Torsion (mechanics)

In solid mechanics, torsion is the twisting of an object caused by an applied torque. It may be described as an angular deformation, measured by the rotation of a cross-section from its undeformed position. Torque is commonly expressed in newton metres (N·m) or foot-pound force (ft·lbf), while the resulting torsional shear stress is expressed in pascals (Pa) or pounds per square inch (psi).

In a circular shaft, the shear stress is tangent to circles centred on the shaft axis. In non-circular cross-sections, twisting is accompanied by warping, so transverse sections do not generally remain plane. Under uniform, unrestrained Saint-Venant torsion, torque and twist are related by

T=GJ_{\mathrm {T} }\Theta ={\frac {GJ_{\mathrm {T} }}{\ell }}\varphi ,

where $T$ is the torque, $G$ is the shear modulus, $J_{\mathrm {T} }$ is the Saint-Venant torsion constant, $\Theta =\mathrm {d} \varphi /\mathrm {d} x$ is the twist rate, $\varphi$ is the total angle of twist over the length $\ell$ , and $GJ_{\mathrm {T} }$ is the torsional rigidity. The geometric quantity $J_{\mathrm {T} }$ has dimensions of length to the fourth power. Only for circular shafts is the stress distribution simply

\tau (r)={\frac {Tr}{J_{\mathrm {T} }}}.

Closed-form solutions are available for only a limited number of cross-sections. Thin open sections are commonly approximated by a sum of rectangular strips, while thin closed cells are commonly treated with Bredt's formula. More general sections are usually analysed numerically or by variational bounds. In 2026, Rocco Ditommaso and Felice Carlo Ponzo presented an open-access exact-arithmetic primal–dual procedure that provides certified lower and upper bounds for solid, open and hollow polygonal sections.