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McGill University Department of Civil Engineering and Applied Mechanics
Solid Mechanics CIVE - 207
Torsion Coupons Experiment Laboratory Write-Up Report
Number of Pages: 28
Group Number: 08 Names: Surname, First Names McGill ID #
February 19th, 2015
Table of Contents:
Page Number
1) List of Figures and Diagrams 3
2) Graphs and Data
A) Aluminum Square Rod....4
B) Softwood Hollow Circular Rod.....6
C) Torsional Characteristics and Sectional Properties. 9
D) Experimental Versus Predicted Values. 10
3) Element principle of stress and strain 12
4) Photos
E) Initial Setup.13
F) Softwood Hollow Circular Rod. 14
G) Aluminum Square Rod. .15
5) Sample Calculations .16
6) Discussion .. 22
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7) Sources of error and Improvements 23
8) Conclusion ..25
9) Appendix 26
10) DVD (included)
List of Tables And Figures:
Page
Graphs:
1) Aluminum Square Rod
A) Elastic Region 4
B) Torque (N.m) versus Twist (0) 4
C) Torque (N.m) versus Twist (rad) 5
2) Softwood Hollow Circular Rod
A) Torque Versus Twist. 6
B) Torsional Shear Stress Versus Shear Stress. 8
Tables:
1) Torsional Characteristics and Sectional Properties 9
2) Experimental Maximum Elemental Stresses and Strains10
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3) Experimental Versus Predicted Values.. 10
Figures:
1) Aluminum Circular Rod .. 15
2) Softwood Hollow Circular Rod ..14
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Proportionality Limit
(24, 12.75N.m)
Yield point(45, 14.15N.m)
Ultimate strength(720, 17.60N.m)
Failure fracture(1080, 17.40N.m)
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 200 400 600 800 1000 1200
Torq
ue(
N.m
)
Twist()
Aluminium-Alloy- Sold Square Bar Torque(N.m) vs Twist()
G = 15.7 GPa
(2)
(3)(4)
(1)
X
(1) Linear Elastic(2) Yielding(3) Strain Hardening(4) Strain Sofening
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0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0.000 2.000 4.000 6.000 8.000 10.000 12.000 14.000 16.000 18.000 20.000
Torq
ue(
N.m
)
Applied Twist(radian)
Aluminium-Alloy- Sold Square Bar Torque(N.m) vs Twist(radian)
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Proportionality limit & Ultimate load(20, 8.550N.m)
Failure fracture
90, 1.900N.m
0.000
1.000
2.000
3.000
4.000
5.000
6.000
7.000
8.000
9.000
0 10 20 30 40 50 60 70 80 90 100
Torq
ue
(N
m)
Twist ()
Softwood- Hollow Circular RodTorque(N.m) vs Twist()
X
(1)
(1) Linear Elastic
G= 0.361 GPa
Torsional Failure Longitudinal Shear 0Brittle Behaviour
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0.000
1.000
2.000
3.000
4.000
5.000
6.000
7.000
8.000
9.000
0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.800
Torq
ue(
N.m
)
Twist(radian)
Softwood- Hollow Circular RodTorque(N.m) vs Twist(radian)
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0
2
4
6
8
10
12
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
Shea
r St
ress
(Mp
a)
Shear strain(radian)
Softwood- Hollow Circular RodShear Stress vs Shear Strain
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Torsional Characteristics and Sectional Properties for Aluminium and Softwood Hollow Specimens
Aluminium Alloy - Solid Square Bar Dimensions (mm) Average(mm) Ratio(b/a)
Width(d) 6.44 6.45 6.43 6.44
1 0.208 0.1406 Depth(t) 6.43 6.45 6.44 6.44
Length(mm) 124.46 124.39 124.43
J(mm4) 241.84
G(Gpa) 15.7
Note: See Appendix and Sample Calculation for reference
Softwood - Hollow Circular Rod Dimensions Average(mm)
Neck Outlet(mm) 16.49 16.48 16.5 16.49
Neck Inlet(mm) 8.28 8.05 8.17
Neck Length(mm) 85.17 85.4 85.29
J (mm4) 6821.68
G(Gpa) 0.3613
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Experimental Versus Predicted Values
Aluminum Solid Square Bar Softwood Hollow Circular
Rod
Experimental
Value
Predicted
Value
Experimental
Value
Predicted
Value
Linear Elastic Region, Proportional Limit, and Yield Point
Proportional Limit Torque (Tpl) 12.75Nm 15.3Nm 8.550Nm 8.73Nm
Proportional Limit Angle of Twist (pl) 24 17.4 20 10.4
