Torsion Lab
By
Charisse Lewis
0729543
ME 354 AC
Date of this Lab Exercise:
February 11, 2010
Date of Lab Report Submission:
February 25, 2010
1
Executive Summary:
Torsion testing is one of the basic mechanical tests that are conducted to obtain material
properties, such as the elastic modulus, strength, shear modulus, and shear strength. During this
torsion lab the Technovate model 9041 Torsion Tester was used to conduct the testing. The
materials used in this lab were 6061-T6 Aluminum, and A36 Steel. The setup for the lab took a
lot of attention to detail, making sure that the specimen was tightened into the grip just right, and
time, if the specimen was not clamped properly or failed inside the grip the test would have to be
redone.
The main purposes of the lab were to observe how the specimens of two materials that were used
reacted under torsion, and to verify the accuracy of the Power-Hardening and Ramberg-Osgood
applied torque predicting models. While observing the specimens of Aluminum and Steel data
was collected as well, so as to compare the two samples numerically to each other and to the
torque predictor models.
When the data for aluminum and steel are compared, the aluminum sample is more ductile than
steel is, which is shown in Figure C.1 in the results. This is evident because the aluminum
sample underwent more angle of twist per unit length than the steel specimen did.
The results show that the Power-Hardening and Ramber-Osgood torque predicting models work
well for Aluminum, which was the only material that had data applied to the models. The
maximum percent error, shown in Table C.3 in the results section, % Error, for the PowerHardening model was 0.706 % and for the Ramberg-Osgood it was 1.40 %. These two models
had the % Error maximums at opposite ends of the data set. The Power-Hardening model had
the greatest % Error when θ/L was a maximum, and the Ramberg-Osgood had the greatest %
Error when θ/L was a minimum.
2
A. Introduction
Torsion testing is widely used for evaluating the elastic modulus, strength, shear modulus, shear
strength and other properties of materials. The main difference between torsion testing and
tensile testing, as discussed in previous labs, is that the stress is not uniform over the cross
section of the test specimen. The only useful exception to that is a round thin-walled tube, when
the wall is sufficiently thin, but this lab did not use thin walled specimens, so that is that will be
said about that.
In this lab the apparatus that was used was the Technovate model 9041 Torsion Tester
(Appendix A) with the specimen being held in by grips. The force that was applied was
measured at the force gage via a lever arm. A threaded horizontal drive rod was used to adjust
the angle of twist in small increments, and when it was necessary to adjust the angle in large
increments there was a lever arm, that could adjust the angle up to 90 degrees. The specimens
that were used for this lab were constant-diameter gage section torsion specimens of 6061-T6
Aluminum, and A36 Steel. The diameters of the specimens were approximately 4.77 mm (about
3/16 inch).
There were many purposes for doing this lab, the most important ones being to examine the
responses of metals, like the steel and aluminum that were used, to applied torsion, and to verify
the accuracy Power-Hardening and Ramberg-Osgood models in predicting the applied torque.
Also, this lab was conducted so that students learned about torsion testing, the apparatus that is
used, and the information that can be obtained from the tests.
At the beginning of the experiment a straight line had to be drawn on the specimen. This was
done so that as the lab was progressing the students were able to tell if the set of data that was
being collected was a good set or not. If it was a good set, then the line that was drawn on the
specimen at the beginning of the lab would wrap around the sample in a uniformly spaced
progression. The less uniform the spacing the worse the data set was. This is known to hold true
because a radial line which is straight before loading remains a straight radial line after loading.
3
B. Procedure
The first thing that was done in the lab was the measuring of the diameter of the specimen gage
section using calipers, and to record that value for later calculations. After that, it was necessary
to draw a straight longitudinal line on the specimen so that the angle of twist of the specimen can
be observed during the test.
When installing the specimen in the machine it must first be installed in the bottom grip, and the
specimen has to be very tightly camped in the grip to ensure that it will not slip. Next, the pin
behind the lower grip that engages the ratchet mechanism must be located and make sure the pin
is pulled forward (this ensures that the ratchet is engaged). Then, the horizontal drive rod must
be unthreaded until the end of the threaded region has reached the threaded nut, just before the
threaded rod is disconnected from the base. Then the lever arm must be rotated counterclockwise, moved to the right, as far as possible.
