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 = G J T Θ = G J T ℓ φ , {\displaystyle T=GJ_{\mathrm {T} }\Theta ={\frac {GJ_{\mathrm {T} }}{\ell }}\varphi ,}
where T {\displaystyle T} is the torque, G {\displaystyle G} is the shear modulus, J T {\displaystyle J_{\mathrm {T} }} is the Saint-Venant torsion constant, Θ = d φ / d x {\displaystyle \Theta =\mathrm {d} \varphi /\mathrm {d} x} is the twist rate, φ {\displaystyle \varphi } is the total angle of twist over the length ℓ {\displaystyle \ell } , and G J T {\displaystyle GJ_{\mathrm {T} }} is the torsional rigidity. The geometric quantity J T {\displaystyle J_{\mathrm {T} }} has dimensions of length to the fourth power. Only for circular shafts is the stress distribution simply
τ ( r ) = T r J T . {\displaystyle \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.
Properties For a circular shaft, the shear stress at radius r {\displaystyle r} is
τ ( r ) = T r J T . {\displaystyle \tau (r)={\frac {Tr}{J_{\mathrm {T} }}}.}
The maximum shear stress therefore occurs at the outer surface. Surface stress concentrations caused by notches, roughness or abrupt geometric changes can further increase the local stress. For a prismatic member of constant cross-section and constant torque, the angle of twist is
φ = T ℓ G J T . {\displaystyle \varphi ={\frac {T\ell }{GJ_{\mathrm {T} }}}.}
For non-circular sections, J T {\displaystyle J_{\mathrm {T} }} is generally not equal to the polar second moment of area, and the shear-stress distribution is not proportional to the distance from the axis.
Torsion constants for common cross-sections The following tables distinguish exact solutions, engineering asymptotic formulas and certified bounds. The symbol J T {\displaystyle J_{\mathrm {T} }} denotes the geometric torsion constant; the physical torsional rigidity is G J T {\displaystyle GJ_{\mathrm {T} }} .
Exact and engineering formulas
Certified primal–dual method In 2026, Rocco Ditommaso and Felice Carlo Ponzo proposed an exact-arithmetic primal–dual procedure for certifying the Saint-Venant torsion constant of polygonal sections. For a convex polygon with rational vertex coordinates, the boundary function B 0 = ∏ i ℓ i {\displaystyle B_{0}=\prod _{i}\ell _{i}} is multiplied by a polynomial q p {\displaystyle q_{p}} to form a kinematically admissible field v p = B 0 q p {\displaystyle v_{p}=B_{0}q_{p}} . A statically admissible field is constructed as σ p = ( − x , − y ) + ∇ ⊥ ψ p {\displaystyle {\boldsymbol {\sigma }}_{p}=(-x,-y)+\nabla ^{\perp }\psi _{p}} . Optimising the two fields gives a lower and an upper bound,
J p − = 4 g T K − 1 g ≤ J T ≤ J p + = ∫ Ω ( x 2 + y 2 ) d A − b T M r e d − 1 b . {\displaystyle J_{p}^{-}=4\mathbf {g} ^{\mathsf {T}}K^{-1}\mathbf {g} \leq J_{\mathrm {T} }\leq J_{p}^{+}=\int _{\Omega }(x^{2}+y^{2})\,\mathrm {d} A-\mathbf {b} ^{\mathsf {T}}M_{\mathrm {red} }^{-1}\mathbf {b} .}
Because the required integrals reduce to polynomial moments over rational triangles, the endpoints and the interval width are exact rational numbers rather than floating-point estimates. The width is the energy mismatch
J p + − J p − = ∫ Ω | σ p − ∇ v p | 2 d A . {\displaystyle J_{p}^{+}-J_{p}^{-}=\int _{\Omega }\left|{\boldsymbol {\sigma }}_{p}-\nabla v_{p}\right|^{2}\,\mathrm {d} A.}
The same construction extends to non-convex open sections by using piecewise polynomial spaces on rational triangulations, and to multiply connected hollow sections by adding constants and flux constraints on the internal boundaries. The method does not replace the classical exact or thin-wall formulas; instead, it brackets the true value when no elementary formula exists and quantifies the error and sign of engineering approximations.
Examples of certified relations
Reliability of commonly used relations
Sample calculation
Consider a solid circular shaft transmitting 1000 megawatts at a rotational frequency of 50 hertz. Let the allowable shear stress initially be 250 MPa. The angular frequency is
ω = 2 π f , {\displaystyle \omega =2\pi f,}
and the transmitted torque is
T = P ω . {\displaystyle T={\frac {P}{\omega }}.}
For P = 1.0 × 10 9 {\displaystyle P=1.0\times 10^{9}} W and f = 50 {\displaystyle f=50} Hz, ω = 314.16 {\displaystyle \omega =314.16} rad/s and T ≈ 3.1831 × 10 6 {\displaystyle T\approx 3.1831\times 10^{6}} N·m. For a solid circular shaft,
J T = π D 4 32 {\displaystyle J_{\mathrm {T} }={\frac {\pi D^{4}}{32}}}
and
τ max = 16 T π D 3 . {\displaystyle \tau _{\max }={\frac {16T}{\pi D^{3}}}.}
Solving for the diameter gives
D = ( 16 T π τ max ) 1 / 3 . {\displaystyle D=\left({\frac {16T}{\pi \tau _{\max }}}\right)^{1/3}.}
The resulting diameter is approximately 0.40 m. With a factor of safety of 5, so that the allowable stress is 50 MPa, the diameter increases to approximately 0.69 m.
Failure modes The shear stress in a shaft under pure torsion may be resolved into principal stresses using Mohr's circle. The principal tensile and compressive stresses act on planes oriented at approximately 45° to the shaft axis. Brittle materials may therefore fail along a helical fracture surface inclined at about 45°, as can be demonstrated by twisting a piece of chalk. A ductile circular shaft may instead undergo substantial plastic shear deformation before fracture. Because an axisymmetric shaft can retain its overall external shape while accumulating permanent twist, overload damage may not be visually obvious. Thin-walled hollow shafts may also fail by torsional buckling, producing diagonal wrinkles typically oriented at approximately 45° to the shaft axis.
Torsional resonator A torsional resonator uses rotational motion to investigate the elastic or viscoelastic behaviour of a fibre or other slender specimen. A typical system consists of a fibre fixed at one end and connected at the other end to a rigid rod or inertial mass. The deformation and damping of the fibre provide information about energy dissipation and viscoelasticity. A simple rotational equation of motion is
T e x t = K t θ + I d 2 θ d t 2 , {\displaystyle T_{\mathrm {ext} }=K_{t}\theta +I{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}},}
where T e x t {\displaystyle T_{\mathrm {ext} }} is the applied torque, θ {\displaystyle \theta } is the angular displacement, K t {\displaystyle K_{t}} is the torsional spring stiffness and I {\displaystyle I} is the rotational moment of inertia. For a cylindrical fibre of diameter d {\displaystyle d} and length l {\displaystyle l} ,
K t = π G d 4 32 l . {\displaystyle K_{t}={\frac {\pi Gd^{4}}{32l}}.}
The undamped natural angular frequency is
ω n = K t I . {\displaystyle \omega _{n}={\sqrt {\frac {K_{t}}{I}}}.}
For an ideal elastic material undergoing free undamped oscillation,
θ = θ 0 cos ( ω n t ) . {\displaystyle \theta =\theta _{0}\cos(\omega _{n}t).}
Viscoelastic behaviour may be represented by a complex shear modulus
G ∗ = G ′ + i G ″ , {\displaystyle G^{*}=G'+iG'',}
where G ′ {\displaystyle G'} is the shear storage modulus and G ″ {\displaystyle G''} is the shear loss modulus. The corresponding complex stiffness and frequency describe both the elastic response and damping. Valtorta and Mazza used a torsional resonator device to measure the viscoelastic properties of soft tissue by characterising its complex shear modulus.
See also List of area moments of inertia Saint-Venant's theorem Second moment of area Structural rigidity Torque tester Torsion siege engine Torsion spring or torsion bar Torsional vibration
References
External links The dictionary definition of torsion at Wiktionary Solid Mechanics at Wikibooks