Shaft Combined Bending and Torsion Calculator

How are bending and torsion combined?

A shaft carrying a pulley, gear or sprocket bends and twists at the same time. For a solid circular section the bending stress is σ = 32 × M / (π × d³) and the torsional shear stress is τ = 16 × T / (π × d³). Both peak at the surface and act at the same point.

The two stresses are not simply added. Two criteria are used to predict yielding in ductile materials:

• Maximum shear stress: τmax = 16 / (π × d³) × √(M² + T²). Yielding starts when τmax reaches half the yield strength.
• Von Mises: σ' = 16 / (π × d³) × √(4 × M² + 3 × T²). Yielding starts when σ' reaches the yield strength.

The first is slightly more conservative. The calculator reports both.

Example

At one section the bending moment is 300 N·m, the torque 400 N·m, the yield strength 300 MPa and the required safety factor 2. Then √(300² + 400²) = 500 N·m, d³ = 32 × 2 × 500 000 / (π × 300) = 33 953 mm³ and d = 32.4 mm. The von Mises criterion gives 31.5 mm. A standard 35 mm diameter could be chosen.

Limits

This is a static check. In rotating shafts the bending stress swings from tension to compression every turn, so failures are usually fatigue failures and a fatigue calculation calls for a larger diameter. Keyways and shoulders also raise the stress locally. Use the result for preliminary sizing and complete the final design with fatigue, deflection and critical speed checks.