At the top of the apparatus identify the wire ropes that transfer the torque to the top grip and
specimen, and loosen the two nuts as far as possible. Rotate the top grip clockwise, so as to
remove the ‘slack’ in the wire ropes, and while holding the top grip in this position as firmly as
possible, tighten the grip. Make sure that the specimen is very tightly fastened in the top grip.
Remove any remaining slack in the wire ropes by retightening the two nuts that were loosened
before, but monitoring the force sensor while that is being done. If the nuts are completely
tightened and slack still remains, tighten the threaded horizontal drive rod until an increase in
load is sensed.
After the specimen is tightly clamped into the machine, measure and record the distance between
the grips, but make sure to account for how far the bottom face of the upper jaw is recessed into
the upper plate in the measurement. Adjust the pointer so that it indicates “zero” degrees, and
zero the output of the force sensor.
All of the prior steps were the setup for the lab, and with those complete the data collection for
the lab may begin. Start applying known angles of twist to the lower grip, beginning with 2o
4
using the threaded drive rod, and record the force displayed on the force gage. After that
increase the angle of twist by 2o until the total angle of twist is 30o, recording the force shown on
the force gage at the end of each increase. Next, increase the angle of twist by 5o. Continue to
take measurements at this increment until the ratchet in the lower grip “clicks”. This usually
occurs around 45o, but it can occur at a total angle of twist larger than that, up to about 60o, just
keep on increasing the angle of twist until the “click” occurs. If the “click” doesn’t occur until
well after 60o, it may be that the specimen was not correctly installed, and therefore may not give
a good data set, so it might be advisable to start over.
After hearing the “click”, record four measurements at increased angles of twist, in increments of
45o. Record four more measurements, this time at increments of 90o, and four more
measurements at increments of 180o. Finally, record measurements for increased angles of twist
in increments of 360o, until specimen failure occurs.
After failure remove the broken halves of the specimen. Carefully examine the specimen and the
fracture surface, record what is necessary to discuss later if it is required.
C. Results
The data that was collected during the torsion testing lab was force (kg), and angle of twist.
Comparison of the measured values for torque (Appendix B) versus the angle of twist divided by
the length, θ/L, is shown in Figure C.1 below:
5
Measured Torque vs. θ/L
12
Measured Torque (N-M)
10
8
A36 Steel
6
6061-T6 Aluminum
4
2
0
0
100
200
300
400
500
600
700
θ/L (rad/m)
Figure C.1: Measured Torque vs. θ/L for the A36 Steel and 6061-T6 Aluminum specimens that
were tested.
On the next page, in Figure C.2, is an example of what one of the A36 Steel specimens looked
like after fracture. This example was representative of most of the steel specimens, so it is used
for determining how brittle or ductile the material was just based on the appearance of the
fracture.
6
Figure C.2: Steel specimen at the point of failure, after fracture.
The values for force converted to Newtons, and the θ/L for aluminum is shown in Tables C.1
and C.2. The other data that is shown in Table C.1 is for predicting torque according to the
Power-Hardening model. The data that is shown in Table C.2 other than the θ/L is the calculated
values necessary for the Ramberg-Osgood model for predicting applied torque.
Table C.1: The Power-Hardening model (Appendix D) for predicting torque with values that
are necessary for calculation
Angle (deg)
Force (N)
0
2
4
6
8
10
12
14
0
10.3
10.5
11.3
12.0
13.0
14.0
14.5
Angle (rad)
0
0.0349
0.0698
0.105
0.140
0.175
0.209
0.244
𝜃
𝐿
(rad/m)
0
0.137
0.275
0.412
0.550
0.687
0.825
0.962
7
Measure Torque (Nm)
0
1.35
1.38
1.48
1.57
1.70
1.83
1.90
Predicted Torque
(P-H) (N-m)
0
0.179
0.357
0.536
0.714
0.893
1.07
1.25
16
18
20
22
24
26
28
30
35
40
45
90
135
180
225
315
405
495
585
765
945
1125
1305
1665
2025
2385
2745
3105
3465
3825
4185
4545
4905
5265
5625
5985
6345
6705
7065
7425
7785
8145
8505
8865
9225
16.8
16.9
17.4
18.1
19.6
20.4
21.6
22.4
24.5
26.5
28.4
35.6
37.5
40.2
40.7
41.7
41.7
41.7
41.7
42.7
42.9
44.4
44.4
45.2
46.2
46.6
47.1
47.1
47.2
47.6
47.6
47.9
48.2
48.3
48.8
49.1
49.1
49.1
49.1
49.1
49.1
49.5
49.3
50.0
50.0
0.279
0.314
0.349
0.384
0.419
0.454
0.489
0.524
0.611
0.698
0.785
1.57
2.36
3.14
3.93
5.50
7.07
8.64
10.2
13.4
16.5
19.6
22.8
29.1
35.3
41.6
47.9
54.2
60.5
66.8
73.0
79.3
85.6
91.9
98.2
104
111
117
123
130
136
142
148
155
161
1.10
1.24
1.37
1.51
1.65
1.79
1.92
2.06
2.40
2.75
3.09
6.18
9.28
12.4
15.5
21.6
27.8
34.0
40.2
52.6
64.9
77.3
89.7
114
139
164
189
213
238
263
288
312
337
362
387
411
436
461
485
510
535
560
584
609
634
8
2.20
2.22
2.28
2.38
2.57
2.67
2.83
2.93
3.21
3.47
3.73
4.66
4.92
5.27
5.34
5.46
5.46
5.46
5.46
5.59
5.62
5.82
5.82
5.93
6.06
6.11
6.17
6.17
6.18
6.24
6.24
6.27
6.31
6.33
6.40
6.43
6.44
6.44
6.44
6.44
6.44
6.49
6.47
6.56
6.56
1.43
1.61
1.79
1.96
2.14
2.32
2.50
2.68
3.12
3.57
3.93
4.81
5.01
5.12
5.19
5.31
5.40
5.47
5.52
5.62
5.69
5.76
5.81
5.90
5.98
6.04
6.09
6.14
6.18
6.22
6.26
6.29
6.32
6.35
6.38
6.40
6.42
6.45
6.47
6.49
6.51
6.53
6.54
6.56
6.58
Table C. 2: The Ramberg-Osgood model (Appendix E) for predicting torque with values that
are necessary for calculation
𝜏̃ 𝑐 (MPa)
0
50
100
150
200
250
270
280
290
300
305
310
315
320
325
330
335
340
345
350
355
𝛾̃𝑝𝑐
0
3.33E-14
4.64E-08
0.000182
0.0646
6.13
29.5
62.0
127
253
355
495
686
945
1280
1770
2410
3260
4390
5890
7860
𝛾̃𝑒𝑐
0
0.00196
0.00391
0.00587
0.00782
0.00978
0.0106
0.0110
0.0113
0.0117
0.0119
0.0121
0.0123
0.0125
0.0127
0.0129
0.0131
0.0133
0.0135
0.0137
0.0139
𝛾̃𝑐
0
0.00196
0.00391
0.00605
0.0724
6.14
29.5
62.0
127
253
355
495
686
945
1300
1770
2410
3260
4390
5890
7860
𝜃/𝐿 (rad/m)
0
0.820
1.64
2.54
30.3
2580
12400
26000
53200
106000
149000
207000
287000
396000
544000
743000
1010000
1370000
1840000
2470000
3300000
Predicted Torque
(R-O) (N-m)
0
1.07
2.13
3.27
5.57
6.99
7.55
7.83
8.11
8.39
8.53
8.67
8.81
8.95
9.09
9.23
9.37
9.51
9.65
9.78
9.92
The comparison of the two predictive models, Power-Hardening and Ramberg-Osgood, to the
measured values of applied torque is in Figure C.3. This was done so that the students could
visually see how the predicted models compared to the measured values, instead of just seeing
data values.
9
Comparison of Measured Torque to Predictive Models for 6061-T6
Aluminum
7
(θ/L) ( rad/m)
6
5
4
Ramberg Osgood
Power Hardening
3
Measured Torque
2
1
0
0
200
400
600
800
Torque (N-m)
Figure C.3: The comparison of measured torque to the predictive models for the 6061-T6
Aluminum specimen.
Data values are still an excellent way of representing the differences between the measured
values and the predicted values. This is done in Table C.3 below, which shows the values for the
measured torque at four different values of θ/L, the corresponding predicted torque for the
Power-Hardening and Ramberg-Osgood, and the % Error corresponding to differences from the
measured values.
