obtaining test data using the data
obtained from the Tinius Olsen 1000 Universal Digital Testing Machine which grips the specimen and
applies tensile load to the tips. Using digital readouts and the knob tools to increase the load in
incremental steps until the elastic range of 320 lbs or 420 lbs for aluminum and steel is reached
respectively, strain values from the strain indicator can be plotted against the load over a variety of
points. (the load is represented as engineering stress by dividing by the original cross sectional area)
Looking at the trial data for aluminum under tensile loading in the elastic region in Table (the
one that summarizes Eexp vs Eal for trial 1,2,3) the experimental E’s obtained were 9.70x10^6 lb/in2,
9.68x10^6 lb/in2, and 9.69x10^6 lb/in2 . This is compared to a theoretical E of 1.00x10^7 lb/in2. The
accuracy obtained for the three trials were 97.04%, 96.83%, and 96.92%. There appears to be a very
strong correlation of experiment with theory for aluminum in measuring Young’s Modulus. However,
there are a few issues that explain the 3% inaccuracy which will be discussed in the case of the larger 6%
inaccuracy that is shown for steel.
The trial data for steel under the same conditions is given in Table (the one that summarizes
Eexp vs Est for trial 1,2,3) is as follows: the experimental E’s obtained were 2.79x10^7 lb/in2,
2.78x10^7 lb/in2, and 2.79x10^7 lb/in2 . This is compared to a theoretical E of 3.00x10^7 lb/in2. The
accuracy obtained for the three trials were 93.00%, 92.70%, and 93.00%. Firstly, in both the
experimental and theoretical values it appears that steel has a far larger E value, almost 3x that of
aluminum. This indicates for steel’s linear region is very steep and high stresses result in very small
strains. This can be seen in a more qualitative manner by looking at a sample trial stress-strain curve for
aluminum (Any trial graph stress-strain for aluminum) and comparing it’s elastic region with that of a
sample trial stress-strain curve for steel (Any trial graph stress-strain for steel). The linear region for
aluminum is far shallower in slope than for steel, indicating that aluminum is a more ductile metal.
Secondly, the accuracy for steel is a bit lower than for aluminum which raises some questions.
One possible reason such for inaccuracy in this experiment in general which apply both to the
aluminum and the steel but more so for the latter, is the usage of engineering stress. Using engineering
stress ignores the change in cross-sectional area as the specimen is being stressed, therefore ignoring
the true stress in the material. The engineering stress divides the load values using the same initial
cross-sectional area, meaning that it always reports lower stress values for a given strain than usual. In
the tensile case, uniaxial tensile loading results in true stresses always greater than engineering stress.
Since Young’s Modulus is calculated as the stress divided by the strain, if true stress is used, a higher and
more accurate Young’s Modulus can be calculated. This can be done by equating the initial volume with
the volume at a given loading and solving for the cross-section at that loading point given the strain
data.
Stress-strain curve allusion – refer and discuss how this one has a shallower linear region than steel,
what is the proportional limit, yield point, ultimate strength
Determining Poisson’s Ratio “v”:
Poisson’s ratio is derived from the fact that axial tensile forces will also induce compressive
strains in the other two directions, in addition to the expected axial strain. This operates under the
assumption that the material doesn’t change significantly in density, maintaining a fixed volume for its
given mass. It can be calculated from Eqn 1-7 (in the manual) and using data obtained from the Tinius
apparatus, using the strain indicator which picks up the axial and transverse strains. Note that the
transverse strain in the z-direction is not obtained, theoretically the axial strain and one of the two
transverse directions are needed for the Poisson ratio.
Looking at the trial data for aluminum under tensile loading in the elastic region in Table (the
one that summarizes POISSON Vexp vs Val for trial 1,2,3) the experimental v’s obtained were
0.32820763, 0.32339856, and 0.32111065. This is compared to a theoretical v of 0.333. The accuracy
obtained for the three trials were 98.56%, 97.12%, and 96.43%. The correlation between experimental
and theoretical values is very strong in the case of aluminum.
For steel under the same conditions in Table (the one that summarizes POISSON Vexp vs Vst
for trial 1,2,3) ) the experimental v’s obtained were 0.27316959, 0.26844712, and 0.27090438. This is
compared to a theoretical v of 0.333. The accuracy obtained for the three trials were 93.90%, 92.20%,
and 93.10%. Here, similar to the issue with Young’s modulus, steel appears to produce more error. One
explanation is that for given stresses, in reality, higher strains are sustained – this is in fact observed.
This kind of behavior is the kind seen in the plastic region where for small changes in stresses, large
changes in the elongation of the material can be observed. This is less the case for aluminum which
experiences loading over a large range of strain values, meaning that the elastic region is fairly large and
not influenced by the plastic region. However, for steel which enters the plastic region over a small
range of strain values, the influence of plastic behavior may have some effect in reality on the elastic
region, which results in a lower experimental Young’s modulus.
Determining Torsional Shear Modulus “G”:
Shear modulus “G” from Eqn 1-8 (in the manual) relates the shearing stress and the shearing
strain which are connected by a constant of proportionality. Looking at Table (the one that summarizes
torsional G data for aluminum for all trials) and Table (the one that summarizes torsional G data for
steel for all trials) the experimental G’s obtained for aluminum were 4.80x10^6 lb/in2, 4.64x10^6 lb/in2,
and 3.70x10^6 lb/in2 . This is compared to a theoretical E of 4.00x10^6 lb/in2 while the experimental G’s
obtained for steel were 1.06x10^7 lb/in2, 1.13x10^6 lb/in2, and 1.12x10^6 lb/in2. This is compared to a
theoretical G of 1.20x10^7 lb/in2.
It should be pointed out that the number of data points is very low given the lack of accuracy in
the torsiometer for very small applied torques. In fact, it is only good for one data point at the elastic
limit of 2.5 lb-ft or 4.5 lb-ft for aluminum and steel. However, to be safe 3 trials were done for this one
data point. Even then, for aluminum there is some major inconsistency as for Trial 1 and 2, the G values
obtained are actually larger than the theoretical while the third is more sensible. Such behavior does not
occur at all for the steel nor anywhere else. Thus variance from the first two trials must be discarded and
subsequent analysis will be using the third trial for aluminum only.
