## 3 Application of the ORW Optimization Model--Case Study
### Optimization of the Embedded Retaining Wall with Partial Safety Factors (Methodology 1)  
For the optimization of the deterministic model, the Eurocode standard [35] prescribes a statistical method for determining the characteristic values. This method is used due to the adequacy of laboratory tests and the high quality of available soil data. According to the Eurocode, the characteristic values are calculated using Equation (9), where \(X_{k}\) is the characteristic value, \(X_{m}\) is the mean value of the soil property, \(\sigma_{N}\) is the standard deviation of the property and \(\kappa_{N}\) is a coefficient related to the sample population. This approach ensures that the optimization model reflects the true variability and reliability of the soil properties. \[\mathrm{X_{k}}=\mathrm{X_{m}}\cdot[1\mp\kappa_{N}\sigma_{N}] \tag{9}\]  
\[\kappa_{N}=1.645\cdot\sqrt{\frac{1}{N}+1} \tag{10}\]  
In accordance with the given equations, the characteristic values for the presented deterministic optimization model (ORW) are calculated. \[\mathit{\varphi^{\prime}}_{k}=\mathit{\varphi^{\prime}}_{mean}\mp 1.645\cdot \sqrt{\frac{1}{N}+1}\cdot\mathit{\sigma_{\varphi^{\prime},k}}=35.15^{\circ}-1.645\cdot 0.82^{\circ}=33.80^{\circ} \tag{11}\]  
\[\gamma_{k}=\gamma_{mean}\mp 1.645\cdot\sqrt{\frac{1}{N}+1}\cdot\mathit{\sigma_{ \gamma,k}}=20.55\frac{\mathrm{kN}}{\mathrm{m}^{3}}-1.645\cdot 0.24\frac{ \mathrm{kN}}{\mathrm{m}^{3}}=20.16\ \mathrm{kN}/\mathrm{m}^{3} \tag{12}\]  
\[c^{\prime}_{k}=c^{\prime}_{mean}\mp 1.645\cdot\sqrt{\frac{1}{N}+1}\cdot \mathit{\sigma_{c^{\prime},k}}=6.6\ \mathrm{kPa}-1.645\cdot 1.05\ \mathrm{kPa}=4.87\ \mathrm{kPa} \tag{13}\]  
The obtained values from Equations (11)-(13) serve as characteristic values and the input data for the deterministic optimization model to obtain the optimal design based on the partial safety factor approach. Further input data can be found in Table 5.  
Figure 4: Probability density function of random variables. After optimizing the deterministic model according to the Eurocode standard [4], and using design approach 2 using partial safety factors, we successfully determined the optimal embedment depths for the embedded retaining wall based on the provided input data. This optimization process was performed under four different scenarios to account for different conditions: one scenario without groundwater, another with groundwater present (hydrostatic conditions), a third scenario where seepage effects (groundwater-flow) were considered, and a fourth scenario where hydraulic heave had to be prevented. The aim of the analysis was to assess how each condition affects the required embedment depths for stability and performance. By comparing these scenarios, we can evaluate the impact of groundwater and seepage on the design and ensure that the wall meets the required stability criteria under different environmental conditions. The results of the optimization for each scenario, including the optimal embedment depths, are detailed and summarized in Table 6 below. The results show a significant influence of groundwater on the embedment depth of the retaining wall. In the presence of groundwater, an increase of 240% was observed. In contrast, the embedment depth increases only slightly by 0.94% if seepage is considered in addition to groundwater. These results illustrate the considerable influence of groundwater on construction planning, while the influence of seepage and hydraulic failure is relatively small in this case study. Hydraulic failure through the piping was also considered, but this condition did not increase the penetration of the retaining wall, which means that the mechanism of rotational failure was crucial in the design. Piping failure in embedded retaining walls occurs when seepage forces exceed the soil's resistance, leading to soil erosion and instability.