The torsion constant or torsion coefficient is a geometrical property of a bar's cross-section. It is involved in the relationship between angle of twist and applied torque along the axis of the bar, for a homogeneous linear elastic bar. The torsion constant, together with material properties and length, describes a bar's torsional stiffness. The SI unit for torsion constant is m<sup>4</sup>.
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<sub>zz</sub>, 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.
Non-circular cross-sections always have warping deformations that require more complex methods calculate the torsion constant. Few non-circular cross-sections have exact analytical solutions for the torsion constant but approximate numerical solutions exist for many shapes.
The torsional stiffness of short beams with open cross sections is significantly increased if the warping of the end sections is restrained by, for example, stiff end blocks.
For a beam of uniform cross-section along its length, the angle of twist (in radians) is:
where:
Inverting the previous relation, we can define two quantities; the torsional rigidity,
And the torsional stiffness,
Bars with given uniform cross-sectional shapes are special cases.
where
This is identical to the second moment of area J<sub>zz</sub> and is exact.
alternatively write: where
where
where
where
Alternatively, the following equation can be used with an error of not greater than 4%:<br>
where
This is a tube with a slit cut longitudinally through its wall. Using the formula above: