# Effect of Dimensional Variables on the Behavior of Trees for Biomechanical Studies
## 2 Materials and Methods
### Horizontal Load
We started the procedure testing a range of wind speeds covering speeds below, within, and above basic wind speeds (peak gust speed averaged over a short time interval of about 3 s duration and measured at 10 m above ground) used in the Brazilian standard, [37] which are 30 m\(\cdot\)s\({}^{-1}\) to 50 m\(\cdot\)s\({}^{-1}\). We calculated the wind pressure (Equation (2)) using this wind-speed range (from 30 m\(\cdot\)s\({}^{-1}\) and 50 m\(\cdot\)s\({}^{-1}\)), and the horizontal load (Equation (1)) was calculated, with frontal canopy area (Ac in Table 2). As we need to apply the same horizontal load to all scenarios (Table 1), the second analysis was to use the horizontal-load range (obtained for the different areas of canopy) to calculate the respective wind pressure (Equation (1)) and so the respective wind speeds capable of producing this level of load (Equation (3)). Since below 5 kN and above 15 kN the wind speed was, respectively, far below (for the largest canopy) and far over (for scenario 1) the proposed wind-speed range, we choose the three load levels (5 kN, 10 kN, and 15 kN) to be applied to all scenarios. According to the Beaufort scale, at speeds from 5.5 to 7.9 m\(\cdot\)s\({}^{-1}\) the wind moves small branches; from 8 to 10.7 m\(\cdot\)s\({}^{-1}\) it moves large branches and small trees; from 13.9 to 17.1 m\(\cdot\)s\({}^{-1}\) it moves large trees; from 17.2 to 20.7 m\(\cdot\)s\({}^{-1}\) it can break branches; from 20.8 to 24.4 m\(\cdot\)s\({}^{-1}\) it can damage trees; and from 24.5 m\(\cdot\)s\({}^{-1}\) it can uproot trees. The chosen load levels allow us to simulate wind speeds from 4.4 m\(\cdot\)s\({}^{-1}\) to 58.8 m\(\cdot\)s\({}^{-1}\) (Table 2), but with the exception of trees with H = 5 m where the speeds are greater than 30 m\(\cdot\)s\({}^{-1}\), all other conditions are simulating at speeds below 22 m\(\cdot\)s\({}^{-1}\) (Table 2), i.e., in the range where large tree damage is not expected by the Beaufort scale. Furthermore, for shorter trees, greater stability is expected due to their lesser slenderness, and, therefore, these short trees will likely be better able to withstand greater wind forces. The same wind-speed level was used by [10], for example, who simulated wind speeds of 10, 20, and 50 m\(\cdot\)s\({}^{-1}\). These horizontal loads were applied to the tree model in a position coincident with the gravitational center of the crown (Table 1). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & & & & \multicolumn{3}{c}{**5 kN**} & \multicolumn{3}{c}{**10 kN**} & \multicolumn{3}{c}{**15 kN**} \\ \hline \multirow{2}{*}{**Scenario**} & **H** & **\(D_{c}\)** & **\(A_{c}\)** & **\(q\)** & **\(V\)** & **\(q\)** & **\(V\)** & **\(q\)** & **\(V\)** \\  & **m** & **m** & **m\({}^{2}\)** & **N/m\({}^{2}\)** & **m/s** & **N/m\({}^{2}\)** & **m/s** & **N/m\({}^{2}\)** & **m/s** \\ \hline \multirow{4}{*}{1} & 5 & 3 & 7.1 & 707 & 34.0 & 1415 & 48.0 & 2122 & 58.8 \\  & 10 & 8 & 50.3 & 99 & 12.7 & 199 & 18.0 & 298 & 22.1 \\ \cline{1-1}  & 15 & 13 & 132.7 & 38 & 7.8 & 75 & 11.1 & 113 & 13.6 \\ \cline{1-1}  & 20 & 18 & 254.5 & 20 & 5.7 & 39 & 8.0 & 59 & 9.8 \\ \cline{1-1}  & 25 & 23 & 415.5 & 12 & 4.4 & 24 & 6.3 & 36 & 7.7 \\ \hline
2 and 3 & 7.5 & 5.5 & 23.8 & 210 & 18.5 & 421 & 26.2 & 631 & 32.1 \\ \hline Validation & 9.4 & 6.4 & 32.2 & 155 & 15.9 & 311 & 22.5 & 466 & 27.6 \\ \hline \end{tabular}  
* 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. Initial crown height fixed at 2.0 m (crown heights = H \(-\) 2.0).  
\end{table}