# Effect of Dimensional Variables on the Behavior of Trees for Biomechanical Studies
## 2 Materials and Methods
### Horizontal Load  
For simplification purposes, the dynamic wind loads given by the wind pressure (\(q\)) were transformed into a horizontal load (\(P_{HV}\)) (Equation (1)), applied to the center of gravity of the crown. This simplification is common in the literature [3]. \[P_{HV}=qA_{c} \tag{1}\]  
where \(A_{c}\) is the area of the front surface of the crown. Wind pressure (\(q\)) depends on air density (\(\rho\)) and wind speed (\(V\)) according to Equation (2). \[q=\
u_{2}\,\rho\,V\,^{2} \tag{2}\]  
Assuming that the air density is approximately 1225 kg\(\cdot\)m\({}^{-3}\) (at sea level and 15 \({}^{\circ}\)C) and considering the acceleration of gravity (m\(\cdot\)s\({}^{-2}\)), the value of the wind pressure (\(q\) in N\(\cdot\)m\({}^{-2}\)) is obtained (Equation (3)). \[q=0.613\,V^{2} \tag{3}\]  
Therefore, horizontal loads depend on wind pressure and canopy area (Equation (1)). The wind pressure depends on the wind speed, while the crown area depends on the size and architecture of the crown. As [35] shows that the most common tree architecture was the monopodial shaped like a globose (round) crown, the crown was simplified to a circumference (Figure 1) with a diameter equal to the height of the tree minus the height \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \(\mathbf{H}\) & \multirow{2}{*}{\(\mathbf{\lambda}\)} & \multirow{2}{*}{\(\mathbf{T}\)} & \(\mathbf{DBH}\) & \(\mathbf{Db}\) & \(\mathbf{Dt}\) & \(\mathbf{YM_{1}}\) & \(\mathbf{YM_{2}}\) & \(\mathbf{YM_{3}}\) & \(\mathbf{YTH}\) \\  & & & & \(\mathbf{m}\) & \(\mathbf{m}\) & \(\mathbf{m}\) & \(\mathbf{m}\) & \(\mathbf{m}\) & \(\mathbf{m}\) & \(\mathbf{m}\) & \(\mathbf{m}\) \\ \hline \multirow{6}{*}{1} & 5 & 22 & 0.030 & 0.23 & 0.27 & 0.116 & 2.00 & 3.00 & 4.00 & 3.50 \\  & 10 & 22 & 0.030 & 0.45 & 0.49 & 0.194 & 2.00 & 4.67 & 7.33 & 6.0 \\  & 15 & 22 & 0.030 & 0.68 & 0.72 & 0.271 & 2.00 & 6.33 & 10.67 & 8.50 \\  & 20 & 22 & 0.030 & 0.91 & 0.95 & 0.348 & 2.00 & 8.00 & 14.00 & 11.00 \\  & 25 & 22 & 0.030 & 1.14 & 1.18 & 0.425 & 2.00 & 9.67 & 17.33 & 13.5 \\ \hline \multirow{6}{*}{2} & 7.5 & 8 & 0.030 & 0.94 & 0.98 & 0.752 & 2.00 & 3.83 & 5.67 & 4.75 \\  & 7.5 & 16 & 0.030 & 0.47 & 0.51 & 0.283 & 2.00 & 3.83 & 5.67 & 4.75 \\  & 7.5 & 24 & 0.030 & 0.31 & 0.35 & 0.127 & 2.00 & 3.83 & 5.67 & 4.75 \\  & 7.5 & 32 & 0.030 & 0.23 & 0.27 & 0.048 & 2.00 & 3.83 & 5.67 & 4.75 \\  & 7.5 & 40 & 0.030 & 0.19 & 0.23 & 0.002 & 2.00 & 3.83 & 5.67 & 4.75 \\ \hline \multirow{6}{*}{3} & 7.5 & 22 & 0.010 & 0.34 & 0.35 & 0.279 & 2.00 & 3.83 & 5.67 & 4.75 \\  & 7.5 & 22 & 0.020 & 0.34 & 0.37 & 0.217 & 2.00 & 3.83 & 5.67 & 4.75 \\ \cline{1-1}  & 7.5 & 22 & 0.030 & 0.34 & 0.38 & 0.155 & 2.00 & 3.83 & 5.67 & 4.75 \\ \cline{1-1}  & 7.5 & 22 & 0.050 & 0.34 & 0.41 & 0.031 & 2.00 & 3.83 & 5.67 & 4.75 \\ \hline Validation & 9.4 & 39 & na & 0.24 & 0.32 & na & 3.00 & 5.13 & 7.23 & 6.20 \\ \hline \hline \end{tabular} \(\gamma\) measured from base to top; moments (\(M_{1}\), \(M_{2}\), and \(M_{3}\)) to simulate the branches in three levels. na = not available. Scenario 1: slenderness 22; taper 0.030; and total tree height ranging between 5 m, 10 m, 15 m, 20 m, and 25 m. Scenario 2: tree height 7.5 m; taper 0.030; and slenderness ranging between 8, 16, 24, 32, and 40. Scenario 3: tree height 7.5 m; slenderness 22; and taper ranging between 0.010, 0.020, 0.030, and 0.050.  
\end{table} 
Table 1: Scenarios with total tree height (\(H\)), slenderness (\(\lambda\) = \(H\)/\(DBH\)), taper (\(T\) = (\(Db\)-\(Dt\))/\(H\)), diameter at breast height (\(DBH\)), diameter at the base (\(Db\)) and at the top (\(Dt\)) of the trunk, and position (\(\gamma\)) of the moments (\(YM_{1}\), \(YM_{2}\), \(YM_{3}\)) and of the horizontal forces (\(YTH\)) applied in the numerical simulations. of the first fork assumed in this work as the average value obtained by [35], which was two meters. Assuming a range of wind speeds, we also obtain a range of wind pressures (Equation (3)) and, therefore, horizontal loads (Equation (1)).