Consider a bar to be rigidly attached at one end and twisted at the other end by a torque or twisting moment T equivalent to F × d, which is applied perpendicular to the axis of the bar, as shown in the figure. Such a bar is said to be in torsion.

Bar in torsion


Torsional Shearing Stress, τ

For a solid or hollow circular shaft subject to a twisting moment T, the torsional shearing stress τ at a distance ρ from the center of the shaft is

$\tau = \dfrac{T \rho}{J}$   and   $\tau_{max} = \dfrac{Tr}{J}$

where J is the polar moment of inertia of the section and r is the outer radius.

For solid cylindrical shaft:
Section of a solid shaft$J = \dfrac{\pi}{32} D^4$

$\tau_{max} = \dfrac{16T}{\pi D^3}$

For hollow cylindrical shaft:
Hollow cylindrical shaft$J = \dfrac{\pi}{32}(D^4 - d^4)$

$\tau_{max} = \dfrac{16TD}{\pi(D^4 - d^4)}$

Angle of Twist

The angle θ through which the bar length L will twist is

$\theta = \dfrac{TL}{JG} \, \text{in radians}$

where T is the torque in N·mm, L is the length of shaft in mm, G is shear modulus in MPa, J is the polar moment of inertia in mm4, D and d are diameter in mm, and r is the radius in mm.

Power Transmitted by the Shaft

A shaft rotating with a constant angular velocity ω (in radians per second) is being acted by a twisting moment T. The power transmitted by the shaft is

$P = T\omega = 2\pi T f$

where T is the torque in N·m, f is the number of revolutions per second, and P is the power in watts.