Torsion constant

The torsion constant is a geometrical property of a beam's cross-section which determines the relationship between angle of twist and applied torque.

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

$\theta ={\frac {TL}{JG}}$

$\theta$ is the angle of twist in radians
T is the applied torque
L is the beam length
J is the torsion constant
G is the modulus of rigidity of the material

For non-circular cross-sections, there are no exact analytical equations for finding J. Approximate solutions have been found for many shapes.

Examples for specific cross-sectional shapes

Circle

$J={\frac {\pi r^{4}}{2}}$
r is the radius
This is identical to the polar moment of inertia and is exact.

Hollow concentric circular tube

$J={\frac {\pi }{2}}\left(r_{o}^{4}-r_{i}^{4}\right)$
$r_{o}$ is the outer radius
$r_{i}$ is the inner radius
This is identical to the polar moment of inertia and is exact.

Square

$J\approx {\frac {a^{4}}{7.10}}$
a is the side length

Rectangle

$J=\beta ab^{3}$
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%:
$J\approx ab^{3}\left({\frac {1}{3}}-0.210{\frac {b}{a}}\left(1-{\frac {b^{4}}{12a^{4}}}\right)\right)$

Thin walled closed tube of uniform thickness

$J={\frac {4A^{2}t}{U}}$
A is the mean of the areas enclosed by the inner and outer boundaries
t is the wall thickness
U is the length of the median boundary

Thin walled open tube of uniform thickness

$J={\frac {1}{2}}Ut^{3}$
t is the wall thickness
U is the length of the median boundary

Circular thin walled open tube of uniform thickness

This is a tube with a slit cut longitudinally through its wall.
$J={\frac {2}{3}}\pi rt^{3}$
t is the wall thickness
r is the mean radius
This is derived from the above equation for an arbitrary thin walled open tube of uniform thickness.