# Effect of Dimensional Variables on the Behavior of Trees for Biomechanical Studies
## 3 Results and Discussion
### Normal Stresses  
For maximum normal stresses (positive and negative) (Figure 6), the influence of slenderness is the most significant, followed by tapering. Regarding tapering, ref. [24] found stress influence for trees with heights bigger than four meters, corresponding to all of our scenarios. Height is the least influencing factor. Height variation had a small impact mainly on trees with up to 10 m (Figure 6). Statistically, height had less influence on normal negative stresses (nonsignificant regression model at all load levels), and its effect on normal positive stresses depended on the load level (Table 7). Figure 5: Behavior variation of the taper versus tree height (m) for the scenario 3 (tree height 7.5 m; slenderness 22) and for simulated horizontal forces (5 kN, 10 kN and 15 kN). Taper 1 = 0.010, Taper 2 = 0.020, Taper 3 = 0.030, and Taper 4 = 0.050. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variables** & **Regression Model** & \begin{tabular}{c} **Load Level** \\ **kN** \\ \end{tabular} & \
\begin{tabular}{c} _p-Value_ \\ \end{tabular} &
\begin{tabular}{c} **R\({}^{2}\)** \\ **(\%)** \\ \end{tabular} \\ \hline \multicolumn{5}{c}{Negative Normal Stresses (\(\sigma_{-}\))} \\ \hline \multirow{3}{*}{\(\sigma_{-}\times H\)} & - & 5 & 0.2498 * & - \\  & - & 10 & 0.2121 * & - \\  & - & 15 & 0.0634 * & - \\ \hline \multirow{3}{*}{\(\sigma_{-}\times\lambda\)} & \(\sigma_{-}=0.54-0.006\;\lambda^{2}\) & 5 & 0.0047 & 95.1 \\  & \(\sigma_{-}=0.54-0.006\;\lambda^{2}\) & 10 & 0.0034 & 96.1 \\  & \(\sigma_{-}=0.54-0.006\;\lambda^{2}\) & 15 & 0.0032 & 96.2 \\ \hline \multirow{3}{*}{\(\sigma_{-}\times T\)} & \(\sigma_{-}=0.54-0.006\;T^{2}\) & 5 & 0.0014 & 99.7 \\  & \(\sigma_{-}=0.54-0.006\;T^{2}\) & 10 & 0.0017 & 99.7 \\  & \(\sigma_{-}=0.54-0.006\;T^{2}\) & 15 & 0.0020 & 99.6 \\ \hline \multicolumn{5}{c}{Positive Normal Stresses (\(\sigma_{+}\))} \\ \hline \multirow{3}{*}{\(\sigma_{+}\times H\)} & - & 5 & 0.5016 * & - \\  & \(\sigma_{+}=sqrt(-6.4+65/H)\) & 10 & 0.0474 & 77.9 \\  & \(\sigma_{+}=sqrt(-20.7+411/H)\) & 15 & 0.0209 & 86.9 \\ \hline \multirow{3}{*}{\(\sigma_{+}\times\lambda\)} & \(\sigma_{+}=-1.25+0.07\;\lambda^{2}\) & 5 & 0.0042 & 95.4 \\  & \(\sigma_{+}=-2.56+0.013\;\lambda^{2}\) & 10 & 0.0039 & 95.6 \\  & \(\sigma_{+}=-3.83+0.020\;\lambda^{2}\) & 15 & 0.0039 & 95.6 \\ \hline \multirow{3}{*}{\(\sigma_{+}\times T\)} & \(\sigma_{+}=sqrt(0.35+0.0017\;T^{2})\) & 5 & 0.0409 & 92.0 \\  & \(\sigma_{+}=sqrt(1.61+0.0066\;T^{2})\) & 10 & 0.0329 & 93.5 \\ \cline{1-1}  & \(\sigma_{+}=sqrt(3.81+0.015\;T^{2})\) & 15 & 0.0303 & 94.0 \\ \hline \hline \end{tabular}  
* not significant with 95% confidence level.  
\end{table} 
Table 7: Regression models considering maximum (negative and positive) normal stresses (\(\sigma\)) as a function of height (\(H\)), slenderness (\(\lambda\)), and taper (\(T\times 10^{2}\)) for each level of horizontal load adopted. Figure 6: Behavior variation of the maximum positive and negative normal stresses versus tree height (m), slenderness, and taper (\(\times 1000\)) for the three levels of simulated horizontal forces (5 kN, 10 kN, and 15 kN). In scenario 1, for the trees with heights ranging from 15 m to 25 m, the maximum normal stresses (positive and negative) always occurred at the base (Tables 4-6). However, for the shortest trees (5 m and 10 m), the maximum normal stresses rose to higher trunk positions, reaching more central regions (Tables 4-6). Since our model considered the tapered trunk, the geometric parameters that go into calculating the stresses varied, even though the moments generated by the horizontal load are greater at the base. In ref. [45], using mathematical models, showed that the variation in dimension and mass along the trunk influences the stress distribution. In ref. [10] simulated a 13 m tree composed of three cylindrical elements with successive diameter reduction, without branches, subjected to horizontal-static loads to simulate wind speed of 10, 20, and 50 m\(\cdot\)s\({}^{-1}\).