# 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  
Obtaining non-dominated designs by applying the genetic algorithm in the RGD procedure can be optimised by reducing the number of random variables included in the reliability analysis. In the reliability calculation, failure is a probabilistic event, and its probability is given by:  
\[P_{f}=Prob\ \{g(X)\leq 0\}, \tag{8}\]  
where \(X\) is a random vector and \(g(. )\) is a limit state function. In the case of a shallow foundation loaded by permanent and variable vertical action in cohesionless soil, the function of the limit state for ULS can be expressed as follows:  
\[g_{ULS}=q_{u}\cdot B\cdot L-V_{G}-V_{Q}=R_{ULS}-V_{g}-V_{q}=q^{\prime}N_{q}s_{ q}+0.5\ \gamma\ tB^{\prime}N_{Y}s_{Y}-V_{G}-V_{Q}, \tag{9}\]  
where \(N_{q}\) i \(N_{Y}\) are load capacity coefficients, \(s_{q}\) and \(s_{Y}\) are the coefficients of foundation shape, \(\gamma^{\prime}\) is the effective unit weight of soil, \(B^{\prime}\) is the effective foundation width, \(q^{\prime}\) is the effective overburden pressure at the foundation base level, \(V_{g}\) is the permanent vertical load, \(V_{q}\) is the variable vertical load. The function of the serviceability limit state can be defined by applying a formula developed by [PERSON] and [PERSON] [20], which is derived from load-settlement behaviour of a shallow foundation under axial compression loading  
\[g_{SLS}=\frac{R_{ULS}(s_{t}/B)}{a(s_{t}/B)+b}-V_{g}-V_{q}, \tag{10}\]  
where \(s_{t}\) is the tolerated settlement, \(B\) is the foundation width, and coefficients \(a\) and \(b\) are the parameters of the hyperbolic model that fit the normalised-settlement curve. Random vectors \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)** & \(\beta_{\text{IPEM}}\) & \(\beta_{\text{MC}}\) & 1\(\beta\beta\)1 & \(\beta_{\text{IPEM}}\) & \(\beta_{\text{MC}}\) & 1\(\beta\beta\)1 & \(\beta_{\text{IPEM}}\) & \(\beta_{\text{MC}}\) & 1 \(\beta\beta\)1 \\ \hline
2 & 0.854 & 0.845 & 0.009 & 0.724 & 0.718 & 0.006 & 0.797 & 0.783 & 0.014 \\
2 & 1.338 & 1.324 & 0.014 & 1.107 & 1.095 & 0.012 & 1.137 & 1.122 & 0.015 \\
2.4 & 1.831 & 1.812 & 0.019 & 1.494 & 1.48 & 0.014 & 1.479 & 1.475 & 0.004 \\
2.6 & 2.334 & 2.309 & 0.025 & 1.887 & 1.872 & 0.015 & 1.823 & 1.788 & 0.035 \\
2.8 & 2.845 & 2.815 & 0.030 & 2.284 & 2.266 & 0.018 & 2.171 & 2.166 & 0.005 \\
3 & 3.364 & 3.334 & 0.030 & 2.686 & 2.663 & 0.023 & 2.521 & 2.496 & 0.025 \\ \hline \hline \end{tabular}  
* \(\beta_{IPEM}\), \(\beta_{\text{MC}}\)—reliability indexes calculated using the IPEM and Monte Carlo methods.  
\end{table} 
Table 4: The errors of the IPEM method for SLS. \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)** & \(\beta_{\text{IPEM}}\) & \(\beta_{\text{MC}}\) & 1\(\beta\beta\)1 & \(\beta_{\text{IPEM}}\) & \(\beta_{\text{MC}}\) & 1\(\beta\beta\)1 & \(\beta_{\text{IPEM}}\) & \(\beta_{\text{MC}}\) & 1 \(\beta\beta\)1 \\ \hline
2 & 2.903 & 2.869 & 0.034 & 2.321 & 2.298 & 0.023 & 2.201 & 2.173 & 0.028 \\
2.2 & 4.026 & 3.988 & 0.038 & 3.184 & 3.163 & 0.021 & 2.949 & 2.83 & 0.119 \\
2.4 & 5.241 & 5.196 & 0.045 & 4.114 & 4.077 & 0.037 & 3.75 & 3.753 & 0.003 \\
2.6 & 6.548 & 6.478 & 0.07 & 5.108 & 5.072 & 0.036 & 4.603 & 4 from Equation (8) for the ultimate limit state can be expressed as \(X_{ULIS}=\left(\phi^{\prime},\ \gamma\ t,V_{g},V_{q}\right)^{T}\); for the serviceability limit state, the following applies: \(X_{SLS}=\left(\phi^{\prime},\ \gamma\ t,a,b,V_{g},V_{q}\right)^{T}\).