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 σ 0 {\displaystyle \sigma _{0}} parallel to the grain and σ 90 {\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 α {\displaystyle \alpha } to the grain is given by
σ α = σ 0 σ 90 σ 0 sin 2 α + σ 90 cos 2 α {\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
σ α = σ 0 σ 90 σ 0 sin n α + σ 90 cos n α {\displaystyle \sigma _{\alpha }={\cfrac {\sigma _{0}~\sigma _{90}}{\sigma _{0}~\sin ^{n}\alpha +\sigma _{90}~\cos ^{n}\alpha }}}
where the exponent n {\displaystyle n} can take values between 1.5 and 2. The stress wave velocity at angle α {\displaystyle \alpha } to the grain at the elastic limit can similarly be obtained from the Hankinson formula
V ( α ) = V 0 V 90 V 0 sin 2 α + V 90 cos 2 α {\displaystyle V(\alpha )={\frac {V_{0}V_{90}}{V_{0}\sin ^{2}\alpha +V_{90}\cos ^{2}\alpha }}}
where V 0 {\displaystyle V_{0}} is the velocity parallel to the grain, V 90 {\displaystyle V_{90}} is the velocity perpendicular to the grain and α {\displaystyle \alpha } is the grain angle.
See also
Material failure theory Linear elasticity Hooke's law Orthotropic material Transverse isotropy
References