## 2 Materials and Methods  
Reliability methods suitability assessments were performed for the ULS (bearing capacity resistance) of a 1.0 \(\times\) 1.0 m centrally loaded square footing embedded in a non-cohesive soil at a depth of 1.0 m. The footing is subjected to a permanent vertical load. The model geometry is shown in Figure 2. In a shallow foundation bearing capacity analysis, compared to the other parameters included in the analysis, the angle of internal friction is the dominant parameter that influences the outcome. Thus, a method suitability assessments were performed based on the results of parametric reliability analyses with variations of its value. All analyses were performed according to Eurocode 7 (EC7) [12], design approach 3 (DA3), such that the design value of the vertical action equals to the design value of the resistance (V\({}_{\text{d}}=\text{R}_{\text{d}}\)). The value of V\({}_{\text{d}}\) is calculated using the following equation [12]:  
\[\text{V}_{\text{d}}=\gamma_{\text{G}}.\text{V}_{\text{g}}, \tag{5}\]  
where:  
V\({}_{\text{g}}\)--is the permanent vertical force  
\(\gamma_{\text{G}}\)--is the partial factor for permanent action, according to EC7, DA3 \(\gamma_{\text{G}}=1.35\)  
In reliability analyses, the following assumptions were introduced:  
* \(\phi^{\prime},\gamma^{\prime}\) and V\({}_{\text{g}}\) are random variables that follow normal distribution [13; 14]  
Figure 1: Visualization of a simple reliability problem consisting of 2 random variables, X1 and X2, integrand f\({}_{\text{X}}\)(x) and performance function g(X1, X2). Figure 2: Reliability analyses geometry model. \(\phi^{\prime}\) and \(\gamma^{\prime}\) are positive correlated, \(V_{g}\) is uncorrelated with \(\phi^{\prime}\) and \(\gamma\)
* foundation soil is non-cohesive, homogeneous and isotropic
* no presence of groundwater in the soil  
Since the statistical scatter of geometric parameters is negligible compared to the scatter of geotechnical parameters and actions [15; 16], they are introduced into the calculation as deterministic variables. According to [13; 14], the \(\phi^{\prime}\) and \(\gamma\), and according to [17], \(V_{g}\) can be considered as random variables with normal PDF. Statistical characteristics of random variables are shown in Table 3, and their mean values used in parametric reliability analyses in Table 4. The values of the coefficients of variation for random variables (\(\text{COV}_{\phi^{\prime}}\),\(\text{COV}_{\text{vg}}\),\(\text{COV}_{\text{v}^{\prime}}\)) were chosen from literature. [PERSON] and [PERSON] [18] analyse variability in soil parameters, and they conclude that for soils with internal friction angles between 20\({}^{\circ}\) and 40\({}^{\circ}\), \(\text{COV}_{\phi^{\prime}}\), is within the range of 0.05-0.15. A slightly wider \(\text{COV}_{\phi^{\prime}}\) range, 0.05-0.2, is presented by [PERSON] [19], with the value depending on how the parameter is determined. [PERSON] et al. recommend the use of \(\text{COV}_{\phi^{\prime}}\) = 0.1 [20] and the same value is used by [PERSON] et al. [21] in reliability analyses of cantilever reinforced concrete (RC) retaining walls. The typical value range for the coefficient of variation of soil unit weight (\(\text{COV}_{\gamma}\)) is 0.03-0.07 [6]. Regarding the permanent load variability, various authors [9; 22; 23; 24] use the value \(\text{COV}_{\text{v}_{g}}=0.1\) in reliability analyses, therefore the same value was chosen for the analyses at issue. In the parametric reliability analysis, the varied value is \(\phi_{\text{m}}\)\({}^{\prime}\), and the \(\gamma_{\text{m}}\)\({}^{\prime}\) values are calculated from the correlative relation of \(\gamma^{\prime}\) and \(\phi^{\prime}\) according to [25]. The \(V_{\text{g,m}}\) values is determined from the condition that \(V_{\text{d}}=R_{\text{d}}\). In calculations carried out according to Eurocode 7, the values of geotechnical parameters and actions will be characteristic or design.