# Modified, Reliability-Based Robust Geotechnical Design Method, in accordance with Eurocode 7
## 2 Materials and Methods
### A Modification of the RGD Procedure
#### 2.2.3 The Optimisation of the Number of Random Variables Using a Sensitivity Analysis
Consequently, we propose conducting reliability analyses for ULS and SLS in which the internal friction angle is the only random variable. An analysis of the influence of ODF values on first order Sobol indices was also conducted. The results of this analysis are not presented, but it was confirmed that ODF has no significant influence on the value of the first order Sobol indexes for both ULS and SLS. We would like to point out that the limit state functions for shallow foundations can also be determined by using other calculation models. This paper used the analytical expression for estimating the bearing capacity of a shallow foundation which is recommended in Eurocode 7, Annex D (2018) (Equation (9)). This expression is based on a theory extrapolated from the results of laboratory tests on the behaviour of small footings in dense sand and under 1 g condition (2018). [PERSON] and [PERSON] (2018) are researching soil behaviour in such conditions and conclude that it has little relevance to the behaviour of a full-scale prototype. The main reasons for this are inadequate stress levels in soil and non-linear stress-strain soil behaviour. With small-scale tests, the depth of model influence is relatively small, so soil behaves as if unconsolidated--after an initial volume assessment, soil dilates and then contracts. This results in a stress-strain curve which suggests ultimate resistance; this has not been proven in full-scale tests to this day. Due to all of the above, [PERSON] (2018) recommends bearing capacity analysis to be used only as a simple estimate for comparing the design with previous designs. Despite the mentioned shortcomings and simplifications, Eurocode 7 recommends using the expression from Equation (9) for determining the bearing capacity of a shallow foundation; it is consequently used in everyday engineering practice. Since the aim of this paper is to present the procedure of the modified RGD method whose results have been harmonised with EC7, the bearing capacity of a shallow foundation is determined according to the recommendations from this technical standard. Limit state functions can \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Foundation**} & \multicolumn{3}{c}{**COV\(\varphi\) = 0.05**} & \multicolumn{3}{c}{**COV\(\varphi\) = 0.10**} & \multicolumn{3}{c}{**COV\(\varphi\) = 0.15**} \\
**Width (m)** & \(\beta\)**1** & \(\beta\)**2** & 1A\(\beta\)1 & \(\beta\)**1** & \(\beta\)**2** & 1A\(\beta\)1 & \(\beta\)**1** & \(\beta\)**2** & 1A\(\beta\)1 \\ \hline
2 & 2.970 & 2.903 & 0.067 & 2.303 & 2.321 & 0.018 & 2.179 & 2.201 & 0.022 \\
2.2 & 4.121 & 4.026 & 0.095 & 3.164 & 3.184 & 0.020 & 2.923 & 2.949 & 0.026 \\
2.4 & 5.367 & 5.241 & 0.126 & 4.091 & 4.114 & 0.023 & 3.719 & 3.75 & 0.031 \\
2.6 & 6.706 & 6.548 & 0.158 & 5.083 & 5.108 & 0.025 & 4.567 & 4.603 & 0.036 \\
2.8 & 8.147 & 7.943 & 0.204 & 6.139 & 6.167 & 0.028 & 5.466 & 5.508 & 0.042 \\
3 & 9.657 & 9.427 & 0.230 & 7.258 & 7.289 & 0.031 & 6.417 & 6.464 & 0.047 \\ \hline \hline \end{tabular}  
* \(\beta\)1, \(\beta\)2 —reliability indexes calculated for 1 and 4 random variables.  
\end{table} 
Table 7: Comparison of the results of ULS reliability analyses with different numbers of random variables. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Foundation**} & \multicolumn{3}{c}{**COV\(\varphi\) = 0.05**} & \multicolumn{3}{c}{**COV\(\varphi\) = 0.10**} & \multicolumn{3}{c}{**COV\(\varphi\) = 0.15**} \\
**Width (m)** & \(\beta\)**1** & \(\beta\)**2** & 1A\(\beta\)1 & \(\beta\)**1** & \(\beta\)**2** & 1A\(\beta\)1 & \(\beta\)**1** & \(\beta\)**2** & 1A\(\beta\)1 \\ \hline
2 & 0.870 & 0.845 & 0.025 & 0.709 & 0.718 & 0.009 & 0.780 & 0.783 & 0.003 \\
2.2 & 1.367 & 1.324 & 0.043 & 1.090 & 1.095 & 0.005 &  
also be defined using other calculation models following the same principle, which will result in different mathematical expressions of limit state functions.