Ignoring that anomaly and utilizing trial 3 for aluminum, still there is far more variance for
aluminum, far higher variance than what we see in steel, where the experimental values come much
closer to the theoretical value. Assuming trial 3 is correct, it appears it shears at a lower shearing stress
or at a higher shearing strain. There is an assumption that if the angle of twist is small, the circular cross
sections of the shaft remain circular during the twist and their diameters and distances between them do
not change. It appears this assumption may not hold very well for softer, more twistable metals which
experience higher shearing strain and therefore higher twist angles which make this assumption invalid
compared to stiffer metals that experience smaller changes in strain and twist angle, for which it may
hold better. This is curious because in tensile loading, it appeared that the stiffer steel produced larger
inaccuracies in Young’s Modulus G due to influence of the plastic region on the sharp elastic region
while in the torsional testing, stiffer metals are preferred because they produce smaller angles of twist
that make the assumption for elastic torsional shear theory work
Error Analysis
In the determination of Young's Modulus linear regression of the Stress-Strain measurements taken
rendered expressions of (1) y = 82.44 + (9.70 x 106)x, (2) y = 17.1+ (9.68 x 106)x, and (3) y = 12.36 + (9.69
x 106)x. These linear expressions take the form y = a+bxi with ucertainties in the constants a and b of ua
= 12 and ub = 37.6x103 in (1), ua =6.83 and ub = 20.3x103 in (2), and ua=10.349 and ub = 31.9x103 in (3).
We consider the constant b of the linear expressions to represent Young's Modulus, which in all cases is
very similar to the known value of 10 x 106 psi [1]. The uncertainties associated with these values while
seemingly high, must be considered with the magnitudes of the quantities being measured. In this case,
we are considering stresses on the order of 106 psi and errors on the order of 103 while not favorable,
are reasonable. Similarly, analysis of the stress-strain measurements for steel rendered expression of (4)
37.105 + (27.9x106)x, (5) 28.104 + (27.8x106)x, and (6) 52.701 + (27x106)x. The associated uncertainties
in the variables comprising these equations were found to be ua = 23.727 and ub=132,281.42 in (4), ua =
13.208 and ub=158,755.86 in (5), and ua = 29.087 and ub = 158,755.86 in (6). Again, the values calculated
for the linear regression constant b were considered to represent the Young's Modulus for Steel and
possessed relative accuracies nearing unity when considering the known value of 30 x 106 psi for steel
[1]. Again, while the uncertainties ub may initially seem alarming, we consider the magnitudes of the
stress measurements taken during testing.
Poisson's Ratio for both materials was also determined via linear regression of transverse and
axial strain measurements taken during testing. For Aluminum, this rendered linear expressions of the
form (7) -7.325x10^-6 - 0.3282x, (8) -2.52x10-7 - 0.3234, and (9) -2.67x10-7 - 0.3211x. The uncertainties
associated with the constants a and b comprising these linear expressions were found to be ua=6.94x107 and ub = 2.11x10-3 in (7), ua = 1.19x10-7 and ub=3.53x10-4 in (8), and ua=1.6x10-7 and ub = 4.94x10-4 in
(9). The absolute value of the constant b representing the slope of each of these expressions is
considered to be Poisson's Ratio, with relative accuracies of our experimentally determined results
nearing unity with respect to the known value of Poisson's ratio for Aluminum of 0.33 [1]. The small
uncertainties associated with both linear regression constants are further verification in the validity of
the method used to determine Poison's ratio. Similarly, linear regression of the lateral-axial strain
relationship for steel rendered equations of the form (10) 2.11x10-6 - 0.2732x, (11) 1.86x10-6 -.2685x,
and (12) 9.19x10^-7 -0.2709x.Again, the absolute value of the constant b comprising the slopes of these
equations can be interpreted as the Poisson's ratio for the steel. We can again consider the accuracy of
these results with respect to the known value of 0.291 for steel, with results slightly less accurate than
those found for aluminum averaging to approximately 0.93. The uncertainties associated with the
constants comprising the linear expressions were found to be ua=2.95x10-7 and ub=1.64x10-3 in (10), ua
= 2.5x10-7 and ub = 1.36x10-3 in (11), and ua = 9.1x10-7 and ub = 4.97x10-3 in (12). We can again
consider both the magnitude of the measurements being taken, with strains of fractional order, as well
as the accuracy of the linear regression constants with respect to the known value for Poisson's Ratio for
steel of carbon-based steel of 0.292 [1].
Linear Regression of the Axial Stress-Axial Strain data obtained from the second aluminum
tensile test rendered a linear approximation expression of the form y = (1.6 x 107)x + 2.8 x 104 psi, with
uncertainties of ua = 2.05 x 105 in the intercept and ub = 2.7 x 108 in the slope. While these uncertainties
are considerably large we must take the relative magnitudes of the quantities used in the regression,
with computed stresses ranging from one kpsi to nearly fifty kpsi against deflections ranging from the
micro-strain scale to mill-strain scale. Similarly, we must take careful consideration to the validity of the
linear approximation, as it is known that the stress-strain relationship is linearly only within the elastic
region of the stress strain plot. These trials sought to not only observe the behavior of the material in
the elastic region, but also those associated with the ultimate tensile strength and fracture point within
the plastic region of the stress-strain plot. These considerations in mind, only a small number of
measurements of the population were taken within the elastic range, with substantially more
measurements being taken in the plastic region where linear approximations are not valid. While the
values obtained set a linear precedent that approximates well within the elastic region, it is the
subsequent measurements taken in the plastic region which give rise to error in the complete linear
approximation.
In the determination of Shear Modulus of Aluminum via linear regression, we get expressions of (1) y = 1.77786 + (4.80 x 106)x, (2) y = 0.02490+ (4.64 x 106)x, and (3) y = 3.2429 + (3.70 x 106)x. These linear
expressions take the form y = a+bxi with ucertainties in the constants a and b of ua = 2.5157 and ub =
291062 in (1), ua =0.75257 and ub = 89025 in (2), and ua=1.0068 and ub = 102517 (3). We consider the
constant b of the linear expressions to represent Shear Modulus, which in all cases is very similar to the
known value of 4 x 106 psi [1]. The uncertainties associated with these values while seemingly high, must
be considered with the magnitudes of the quantities being measured. In this case, we are considering
stresses on the order of 106 psi and errors on the order of 103 while not favorable, are reasonable.
