# Effect of Dimensional Variables on the Behavior of Trees for Biomechanical Studies
## 2 Materials and Methods  
### Finite-Element Model  
The dendrometric characteristics of the trees, such as height, slenderness, and taper, play an essential and complex role in the risk of falling trees. Therefore, to make more viable the analysis of the influence of these dendrometric parameters in the deflection and in normal and shear stresses, this study proposed to create a numerical modeling using the finite-element method (FEM). The model was developed using the Ansys(r) Academic Research Mechanical, Release 18.2 software. The system was composed of the trunk, root, and soil bin (Figure 1). The FEM model was created using a mesh with solid and quadratic 3D elements (Figure 1), varying in number according to the size of the tree and the type of element (trunk, root, or soil bin), with a maximum size of 200 mm for the tree and 300 mm for the soil bin. The displacement and stress fields are computed considering the linear static analysis. The root was simulated as a 0.15 m deep disc, as proposed by [34]. The root diameter varied according to the crown. The crown diameter varied according to the tree's height and was fixed at 5.5 m in the scenario where the height is fixed. The soil bin proposed by [26] also had a fixed dimension (16 m \(\times\) 16 m), but in our model, the cubic-shape soil bin was simulated with a dimension proportional to the root (2.62 times). In addition, as proposed by [26], frictions between soil and root were ignored, and for boundary conditions, only the base was considered fixed. The trunk (without crown and branches) was created as conical (larger at the bottom) instead of cylindrical, as proposed by [10] (Figure 1). Figure 1: _Cont._  
To propose the tree model (architecture) and the range of the dimensions of the trunk and branches, we used some results from [35] as a reference. The study [35] was carried out with six species of trees (_Handraonthus pentaphylla_ (Ipe Rosa), _Cenostigma pluviosum_ (Sibipiriruna), _Tipuana tipu_ (Tipuana), _Schinus molle_ (Aroeira Salsa), _Casesalpinia ferrea_ (Pau Ferro), and _Schinus terenbinthifolius_ (Aroeira pimenteira), which are all commonly planted in the state of Sao Paulo, Brazil. As the proposed model uses only the conical trunk (without canopy), it was necessary to consider, in a simplified way, the effect of the branches and leaves. Thus, the weight of branches and leaves was considered as a concentrated load at the end of the branch, allowing the calculation of a moment, which was applied to the trunk. To obtain reference data for calculating the weight of branches, we also used the results (diameter, length, tapering, and density of tree branches) obtained in previous research [3; 35]. We added 4.0% of the weight of the branch to consider the weight of the leaves, as proposed by [36]. The mean value of calculated moments (969 N\(\cdot\)m) was applied at three levels, representing the most common tree architecture obtained in the previous study [35]. The height levels varied according to the total height of the tree (Table 1). The first level was always at the first fork (the point where the canopy begins), which was set at 2.0 m (average value obtained in [35]). From that point, the canopy height (H-h) was divided into three parts to place the remaining levels of branches (Table 1). For each level, four branches were adopted around the trunk: two in the X direction, with moments around Z; and two in the Z direction, with moments around X (Figure 1). Moments on opposite faces were considered with opposite signs (X1 positive and X2 negative and Z1 positive and Z2 negative).