How to use this calculator
- Choose solid or hollow. Pick solid round shaft or hollow round shaft. Hollow shafts reveal an inside diameter input.
- Enter torque and length. Use the torque carried by the shaft and the uniform length over which twist is calculated.
- Enter shaft diameters. Enter the outside diameter, and the inside diameter if the shaft is hollow.
- Set material and allowable stress. Enter shear modulus G for twist and allowable shear stress for capacity.
- Review stress, twist and capacity. Compare maximum shear stress and safety factor with your allowable, then check angle of twist against the machine limit.
How it works
This calculator uses the elastic torsion equation for uniform circular shafts:
T / J = tau / r = G*theta / L
where T is torque, J is the polar moment of inertia,
tau is shear stress, G is shear modulus and
theta is angle of twist.
For a solid shaft, J = pi*D^4/32. For a hollow shaft,
J = pi*(Do^4 - Di^4)/32. Maximum stress is evaluated at the
outside surface:
tau_max = T*(Do/2)/J. The twist is
theta = T*L/(G*J).
If you need the diameter from an allowable stress, use the shaft diameter calculator. If the shaft also carries transverse load, check bending with the shaft deflection calculator.
Worked example
Verified against the live calculator
A solid steel shaft with D = 50 mm, L = 1000 mm,
T = 1000 N*m and G = 79 GPa has
J = pi*50^4/32 = 613,592 mm^4.
The maximum shear stress is
tau = 1,000,000*25/613,592 = 40.74 MPa. The angle of twist is
theta = T*L/(G*J) = 0.0206 rad = 1.18 deg. With an allowable shear
stress of 80 MPa, the stress safety factor is about 1.96.
Frequently asked questions
What is the torsion formula for a circular shaft?
For a round shaft in elastic torsion, T/J = tau/r = G*theta/L. The calculator uses tau_max = T*(Do/2)/J for surface shear stress and theta = T*L/(G*J) for angle of twist.
How do you calculate the polar moment of inertia of a shaft?
For a solid round shaft, J = pi*D^4/32. For a hollow round shaft, J = pi*(Do^4 - Di^4)/32, where Do is outside diameter and Di is inside diameter.
Where is maximum shear stress in a shaft?
Maximum torsional shear stress occurs at the outside surface of a circular shaft. In the elastic model, stress varies linearly from zero at the centerline to tau_max at the outer radius.
Why can a hollow shaft be efficient in torsion?
A hollow shaft removes material near the centerline, where torsional shear stress is low. For the same outside diameter it loses some J, but often saves much more weight than torsional stiffness.
What shear modulus should I use?
Use the material value for G. Typical room-temperature values are about 79 GPa for steel and about 26 GPa for aluminium. Supplier data should be used for production checks.
Does this include keyways or fatigue?
No. This is a smooth, uniform, circular-shaft elastic torsion check. Keyways, shoulders, splines, welds, surface finish, fatigue and stress concentrations require separate factors or a detailed shaft design check.
Method & assumptions
- Uniform circular shaft, solid or concentric hollow, in linear elastic torsion.
- Plane sections are assumed to remain plane; the formula is for circular shafts, not open channels or rectangular bars.
- Stress is evaluated at the outside surface; hollow-shaft bore stress is lower than the outside stress for the same torque.
- Static first-pass check only: fatigue, shock, keyways, shoulders, splines, welds, stress concentration factors and non-circular warping are not modeled.