Similarly, analysis of the Torsion Testing measurements for steel rendered expression of (4) -5.66321 +
(11.2x106)x, (5) -0.5074 + (10.58x106)x, and (6) -5.23017 + (11.19x106)x. The associated uncertainties in
the variables comprising these equations were found to be ua = 4.3616and ub=531640 in (4), ua = 3.8743
and ub=470245 in (5), and ua = 1.109 and ub = 136805 in (6). Again, the values calculated for the linear
regression constant b were considered to represent the Young's Modulus for Steel and possessed
relative accuracies nearing unity when considering the known value of 10 x 106 psi for steel [1]. Again,
while the uncertainties ub may initially seem alarming, we consider the magnitudes of the stress
measurements taken during testing.
Conclusion
This experiment involved tensile loading in the elastic region for steel and aluminum, tensile
loading until fracture for steel to map it’s stress strain curve, and torsional loads. All three are used to
calculate three major material properties – Young’s Modulus “E”, Poisson’s Ratio “V”, and the Torsional
Shear Modulus “G”.
A major experimental flaw is in the torsional testing which requires a more accurate apparatus
in future experiments to provide a range of values over several angles of twist in order to see if the
assumption of small twist angles truly skews the data as for those cases the use of an elastic torsional
shear modulus does not make sense since it will then exhibit plastic behavior. This is supported in this
experiment, where only one reasonable trial existed to evaluate aluminum against steel due to few data
points being obtainable – steel appeared to conform well, which says something about stiff metals and
small angle of twist allowing for a more accurate G. However, softer metals like the aluminum appear to
shear at far lower stresses or far higher strains which means significant changes in angles and the
assumption used for the elastic shear modulus theory becomes more invalid as we saw for aluminum.
For the tensile loading tests to verify results, it might be useful to compare results of the Poisson
ratio calculated from both transverse directions not just from the y-direction but also from the z-
direction. Although theoretically they should be equivalent, it may actually account for some of the
inaccuracy by contributing to the strain in the experiment.
Lastly, for the tensile loading until fracture testing to obtain the stress-strain curve, there is a
serious issue in that loading in the plastic region often doesn’t stay constant for a reading once the load
is halted. In fact, the load value immediately begins to drop which might be due to softening of the
metal as it is being stressed, “giving way” as the load is kept for a certain amount of time. As a result,
there is a great uncertainty from the range of values that are associated with a given strain and it makes
it difficult to produce an accurate stress-strain curve, where the proportional limits and yield points may
not truly be as calculated. This was mitigated as much as possible by systematically taking the
immediate reading but it may actually overshoot the true value in some way.
Reference
[1] Budynas, Richard G (2010-01-29). Shigley's Mechanical Engineering Design (Mcgraw-Hill Series in
Mechanical Engineering) (Page 1007). Science Engineering & Math. Kindle Edition.
Beer, Ferdinand , E. Russell Johnston, Mechanics Of Materials. (McGraw-Hill in Mechanical Engineering)
(Page 144-147) cience Engineering & Math. Kindle Edition.
Appendices
APPENDIX A
Calculation of Young’s Modulus for Aluminum via Linear Regression
-6
εxx (in/in)
2.80E-05
1.11E-04
1.95E-04
2.81E-04
3.67E-04
4.53E-04
5.39E-04
2
σxx (lb/in )
330.578512
1161.98347
1983.47107
2826.44628
3646.28099
4462.80992
5302.47934
xixi
7.84E-10
1.22E-08
3.80E-08
7.87E-08
1.34E-07
2.05E-07
2.91E-07
xiyi
9.26E-03
1.28E-01
3.87E-01
7.93E-01
1.34E+00
2.02E+00
2.86E+00
a+bxi
354.2
1154.8
1974.8
2804.5
3639.1
4473.7
5313.1
(A1)
[y-(a+bx)]2
556.24607
52.014698
75.395868
481.2452
51.82356
117.62205
112.38697
N
Sx
Sy
Sxx
Sxy
a
b
θ
ua
ub
(A2)
7
1.97E-03
19714.05
7.59E-07
7.53E+00
82.44189
9704339
1446.734
12.41106
37683.53
Tables A1, A2; Aluminum Tensile Test One Experimental Data used in determination of Young’s
Modulus via Linear Regression Method and Calculated Linear Regression Variables
(1) Table giving the measured axial strains and calculated axial stresses for each induced load, as
well as the products used in Linear Regression Calculations
(2) The variables N, Sx, Sy, Sxx, and Sxy used to calculate the constants a = 82.44189 and b =
𝑙𝑏
9,704,3392709 and their uncertainties via Linear Regression. The value 𝑏 = 9.70𝑥106 𝑖𝑛2
represents the Young’s Modulus of the Aluminum as per trial one.
-6
εxx (in/in)
3.65E-05
1.14E-04
2.01E-04
2.88E-04
3.74E-04
2
σxx (lb/in )
360.330579
1127.27273
1966.94215
2793.38843
3638.01653
xixi
1.33E-09
1.29E-08
4.02E-08
8.27E-08
1.40E-07
xiyi
1.32E-02
1.28E-01
3.94E-01
8.03E-01
1.36E+00
a+bxi
370.5
1116.1
1958.5
2801.0
3638.6
(A3)
[y-(a+bx)]2
102.4
125.6
71.1
57.3
0.3
N
Sx
Sy
Sxx
Sxy
a
(A4)
7
2.02E-03
19679.34
7.89E-07
7.67E+00
17.00948
4.62E-04
5.47E-04
4479.33884
5314.04959
2.13E-07
2.99E-07
2.07E+00
2.90E+00
4485.9
5308.9
42.4
26.2
b
9683303
θ
425.3265
ua 6.825904
ub 20336.82
Tables A3, A4; Aluminum Tensile Test Two Experimental Data used in determination of Young’s
Modulus via Linear Regression Method and Calculated Linear Regression Variables
(3) Table giving the measured axial strains and calculated axial stresses for each induced load, as
well as the products used in Linear Regression Calculations
(4) The variables N, Sx, Sy, Sxx, and Sxy used to calculate the constants a = 17.00948 and b =
𝑙𝑏
9,683,3032709 and their uncertainties via Linear Regression. The value 𝑏 = 9.68𝑥106 𝑖𝑛2
represents the Young’s Modulus of the Aluminum as per trial two.
