# 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 m$^4$.

## 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 J$_{zz}$, 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⋅m$^2$/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 J$_{zz}$ 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

    $J\approx \,2.25a^{4}$
where
    a is half the side length.

### Rectangle

    $J\approx \beta ab^{3}$
where
    a is the length of the long side
    b is the length of the short side
    $\beta$ is found from the following table:
| a/b | $\beta$ |
| 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 |
| $\infty$ | 0.333 |
Alternatively the following equation can be used with an error of not greater than 4%:
    $J\approx {\frac {ab^{3}}{16}}\left({\frac {16}{3}}-{3.36}{\frac {b}{a}}\left(1-{\frac {b^{4}}{12a^{4}}}\right)\right)$
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:
    $U=2\pi r$
    $J={\frac {2}{3}}\pi rt^{3}$
    t is the wall thickness
    r is the mean radius