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 m4.

[Main quantities involved in bar torsion:

\theta is the angle of twist; T is the applied torque; L is the beam length. | upload.wikimedia.org/wikipedia/commons/1/1b/TorsionConstantBar.svg]

History

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 Jzz, 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. 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. 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.

Formulation

For a beam of uniform cross-section along its length, the angle of twist (in radians) \theta is: where: Inverting the previous relation, we can define two quantities; the torsional rigidity, And the torsional stiffness,

Examples

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

Circle

where This is identical to the second moment of area Jzz and is exact. alternatively write: where

Ellipse

where

Square

where

Rectangle

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

Thin walled open tube of uniform thickness

Circular thin walled open tube of uniform thickness

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