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 plane 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-section always have warping deformations that require numerical methods to allow the exact calculation of the torsion constant.[2]

The torsional stiffness of a 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:

\theta = \frac{TL}{JG}

where

\theta is the angle of twist in radians
T is the applied torque
L is the beam length
J is the moment of inertia
G is the Modulus of rigidity (shear modulus) of the material

Examples for specific uniform cross-sectional shapes

Circle

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 Jzz and is exact.

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

D is the Diameter

Ellipse

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

J \approx \,2.25 a^4[7]

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
\infty 0.333

[8]

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

J \approx a b^3 \left ( \frac{1}{3}-0.21 \frac{b}{a} \left ( 1- \frac{b^4}{12a^4} \right ) \right )[5]

Thin walled open tube of uniform thickness

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

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

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. ↑ 4.0 4.1 "Area Moment of Inertia." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/AreaMomentofInertia.html
  5. ↑ 5.0 5.1 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