## 3 Numerical Modeling and Validation
### Soil Constitutive Model, Model Parameters, and Modeling Procedure  
Many researchers have emphasized the significance of accounting for the small strain behavior of soil in geotechnical engineering problems [42; 43; 44; 45]. Neglecting the small strain stiffness of the soil may lead to overestimations of the deformation of retaining walls and ground surface settlement. The hardening soil with the small strain model (HSS), initially formulated by [PERSON] [46; 47] and further developed by [PERSON] [48] to consider small strains, incorporates Mohr-Coulomb failure criterion, is one of the most advanced constitutive models for soils, and has been widely applied to deep excavations [49; 50; 51]. Hence, the HSS model was used to simulate the small strain behavior of the soil layers. Table 1 summarizes the soil parameters of the HSS model. The relationship of the stiffness parameters (\(E_{s}:E_{ood}^{ref}:E_{50}^{ref}:E_{ur}^{ref}=1:1:1:3\)) is used in this paper [52; 53; 54]. The shear modulus at very small strains (\(G_{0}\)) is estimated using the relationship proposed by [PERSON] and [PERSON] [55] (Equation (1)), while the threshold shear strain (\(\gamma_{0.7}\)) is determined using the formula presented by [PERSON] [48] (Equation (2)). \[G_{0}=33\frac{(2.97-e)^{2}}{1+e}\sqrt{\frac{p^{\prime}}{p_{ref}}} \tag{1}\]  
\[\gamma_{0.7}=\frac{3}{28G_{0}}\big{[}2c^{\prime}\big{(}1+\cos\big{(}2\varphi^{ \prime}\big{)}\big{)}+\sigma^{\prime}{}_{1}(1+K_{0})\sin\big{(}2\varphi^{ \prime}\big{)}\big{]} \tag{2}\]  
In the simulation, the supporting piles, steel plates, and walings were assumed to have linear elastic behaviour. Detailed information on their characteristics and parameters is given in Table 2. The supporting piles assembled by HW350 and L125 are simplified as solid piles with a cross-section of a 350 \(\times\) 350 square considering the equivalent of bending stiffness (\(EI\)). In addition, zero-thickness interface elements were generated to replicate the interaction between the structure and the soil, which followed the Mohr-Coulomb criterion. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Soil Layer** & \(c^{\prime}\) & \(\varphi^{\prime}\) & \(E_{ood}^{ref}\) & \(E_{50}^{ref}\) & \(E_{ur}^{ref}\) & \(v_{ur}\) & \(G_{0}^{ref}\) & \(\gamma_{0.7}\) & \(m\) & \(R_{f}\) \\  & **[kPa]** & [\({}^{\circ}\)] & **[MPa]** & **[MPa]** & **[MPa]** & **[kN/m\({}^{2}\)]** & **[10\({}^{-4}\)]** & **[**\({}^{-4}\)**]** & **[**\({}^{-4}\)**]** \\ \hline Silty clay-1 & 28.5 & 13.3 & 7.76 & 7.76 & 23.28 & 0.2 & 97.27 & 2.59 & 1.0 & 0.9 \\ Silty clay-2 & 31.6 & 14.2 & 8.09 & 8.09 & 24.27 & 0.2 & 99.44 & 2.48 & 1.0 & 0.9 \\ Silty clay-3 & 30.8 & 13.4 & 8.37 & 8.37 & 25.11 & 0.2 & 94.59 & 2.62 & 1.0 & 0.9 \\ Silty clay-4 & 31.5 & 14.6 & 9.01 & 9.01 & 27.03 & 0.2 & 99.73 & 2.87 & 1.0 & 0.9 \\ Silty clay-5 & 29.8 & 13.7 & 9.95 & 9.95 & 29.85 & 0.2 & 101.21 & 2.87 & 1.0 & 0.9 \\ Silty clay-6 & 31.1 & 13.8 & 9.52 & 9.52 & 28.56 & 0.2 & 97.99 & 3.28 & 1.0 & 0.9 \\ Silty clay-7 & 33.0 & 14.2 & 9.79 & 9.79 & 29.37 & 0.2 & 97.85 & 3.68 & 1.0 & 0.9 \\ \hline \hline \end{tabular} Note: \(c^{\prime}\) represents effective cohesion; \(\varphi^{\prime}\) represents effective internal friction angle; \(E_{ood}^{ref}\) represents tangent referential stiffness for primary oedometer loading; \(E_{50}^{ref}\) represents secant referential stiffness in standard drained triaxial test; \(E_{eff}^{ref}\) represents unloading/reloading referential stiffness; \(\gamma_{ur}\) represents [PERSON]’s ratio for unloading-reloading; \(G_{0}^{ref}\) represents reference shear modulus at very small strains (\(e<10^{-6}\)); \(\gamma_{0.7}\) represents shear strain for 70% of \(G_{0}\); \(m\) represents power for stress-level dependency of stiffness; \(R_{f}\) represents failure ratio.  
\end{table} 
Table 1: Soil parameters for the HSS model.