How to use this calculator
- Enter torque. Enter the transmitted torque at the critical shaft section.
- Add bending if needed. Enter the bending moment at that section, or leave it at 0 for pure torsion.
- Enter allowable stress. Use the material allowable shear stress before safety factor unless it is already derated.
- Set safety factor. The calculator divides the allowable shear by this factor to get the design stress.
- Read diameter. Round the theoretical diameter up to an available stock size and recheck stress, deflection and details.
How it works
A solid circular shaft in torsion has maximum surface shear stress:
tau = 16T / (pi d^3)
where T is torque and d is diameter. Solving for
diameter gives d = [16T/(pi tau_d)]^(1/3), using the design
shear stress tau_d = tau_allow / N.
If the shaft also carries bending moment M, the bending normal stress is
sigma_b = 32M/(pi d^3) and the torsional shear stress is
tau_t = 16T/(pi d^3). The maximum-shear stress at the surface is
sqrt((sigma_b/2)^2 + tau_t^2), which rearranges to the same diameter
formula using an equivalent torque T_e = sqrt(T^2 + M^2). If the
bending comes from a pulley, sprocket, pinion or gear outside the bearings, the
overhung load calculator will estimate
the peak bending moment to enter here.
Worked example
Verified against the live calculator
A shaft carrying 100 N*m torque with no bending, allowable shear
80 MPa and safety factor 2 has design shear
tau_d = 40 MPa. Convert torque to 100,000 N*mm, then:
d = [16 x 100000 / (pi x 40)]^(1/3) = 23.35 mm.
If the same section also has 100 N*m bending moment, the equivalent
torque becomes sqrt(100^2 + 100^2) = 141.4 N*m, so the required
diameter increases to about 26.21 mm. That diameter increase is why
gear and pulley shaft sections should be checked with bending included.
Frequently asked questions
How do I calculate shaft diameter from torque?
For a solid round shaft in pure torsion, maximum shear stress is tau = 16T/(pi d^3). Rearranging gives d = [16T/(pi tau)]^(1/3), with torque in N*mm and stress in MPa. The calculator also applies your safety factor by using tau_d = tau_allow / N.
How does bending moment change the shaft diameter?
Bending adds normal stress at the surface. This calculator uses the maximum-shear relation tau_max = sqrt((sigma_b/2)^2 + tau_t^2), where sigma_b = 32M/(pi d^3) and tau_t = 16T/(pi d^3). That collapses to an equivalent torque sqrt(T^2 + M^2).
Can I leave bending moment at zero?
Yes. With bending moment set to 0, the result is the pure torsion shaft diameter. Use a nonzero bending moment when gears, pulleys, sprockets, belts, overhung loads or bearing reactions bend the same critical section.
What allowable shear stress should I enter?
Use the allowable shear stress for the material and loading case, then choose a safety factor. If your allowable value is already derated by a design code or company standard, set the safety factor to 1 so it is not divided twice.
Does this check fatigue or keyways?
No. It is a static solid-shaft screen. Real shafts often govern by fatigue, shoulders, keyways, splines, fillets, press fits, surface finish, deflection or critical speed, so round up and do the downstream checks.
Does it work in metric and imperial?
Yes. Enter torque in N*m or lbf*ft, moments in the same torque unit, and stress in MPa or ksi. The math converts everything to fixed internal units before solving.
Method & assumptions
- Solid round shaft, static loading, elastic stress formulas and maximum-shear stress theory.
- Torque and bending moment are assumed to act at the same critical section.
- Stress concentrations from keyways, shoulders, fillets, splines, grooves and press fits are not included.
- Fatigue, surface finish, size factors, residual stress, shaft deflection, critical speed and bearing reactions require separate checks.
- The solved diameter is theoretical; round up to stock and rerun the final shaft geometry.