Torsion constant

The torsion constant is a geometrical property of a bar's cross-section which 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 Jzz, 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]

Partial Derivation

For a beam of uniform cross-section along its length:

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

where

${\displaystyle \theta }$ is the angle of twist in radians
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

Torsional Rigidity (GJ) and Stiffness (GJ/L)

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 for specific uniform cross-sectional shapes

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

This is identical to the second moment of area Jzz 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

Square

${\displaystyle J\approx \,2.25a^{4}}$ [7]

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

[8]

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

${\displaystyle J\approx ab^{3}\left({\frac {16}{3}}-3.36{\frac {b}{a}}\left(1-{\frac {b^{4}}{12a^{4}}}\right)\right)}$ [5]

In the formula above, a and b are half the length of the long and short sides, respectively.

Thin walled open tube of uniform thickness

${\displaystyle J={\frac {1}{3}}Ut^{3}}$ [9]
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 (approximation)

This is a tube with a slit cut longitudinally through its wall.

${\displaystyle J={\frac {2}{3}}\pi rt^{3}}$ [10]
t is the wall thickness

This is derived from the above equation for an arbitrary thin walled open tube of uniform thickness.

References

1. Archie Higdon et al. "Mechanics of Materials, 4th edition".
2. Advanced structural mechanics, 2nd Edition, David Johnson
3. The Influence and Modelling of Warping Restraint on Beams
4. "Area Moment of Inertia." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/AreaMomentofInertia.html
5. Roark's Formulas for stress & Strain, 7th Edition, Warren C. Young & Richard G. Budynas
6. Continuum Mechanics, Fridtjov Irjens, Springer 2008, p238, ISBN 978-3-540-74297-5
7. Torsion Equations, Roy Beardmore, http://www.roymech.co.uk/Useful_Tables/Torsion/Torsion.html
8. Advanced Strength and Applied Elasticity, Ugural & Fenster, Elsevier, ISBN 0-444-00160-3
9. Advanced Mechanics of Materials, Boresi, John Wiley & Sons, ISBN 0-471-55157-0
10. Roark's Formulas for stress & Strain, 6th Edition, Warren C. Young