Science & Engineering
Shaft torsion stress.
Maximum shear stress τ = 16T ÷ (πd³) in a solid round shaft (and the hollow-shaft version), the angle of twist, or the smallest diameter for an allowable shear stress. Runs in your browser.
Shaft
Result
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Worked example
A 25 mm solid steel shaft carries 100 N·m: τ = 16T ÷ (πd³) = 16 × 100 ÷ (π × 0.025³) = 32.59 MPa. Over 1 m with G = 79.3 GPa (a handbook figure for steel) it twists φ = TL ÷ GJ = 1.88°. Going the other way, for 100 N·m and an allowable 40 MPa the smallest solid shaft is d = (16T ÷ πτ)^(1/3) = 23.4 mm.
How it works
Twisting a round shaft produces shear stress that is zero on the axis and greatest at the outer surface. For a solid shaft of diameter d carrying torque T, τmax = 16T ÷ (πd³), using the polar moment J = πd⁴ ÷ 32. For a hollow shaft, J = π(do⁴ − di⁴) ÷ 32 and τ = T·(do ÷ 2) ÷ J. The angle of twist over length L is φ = TL ÷ GJ, with G the shear modulus of the material.
The minimum-diameter mode inverts the solid-shaft formula: d = (16T ÷ πτallow)^(1/3), and for a hollow shaft with inner-to-outer ratio k, do = (16T ÷ (πτallow(1 − k⁴)))^(1/3). The allowable stress is your input; it depends on the material, the duty and the safety factor, so this page does not supply one. For torque from power and speed, see the Torque to Horsepower Calculator.
Not for safety-critical design
Planning and learning aid only. It is not for structural, lifting, pressure-system or other safety-critical design, and it is not professional engineering advice. Have a qualified engineer check anything that carries people or could fail dangerously.
Sources and basis: Wikipedia: Torsion (mechanics); Wikipedia: Shear modulus. Checked 2026-10-16. Planning aid only; see the note above.
FAQ
What is the torsional shear stress formula for a shaft?
For a solid round shaft τ = 16T ÷ (πd³), where T is torque and d is diameter. For a hollow shaft τ = 16 T do ÷ (π (do⁴ − di⁴)).
How do I find the shaft diameter for a given torque?
Rearrange to d = (16T ÷ (π τallow))^(1/3), with your allowable shear stress. Choose the allowable stress from the material data with a safety factor.
Why is a hollow shaft more efficient?
Shear stress is highest at the surface and zero at the centre, so material near the axis does little work. A hollow shaft carries nearly the same torque with less weight.
What is the angle of twist?
φ = T L ÷ (G J), where L is the length, G the shear modulus and J the polar moment of the section. Longer or more flexible shafts twist more.
Is torsion the only load I need to check?
No. Real shafts also see bending, shock loads, keyway stress concentration and fatigue. This calculator covers steady torsion of a round section only.