Learning topic
Torsion
Internal torque, torque diagrams, torsional shear stress, angle of twist, strength and stiffness of circular shafts, and advanced section types.
Torsion is loading by couples acting about a member's longitudinal axis. An internal torque \(T\) develops in each cross-section and the material is subjected primarily to shear deformation.
Internal torque
The torque at a section is found by cutting the shaft and applying rotational equilibrium to either segment. A torque diagram \(T(x)\) identifies critical regions and changes of sign.
Solid and hollow circular shafts
For a circular shaft, cross-sections remain plane and rotate as rigid discs. Shear stress varies linearly with radius:
\[\tau(\rho)=\frac{T\rho}{J},\qquad \tau_{max}=\frac{T}{W_p}.\]
For a uniform segment, the angle of twist is
\[\varphi=\frac{TL}{GJ}.\]
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 $\varphi=\sum_i T_iL_i/(G_iJ_{p,i})$.
Strength and stiffness
Shaft dimensions must satisfy both \(\tau_{max}\le\tau_{allow}\) and \(|\varphi|\le\varphi_{allow}\). Adequate strength alone does not guarantee acceptable torsional stiffness.
Power transmission
For a rotating shaft, power, torque, and angular velocity are related by \(P=T\omega\). This converts motor power and rotational speed into design torque.
Special cases
- statically indeterminate torque-loaded systems;
- stepped and composite shafts;
- stress concentrations at grooves, fillets, and holes;
- inelastic torsion and residual stresses;
- helical cylindrical springs.
More complex cross-sections
The circular-shaft formula cannot be applied directly to rectangular, elliptical, or other shapes. See Torsion of Solid Noncircular Shafts for an overview of warping and the torsion constant. Detailed thin-walled torsion is treated as a separate advanced section.
Calculation route
Applied torques → reaction torques → torque diagram \(T\) → theory selected by section shape → shear stress → angle of twist → strength and stiffness checks.
About this topic
Learn shaft torsion through internal torque, torque diagrams, circular-shaft shear stress, angle of twist, power transmission, strength and stiffness checks.