The torsion constant is a geometrical property of a bar's cross-section which 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.
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.
Partial Derivation
For a beam of uniform cross-section along its length: where
Inverting the previous relation, we can define two quantities: the torsional rigidity with SI units N.m2/rad And the torsional stiffness: with SI units N.m/rad
Examples for specific uniform cross-sectional shapes
Circle
where This is identical to the second moment of area Jzz and is exact. alternatively write: where
Ellipse
where
Square
where
Rectangle
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%: In the formula above, a and b are halfthe length of the long and short sides, respectively.
Circular thin walled open tube of uniform thickness (approximation)
This is a tube with a slit cut longitudinally through its wall. This is derived from the above equation for an arbitrary thin walled open tube of uniform thickness.
