This calculator determines the nominal maximum torsional shear stress in a solid circular shaft. Choose the variable to calculate, then enter the other three. It can solve for shaft shear stress, transmitted horsepower or power, rotational speed, or the solid-shaft diameter required for the entered nominal stress.
The formulas
For a solid circular shaft under pure torque, the polar moment of inertia is J = πD⁴/32. The torsion relation τ = Tr/J, evaluated at the outside radius r = D/2, gives the maximum surface stress Ss = τmax = 16T/(πD³). Shaft torque follows from power and angular speed: T = P/ω and ω = 2πN/60.
Solve in any direction
- Shaft shear stress (Ss) — enter power, rotational speed, and solid-shaft diameter.
- Shaft power (HP or P) — enter nominal shear stress, speed, and diameter.
- Rotational speed (N) — enter nominal shear stress, power, and diameter.
- Shaft diameter (D) — enter nominal shear stress, power, and speed.
Worked example
A solid 2 in (50.8 mm) shaft transmitting 100 hp at 1,750 rpm carries 406.9 N·m of torque. The nominal maximum surface shear stress is 2,293 psi, or 15.81 MPa. Solving backward with that stress, speed, and diameter returns 100 hp.
Assumptions
- The shaft is solid, circular, straight, and experiencing pure torque.
- There is no bending, axial load, transverse shear, shock, fatigue, or residual stress.
- There are no stress risers such as keyways, shoulders, grooves, splines, holes, or surface defects.
- Power and rotational speed describe the same operating condition and neglect drivetrain losses between the stated power point and shaft section.
Accounting for a stress riser
The calculator reports the nominal smooth-shaft stress. If an applicable torsional stress concentration factor Kt is known, estimate the local elastic peak as τlocal = Kt × Ss. Obtain Kt from a validated geometry-specific reference; do not assume one generic multiplier covers every keyway, shoulder, groove, or spline. Fatigue design may require a fatigue stress concentration factor rather than the theoretical elastic Kt.
Diameter sensitivity
Nominal shear stress varies with 1/D³. At the same transmitted torque, a 10% reduction in diameter raises nominal stress by about 37%. Use the actual minimum load-carrying diameter at the section being checked, not a nominal bearing or coupling size.
Limits
- This solid-shaft formula does not apply to hollow shafts; use their polar moment of inertia instead.
- The result is stress, not allowable capacity. Compare it with an appropriate material limit and design factor for static or fatigue service.
- Transient startup, trip, jam, and torsional-vibration torque can exceed the steady torque inferred from horsepower and rpm.
- Final shaft design should include combined loading, stress concentrations, fatigue, deflection, critical speed, and applicable code or manufacturer requirements.