How to use it
- Choose a solid or hollow shaft.
- Enter the torque in N·m, the outer diameter and, for a tube, the inner diameter in mm.
- Enter the length in mm and the shear modulus G in GPa.
- Read the maximum shear stress, the twist angle, J and J / r.
Formulas
- Polar moment of inertia: J = π(D⁴ − d⁴) / 32 (d = 0 for a solid shaft).
- Maximum shear stress (at the surface): τ = T × (D/2) / J.
- Angle of twist: θ = T × L / (G × J) radians.
- Polar section modulus: Wp = J / (D/2), so τ = T / Wp.
Units: torque in N·mm, lengths in mm and G in N/mm² give τ in MPa.
Worked example
A solid 20 mm steel shaft, 1,000 mm long, carries 100 N·m = 100,000 N·mm, G = 80 GPa:
- J = π × 20⁴ ÷ 32 = 15,708 mm⁴;
- τ = 100,000 × 10 ÷ 15,708 = 63.662 MPa (9,233 psi);
- θ = 100,000 × 1,000 ÷ (80,000 × 15,708) = 0.0796 rad = 4.5595°.
Bored to a 20/10 mm tube: J = 14,726.2 mm⁴, τ = 67.906 MPa, the twist rises only a little.
Tips
- Stress concentrations at keyways, shoulders and grooves raise the local stress well above τ; apply the factor for your detail.
- Combined loading: shafts that also bend need a combined stress check (for example von Mises).
- Power to torque: T = P / ω; at 1,500 rpm, 15 kW is 15,000 ÷ (2π × 25) = 95.5 N·m.