# Modified, Reliability-Based Robust Geotechnical Design Method, in accordance with Eurocode 7
## 3 Results
### Illustrative Example 1--Design of Shallow Foundation
This fact is a consequence of a relatively high value of the internal friction angle's standard deviation, along with the fact that the prescribed condition \(ODF_{ULS}\geq 1\) is more critical than the criteria associated with the reliability indexes. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Symbol** & **Value** \\ \hline Permanent load & \(V_{G}\) & 2000 kN \\ Variable load & \(V_{Q}\) & 1000 kN \\ Soil unit weight & \(\gamma\) & 18.5 kN/m3 \\ Fitting parameters of the & \(a\) & 0.6992 \\ load-settlement curve & \(b\) & 1.7675 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The characteristics of the deterministic variables used in foundation optimisation. We analysed the influence of individual constraints on the position of non-dominated designs in the decision space, and the results are shown in Figure 12. The four broken lines represent the boundary between feasible and infeasible designs for different constraints. Left of the line are infeasible designs, with feasible designs being right of the line and on the line itself. Figure 12 clearly shows that the constraint \(ODF_{ULS}\geq 1\) defines the position of the Pareto front in the design space. Comparing to the other constraints, to meet \(ODF_{ULS}\geq 1\) constraint, a wider foundation width at the same depth is needed. Such a relation among constraints is the consequence of a relatively low value of the standard deviation of the soil's internal friction angle (\(\sigma_{\phi}\)). Reliability indexes are more sensitive to a change in standard deviations of random variables than ODF. To illustrate this point further, Figure 12--on the right--shows lines that represent the boundaries of the distribution of design space into feasible/infeasible designs, in the case of a larger standard deviation of the internal friction angle, \(\sigma_{\phi}=3.44\). In that case, the number of feasible designs is greatly decreased (from 311 to 236), and the main reason behind the decrease is a significant reduction of the reliability index, which led to an additional filtering of the designs. In this case, the position of the Pareto front is defined by the constrains \(\beta_{ULS}\geq 3.8\) and \(\beta_{SLS}\geq 1.5\), while the constraints relating to \(ODF_{ULS}\) and \(ODF_{SLS}\) are not as critical. Figure 11: The projections of the Pareto front, the dominated and infeasible designs onto planes \(\beta_{ULS}-cost\) and \(\beta_{SLS}-cost\). Figure 10: The Pareto front, along with the dominated and infeasible designs in objective space (**left**) and decision space (**right**). A comparison was made between the optimisation results yielded from the original and the modified RGD procedure. The feasible region of the original RGD procedure is composed of 331 designs, while the modified RGD procedure includes 310 designs. The results of the comparison are presented in Figure 13. The modified RGD procedure resulted in more conservative solutions, as seen from the boundary between the feasible and infeasible region, which is shifted slightly towards the right (Figure 13). A reduced number of designs in the feasible region in the modified RGD procedure, as opposed to the original procedure, is the consequence of additionally introduced constrains, according to the criteria of limit states prescribed in Eurocode 7. Figure 14 shows the comparison of the Pareto fronts in the decision space, which illustrates how the non-dominated designs of the modified method are moved to the right, when compared to the original method. The Pareto front of the original method includes  
Figure 12: The analysis of constraints on the position of feasible/infeasible design in design space for \(\sigma_{\phi}=0.59\) (**left**) and for \(\sigma_{\phi}=3.4\) (**right**). Figure 13: The result comparison of the original and modified RGD procedure for the decision space. a total of 63 non-dominated designs, with 72 in the modified one; 54 of them are present in both Pareto fronts.