# Modified, Reliability-Based Robust Geotechnical Design Method, in accordance with Eurocode 7
## 2 Materials and Methods
### A Modification of the RGD Procedure
#### 2.2.2 Simplification of the RGD Procedure by Substituting the Robustness Measure  
The most demanding part of the RGD procedure is the outer loop from Figure 1, which is composed of M repetitions of FORM with integrated PEM, where M is the total number of the designs in the design domain. Every design requires seven FORM and one PEM analysis, done separately for ULS and SLS. The number seven refers to the number of estimating points in PEM [14]. The aim of the mentioned procedure is the calculation of the mean value and standard deviation of \(\beta\), which are used to determine the design robustness measure \(\beta_{\beta}\). Such a procedure is extremely computationally expensive, and includes differentiations of limit states' functions, which are mathematically complex in the case of shallow foundations. Furthermore, another shortcoming of applying \(\beta_{\beta}\) is the fact that we need to estimate the probability density function (PDF) of \(\beta\), which is unknown. Making a wrong assumption considering the PDF of \(\beta\) can lead to a significant mistake in the estimation of \(\beta_{\beta}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Foundation Width (m)** & **Foundation Base (m)** & **ODF (EC7, DA3)** \\ \hline
1.9 & 1.9 & 0.79 \\
2.0 & 2.0 & 0.93 \\ \hline
2.3 & 1.9 & 1.21 \\
2.7 & 2.0 & 1.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The ULS analysis results for the selected final designs from [1]. Figure 2: A schematic illustration of the existing and the suggested ULS feasible region. This paper explores the possibility of substituting the robustness measure \(\beta_{\beta}\) with a generalised reliability index \(\beta\)[15], which is defined as follows:  
\[\beta=-\Phi^{-1}\Big{(}P_{f}\Big{)}, \tag{4}\]  
In Equation (4), \(\Phi^{-1}\) is the inverse Gaussian distribution and \(P_{f}\) is the probability of failure. Defining \(\beta\) is a part of the RGD procedure within the constraint verification part; consequently, by applying it as the robustness measure, the whole procedure is significantly simplified. We consider \(\beta\) to be a good indicator of the design's robustness, since its value is directly related to the standard deviation of the considered system response. Designs with a smaller standard deviation of the system response are, by definition, more robust. Figure 3 illustrates the relation of \(\beta\) to different values of the standard deviation of system response in shallow foundations in cohesionless soil. In the given example, a factor of safety (FS) was chosen for system response, with the assumption that it is lognormally distributed. The statistical distribution of the factor of safety in shallow foundations was studied by [PERSON] et al. [16]; based on comprehensive statistical analyses, they concluded that such an assumption was justified. Figure 3 clearly shows that designs with higher \(\beta\) favour lesser values of the standard deviation of system response, making them less sensitive to the variations of input parameters--i.e., more robust. In order to compare the results yielded from the application of the original robustness measure, we analysed the relation \(\beta_{\beta}-\beta\) for different foundation widths and coefficients of variations of the soil's internal friction angle (\(COV_{\phi}\)). The results of the analysis are shown in Figure 4. Due to a linear relation with a high correlation coefficient between \(\beta_{\beta}\) and \(\beta\), we conclude that the design within the Pareto front--but the shape of the Pareto front as well--will be very similar for examples where the robustness measure is \(\beta\) or \(\beta_{\beta}\). It is possible to calculate the reliability index by using the PEM method, which is significantly simpler and computationally less demanding than the other reliability methods (FORM, SORM, FOSN, Monte Carlo).