εxx (in/in)
-6
3.35E-05
1.20E-04
2.00E-04
2.70E-04
3.58E-04
4.42E-04
5.31E-04
2
σxx (lb/in )
328.925619
8
1158.67768
6
1971.90082
6
2634.71074
4
3469.42148
8
4302.47933
9
5148.76033
1
xixi
xiyi
a+bxi
(A5)
[y-(a+bx)]2
N
(A6)
7
1.12E-09
1.10E-02
337.0
65.8
Sx
1.95E-03
1.43E-08
1.38E-01
1170.5
140.8
Sy
19014.876
4.00E-08
3.94E-01
1950.7
447.5
Sxx
7.33E-07
7.26E-08
7.10E-01
2624.3
107.6
Sxy
7.13E+00
1.28E-07
1.24E+00
3477.2
61.0
a
1.95E-07
1.90E+00
4296.2
39.4
b
2.82E-07
2.73E+00
5158.8
100.4
θ
12.357102
9691948.9
6
962.56752
9
10.348645
6
31976.576
6
ua
ub
Tables A5, A6: Aluminum Tensile Test Three Experimental Data used in determination of Young’s
Modulus via Linear Regression Method and Calculated Linear Regression Variables
(a) Table giving the measured axial strains and calculated axial stresses for each induced load, as
well as the products used in Linear Regression Calculations.
(b) The variables N, Sx, Sy, Sxx, and Sxy used to calculate the constants a = 12.3571 and b =
𝑙𝑏
9,691,9492709 and their uncertainties via Linear Regression. The value 𝑏 = 9.69𝑥106 𝑖𝑛2
represents the Young’s Modulus of the Aluminum as per trial three.
APPENDIX B
Calculation of Poisson’s Ratio for Aluminum via Linear Regression
(B2)
εxx (in/in)
εyy (in/in)
xixi
xiyi
a+bxi
N
7
2.80E-05
-1.50E-05
7.84E-10 -4.20E-10 -1.65E-05
Sx
1.97E-03
1.11E-04
-4.45E-05
1.22E-08 -4.92E-09 -4.36E-05
Sy
-0.0007
1.95E-04
-7.20E-05
3.80E-08 -1.40E-08 -7.13E-05
Sxx
7.59E-07
2.81E-04
-1.00E-04
7.87E-08 -2.81E-08 -9.94E-05
Sxy -2.64E-07
3.67E-04
-1.28E-04
1.34E-07 -4.67E-08 -1.28E-04
a
-7.3E-06
4.53E-04
-1.56E-04
2.05E-07 -7.06E-08 -1.56E-04
b
-0.32821
5.39E-04
-1.84E-04
2.91E-07 -9.89E-08 -1.84E-04
θ
4.52E-12
ua
6.94E-07
ub
0.002107
Tables B1, B2: Aluminum Tensile Test One Experimental Data used in determination of Poisson’s Ratio
-6
-6
(B1)
[y-(a+bx)]2
2.22E-12
8.701E-13
4.887E-13
4.059E-13
7.915E-15
3.422E-14
4.973E-13
via Linear Regression Method and Calculated Linear Regression Variables
(B1) Table giving the measured axial and lateral strains for each induced load, as well as the products
used in Linear Regression Calculations.
(B2) The variables N, Sx, Sy, Sxx, and Sxy used to calculate the constants 𝑎 = −7.3 𝑥 10−6 and 𝑏 =
−0.32821 2709 and their uncertainties via Linear Regression. The value 𝑏 = −0.32821 is used to
𝜀
compute Poisson’s Ratio via the relation 𝜐 = − 𝜀𝑦𝑦 = −𝑏.
𝑥𝑥
εxx (in/in)
-6
-6
εyy (in/in)
3.65E-05
-1.15E-05
1.14E-04
-3.65E-05
2.01E-04
-6.45E-05
2.88E-04
-9.30E-05
3.74E-04
-1.21E-04
4.62E-04
5.47E-04
-1.49E-04
-1.77E-04
xixi
xiyi
a+bxi
1.33E-09
-4.20E-10
-1.16E-05
1.29E-08
-4.14E-09
-3.65E-05
4.02E-08
-1.29E-08
-6.46E-05
8.27E-08
-2.67E-08
-9.27E-05
1.40E-07
2.13E-07
2.99E-07
-4.51E-08
-6.88E-08
-9.65E-08
-1.21E-04
-1.49E-04
-1.76E-04
(B3)
[y-(a+bx)]2
2.71441E15
2.12521E15
8.04609E15
7.53502E14
3.98402E14
8.41E-18
1.9321E-16
N
(B4)
7
Sx
2.02E-03
Sy
-0.000652
Sxx
7.89E-07
Sxy
-2.55E-07
a
2.52E-07
b
-0.323399
θ
1.28E-13
ua
1.19E-07
ub
0.000353
Tables B3, B4: Aluminum Tensile Test Two Experimental Data used in determination of Poisson’s Ratio
via Linear Regression Method and Calculated Linear Regression Variables
(B3) Table giving the measured axial and lateral strains for each induced load, as well as the products
used in Linear Regression Calculations.
(B4) The variables N, Sx, Sy, Sxx, and Sxy used to calculate the constants 𝑎 = 2.52 𝑥 10−7 and 𝑏 =
−0.3234 2709 and their uncertainties via Linear Regression. The value 𝑏 = −0.3234 is used to
𝜀𝑦𝑦
compute Poisson’s Ratio via the relation 𝜐 = − 𝜀
εxx (in/in)-6
εyy (in/in)-6
3.35E-05
-1.10E-05
1.20E-04
-3.85E-05
2.00E-04
-6.45E-05
𝑥𝑥
= −𝑏.
xixi
xiyi
a+bxi
1.12E-09
-3.69E-10
-1.10E-05
1.43E-08
4.00E-08
-4.60E-09
-1.29E-08
-3.86E-05
-6.45E-05
(a)
[y-(a+bx)]2
7.39024E16
2.03476E14
6.4E-17
N
(b)
7
Sx
1.95E-03
Sy
Sxx
-0.000629
7.33E-07
2.70E-04
-8.70E-05
3.58E-04
-1.15E-04
4.42E-04
-1.43E-04
5.31E-04
-1.71E-04
7.26E-08
-2.34E-08
-8.68E-05
1.28E-07
-4.11E-08
-1.15E-04
1.95E-07
-6.30E-08
-1.42E-04
2.82E-07
-9.05E-08
-1.71E-04
3.64256E14
4.46558E15
8.96284E14
7.80699E14
Sxy
-2.36E-07
a
-2.673E-07
b
-0.3211107
θ
2.2974E-13
ua
1.5988E-07
ub 0.00049401
Tables B5, B6: Aluminum Tensile Test Three Experimental Data used in determination of Poisson’s
Ratio via Linear Regression Method and Calculated Linear Regression Variables
(B5) Table giving the measured axial and lateral strains for each induced load, as well as the products
used in Linear Regression Calculations.
