Angle of Twist Calculator
How far a round shaft twists under a torque — from its length, diameter and shear modulus — with the maximum shear stress at the surface, the polar moment of area, the torsional stiffness, and the diameter that would hold the twist to a limit.
A torque twists a shaft through an angle proportional to the torque and the length and inversely to the shear modulus and the polar moment of area — which goes with the fourth power of diameter, so a small increase in size stiffens a shaft a lot.
How the angle of twist calculator works
A torque twists a shaft through an angle proportional to the torque and the length and inversely to the shear modulus and the polar moment of area — which goes with the fourth power of diameter, so a small increase in size stiffens a shaft a lot. The shear stress peaks at the surface at torque times radius over the polar moment. Drive-shaft design usually limits twist to about a degree per metre and stress to a fraction of the material's shear yield.
Formula: J = π d⁴/32 (solid), π (D⁴ − d⁴)/32 (hollow); θ = T L / (G J); τ_max = T r / J
Worked examples
| Inputs | Angle of twist (°) | Note |
|---|---|---|
| 500 N·m on a 40 mm steel shaft | 1.7098 | under two degrees |
| Hollow | 2.5012 | lighter, somewhat more twist |
| Aluminium | 5.2609 | three times the twist |
FAQFrequently asked questions
What is angle of twist?
How far one end of a shaft rotates relative to the other under torque. It depends on the torque, the length, the material's shear modulus and the polar moment of area of the cross-section.
Why does diameter matter so much?
Because the polar moment goes with the fourth power of diameter. Ten per cent more diameter is 46% more stiffness and 33% less stress.
Why use a hollow shaft?
The material near the axis carries little shear and contributes little stiffness. Removing it saves weight at small cost — a tube with 75% bore keeps 68% of the solid shaft's stiffness at 44% of the mass.
What twist is acceptable?
Machine design commonly limits shafts to about 0.25 to 1° per metre for precision drives and up to 3 for rough ones. Shear stress must also stay below the allowable for the material, typically half its tensile yield.
What shear modulus should I use?
Steel about 80 GPa, aluminium 26, titanium 44, brass 40, cast iron 40 to 50. It is the material's stiffness in shear, distinct from Young's modulus.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Physical Laboratory — the UK's national measurement institute
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.