## 3 Finite Element Model and Structural Analysis
### Structural Analysis of External Fixation Device
maximal interfragmentary displacements at fracture gap R, displacements in x, y and z direction of neighboring points at last planes of proximal and distal segment at fracture gap need to be defined [21; 23]. Relative craniocaudal and latoromedial displacements (x and y direction) and axial displacement (z direction) of selected points are calculated using the following equations:  
\[\mathrm{r_{D(x)}}=\mathrm{D_{p(x)}}-\mathrm{D_{d(x)}}\,\ \mathrm{r_{D(y)}}= \mathrm{D_{p(y)}}-\mathrm{D_{d(y)}},\ \mathrm{r_{D(z)}}=\mathrm{D_{p(z)}}-\mathrm{D_{d(z)}} \tag{2}\]  
where: \(\mathrm{r_{D(x)}}\), \(\mathrm{r_{D(y)}}\) i \(\mathrm{r_{D(z)}}\) are relative displacements of bone segment model at fracture gap in x, y and z direction, \(\mathrm{D_{p(x)}}\), \(\mathrm{D_{p(y)}}\) i \(\mathrm{D_{p(z)}}\) are absolute displacements of points of proximal bone end in x, y and z direction, \(\mathrm{D_{d(x)}}\), \(\mathrm{D_{d(y)}}\) i \(\mathrm{D_{d(z)}}\) are absolute displacements of points of distal bone segment end in x, y and z direction. Intensity of maximal interfragmentary displacement vector at the fracture gap R is defined as:  
\[\mathrm{R}=\sqrt{\left(\mathrm{r_{D(x)}}\right)^{2}+\left(\mathrm{r_{D(y)}} \right)^{2}+\left(\mathrm{r_{D(z)}}\right)^{2}} \tag{3}\]  
The values of displacements are shown in Table 3.  
The full numerical structural analysis of external fixation device, except the analysis of displacements at fracture gap, includes also the analysis of principal stress at characteristic places of design [14; 20]. During FEM and experimental analysis, values and directions of principal stress are calculated at two control places at the middle part of the fixator tree. The place which is closer to the bone segment model is marked with MP\(-\), and the place on another side is marked with MP\(+\) (Figure 6, detail A). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{2}{c}{**Displ. of the Prox.**} & \multicolumn{2}{c}{**Displ. of the Distal. Segment at the**} & \multicolumn{2}{c}{**Max. Relat.**} & \multicolumn{2}{c}{**Displ. at the**} \\  & \multicolumn{2}{c}{**Segment at the Fracture Gap,**} & \multicolumn{2}{c}{**Fracture Gap,**} & \multicolumn{2}{c}{**Displ. at the**} & \multicolumn{2}{c}{**Point of Load,**} \\  & \multicolumn{2}{c}{**mm**} & \multicolumn{2}{c}{**mm**} & \multicolumn{2}{c}{**Gap, mm**} & \multicolumn{2}{c}{**mm**} \\ \hline  & \(\mathrm{D_{p(x)}}\) & \(\mathrm{D_{p(y)}}\) & \(\mathrm{D_{p(z)}}\) & \(\mathrm{D_{d(x)}}\) & \(\mathrm{D_{d(y)}}\) & \(\mathrm{D_{d(z)}}\) & \(\mathrm{R}\) & \(\mathrm{d}\) \\ \hline FEA & 0.53 & 4.14 & \(-\)4.36 & 0.53 & 4.29 & 0.22 & 4.58 & 4.18 \\ \hline Exp. & - & - & - & - & - & - & 4.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of displacements under maximum intensity of load. Figure 6: Principal stresses under maximum axial load. Pressure stress is detected at MP\(-\) place, and they have bigger intensity in correspondence with tensile stress at MP+ place. This is because the pressure in the tree of external fixation device is eccentric. The direction of the biggest principal stress \(\sigma_{1}\) at place MP+ is the same as the direction of the lowest principal stress \(\sigma_{3}\) at place MP\(-\), coinciding with the axis of symmetry of the tree of the external fixation device. Direction and intensity of principal stresses at measuring places are shown Figure 6 (view B). It can be seen that, at place MP+, the biggest principal stress is tensile stress, and at place MP\(-\), the smaller principal stress is pressure stress. In additional, it can be noticed that principal stresses (\(\sigma_{1}\) and \(\sigma_{3}\)) are in the bending plane of the fixator. This plane does not match with the plane of half pins (Figure 6, view B). This is because of eccentric pressure load and the geometry of the external fixation device. The intensities of three principal stresses are shown in Table 4.