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
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a b ...
. 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
The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
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.
[Advanced structural mechanics, 2nd Edition, David Johnson]
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:
:
where
:
is the angle of twist in radians
:''T'' is the applied torque
:''L'' is the beam length
:''G'' is the
Modulus of rigidity
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stackrel ...
(shear modulus) of the material
:''J'' is the torsional constant
Torsional Rigidity (GJ) and Stiffness (GJ/L)
Inverting the previous relation, we can define two quantities: the torsional rigidity,
:
with SI units N⋅m
2/rad
And the torsional stiffness,
:
with SI units N⋅m/rad
Examples for specific uniform cross-sectional shapes
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
The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
J
zz and is exact.
alternatively write:
where
:''D'' is the Diameter
Ellipse
:
[Roark's Formulas for stress & Strain, 7th Edition, Warren C. Young & Richard G. Budynas][Continuum 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:
Alternatively the following equation can be used with an error of not greater than 4%:
:
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
{{DEFAULTSORT:Torsion Constant
Continuum mechanics
Structural analysis