🌀 Shaft Torsion

Written and maintained by the PhDino author · Last reviewed 21 September 2026 · Checked against 2 independent reference calculations · how PhDino checks its numbers

Shear stress and angle of twist in a circular shaft carrying torque.

A shaft transmitting torque (from a motor to a load, for instance) develops shear stress that is zero at the center and maximum at the outer surface — torsion loads a circular cross-section very differently from bending. The shaft also twists along its length by an angle proportional to the torque and inversely proportional to both the material's shear stiffness and the shaft's polar moment of inertia.

This is classical torsion theory for a solid circular shaft; non-circular cross-sections (rectangular, for instance) behave quite differently and need a different set of formulas entirely.

Key formula

Polar moment: J = πd⁴/32
Max shear stress: τ_max = T(d/2)/J
Angle of twist: φ = TL/(GJ)

Variables

T

applied torque

d

shaft diameter

L

shaft length

G

shear modulus of the shaft material

Notes & limitations

• Shear stress is zero at the centerline and increases linearly to a maximum at the outer surface — this is why hollow shafts (tubes) can be more material-efficient than solid shafts for the same torque capacity, since the low-stress material near the center is contributing little.
• Stress concentrations at keyways, shoulders, and splines are not included in this basic formula — see the Stress Concentration article for how local geometry raises the actual peak stress.