1. Torsion 
Introduction
-- Analyzing the stresses and strains in machine parts which are subjected to torque T
Circular
-- Cross-section Non-circular
Irregular shapes
-- Material (1) Elastic
(2) Elasto-plastic
-- Shaft (1) Solid
(2) Hollow

2. 3.1 Introduction T is a vector  Two ways of expression
-- Applications:
a. Transmission of torque in shafts, e.g. in automobiles

3. Assumptions in Torque Analysis:
a. Every cross section remains plane and undistorted.
b. Shearing strain varies linearly along the axis of the shaft.

4. 3.2 Preliminary Discussion of the Stresses in a Shaft
Where  = distance (torque arm)
Since dF =  dA
The stress distribution is Statically Indeterminate.
Free-body Diagram

5. -- Must rely on “deformation” to solve the problem.
Analyzing a small element:

6. 3.3 Deformations in a Circular Shaft
 =  (T, L) -- the angle of twist (deformation)
Rectangular cross section warps under torsion

7. A circular plane remains circular plane

8. Determination of Shear Strain 
(in radians)
The shear strain   

9.  = c = radius of the shaft
Since

10. 3.4 Stresses in the Elastic Range
Hooke’s Law 
Therefore,
(3.6)

12. (3.1)
(3.6)
But
(3.9)
Therefore,
Or,

13. Substituting Eq. (3.9) into Eq. (3.6)
(3.10)
(3.9)
These are elastic torsion formulas.
For a solid cylinder:
For a hollow cylinder:

14. (3-13)
Since  

15. 3.5 Angle of Twist in the Elastic Range
(3.3)
(3.15)
Therefore,
Eq. (3.3) = Eq. (3.15) 
Hence,

16. For Multiple-Section Shafts:

17. Shafts with a Variable Circular Cross Section

18. 3.6 Statically Indeterminate Shafts
-- Must rely on both
Torque equations and
(2) Deformation equation, i.e.
Example 3.05

19. 3.7 Design of Transmission Shafts
-- Two Parameters in Transmission Shafts:
a. Power P
b. Speed of rotation
where  = angular velocity (radians/s) = 2
 = frequency (Hz)
[N.m/s = watts (W)]
(3.21)

20. (3.21)
(3.9)
Therefore,
For a Solid Circular Shaft: 