# Stochastic Strength Analyses of Screws for Femoral Neck fractures
## 3 Materials and Methods
The corrosion-resistant, i.e., stainless, steels (for example DIN 1.4441-316 L medical, AISI 316 L, ISO 5832-1, formerly standard CSN 17 350 in the Czech Republic) used nowadays to produce implants are primarily high-alloy austenitic steels with high Cr, Ni, and Mo content and low carbon content; see [33]. Pure titanium and its alloys (for example Ti6 Al4V, ISO 5832-3, see [33; 34]) usually have good mechanical properties and inertness, with a high degree of corrosion resistance--both when exposed to air and in the chemically aggressive environment of the human body. Titanium alloys are more suitable for traumatology/orthopedics implants than stainless steels because in terms of biocompatibility, they are more inert. Titanium alloys can be surface-anodized (i.e., the titanium oxide can be electrolytically produced on the surface of the implant), see [33]. The mass of the human body resting on the hip joint m is the full patient mass without one lower limb (i.e., 78-82% of the full patient mass; see Figure 3). This fact (probabilistic value) is taken into consideration by the coefficient \(\mathrm{k_{m}}\). The formula for the total force acting on the screws is  
\[\mathrm{F_{m}}=\mathrm{m}\;\mathrm{k_{m}}\mathrm{k_{dyn}}\mathrm{g} \tag{4}\]  
including dynamical effects, i.e., the coefficient \(\mathrm{k_{dyn}}\), and gravitational acceleration g. Division of \(\mathrm{F_{m}}\) into three screws (beams) is explained in [1]. In one beam, the force F, see Figure 4, is defined by the expressions  
\[\mathrm{F}=\mathrm{F_{m}}/\mathrm{n},\;\mathrm{F_{1}}=\mathrm{F}\cos(\infty), \;\mathrm{F_{2}}=\mathrm{F}\sin(\infty), \tag{5}\]  
where n is the coefficient of inequality in the division of forces. Coefficient n \(\in(2;3)\) respects possible variations of maximal and minimal values of force \(\mathrm{F_{m}}\). Coefficient n is partially connected with the quality of a bone and with the quality of possible screw insertions. Force \(\mathrm{F_{1}}\) is the tangential force and \(\mathrm{F_{2}}\) is the axial force; see Figure 4. For more information, see [1]. The typical diameter of the femoral screws is the shank diameter D, which is used in the solutions below. However, the cannulated femoral screws also have their inner diameter d; see Figure 2.  
Let us solve one femoral screw of a given length L (i.e., a beam on an elastic foundation) presented in Figures 3 and 4. The vertical displacement \(v_{1}=v(x_{1})\), see Equation (2), must be solved in three defined sections \(x_{i}\), see Figure 3:0 Hence, three differential equations must be solved. This, in turn, requires the solution of twelve constants of integration \(\mathrm{A_{1i}},\;\ldots,\;\mathrm{A_{4i}}\) via twelve boundary conditions at points \(x_{1}=0\) m, \(x_{1}=x_{2}=\mathrm{L_{1}}\), \(x_{2}=x_{3}=\mathrm{L_{2}}\), and \(x_{3}=\mathrm{L}\). The boundary conditions, therefore, are  
Figure 3:1  
\[\  
\[\left. \begin{array}{l}v_{2}(x_{2}=L_{2})-v_{3}(x_{3}=L_{2})=0\;,\\ \frac{dv_{2}}{dx_{2}}(x_{2}=L_{2})-\frac{dv_{2}}{dx_{3}}(x_{3}=L_{2})=0\;,\\ M_{o2}(x_{2}=L_{2})-M_{o3}(x_{3}=L_{2})=0,\\ T_{2}(x_{2}=L_{2})-T_{2}(x_{3}=L_{2})=-F_{1},\\ M_{o3}(x_{3}=L)=0\;,\;T_{3}(x_{3}=L)=0.\end{array} \right\} \tag{8}\]  
Hence, based on Equations (6)-(9), it is possible to derive and solve a set of twelve linear equations which can be expressed in matrix form as  
\[\{\mathbf{A}\}=\left[\mathbf{M}\right]^{-1}\times\{\mathbf{B}\} \tag{10}\]  
The constants \(\{\mathbf{A}\}=\left\{\text{A}_{1i},\,\ldots,\,\text{A}_{i}\right\}^{\text{T}}\) can be acquired by solving a set of linear equations; see [1]. This presented analytical approach is easy to solve and evaluate. Hence, shearing forces \(T_{i}\), bending moments \(M_{oi}\), normal forces \(N\), slopes \(\frac{dv_{1}}{dx_{1}}\), and displacements \(v_{i}\) can be evaluated over the whole length of the femoral screw (beam).