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Hollow cross-sections

Material near the center of a section contributes little to torsional or bending stiffness, since it lies close to the neutral axis. A hollow shaft can therefore give almost the same stiffness as a solid one with much less weight, by moving material outward.

J = \dfrac{\pi(D^4-d^4)}{32}

polar second moment of area, hollow circular shaft

Symbols

$D$	outer diameter	m
$d$	inner diameter	m
$J$	polar second moment of area	m⁴

Example

A hollow shaft $D = 40$ mm, $d = 30$ mm:

$J = \pi(40^4 - 30^4)/32 \approx 214\,000$ mm⁴, more than a solid $d = 30$ mm shaft ( $\approx 79\,500$ ).

Hollow shafts give a lot of stiffness per kilogram – useful where weight is critical, as in aircraft and bicycles.

Practise torsion and bending for free →

← Bending stress · Principal stresses and Mohr's circle →

Part of Strength of Materials: Torsion and bending.