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Torsion

When a shaft is subjected to a torque, shear stress arises that is zero at the center and grows linearly outward toward the surface. The polar second moment of area $J$ describes how well the cross-section resists twisting.

\tau = \dfrac{Tr}{J}

shear stress at distance

r

J = \dfrac{\pi d^4}{32}

polar second moment of area, solid circular shaft

Symbols

$T$	torque	Nm
$\tau$	shear stress	Pa
$J$	polar second moment of area	m⁴
$r$	distance from the center	m

Example

A shaft with $d = 30$ mm, $T = 200$ Nm:

$J = \pi\cdot 30^4/32 \approx 79\,500$ mm⁴, $\tau_{max} \approx 200\,000\cdot 15/79\,500 \approx 37.7$ MPa.

The shear stress is largest at the surface – that is where torsional failure starts.

Practise torsion and bending for free →

← Stress concentration · Angle of twist →

Part of Strength of Materials: Torsion and bending.