Torsion constant

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

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.

History

Template:See also 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.

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.

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.

Formulation

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

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

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,

$GJ=\frac{TL}{\theta }$ with SI units N⋅m²/rad

And the torsional stiffness,Template:Anchor

$\frac{GJ}{L}=\frac{T}{\theta }$ with SI units N⋅m/rad

Examples

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

Circle

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

where

r is the radius

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

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

D is the Diameter

Ellipse

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

where

a is the major radius

b is the minor radius

Square

$J\approx 2.25a^4$

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
$\mathrm{\infty }$	0.333

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

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

where

a is the length of the long side

b is the length of the short side

Thin walled open tube of uniform thickness

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

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:

$U=2\pi r$

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

t is the wall thickness

r is the mean radius

External links

• Torsion constant calculator