# 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
The Sobol sensitivity analysis is conducted, with the aim of determining the contribution of the variance of each individual random variable to the total variance of limit state functions for both ULS and SLS. The Sobol sensitivity analysis is a variance-based method which determines the influence of uncertainties in the model input factors on the uncertainty in the output of the model [21]. By determining the first order Sobol indices, we determine the effect of the variation of individual random variables on the variance of system response. The Sobol indices were calculated in the Python program, using the open source package \"OpenTURNS\" [22]. The variables whose contribution to the system response variance is negligible can be \"frozen\", which will make the RGD procedure less computationally demanding. Tables 5 and 6 show the results of sensitivity analyses for different \(COV_{\phi}\) values, for both ULS and SLS. The presented results demonstrate that the soil's internal friction angle is the dominant variable and contributes most to the total variance of system responses. The value of the first order Sobol index is similar for ULS and SLS, and for different \(COV_{\phi}\) values it is within in the \(\approx 0.84-0.99\) range, depending on the value of the internal friction angle. Due to the high values of first order Sobol indices of the soil's internal friction angle, reliability analyses of the ultimate and serviceability limit states were conducted for two groups of random vectors, which differ in the number of random variables. The first group of random vectors is composed solely of the soil's internal friction angle, and can be expressed as follows: \(X_{ULIS}=\left(\phi^{\prime}\right)^{T}\) and \(X_{SLS}=\left(\phi^{\prime}\right)^{T}\). In the second group the random vectors include all the random variables defined by limit state functions for ULS and SLS: \(X_{ULS}=\left(\phi^{\prime},\ \gamma\ t,V_{g},V_{q}\right)^{T}\) and \(X_{SLS}=\left(\phi^{\prime},\ \gamma\ t,a,b,V_{g},V_{q}\right)^{T}\). The aim of the analysis is to determine the influence of reducing the number of random variables on the reliability analyses results, in order to optimise the RGD procedure. Reliability analyses were conducted using the IPEM method, and the results are presented in Tables 7 and 8. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & \(COV_{\phi}\)**= 0.05** & \(COV_{\phi}\)**= 0.10** & \(COV_{\phi}\)**= 0.15** \\ \hline \(\phi^{\prime}\) & 0.845 & 0.970 & 0.994 \\ \(\gamma^{\prime}\) & 0.032 & 0.006 & 0.001 \\ \(V_{g}\) & 0.060 & 0.012 & 0.002 \\ \(V_{q}\) & 0.048 & 0.009 & 0.002 \\ a & 0.001 & 0.000 & 0.000 \\ b & 0.014 & 0.003 & 0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: First order Sobol indices of the SLS limit state function of shallow foundation in non-coherent soil. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & \(COV_{\phi}\)**= 0.05** & \(COV_{\phi}\)**= 0.10** & \(COV_{\phi}\)**= 0.15** \\ \hline \(\phi^{\prime}\) & 0.840 & 0.960 & 0.990 \\ \(\gamma^{\prime}\) & 0.037 & 0.009 & 0.002 \\ \(V_{g}\) & 0.069 & 0.018 & 0.005 \\ \(V_{q}\) & 0.054 & 0.013 & 0.003 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: First order Sobol indices of the ULS limit state function of shallow foundation in non-coherent soil. The average difference in reliability indexes \(\beta_{1}\) and \(\beta_{2}\) of the results presented in Tables 7 and 8 for ULS is 1.3%, and 1.42% for SLS, with maximum deviations of 2.36% and 3.25%. For the value of \(COV_{\phi}=0.1\), as suggested in literature, deviations are smaller--0.56% for ULS and 0.41% for SLS. Such small deviations are in line with the results of sensitivity analyses presented in Tables 3 and 4. We therefore conclude that the errors in calculating the reliability index, stemming from freezing random variables \(\gamma^{\prime}\), \(V_{g}\) and \(Vq\) in the ULS reliability analysis and \(\gamma^{\prime}\), \(V_{g}\), \(Vq\), \(a\) and \(b\) in the SLS reliability analysis, are negligible.