(B6) The variables N, Sx, Sy, Sxx, and Sxy used to calculate the constants 𝑎 = −2.67𝑥 10−7 and 𝑏 =
−0.3211 2709 and their uncertainties via Linear Regression. The value 𝑏 = −0.3211 is used to
𝜀𝑦𝑦
compute Poisson’s Ratio via the relation 𝜐 = − 𝜀
𝑥𝑥
= −𝑏.
APPENDIX C
Calculation of Young’s Modulus for Steel via Linear Regression
εxx (in/in)-6
1.05E-05
3.80E-05
6.65E-05
9.65E-05
1.27E-04
1.53E-04
1.83E-04
2.15E-04
2.42E-04
2.69E-04
2.98E-04
2
σxx (lb/in )
286.108999
1067.59529
1922.65248
2735.20203
3570.64031
4365.20588
5182.66016
6006.65408
6832.2829
7499.3256
8305.33552
xixi
1.10E-10
1.44E-09
4.42E-09
9.31E-09
1.60E-08
2.34E-08
3.35E-08
4.62E-08
5.86E-08
7.24E-08
8.85E-08
xiyi
3.00E-03
4.06E-02
1.28E-01
2.64E-01
4.52E-01
6.68E-01
9.48E-01
1.29E+00
1.65E+00
2.02E+00
2.47E+00
a+bxi
330.1
1097.4
1892.7
2729.8
3566.8
4306.3
5143.4
6036.3
6789.7
7543.0
8338.3
(C1)
[y-(a+bx)]2
1934.1
889.7
899.5
29.7
14.4
3472.4
1543.6
877.2
1817.2
1910.8
1085.1
N
Sx
Sy
Sxx
Sxy
a
b
θ
ua
ub
(C2)
11
1.70E-03
47773.66326
3.54E-07
9.94E+00
37.10474271
27903099.32
14473.5
23.726956
132291.4213
Tables C1, C2: Steel Tensile Test One Experimental Data used in determination of Young’s Modulus via
Linear Regression Method and Calculated Linear Regression Variables
(C1) Table giving the measured axial strains and calculated axial stresses for each induced load, as well
as the products used in Linear Regression Calculations
(C2) The variables N, Sx, Sy, Sxx, and Sxy used to calculate the constants a = 37.10474271 and b =
𝑙𝑏
27,903,099.32 2709 and their uncertainties via Linear Regression. The value 𝑏 = 27.9 𝑥 106 𝑖𝑛2
represents the Young’s Modulus of the Steel as per trial one.
(a)
(b)
2
εxx (in/in)
σxx (lb/in )
xixi
xiyi
a+bxi [y-(a+bx)]
N
11
1.20E-05
333.521348 1.44E-10 4.00E-03 362.0
808.6
Sx
1.74E-03
4.10E-05
1154.24545 1.68E-09 4.73E-02 1168.8
210.9
Sy 48759.51312
7.00E-05
1978.23937 4.90E-09 1.38E-01 1975.6
7.1
Sxx
3.71E-07
9.90E-05
2787.51911 9.80E-09 2.76E-01 2782.4
26.3
Sxy
1.04E+01
1.28E-04
3611.51303 1.63E-08 4.60E-01 3575.3
1312.2
a 28.10415349
1.59E-04
4443.68149 2.51E-08 7.04E-01 4437.7
35.3
b 27821055.09
1.88E-04
5259.50086 3.52E-08 9.86E-01 5244.6
223.5
θ
4439.3
2.17E-04
6076.95515 4.71E-08 1.32E+00 6065.3
136.5
ua 13.20844617
2.46E-04
6876.42544 6.03E-08 1.69E+00 6858.2
333.1
ub 71913.05122
2.77E-04
7708.5939 7.67E-08 2.14E+00 7734.5
673.0
3.07E-04
8529.318
9.39E-08 2.61E+00 8555.3
672.9
Tables C3, C4: Steel Tensile Test Two Experimental Data used in determination of Young’s Modulus via
-6
2
Linear Regression Method and Calculated Linear Regression Variables
(C3) Table giving the measured axial strains and calculated axial stresses for each induced load, as well
as the products used in Linear Regression Calculations
(C4) The variables N, Sx, Sy, Sxx, and Sxy used to calculate the constants a =28.10415349 and b =
𝑙𝑏
27,821,055.09 2709 and their uncertainties via Linear Regression. The value 𝑏 = 28.1 𝑥 106 𝑖𝑛2
represents the Young’s Modulus of the Steel as per trial two.
εxx (in/in)
-6
1.10E-05
3.95E-05
6.90E-05
9.80E-05
1.28E-04
1.56E-04
1.86E-04
2.18E-04
2.48E-04
2.77E-04
3.05E-04
2
σxx (lb/in )
328.616622
1159.15017
1965.1601
2805.5031
3611.51303
4443.68149
5259.50086
6135.81185
6964.7105
7759.27606
8534.22272
xixi
1.21E-10
1.56E-09
4.76E-09
9.60E-09
1.63E-08
2.43E-08
3.46E-08
4.73E-08
6.13E-08
7.67E-08
9.27E-08
xiyi
3.61E-03
4.58E-02
1.36E-01
2.75E-01
4.60E-01
6.93E-01
9.78E-01
1.33E+00
1.72E+00
2.15E+00
2.60E+00
a+bxi
334.1
1127.0
1947.8
2754.6
3575.3
4368.2
5202.8
6079.2
6913.8
7734.5
8499.6
(a)
[y-(a+bx)]2
30.5
1031.3
302.9
2594.4
1312.2
5699.2
3212.7
3206.8
2590.3
612.1
1197.7
N
Sx
Sy
Sxx
Sxy
a
b
θ
ua
ub
(b)
11
1.73E-03
48967.14651
3.69E-07
1.04E+01
52.70075345
27913145.79
21789.9
29.08650092
158755.8637
Tables C5, C6: Steel Tensile Test Three Experimental Data used in determination of Young’s Modulus
via Linear Regression Method and Calculated Linear Regression Variables
(C5) Table giving the measured axial strains and calculated axial stresses for each induced load, as well
as the products used in Linear Regression Calculations
(C6) The variables N, Sx, Sy, Sxx, and Sxy used to calculate the constants a =52.70075345 and b =
𝑙𝑏
27,913,145.79 2709 and their uncertainties via Linear Regression. The value 𝑏 = 27.9 𝑥 106 𝑖𝑛2
represents the Young’s Modulus of the Steel as per trial three.
