Torsion constant

The torsion constant or torsion coefficient is a geometrical property of a bar's cross-section. It 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.

Main quantities involved in bar torsion: ${\displaystyle \theta }$ is the angle of twist; T is the applied torque; L is the beam length.

History

See also: Torsion balance

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 J_zz, which has an exact analytic equation, by assuming that a plane section before twisting remains planar 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 where warping takes place.^[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-sections always have warping deformations that require numerical methods to allow for the exact calculation of the torsion constant.^[2]

The torsional stiffness of beams with non-circular cross sections is significantly increased if the warping of the end sections is restrained by, for example, stiff end blocks.^[3]

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

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

where

r is the radius

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

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

D is the Diameter

Ellipse

${\displaystyle J\approx {\frac {\pi a^{3}b^{3}}{a^{2}+b^{2}}}}$^[5][6]

where

a is the major radius

b is the minor radius

Square

${\displaystyle J\approx \,2.25a^{4}}$^[5]

where

a is half the side length.

Rectangle

${\displaystyle J\approx \beta ab^{3}}$

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:

a/b	${\displaystyle \beta }$
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
${\displaystyle \infty }$	0.333

^[7]

Thin walled open tube of uniform thickness

${\displaystyle J={\frac {1}{3}}Ut^{3}}$^[8]

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:

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

${\displaystyle J={\frac {2}{3}}\pi rt^{3}}$^[9]

t is the wall thickness

r is the mean radius