## 4 Conclusions  
By optimizing both the deterministic model based on Eurocode 7 and an uncertainty model that includes a probabilistic failure analysis, we have determined the optimal designs for an embedded retaining wall under different conditions. The most important results of the case study presented show that the presence of groundwater has a considerable influence on the required embedment depth (\(d_{d}\)) of the retaining wall (approx. 2.4 times deeper). The design based on partial safety factors, which considers seepage and a combination of seepage and hydraulic heave, leads to a modest increase in the embedment depth of 0.94% compared to a retaining wall design that only takes groundwater into account. The analysis of the probability of failure of designs based on partial safety factors compared to designs based on a full reliability analysis shows that the embedment depth is more dependent on seepage in an analysis based on the reliability analysis than in an analysis with the partial safety factor. The percentage increase in embedment depth due to seepage is between 2.19% and 6.41% for the design based on the probability of failure, depending on the targeted probability of failure. The RBD method also provides valuable information on the required embedment depths as a function of the probability of failure, whereas the Eurocode design method specifies a single embedment depth with a corresponding probability of failure. In this case study, the analysis of the failure mechanisms shows that piping failure never occurs as an independent failure mechanism, but is linked to the overturning moment mechanism. Piping failure in embedded retaining walls happens when seepage forces surpass the soil's resistance, causing soil erosion and instability. Key factors contributing to piping failure include excessive hydraulic gradients, highly permeable soils, insufficient embedment depth, significant hydraulic head differences and weak soil resistance [25]. It should be noted that the RBD method is more sensitive to fluctuating field conditions such as seepage and hydraulic heave. This could allow for a more accurate risk assessment as these uncertainties are explicitly considered. This makes the RBD method a more robust approach to ensure optimal structure in real-world scenarios where environmental conditions are unpredictable and involve a degree of uncertainty. Engineers should consider using the RBD method when there is significant uncertainty in the input data. This method can serve as an additional analysis for conventional approaches based on partial safety factors. In addition, the RBD method allows for potential structural optimization when applied with an appropriate methodology, as shown in this article. Further research is required as this work did not consider structural failure modes, which limits the applicability of the designs obtained. However, different types of retaining walls such as sheet \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Failure Mechanism**} & \multirow{2}{*}{**No Groundwater**} & \multicolumn{2}{c}{**Groundwater Present**} & \multicolumn{2}{c}{**Groundwater and**} & \multirow{2}{*}{
\begin{tabular}{c} **Groundwater, Seepage** \\ **and Hydraulic Heave** \\ \end{tabular} } \\  & & **(Hydrostatic** & **Sepage** & **(Groundwater-Flow)** & **and Hydraulic Heave** \\ \hline \(H_{Ed}-H_{Rd}\leq 0\) & 0 * & 0 & 0 & 0 \\ \hline \(M_{Ed}-M_{Rd}\leq 0\) & 2 & 2 & 8 & 8 \\ \hline \(S_{d,dst}-G^{\prime}_{d,stb}\leq 0\) & 0 & 0 & 0 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Failure mechanisms for each combination of effects on the embedded retaining wall (\(N=100\),000). pile walls, reinforced concrete pile walls or Berliner walls have different structural failure modes, which should be integrated into the optimization model. Most walls are not entirely rigid, meaning they experience both rigid body movement and flexing. Steel sheet piling is especially flexible, but even bored piles can bend significantly when used to support deep excavations.