APPENDIX D
Calculation of Poisson’s Ratio for Steel via Linear Regression
εxx (in/in)
-6
-6
εyy (in/in)
1.05E-05
-1.50E-06
3.80E-05
-8.00E-06
6.65E-05
-1.60E-05
9.65E-05
-2.40E-05
xixi
xiyi
a+bxi
1.10E-10
-1.58E-11
-7.59E-07
1.44E-09
-3.04E-10
-8.27E-06
4.42E-09
9.31E-09
-1.06E-09
-2.32E-09
-1.61E-05
-2.43E-05
(D1)
[y-(a+bx)]2
5.49525E13
7.33675E14
3.15819E15
6.31444E-
N
(D2)
11
Sx
1.70E-03
Sy
-0.0004405
Sxx
Sxy
3.54E-07
-9.31E-08
14
1.99249E1.27E-04
-3.20E-05
1.60E-08 -4.05E-09 -3.24E-05
13
a
2.10958E-06
9.89938E1.53E-04
-4.00E-05
2.34E-08 -6.12E-09 -3.97E-05
14
b
-0.27316959
1.83E-04
-4.80E-05 3.35E-08 -8.78E-09 -4.79E-05 1.4291E-14
θ
2.23597E-12
3.86737E2.15E-04
-5.60E-05
4.62E-08 -1.20E-08 -5.66E-05
13
ua
2.94909E-07
6.44764E2.42E-04
-6.40E-05
5.86E-08 -1.55E-08 -6.40E-05
18
ub 0.001644284
1.39159E2.69E-04
-7.10E-05
7.24E-08 -1.91E-08 -7.14E-05
13
7.08336E2.98E-04
-8.00E-05
8.85E-08 -2.38E-08 -7.92E-05
13
Tables D1, D2: Steel Tensile Test One Experimental Data used in determination of Poisson’s Ratio via
Linear Regression Method and Calculated Linear Regression Variables
(D1) Table giving the measured axial and lateral strains for each induced load, as well as the products
used in Linear Regression Calculations.
(D2) The variables N, Sx, Sy, Sxx, and Sxy used to calculate the constants 𝑎 = 2.109 𝑥 10−6 and 𝑏 =
−0.2732 2709 and their uncertainties via Linear Regression. The value 𝑏 = −0.2732 is used to
compute Poisson’s Ratio via the relation 𝜐 = −
𝜀𝑦𝑦
𝜀𝑥𝑥
= −𝑏.
(D3)
εxx (in/in)
-
(D4)
-
εyy (in/in)
6
6
1.20E-05
-2.00E-06
4.10E-05
-9.00E-06
7.00E-05
9.90E-05
-1.65E-05
-2.45E-05
xixi
xiyi
a+bxi
1.44E-10
-2.40E-11
-1.36E-06
1.68E-09
4.90E-09
9.80E-09
-3.69E-10
-1.16E-09
-2.43E-09
-9.14E-06
-1.69E-05
-2.47E-05
[y-(a+bx)]2
4.12594E13
2.03439E14
1.8284E-13
4.51838E-
N
11
Sx
1.74E-03
Sy
Sxx
Sxy
-4.47E-04
3.71E-07
-9.64E-08
14
1.31993E1.28E-04
-3.20E-05
1.63E-08 -4.08E-09 -3.24E-05
13
a
1.8637E-06
3.42874E1.59E-04
-4.05E-05
2.51E-08 -6.42E-09 -4.07E-05
14
b 0.268447118
2.80757E1.88E-04
-4.90E-05
3.52E-08 -9.19E-09 -4.85E-05
13
θ
1.58512E-12
1.22489E2.17E-04
-5.65E-05
4.71E-08 -1.23E-08 -5.64E-05
14
ua
2.49589E-07
2.11537E2.46E-04
-6.45E-05
6.03E-08 -1.58E-08 -6.40E-05
13
ub 0.001358879
2.46167E2.77E-04
-7.20E-05
7.67E-08 -1.99E-08 -7.25E-05
13
7.16693E3.07E-04
-8.05E-05
9.39E-08 -2.47E-08 -8.04E-05
15
Tables D3, D4: Steel Tensile Test Two Experimental Data used in determination of Poisson’s Ratio via
Linear Regression Method and Calculated Linear Regression Variables
(D3) Table giving the measured axial and lateral strains for each induced load, as well as the products
used in Linear Regression Calculations.
(D4) The variables N, Sx, Sy, Sxx, and Sxy used to calculated the constants 𝑎 = 1.864 𝑥 10−6 and 𝑏 =
−0.2684 2709 and their uncertainties via Linear Regression. The value 𝑏 = −0.2684 is used to
𝜀
compute Poisson’s Ratio via the relation 𝜐 = − 𝜀𝑦𝑦 = −𝑏.
𝑥𝑥
-6
-6
εxx (in/in)
1.10E-05
3.95E-05
εyy (in/in)
-2.50E-06
-1.00E-05
6.90E-05
-1.75E-05
9.80E-05
-2.55E-05
1.28E-04
-3.30E-05
1.56E-04
-4.10E-05
1.86E-04
-4.95E-05
2.18E-04
-5.85E-05
xixi
1.21E-10
1.56E-09
xiyi
-2.75E-11
-3.95E-10
a+bxi
-1.09E-06
-8.74E-06
4.76E-09
-1.21E-09
-1.67E-05
9.60E-09
-2.50E-09
-2.44E-05
1.63E-08
-4.21E-09
-3.24E-05
2.43E-08
-6.40E-09
-4.00E-05
3.46E-08
4.73E-08
-9.21E-09
-1.27E-08
-4.81E-05
-5.65E-05
(D5)
[y-(a+bx)]2
1.9903E-12
1.5877E-12
7.07027E13
1.11489E12
4.05377E13
9.72096E13
2.05216E12
3.90636E-
N
Sx
Sy
(D6)
11
1.73E-03
-4.60E-04
Sxx
3.69E-07
Sxy
-9.84E-08
a
b
9.1934E-07
0.270904379
θ
ua
2.1325E-11
9.0993E-07
12
3.69807E2.48E-04
-6.65E-05
6.13E-08 -1.65E-08 -6.46E-05
12
ub 0.004966452
2.26156E2.77E-04
-7.40E-05
7.67E-08 -2.05E-08 -7.25E-05
12
2.62943E3.05E-04
-8.15E-05
9.27E-08 -2.48E-08 -7.99E-05
12
Tables D5, D6: Steel Tensile Test Three Experimental Data used in determination of Poisson’s Ratio via
Linear Regression Method and Calculated Linear Regression Variables
(D5) Table giving the measured axial and lateral strains for each induced load, as well as the products
used in Linear Regression Calculations.
