deletions | additions       diff --git a/method.tex b/method.tex  index 6c91e30..65d1e6d 100644  --- a/method.tex  +++ b/method.tex 
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equations  Where r s is the radius of the stem (from Equation 3.1), 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. ρ c is the canopy density of 5.6 kg/m 3 estimated from data in Beets and Whitehead (1996) at a stocking of 741 stems per hectare and converted to green density assuming the same relationship as for wood presented in Chapter 2. hectare.  The canopy force due to gravity is only applied to the subdomain Ω 1 , defined in Equation 3.15, where z is the vertical coordinate of any point. V 1 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 Γ 1 (defined by Equation 3.20) a surface subregion of total surface Γ. The common drag model presented in Equation 3.19 has been used previously (Spatz and Bruechert, 2000; Rudnicki et al., 2004; Mayer et al., 1989) and is used to approximate the wind load. It should be noted that more complex models are available (Coutts and Grace, 1995). The drag coefficient ς in Equation 3.16 was produced from data reported by Mayhead (1973), for Scotts pine as no data was available for radiata. The use of the Mayhead (1973) Scotts pine data set has previously been suggested as a suitable substitute (Moore and Gardiner, 2001). 
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T w is the force induced on the stem via the canopy for a given wind speed ω. Air density ρ air is constant at 1.226 kg/m 3 (Mayhead, 1973). The canopy area A c is calculated as per Equation 3.17 and stem area A s is calculated as per Equation 3.18. A 1 s is needed in order to transform the wind induced force into a force per unit stem surface area. T w is imposed on the stem as a boundary force in the sub-domain Γ 1 as described by Equation 3.20.  The constant velocity wind profile used is simplistic, more complex profiles, gust factors and dynamic loading simulations have been suggested by other authors = (Coutts and Grace, 1995; Spatz and Bruechert, 2000; Peltola et al., 1999; James, 2003). A combination of factors such as the wind speed profile and the canopy implementation induce unknown magnitudes of error.  Papesch et al. (1997) reported statistical regressions (reproduced in Equations 3.21 and 3.22) 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 3.21’s prediction for further insight.  ln(M c ) = 2.5578ln(h) − 3.18 (3.21)