## 3 Numerical Simulation of Tunnel Excavation without Grouting
### Mechanical Response Analysis of I-Type Section Tunnel
This is due to the enhanced mechanical properties of the backfill body, resulting in a greater load than the surrounding rock in the backfill body, while the shear stress in the limestone region remains unchanged, the greater shear force is produced. Figure 8: Mechanical response of I-type tunnel on Y = 10 m section (**a**) bending moment (**b**) torque (**c**) deformation and (**d**) max principle of support. Figure 8b shows the torque distribution of Y = 10 m section before and after grouting in the I-type tunnel. The torque in the limestone region is almost 0, while large torque is generated in the backfill and contact region, where the minimum negative torque of \(-\)79.95 kN \(\times\) m is occurred at the right arch foot, and the maximum positive torque is produced at the M with a value of 66.71 kN \(\times\) m. After grouting, the torque distribution at the primary support is almost 0. Calculate the torque distribution by using Kirchhoff thin plate theory  
\[M_{yx}=\int_{-\frac{t}{2}}^{\frac{t}{2}}z\tau_{yx}dz=-\frac{Et^{3}}{12(1-\mu^ {2})}\frac{\partial^{2}w}{\partial x\partial y} \tag{3}\]  
where \(E\) and \(t\) are the elastic modulus and thickness of primary support, \(y\) is the direction along the tunnel excavation, \(z\) is the direction of the plate thickness, \(x\) is the direction determined by the right-hand rule, \(\mu\) is the Poisson's ratio, and \(w\) is the deflection of the plate. Equation (3) shows that the torque of the primary support is related to the deflection. In the current tunnel support design method, the effect of torque is often neglected, because the tunnel is considered as a plane stain problem in the design. Therefore, \(\partial w/\partial x=0\) in the x-direction and the torque is 0, the effect of torque can be ignored in the design process. When the tunnel is excavated in the karst cave backfilled with engineering spoil, the mechanical property of the backfill body is very weak, resulting in complex forces on the primary structure and generating a non-negligible torque. After grouting, the deformation along the x-direction decreases and the torque decreases to negligible, and the effect of torque on the primary support can be disregarded. Figure 8c shows the deformation of the primary support on the Y = 10 m section before and after grouting in the I-type tunnel. The deformation of the primary support in the limestone region is almost 0, indicating that the limestone has good mechanical properties. The surrounding rock in the limestone region can withstand large loads during the stress redistribution caused by the tunnel excavation, which improves the mechanical characteristics of the primary structure. In the backfill region, the maximum deformation generated in the tunnel bottom and second step foot are 23.61 mm and 18.64 mm, respectively. The maximum deformation generated in the first step foot K is 17.25 mm. The reason for the large deformation in the tunnel bottom is that the bearing capacity is insufficient due to the weak mechanical properties of the surrounding rock, and the primary support bears a large load. The large deformation at the excavation step foot is the result of the stress release caused by the lagging construction of the primary support. After grouting, the improvement of the mechanical properties of the surrounding rock improves the bearing capacity, which reduces the primary support deformation at the tunnel bottom to 0. The large deformation of the primary support at the excavation step foot is decrease, the maximum deformation are only 12.35 mm at K.  
Figure 8d shows the max principle distribution on the Y = 10 m section in the I-type tunnel, the negative value means to the compression state of the primary support. The large maximum principle is distributed in the limestone region, the maximum value occurs at the left tunnel shoulder of \(-\)17.40 MPa. The maximum principle in the backfill region is small, the maximum value in the backfill region are only \(-\)5.77 MPa.