Uniform Torsion - torsion of a prismatic bar subjected to torques acting only at the ends

angle of twist

φ = θ L = T L / G J

We can experimentally determine the modulus of rigity by measuring angle of twist, torque, length, and diameter.

Nonuniform Torsion - bar need not be prismatic and the applied torques may act anywhere along the axis of the bar

bar consisting of prismatic segments with constant torque throughout each segment

φ = &Sigma φi = &Sigma (TL / GJ)i

Solution steps:

1. Divide bar into prismatic sections under constant torque.
2. Determine magnitude and direction of internal torques. (equilibrium equations)
3. Determine shear stresses in each segment. The maximum shear stress for the entire bar will be largest of these segment shear stresses. τ_i = (Tc / J)_i
4. Determine angle of twist in each segment and sum to get the total angle of twist for the bar. φ = Σ φ_i = Σ (TL / GJ)_i

Sign convention:

An internal torque is positive when its vector points away from the cut section and negative when its vector points toward the section.  (Use the right-hand-rule.)

 

bar with continuously varying cross sections and constant torque

Solution steps:

1. The maximum shear stress will occur at the smallest cross section.
2. The angle of twist for the entire bar is the sum of the differential angles of rotation.

 

bar with continuously varying cross sections and continuously varying torque (distributed torque)

Solution steps:

1. Determine the maximum shear stress as a function of x (shear stress distribution) and locate the maximum shear stress. τ(x) = T(x) c(x) / J(x)
2. The angle of twist for the entire bar is the sum of the differential angles of rotation.