## 4 Numerical Simulation of Excavation and Support
### Peripheral Displacement of Tunnel  
The model with the optimal bench length of 12 m, was selected for the simulation calculations here. When the upper bench face of the tunnel was excavated to different positions, the relationship between displacement and the time-history of the characteristic points (vault) of the tunnel monitoring section were sampled and recorded. When the tunnel was half excavated, the displacement data of the characteristic points were selected when the distance between the monitoring section and the upper bench face was \(-16\) m (\(-2\) D), \(-8\) m (\(-1\)D), \(-4\) m (\(-0.5\) D), \(-0\) m (D), \(8\) m (\(1\) D), \(16\) m (\(2\) D) and \(32\) m (\(4\) D), respectively. The curves of the relationship between displacement and time-history are shown in Figure 8. Displacement at monitoring points of the tunnel in the trial propulsion stage is shown in Table 4. The negative sign indicates that the monitoring section is in the range of unexcavated rock. Figure 7: Distribution of peripheral displacement under different bench lengths: (a) 20 m bench length; (b) 15 m bench length; (c) 10 m bench length; (d) 5 m bench length. Figure 8: Time-history curve of vault settlement at monitoring points at different distances from upper tunnel face: (**a**) the distance is \(-\)2 D; (**b**) the distance is \(-\)1 D; (**c**) the distance is \(-\)0.5 D; (**d**) the distance is 0 m; (**e**) the distance is 1 D; (**f**) the distance is 2 D; (**g**) the distance is 4 D; (**h**) curve in total time history. As can be seen from Figure 8 and Table 4, the vertical displacement of the vault appears at the monitoring point of \(-2\) D. Displacement reaches almost the maximum at the monitoring point of 4 D. The displacement release rate is the highest at the tunnel face (i.e., the maximum slope of the curve). Most of displacement occurs after the tunnel face is propelled through the monitoring point. The statistical data of displacement and the time-history of the characteristic points of the monitoring sections were analyzed by the same method. When the tunnel face was propelled to the monitoring point, the displacement at this point reached about 11.5% of the total displacement (the displacement release rate was 11.5%). When the tunnel face was propelled at 2 D in front of the monitoring point, the displacement release rate reached about 80%. Meanwhile, analyses found that the change law of the curve was quite different below and above 0.5 D, so it is hard to fit with only one curve. Therefore, the curve is divided into two parts for fitting, using the curve Equations (14) and (15), respectively. \[\text{Part 1}:\ \text{y}=0.0023x^{5}+0.029x^{4}+0.1195x^{3}+0.22x^{2}+02154x+0. 1197 \tag{14}\]  
\[\text{Part 2}:\ \text{y}=0.0033x^{5}-0.0468x^{4}+0.2602x^{3}-0.7558x^{2}+1.30 53x-0.1977 \tag{15}\]  
The fitting curve is shown in Figure 9. Using Equations (14) and (15), the displacement release rate of any point at different distances from the tunnel face can be found. For example, the displacement release rate is 57% at 1 D and 85% at 2 D. Therefore, the displacement can be effectively controlled if the second lining is constructed in time at a position within 2 D (16 m) behind the tunnel face. However, the construction of second lining should not occur too early. If it is constructed too early, it will not make full use of the bearing capacity of the rock mass and this will make the lining bear too much of the rock mass load. This may lead to the lining cracking due to excessive stress, which will affect the durability and safety of the tunnel structure. The surrounding rock of a soft rock tunnel is prone to large deformation after excavation. In order to prevent the initial support from invading the clearance after deformation, the excavation contour must be enlarged along the radial direction. The size of the enlarged radial direction is called the reserved deformation. Theoretically, the reserved deformation should be slightly larger than the total deformation around the tunnel in the process of