Torsion (mechanics)

Torsion constants for common cross-sections

The following tables distinguish exact solutions, engineering asymptotic formulas and certified bounds. The symbol ${\displaystyle J_{\mathrm{T}}}$ denotes the geometric torsion constant; the physical torsional rigidity is ${\displaystyle G{J}_{\mathrm{T}}}$.

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 ${\displaystyle B_0=\prod _i{\ell }_i}$ is multiplied by a polynomial ${\displaystyle q_p}$ to form a kinematically admissible field ${\displaystyle v_p=B_0{q}_p}$. A statically admissible field is constructed as ${\displaystyle {\mathit{\sigma }}_p=(-x,-y)+{\mathrm{\nabla }}^{\perp }{\psi }_p}$. Optimising the two fields gives a lower and an upper bound,

${\displaystyle J_p^{-}=4{\mathbf{g}}^{\mathsf{T}}{K}^{-1}\mathbf{g}\le J_{\mathrm{T}}\le J_p^{+}={\int }_{\mathrm{\Omega }}(x^2+y^2)\mathrm{d}A-{\mathbf{b}}^{\mathsf{T}}{M}_{\mathrm{r}\mathrm{e}\mathrm{d}}^{-1}\mathbf{b}.}$

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

${\displaystyle J_p^{+}-J_p^{-}={\int }_{\mathrm{\Omega }}{\left|{\mathit{\sigma }}_p-\mathrm{\nabla }{v}_p\right|}^2\mathrm{d}A.}$

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

${\displaystyle \omega =2\pi f,}$

and the transmitted torque is

${\displaystyle T=\frac{P}{\omega }.}$

For ${\displaystyle P=1.0\times {10}^9}$ W and ${\displaystyle f=50}$ Hz, ${\displaystyle \omega =314.16}$ rad/s and ${\displaystyle T\approx 3.1831\times {10}^6}$ N·m.

For a solid circular shaft,

${\displaystyle J_{\mathrm{T}}=\frac{\pi D^4}{32}}$

and

${\displaystyle {\tau }_{max}=\frac{16T}{\pi D^3}.}$

Solving for the diameter gives

${\displaystyle D={\left(\frac{16T}{\pi {\tau }_{max}}\right)}^{1/3}.}$

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

${\displaystyle T_{\mathrm{e}\mathrm{x}\mathrm{t}}=K_t\theta +I\frac{{\mathrm{d}}^2\theta }{\mathrm{d}{t}^2},}$

where ${\displaystyle T_{\mathrm{e}\mathrm{x}\mathrm{t}}}$ is the applied torque, ${\displaystyle \theta }$ is the angular displacement, ${\displaystyle K_t}$ is the torsional spring stiffness and ${\displaystyle I}$ is the rotational moment of inertia.

For a cylindrical fibre of diameter ${\displaystyle d}$ and length ${\displaystyle l}$,

${\displaystyle K_t=\frac{\pi G{d}^4}{32l}.}$

The undamped natural angular frequency is

${\displaystyle {\omega }_n=\sqrt{\frac{K_t}{I}}.}$

For an ideal elastic material undergoing free undamped oscillation,

${\displaystyle \theta ={\theta }_0\cos ({\omega }_{n}t).}$

Viscoelastic behaviour may be represented by a complex shear modulus

${\displaystyle G^{\ast }=G^{\prime }+i{G}^{″},}$

where ${\displaystyle G^{\prime }}$ is the shear storage modulus and ${\displaystyle G^{″}}$ 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.