Torsion constant



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.

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:
 * $$\theta = \frac{TL}{GJ}$$

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,


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

And the torsional stiffness,


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

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

Circle

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

where
 * r is the radius

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

alternatively write: $$J = \frac{\pi D^4}{32}$$ where
 * D is the Diameter

Ellipse

 * $$J \approx \frac{\pi a^3 b^3}{a^2 + b^2}$$

where
 * a is the major radius
 * b is the minor radius

Square

 * $$J \approx \,2.25 a^4$$

where
 * a is half the side length.

Rectangle

 * $$J \approx\beta a b^3$$

where
 * a is the length of the long side
 * b is the length of the short side


 * $$\beta$$ is found from the following table:

Alternatively the following equation can be used with an error of not greater than 4%:
 * $$J \approx \frac {a b^3}{16} \left ( \frac{16}{3}- {3.36} \frac{b}{a} \left ( 1- \frac{b^4}{12a^4} \right ) \right )$$

where
 * a is the length of the long side
 * b is the length of the short side

Thin walled open tube of uniform thickness

 * $$J = \frac{1}{3}Ut^3$$
 * t is the wall thickness
 * U is the length of the median boundary (perimeter of median cross section)

Circular thin walled open tube of uniform thickness
This is a tube with a slit cut longitudinally through its wall. Using the formula above:
 * $$U = 2\pi r$$
 * $$J = \frac{2}{3} \pi r t^3$$
 * t is the wall thickness
 * r is the mean radius