Table C.3: The percent error of the predictive models, Power-Hardening (P-H) and RambergOsgood (R-O), to the measured values for torque
θ/L (rad/m)
100
250
400
550
Measured
Torque (N-m)
5.86
6.21
6.41
6.47
Predicted
Torque (P-H)
(N-m)
5.85
6.20
6.39
6.52
10
% Error
0.222 %
0.111 %
0.391 %
0.706 %
Predicted
Torque (R-O)
(N-m)
5.95
6.23
6.38
6.45
% Error
1.40 %
0.307 %
0.596 %
0.403 %
D. Discussion
In this experiment the main goals were to observe the metals specimens to see how they act
when under torsion, and to verify the accuracy of the Power-Hardening and Ramberg-Osgood
predictive models for determining torque.
Observing the data collected for each of the metals in a graph of measured torque vs. θ/L, Figure
C.1, it can be seen that the Aluminum specimen is much more ductile than the steel specimen.
This is shown by the fact that the aluminum withstood a greater amount of angle of twist
compared to its length, and the steel endured a greater amount of torque prior to fracture.
Although, the steel was more brittle than the aluminum, the steel did behave as mostly ductile.
This is shown in figure C.2 in the Results section, where the steel specimen exhibited mostly
ductile behavior at fracture, which is shown by the flat fracture surface, and the small amount of
tearing at the edge. The aluminum sample looked very similar to that of the steel, with the
biggest difference being around the area of fracture where on the steel specimen it was course
like sand paper along the sides that were originally smooth (Figure C.2 in the Results). Both
samples initially had the straight line that was drawn on them before starting the testing, and by
the end of the lab it formed a spiral around the outside of the sample. The more uniform the
spacing that the spiral had, the better the data for the specimen would be.
As stated above the other main goal was to verify the accuracy of the Power-Hardening and
Ramberg-Osgood models for predicting torque. This can be shown by Figure C.3 in the results
section, which is a graph with the measured torque and the predictive models all on the same
graph. The predictive models for torque appear to be very close to the measured values, and they
are as shown in Table C.3, the percent error, % Error, of the predictive models as compared to
the measured values for torque. The values for the Power-Hardening had an increasing error
percentage as the values for θ/L increased, which means that this model is better for predicting
the torque being applied for smaller values of θ/L. With the Ramberg-Osgood, the opposite
thing happened, where the % Error was greatest in the beginning and then it became smaller as
the value for θ/L increased. Even though the % Errors varied from beginning to end, the values
for the % Error is still very small, with a max error for the Power-Hardening model being .706 %
and for the Ramberg-Osgood it was 1.40 %. Since the max percent errors are so small, it is a
11
good assumption that both models are a good at predicting the applied torque, at least for a
ductile material.
The best way to test how applicable the Power-Hardening and Ramberg-Osgood models are for
predicting torque would be to test them with a lager variety of materials, and run more than one
test for each material. The materials used in this lab were both mostly ductile, and this showed
in the lab results, but this only gives results for ductile materials for the behavior of the applied
torque predictive models. If there was a more brittle material used in the lab alongside the
ductile materials it could be seen whether or not the predictive models apply. The minimum that
could have been done was try to apply the predictive models to steel, which as discussed is more
brittle than the aluminum.
Another problem that many groups experienced was tightening the specimens in the grips. If the
grips were too tight, the material would get scratched, and would tend to fail in that region. On
the other hand if the specimen wasn’t tightened into the grips tightly enough, then the specimen
would slip, and the data set would be poor. The best way to resolve this would be to find a way
to gage how tight the specimen is in the grip, and have a specified grip strength (how tight the
grip is on the specimen) so that students know how much tighter it has to be, or if they have gone
too far.
Also, the machines that were used were very old, and the force gage was in kilograms (kg). The
force gage being in kg, which is a unit of mass and not force, was not terrible because it is not
difficult to convert kg to N, but it was also very touchy, probably due to how old it was. If
anyone hit the table at all the gage usually changed values, whether it increased or decreased the
value was not consistent either. This could also be a source of error between the measured and
predicted values.
Other possible sources of error include; not tightening the specimen into the grip enough,
tightening the specimen into the grip too much (as discussed above), the specimen being bent at
all when installed, and adding angle of twist at too rapid a pace. The first two sources of error
were already discussed above, but if the specimen was bent at all, this would create other forces
12
other than the shear stress and shear strain. The predictive models would not be as accurate
because the initial assumption that was made about there only being shear stresses and forces
would be incorrect; all of the equations would be off and would not account for the additional
forces that are present. Also, if the angle of twist was added to the specimen too rapidly, then the
planes in the crystalline structure of the material would not have as much time to slide past each
other. Therefore the material would fracture sooner than a specimen that had the angle of twist
applied at a slower rate.
