Torsion constant

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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 m ⁴.

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

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.^[3]

For a beam of uniform cross-section along its length, the angle of twist (in radians) ${\displaystyle \theta }${\displaystyle \theta } is:

${\displaystyle \theta ={\frac {TL}{GJ}}}${\displaystyle \theta ={\frac {TL}{GJ}}}

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,

${\displaystyle GJ={\frac {TL}{\theta }}}${\displaystyle GJ={\frac {TL}{\theta }}} with SI units N⋅m²/rad

And the torsional stiffness,

${\displaystyle {\frac {GJ}{L}}={\frac {T}{\theta }}}${\displaystyle {\frac {GJ}{L}}={\frac {T}{\theta }}} with SI units N⋅m/rad

Bars with given uniform cross-sectional shapes are special cases.

${\displaystyle J_{zz}=J_{xx}+J_{yy}={\frac {\pi r^{4}}{4}}+{\frac {\pi r^{4}}{4}}={\frac {\pi r^{4}}{2}}}${\displaystyle 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 J_zz and is exact.

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

D is the Diameter

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

${\displaystyle J\approx ,2円.25a^{4}}${\displaystyle J\approx ,2円.25a^{4}}^[5]

where

a is half the side length.

${\displaystyle J\approx \beta ab^{3}}${\displaystyle J\approx \beta ab^{3}}

where

a is the length of the long side

b is the length of the short side

${\displaystyle \beta }${\displaystyle \beta } is found from the following table:

a/b	${\displaystyle \beta }${\displaystyle \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
${\displaystyle \infty }${\displaystyle \infty }	0.333

^[7]

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

${\displaystyle J\approx {\frac {ab^{3}}{16}}\left({\frac {16}{3}}-{3.36}{\frac {b}{a}}\left(1-{\frac {b^{4}}{12a^{4}}}\right)\right)}${\displaystyle J\approx {\frac {ab^{3}}{16}}\left({\frac {16}{3}}-{3.36}{\frac {b}{a}}\left(1-{\frac {b^{4}}{12a^{4}}}\right)\right)}^[5]

where

a is the length of the long side

b is the length of the short side

${\displaystyle J={\frac {1}{3}}Ut^{3}}${\displaystyle J={\frac {1}{3}}Ut^{3}}^[8]

t is the wall thickness

U is the length of the median boundary (perimeter of median cross section)

This is a tube with a slit cut longitudinally through its wall. Using the formula above:

${\displaystyle U=2\pi r}${\displaystyle U=2\pi r}

${\displaystyle J={\frac {2}{3}}\pi rt^{3}}${\displaystyle J={\frac {2}{3}}\pi rt^{3}}^[9]

t is the wall thickness

r is the mean radius

• Torsion constant calculator

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