Shaft Design for Combined Bending and Torsion
Shaft Design for Combined Bending and Torsion
Power-transmission shafts rarely see pure torsion alone — a shaft carrying a pulley, gear, or coupling almost always also bends under the belt tension or gear mesh force acting perpendicular to its axis, so bending and torsion act together. Combining the two into a single equivalent stress is the standard way to size a shaft against yielding under this combined state.Ignoring the bending component and sizing a shaft for torque alone is a common under-design mistake, since even a modest bending moment can add meaningfully to the equivalent stress once combined with torsion through the square-root-of-sum-of-squares relationship — the two do not simply add linearly, but neither can safely be ignored.
The equivalent bending moment combining torque and bending is M_e = sqrt(M_b^2 + T_t^2), and the resulting equivalent bending stress in a solid circular shaft is sigma_e = 32·M_e/(pi·d^3). where M_b is the applied bending moment, T_t is the applied torque, d is the shaft diameter, M_e is the resulting equivalent bending moment, and sigma_e is the equivalent bending stress used to size the shaft.
Inputs
M_b := 500 N·m=500 Nm
T_t := 350 N·m=350 Nm
d := 40 mm=40 mm
Combining bending and torsion into a single equivalent moment through this square-root relation reflects how the two stress states interact under a maximum-shear-stress-based failure theory.
M_e := sqrt(M_b^2 + T_t^2)=610.33 Nm
Applying the standard solid-shaft flexure formula to the equivalent moment converts it into an equivalent bending stress that can be compared directly against the shaft material's allowable stress.
sigma_e := 32·M_e/(pi·d^3)=97.14 MPa
Results
An equivalent stress of about 97 MPa for this 40 mm shaft would need to be checked against the material's allowable bending stress (with an appropriate safety factor) — for a typical medium-carbon steel shaft with a yield strength around 250–350 MPa, this leaves reasonable margin. Because stress scales with the cube of diameter in the denominator, even a modest increase in shaft diameter produces a large reduction in stress, making diameter the most powerful lever for a shaft that is found to be under-designed. This static equivalent-stress check does not address fatigue, which typically governs shaft sizing in practice since most transmission shafts see fully reversed bending stress as they rotate.
Assumptions & limitations
- Solid circular shaft cross-section (no keyway or shoulder stress concentration included).
- Static equivalent stress check only — fatigue, which usually governs rotating shafts, is a separate calculation.
- Bending and torque assumed as their peak simultaneous values at the same shaft location.
- Combined-stress formula follows a maximum-shear-stress (Tresca-style) approach; other theories give slightly different combinations.