## 3 Numerical Simulation Model
### Material Constitutions and Parameters  
The numerical simulation model consists of the stratum and retaining structure. In the assumptions above, the stratum is assumed to be isotropic and continuous, and its constitutive model adopts the Mohr-Coulomb model [18; 19]. Stratum material parameters are determined according to geological exploration data and laboratory test, as shown in Table 2. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Serial** & **Angle (\({}^{\circ}\))** & **Length (m)** & **Preload** & **Diameter** & **Anchor** \\
**Number** & & & **Force (kN)** & **(m)** & **Point Depth** \\ \hline
1 & 8 & 55.5 & 450 & 0.15 & \(-\)3.0 \\
2 & 8 & 63.5 & 450 & 0.15 & \(-\)5.0 \\
3 & 8 & 56 & 450 & 0.172 & \(-\)7.0 \\
4 & 8 & 22 & 450 & 0.15 & \(-\)9.0 \\
5 & 15 & 29.5 & 650 & 0.172 & \(-\)11.0 \\
6 & 15 & 26 & 650 & 0.172 & \(-\)13.0 \\
7 & 15 & 23 & 650 & 0.172 & \(-\)15.0 \\
8 & 15 & 22 & 650 & 0.15 & \(-\)17.0 \\
9 & 15 & 20 & 650 & 0.15 & \(-\)20.0 \\
10 & 15 & 20 & 650 & 0.15 & \(-\)23.0 \\
11 & 15 & 20 & 650 & 0.15 & \(-\)26.0 \\
12 & 15 & 20 & 650 & 0.15 & \(-\)29.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of anchor cables. Figure 5: Numerical simulation model. (**a**) Overall model. (**b**) Support structure model. The retaining structure includes the cast-in-place pile, waist beam, ribbed beam, crown beam and anchor cable. The cable element of the software is used to simulate the anchor cable. Conventional elastic or elastic-plastic models cannot clarify the constitutive model of concrete, while the constitutive model combined with damage mechanics can clarify the physical properties of concrete. In Abaqus, the Concrete Damaged Plasticity model was developed by [PERSON], [PERSON] and [PERSON] [19; 20; 21; 22]; its abbreviated form is the CDP model. A large number of studies show that the CDP model is applicable to concrete materials and can reflect the characteristics of the loading curve of concrete. Therefore, the CDP model is adopted for the concrete constitutive model in this study [23; 24]. The stiffness degradation of concrete materials under uniaxial tension and compression is defined by two independent uniaxial damage variables, namely the tensile damage factor \(D_{\rm t}\) and the compressive damage factor \(D_{\rm c}\). The stress-strain relationship for concrete materials in tension and compression are shown in Figure 6, and the corresponding mathematical expressions of the intrinsic model are as follows:  
\[\sigma_{t}\ =\ (1\ -\ D_{\rm t})E_{0t}(\varepsilon_{t}\ -\ \varepsilon_{t}{}^{p}) \tag{1}\]  
\[\sigma_{c}\ =\ (1\ -\ D_{\rm c})E_{0c}(\varepsilon_{c}\ -\ \varepsilon_{c}{}^{p}) \tag{2}\]  
where \(\sigma_{t}\) and \(\sigma_{c}\) are the tension stress and compressive stress, respectively; \(D_{\rm t}\) and \(D_{\rm c}\) are tensile and compressive damage parameters, respectively; \(E_{0t}\) and \(E_{0c}\) are initial tensile and compressive elasticity tensor, respectively; \(\varepsilon_{t}\) and \(\varepsilon_{c}\) are total tension strain tensor and total compressive strain tensor, respectively; \(\varepsilon_{t}{}^{p}\) and \(\varepsilon_{c}{}^{p}\) are tensile and compressive plastic strain tensors, respectively. This constitutive model is often used to simulate the concrete materials in the structure. The parameters of the retaining structure are shown in Table 3. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & **Material** & \
\begin{tabular}{c} **Elastic Modulus** \\ **(MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Natural Heavy \(\gamma\)** \\ **(KN/m3)** \\ \end{tabular} & **Poisson’s Ratio** & **Size (m)** \\ \hline Waist beam & Concrete & 35 & 23.5 & 0.28 & 1.0 \(\times\) 1.4 \\ Ribbed beam & Concrete & 35 & 23.5 & 0.28 & 0.4 \(\times\) 0.4 \\ Crown beam & Concrete & 35 & 23.5 & 0.28 & 1.0 \(\times\) 1.4 \\ Cast-in-place pile & Concrete & 35 & 23.5 & 0.28 & Diameter 1.4 \\ Anchor cable & Cable & \(200\times 10^{3}\) & - & 0.20 & Diameter 0.15 or \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of retaining structure.