# Modified, Reliability-Based Robust Geotechnical Design Method, in accordance with Eurocode 7
## 3 Results
### Illustrative Example 2--Design of Axially Loaded Pile  
The modified RGD method is applied for the design of an axially loaded pile in a stiff clay, whose geometry is shown in Figure 15. The soil parameters and external action are taken from the example given in [34]. It is necessary to optimise pile length (D) and diameter (B), with the aim of maximising robustness while minimising pile cost. The \"\(\alpha\) method\" [35] is used for determining the bearing capacity of the pile, and the ULS limit state function is defined as follows:  
\[g_{ULS}=R_{b}+R_{s}-W_{p}-V_{G}-V_{Q}=(N_{c}c_{u}+\sigma_{v0})\;D^{2}\;\pi/4+ \alpha_{u}c_{u}D\pi\mathrm{L}-V_{G}-V_{Q} \tag{12}\]  
where \(R_{b}=(N_{c}c_{u}+\sigma_{v0})\;D^{2}\;\pi/4\) and \(R_{s}=\alpha_{u}c_{u}D\pi\mathrm{L}\) are the pile base and shaft resistances [35], \(W_{p}\) is the weight of the pile, \(N_{c}=9\), \(c_{u}\) is undrained shear strength, \(\sigma_{v0}\) is total  
Figure 14: The comparison of Pareto fronts obtained by applying the original and the modified RGD method in the decision space. Figure 15: Pile geometry. vertical stress at the depth of the pile base, \(\alpha_{u}\) is the coefficient used to relate \(c_{u}\) to the adhesive stress along the pile shaft. The SLS limit state function is defined by using the equation which connects the bearing capacities of ULS and SLS, attained through the statistical analyses of the results of pile load tests [36]:  
\[\frac{Q_{SLS}}{Q_{ULS}}=\frac{s_{a}}{a+bs_{a}} \tag{13}\]  
where \(Q_{ULS}\) and \(Q_{SLS}\) are the bearing capacities for ULS and SLS, \(s_{a}\) is allowable settlement (20 mm), \(a\) and \(b\) are hyperbolic curve-fitting parameters for the normalised load-settlement curve. Using Equation (13), the SLS limit state function can be expressed as follows:  
\[g_{SLS}=\bigg{(}\frac{s_{a}}{a+bs_{a}}\bigg{)}Q_{ULS}-V_{G}-V_{Q} \tag{14}\]  
The cost of the pile (C) is related to the concrete volume. In the presentation of applying of the RGD method to the case of an axially loaded pile, Equation (15) was used, in which the cost is estimated by the following expression [34]:  
\[C=\bigg{(}\frac{\pi\ \bar{B}^{2}D}{4}\bigg{)}unit \tag{15}\]  
The pile is subjected to axial permanent (\(V_{G}\)) and variable (\(V_{Q}\)) loads. Undrained shear strength \(c_{u}\) is determined from a triaxial test, and its mean value increases linearly with depth, and according to the following expression: \(c_{u}=13.0z\), where \(z\) is depth. The characteristics of random variables are shown in Table 11.  
In order to determine the influence of individual random variables on the variance of ULS and SLS limit state functions, a sensitivity analysis was conducted using Soboli indices. The analysis results are given in Table 12.  
Based on the conducted sensitivity analysis, the variables with small contributions to the system response variances are frozen. The reliability analyses are conducted using random vectors with a reduced number of random variables, defined as follows: \(X_{ULS}=(c_{u},\ a)^{T}\) and \(X_{SLS}=(c_{u},\ a,\ b)^{T}\). The next step in the implementation of the RGD method is the identification of decision variables and their lower and upper boundaries. In this case, the following decision variables were chosen:  
* length of the pile: 9 \(\leq L\ \leq\ 18\) m; \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable/Analysis** & \(c_{u}\) & \(\gamma\) & \(\alpha\) & \(V_{G}\) & \(V_{Q}\) & \(a\) & \(b\) \\ \hline ULS & 0.95 & \(\approx\)0 & 0.045 & \(\approx\)0 & \(\approx\)0 & - \\ SLS & 0.64 & \(\approx\)0 & 0.03 & \(\approx\)0 & \(\approx\)0 & 0.19 & 0.14 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: First order Sobol indices for ULS and SLS analyses. diameter of the pile: 0.5 \(\leq D\)\(\leq\) 0.8 m.  
After defining limit state functions, random and deterministic variables, and decision variables, the problem of optimisation according to the modified RGD method can be expressed as follows: