Torsion constant

Short description: Geometrical property of a bar's cross-section

Main quantities involved in bar torsion: [math]\displaystyle{ \theta }[/math] 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

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) [math]\displaystyle{ \theta }[/math] is:

[math]\displaystyle{ \theta = \frac{TL}{GJ} }[/math]

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,

[math]\displaystyle{ GJ = \frac{TL}{\theta} }[/math] with SI units N⋅m²/rad

And the torsional stiffness,

[math]\displaystyle{ \frac{GJ}{L} = \frac{T}{\theta} }[/math] with SI units N⋅m/rad

Examples

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

Circle

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

where

r is the radius

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

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

D is the Diameter

Ellipse

[math]\displaystyle{ J \approx \frac{\pi a^3 b^3}{a^2 + b^2} }[/math]

where

a is the major radius

b is the minor radius

Square

[math]\displaystyle{ J \approx \,2.25 a^4 }[/math]

where

a is half the side length.

Rectangle

[math]\displaystyle{ J \approx\beta a b^3 }[/math]

where

a is the length of the long side

b is the length of the short side

[math]\displaystyle{ \beta }[/math] is found from the following table:

a/b	[math]\displaystyle{ \beta }[/math]
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
[math]\displaystyle{ \infty }[/math]	0.333

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

[math]\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 ) }[/math]

where

a is the length of the long side

b is the length of the short side

Thin walled open tube of uniform thickness

[math]\displaystyle{ J = \frac{1}{3}Ut^3 }[/math]

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:

[math]\displaystyle{ U = 2\pi r }[/math]

[math]\displaystyle{ J = \frac{2}{3} \pi r t^3 }[/math]

t is the wall thickness

r is the mean radius

External links

• Torsion constant calculator