# Estimation of Rock Load of Multi-Arch Tunnel with Cracks Using Stress Variable Method
## 3 Numerical Parametric Study of Three-Arch Tunnel Excavation for Rock Load Estimation
### Numerical Modeling for Three-Arch Tunnel  
Through numerical simulations, the rock load was estimated from various overburden heights and compared with the empirically calculated load. Two-dimensional numerical analyses in plane-strain conditions were conducted to analyze the development of rock load according to rock mass classification, overburden, and excavation sequence. The commercial finite-difference software, FLAC (version 7.0), was utilized with an explicit solution method. Note that the tunnel was sufficiently long to ignore the strain in the longitudinal direction. A single homogeneous rock mass was applied to the surrounding rock to investigate the difference in rock load according to the rock conditions and overburden. Figure 6 presents the mesh and boundary conditions for the analysis. The boundary condition at the bottom was fixed in the x and y directions, and both sides were fixed only in the x direction. A sufficient distance from the tunnel to the end of the model was used to eliminate the boundary effect. Table 1 presents the mechanical properties of each rock mass class that were obtained from field investigations and laboratory tests [16]. The rock mass was modeled as an isotropic linear elastic material with the Mohr-Coulomb failure criterion to describe the plastic behavior. In the plastic behavior, associated flow rule was employed. Note that no external load was applied and only gravity force was continuously imposed during the whole simulation process. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Rock Mass Class.** & **Unit Weight (kN/m3)** & **Cohesion (kPa)** & **Friction Angle (\({}^{\circ}\))** & **Deformation Modulus (MPa)** & **Poisson’s Ratio** \\ \hline III & 25.0 & 1020 & 38.0 & 5000 & 0.25 \\ IV & 23.0 & 500 & 35.0 & 2500 & 0.28 \\ V & 21.0 & 100 & 33.0 & 1000 & 0.30 \\ \hline \end{tabular}
\end{table} 
Table 1: Mechanical material properties of different rock mass classes. Figure 5: Three-arch tunnel: (**a**) air view (**b**) cross-sectional view. Figure 6: Two-dimensional model showing the finite mesh and boundary conditions for the analysis. Figure 7 presents a schematic of the parameters considered in this study, where the rock load, overburden, and tunnel width are represented by h, H, and D, respectively. A total of 21 cases of numerical simulations were conducted, involving three rock mass classes with seven overburdens ranging from 10 to 70 m. The rock load was estimated based on a factor of safety of 3.0 [8] and a stress variable of 10% [9], as suggested in previous literature. The numerical simulation process contains 19 construction steps, as presented in Table 2. All the tunnels in the multi-arch tunnel were excavated using the partial face excavation method, starting with the excavation of the central tunnel, followed by the excavations of the left and right tunnels. The concrete lining was applied only to the central tunnel in the simulation. The shotcrete being used as the tunnel support was simulated in two stages--soft shotcrete (approximately three-hour strength) and hard shotcrete (one-day strength)--depending on its hardening with respect to time. The shotcrete was modeled as a beam element behaving in an elastic manner. To consider the three-dimensional effect in a two-dimensional simulation of tunnel excavation, the load distribution ratio was used as 45%, 35%, and 20% for the three consecutive stages: excavation, soft shotcrete, and hard shotcrete, respectively. As the compressive strength of the shotcrete increased with time, two different elastic moduli were adopted, as presented in Table 3, where the elastic moduli of the soft and hard shotcretes were three-hour and one-day compressive strengths, respectively.