In 1820, the French engineer A. Duleau derived analytically that the torsion constant of a beam is identical to the second moment of area normal to the section J_zz, which has an exact analytic equation, by assuming that a plane section before twisting remains planar after twisting, and a diameter remains a straight line. Unfortunately, that assumption is correct only in beams with circular cross-sections, and is incorrect for any other shape where warping takes place.^[1]

For non-circular cross-sections, there are no exact analytical equations for finding the torsion constant. However, approximate solutions have been found for many shapes. Non-circular cross-sections always have warping deformations that require numerical methods to allow for the exact calculation of the torsion constant.^[2]

The torsional stiffness of beams with non-circular cross sections is significantly increased if the warping of the end sections is restrained by, for example, stiff end blocks.^[3]

For a beam of uniform cross-section along its length, the angle of twist (in radians) ${\displaystyle \theta }$  is:

${\displaystyle \theta ={\frac {TL}{GJ}}}$ 

where:

T is the applied torque

L is the beam length

G is the modulus of rigidity (shear modulus) of the material

J is the torsional constant

Inverting the previous relation, we can define two quantities; the torsional rigidity,

${\displaystyle GJ={\frac {TL}{\theta }}}$  with SI units N⋅m²/rad

And the torsional stiffness,

${\displaystyle {\frac {GJ}{L}}={\frac {T}{\theta }}}$  with SI units N⋅m/rad

Bars with given uniform cross-sectional shapes are special cases.

${\displaystyle J_{zz}=J_{xx}+J_{yy}={\frac {\pi r^{4}}{4}}+{\frac {\pi r^{4}}{4}}={\frac {\pi r^{4}}{2}}}$ ^[4]

where

r is the radius

This is identical to the second moment of area J_zz and is exact.

alternatively write: ${\displaystyle J={\frac {\pi D^{4}}{32}}}$ ^[4] where

D is the Diameter

${\displaystyle J\approx {\frac {\pi a^{3}b^{3}}{a^{2}+b^{2}}}}$ ^[5][6]

where

a is the major radius

b is the minor radius

${\displaystyle J\approx \,2.25a^{4}}$ ^[5]

where

a is half the side length.

${\displaystyle J\approx \beta ab^{3}}$ 

where

a is the length of the long side

b is the length of the short side

${\displaystyle \beta }$  is found from the following table:

^[7]

Alternatively the following equation can be used with an error of not greater than 4%:

${\displaystyle J\approx {\frac {ab^{3}}{16}}\left({\frac {16}{3}}-{3.36}{\frac {b}{a}}\left(1-{\frac {b^{4}}{12a^{4}}}\right)\right)}$ ^[5]

where

a is the length of the long side

b is the length of the short side

${\displaystyle J={\frac {1}{3}}Ut^{3}}$ ^[8]

t is the wall thickness

U is the length of the median boundary (perimeter of median cross section)

This is a tube with a slit cut longitudinally through its wall. Using the formula above:

${\displaystyle U=2\pi r}$ 

${\displaystyle J={\frac {2}{3}}\pi rt^{3}}$ ^[9]

t is the wall thickness

r is the mean radius

• Torsion constant calculator