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

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

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

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

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

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

A circular plane remains circular plane

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

 = c = radius of the shaft Since

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

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

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:

(3-13) Since  

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

For Multiple-Section Shafts:

Shafts with a Variable Circular Cross Section

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

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)

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