Hankinson's equation

Hankinson's equation (also called Hankinson's formula or Hankinson's criterion) 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 or the stress wave velocity at the elastic limit as a function of grain angle in wood. For a wood that has uniaxial compressive strengths of [math]\displaystyle{ \sigma_0 }[/math] parallel to the grain and [math]\displaystyle{ \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]\displaystyle{ \alpha }[/math] to the grain is given by

[math]\displaystyle{ \sigma_\alpha = \cfrac{\sigma_0~\sigma_{90}}{\sigma_0~\sin^2\alpha + \sigma_{90}~\cos^2\alpha} }[/math]

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

[math]\displaystyle{ \sigma_\alpha = \cfrac{\sigma_0~\sigma_{90}}{\sigma_0~\sin^n\alpha + \sigma_{90}~\cos^n\alpha} }[/math]

where the exponent [math]\displaystyle{ n }[/math] can take values between 1.5 and 2.

The stress wave velocity at angle [math]\displaystyle{ \alpha }[/math] to the grain at the elastic limit can similarly be obtained from the Hankinson formula

[math]\displaystyle{ V(\alpha) = \frac{V_0 V_{90}}{V_0 \sin^2\alpha + V_{90} \cos^2\alpha} }[/math]

where [math]\displaystyle{ V_0 }[/math] is the velocity parallel to the grain, [math]\displaystyle{ V_{90} }[/math] is the velocity perpendicular to the grain and [math]\displaystyle{ \alpha }[/math] is the grain angle.

See also

• Material failure theory
• Linear elasticity
• Hooke's law
• Orthotropic material
• Transverse isotropy