## 4 Discussion  
The paper shows that the FS for the ULS of shallow foundation is neither lognormally nor normally distributed (Figures 5-7), i.e., its actual PDF is unknown. Nevertheless, due to the similarity of the lognormal PDF and the actual PDF of FS, the claim from [1; 2; 3; 4; 5; 6] according to which it is reasonable to assume a lognormal distribution of FS may be acceptable for use in everyday engineering practice, under certain conditions--e.g., when used in combination with the Monte Carlo method or Analytical FOSM method (Figures 8 and 9). The reason for this is a relatively good match between the actual PDF of FS and the PDF of lognormal distribution. For different internal friction angles, good matching of the two PDF curves ranges up to the 98 th percentile for \(\Phi^{\prime}=28^{\circ}\), and the 93 rd percentile for \(\Phi^{\prime}=44^{\circ}\) (Figure 7). These findings are in agreement with the results of the study which analysed the statistic characteristics of the performance functions in the case of ULS and SLS piles [4]. Much like this study, most of the theoretical curve and the actual PDF of FS match well, with larger deviations visible only in their tails. In the case of normally distributed FS, which was used for reliability analyses [1; 2; 6], the error size is significantly larger. Such a result is expected, since the deviations between the actual PDF of FS and the fitted normal PDF are large (Figures 5 and 6). This leads to a conclusion that the dominant cause for error in the Taylor Series and PEM methods arises from the simplification in the mean values and variances of the performance function estimations (Figure 8), and not from an incorrect approximation of the distribution of the performance function. The claim from [3], according to which a lognormal approximation leads to a conservative estimate of the reliability index, partially matches the results of this study, but only with friction angles larger than \(30^{\circ}\). Contrary to this, in all analysed cases a normally distributed FS leads to a conservative approximation of \(\beta\) (Figure 8). Figure 9 clearly shows a positive correlation between the reliability index calculated using the methods from the FOSM group and FORM, which matches the data from the study using this correlation for the modification of the RBO procedure by replacing the FOSM method for FORM [9]. The results of the reliability methods suitability assessments in the case of the ULS of shallow foundation are shown in Table 8. The Monte Carlo method, FORM and Simplified FORM methods result in a negligible error up to 0.05 and FOSM analytical-LN in a slightly \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**Bearing Capacity (Undrained)**} \\ \cline{2-4}  & **Error (Absolute Value)** & **Acceptable Error** & **Meet the Criteria** \\ \hline Simplified FORM & 0.02 & & Yes \\ Taylor Series-N & 0.19 & & Yes \\ PEM-N & 0.26 & & No \\ PEM-LN & 0.31 & & No \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Reliability methods suitability assessment in the case of retaining wall [2] according to the proposed criteria. greater error--up to 0.25 (Figure 10). According to the proposed criterion (Section 2.4), they are suitable for a reliability assessment of the footing researched in this paper. Introducing additional variables, e.g., load eccentricity or soil cohesion, might result in larger errors; with the exception of the Monte Carlo method, their suitability should be checked for shallow foundation reliability analysis in general. The Taylor series, PEM and FOSM analytical with normally distributed FS result in a large error, ranging from 0.56 to 1.45. Based on the reliability method suitability assessment, it is concluded that they are not suitable for reliability assessment of the shallow footing researched in this paper.