# Stochastic Strength Analyses of Screws for Femoral Neck fractures
## 5 Probabilistic Solution and Probabilistic Reliability Assessment  
Based on the parameters presented in Table 2, the probabilistic (stochastic) solution was performed for \(5\times 10^{6}\) random Monte Carlo simulations using the Anthill sw. Some examples of calculated output histograms are presented in Figures 10 and 11. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Input** & **Minimum** & **Maximum** & **Mean** & **Median** & **Standard** & **Bounded (Truncated) Histogram** \\ \hline m/kg/ & & & & & & & \\ Entire mass of & 70 & 145 & 107.50 & 107.50 & 12.33 & & \\ a patient & & & & & & & \\ \hline k\({}_{\rm m}\)/1/ & & & & & & & \\ Mass reduction coefficients & 0.78 & 0.82 & 0.80 & 0.80 & \(6.57\times 10^{-3}\) & & \\  & & & & & & & \\ \hline k\({}_{\rm dyn}\)/1/ & & & & & & & \\ Dynamic & 1 & 4 & 1.57 & 1.51 & 0.301 & & \\ force coefficient & & & & & & & \\ \hline n/1/ & & & & & & & \\ Coefficient of & & & & & & & \\ inequality in & 2 & 3 & 2.5 & 2.5 & 0.289 & & \\ the division of & & & & & & & \\ force F & & & & & & & \\ \hline R\({}_{\rm e}\)/MPa/ & & & & & & & \\ Yield stress of & & & & & & & \\ Ti6A14V & 775 & 1110 & 948.07 & 956.47 & 71.84 & & \\ material of & & & & & & & \\ cellous screws & & & & & & & \\ \hline \hline \end{tabular} Other input parameters are constant, i.e., D = 5 mm, d = 1.8 mm, \(E=1.138\times 10^{11}\) Pa, \(k=2.2222\times 10^{7}\) Pa, L\({}_{1}\) = 15 mm, L\({}_{2}\) = 68 mm, L = 90 mm, g = 9.807 ms\({}^{-2}\).  
\end{table} 
Table 2: _Cont._Using the SBRA method (Anthill sw), the probabilistic reliability function \(RF\) can be defined as  
\[RF=\text{R}_{\text{e}}-\sigma_{MAX} \tag{12}\]  
where maximal global stress  
\[\sigma_{MAX}=|\sigma_{MAX2}|\in\ (35.49;\ 1129.21)\ \text{MPa} \tag{13}\]  
is defined in Figure 5 and in Equation (11). The histogram of \(RF\) is presented in Figure 12.  
Figure 11: Histogram of the calculated maximal global compression stress \(\sigma_{MAX2}\in\ (-1129.21;-35.49)\ \text{MPa}\) acting on a cancellous screw (Anthill software, cannulated screw, TieA14V material). Figure 12: Histogram of the calculated reliability function \(RF\in\ (-792.56;1063.15)\ \text{MPa}\) for a cancellous screw (Anthill software, cannulated screw, TieA14V material). Figure 10: Histogram of the calculated force \(\text{F}\in(188.3;2129.2)\ \text{N}\) acting on a cancellous screw (Anthill software, cannulated screw, TieA14V material). If situations when \(RF\leq 0\) occur, the yield limit is reached in the cancellous screw (i.e., an undesirable or unsafe situation occurs), while as long as \(RF>0\), the situations are safe. In physics, mechanics, biomechanics, and structural reliability assessment, the concepts of limit states separating multidimensional domains of random (stochastic) variables into \"safe\" and \"unsafe\" domains have been generally accepted. Therefore, the probability of failure or of an undesirable situation (in other words, the probability that the yield limit R\({}_{\text{e}}\) exceeds \(\sigma_{MAX}\)) is the probability of \(P_{RF\leq 0}\); see Figures 12 and 13 and [24]. The presented results for a cannulated cancellous screw inserted in the _collum femoris_ imply the biomechanical probability of failure to be \(P_{RF\leq 0}=2.51\times 10^{-5}=0.00251\%\). It should be, however, noted that the calculated probability \(P_{RF\leq 0}\) is caused mainly by the _collum femoris_ overloading (i.e., it accounts for mechanical/ biomechanical problems). Other possible problems, such as the development of pathological problems during the treatment, surgical mistakes, etc., are not taken into the account. Hence,  
\[P_{unsuccessful}\geq P_{RF\leq 0} \tag{14}\]  
In other words, the total probability of unsuccessful treatment \(P_{unsuccessful}\) of _collum femoris fractura_ is greater than our calculated \(P_{RF\leq 0}\). Calculation of the overall probability of unsuccessful treatment encompassing all possible nonmechanical effects, i.e., \(P_{unsuccessful}\), is, however, not the aim of this article.