# 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 plane 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.[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-section always have warping deformations that require numerical methods to allow the exact calculation of the torsion constant.[2]

## Partial Derivation

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

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

where

$\theta$ is the angle of twist in radians
T is the applied torque
L is the beam length
J is the moment of inertia
G is the Modulus of rigidity (shear modulus) of the material

## Examples for specific uniform cross-sectional shapes

### Circle

$J_{zz} = J_{xx}+J_{yy} = \frac{\pi r^4}{4} + \frac{\pi r^4}{4} = \frac{\pi r^4}{2}$[3]

where

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

alternatively write: $J = \frac{\pi D^4}{32}$[3] where

D is the Diameter

### Ellipse

$J \approx \frac{\pi a^3 b^3}{a^2 + b^2}$[4][5]

where

### Square

$J \approx \,2.25 a^4$[6]

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:
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

[7]

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

$J \approx a b^3 \left ( \frac{1}{3}-0.21 \frac{b}{a} \left ( 1- \frac{b^4}{12a^4} \right ) \right )$[4]

### Thin walled closed tube of uniform thickness

$J = \frac{4A^2t}{U}$[8]
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

$J = \frac{1}{3}U t^3$[9]
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 (approximation)

This is a tube with a slit cut longitudinally through its wall.

$J = \frac{2}{3} \pi r t^3$[8]
t is the wall thickness

This is derived from the above equation for an arbitrary thin walled open tube of uniform thickness.

## Commercial Products

There are a number specialized software tools to calculate the torsion constant using the finite element method.

## References

1. ^ Archie Higdon et al. "Mechanics of Materials, 4th edition".
2. ^ Advanced structural mechanics, 2nd Edition, David Johnson
3. ^ a b "Area Moment of Inertia." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/AreaMomentofInertia.html
4. ^ a b Roark's Formulas for stress & Strain, 7th Edition, Warren C. Young & Richard G. Budynas
5. ^ Continuum Mechanics, Fridtjov Irjens, Springer 2008, p238, ISBN 978-3-540-74297-5
6. ^ Torsion Equations, Roy Beardmore, http://www.roymech.co.uk/Useful_Tables/Torsion/Torsion.html
7. ^ Advanced Strength and Applied Elasticity, Ugural & Fenster, Elsevier, ISBN 0-444-00160-3
8. ^ a b Roark's Formulas for stress & Strain, 6th Edition, Warren C. Young
9. ^ Advanced Mechanics of Materials, Boresi, John Wiley & Sons, ISBN 0-471-55157-0