# Torsion constant

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.[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]

## Partial Derivation

For a beam of uniform cross-section along its length:

${\displaystyle \theta ={\frac {TL}{GJ}}}$

where

${\displaystyle \theta }$ is the angle of twist in radians
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

## Torsional Rigidity (GJ) and Stiffness (GJ/L)

Inverting the previous relation, we can define two quantities: the torsional rigidity

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

And the torsional stiffness:

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

## Examples for specific uniform cross-sectional shapes

### Circle

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

where

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

alternatively write: ${\displaystyle J={\frac {\pi D^{4}}{32}}}$[4] where

D is the Diameter

### Ellipse

${\displaystyle J\approx {\frac {\pi a^{3}b^{3}}{a^{2}+b^{2}}}}$[5][6]

where

### Square

${\displaystyle J\approx \,2.25a^{4}}$[5]

where

a is half the side length.

### Rectangle

${\displaystyle J\approx \beta ab^{3}}$

where

a is the length of the long side
b is the length of the short side
${\displaystyle \beta }$ is found from the following table:
a/b ${\displaystyle \beta }$
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
${\displaystyle \infty }$ 0.333

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

${\displaystyle J\approx ab^{3}\left({\frac {16}{3}}-3.36{\frac {b}{a}}\left(1-{\frac {b^{4}}{12a^{4}}}\right)\right)}$[5]

In the formula above, a and b are half the length of the long and short sides, respectively.

### Thin walled open tube of uniform thickness

${\displaystyle J={\frac {1}{3}}Ut^{3}}$[8]
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:

${\displaystyle U=2\pi r}$
${\displaystyle J={\frac {2}{3}}\pi rt^{3}}$[9]
t is the wall thickness