# Modified, Reliability-Based Robust Geotechnical Design Method, in accordance with Eurocode 7
## 3 Results  
### Illustrative Example 1--Design of Shallow Foundation  
The problem of optimising a shallow foundation on non-coherent soil was considered, similar to the one presented by [PERSON] et al. [1], from which we took the characteristics of the used variables. As shown in Figure 9, the foundation is subjected to vertical permanent (\(V_{C}\)) and variable (\(V_{Q}\)) loads. Aside from the soil's internal friction angle, all other random variables are \"frozen\", since their contribution to the total system response variance is negligible, as shown in Section 2.2.3. A three-objective, constrained optimisation was conducted (ULS, SLS, foundation cost) with two decision variables: foundation width (B) and depth (D). A square foundation layout is assumed (B = L). Figure 8: A flowchart of the modified RGD method. Figure 9: The geometry of the computation model. The ultimate and serviceability limit states are defined in Equations (5) and (6), and maximum tolerated settlement is set at 25 mm. The estimation of foundation cost is conducted using the expression defined by [PERSON] and [PERSON] [33]:  
\[Z=Q_{e}c_{e}+Q_{f}c_{f}+Q_{c}c_{c}+Q_{b}c_{b}, \tag{11}\]  
where \(Q_{e},Q_{f},Q_{c},Q_{b}\) signify the quantities of excavation, formwork, concrete and compacted backfill, respectively, and \(c_{e},c_{f},c_{c},c_{b}\) the associated unit prices. More on the cost calculation for a shallow foundation can be found in [33]. The foundation soil is dry sand with the following statistical characteristics of the internal friction angle: \(\mu_{\phi}=34.4^{\circ}\) and \(\sigma_{\phi}=0.59\). The characteristics of the deterministic variables are shown in Table 10.  
The problem of shallow foundation optimisation in the case in question can be written out as follows: \begin{tabular}{l l l} \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}   
The problem of shallow foundation optimisation in the case in question can be written out as follows: \begin{tabular}{l l} \hline \hline Find: & d = [B,D] \\ Subject to: & \(B\in\{1.0\text{ m},\,1,1\text{ m},\,\dots,\,5.0\text{ m}\}\) and \\  & \(D\in\{1.0\text{ m},\,1,1\text{ m},\,\dots,\,2.0\text{ m}\}\) \\  & \(\beta_{ULS}\leq 3.8\) \\  & \(\beta_{SSL}\leq 1.5\) \\  & \(ODF_{ULS}\leq 1.0\) \\  & \(ODF_{SLS}\leq 1.0\) \\ \multicolumn{2}{l}{Objectives: Maximising \(\beta_{ULS}\) and \(\beta_{SLS}\)} \\ \multicolumn{2}{l}{Minimising cost of shallow foundation} \\ \hline \hline \end{tabular}   
The solutions for the optimisation are shown in Figures 10 and 11. A set of non-dominated designs (Pareto front) in objective space is shown in Figure 10 on the left, and in decision space in Figure 10 on the right. Aside from the Pareto front (magenta), also shown are the dominated feasible designs (green) and infeasible designs (black). The design space is composed of a total of 451 designs, 311 of which are feasible and 140 infeasible. The optimisation resulted in 71 non-dominated designs which make up the Pareto front. The minimal foundation dimensions from the Pareto front are B \(\times\) D = 2.1 \(\times\) 1.8 m, with the maximum being 5.0 \(\times\) 2.0 m, at the costs of USD 1160 and USD 5687, respectively. Figure 10, on the right, shows the grouping of non-dominated designs in the vicinity of the maximum prescribed depth of D = 2.0 m, which is in accordance with the original method's solutions presented by [PERSON] et al. in [1]. Figure 11 shows the projections of the Pareto front onto axes \(\beta_{ULS}\) and \(\beta_{SLS}\). The minimum values of the reliability index for ULS and SLS are 9.4 and 3.2, respectively, which is significantly higher than the prescribed 3.8 and 1.5, respectively.