# Effect of Dimensional Variables on the Behavior of Trees for Biomechanical Studies
## 3 Results and Discussion
### Shear Stresses  
The behavior of the position for maximum shear stresses along the trunk, simulated in Scenario 1 (trees of different heights), was similar to that of normal stress (Figure 7); that is, maximum stresses (positive and negative) at the base for trees ranging from 15 m to 25 m high, and in higher positions for lower trees (\(H\) = 5 m and 10 m) (Tables 4-6). In scenario 2, the maximum shear stress occurs at the base only for the minimum \(\lambda\) (correspondent to the maximum \(DBH\)) for all levels of load (Tables 4-6). In scenario 3, the maximum shear stresses (positive and negative) increase with taper growth, while the position of the maximum varies from 26% \(H\) to 51% \(H\). Statistically, height had an influence on the negative shear stresses only for the lowest horizontal load (5 kN) and on the shear positive stress for the two highest horizontal-load levels (5 and 10 kN) (Table 8). Slender and tapper had a statistical influence on the shear stress for all load levels (Table 8).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variables** & **Regression Model** & **Load Level** & \(p\)**-Value** & \(\mathbf{R^{2}}\) \\  & & **kN** & **(\%)** \\ \hline \multicolumn{5}{c}{Negative Normal Stresses (\(\sigma_{-}\))} \\ \hline \multirow{3}{*}{\(\tau_{-}\times H\)} & \(\tau_{-}=-0.31-0.0032\;H^{2}\) & 5 & 0.0142 & 89.8 \\  & - & 10 & 0.0631 * & - \\  & - & 15 & 0.2136 * & - \\ \hline \multirow{3}{*}{\(\tau_{-}\times\lambda\)} & \(\tau_{-}=0.07-0.0014\;\lambda^{2}\) & 5 & 0.0135 & 90.2 \\  & \(\tau_{-}=0.42-0.0031\;\lambda^{2}\) & 10 & 0.0084 & 92.8 \\  & \(\tau_{-}=0.72-0.0046\;\lambda^{2}\) & 15 & 0.0072 & 93.5 \\ \hline \multirow{3}{*}{\(\tau_{-}\times T\)} & \(\tau_{-}=-0.29-0.00009\;T^{2}\) & 5 & 0.0321 & 93.7 \\  & \(\tau_{-}=-0.44-0.00024\;T^{2}\) & 10 & 0.0332 & 93.5 \\  & \(\tau_{-}=-0.67-0.00037\;T^{2}\) & 15 & 0.0330 & 93.5 \\ \hline \multicolumn{5}{c}{Positive Normal Stresses (\(\sigma_{+}\))} \\ \hline \multirow{3}{*}{\(\tau_{+}\times H\)} & \(\tau_{+}=sqrt\;(-0.42+0.008\;H^{2})\) & 5 & 0.0027 & 96.6 \\  & \(\tau_{+}=sqrt\;(-0.11+0.007\;H^{2})\) & 10 & 0.0148 & 89.6 \\ \cline{1-1}  & - & 15 & 0.0810 * & - \\ \hline \multirow{3}{*}{\(\tau_{+}\times\lambda\)} & \(\tau_{+}=sqrt\;(-1.55+0.0016\;\lambda^{2})\) & 5 & 0.0418 & 97.4 \\  & \(\tau_{+}=sqrt\;(-1.49+0.0021\;\lambda^{2})\) & 10 & 0.0024 & 95.6 \\ \cline{1-1}  & \(\tau_{+}=sqrt\;(-1.38+0.0023\;\lambda^{2})\) & 15 & 0.0022 & 94.2 \\ \hline \multirow{3}{*}{\(\tau_{+}\times T\)} & \(\tau_{+}=sqrt\;(0.054+0.00007\;T^{2})\) & 5 & 0.0409 & 92.0 \\ \cline{1-1}  & \(\tau_{+}=sqrt\;(0.17+0.00029\;T^{2})\) & 10 & 0.0329 & 99.5 \\ \cline{1-1}  & \(\tau_{+}=sqrt\;(0.41+0.00065\;T^{2})\) & 15 & 0.0303 & 99.6 \\ \hline \hline \end{tabular}
* not significant with 95% confidence level.  
\end{table}
Table 8: Regression models considering maximum (negative and positive) shear stresses (\(\tau\)) as a function of height (\(H\)), slenderness (\(\lambda\)), and taper (\(T\times 10^{2}\)) for each level of horizontal load adopted.  
Figure 7: Behavior variation of the maximum positive and negative shear 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).