## 3 Finite Element Model and Structural Analysis of Fixators
During its application, fixation device is influenced by complex stress state, i.e., combination of a pressure, bending and torsion. Most of the conducted studies, with usage of FEM or an experimental method, are based on separated load conditions, meaning research of the fixation device is carried out separately for pressure force, AP bending and torsion [2; 14; 16; 28]. This study considers pressure force load. A developed FEM model \"mirrors\" the experimental investigation of the fixation device under the axial pressure load. During examination, bone models are supported with spherical joints, while the axial pressure force is variable and has a value of \(F_{p}=0-600\) N with an increment rate of 2 N/s. Load intensity corresponds to the actual load applied on the fixation device, for the case of patient treatment, and provided value is defined with the results obtained from in vivo examinations [10; 29]. Structural analysis was performed with following assumptions: the load is quasistatic, supports are modeled with frictionless joints, fastened connection property is established between the half-pins and the bone segments and bone material characteristics correspond to beech tree mechanical properties. While performing the structural analysis, deflection values in the load zone were monitored (Figure 4). These values are used to define the stiffness of the fixation device construction as a ratio of the applied load and the deflection values. Stiffness of the device construction under the impact of the axial pressure is calculated with following relation [25; 26; 27; 28]:  
\[C_{p}=\frac{F_{p}}{\delta_{p}}\left[\frac{\text{N}}{\text{mm}}\right] \tag{1}\]  
where:  
\(F_{p}\)--axial pressure force [N];  
\(\delta_{p}\)--axial displacement of the bone segments in the load zone [mm]. In order to define the stiffness of the fracture, displacements for the adjacent pair of points on the distal and proximal fracture planes are determined in the \(x\), \(y\) and \(z\) directions. For these points, the resultant vector of the relative displacement (\(R\)) has the maximum value. Intensity of this vector is described with the following [2; 14; 16]:  
\[R=\sqrt{\left(r_{D(x)}\right)^{2}+\left(r_{D(y)}\right)^{2}+\left(r_{D(z)} \right)^{2}} \tag{2}\]Relative displacements for the pair of points on proximal and distal bone segment planes in \(x\), \(y\) and \(z\) directions are determined as [30,31]:  
\[r_{D(x)}=D_{p(x)}-D_{d(x)}r_{D(y)}=D_{p(y)}-D_{d(y)}r_{D(z)}=D_{p(z)}-D_{d(z)} \tag{3}\]  
where:  
\(r_{D(x)}\), \(r_{D(y)}\) i \(r_{D(z)}\)--relative displacements of the bone segment points in \(x\), \(y\) and \(z\) directions (mm);  
\(D_{p(x)}\), \(D_{p(y)}\) i \(D_{p(z)}\)--displacement of the proximal bone segment in the \(x\), \(y\) and \(z\) direction (mm);  
\(D_{d(x)}\), \(D_{d(y)}\) i \(D_{d(z)}\)--displacement of the distal bone segment in the \(x\), \(y\) and \(z\) directions (mm). Stiffness of the fracture is defined as a ratio of the applied load and the resultant interfragmental displacement of the bone segments [32,33]:  
\[C_{pp}=\frac{F_{p}}{R_{max}}=\frac{F_{p}}{\sqrt{\left(r_{D(x)}\right)^{2}+ \left(r_{D(y)}\right)^{2}+\left(r_{D(z)}\right)^{2}}} \tag{4}\]  
Figure 4 shows the FEM model of the B50 device configuration after preprocessing. The end of the upper bone segment is loaded with an axial force, i.e., surface force in the \(z\) direction. Translations in directions \(x\) and \(y\) are restricted in order to simulate the device load via the spherical joints. Movement of the lower part of the fixation device model is restricted with the ball joint property, where the lower bone segment is connected with the bone model via the  
Figure 4: FEM model (B50) in structural analysis; (a) preprocessing, (b) postprocessing. smooth virtual part property. Displacements of the described FEM model setup under the impact of axial load are show in Figure 4b.