How this calculation works
A shaft can meet a nominal stress comparison yet twist too far for a mechanism. Stress follows torque and cross-section, while angular twist also follows length and shear modulus. Reporting both makes those separate questions visible.
J = π(D⁴ − d⁴)/32; τ = TD/(2J); θ = TL/(GJ)A worked example
A 40 mm outside diameter with a 20 mm bore gives J = 235619.449 mm⁴. At 200 N m, the surface shear is 16.9765 MPa. One metre of this shaft at 80 GPa shear modulus twists about 0.608 degrees.
Good to know
Pure elastic static torsion of a straight uniform circular shaft only. A keyway, shoulder, spline or cross-hole changes local stress. The page does not choose material strength, fatigue life or an allowable twist for your equipment.
Reference: Boston University — circular shaft torsion (opens in a new tab) ↗
What this model answers
A shaft can meet a nominal stress comparison yet twist too far for a mechanism.
Stress follows torque and cross-section, while angular twist also follows length and shear modulus. Reporting both makes those separate questions visible.
For a keyed connection, the Rectangular shaft key shear and bearing stress calculator examines the key shear and projected contact face; shaft surface torsion alone cannot replace those checks.
Keep the input conventions consistent
Enter outside and bore diameters, not radii.
The calculation converts torque from N m into N mm and modulus from GPa into N/mm² before using millimetre geometry. A zero bore represents a solid circular shaft.
Read the result carefully
The fourth-power polar moment explains why material near the outside contributes strongly to torsional stiffness.
Removing a small central core does not remove stiffness in direct proportion to removed area. Compare at fixed outside diameter before comparing by mass.
A useful comparison holds torque, outside diameter and material fixed while changing only the bore. This isolates the section effect. Changing shaft length separately isolates angular compliance, so a change in twist can be traced to geometry or span rather than confused with a change in nominal stress.
Work through the numbers
A 40 mm outside diameter with a 20 mm bore gives J = 235619.449 mm⁴.
At 200 N m, the surface shear is 16.9765 MPa. One metre of this shaft at 80 GPa shear modulus twists about 0.608 degrees.
Where the calculation stops
Pure elastic static torsion of a straight uniform circular shaft only.
A keyway, shoulder, spline or cross-hole changes local stress. The page does not choose material strength, fatigue life or an allowable twist for your equipment.
A few questions, answered
Will doubling the shaft length double shear stress?
No. At fixed torque and section it doubles twist but leaves nominal shear stress unchanged.
What should I change in the comparison?
Double the shaft length; watch twist change while surface stress stays fixed.