Torsional Rigidity

The torsional rigidity of a homogeneous, isotropic, linearly elastic prismatic shaft under uniform torsion is , the ratio of applied torque to twist per unit length (Baksa 2020, p. 85). The angular twist of the shaft is given by

(Roark 1954, p. 174), where is the twisting moment (commonly measured in units of inch-pounds-force), is the length (inches), is the modulus of rigidity (pounds-force per square inch), and (sometimes also denoted ) is the torsional rigidity multiplier for a given geometric cross section (inches to the fourth power). Note that the quantity is sometimes denoted (e.g., Timoshenko and Goodier 1951, p. 264).

For a bounded, connected plane domain with a piecewise smooth boundary, let be the outward unit normal vector. A harmonic function , satisfying in , is determined up to an additive constant by the Neumann boundary conditions

on . Here is the Laplacian and is the gradient of . The geometric multiplier is the double integral

where (Lipton 1998, pp. 87-89; Freddi et al. 2010). This formulation also applies to cross sections with holes, such as an annulus. For a slit disk, the Neumann boundary conditions apply on each bank with its own outward unit normal vector.

For a simply connected cross section, an equivalent formulation uses a function satisfying

with zero Dirichlet boundary conditions on . Then

With holes, the boundary values of need not all be zero (Freddi et al. 2010).

Values of are known exactly only for a small number of cross sections, and as closed-form solutions for even fewer. The following table lists exact expressions and approximate values for some common shapes (Timoshenko and Goodier 1951, pp. 258-280; Roark 1954, pp. 174-179). Here, is the radius for circular cross sections and circular sectors, the side length for the equilateral triangle and square, and the equal leg length for the isosceles right triangle. For an annulus, and are the outer and inner radii with ; for an ellipse, they are positive semiaxes; and for a rectangle, they are positive full side lengths. The exact expressions in the table use the convergent series

Here . The table gives the full geometric multiplier , so the physical torsional rigidity is . Timoshenko and Goodier (1951, p. 279) tabulate approximate coefficients calculated by Saint-Venant (1878ab) from the classical circular sector solution. The closed-form expressions in the table for the half-disk and slit disk were obtained by E. Weisstein (Aug. 27, 2010).

Exact series representations are known for the rectangle and square

(Timoshenko and Goodier 1951, pp. 275-277), isosceles right triangle

(Galerkin 1919, p. 116, correcting to in equation (18)).

Applying Ramanujan's transformation for hyperbolic cotangent sums (Berndt and Straub 2017, p. 6, equation (3.1)), with analytic continuation, gives

and hence the exact relation

where the square has side length and the isosceles right triangle has equal legs of length .

A faster convergent series is obtained by putting

(OEIS A255702) and writing

where is the Riemann zeta function. These are series representations, with exponentially decaying summands in .

For a circular sector, the solution is

where are polar coordinates, , , and

(Dinnik; Saint-Venant 1878ab, Greenhill 1879, Föppl and Föppl 1928; Timoshenko and Goodier 1951, pp. 278-280). Here is the radial coordinate, and is the fixed outer radius. The first term has Laplacian , and the remaining series is a harmonic function. The separated expression for is interpreted by continuous limits at and . At , the cross section is a slit disk formed by cutting a disk along one radius from its center to its circumference. The two sides of the cut remain separate free edges. The ideal slit has zero area, but changes the boundary conditions; its multiplier is therefore smaller than the uncut disk multiplier .

Integrating the circular sector solution gives the finite expression

where is the digamma function and is the trigamma function. This formula is valid for , including the resonant values above. At , , and , it reduces to the elementary quarter-disk, half-disk, and slit disk expressions.

For a quarter-disk, integrating the circular sector solution at opening angle gives

as given in the table above.

The circular sector series for can be written using a generalized hypergeometric function (Cetinkaya et al. 1996, p. 7). Put and

and define

Then

Here and . The complete expression for is extended to the exceptional angles specified above and to by continuous limits.

A sharp lower bound involves distance to the boundary of a convex domain. Let solve in a bounded open convex set , , with zero Dirichlet boundary conditions, and put . Perelmuter (2026) proved

Here denotes the boundary of , and is the distance from to that boundary, measured with the Euclidean norm . Balls attain equality. In dimension two, this normalization of equals the geometric multiplier above, rather than the rigidity .