Definitions

# 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:

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

$theta$ 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

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

### Hollow concentric circular tube

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

### Square

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

### Rectangle

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

### Thin walled closed tube of uniform thickness

$J = frac\left\{4A^2t\right\}\left\{U\right\}$
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

$J = frac\left\{1\right\}\left\{2\right\}U t^3$
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.
$J = frac\left\{2\right\}\left\{3\right\} pi r t^3$
t is the wall thickness