Torsion Constant - Wikipedia

Geometrical property of a bar's cross-section

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

[edit] See also: Torsion balance

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.[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]

Formulation

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

θ = T L G J {\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,

G J = T L θ {\displaystyle GJ={\frac {TL}{\theta }}} with SI units N⋅m2/rad

And the torsional stiffness,

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

Examples

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

Circle

[edit] J z z = J x x + J y y = π r 4 4 + π r 4 4 = π 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 Jzz and is exact.

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

D is the Diameter

Ellipse

[edit] J ≈ π 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

Square

[edit] J ≈ 2.25 a 4 {\displaystyle J\approx \,2.25a^{4}} [5]

where

a is half the side length.

Rectangle

[edit] J ≈ β a b 3 {\displaystyle J\approx \beta ab^{3}}

where

a is the length of the long side b is the length of the short side β {\displaystyle \beta } is found from the following table:

a/b	β {\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 }	0.333

[7]

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

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

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

[edit] J = 1 3 U t 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)

Circular thin walled open tube of uniform thickness

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

U = 2 π r {\displaystyle U=2\pi r} J = 2 3 π r t 3 {\displaystyle J={\frac {2}{3}}\pi rt^{3}} [9] t is the wall thickness r is the mean radius

References

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1. ^ Archie Higdon et al. "Mechanics of Materials, 4th edition".
2. ^ Advanced structural mechanics, 2nd Edition, David Johnson
3. ^ The Influence and Modelling of Warping Restraint on Beams
4. ^ a b "Area Moment of Inertia." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/AreaMomentofInertia.html
5. ^ a b c Roark's Formulas for stress & Strain, 7th Edition, Warren C. Young & Richard G. Budynas
6. ^ Continuum Mechanics, Fridtjov Irjens, Springer 2008, p238, ISBN 978-3-540-74297-5
7. ^ Advanced Strength and Applied Elasticity, Ugural & Fenster, Elsevier, ISBN 0-444-00160-3
8. ^ Advanced Mechanics of Materials, Boresi, John Wiley & Sons, ISBN 0-471-55157-0
9. ^ Roark's Formulas for stress & Strain, 6th Edition, Warren C. Young

External links

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• Torsion constant calculator