13
Appendix A: The Technovate Torsion Test Machine
During this torsion lab the Technovate model 9041 Torsion Tester was used to test specimens
made of A36 Steel, and 6061-T6 Aluminum. In order to know how the lab works, it is necessary
to know the setup that is being used. Therefore, below in Figure A.1 is a picture of the top view
of the torsion tester with the key components labeled.
Wire Rope
Top Lever
Arm
Top Grip
Force Gage
Wire Rope
Figure A.1: To view of the Technovate model 9041 Torsion Tester, with key components
labeled.
There is also a schematic of the above picture in Appendix B, which is used for discussing how
to calculate torque from the force reading collected. Below are two photos that display the
bottom half of the testing apparatus with the key components labeled. These pictures are for
reference throughout the report so that it is clear what is being discussed.
14
Ratchet Pin
Driving Rod
Bottom Grip
Bottom Lever Arm
Figure A.2: A view of the bottom half of the testing apparatus, and the key components labeled.
Bottom Grip
Angle Indicator
Bottom Lever Arm
Driving Rod
Figure A.3: An alternate view of the bottom half of the testing apparatus, with key components
labeled.
15
Appendix B: Converting the Force Reading to Torque
The setup of the torsion test needs to be known in order to be able to calculate the torque being
applied to the specimen, and this is shown in Figure B.1 below:
Figure B.1: Schematic top view of the Technovate Torsion Test Machine.
As shown above the grip and specimen are held in place by the wire ropes, one that is attached to
one side of the machine and one that is attached to the lever arm. The lever arm is held in place
at the pivot point and the force gage, both of which cause a force at either end of the beam. The
wire rope is the only other force acting on the lever arm, which is acting on the lever arm slightly
off center. This is shown in the form of a Free Body Diagram (FBD) in Figure B.2, on the next
page.
16
𝐹𝑤𝑖𝑟𝑒 𝑟𝑜𝑝𝑒
Applied Force
(𝐹𝑓𝑜𝑟𝑐𝑒 𝑔𝑎𝑔𝑒 )
Reaction Force
Figure B.2: FBD for the lever arm with the three forces acting on it.
During the torsion experiment the data that was collected was the force at the force gage, and the
force was in kilograms (kg). First the force in kg had to be converted to Newtons (N), which is
𝑚
done by multiplying the force value, in kg, by 9.81 𝑠2 and then to calculate the torque on the
specimen it was necessary to calculate the force pulling on the wire rope. This was done using
static equilibrium moment equations centered at the pivot point, pp, shown here:
∑ 𝑀𝑃𝑃 = 0 = 𝐹𝑤𝑖𝑟𝑒 𝑟𝑜𝑝𝑒 ∗ .085𝑚 − 𝐹𝑓𝑜𝑟𝑐𝑒 𝑔𝑎𝑔𝑒 ∗ .213𝑚
(Equation B.2)
With Equation B.2, 𝐹𝑤𝑖𝑟𝑒 𝑟𝑜𝑝𝑒 is able to be obtained since 𝐹𝑓𝑜𝑟𝑐𝑒 𝑔𝑎𝑔𝑒 is the measured data from
the lab that was converted to N. With the force in the wire rope the torque applied to the
specimen can be calculated by multiplying the force in the wire rope times the radius of the top
grip, (the distance to the application of the torque). This is shown in Equation B.3 below.
𝑇𝑤𝑖𝑟𝑒 𝑟𝑜𝑝𝑒 = 𝐹𝑤𝑖𝑟𝑒 𝑟𝑜𝑝𝑒 ∗ 0.02615𝑚
(Equation B.3)
The FBD for the forces acting on and the torque being applied to the top grip are shown in
Figure B.3 below.
17
Torque
Figure B.3: FBD of the grip with the wire ropes as the forces causing torque.
