The torsion constant or torsion coefficient is a geometrical property of a bar's cross-section. It 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.Archie Higdon et al. "Mechanics of Materials, 4th edition".
Non-circular cross-sections always have warping deformations that require more complex methods calculate the torsion constant. Few non-circular cross-sections have exact analytical solutions for the torsion constant but approximate numerical solutions exist for many shapes.Advanced structural mechanics, 2nd Edition, David Johnson
The torsional stiffness of short beams with open cross sections is significantly increased if the warping of the end sections is restrained by, for example, stiff end blocks.The Influence and Modelling of Warping Restraint on Beams
Formulation For a beam of uniform cross-section along its length, the angle of twist (in radians) is:
where: 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
Inverting the previous relation, we can define two quantities; the torsional rigidity,
with SI units N⋅m2/rad
And the torsional stiffness,
with SI units N⋅m/rad
Examples Bars with given uniform cross-sectional shapes are special cases.
Circle "Area Moment of Inertia." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/AreaMomentofInertia.html where r is the radius This is identical to the second moment of area Jzz and is exact.
alternatively write: where D is the Diameter
Ellipse Roark's Formulas for stress & Strain, 7th Edition, Warren C. Young & Richard G. BudynasContinuum Mechanics, Fridtjov Irjens, Springer 2008, p238, where a is the major radius b is the minor radius
Square
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 0.333Advanced Strength and Applied Elasticity, Ugural & Fenster, Elsevier,
Alternatively, the following equation can be used with an error of not greater than 4%:
where a is half the length of the long side b is half the length of the short side
Thin walled open tube of uniform thickness Advanced Mechanics of Materials, Boresi, John Wiley & Sons, 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 This is a tube with a slit cut longitudinally through its wall. Using the formula above:
Roark's Formulas for stress & Strain, 6th Edition, Warren C. Young t is the wall thickness r is the mean radius
References
External links Torsion constant calculator