Stress and Angle of Twist of a Circular Shaft

For a straight solid or hollow circular shaft under elastic torsion:

$$\tau(\rho)=\frac{T\rho}{J_p},\qquad \tau_{\max}=\frac{T}{W_p}.$$

If $T$, $G$, and $J_p$ are constant over a segment, the angle of twist is:

$$\varphi=\frac{TL}{GJ_p}.$$

• $T$ — torque;
• $\rho$ — radial distance from the shaft axis;
• $J_p$ — polar second moment of area;
• $W_p$ — polar section modulus;
• $G$ — shear modulus;
• $L$ — segment length;
• $\varphi$ — angle of twist, rad.

For a stepped shaft, the total twist is the algebraic sum $\varphi=\sum_i T_iL_i/(G_iJ_{p,i})$.