\(T_{w}\) is the force induced on the stem via the canopy for a given wind speed \(\omega\). Air density \(\rho_{air}\) air is constant at 1.226 \(kg/m^{3}\) \citep{Mayhead_1973}. The canopy area \(A_{c}\) is calculated as per Equation \ref{eq:Ac} and stem area \(A_{s}\) is calculated as per Equation \ref{eq:As}. \(\frac{1}{A_{s}}\) is needed in order to transform the wind induced force into a force per unit stem surface area. \(T_{\omega}\) is imposed on the stem as a boundary force in the sub-domain \(\Gamma_{1}\).
\citet{papesch_mechanical_1997} reported statistical regressions (reproduced in Equations \ref{eq:Mc} and \ref{eq:DMc}) in order to predict the maximum bending moment and the angle of deflection at the maximum applied bending moment. Assuming the deflection when a stem first reaches its proportionality limit stress coincides with the angle of deflection at the maximum bending moment, by calculating the expected deflection at the maximum bending moment and comparing with the results the model produces gives insight into the general accuracy of the model. The force imposed by the canopy can also be converted into a bending moment at the first wind speed which breaks proportionality and compared to Equation \ref{eq:Mc}’s prediction for further insight.
Where \(M_{c}\) is the maximum bending moment, \(h\) is the height and \(D\) is deflection.
Growth stresses are represented in a simplified fashion. \citet{gillis_elastic_1979} provide Equation \ref{eq:GS} to describe growth stress profiles. The growth stress profile presented in Equations \ref{eq:rcore} to \ref{eq:GS} are imposed on the stem at each growth step, and the new growth added to the pre-stressed stem. Growth stresses are imposed by adding the growth stress vector directly onto the stress vector during its calculation (in a similar way to how temperature dependent stresses are often represented) the growth stress vector, \(\boldsymbol{\sigma_{gs}}\) is presented in Equation \ref{eq:stress_gs}. It should be noted that the growth stresses are only added into the longitudinal direction, it is assumed that the resulting deformation through the interaction of the stiffness matrix will result in an appropriate stress profile in the other material directions. At the periphery of the stem in a 15 year old tree the tensile growth stresses calculated here ranges from \(0.8\) to \(2MP_{a}\) depending on the outerwood used, this is similar to the values reported by \citet{timell_compression_1986-3} for Pinus taeda of \(1.2MP_{a}\) in tension at the periphery, however they also report similar values for compression at the centre of the stem, which is somewhat lower than the assumption used here.
\(r_{core}\) is the radius which the growth stress model assumes the stress strain relationship is flat at the limit of proportionality under compression. \(Y_{l}\) is the yield limit set to \(0.1\) as presented by \citet{gillis_elastic_1979} and is the radial proportion of the stem at the compressive proportionality limit. \(R\) is the maximum radius of the stem for the height of the point being evaluated. When the point being evaluated is the apex, \(R\) is set to \(0\), as the apical growth from the current time step is assumed not to have gone through the maturation process.