Torsion of circular shafts: T/J = τ/r = Gθ/L
How twisting moment creates shear stress in a shaft, why hollow shafts are efficient, and how to size a shaft for power.
A shaft transmitting power is twisted by a torque T. That twist produces shear stress that is zero at the centre and largest at the outer surface.
The torsion equation
TJ = τr = G θL
T is the torque, J the polar moment of inertia, τ the shear stress at radius r, G the shear modulus, θ the angle of twist in radians and L the length.
Polar moment of inertia
Solid: J = π D⁴32 Hollow: J = π (D⁴ − d⁴)32
Maximum shear stress
For a solid shaft, putting r = D/2:
τmax = 16 Tπ D³
Power and torque
A shaft running at N rpm transmits power
P = 2 π N T60
with P in watts and T in N·m.
Worked example
Find the diameter of a solid shaft to transmit 30 kW at 200 rpm if τ must not exceed 60 N/mm².
T = 60 × 30 × 10³2 π × 200 = 1432 N·m = 1.432 × 10⁶ N·mm
D³ = 16 × 1.432 × 10⁶π × 60 = 1.216 × 10⁵ mm³
So D ≈ 49.5 mm. Adopt a 50 mm shaft.
Hollow versus solid
Material near the centre carries almost no stress. Removing it saves weight while losing little strength, which is why vehicle propeller shafts are hollow tubes.
Quick check
For the same material and weight, which shaft carries more torque?