Torsion constant

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.

Partial Derivation

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

θ = T L J G

where

θ is the angle of twist in radiansT is the applied torqueL is the beam lengthJ is the torsional constantG is the Modulus of rigidity (shear modulus) of the material

Circle

J z z = J x x + J y y = π r 4 4 + π r 4 4 = π r 4 2

where

r is the radius

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

alternatively write: J = π D 4 32 where

D is the Diameter

Ellipse

J ≈ π a 3 b 3 a 2 + b 2

where

a is the major radiusb is the minor radius

Square

J ≈ 2.25 a 4

where

a is half the side length

Rectangle

J ≈ β a b 3

where

a is the length of the long sideb is the length of the short side β is found from the following table:

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

J ≈ a b 3 ( 1 3 − 0.21 b a ( 1 − b 4 12 a 4 ) )

Thin walled open tube of uniform thickness

J = 1 3 U t 3 t is the wall thicknessU 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 = 2 3 π r t 3 t is the wall thicknessr 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