1. FME201 Solid & Structural Mechanics I
Dr. Hussein Jama
Office 414
Lecture: Mon 11am -1pm (CELT)
Tutorial Tue 12-1pm (E207)
12/4/2018

2. Lecture Outline
This lecture is based on the book by R C Hibbeler Chapter 5
5.1Torsional deformation of a circular shaft
5.2 The Torsion Formula
5.3 Power of Transmission
5.4 Angle of twist
5.5 Statically indeterminate Torque Loaded Members*
5.8 Stress concentration
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3. Engineering Professor of the Year
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4. Engineering Professor of the Year
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5. Student complaints
I'm currently studying for my mechanical engineering exam on this Friday and I'd like to hear stories of bad professors to lighten my frustration. The professor for this class is the least liked professor in the ME department. The only preparation for what's on the exam is basically the table of contents that he typed out. He also noted that it's an incomplete list so we have to figure out what else to study. Roughly 30% of the class fails and he doesn't care and has said so happily.
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6. Torsion
Torque is a moment that twists a member about its longitudinal axis.
External loads (T) produce internal loads which produce deformation, strain and stress.
Application
Design of shafts in machinery
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7. Introduction Assumption: The angle of rotation is small
The length of the shaft will remain unchanged
The radius of the shaft will remain unchanged.
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8. Torsion
Before Torque
After Torque
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9. 5.1 Torsional deformation of a circular shaft
f(x) = angle of twist (varies linearly along the length, 0 at x = 0, max at x = L)
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10. Torsional deformation cont..
Recall g = shear strain (rad)
Notice, shear strain, g varies linearly with radial distance, r, and is max on the outer surface!!
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11. Torsional deformation cont..
Notice, shear strain, g varies linearly with radial distance, r, and is max on the outer surface!!
Distance from center to point of interest
Distance from center to outer fiver (i.e. outer radius)
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12. Stress due to torsion Deformation = shear strain
Shear stress is proportional to shear strain (Hooke’s Law)
If you can visualize deformation, you can visualize stress
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13. Recap
f = angle of twist varies from zero at fixed support to max at end.
g = shear strain varies from zero at center to max at outer fiber
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14. 5.2 The Torsion formula If linear elastic, Hooke’s law applies, t = Gy
Therefore, stress follows same profile as strain!!
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15. Torsion formula
Derivation – simple Torque balance. The torque produced by the stress distribution over the entire cross section must be equal to the resultant internal torque, or:
This is simply polar moment of inertia, J (an area property)
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16. Torsion formula or Torque (N-m, N-mm or lb-in, lb-ft, etc)
Outer radius of shaft (m or in)
Polar moment of inertia (m4 or in4)
Max shear stress in shaft (MPa, psi/ksi, etc.) or
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17. Polar moment of inertia
This is a geometric property and is always positive.
It has units of m^4 of mm^4
Solid shaft:
Hollow shaft:
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18. Stress profile
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19. Stress profile – wood failure
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20. Example 5.3
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21. Solution Example 5.3
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22. Example 5.4
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23. Solution example 5.4
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24. 5.3 Power transmission P = Tw
Nothing new, just calculate Torque, T, from power equation:
P = Tw
Careful with units!
Power (watts, ft-lb/s or hp)
Angular velocity (rad/s or Hz)
Torque (N-m, lb-ft)
f = Hz or rev/s
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25. Power transmission cont..
Shaft powered by 500 W electric motor spins at 10 Hz, find Torque in shaft.
P = Tw
10 Hz (2p rad/rev) = rad/s
T = 500 N-m/s
62.83 rad/s
= 7.96 N-m
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26. Class example Determine: Torque throughout shaft
Stress throughout shaft
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27. Homework
5.5
5.7
5.15
5.19
5.25
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28. Recommended Texts
Mechanics of Materials – 2nd Edition, Madhukar Vable – available online FREE
Engineering Mechanics – Statics, R.C. Hibbler,
Engineering Mechanics – Statics, D.J. McGill & W.W. King
Mechanics of Materials , J.M. Gere & S.P. Timoshenko
Mechanics of solids, Abdul Mubeen, Pearson Education Asia
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