## 3 Application of the ORW Optimization Model--Case Study
### Optimization of the Embedded Retaining Wall with Target Failure Probability (Methodology 2)  
The original deterministic optimization model, which was formulated according to the Eurocode 7 standard, was expanded into an uncertainty model that includes a certain target probability of failure. In this extended model, the partial safety factors are set to 1.0, effectively eliminating additional safety margins. This approach allows for the inclusion of variability in the input data, represented by random values for the parameters listed in Table 4. The uncertainty model is based on the principle that failure occurs if one of the criteria specified in Equation (14) is not met. This equation defines the failure conditions based on the design parameters:  
\[F= \begin{cases}M_{1}\leq 0\\ M_{2}\leq 0\\ M_{3}\leq 1\end{cases} = \begin{cases}H_{Ed}-H_{Rd}\leq 0\\ M_{Ed}-M_{Rd}\leq 0\\ S_{d,dst}-G^{\prime}_{d,stb}\leq 0\end{cases}  \tag{14}\]  
To investigate different failure probabilities, ranging from 1\(\cdot\)10\({}^{-5}\) to 1.0, the optimization was performed step by step. The algorithm systematically evaluated different design dimensions based on these probabilities and used Equation (6) to determine if the conditions were met. It assigned a score of 1 for failures and 0 for successful outcomes. With a target failure probability of 1 \(\times\) 10\({}^{-4}\), the algorithm identified dimensions that resulted in failure in only 10 out of 100,000 samples. This process was repeated for other failure probabilities and similar results were observed. The presence of groundwater significantly increases the embedment depth at each TFP, with the percentage increase ranging from approximately 113.8% at the highest TFP to over 146% at the lowest TFP. This indicates that the groundwater significantly increases the embedment depth, but its relative effect decreases with increasing TFP. The addition of seepage to groundwater has a modest effect on the embedment depth. The percentage increase in embedment depth due to seepage ranges from around 2.19% to 6.41%, depending on the TFP. This shows that the effect of seepage is small compared to the more significant increase observed when groundwater is first introduced, but it still slightly increases the embedment depth, especially at lower TPFs. These results are summarized in Table 7, which illustrates how the embedment depth of the retaining wall changes with changes in the target probability of failure. The characteristic values for the deterministic model were derived using a statistical method based on mean values, as prescribed in the Eurocode. In this approach, characteris \begin{table}
\begin{tabular}{l l l l l} \hline \hline  & \multicolumn{2}{c}{**Embedment Depth \(d_{d}\) [m]**} & \\ \cline{2-5}
**TFP** & **No Groundwater** & **Groundwater Present** & **Groundwater and Seepage** & **Groundwater, Seepage** \\  & & **(Hydrostatic Conditions)** & **(Groundwater-Flow)** & **and Hydraulic Heave** \\ \hline
0 & 2.72 & 6.71 & 7.14 & 7.14 \\ \hline
0.00001 & 2.71 & 6.59 & 6.90 & 6.90 \\ \hline
0.00005 & 2.68 & 6.34 & 6.57 & 6.57 \\ \hline
0.0001 & 2.62 & 6.06 & 6.32 & 6.32 \\ \hline
0.0005 & 2.49 & 5.89 & 6.09 & 6.09 \\ \hline
0.001 & 2.45 & 5.78 & 5.98 & 5.98 \\ \hline
0.005 & 2.33 & 5.51 & 5.67 & 5.67 \\ \hline
0.01 & 2.27 & 5.36 & 5.51 & 5.51 \\ \hline
0.05 & 2.1 & 5.02 & 5.13 & 5.13 \\ \hline
0.1 & 2.02 & 4.79 & 4.93 & 4.93 \\ \hline
1 & 1.23 & 2.62 & 2.82 & 2.82 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Optimal designs obtained using methodology 2 for each target failure probability (_TFP_). tic values are calculated by applying a statistical analysis to mean soil properties, which provides a basis for determining design parameters under assumed average conditions. By integrating these characteristic values into the uncertainty model, we were able to accurately assess the failure probabilities associated with the optimal designs determined based on the partial safety factor approach.