Torsion of shafts

torsional stressangle of twistpolar moment of inertiadrive shaft design

Twisting a round shaft: the polar moment of inertia, the shear stress it produces, and how far the far end rotates under load.

Polar Moment of Inertia — Solid Shaft

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

Polar moment of inertia of a solid round shaft, J = πd⁴/32, in m⁴ — exactly twice the diametral moment of inertia.

Torsional Shear Stress (τ = Tr/J)

\tau = \frac{T r}{J}

Torsional shear stress at radius r in a round shaft, τ = Tr/J, peaking at the surface, with J entered in m⁴ as a plain number.

Angle of Twist (φ = TL/JG)

\varphi = \frac{T L}{J G}

Angle of twist of a round shaft under torque, φ = TL/JG, the stiffness check that governs long drive and torque shafts.

Shear Modulus (G = τ/γ)

G = \frac{\tau}{\gamma}

Shear (rigidity) modulus as shear stress divided by shear strain, roughly 0.38 of Young's modulus for common metals.

How they fit together

A shaft carrying torque develops shear stress that is zero on the centreline and greatest at the surface, which is why hollow shafts are so efficient — removing the core takes away material that was barely working. The polar moment J plays the role here that the area moment I plays in bending, and it rises with the fourth power of diameter, so a 10% larger shaft is nearly 50% stiffer in torsion.

Design usually splits on which limit binds first. A short, heavily loaded shaft fails on stress; a long one is governed by angle of twist, because a machine tool or driveline that winds up several degrees under load loses accuracy long before anything breaks. Check both — and note the twist depends on the shear modulus G, not Young's modulus E. Substituting one for the other overstates stiffness by about a factor of 2.6 in steel.