This is just the torque caused by one of the wire ropes, therefore the 𝑇𝑠𝑝𝑒𝑐𝑖𝑚𝑒𝑛 has to be doubled
in order to obtain the actual value for the torque applied to the specimen. In other words;
𝑇𝑠𝑝𝑒𝑐𝑖𝑚𝑒𝑛 = 2 ∗ 𝑇𝑤𝑖𝑟𝑒 𝑟𝑜𝑝𝑒
(Equation B.4)
This is the measured torque in the specimen that is compared to the predicted torque from the
Power-Hardening model (Appendix D), and the Ramberg-Osgood model (Appendix E).
18
Appendix C: True and Effective Stress and Strain
The equation for effective stress is listed below as Equation C.1
𝜎̅ =
1
√2
√(𝜎𝑥 − 𝜎𝑦 )2 + (𝜎𝑦 − 𝜎𝑧 ) + (𝜎𝑧 − 𝜎𝑥 ) + 6(𝜏𝑥𝑦 2 + 𝜏𝑦𝑧 2 + 𝜏𝑧𝑥 2 )
(Equation C.1)
Since this lab was interested in finding predicted values for torque beyond yielding, the use of
true stress was used in calculations. Thus the equation for effective stress becomes:
𝜎̅ =
1
√2
2
√(𝜎̃𝑥 − 𝜎̃𝑦 ) + (𝜎̃𝑦 − 𝜎̃𝑧 ) + (𝜎̃𝑧 − 𝜎̃𝑥 ) + 6(𝜏̃ 𝑥𝑦 2 + 𝜏̃ 𝑦𝑧 2 + 𝜏̃ 𝑧𝑥 2 ) (Equation C.2)
The effective stress equation can be rewritten to get the effective strain equation, as shown in
Equation C.3.
𝜀̅ =
1
√2
2
√(𝜀̃𝑥 − 𝜀̃𝑦 ) + (𝜀̃𝑦 − 𝜀̃𝑧 ) + (𝜀̃𝑧 − 𝜀̃𝑥 ) + 6(𝛾̃𝑥𝑦 2 + 𝛾̃𝑦𝑧 2 + 𝛾̃𝑧𝑥 2 )
(Equation C.3)
During the torsion test there was only one stress, 𝜏̃𝑥𝑦 , and one strain, 𝛾̃𝑥𝑦 , applied. Knowing that
there is only uniaxial stress, and uniaxial shear strain, Equation C.2 and Equation C.3 become:
𝜎̅ = √3 𝜏̃ 𝑥𝑦
𝜀̅ =
1
𝛾̃
√3 𝑥𝑦
(Equation C.4)
(Equation C.5)
Equation C.4 an Equation C.5 are important for the application of the power-hardening and
Ramberg-Osgood torque predicting models as discussed in Appendix D and Appendix E,
respectively.
19
Appendix D: Power-Hardening Model
Using the data collected from the torsion test that was conducted, a power-hardening model,
(Equation D.1), was fit to the measured force, (assuming true axial stress and true axial strain),
response for 6061-T6 Aluminum.
𝜎̅𝑥 = 𝐻𝜀̅𝑥𝑛
(Equation D.1)
The coefficients in Equation D.1 are H, the strength coefficient, which is 413 MPa, and n, the
hardening exponent, which is .0633. Also, assuming the use of true stress and strain, as
discussed in Appendix A, and using Equations C.4 and C.5, a new equation for 𝜏̃ 𝑥𝑦 can be
developed that relates it to 𝛾̃𝑥𝑦 using the coefficients H and n, as shown in Equation D.2.
𝜏̃ 𝑥𝑦 =
𝐻
√3
̃𝑥𝑦 𝑛
𝛾
(
√3
)
(Equation D.2)
With this model the goal is to predict torque versus (𝜃/L) response, and for this it is necessary to
find equations relating the two. The total applied torque is equal to the sum of the torque acting
on the elastic inner-region and the torque acting on the plastically-deformed outer-region, shown
below:
𝑇𝑡𝑜𝑡𝑎𝑙 = 𝑇𝑒𝑙𝑎𝑠𝑡𝑖𝑐 + 𝑇𝑝𝑙𝑎𝑠𝑡𝑖𝑐
(Equation D.3)
𝑟
𝜎𝑜
𝑦
√3
The elastic inner-region extends over 0 < 𝑟 < 𝑟𝑦 , and in this region 𝜏̃ 𝑥𝑦 = 𝑟 𝜏̃𝑜 , where 𝜏̃ 𝑜 =
.
