Torsion constant - Citizendia

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:

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

$θ$ 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.

1 Examples for specific cross-sectional shapes
2 References

Examples for specific cross-sectional shapes

Circle

$J = \frac{\pi r^4}{2}$ ^[1]
r is the radius
This is identical to the polar moment of inertia and is exact. Polar moment of inertia is a quantity used to predict an object's ability to resist torsion, in objects (or segments of objects with an invariant circular Cross-section

Hollow concentric circular tube

$J = \frac{\pi}{2} \left ( r_o^4 - r_i^4 \right )$ ^[1]
$r o$ is the outer radius
$r i$ is the inner radius
This is identical to the polar moment of inertia and is exact. Polar moment of inertia is a quantity used to predict an object's ability to resist torsion, in objects (or segments of objects with an invariant circular Cross-section

Square

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

Rectangle

$J = β a b 3$
a is the length of the long side
b is the length of the short side
$β$ is found from the following table:

a/b	$β$
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

^[2]

Alternatively the following equation can be used with an error of not greater than 4%:
$J \approx a b^3 \left ( \frac{1}{3}-0.210 \frac{b}{a} \left ( 1- \frac{b^4}{12a^4} \right ) \right )$ ^[1]

Thin walled closed tube of uniform thickness

$J = \frac{4A^2t}{U}$ ^[1]
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

$J = \frac{1}{2}U t^3$ ^[1]
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.
$J = \frac{2}{3} \pi r t^3$ ^[1]
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.

References

^ ^a ^b ^c ^d ^e ^f Roark's Formulas for stress & Strain, 6th Edition, Warren C. Young
^ Advanced Strength and Applied Elasticity, Ugural & Fenster, Elsevier, ISBN 0-444-00160-3

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