A torque applied about the longitudinal axis of a shaft produces torsion: the shaft twists and develops shear stress and shear strain.

For a circular shaft under elastic torsion,

$$\tau(r)=\frac{Tr}{J},$$

where $T$ is the applied torque, $r$ is distance from the shaft axis and $J$ is the polar second moment of area.

Stress distribution

Shear stress is zero at the centre and increases linearly with radius, reaching its maximum value at the outer surface:

$$\tau_{\max}=\frac{Tc}{J}.$$

Angle of twist

For a uniform shaft of length $L$ and shear modulus $G$,

$$\phi=\frac{TL}{JG}.$$

The quantity $JG$ is the torsional analogue of flexural rigidity.

Polar second moment of area

For a solid circular shaft of radius $R$,

$$J=\frac{\pi R^4}{2}.$$

Geometry therefore has a very strong effect on resistance to twisting.