# 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 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.

## References

1. 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.
2. Clouston, P., 1995, The Tsai-Wu strength theory for Douglas fir laminated veneer, M. S. Thesis, The University of British Columbia.