Yield Torque (Ty) 14.15Nm 15.3Nm 8.550Nm 8.73Nm
Yield Angle of Twist (y) 45 17.4 20 10.4
Shear Modulus (G) 15.7GPa 26GPa 0.361GPa 0.602GPa
Prop. Limit Torsion Shear Stress (pl) 230MPa 276MPa 10.33MPa 10.6MPa
Prop. Limit Torsion Shear Strain (pl) 0.0147 0.011 0.0286 0.0175
Yield Torsional Shear Stress (y) 255MPa 276MPa 10.33MPa 10.6MPa
Yield Torsional Shear Strain (y) 0.0163 0.011 0.0286 0.0175
Maximum Normal Tensile and Compressive Stresses and Strains at 45
Maximum/Minimum Normal Stress () - - 10.33MPa 10.6MPa
Maximum/Minimum Normal Strain () - - 0.0143 0.00876
Yielding Plastic Region Plateau
Plastic Torque (Tp) 17.60Nm 24.6Nm 8.550Nm 12.4Nm
Angle of Twist (p) 720 27.9 20 14.7
Ratio of Plastic Torque to Yield Torque 1.24 1.60 1 1.42
Ultimate Load and Fracture Load Levels
Ultimate Torque (Tult) 17.60Nm 17.2Nm 8.550Nm 6.91Nm
Ultimate Angle of Twist (ult) 720 19.5 20 8.22
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Ultimate Torsional Shear Stress (ult) 317MPa 310MPa 10.33MPa 8.35MPa
Ultimate Torsional Shear Strain (yult) 0.0202 0.012 0.0286 0.0139
Fracture Torque (Tfrac) 17.40Nm - 1.900Nm -
Fracture Angle of Twist (frac) 1080 - 90 -
Ductility Ratio (ult/y) 16 1.12 1 0.791
Failure Type and Angle Ductile,1080 Brittle, 90
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Element principle of stress and strain
An element that experiences equal shear stress on all sides is called pure shear. The element below is
experiencing pure shear at 0 degrees. The shear is parallel to all edges and meets at the corners of the
element. The effect of pure shear causes a shift in the plane of the element as seen below.
If the element is ductile, the plane at which it fractures is 90 degrees to due failure under shear stress.
Whereas if an element is brittle, the plane at which it fails under shear stress will be at 0 degrees.
An element that experiences shear at a plane of 45 degrees behaves differently because the shear is
orthogonal to all the faces of the element. This results in compression and two opposite sides and
tension on the other two opposite sides.
If the element is brittle and at 45 degrees, under pure shear force the element will fail at 45 degrees.
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Section 2: Photographs
Figure.2.1 Initial Setup
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Softwood Hollow Circular Rod
Initial setup of specimen First fracture at 22.5 applied twist
Second fracture at 24
Failure
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Aluminium Rod
Initial setup Wrapping
Failure
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Sample Calculation
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Discussion
This lab was conducted to observe torsional behavior of brittle and ductile, square and
circular cross-sectional specimens. A wooden shaft with a hollow circular cross-section,
and an aluminum shaft with a square cross-section were used. During the lab, data
points for degree of twist and the corresponding torque were recorded to produce
stress-strain curves. From these curves, we can determine experimental shear
modulus, proportionality limit, yield limit, strain hardening, ultimate strength, and fracture
point.
Ductile and brittle materials differ visually by their fracture degrees. Ductile materials will
fracture at 90 degree angles because the maximum torsional stress occurs
perpendicular to the shaft. Brittle materials will fracture at 45 degree angles because
maximum torsional stress is on a helix line along the shaft.
Experimental Results
As the norm with ductile materials, the aluminum shaft fractured at the normal plane.
With an experimental shear modulus of 15.7 GPa, this is lower than the average range
of 26-28 GPa for aluminum alloys. However the experimental yield torsional strength of
255MPa is relatively close to the average of 276MPa for aluminum alloys. The ultimate
strength of this aluminum alloy shaft is 317MPa at 720 degree of twist. The final fracture
point happened at 1080 degree angle of twist with 313MPa of shear stress.
