## 3 Design Analysis of Curtain Grouting  
### Calculation Model and Operating Conditions  
Curtain grouting uses medium and low pressure to inject pressure. The pressure of grouting slurry enters the fractures, cracks, and voids of the broken rock stratum, soft sand rock layer, and other strata. Meanwhile, certain water will permeate through the advanced grouting holes on the working face. After solidification, the rocky soil or particles are cemented into a whole to form a plugging layer of grouting water. The Yongfuttun tunnel section with the stake number of YK66+490 is selected for the establishment of the 3D model. The 3D model is divided into element grids according to the 1 m element size. The solid elements are hexahedrons, and the grids are set in combination with the calculation steps. The surrounding rock, grouting area, and initial support are simulated by the solid elements, and the secondary lining is simulated by the shell element. With Mohr-Coulomb (M-C) adopted as the constitutive relationship between the surrounding rock and grouting area, other materials are all elastic. The material parameters of the initial support, which are composed of shotcrete and a steel arch frame, are obtained by the equivalent stiffness method (E\({}_{1}\)I\({}_{1}\) = E\({}_{2}\)I\({}_{2}\)). As for the mechanical boundaries (see Figure 6), the front, rear, left, and right boundaries are applied with normal constraints. The lower boundary is applied with a fixed constraint. The bottom of the seepage boundary is impermeable, and the other boundaries are permeable. The model is a 3D finite element stratum model. Its width is 45 m in the X direction, 45 m in the Y direction, and 130 m in the Z direction. The net height and net width of the simulated tunnel are 10.02 m and 6.44 m, respectively (symmetrical); \begin{table}
\begin{tabular}{c c c} \hline \hline
**Grouting Range (m)** & **Bending Moment (kN.m)** & **Axial Force (kN)** \\ \hline
0 & 219 & 1602.26 \\
2 & 148 & 1079.51 \\
4 & 85.4 & 614.95 \\
6 & 75.3 & 531.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Minimum and maximum internal forces of secondary lining with different grouting range. 4.41 times the maximum span of the tunnel is taken in the X direction. The tunnel height (approximately 30 m) is taken 3 times vertically downward and 130 m of the tunnel height is taken vertically upward. Moreover, the simulated hydrostatic head is 90 m.  
In the finite element model, 1.5 m is selected as the cyclical footage. The bench length is 3.0 m. The longitudinal length of the tunnel is 36 m. The analysis model adopts the bench method according to the construction scheme; the upper bench is excavated every 1.5 m, and the analysis load step is gradually simulated and established according to the excavation and support conditions. The calculation is completed during simulation of one board lining and the secondary lining. The bench method includes the following procedures: firstly, the front soil is conducted with full curtain grouting for the construction of an advanced anchor bolt. After the excavation of the upper part, the arch of the upper bench is conducted with initial support (two series of initial support are carried out separately). In this case, the upper bench is 12 m long. The side walls are conducted with initial support during the lower bench excavation. In this way, initial support is implemented close to a ring. The secondary lining can be conducted when the heading face is pushed to a certain distance. This is carried out with a stable initial support. The displacement boundary and stress boundary are two parts of the setting for model boundary conditions. The displacement boundary condition is used in this simulation. In other words, the longitudinal displacement of the model is fixed at the front and back along the tunnel excavation. The horizontal displacement of the model is horizontally fixed on the left and right sides. The vertical displacement is fixed vertically on the lower surface.