Torsion constant

Formulation

For a beam of uniform cross-section along its length, the angle of twist (in radians) ${\displaystyle \theta }$ is:

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

where:

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

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

Bars with given uniform cross-sectional shapes are special cases.

Circle

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

where

r is the radius

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

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

D is the Diameter

Ellipse

${\displaystyle J\approx \frac{\pi a^3{b}^3}{a^2+b^2}}$

where

a is the major radius

b is the minor radius

Square

${\displaystyle J\approx 2.25a^4}$

where

a is half the side length.

Rectangle

${\displaystyle J\approx \beta a{b}^3}$

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:

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

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

where

a is the length of the long side

b is the length of the short side

Thin walled open tube of uniform thickness

${\displaystyle J=\frac{1}{3}U{t}^3}$

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:

${\displaystyle U=2\pi r}$

${\displaystyle J=\frac{2}{3}\pi r{t}^3}$

t is the wall thickness

r is the mean radius