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 ${\sigma }_0$ parallel to the grain and ${\sigma }_{90}$ perpendicular to the grain, Hankinson's equation predicts that the uniaxial compressive strength of the wood in a direction at an angle $\alpha$ to the grain is given by

${\sigma }_{\alpha }=\frac{{\sigma }_0{\sigma }_{90}}{{\sigma }_0{\sin }^2\alpha +{\sigma }_{90}{\cos }^2\alpha }$

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

${\sigma }_{\alpha }=\frac{{\sigma }_0{\sigma }_{90}}{{\sigma }_0{\sin }^n\alpha +{\sigma }_{90}{\cos }^n\alpha }$

where the exponent $n$ can take values between 1.5 and 2.

The stress wave velocity at angle $\alpha$ to the grain at the elastic limit can similarly be obtained from the Hankinson formula

$V(\alpha )=\frac{V_0{V}_{90}}{V_0{\sin }^2\alpha +V_{90}{\cos }^2\alpha }$

where $V_0$ is the velocity parallel to the grain, $V_{90}$ is the velocity perpendicular to the grain and $\alpha$ is the grain angle.

See also

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