## 3 Investigation of the Mechanisms Underlying the HRS
### Results and Discussion  
#### 3.3.1 Behavior of the Retaining Wall (Vertical Piles)  
Figure 10 presents the results obtained for the deflection of the wall (vertical piles) at different depths after different stages of the excavation process (when the HRS is employed). The wall deflections shown are those experienced by the vertical center line of the wall (in order to reduce boundary effects). A positive value corresponds to an inward movement of the wall (piles). As demonstrated in Figure 10, the deformation of the retaining wall (piles) gradually increased during the excavation of the 0 to \(-5\) m layer of soil. The maximum lateral deformation developed at the top of the wall (piles) and formed a triangular distribution (similar to that encountered when a cantilever retaining structure was used). During the excavation of the soil layer from \(-5\) to \(-10\) m, the top of the retaining wall (piles) rebounded outwards. At the same time, its lower parts, i.e., below the wake, continued to move inwards slightly. At the final excavation depth of \(-10\) m, the deflection profile between the wake and toe of the wall (piles) is locally convex with the largest deflection occurring 1.5 m above the bottom. However, the maximum value of the deflection is 16 mm (i.e., 0.16% of H where H is the excavation depth) and still occurs at the top of the retaining wall (piles) which is \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Structure Properties**} & \multicolumn{4}{c}{**Contact Properties**} \\ \cline{2-10}
**Structure Type** & \(\rho_{s}\) & \(E_{s}\) & \(t\) & \(k_{n}\) (Pa/m) & \(k_{s}\) (Pa/m) & \(\sigma_{\tau}\) & \(c_{\tau}\) & \(c_{s}\) & \(\varphi_{s}\) \\  & **(kg/m\({}^{3}\))** & **(GPa)** & **(m)** & **\(k_{n}\) (Pa/m)** & **(kPa)** & **(kPa)** & **(kPa)** & **(\({}^{\circ}\))** \\ \hline \hline \begin{tabular}{l} Capping beam \\ \end{tabular} & 2600 & 30 & 0.2 & - & - & - & - & - & - & - \\ \begin{tabular}{l} Wale \\ \end{tabular} & 2600 & 30 & 0.2 & - & - & - & - & - & - \\ \
\begin{tabular}{l} Retaining \\ walls \\ \end{tabular} & 2600 & 30 & 0.2 & 0.53 & \(4.59\times 10^{8}\) & \(4.59\times 10^{8}\) & 0 & 0 & 0 & 30 \\
\begin{tabular}{l} Batter piles \\ \end{tabular} & 2600 & 30 & 0.2 & - & \(4.59\times 10^{8}\) & \(4.59\times 10^{8}\) & 0 & 0 & 0 & 30 \\ \hline \hline \end{tabular} Note: \(\rho_{s}\) = density of structures; \(E_{s}\) = [PERSON]’s modulus of structures; \(v_{s}\) = [PERSON]’s ratio of structures; \(t\) = the equivalent thickness of the retaining wall; \(k_{n}\), \(k_{s}\) = normal stiffness and shear stiffness, respectively; \(\sigma_{\tau}\) = normal coupling spring tensile strength; \(c_{\tau}\), \(c_{s}\) = residual cohesion and cohesion of shear coupling spring, respectively; \(\varphi_{s}\) = shear coupling spring friction angle.  
\end{table} 
Table 3: Summary of the main properties of the structural elements used in the numerical analysis. Figure 10: Variation of the horizontal deflection of the wall (vertical piles) of the HRS at different excavation depths. consistent with results obtained as H \(\leq\)5 m. Overall, Figure 10 shows that the deflection of the retaining wall (piles) is small when it is supported by the batter piles. Furthermore, the retaining wall (piles) rotates counterclockwise around the wake, causing the top of the retaining wall (piles) to rebound outwards. This is because the batter piles act as a lateral restraint on the movement of the retaining wall (piles) at the level of the wake. The soil pressure in front of the retaining wall (piles) decreases during the excavation process, reducing the horizontal constraint on the retaining wall (piles). As a result, the deflection profile of the wall (piles) between the wake and the bottom surface bulged inwards during excavation and the top of the retaining wall (piles) rebounded outwards. Figure 11 shows the variation of the bending moment acting on the retaining wall (piles) at the different excavation depths when it is supported using HRS.