Shaft Torsion Stress Calculator
Torsional shear stress is the internal stress that develops in a shaft when a torque (twisting moment) is applied. This calculator uses the standard torsion formula: tau = T * c / J, where T is the applied torque, c is the distance from the neutral axis to the outer surface, and J is the polar moment of inertia of the cross-section. Choose solid or hollow shaft, enter the dimensions and torque, and the calculator returns maximum shear stress and the angle of twist per unit length.
Torsion formula
tau = T × c / J
J (solid) = pi × d4 / 32
J (hollow) = pi × (do4 - di4) / 32
theta/L = T / (G × J)
Where tau is shear stress (Pa), T is torque (N.m), c = d/2 is the outer radius (m), J is polar moment of inertia (m4), G is shear modulus (Pa), and theta/L is angle of twist per unit length (rad/m).
Design guidance
- The maximum shear stress occurs at the outer surface of the shaft.
- ASME B106.1M recommends allowable shear stress of 55 MPa for steel shafts without keyways (41 MPa with keyways).
- For combined bending and torsion, use the von Mises equivalent stress: sigma_e = sqrt(sigma_b^2 + 3 * tau^2).
- A hollow shaft provides similar torsional strength to a solid shaft with less weight, making it preferred in weight-sensitive applications.
Shaft torsion calculator: frequently asked questions
What is the torsion formula for a circular shaft?
The torsional shear stress is tau = T * c / J, where T is the applied torque (N.m), c is the outer radius of the shaft (m), and J is the polar moment of inertia (m^4). For a solid shaft, J = pi * d^4 / 32. For a hollow shaft, J = pi * (d_o^4 - d_i^4) / 32.
What is the polar moment of inertia?
The polar moment of inertia J measures a cross-section's resistance to torsional deformation. For a solid circular shaft of diameter d, J = pi * d^4 / 32. For a hollow shaft with outer diameter d_o and inner diameter d_i, J = pi * (d_o^4 - d_i^4) / 32.
What units does this calculator use?
Enter torque in Newton-metres (N.m) and shaft diameter in millimetres (mm). The calculator converts internally to SI base units (m) and outputs shear stress in megapascals (MPa). 1 MPa = 1 N/mm^2.
What are typical allowable shear stresses for steel shafts?
ASME guidelines for solid steel transmission shafts recommend an allowable shear stress of 8,000 psi (55 MPa) for shafts without keyways and 6,000 psi (41 MPa) for shafts with keyways. These values apply to shafts loaded in pure torsion.
How do I find the torque from motor power and speed?
Torque (N.m) = Power (W) / Angular velocity (rad/s). Angular velocity = 2 * pi * RPM / 60. For example, a 15 kW motor at 1,450 RPM has omega = 151.8 rad/s and T = 15,000 / 151.8 = 98.8 N.m.
Official sources
- ASME B106.1M Design of Transmission Shafting: ASME B106.1M.
- NIST Engineering Laboratory: nist.gov/el.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.