Deviational behaviour was observed in the softwood shaft. The degree of twist was at
22.5 degrees for the first fracture. This initial fracture did happen diagonally at around
45 degrees as seen on the specimen. However, contrary to the expectation of a final 45
degree fraction angle, the shaft fractured in a 90 degree fashion when it finally split into
two at 90 degree angle of twist. This behaviour can be explained by a fundamental
property of wood. At closer inspection, the wooden shaft looked to be pieced together
with interlacing wooden fibres at the fracture point. Wood is anisotropic, meaning it is
constructed unidirectional with layers and fibres. When anisotropic materials fracture,
they will fail through splintering or delamination across the layers or fibres. This is what
happened to the wood hollow shaft.
The experimental shear modulus for the softwood shaft is 0.361GPa with a yield
torsional strength of 10.33MPa. These values fall comfortably into the range of 0.023-
1.18 GPa for the shear modulus and 5.2-15.9MPa for the yield torsional strength.
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SOURCES OF ERROR
There were numerous sources of error that have occurred during this experiment, these errors cause the fluctuations between our experimental values in comparison with the theoretical. Some of these errors were human mistakes, others were technical. Human errors occur while recording the torque at different intervals, especially in materials such as Aluminum which required a large number of rotations. Thus, the twisting of the wheel cannot be perfectly rotated and stopped at our desired value; furthermore, we were unable to stop the rotation at the instant the material fails.
.The errors incurred in this experiment can be divided into two broad types:
Random Errors:
These could have included:
1. Human errors: occur while recording the torque at different intervals, especially in materials such as Aluminum which required a large number of rotations. Thus, the twisting of the wheel cannot be perfectly rotated and stopped at our desired value; furthermore, we were unable to stop the rotation at the instant the material fails.
2. Equipment/Machine/sample error: The errors mentioned here involve the
machinery used in the experimental process as well as errors present in the test sample itself. Due to the natures and shapes of the specimens provided, some did not fit perfectly into the jaws and jigs of the apparatus. Because of this, the specimens remained slightly loose within the jaws and were not completely fixed at all times. As a result, accuracy of the measurements provided was compromised. This was especially true in the cases of the plastic acrylic square tube. Furthermore, there could be a chance that some of the samples themselves could have been defective. The impurities and flaws of the materials may affect our results
3. Adjustment errors: not noting down the readings quickly at the proper torque
would result in a lower torque as the material is adjusting itself at a lower torque, introducing more errors.
4. Backlash: When stronger twists were applied to the specimens, they tended to
attempt to revert to their original states and untwist very quickly in order to
reduce the strain put on them. As a result of this, it became difficult to read the
true values on the torsion meter before they artificially went down as the
specimens attempted to untwist themselves.
The impurities and flaws of the materials may affect our results
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Systematic Errors:
These could have included:
1. Zero errors: The measurement devices may not have been set properly to 0 before using them for measurement
2. Calibration errors: The measurement devices may have an inherent error
induced during its calibration. Therefore, the torque given by the torsiometer and the given angle may not be correct.
Improvements : A few suggestions to improve and give more accurate results for the experiment are:
1. Increase the number of samples size for each material. This would give more data to use in the average.
2. With more number of data points we can apply principle of statistics and do a
statistical analysis for greater accuracy. This would reduce the deviation in the system significantly.
3. Making sure that the readings are only taken once the specimen is properly positioned in the apparatus.
4. Avoid a time lag by trying to take quickly take readings or have a computer
record data directly
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Conclusion
1. Aluminum Square Bar
The Aluminum rod fails under shear by fracturing along a 90 degree vertical
plane; which is consistent with failure due to torsion of a ductile material.
The location of the fracture of the aluminum rod should also be noted as the failure occurred closed to its end consistent with the theory.
The shear modulus of elasticity, yield torsional shear stress and ultimate torsional shear stress and strain are within the predicted value range.
Ductile materials can support a much greater torque than brittle materials, leading to a larger distortion prior to failure.
2. Hollow Circular Softwood Rod
Contrary to the expectation of a final 45 degree fraction angle, the shaft fractured in a 90 degree fashion when it finally split into two at 90 degree angle of twist. This behaviour can be explained by a fundamental property of wood (anisotropic) as mentioned on the discussion.
The discrepancy in theoretical and experimental shear modulus of elasticity values can be attributed to random and systematic error.
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Appendix
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