## 3 Numerical Modeling
### Support Conditions  
The support structure in the shallow-buried eccentric pressure section is a composite lining structure consisting of a system anchor rod, shotcrete, grid steel frame as the initial support, and secondary lining concrete. D25 hollow grouting anchors (L = 4 m/6 m (ring) and 75 \(\times\) 60 (longitudinal)) are installed in the tunnel arch and side walls. The \(\varphi\)8 steel mesh (20 \(\times\) 20 cm spacing), C25 shotcrete (30 cm thick), and 20a I-steel support (60 cm spacing) are also used. The secondary lining is a 55 cm-thick C35 reinforced concrete. The supporting structure design is shown in Figure 5.  
The stratigraphy of the portal section of the Guanyin Mountain Tunnel is V-grade surrounding rock. The left tunnel adopted the circular reserved core soil method, and the right tunnel adopted the CD method. The stagger distance between the left and right tunnel is less than 45 m. The secondary lining inverted arch and the inverted arch backfill layer must follow the excavation face. The secondary lining should be constructed in time after the initial support falls to the bottom. The distance from the secondary lining to the excavation surface should not exceed 20 m. The tunnel construction sequence is shown in Figure 6. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Material Properties**} & \multirow{2}{*}{**Clay**} & \multicolumn{2}{c}{**Strongly**} & **Moderately** \\  & & **Weathered** & **Weathered** \\  & & **Rock** & **Rock** \\ \hline Unit weight, \(\gamma\) (kN/m\({}^{3}\)) & 18.0 & 23.5 & 25.6 \\ [PERSON]’s modulus/\(E\) (MPa) & 16 & 255 & 986 \\ Poisson’s ratio/\(v\) & 0.40 & 0.35 & 0.30 \\ Cohesion/\(c\) (kPa) & 25 & 50 & 200 \\ Internal friction angle/(c) & 16 & 35 & 50 \\ Hoek-Brown parameter/\(a\) & / & 0.5313 & 0.5114 \\ Hoek–Brown parameter/\(m_{b}\) & / & 0.287 & 0.574 \\ Hoek–Brown parameter/\(s\) & / & \(4.54\times 10^{-5}\) & \(3.35\times 10^{-4}\) \\ Saturated uniaxial compressive strength/(MPa) & / & 8.5 & 26.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic parameters of tunnel rock formations. Figure 5: Tunnel section and lining structure design. According to the structural design, the initial support was simulated by the built-in shell structure unit. The structural and mechanical parameters were equivalent by the equivalent principle of compressive stiffness [27]. The secondary lining structure was simulated by the zone element and isotropic elastic constitutive model, with a thickness of 0.55 m. The parameters were selected according to the concrete material properties. The equivalent elastic modulus of initial support is based on Equation (6):  
\[E_{eq}=\frac{E_{c}A_{c}+E_{s}A_{s}}{A_{c}+A_{s}} \tag{6}\]  
where \(E_{c}\) and \(E_{s}\) are the elastic modulus of the shotcrete and the steel arch, respectively; and \(A_{c}\) and \(A_{s}\) are the equivalent unit cross-sectional areas of shotcrete and the steel arch, respectively. The density equivalent relation for the initial support is based on Equation (7):  
\[\rho_{eq}=\frac{\rho_{c}V_{c}+\rho_{s}V_{s}}{V_{c}+V_{s}}=\frac{\rho_{c}A_{c}+ \rho_{s}A_{s}}{A_{c}+A_{s}} \tag{7}\]  
where \(\rho_{c}\) and \(V_{c}\) are the density and the equivalent unit volume of the shotcrete, respectively; and \(\rho_{s}\) and \(V_{s}\) are the density and the equivalent unit volume of the steel arch, respectively. The parameters of initial support and secondary lining are given in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Support Items**} & **Equivalent** & **Equivalent** & \multirow{2}{*}{**Thickness/m**} & **Poisson’s** \\  & **Density** & **Elastic Modulus** & & **Ratio/\(v\)** \\  & **\(\rho_{eq}\)/(kg/m\({}^{3}\))** & **\(E_{eq}\)/(GPa)** & & \\ \hline Initial support & 2350 & 28.0 & 0.30 & 0.25 \\ Secondary lining & 2500 & 32.5 & 0.35 & 0.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic parameters of tunnel structure material after equivalence. Figure 6: Construction sequence of grade V surrounding rock section of tunnel.