# Hankinson's equation

Hankinson's equation (also called Hankinson's formula or Hankinson's criterion)[1] 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 ${\displaystyle \sigma _{0}}$ parallel to the grain and ${\displaystyle \sigma _{90}}$ perpendicular to the grain, Hankinson's equation predicts that the uniaxial compressive strength of the wood in a direction at an angle ${\displaystyle \alpha }$ to the grain is given by

${\displaystyle \sigma _{\alpha }={\cfrac {\sigma _{0}~\sigma _{90}}{\sigma _{0}~\sin ^{2}\alpha +\sigma _{90}~\cos ^{2}\alpha }}}$

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[2]

${\displaystyle \sigma _{\alpha }={\cfrac {\sigma _{0}~\sigma _{90}}{\sigma _{0}~\sin ^{n}\alpha +\sigma _{90}~\cos ^{n}\alpha }}}$

where the exponent ${\displaystyle n}$ can take values between 1.5 and 2.

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

${\displaystyle V(\alpha )={\frac {V_{0}V_{90}}{V_{0}\sin ^{2}\alpha +V_{90}\cos ^{2}\alpha }}}$

where ${\displaystyle V_{0}}$ is the velocity parallel to the grain, ${\displaystyle V_{90}}$ is the velocity perpendicular to the grain and ${\displaystyle \alpha }$ is the grain angle.