(D6) The variables N, Sx, Sy, Sxx, and Sxy used to calculate the constants 𝑎 = 9.193 𝑥 10−7 and 𝑏 =
−0.2709 and their uncertainties via Linear Regression. The value 𝑏 = −0.2709 is used to compute
Poisson’s Ratio via the relation 𝜐 = −
𝜀𝑦𝑦
𝜀𝑥𝑥
= −𝑏.
APPENDIX E
Linear Regression Uncertainty Calculations for Aluminum Tensile Testing Stress-Strain Plot
εxx (in/in)
0.00E+00
6.01E-07
3.76E-06
7.06E-06
1.07E-05
1.46E-05
1.87E-05
2.32E-05
2.78E-05
3.30E-05
3.85E-05
4.71E-05
6.26E-05
1.50E-04
2.22E-04
σxx
(lb/in2)
xixi
xiyi
1128.2 0.00E+00 0.00E+00
3954.4 3.61E-13 2.38E-03
6831.9 1.41E-11 2.57E-02
9709.4 4.98E-11 6.85E-02
12535.6 1.14E-10 1.34E-01
15424.5 2.12E-10 2.25E-01
18250.7 3.50E-10 3.41E-01
21151.0 5.39E-10 4.91E-01
23920.2 7.72E-10 6.65E-01
26797.7 1.09E-09 8.86E-01
29618.2 1.48E-09 1.14E+00
32507.1 2.22E-09 1.53E+00
35339.0 3.91E-09 2.21E+00
38205.1 2.25E-08 5.73E+00
38940.2 4.95E-08 8.66E+00
a+bxi
28.1
44.8
132.6
224.5
324.8
433.4
548.4
673.7
801.2
947.4
1097.9
1338.2
1768.6
4198.6
6216.9
[y-(a+bx)]2
1.2E+06
1.5E+07
4.5E+07
9.0E+07
1.5E+08
2.2E+08
3.1E+08
4.2E+08
5.3E+08
6.7E+08
8.2E+11
9.0E+11
9.8E+11
1.1E+12
1.1E+12
2.31E-04
2.39E-04
2.46E-04
2.54E-04
2.65E-04
2.73E-04
2.84E-04
2.91E-04
2.99E-04
3.08E-04
3.15E-04
3.24E-04
3.33E-04
3.57E-04
3.66E-04
3.72E-04
3.89E-04
4.05E-04
4.35E-04
4.64E-04
4.93E-04
5.26E-04
5.55E-04
5.89E-04
6.28E-04
6.66E-04
7.02E-04
7.41E-04
7.79E-04
8.17E-04
8.53E-04
9.09E-04
9.46E-04
9.88E-04
1.03E-03
1.07E-03
1.11E-03
1.15E-03
1.19E-03
1.23E-03
1.27E-03
39504.3
39282.1
39202.3
38792.0
40227.9
40114.0
40455.8
40057.0
40512.8
40472.9
40438.7
40541.3
40655.3
40621.1
40945.9
40968.7
41105.4
40997.2
40985.8
41270.7
41168.1
41737.9
42005.7
42222.2
42524.2
40757.8
43190.9
42279.2
42797.7
43076.9
43646.7
43441.6
43646.7
44387.5
44444.4
44558.4
44792.0
44735.0
45185.2
45299.1
45242.2
5.32E-08
5.70E-08
6.05E-08
6.46E-08
7.03E-08
7.47E-08
8.09E-08
8.46E-08
8.97E-08
9.46E-08
9.92E-08
1.05E-07
1.11E-07
1.28E-07
1.34E-07
1.38E-07
1.52E-07
1.64E-07
1.89E-07
2.15E-07
2.43E-07
2.77E-07
3.08E-07
3.47E-07
3.94E-07
4.44E-07
4.92E-07
5.49E-07
6.06E-07
6.68E-07
7.27E-07
8.26E-07
8.94E-07
9.77E-07
1.05E-06
1.13E-06
1.23E-06
1.33E-06
1.42E-06
1.51E-06
1.61E-06
9.11E+00
9.38E+00
9.65E+00
9.86E+00
1.07E+01
1.10E+01
1.15E+01
1.17E+01
1.21E+01
1.24E+01
1.27E+01
1.31E+01
1.35E+01
1.45E+01
1.50E+01
1.52E+01
1.60E+01
1.66E+01
1.78E+01
1.91E+01
2.03E+01
2.20E+01
2.33E+01
2.49E+01
2.67E+01
2.72E+01
3.03E+01
3.13E+01
3.33E+01
3.52E+01
3.72E+01
3.95E+01
4.13E+01
4.39E+01
4.56E+01
4.75E+01
4.97E+01
5.17E+01
5.39E+01
5.57E+01
5.74E+01
6442.6
6668.2
6873.0
7098.6
7405.8
7633.5
7942.8
8122.5
8358.6
8584.2
8789.0
9037.6
9292.5
9973.7
10211.9
10370.6
10857.5
11287.9
12119.5
12936.4
13738.8
14660.2
15458.3
16425.7
17487.2
18569.5
19547.3
20650.5
21686.9
22762.9
23747.0
25312.0
26335.8
27528.8
28579.8
29662.1
30896.9
32150.6
33191.1
34223.3
35301.4
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.1E+12
1.2E+12
1.2E+12
1.2E+12
1.2E+12
1.1E+12
1.2E+12
1.2E+12
1.2E+12
1.2E+12
1.2E+12
1.2E+12
1.2E+12
1.2E+12
1.2E+12
1.2E+12
1.2E+12
1.2E+12
1.3E+12
1.3E+12
1.3E+12
1.30E-03
45071.2 1.70E-06 5.87E+01 36266.7
1.3E+12
1.34E-03
45014.2 1.80E-06 6.04E+01 37369.9
1.3E+12
1.38E-03
45584.0 1.90E-06 6.29E+01 38414.6
1.3E+12
1.42E-03
45652.4 2.00E-06 6.46E+01 39415.4
1.3E+12
1.45E-03
45812.0 2.11E-06 6.66E+01 40479.0
1.3E+12
1.48E-03
45829.1 2.20E-06 6.80E+01 41312.6
1.3E+12
1.52E-03
45641.0 2.30E-06 6.92E+01 42223.6
1.3E+12
Table E1: Aluminum Tensile Testing Data used in Linear Regression Uncertainty Analysis
Table giving the calculated axial strains and stresses, the products used in linear regression calculations,
the axial stress as per the approximation function calculated using linear regression, and the error
propagation term associated with each measurement.
