# Stochastic Strength Analyses of Screws for Femoral Neck fractures
## 3 Materials and Methods  
As mentioned above, the beams on elastic foundations are frequently used in many types of engineering applications. The linear or nonlinear elastic foundation can also be applied if a physical object, such as an implant or bone, is supported or embedded in a continuum. This leads to a suitable approximation/simplification of mechanical contacts; see [1; 19; 20; 21; 22; 25; 28; 29]. Therefore, from the biomechanical perspective, the cancellous screws are, in this paper, described and solved as beams on an elastic foundation. From the engineering/mechanical/biomechanical point of view, the new numerical model is derived from and based on 2 nd-order theory and on the theory of 2D beams on an elastic bilateral (Winkler's) foundation. This leads to a set of three 4 th-order linear differential equations:  
\[E\textsl{J}_{\text{ZT}}\,\frac{d^{4}v_{i}}{dx_{i}^{4}}-N\,\frac{d^{2}v_{i}}{ dx_{i}^{2}}+kv_{i}=0 \tag{1}\]  
for calculated deflections (displacements) \(v_{i},i=1,\ 2,\ \text{and}\ 3\) in global coordinate systems \(x_{1}\in(0;\text{L}_{1})\), \(x_{2}\in(\text{L}_{1};\text{L}_{2})\); \(x_{3}\in(\text{L}_{2};\text{L}_{3})\), see Figure 4, where the bone-screw interaction is approximated by the elastic foundation via parameter \(k\). Figure 3: Loading of femoral (cancellous) screws and an example of three femoral screws and their approximation via parallel beams on an elastic foundation (dimensions are in millimeters). According to [21] and general mathematical solution of linear equations, the general solutions of differential Equation (1) are  
\[v_{1}=e^{\omega_{1}x_{1}}[\mathrm{A}_{11}\cos(\omega_{\mathrm{R}}x_{i})+ \mathrm{A}_{21}\sin(\omega_{\mathrm{R}}x_{i})]+e^{-\omega_{1}x_{1}}[\mathrm{A} _{31}\cos(\omega_{\mathrm{R}}x_{i})+\mathrm{A}_{41}\sin(\omega_{\mathrm{R}}x_{ i})] \tag{2}\]  
where parameters  
\[\omega_{\mathrm{R}}=\sqrt{\omega^{2}+\frac{|N|}{4 EJ_{\mathrm{ZT}}}},\ \omega_{1}=\sqrt{\omega^{2}-\frac{|N|}{4 EJ_{\mathrm{ZT}}}}\ \mathrm{and}\ \omega=\sqrt[4]{\frac{k}{4 EJ_{ \mathrm{ZT}}}} \tag{3}\]  
Equation (2) contain twelve integral constants \(\mathrm{A}_{11},\ \ldots,\ \mathrm{A}_{41}\). Hence, the femoral screw is resting on an elastic foundation prescribed by elastic stiffness \(k\); see [1,21]. By changing the stiffness \(k\), it is simply possible to consider a fracture or a healthy bone or even to fit it to specific bones, such as healthy or osteoporonic bone. Three screws of the length L were applied in parallel positions on the elastic foundation, i.e., in the _os femoris_, and were loaded by the total quasi-dynamical force \(\mathrm{F}_{\mathrm{m}}\) acting on the direction of the femoral screw angle \(\varpi\); see Figure 3.  
However, the real variability of all inputs/outputs is taken into account by the probabilistic/stochastic/statistical approach, i.e., the SBRA method. The SBRA method (in our case, the application of Anthill sw), was developed to use available personal computers for a qualitative new improvement of the reliability assessment of structures. Its application allows desired transitioning from the deterministic to the probabilistic concept of reliability assessment of the natural phenomena, i.e., transitioning from deterministic reliability assessment to probabilistic assessment of structures or sys  
Figure 4: Loading of a single femoral screw and defined global coordinate systems \(x_{i}\) and dimensions \(\mathrm{L}_{i}\). tems. With the SBRA method, the calculated probability of failure is estimated according to the theory of limit states [21; 23; 24; 25; 26; 27; 30; 31]. Reference [32] also investigated femoral screws as beams without elastic foundations. However, the solution in that paper is different, being performed for only one loading force and neglecting the influence of axial forces; besides, the elastic foundation is only mentioned but not used. The accuracy of this solution is, in our opinion, not sufficient. Typical shapes and dimensions of the cancellous screws are presented in [13].