## 3 Results  
### Statistical Distribution of Factor of Safety (FS)  
The statistical characteristics of FS as a random variable, are analysed using the Monte Carlo method. Results for different internal friction angles are shown in Figure 5. The figure shows histograms based on \(10^{5}\) data, along with the best-fit curves of normal and lognormal PDFs. All curves are skewed to the right with a single distinct peak. This type of histogram shape indicates that the data are unlikely to follow a normal distribution. The deviation of FSs PDF from normal distribution PDF can be seen even more clearly in Figure 6, which shows the relationships between the sample quantiles and quantiles of normal distribution (Q-Q plots). What is apparent is that the points of all the charts shown in Figure 6 follow a distinctly non-linear pattern. The trend of increasing curvature by an increase in the angle of internal friction can also be seen. Such non-linear curve shapes can be explained by the differences between the PDF of normal distribution and the actual PDF of the factor of safety. Normal distribution PDF is symmetric and is defined on the interval \(\langle-\infty,\ +\infty\rangle\). The factor of safety can theoretically take values only from the right half of that interval, from which it follows that the shape of its PDF is necessarily asymmetric. Compared to the PDF of normal distribution, the best fit lines of lognormal PDF generally better fit the histogram shapes of the FS. This is also visible in Figure 7, which shows the Q-Q plots of lognormal distribution. As in the previous case, a trend of increasing data deviation from the quantile of lognormal distribution with an increase in the internal friction angle can be observed. Plots shown in Figure 7 also show that, up to a specific quantile, points form a linear pattern with the inclination of \(45^{\circ}\), which passes through the origin. Within these intervals, sample quantiles match the quantiles of lognormal distribution. Quantiles to which the lognormal Q-Q plots are linear are shown in Table 6.  
Figure 5: The factor of safety histograms for different \(\phi^{\prime}\), along with theoretical best-fit curves of normal and lognormal distribution. Figure 6: Quantiles and quantiles (Q-Q) plots of normal distribution. Table 6 illustrates that the quantiles to which the lognormal Q-Q plot is linear are within the range of FSs commonly used in everyday engineering practices. It also shows that the cumulative density function (CDF) values at X are within the interval of 0.93-0.98. After the linear part, the points form a convex curve located above the line. This phenomenon is the result of a thicker tail of the actual PDF of the FS compared to the PDF of lognormal distribution. It can be concluded that the actual PDF of FS is not lognormal, but it shows a good agreement with it for a significant part of the curve. Figure 8 shows the results of reliability analysis obtained using lognormal and normal PDFs of FS from Figure 6 along with results of Direct Integration. The results of Direct Integration are presented to evaluate errors caused by an incorrect assumption of the FS distribution. The pf values of the CP1 and CP2 curves represent the areas below the best-fit PDFs (normal and lognormal) shown in Figure 5, considering only the following interval \(\langle-\infty,\ 1\rangle\). The common feature of all three curves is the trend of increasing probability of failure with increasing \(\phi^{\prime}\), or the reverse relationship in the case of the reliability index. The average error value expressed by \(\beta\) in the case of normal distribution is 1.40, and for lognormal considerably less, 0.06. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\phi\)\(\phi^{\text{s}}\) & **Quantiles to which the Lognormal Q-Q** & **Cumulative Density** \\  & **Plots are Linear (X)** & **Function at X** \\ \hline
28 & 7.0 & 0.98 \\
32 & 7.3 & 0.95 \\
36 & 9.2 & 0.94 \\
40 & 11.9 & 0.93 \\
44 & 17.1 & 0.93 \\ \hline \hline \end{tabular}
\end{table}