N
63
Sx
3.52E-02
Sy 2351208.0
Sxx
3.33E-05
Sxy
1.53E+03
a
2.8E+04
b
1.6E+07
θ
6.2E+13
ua
2.0E+05
ub
2.7E+08
Table E2: Linear Regression Variables and Uncertainties
Table giving the parameters N, Sx, Sy, Sxx, and Sxy used in linear regression calculations, the constants a
and b of the equation 𝑦 = 𝑎 + 𝑏𝑥𝑖 calculated using linear regression, the error propagation θ of the
experimental results, and the uncertainties of the linear regression constants a and b.
APPENDIX F
Calculation of Shear Modulus for Aluminum via Linear Regression
F1
Mt*L y
14.4
26.4
φ*I
x
3.6432E-06
5.52233E-06
xixi
1.32729E-11
3.04961E-11
xiyi
5.24621E-05
0.00014579
a+bxi
15.72755639
24.75666564
[y-(a+bx)]2
1.762405966
2.700547835
62.4
1.34223E-05
1.80159E-10
Mt*L y
12
24
62.4
φ*I x
2.49272E-06
5.29223E-06
1.34223E-05
xixi
6.21365E-12
2.80077E-11
1.80159E-10
Mt*L y
12
40.8
57.6
φ*I
x
2.30097E-06
1.03544E-05
1.45728E-05
xixi
5.29447E-12
1.07213E-10
2.12367E-10
0.000837554
62.71577798
0.09971573
a+bxi
11.58211988
24.5617728
62.25610732
[y-(a+bx)]2
0.174623795
0.315588682
0.020705104
a+bxi
11.75031526
41.52635561
57.12332913
[y-(a+bx)]2
0.06234247
0.527592475
0.227215118
F3
xiyi
2.99126E-05
0.000127014
0.000837554
F5
xiyi
2.76117E-05
0.000422458
0.000839394
F2
F4
F6
N
3
N
3
N
3
Sx
2.25879E-05
Sx
2.12073E-05
Sx
2.72282E-05
Sy
103.2
Sy
98.4
Sy
110.4
Sxx
2.23928E-10
Sxx
2.1438E-10
Sxx
3.24874E-10
Sxy
0.001035805
Sxy
0.00099448
Sxy
0.001289464
a
-1.777859496
a
0.024894675
a
3.242875158
b
4804950.286
b
4636393.522
b
3697325.794
θ
4.562669531
θ
0.51091758
θ
0.817150063
ua
2.51466476
ua
0.75257266
ua
1.066834805
ub
291062.6339
ub
89025.9359
ub
102517.9743
Tables F1, F3, F5 represent Aluminum Torsion Test results or trial 1, trial 2 and trial 3 respectively.
Experimental Data used in determination of Shear Modulus via Linear Regression Method and
Calculated Linear Regression Variables
(5) Table F1, F3 and F5 giving the measured torques and twisted angles, as well as the products
such asφ*I and Mt*L that are used in Linear Regression Calculations for trial 1, trial 2, and trial
3 respectively.
(6) In Table F2, F4 and F6, the variables N, Sx, Sy, Sxx, and Sxy are used to calculate the constants a,
b and their uncertainties via Linear Regression for trial 1, trial 2, and trial 3 respectively. The
value 𝑏 represents the Shear Modulus of the Aluminum as per trial one.
APPENDIX G
Calculation of Shear Modulus for Steel via Linear Regression
G1
Mt*L y
62.4
86.4
105.6
φ*I x
5.90583E-06
8.32185E-06
9.97088E-06
xixi
3.48788E-11
6.92531E-11
9.94184E-11
xiyi
0.000368524
0.000719007
0.001052924
a+bxi
61.9521989
87.50388179
104.9439193
[y-(a+bx)]2
0.200525827
1.218555
0.430441865
a+bxi
59.47029973
87.69482289
107.2348774
[y-(a+bx)]2
0.280582379
1.676566312
0.585412617
a+bxi
57.7393353
86.06074884
105.7999159
[y-(a+bx)]2
0.019414325
0.115091347
0.039966351
G3
Mt*L y
60
86.4
108
φ*I x
5.75243E-06
8.24515E-06
9.97088E-06
xixi
3.30904E-11
6.79824E-11
9.94184E-11
xiyi
0.000345146
0.000712381
0.001076855
G5
Mt*L y
57.6
86.4
105.6
φ*I x
5.63738E-06
8.16845E-06
9.93253E-06
xixi
3.178E-11
6.67235E-11
9.86551E-11
G2
xiyi
0.000324713
0.000705754
0.001048875
G4
G6
N
3
N
3
N
3
Sx
2.41985E-05
Sx
2.39684E-05
Sx
2.37384E-05
Sy
254.4
Sy
254.4
Sy
249.6
Sxx
2.0355E-10
Sxx
2.00491E-10
Sxx
1.97159E-10
Sxy
0.002140455
Sxy
0.002134381
Sxy
0.002079342
a
-0.507470388
a
-5.663215259
a
-5.340176693
b
10575941.27
b
11322786.74
b
11189509.8
θ
1.849522693
θ
2.542561308
θ
0.174472024
ua
3.874294166
ua
4.346157251
ua
1.10905045
ub
470345.9416
ub
531640.8955
ub
136805.6434
Tables G1, G3, G5 represent Aluminum Torsion Test results or trial 1, trial 2 and trial 3 respectively.
Experimental Data used in determination of Shear Modulus via Linear Regression Method and
Calculated Linear Regression Variables
(1) Table G1, G3 and G5 giving the measured torques and twisted angles, as well as the products
such asφ*I and Mt*L that are used in Linear Regression Calculations for trial 1, trial 2, and trial
3 respectively.
(2) In Table G2, G4 and G6, the variables N, Sx, Sy, Sxx, and Sxy are used to calculate the constants
a, b and their uncertainties via Linear Regression for trial 1, trial 2, and trial 3 respectively. The
value 𝑏 represents the Shear Modulus of the Aluminum as per trial one.