deletions | additions       diff --git a/method.tex b/method.tex  index 84d98a3..e5172aa 100644  --- a/method.tex  +++ b/method.tex 
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(λ i − λ o ) + λ i  r t  (3.2)  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 in Chapter 2 at r = 0 and r = r t respectively. Due to the microstructure 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 Chapter 2 is used in combination with the interpolations presented in Equation 3.2 and presented as the stiffness matrix in Equation 3.4 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 3.5 provides the transformation matrix. The use of Voigt notation allows for transformation of the elastic constants into the global system, via Equations 3.6 and 3.8. 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.  Where a ji is the directional cosine from j to i (global to local), j = x, y, z and i = r, t, l.  The stiffness matrix C, stress vector σ and strain vector are in the global coordinate system while C l ,σ l , l are in the local system.  The forces being applied in the global system (through the transformation of the elastic constants) results in stresses being calculated in the global system. As the experimental work was conducted in the local system the proportionality limit stresses are in the local system. Equation 3.8 was used in order to transform the stresses in the global system (calculated by the model) back into the local system in order to evaluate failure.