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| '''Hankinson's equation''' (also called '''Hankinson's formula''' or '''Hankinson's criterion''')<ref>Hankinson, R. L., 1921, ''Investigation of crushing strength of spruce at varying angles of grain'', '''Air Force Information Circular No. 259''', U. S. Air Service.</ref> is a mathematical relationship for predicting the off-axis uniaxial compressive strength of wood. The formula can also be used to compute the fiber [[stress (physics)|stress]] or the [[Stress (physics)|stress]] wave velocity at the elastic limit as a function of [[Wood grain|grain]] angle in [[wood]]. For a wood that has uniaxial compressive strengths of <math>\sigma_0</math> parallel to the grain and <math>\sigma_{90}</math> perpendicular to the grain, Hankinson's equation predicts that the uniaxial compressive strength of the wood in a direction at an angle <math>\alpha</math> to the grain is given by
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| :<math>
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| \sigma_\alpha = \cfrac{\sigma_0~\sigma_{90}}{\sigma_0~\sin^2\alpha + \sigma_{90}~\cos^2\alpha}
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| </math>
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| Even though the original relation was based on studies of [[spruce]], Hankinson's equation has been found to be remarkably accurate for many other types of wood. A generalized form of the Hankinson formula has also been used for predicting the uniaxial tensile strength of wood at an angle to the grain. This formula has the form<ref>Clouston, P., 1995, '''[http://hdl.handle.net/2429/4154 The Tsai-Wu strength theory for Douglas fir laminated veneer]''', M. S. Thesis, The University of British Columbia.</ref>
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| :<math>
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| \sigma_\alpha = \cfrac{\sigma_0~\sigma_{90}}{\sigma_0~\sin^n\alpha + \sigma_{90}~\cos^n\alpha}
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| </math>
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| where the exponent <math>n</math> can take values between 1.5 and 2.
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| The stress wave velocity at angle angle <math>\alpha</math> to the grain at the elastic limit can similarly be obtained from the Hankinson formula
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| :<math>
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| V(\alpha) = \frac{V_0 V_{90}}{V_0 \sin^2\alpha + V_{90} \cos^2\alpha}
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| </math>
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| where <math>V_0</math> is the velocity parallel to the grain, <math>V_{90}</math> is the velocity perpendicular to the grain and <math>\alpha</math> is the grain angle.
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| == See also ==
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| {{continuum mechanics|cTopic=[[Solid mechanics]]}}
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| * [[Material failure theory]]
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| * [[Linear elasticity]]
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| * [[Hooke's law]]
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| * [[Orthotropic material]]
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| * [[Transverse isotropy]]
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| ==References==
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| <references/>
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| == External links ==
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| * [http://www.worldwideflood.com/ark/design_calculations/wood_strength.htm DESIGN CALCULATIONS - WOOD STRENGTH]
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| [[Category:Materials science]]
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| [[Category:Solid mechanics]]
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| [[Category:Equations]]
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