## 3 Numerical Analysis
### Failure Criterion, Loading Condition, and Parameters Used in the Numerical Computation  
In terms of the assumption of the constitutive law of the gravel layer in the numerical analysis, this study adopted the Mohr-Coulomb failure criterion with elastic-perfectly-plastic behavior. The two-dimensional equation of this criterion can be expressed as follows:  
\[f(\sigma_{3},\sigma_{1})=\sigma_{1}-\sigma_{3}K_{p}-\sigma_{c}=0 \tag{1}\]  
where _Kp_ is a coefficient of the passive lateral pressure, and \(\sigma_{c}\) is the uniaxial compression strength (UCS), which can be expressed as  
\[\sigma_{c}=2c\sqrt{K_{p}} \tag{2}\]  
\[K_{p}=tan^{2}\left(45^{\circ}+\frac{\phi}{2}\right) \tag{3}\]  
where \(c\) and \(\phi\) are the cohesion and internal friction angle of the material, respectively. In addition, the three-dimensional equation can be expressed as  
\[f\left(I_{1},\sqrt{J_{2}}\right)=\frac{I_{1}}{3}sin(\phi)+\sqrt{J_{2}}\left[ cos(\theta)-\frac{1}{\sqrt{3}}sin(\theta)sin(\phi)\right]-c\cdot cos(\phi)=0 \tag{4}\]  
where \(\theta\) is the Lode angle, which can be considered a measure of the loading type. The Lode angle varies with respect to the middle eigenvalue of the stress. There are many definitions of the Lode angle, and each utilizes different trigonometric functions [31]. Furthermore, \(I_{1}\) and \(\sqrt{J_{2}}\) are the first stress invariant and second deviatoric stress invariant, respectively, and can be explicitly written as  
\[I_{1}=\sigma_{1}+\sigma_{2}+\sigma_{3} \tag{5}\]  
\[\sqrt{J_{2}}=\sqrt{\frac{1}{6}\left[\left(\sigma_{1}-\sigma_{2}\right)^{2}+ \left(\sigma_{2}-\sigma_{3}\right)^{2}+\left(\sigma_{3}-\sigma_{1}\right)^{2 }\right]} \tag{6}\]  
where \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) are the major, intermediate, and minor principal stresses, respectively. The surface loading of the design included a working live load of 10 kN/m\({}^{2}\) and a train live load of 80 kN/m\({}^{2}\) (using the KS-18 standard live load), as shown in Figure 5. The stratum could mainly be divided into three layers; from top to bottom, these were the backfill layer, the yellow sandy silt, and the gravel layer. Regarding the values of parameters such as the cohesion and friction angle, the parameters of the backfill layer and the yellow sandy silt were obtained through a standard penetration test (SPT) and thin-wall tube samplers, and they were taken to the laboratory for uniaxial compression tests and direct shear tests. However, the parameters of the gravel layer were estimated by using empirical formulas. The physical properties of each layer were simplified, as shown in Table 1, and the provision of the site borehole log is shown in Figure 6. The red part in the figure represents the gravel layer, and the others are backfill layers. The design spacing of the retaining piles was 2.25 m, and each retaining pile was 120 cm long and 60 cm wide. The parameters of the concrete used in the numerical calculation included a unit weight of 24.53 Kpa/m, an elastic modulus of 24.63 GPa, Poisson's ratio of 0.2, and a uniaxial compressive strength of 27.47 MPa. In addition, the horizontal angle of the ground anchor  
simulate the excavation sequence. The calculation's contents mainly included: (a) the first stage of excavation to GL -3.5 m below the surface and (b) the second stage of excavation to GL -6.75 m below the surface. In the 2D finite element analysis, the mesh used for the calculation contained six groups (three soil layers, the excavation, and the support), 4566 nodes, 2210 elements (2209 triangles with 6 points, T6 elements, and 1 bar element), as shown in Figure 7.  
Figure 7a shows the finite element mesh of the completed vertical wall and retaining pile and of the not-yet-excavated gravel layer. In addition, the finite element meshes of the first-stage excavation and the second-stage excavation are shown in Figure 7b,c, respectively.