## 3 Investigation of the Mechanisms Underlying the HRS
### Construction of the Numerical Model
#### 3.2.3 Structure Model and Input Properties  
Several types of structure-support members are available in FLAC3D and structures with arbitrary geometries, properties, and soil interactions can be modeled [20]. The vertical piles are essentially working as diaphragm walls that are subject to bending deformation (resulting from the active earth pressure acting behind the vertical piles, passive earth pressure in front of the vertical piles, and load from the batter piles). Therefore, the vertical piles can be converted into retaining walls of a certain thickness, \(t\), based on the principle of equivalent flexural rigidity [19], and given by:  
\[t=\frac{D}{2}\sqrt{\frac{3\pi D}{2B}}, \tag{4}\]  
where \(D\) is the diameter of a vertical pile and \(B\) is the center-center spacing of the vertical piles.  
Liner structural elements are attached to the surface of the grid and resist bending loading as linearly elastic shells. They can also simulate the behavior of shear-directed frictional interactions acting between themselves and the grid. Therefore, liner structural elements were used to simulate the behavior of the vertical piles after their conversion to retaining walls.  
The batter piles were simulated using pile structural elements with skin-friction and end-bearing effects. The strength properties of the pile-soil interface were assumed to be equal to that of the soil mass. These assumptions have all been used in previous studies [20] and are also valid in this paper. The normal coupling spring tensile strength (\(\sigma_{\tau}\)), shear coupling spring cohesion (\(c_{s}\)), and shear coupling spring residual cohesion (\(c_{r}\)) of the contact between the liner-grid (pile-grid) and soil around the interfaces were set to zero. The shear coupling spring friction angle (\(\sigma_{s}\)) was set to be 30\({}^{\circ}\). It is recommended by FLAC3D that the normal stiffness, \(k_{n}\), and shear stiffness, \(k_{s}\), were calculated using the expression [20]:  
\[k_{n}=k_{s}=10\times\max\left[\left(K+\frac{4}{3}G\right)/\Delta z_{\min} \right], \tag{5}\]  
where \(\Delta z_{\min}\) is the smallest dimension of a neighboring zone in the normal direction. According to Equations (1)-(3), \(K\) and \(G\) increase linearly with \(z\). However, these parameters are assumed to be constant in this numerical analysis in order to simplify and improve the calculation efficiency. Thus, an average value was calculated (0.46 GPa/m) and used for \(K\) and \(G\).  
The capping beam and wake components were both simulated using beam structural elements. All the structural elements were modeled using anisotropic liner elastic models. The retaining walls (vertical piles), batter piles, and beams were all taken to have a Young's modulus, \(E_{s}\), of 30 GPa. Their Poisson's ratios, \(\
u_{s}\), were taken to be 0.2. The connections between the beams and liner elements were set to be rigid, however, the heads of the batter piles were hinged to the wake in order to prevent there being a bending moment between them. For convenience, the main properties of the structural elements used in the analysis are summarized in Table 3. The geometric parameters used for the supporting structure are consistent with those of the prototype given in Section 3.1.1.  
We are only interested in the forces acting on the retaining structures that arise due to excavation (rather than those induced by the installation process). Therefore, the installation of the retaining structures was modeled using a 'wished-in-place' method [23, 24, 25]. We also note that the analyses were conducted in drained conditions using effective stress and so we did not take into account the effect of the curtain.