What is the torsion formula?

The torsion formula τ = T·r/J relates shear stress τ (Pa) to torque T (N·m), radial distance r (m), and polar moment of inertia J (m⁴). For a circular shaft the maximum stress occurs at the outer surface where r equals the outer radius.

What is the polar moment of inertia J?

For a solid circle of diameter d, J = π·d⁴/32. For a hollow circle with outer diameter D and inner diameter d, J = π·(D⁴ − d⁴)/32. J captures how the cross-section resists twisting.

What units does this calculator use?

Inputs can be entered in mm or inches for diameter/length, N·m or lbf·in for torque, and GPa or psi for shear modulus. The tool converts everything to SI internally for consistency.

What shear modulus should I use?

Common values: steel ≈ 79 GPa, aluminum ≈ 26 GPa, brass ≈ 38 GPa, titanium ≈ 44 GPa. Use the value for your specific alloy from a materials reference for accurate results.

Does this apply to non-circular shafts?

No. The torsion formula τ = T·r/J is exact only for circular cross-sections. Non-circular sections (rectangular, I-beam) require Prandtl's stress function or finite element methods.

What is the angle of twist?

The angle of twist φ = T·L/(G·J) is the total rotation (in radians, converted to degrees here) between the two ends of the shaft. It depends on shaft length L and shear modulus G in addition to the torque and cross-section.

Torsional Stress Calculator

Name: Torsional Stress Calculator
Availability: InStock
Author: Nham Vu

Enter torque, shaft dimensions, length, and shear modulus to instantly compute shear stress and angle of twist for solid or hollow circular shafts.

Shaft Type

Solid Hollow

Unit System

Outer Diameter (mm)

Inner Diameter (mm)

Torque (N·m)

Shaft Length (mm)

Shear Modulus (G) (GPa)

Fill in the inputs and click Calculate to see results.

Summary

Enter torque, shaft dimensions, length, and shear modulus to instantly compute shear stress and angle of twist for solid or hollow circular shafts.

How it works

Select shaft type: solid circle or hollow circle (tube).
Enter the outer diameter (and inner diameter for hollow shafts).
Input the applied torque, shaft length, and shear modulus of the material.
The polar moment of inertia J is computed from the geometry.
Maximum shear stress τ = T · r / J is evaluated at the outer radius.
Angle of twist φ = T · L / (G · J) gives the rotation in degrees.

Use cases

Sizing drive shafts in mechanical transmissions.
Checking whether a propeller shaft meets allowable stress limits.
Verifying torsion bar spring designs in suspension systems.
Teaching the torsion formula in undergraduate mechanics of materials courses.
Comparing solid vs. hollow shaft performance for a given weight budget.
Quick feasibility checks before running FEA on a shaft assembly.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu