\begin{equation}
\label{eq:Sc}
\label{eq:Sc}S_{c}=−123.4r_{s}+20.6\\
\end{equation}
\begin{equation}
\label{eq:rc}
\label{eq:rc}r_{c}=34.4r_{s}−2.07\\
\end{equation}
\begin{equation}
\label{eq:Bs}
\label{eq:Bs}B_{s}=\rho g\\
\end{equation}
\begin{equation}
\label{eq:Vc}
\label{eq:Vc}V_{c}=\pi r^{2}_{c}\Bigg{(}z_{1}\Bigg{(}\frac{1-z^{2}_{1}}{3(h-S_{c})^{2}}\Bigg{)}-z_{0}\Bigg{(}\frac{1-z^{2}_{0}}{3(h-S_{c})^{2}}\Bigg{)}\Bigg{)}\\
\end{equation}
\begin{equation}
\label{eq:Vs}
\label{eq:Vs}V_{s}=\frac{\pi}{3}\Big{(}r^{2}_{\|_{z_{0}}}z_{0}-r^{2}_{\|_{z_{1}}}z_{1}\Big{)}\\
\end{equation}
\begin{equation}
\label{eq:Bc1}
\label{eq:Bc1}B_{c}|_{\boldsymbol{\Omega_{1}}}=V_{c}\rho_{c}g\frac{1}{V_{s}}\\
\end{equation}
\begin{equation}
\label{eq:O1}
\label{eq:O1}\boldsymbol{\Omega_{1}}=\Big{\{}(x,y,z)\in\boldsymbol{\Omega}:z>S_{c}\Big{\}}\\
\end{equation}
Where \(r_{s}\) is the radius of the stem (from Equation \ref{eq:slenderness}), \(r_{z_{0}}\) is the radius of the stem at height \(z_{0}\) , \(r_{z_{1}}\) is the radius of the stem at height \(z_{1}\) and \(g\) is gravity. \(\rho_{c}\) is the canopy density of \(5.6kg/m^{3}\) estimated from data in \citet{Beets_1996} at a stocking of 741 stems per hectare. The canopy force due to gravity is only applied to the sub-domain \(\boldsymbol{\Omega_{1}}\) , defined in Equation \ref{eq:O1}, where \(z\) is the vertical coordinate of any point. \(\frac{1}{V_{s}}\) is needed in order transform the canopy’s gravitational force into a force per unit stem volume.
In order to stress the stem, a constant wind profile was applied to the canopy. The crown sail area was assumed to be the upper half of an ellipse attached to the stem on the surface \(\Gamma_{1}\) (defined by Equation \ref{eq:T1}) a surface subregion of total surface \(\Gamma\). The common drag model presented in Equation \ref{eq:Tw} has been used previously \citep{spatz_basic_2000, rudnicki_wind_2004, mayer_windthrow_1989} and is used to approximate the wind load. It should be noted that more complex models are available \citep{coutts_wind_1995}. The drag coefficient \(\varsigma\) in Equation \ref{eq:DC} was produced from data reported by \citet{Mayhead_1973}, for Scotts pine as no data was available for radiata. The use of the \citet{Mayhead_1973} Scotts pine data set has previously been suggested as a suitable substitute \citep{moore2001relative}.
\begin{equation}
\label{eq:DC}
\label{eq:DC}\varsigma=e^{-0.377\omega-0.306}\\
\end{equation}
\begin{equation}
\label{eq:Ac}
\label{eq:Ac}A_{c}=\int_{z_{0}}^{z_{1}}2r_{c}\sqrt{1-\frac{z^{2}}{(h-S_{c})^{2}}}dz\\
\end{equation}
\begin{equation}
\label{eq:As}
\label{eq:As}A_{s}=\frac{\pi}{2}\Big{(}r_{\|_{z_{0}}}\sqrt{z_{0}^{2}+r_{\|_{z_{0}}}^{2}}-r_{\|_{z_{1}}}\sqrt{z_{1}^{2}+r_{\|_{z_{1}}}^{2}}\Big{)}\\
\end{equation}
\begin{equation}
\label{eq:Tw}
\label{eq:Tw}T_{\omega}|_{\boldsymbol{\Gamma_{1}}}=\frac{1}{2}\rho_{air}\varsigma\omega^{2}A_{c}\frac{1}{A_{s}}\\
\end{equation}
\begin{equation}
\label{eq:T1}
\label{eq:T1}\boldsymbol{\Gamma_{1}}=\{(x,y,z)\in\boldsymbol{\Gamma}:x>0,z>S_{c}\}\\
\end{equation}
