## 1 Introduction
It is assumed that the good agreement is the consequence of the mathematically simple performance functions in the case studied. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Case 1**} & \multicolumn{2}{c}{**Case 2**} \\ \cline{2-5}
**Method** & **Pr[1]** & \(\beta\) & **Pr[1]** & \(\beta\) \\ \hline Direct Integration & 0.032 & 1.85 & \(9\times 10^{-5}\) & 3.75 \\ Taylor Series, lognormal dist. of FS & 0.0036 & 2.69 & \(3\times 10^{-6}\) & 4.53 \\ Taylor Series, normal dist. of FS & 0.01 & 2.33 & 0.003 & 2.75 \\ PEM, normal dist. of FS & 0.017 & 2.12 & 0.0053 & 2.56 \\ Simplified FORM & 0.024 & 1.98 & \(10^{-4}\) & 3.72 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of the reliability analyses of embankments stability for Cases 1 and 2 [1]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Sliding on**} & \multicolumn{2}{c}{**Sliding in**} & \multicolumn{2}{c}{**Bearing Capacity**} \\ \cline{2-7}
**Method** & **Sand** & **Clay** & **(Undrained)** & **(Undrained)** \\ \cline{2-7}  & **Pr\({}_{\text{f}}\)** & \(\beta\) & **Pr\({}_{\text{f}}\)** & \(\beta\) & **Pr\({}_{\text{f}}\)** & \(\beta\) \\ \hline Monte Carlo, 20,000 simulations & 0.024 & 1.98 & 0.022 & 2.01 & 0.018 & 2.10 \\ Simplified FORM & 0.024 & 1.98 & 0.025 & 1.96 & 0.023 & 2.00 \\ Taylor Series, normal dist. of FS & 0.055 & 1.60 & 0.028 & 1.91 & 0.015 & 2.17 \\ PEM, normal dist. of FS & 0.045 & 1.70 & 0.05 & 1.64 & 0.043 & 1.72 \\ PEM, lognormal dist. of FS & 0.026 & 1.94 & 0.016 & 2.14 & 0.011 & 2.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of reliability analyses of retaining wall [2]. Considering the Monte Carlo method, [PERSON] [7] claims that it is necessary to conduct 10/P\({}_{\text{f}}\) simulations for the coefficient of variation of p\({}_{\text{f}}\) (COV\({}_{\text{pf}}\)) to be less than 0.3. This paper confirms the stated hypothesis, and presents a graphic representation of the relation between the number of simulations and COV\({}_{\text{pf}}\). The statistic characteristics of the performance function in the case of the SLS of the foundation are examined by [PERSON] and [PERSON] [8]. They analyse 375 bridge and 300 building settlement examples (SLS). They conclude that the performance function of the SLS of the structures follows a lognormal distribution. This information, along with the information on the distribution of the performance function in the case of ULS of the foundation, can be used for a more comprehensive estimate of foundation reliability. Except in direct reliability calculations, reliability methods are used as parts of more complex procedures, e.g., in reliability-based optimization (RBO). To simplify the calculation work in RBO, [PERSON] et al. [9] suggest replacing FORM with the simpler FOSM method. FOSM gives acceptably accurate results in linear performance functions and multivariant normal distribution. In other cases, the error can be large, but they believe that this can be rectified by finding the functional relationship between reliability indexes obtained using FORM (\(\beta^{\text{F}}\)) and FOSM (\(\beta^{\text{M}}\)), which they prove to be highly correlated.