torsion constant wiki

7/28/2019 Torsion Constant Wiki

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Torsion constant 1

Torsion constant

The torsion constant is a geometrical property of a beam'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. That is, the

torsion constant describes a beam's torsional stiffness.

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 plane 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.[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-section always have warping

deformations that require numerical methods to allow the exact calculation of the torsion constant.[2]

Partial Derivation

For a beam of uniform cross-section along its length:

where

is the angle of twist in radians

Tis the applied torque

L is the beam length

Jis the 2nd Polar Moment of Area

G is the Modulus of rigidity (shear modulus) of the material
http://en.wikipedia.org/w/index.php?title=File:TorsionConstantBar.svghttp://en.wikipedia.org/w/index.php?title=Modulus_of_rigidityhttp://en.wikipedia.org/w/index.php?title=Second_moment_of_areahttp://en.wikipedia.org/w/index.php?title=Stiffness


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Examples for specific uniform cross-sectional shapes

Circle

[3]

where

ris the radius

This is identical to the second moment of area Jzz

and is exact.

alternatively write:[3]

where

D is the Diameter

s

Ellipse

[4][5]

where

a is the major radius

b is the minor radius

Square

[6]

where

2a is 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
http://en.wikipedia.org/w/index.php?title=Second_moment_of_area


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0.333

[7]

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

[4]

Thin walled closed tube of uniform thickness

[8]

A is the mean of the areas enclosed by the inner and outer boundaries

tis the wall thickness

Uis the length of the median boundary

Thin walled open tube of uniform thickness[9]


Uis 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.

[8]


ris the mean radius

This is derived from the above equation for an arbitrary thin walled open tube of uniform thickness.

Commercial Products

There are a number specialized software tools to calculate the torsion constant using the finite element method.

ShapeDesigner[10]

by Mechatools Technologies[11]

ShapeBuilder[12]

by IES Web[13]

STAAD SectionWizard[14]

by Bentley[15]

SectionAnalyzer[16]

by Fornamagic Ltd[17]

Strand7 BXS Generator[18]

by Strand7 Pty Limited[19]
http://www.strand7.com/http://strand7.com/http://www.fornamagic.com/http://fornamagic.com/download.php?view.15http://www.bentley.com/http://www.bentley.com/en-US/Products/STAAD.Pro/Section-wizard.htmhttp://www.iesweb.com/http://www.iesweb.com/products/shapebuilder/index.htmhttp://www.mechatools.com/http://www.mechatools.com/en/shapedesigner.html


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References

[1][1] Archie Higdon et al. "Mechanics of Materials, 4th edition".

[2][2] Advanced structural mechanics, 2nd Edition, David Johnson

[3] "Area Moment of Inertia." From MathWorld--A Wolfram Web Resource. http://mathworld. wolfram.com/AreaMomentofInertia. html

[4] Roark's Formulas for stress & Strain, 7th Edition, Warren C. Young & Richard G. Budynas

[5][5] Continuum Mechanics, Fridtjov Irjens, Springer 2008, p238, ISBN 978-3-540-74297-5

[6] Torsion Equations, Roy Beardmore, http://www.roymech. co.uk/Useful_Tables/Torsion/Torsion. html[7] Advanced Strength and Applied Elasticity, Ugural & Fenster, Elsevier, ISBN 0-444-00160-3

[8] Roark's Formulas for stress & Strain, 6th Edition, Warren C. Young

[9] Advanced Mechanics of Materials, Boresi, John Wiley & Sons, ISBN 0-471-55157-0

[10] http://www.mechatools.com/en/shapedesigner. html

[11] http://www.mechatools.com

[12] http://www.iesweb.com/products/shapebuilder/index. htm

[13] http://www.iesweb.com

[14] http://www.bentley.com/en-US/Products/STAAD.Pro/Section-wizard.htm

[15] http://www.bentley.com

[16] http://fornamagic. com/download. php?view. 15

[17] http://www.fornamagic. com

[18] http://strand7. com

[19] http://www.strand7.com
http://www.strand7.com/http://strand7.com/http://www.fornamagic.com/http://fornamagic.com/download.php?view.15http://www.bentley.com/http://www.bentley.com/en-US/Products/STAAD.Pro/Section-wizard.htmhttp://www.iesweb.com/http://www.iesweb.com/products/shapebuilder/index.htmhttp://www.mechatools.com/http://www.mechatools.com/en/shapedesigner.htmlhttp://www.roymech.co.uk/Useful_Tables/Torsion/Torsion.htmlhttp://mathworld.wolfram.com/AreaMomentofInertia.html


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Article Sources and Contributors 5

Article Sources and ContributorsTorsion constant Source: http://en.wikipedia.org/w/index.php?oldid=563815380 Contributors: Bassplr19, BenFrantzDale, Bradv, Closedmouth, Davius, GTi16V, Greglocock, Kallog, Lockley,

RockMagnetist, Suffusion of Yellow, The wub, Zerodamage, 58 anonymous edits

Image Sources, Licenses and ContributorsFile:TorsionConstantBar.svg Source: http://en.wikipedia.org/w/index.php?title=File:TorsionConstantBar.svg License: Creative Commons Attribution-Sharealike 3.0 Contributors:

TorsionConstantBar.png: Kallog (talk). derivative work: Zerodamage

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