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
 |
(1)
|
(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).
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 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.
| cross section |  approx. | OEIS |
| circle | 1.570796... | A019669 |
| equilateral triangle | 0.021650... | A180317 |
| half-disk | 0.297556... | A180310 |
| isosceles right triangle | 0.026089... | A180314 |
| quarter-disk | 0.082327... | |
| slit full disk | 0.878055... | A180311 |
| square | 0.140577... | A180309 |
Closed-form solutions are known for the annulus
 |
(2)
|
(Roark 1954, p. 175), circle
 |
(3)
|
(Roark 1954, p. 174), ellipse
 |
(4)
|
(Timoshenko and Goodier 1951, pp. 263-265; Roark 1954, p. 174), equilateral triangle
 |
(5)
|
(Timoshenko and Goodier 1951, pp. 265-267; Roark 1954, p. 175), and half-disk and slit full disk (i.e., circular sector from
0 to 
)
 |  |  |
(6)
|
 |  |  |
(7)
|
(E. Weisstein, Aug. 27, 2010; given approximately by Saint-Venant 1878ab; Timoshenko and Goodier 1951, pp. 263-265; Roark 1954, p. 174).
Exact solutions expressed as sums (with no known closed-form solution) are known for the rectangle and square
 |  | ![(a^3b)/3[1-(192)/(pi^5)a/bsum_(n=1)^(infty)1/((2n-1)^5)tanh((pi(2n-1)b)/(2a))]](/images/equations/TorsionalRigidity/Inline25.svg) |
(8)
|
 |  | ![(a^4)/3[1-(192)/(pi^5)sum_(n=1)^(infty)1/((2n-1)^5)tanh((pi(2n-1))/2)]](/images/equations/TorsionalRigidity/Inline28.svg) |
(9)
|
(Timoshenko and Goodier 1951, pp. 275-277), isosceles right triangle
![K_(isos. rt. tri.)=a^4[1/(12)-(16)/(pi^5)sum_(n=1)^infty1/((2n-1)^5)coth((pi(2n-1))/2)]](/images/equations/TorsionalRigidity/NumberedEquation6.svg) |
(10)
|
(Galerkin 1919; correcting the typo 1/2 for 1/12), and circular sector
 |
(11)
|
where 
,
,
and
![f(r,psi)=-r^2[1-(cos(2psi))/(cosalpha)]+(16a^2alpha^2)/(pi^3)sum_(n=1,3,5,...)(-1)^((n+1)/2)(r/a)^(npi/alpha)(cos((npipsi)/alpha))/(n(n+(2alpha)/pi)(n-(2alpha)/pi))](/images/equations/TorsionalRigidity/NumberedEquation8.svg) |
(12)
|
(Dinnik; Saint-Venant 1878ab, Greenhill 1879, Föppl and Föppl 1928; Timoshenko and Goodier 1951, pp. 278-280). The separated expression for 
is interpreted by continuous limits at 
and 
. At 
, the cross section
is the full disk with a radial slit.
