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.
See also: Torsion balance |
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.[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) 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 Jzz and is exact.
alternatively write: [4] where
where
where
where
a/b | |
---|---|
1.0 | 0.141 |
1.5 | 0.196 |
2.0 | 0.229 |
2.5 | 0.249 |
3.0 | 0.263 |
4.0 | 0.281 |
5.0 | 0.291 |
6.0 | 0.299 |
10.0 | 0.312 |
0.333 |
Alternatively the following equation can be used with an error of not greater than 4%:
where
This is a tube with a slit cut longitudinally through its wall. Using the formula above: