Shaft Torsion Calculator
Enter shaft diameter and torque to get surface shear stress (τ = Tr/J), or leave torque blank to size the shaft from an allowable stress.
Last updated: 2026-09-28
How the calculation works
- Polar moment J = πd⁴/32 — diameter enters to the fourth power, so small diameter increases buy huge capacity.
- Surface shear stress follows τ = T·r/J; it scales linearly from zero at the center to maximum at the surface.
- With torque blank, the tool inverts the relation: the largest torque the allowable stress permits.
Formula
τ = T·r / J J = π·d⁴ / 32 T_capacity = τ_allow × J / r
| Symbol | Meaning | Unit |
|---|---|---|
τ | Shear stress at the outer surface | MPa |
T | Torque | N·m |
J | Polar moment of inertia | mm⁴ |
Worked example
Interpreting the result
The fourth-power diameter sensitivity is why shafts are cheap insurance: one size up roughly doubles torque capacity. Two practical deratings come before the theoretical number fails: keyways and shoulders concentrate stress (count on 60-80% of the theoretical capacity), and reversed or shock torsion needs a fatigue allowable well below the static one. Hollow shafts beat solid ones per kilogram — removing the lightly-stressed core barely reduces J.
Assumptions
- Solid circular shaft, linear-elastic, static torsion.
- No keyway, shoulder or hole stress concentrations applied.
- Allowable stress is user-set — typical machine-design practice is 40% of yield for static, less for fatigue.
Limitations
- Not valid for hollow shafts (use J = π(D⁴−d⁴)/32) or non-circular sections (warping complicates the math).
- Critical speeds, torsional vibration and combined bending+torque need full shaft design.
Frequently asked questions
What size shaft for a given torque?
Solve d from τ = 16T/(πd³): d = (16T/πτ)^⅓. For 100 N·m at 100 MPa allowable: d ≈ 17 mm — choose 20 mm for margin and keyway effects.
Why does diameter matter so much?
Torque capacity scales with d³ (stress) or d⁴ (stiffness): material added at the outer surface, where stress is highest, works hardest. That's also why hollow shafts are efficient.