1. Torsion: Shear Stress & Twist (3.1-3.5)
MAE 314 – Solid Mechanics
Yun Jing
Torsion: Shear Stress & Twist

2. Torsion of Circular Shafts
In this chapter, we will examine uniaxial bars subject to torque.
Where does this occur?
Transmission Shaft
Force Couples
Torsion: Shear Stress & Twist

3. Torsion of Circular Shafts
We assume
Bar is in pure torsion
Small rotations (the length and radius will not change)
How does the bar deform?
Cross-section of the bar remains the same shape, bar is simply rotating.
Cross-section remains perpendicular to axis of cylinder (cylinder does not warp).
Not true for most non-circular bars
Torsion: Shear Stress & Twist

4. Torsion: Shear Stress & Twist
Angle of Twist
Deformation of a circular shaft subjected to pure torsion
Fix left end of shaft
A moves to A’
 = angle of twist (in radians)
What are the boundary conditions on ?
(x) = 0 at x = 0
(x) =  at x = L
For pure torsion,  is linear. x
Torsion: Shear Stress & Twist

5. Torsion: Shear Stress & Twist
Shearing Strain
Calculate the surface shear strain in the cylinder.
For pure torsion (x) = x / L, so
Torsion: Shear Stress & Twist

6. Torsion: Shear Stress & Twist
Shearing Strain
Maximum shear strain on surface
The maximum shear strain on the surface of the cylinder occurs when ρ=c.
We can express the shearing strain at any distance from the axis of the shaft as
Torsion: Shear Stress & Twist

7. Torsion: Shear Stress & Twist
Shearing Strain
We can also apply the equation for maximum surface shear strain to a hollow circular tube.
This applies for all types of materials: elastic, linear, non-linear, plastic, etc. c c
Torsion: Shear Stress & Twist

8. Elastic Shearing Stress
Calculate shear stress in a bar made of linearly elastic material.
Recall Hooke’s Law for shearing stress: τ=Gγ
Torsion: Shear Stress & Twist

9. Torsion: Shear Stress & Twist
Torque
We still need to relate τ to the applied torque T, which is generally the known, applied load.
First, find the resultant moment acting on a cross-section and set this equal to T. c
Torsion: Shear Stress & Twist

10. Torsion: Shear Stress & Twist
Torque
Continuing from previous slide:
Where J is the polar moment of inertia of the cross section of the bar (see Appendix A.3 in your textbook).
Plug this into the equation for τmax.
Torsion: Shear Stress & Twist

11. Torsion: Shear Stress & Twist
Torque
For a non-uniform bar
For a continuously varying bar
Torsion: Shear Stress & Twist

12. Torsion: Shear Stress & Twist
Inclined Plane
Cut a rectangular element along the plane at an angle θ.
Torsion: Shear Stress & Twist

13. Torsion: Shear Stress & Twist
Inclined Plane y x
Sum forces in x-direction.
Sum forces in y-direction.
Torsion: Shear Stress & Twist

14. Torsion: Shear Stress & Twist
Inclined Plane
τmax occurs at θ = 0º, ±90º
σmax occurs at θ = ±45º
τmax = σmax
When σθ is max, τθ = 0, and when τθ is max, σθ =0.
Torsion: Shear Stress & Twist

15. Torsion: Shear Stress & Twist
Example Problem
Part 1. For the 60 mm diameter solid cylinder and loading shown,
determine the maximum shearing stress.
Part 2. Determine the inner diameter of the hollow cylinder , of 80 mm
outer diameter, for which the maximum stress is the same as in part 1.
Torsion: Shear Stress & Twist

16. Torsion: Shear Stress & Twist
Example Problem
Part 1. For the aluminum shaft shown (G = 27 GPa), determine the torque
T that causes an angle of twist of 4o.
Part 2. Determine the angle of twist caused by the same torque T in a solid
cylindrical shaft of the same length and cross-sectional area.
Torsion: Shear Stress & Twist

17. MAE 314 – Solid Mechanics Yun Jing
Torsion: Statically Indeterminate Problems and Transmission Shafts ( )
MAE 314 – Solid Mechanics
Yun Jing
Torsion: Statically Indeterminate Problems and Transmission Shafts

18. Statically Determinate Problems
Find the maximum shearing stress in each bar. T3 T2 T1
Torsion: Statically Indeterminate Problems and Transmission Shafts

19. Statically Indeterminate Problems
Method for torsion is the same as the method for statically indeterminate axial load deflection problems.
Apply what you’ve already learned:
M = R – N
M = number of compatibility equations needed
R = number of unknown reactions (or internal stresses)
N = number of equilibrium equations
Compatibility equations for a torsion problem are based on angle of twist.
Torsion: Statically Indeterminate Problems and Transmission Shafts

20. Statically Indeterminate Problems
Find the largest torque T0 that can be applied to the end of shaft
AB and the angle of rotation of the end A of shaft AB. Allowable shearing stress is
LCD
dCD
dAB
LAB rB rC
Torsion: Statically Indeterminate Problems and Transmission Shafts

21. Torsion: Shear Stress & Twist
A circular shaft AB consists of a 10-in.-long, 7/8-in.-diameter steel cylinder, in which a 5-in.long,5/8-in.-diameter cavity has been drilled from end B. The shaft is attached to fixed supports at both ends, and a 90 lb.ft torque is applied at its midsection. Determine the torque exerted on the shaft by each of the supports.
Torsion: Shear Stress & Twist

22. Torsion: Statically Indeterminate Problems and Transmission Shafts
In a transmission, a circular shaft transmits mechanical power from one device to another.
ω = angular speed of rotation of the shaft
The shaft applies a torque T to another device
To satisfy equilibrium the other device applies torque T to the shaft.
The power transmitted by the shaft is
Generator
Turbine
Torsion: Statically Indeterminate Problems and Transmission Shafts

23. Torsion: Statically Indeterminate Problems and Transmission Shafts
Units for P=Tω
ω = rad/s
T = N·m (SI)
T = ft·lb (English)
P = Watts (1 W = 1 N·m/s) (SI)
P = ft·lb/s (1 horsepower = hp = 550 ft·lb/s) (English)
We can also express power in terms of frequency.
Torsion: Statically Indeterminate Problems and Transmission Shafts

24. Torsion: Statically Indeterminate Problems and Transmission Shafts
Example Problem
A 1.5 meter long solid steel shaft of 22 mm diameter is to
transmit 12 kW. Determine the minimum frequency at which the
shaft can rotate, knowing that G = 77.2 GPa, that the allowable
shearing stress is 30 MPa, and that the angle of twist must not
exceed 3.5o.
Torsion: Statically Indeterminate Problems and Transmission Shafts

25. Stress Concentrations in Circular Shafts
Up to now, we assumed that transmission shafts are loaded at the ends through solidly attached, rigid end plates.
In practice, torques are applied through flange couplings and fitted keyways, which produce high stress concentrations.
One way to reduce stress concentrations is through the use of a fillet.
Fitted keyway
Flange coupling
Torsion: Statically Indeterminate Problems and Transmission Shafts

26. Stress Concentrations in Circular Shafts
Maximum shear stress at the fillet
Tc/J is calculated for the smaller-diameter shaft
K = stress concentration factor
Fillet
Torsion: Statically Indeterminate Problems and Transmission Shafts

27. Torsion: Statically Indeterminate Problems and Transmission Shafts
Example Problem
The stepped shaft shown rotates at 450 rpm. Knowing that r = 0.25 in,
determine the maximum power that can be transmitted without
exceeding an allowable shearing stress of 7500 psi.
Torsion: Statically Indeterminate Problems and Transmission Shafts