# 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 m^{4}.

## 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.^{[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:

where

- is the angle of twist in radians
*T*is the applied torque*L*is the beam length*J*is the torsional constant*G*is the Modulus of rigidity (shear modulus) of the material

## Examples for specific uniform cross-sectional shapes

### Circle

^{[4]}

where

*r*is the radius

This is identical to the second moment of area J_{zz} and is exact.

alternatively write: ^{[4]}
where

*D*is the Diameter

### Ellipse

^{[5]}^{[6]}

where

*a*is the major radius*b*is the minor radius

### Square

^{[7]}

where

*a*is half the side length

### Rectangle

where

*a*is the length of the long side*b*is the length of the short side- is found from the following table:

a/b | |
---|---|

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 |

0.333 |

^{[8]}

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

^{[5]}

### Thin walled open tube of uniform thickness

^{[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.

^{[10]}*t*is the wall thickness*r*is the mean radius

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

## Commercial Products

There are a number specialized software tools to calculate the torsion constant using the finite element method.

- ShapeDesigner by Mechatools Technologies
- ShapeBuilder by IES, Inc.
- STAAD SectionWizard by Bentley
- SectionAnalyzer by Fornamagic Ltd
- Strand7 BXS Generator by Strand7 Pty Limited

## References

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