Therefore:
𝑟
𝑇𝑒𝑙𝑎𝑠𝑡𝑖𝑐 = ∫0 𝑦( 𝜏̃ 𝑥𝑦 )(𝑟)(2𝜋)𝑑𝑟 =
𝜋𝜏̃𝑜 𝑟𝑦3
2
(Equation D.4)
The plastic outer-region extends over 𝑟𝑦 < 𝑟 < 𝑐, and in this region Equation D.2 becomes
𝜏̃ 𝑥𝑦 =
𝑐
𝐻
(
𝑟𝜃
√3 √3𝐿
2𝜋𝐻
)
𝑛
(Equation D.5)
𝜃
𝑛
𝑇𝑝𝑙𝑎𝑠𝑡𝑖𝑐 = ∫𝑟 ( 𝜏̃ 𝑥𝑦 )(𝑟)(2𝜋)𝑑𝑟 = √
( ) [𝑐𝑛+3 − 𝑟𝑛+3
𝑦 ]
3(𝑛+3) √3𝐿
𝑦
(Equation D.6)
Using Equation D.4 and D.6 the torque vs. (𝜃/L) response can now be predicted based on the
Power-Hardening model. During this lab the Power-Hardening model was only applied to the
Aluminum 061–T6 sample data.
20
Appendix E: Ramberg Osgood Model
An alternate approach to the Power-Hardening model for fitting data is the Ramberg-Osgood
model, which uses Equation E.1 as a base equation for relating the predicted torque to the (𝜃/L)
response.
̃
𝜎
̃
𝜎
𝜀̃ = 𝐸 + (𝐻)
1⁄
𝑛
(Equation E.1)
Just like the Power-Hardening model (Appendix C) the coefficients H and n are the strength
coefficient and the strain hardening exponent, respectively. The values for H and n are: H= 407
MPa, and n=0.0490. As discussed in Appendix C, the use of effective stress and effective strain
changes Equation E.1 into:
𝛾̃𝑥𝑦 =
𝜏̃𝑥𝑦
𝐺
+ √3 (
√3𝜏̃𝑥𝑦
𝐻
)
1⁄
𝑛
(Equation E.2)
Where:
𝐺=
𝐸
(Equation E.3)
2(1+𝜈)
The modulus of elasticity, E, for Aluminum 6061-T6 is 68.5 MPa, and the Poisson’s ration, 𝜈, is
0.34 . Also, the equation becomes less daunting when a “shear” strength coefficient is defined,
𝐻𝜏 , as shown below:
𝐻𝜏 =
𝐻
(Equation E.4)
(𝑛+1)⁄
2
3
This changes Equation E.2 into:
𝛾̃𝑥𝑦 =
𝜏̃𝑥𝑦
𝐺
𝜏̃𝑥𝑦
+(
𝐻𝜏
1⁄
𝑛
)
(Equation E.5)
With all of these equations an equation for torque is made as shown below in Equation E.6.
3
𝑇 = 2𝜋𝑐 𝜏̃ 𝑚𝑎𝑥 [
1 2𝑛+1
𝑛+2
1
+
𝛽 +
𝛽 2+
𝛽 3
4 3𝑛+1 𝜏 2𝑛+2 𝜏
𝑛+3 𝜏
(1+𝛽𝜏 )3
]
(Equation E.6)
Where 𝜏̃𝑚𝑎𝑥 = 𝜏𝑐 , meaning that the max shear stress occurs where the radius is the largest.
Also:
𝛽𝜏 =
21
̃𝑝𝑐
𝛾
̃𝑒𝑐
𝛾
(Equation E.7)
1⁄
𝑛
𝜏
𝛾̃𝑝𝑐 = ( 𝑐 )
𝐻
(Equation E.8)
𝜏
𝛾̃𝑒𝑐 =
𝜏𝑐
𝐺
𝛾̃𝑐 = 𝛾̃𝑒𝑐 + 𝛾̃𝑝𝑐
(Equation E.9)
(Equation E.10)
Equations E.6 through Equation E.10 are used to predict the torque vs. (𝜃/L) response based on
the Ramberg-Osgood model, with (𝜃/L) being equal to:
𝜃
=
𝛾𝑐
(Equation E.11)
Just like the Power-Hardening model the Ramberg-Osgood model was only fit to the Aluminum
6061-T6 data set.
𝐿
𝑐
22