## 3 Application of the ORW Optimization Model--Case Study
### Optimization of the Embedded Retaining Wall with Target Failure Probability (Methodology 2)
The partial safety factor approach applies safety factors to account for variability and uncertainties. In contrast, the RBD approach directly considers the probabilistic nature of the design parameters and the associated uncertainties. RBD methods use probability distributions and statistical analysis to more explicitly quantify and manage risk and provide a more nuanced assessment of failure probabilities. Because a gaussian (normal) distribution of random variables was used, a direct comparison between the two methodologies can be made. The results are shown in Table 8 below. The results of the study show that although the embedment depths determined using the partial safety factor approach based on the Eurocode are within acceptable limits for the probability of failure [36], alternative designs for retaining structures could offer significantly greater safety margins. The most notable observation arises when the seepage flow is considered in the optimal design according to the Eurocode standard, as shown in Figure 5. This comparison shows that the partial safety factors may not fully account for the influence of seepage and hydraulic heave on the stability of the structure. It should also be noted that the results of the RBD method for the given input parameters show that failure due to hydraulic heave never occurs as an independent failure mechanism, but only in combination with the overturning moment (see, Table 9). Therefore, the results are identical when only the seepage is considered and when the seepage is considered in combination with the hydraulic heave. Figure 5: Relationship between embedment depth and probability of failure. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No Groundwater**} & \multicolumn{2}{c}{**Groundwater Present**} & \multicolumn{2}{c}{**Groundwater and**} \\  & \multicolumn{2}{c}{**(Hydrostatic Conditions)**} & \multicolumn{2}{c}{**Seepage**} \\  & \multicolumn{2}{c}{**(Groundwater-Flow)**} & \multicolumn{2}{c}{**(Groundwater, Seepage and Hydraulic Heave**} \\ \hline Embedment depth & & & & \\ \(d_{d}\) [m] & 2.691 & 6.459 & 6.520 & 6.520 \\ \hline Actual failure & & & & \\ probability (AFP) & 0.00002 & 0.00002 & 0.00008 & 0.00008 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Failure probability of optimal designs obtained using the partial safety factor approach. A direct comparison with the reliability-based design approach, which targets a specific probability of failure, underlines the limitations of the Eurocode method. Although the Eurocode-based design remains within the acceptable failure range, the analysis shows that increasing the embedment depth by only 0.94% does not sufficiently reduce the risk, as the probability of failure under seepage and hydraulic heave increases by a factor of 4. This underlines the usefulness of the RBD method, which allows for a more accurate risk assessment by explicitly considering uncertainties and probabilistic variables, which could lead to safer and more reliable structural designs.