Torsion constant

The torsion constant is a geometrical property of a beam'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. That is, the torsion constant describes a beam's torsional stiffness.

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 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.

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.

Partial Derivation

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

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

where

θ is the angle of twist in radians

T is the applied torque

L is the beam length

J is the torsion constant (also referred to as torsional stiffness)

G is the Modulus of rigidity (shear modulus) of the material

Examples for specific uniform cross-sectional shapes

Circle

$J = \frac{\pi r^4}{2}$

where

r is the radius

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

Ellipse

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

where

a is the major radius

b is the minor radius

Square

$J = \,2.25 a^4$

where

2a is the side length

Thin walled closed tube of uniform thickness

$J = \frac{4A^2t}{U}$

A is the mean of the areas enclosed by the inner and outer boundaries

t is the wall thickness

U is the length of the median boundary

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

Circular thin walled open tube of uniform thickness

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

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

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 Web
• STAAD SectionWizard by Bentley
• SectionAnalyzer by Fornamagic Ltd
• Strand7 BXS Generator by Strand7 Pty Limited