deletions | additions       diff --git a/method.tex b/method.tex  index 085c514..5318809 100644  --- a/method.tex  +++ b/method.tex 
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The pre-stressed state of the cambium from gravity and growth stresses produces the need to successively ’grow’ the mesh which is to be used so that each mesh addition is added in a non stressed state on top of a stressed surface. Mesh growth is achieved by first defining an initial state, this state can be thought of as a seedling. The seedling is subjected to gravitational forces, deformation occurs and an updated mesh is created. The positions of the nodes and vertices of the deformed mesh are used to calculate the positions of new vertices and nodes to be added in order to represent cambial growth in a non stressed state onto the stressed surface along with apical growth to a predefined height. This procedure is repeated for five time steps imposing gravity (and in required cases growth stress), regardless of the final age, i.e. a tree with wind imposed at 5 years utilises 1 year steps, a tree with wind imposed at 15 years has 3 year time steps.  The material constants for \textit{Pinus radiata} reported by --ref-- \citet{Davies_2016}  were used to parametrise the model. Four sets of constants representing extreme values of green density and stiffness for the species were used. Note the abbreviations of the sample type used here, the first letter defines the density as either \(H\) (High density) or \(L\) (Low density) and the second letter defines the MFA as either \(W\) (Wide) or \(S\) (shallow). Linear interpolation was used to approximate material constants between two samples to provide a gradient of material properties from the corewood to the outerwood. The TRP of cellular properties was investigated using the gradient of elastic constants and Poisson ratios from different samples by applying simulated wind loads to stems with different radial profiles. 
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\lambda = \frac{-r}{r_t}(\lambda_i-\lambda_o)+\lambda_i  \end{equation}  Where \(r\) is the stem radius at the current point, \(r_t\) is the total radius at 15 years, \(λ_i\) and \(λ_o\) are the values of the material properties obtained from --ref-- \citet{Davies_2016}  at \(r = 0\) and \(r = r_t\) respectively. Due to the micro-structure of wood the native coordinate system for describing material properties is not the same as the global coordinates system used to impose external forces on the stem. Transformation between the two systems is needed. The \(rtl\) local system used for experimental work presented in --ref-- \citet{Davies_2016}  is used in combination with the interpolations presented in Equation \ref{eq:constants} and presented as the stiffness matrix in Equation \ref{eq:stiffness_mat} using Voigt (engineering) notation at any point in the stem. Because the stiffness matrix is calculated in the local coordinates it needs to be converted into an \(xyz\) system in order to apply wind loadings in a sensible fashion, Equation \ref{eq:trans_mat} provides the transformation matrix. The use of Voigt notation allows for transformation of the elastic constants into the global system, via Equations \ref{eq:cm_trans} to \ref{eq:stress_tans}. It is assumed that there is no spiral grain occurrence within the stem, the local \(l\) axis is always parallel with the global \(z\) axis. Material constants in the longitudinal direction are parallel with the \(z\) axis, i.e. there is no correction angle applied to account for taper. \begin{equation}  S_l = \begin{bmatrix} 
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Failure surfaces were created using Tsai and Wu’s failure criterion \citep{tsai_general_1971} and calculated for every point in the stem for all wind speeds. Each point is evaluated for its safety factor, where a factor of one is on the failure surface, with a lower than one factor being before the limit of proportionality and higher than one factor after the limit. The values are the observed stress over the proportional limit stress given the other five stress states. Due to the dependence of each direction of failure on the other directions, all but the variable in question are held constant at their modelled values. Once passed the proportional limit the linear stress strain curves were still assumed, this is not physically accurate. The proportional limit stresses were calculated from the linear interpolation described by Equation \ref{eq:constants}  From the experimental work presented in --ref-- \citet{Davies_2016}  and the relationships presented in Equation \ref{eq:constants} proportional limit surfaces for each point were calculated using \citet{tsai_general_1971}’s criterion. Because the proportional limit surfaces are defined in the local \(rtl\) coordinate system and the stresses provided by the model are in the global \(xyz\) system Equation \ref{eq:stress_tans} was used to convert each stress vector into the local system at its given global location. The failure criterion was applied through Equation \ref{eq:tw_app} and the maximum and minimum stress values which could be obtained without failure calculated for each point assuming all other stresses stay fixed (Equations \ref{eq:tw_opt_max} and \ref{eq:tw_opt_min}). With the maximum tensile and compressive stresses a safety factor was calculated as per Equations \ref{eq:SFten} and \ref{eq:SFcomp}. The stress bounds can be used to investigate how much redundant strength is available at failure by both position in the stem and direction of stress. \begin{equation}\label{eq:tw_app}  \boldsymbol{\sigma^T q + \sigma^T P \sigma} - 1 = 0