## 3 Three-Dimensional Numerical Model
### Finite Element Calculation Parameters
#### 3.2.3 Existing Tunnel Operational Loads  
The dynamic load of each wheel during train operation is simplified into a series of vertical concentrated forces whose size changes dynamically with time, and the artificial excitation function method is utilized to determine the train vibration load. The excitation function consists of a static load and a series of sinusoidal functions superimposed on the dynamic load [44; 45], whose expression is:  
\[P(t)=P_{0}+\sum_{i=1}^{3}P_{i}\text{sin}\,\omega_{i}t \tag{12}\]  
\[P_{i} =M_{0}a_{i}\omega_{i}^{2} \tag{13}\] \[\omega_{i} =2\pi v/L_{i} \tag{14}\]  
where \(P_{0}\) is the static wheel load; \(P_{1}\), \(P_{2}\), and \(P_{3}\) are the peak loads corresponding to the smoothness of travel, line power additional load, and waveform wear; \(M_{0}\) is the mass of the train under the springs; \(L_{i}\) is the uneven vibration wavelengths of the smoothness of travel, line power additional load, and waveform wear control conditions; \(a_{i}\) is the lost height corresponding to the uneven vibration wavelength of the three control conditions; and \(v\) is the train speed. According to the existing tunnel train operation information and comprehensively considering the relevant parameters of CRH series trainsets, the train parameters are selected as having eight cars, a length of 200 m, a speed of 200 km/h (55.5 m/s), an axle weight of 19 t, an unsprung mass of 2 t, and a static wheel weight of 95 kN on one side. According to the track unevenness laying accuracy standard (Table 7) mentioned in the literature [32], the uneven wavelength and vector height are determined under three control conditions: \(L_{1}\) = 10 m, \(a_{1}\) = 3.5 mm; \(L_{2}\) = 2 m, \(a_{2}\) = 0.4 mm; and \(L_{3}\) = 0.5 m, \(a_{3}\) = 0.1 mm. Formulas (12) and (13) are used to obtain the train dynamic loading time curve, as shown in Figure 9.  
Figure 9: Train load time curve. Figure 10: _Cont._  
By combining Figures 10 and 11, the sensitive area of the existing tunnel impact after S1 excavation is in the 26 m area on the excavation side of the A end of the new tunnel can be seen. The maximum settlement is 0.11 mm at the location of the arch of the existing tunnel and the minimum settlement is 0.05 mm at the location of the tunnel floor at the B end. The maximum settlement of the track floor is 0.09 mm and the minimum settlement is 0.05 mm, which occurs at the tunnel end. The settlement value does not change much in the sensitive area, and gradually decreases as it moves away from the sensitive area. Figure 11: _Cont._  
Figure 10: A: entrance; B: exit. Overall displacement maps of the existing tunnels at different data extraction time steps. Figure 11: _Cont._  
the track floor is 0.23 mm, the minimum settlement is 0.12 mm, and the influence-sensitive area extends to 90 m.  
After S4 excavation, the location of the maximum value of the existing tunnel settlement changed and occurs in the arch waist position on the excavation side of the new tunnel, 80 m from end A, with a settlement value of 0.39 mm, affecting the sensitive area in the 100 m area on the excavation side of the new tunnel at end A. There is no change in the location of the minimum value of the tunnel settlement and the maximum and minimum value of the track floor, with the maximum settlement of the track floor being 0.35 mm and the minimum settlement being 0.19 mm. After S5 excavation, the maximum settlement occurs in the new tunnel excavation side arch waist position, 85 m from end A, and the settlement value is 0.55 mm. The maximum settlement of the track base plate occurs 90 m from end A, and the settlement value is 0.45 mm; the tunnel settlement sensitive area moves from end A to the middle of the tunnel. After S6 excavation, the existing tunnel settlement pattern is the same as S5, the maximum settlement value of the tunnel and track surface moves to the middle of the tunnel, and the sensitive area of the tunnel settlement moves to the middle of the tunnel from the A end, and the settlement changes in the middle of the tunnel are obvious.