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| 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]]. The SI unit for torsion constant is m<sup>4</sup>.
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| == History ==
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| 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<sub>zz</sub>, which has an exact analytic equation, by assuming that a plane section before twisting remains plane after twisting, and a diameter remains a straight line.
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| Unfortunately, that assumption is correct only in beams with circular cross-sections, and is incorrect for any other shape.<ref>
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| Archie Higdon et al.
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| "Mechanics of Materials, 4th edition".
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| </ref>
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| 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.
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| Non-circular cross-section always have warping deformations that require numerical methods to allow the exact calculation of the torsion constant.<ref name="David">Advanced structural mechanics, 2nd Edition, David Johnson</ref>
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| == Partial Derivation ==
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| For a beam of uniform cross-section along its length:
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| :<math>\theta = \frac{TL}{JG}</math>
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| where
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| :<math>\theta</math> is the angle of twist in radians
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| :''T'' is the applied torque
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| :''L'' is the beam length
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| :''J'' is the torsion constant
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| :''G'' is the [[Modulus of rigidity]] (shear modulus) of the material
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| [[File:TorsionConstantBar.svg]]
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| ==Examples for specific uniform cross-sectional shapes==
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| ===Circle===
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| :<math>J_{zz} = J_{xx}+J_{yy} = \frac{\pi r^4}{4} + \frac{\pi r^4}{4} = \frac{\pi r^4}{2}</math><ref name="Weisstein, Eric W.">"Area Moment of Inertia." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/AreaMomentofInertia.html</ref>
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| where
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| :''r'' is the radius
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| This is identical to the [[second moment of area]] J<sub>zz</sub> and is exact.
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| alternatively write: <math>J = \frac{\pi D^4}{32}</math><ref name="Weisstein, Eric W.">"Area Moment of Inertia." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/AreaMomentofInertia.html</ref>
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| where
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| :''D'' is the Diameter
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| ===Ellipse===
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| :<math>J \approx \frac{\pi a^3 b^3}{a^2 + b^2}</math><ref name="Roark7">Roark's Formulas for stress & Strain, 7th Edition, Warren C. Young & Richard G. Budynas</ref><ref name="Irjens">Continuum Mechanics, Fridtjov Irjens, Springer 2008, p238, ISBN 978-3-540-74297-5</ref>
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| where
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| :''a'' is the major radius
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| :''b'' is the minor radius
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| ===Square===
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| :<math>J \approx \,2.25 a^4</math><ref name="RoyMech7">Torsion Equations, Roy Beardmore, http://www.roymech.co.uk/Useful_Tables/Torsion/Torsion.html</ref>
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| where
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| :''a'' is half the side length
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| ===Rectangle===
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| :<math>J \approx\beta a b^3</math>
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| where
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| :''a'' is the length of the long side
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| :''b'' is the length of the short side
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| :<math>\beta</math> is found from the following table:
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| {| class="wikitable"
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| |-
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| ! a/b
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| ! <math>\beta</math>
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| |-
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| | 1.0
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| | 0.141
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| |-
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| | 1.5
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| | 0.196
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| |-
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| | 2.0
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| | 0.229
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| |-
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| | 2.5
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| | 0.249
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| |-
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| | 3.0
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| | 0.263
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| |-
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| | 4.0
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| | 0.281
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| |-
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| | 5.0
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| | 0.291
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| |-
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| | 6.0
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| | 0.299
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| |-
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| | 10.0
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| | 0.312
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| |-
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| | <math>\infty</math>
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| | 0.333
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| |}<ref>Advanced Strength and Applied Elasticity, Ugural & Fenster, Elsevier, ISBN 0-444-00160-3</ref>
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| Alternatively the following equation can be used with an error of not greater than 4%:<br>
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| :<math>J \approx a b^3 \left ( \frac{1}{3}-0.21 \frac{b}{a} \left ( 1- \frac{b^4}{12a^4} \right ) \right )</math><ref name="Roark7" />
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| ===Thin walled closed tube of uniform thickness===
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| :<math>J = \frac{4A^2t}{U}</math><ref name="Roark">Roark's Formulas for stress & Strain, 6th Edition, Warren C. Young</ref>
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| :''A'' is the mean of the areas enclosed by the inner and outer boundaries
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| :''t'' is the wall thickness
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| :''U'' is the length of the median boundary
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| ===Thin walled open tube of uniform thickness===
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| :<math>J = \frac{1}{3}U t^3</math><ref>Advanced Mechanics of Materials, Boresi, John Wiley & Sons, ISBN 0-471-55157-0</ref>
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| :''t'' is the wall thickness
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| :''U'' is the length of the median boundary (perimeter of median cross section) | |
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| ===Circular thin walled open tube of uniform thickness (approximation)===
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| This is a tube with a slit cut longitudinally through its wall.
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| :<math>J = \frac{2}{3} \pi r t^3</math><ref name="Roark" />
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| :''t'' is the wall thickness
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| :''r'' is the mean radius
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| This is derived from the above equation for an arbitrary thin walled open tube of uniform thickness.
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| ==Commercial Products ==
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| There are a number specialized software tools to calculate the torsion constant using the finite element method.
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| * [http://www.mechatools.com/en/shapedesigner.html ShapeDesigner] by [http://www.mechatools.com Mechatools Technologies]
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| * [http://www.iesweb.com/products/shapebuilder/index.htm ShapeBuilder] by [http://www.iesweb.com IES Web]
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| * [http://www.bentley.com/en-US/Products/STAAD.Pro/Section-wizard.htm STAAD SectionWizard] by [http://www.bentley.com Bentley]
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| * [http://fornamagic.com/download.php?view.15 SectionAnalyzer] by [http://www.fornamagic.com Fornamagic Ltd]
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| * [http://strand7.com Strand7 BXS Generator] by [http://www.strand7.com Strand7 Pty Limited]
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Torsion Constant}}
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| [[Category:Continuum mechanics]]
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| [[Category:Structural analysis]]
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Marvella is what you can contact her but it's not the most feminine title out there. Hiring is his profession. Body building is what my family members and I enjoy. Years ago we moved to North Dakota.
my blog: at home std testing