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Angle of twist

A torque does not just cause shear stress in a shaft, it also twists it through an angle $\theta$ over its length. This is analogous to $\delta = FL/(AE)$ for axial elongation, with $G$ and $J$ in place of $E$ and $A$ .

\theta = \dfrac{TL}{GJ}

angle of twist (radians)

Symbols

$\theta$	angle of twist	rad
$L$	length of the shaft	m
$G$	shear modulus	Pa
$J$	polar second moment of area	m⁴

Example

A steel shaft ( $G = 80$ GPa), $J = 79\,500$ mm⁴, $L = 1$ m, $T = 200$ Nm:

$\theta = 200\,000/(80\,000\cdot 79\,500) \approx 0.0314$ rad $\approx 1.8^\circ$ .

Remember to compute

\theta

in radians before converting to degrees.

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