Equivalent Stress under Combined Bending and Torsion

At a critical point of a circular shaft subjected to bending and torsion, the bending moment produces normal stress $\sigma_b$ and the torque produces shear stress $\tau_t$:

$$\sigma_b=\frac{M_b}{W},\qquad \tau_t=\frac{T}{W_p}.$$

For this plane stress state, the von Mises equivalent stress is:

$$\sigma_{\mathrm{eq,VM}}=\sqrt{\sigma_b^2+3\tau_t^2}.$$

According to the Tresca criterion:

$$\sigma_{\mathrm{eq,T}}=\sqrt{\sigma_b^2+4\tau_t^2}.$$

• $M_b$ — resultant bending moment;
• $T$ — torque;
• $W$ — bending section modulus;
• $W_p$ — polar section modulus.

The resulting equivalent stress is compared with an allowable value appropriate to the material and the adopted design method.

The calculator uses the von Mises criterion for the bending-plus-torsion plane stress state.