# Analysis of the Interaction Effects of Shield Structure Oblique Passing under an Existing Tunnel
## 6 Effect of Spatial Intersection Angle between New and Existing Tunnels
### Existing Tunnel Settlement  
Figure 19 shows the settlement of the existing tunnel vault and arch bottom at different spatial intersection angles. From Figure 19a, it is evident that with the increase in the spatial intersection angle \(\beta\), the maximum change in the settlement value of the vault position of the existing tunnel is 2.06 mm, with an increase ratio of about 341.4%. This indicates that the settlement of the existing tunnel vault, due to the excavation of the new shield tunnel, is greatly influenced by the spatial intersection angle \(\beta\). As seen in Figure 19b, as the spatial intersection angle \(\beta\) increases from 0\({}^{\circ}\) to 90\({}^{\circ}\), the minimum settlement at the bottom of the existing tunnel arch appears at \(\beta\) = 90\({}^{\circ}\), while the maximum settlement is located at \(\beta\) = 0\({}^{\circ}\). The maximum change in settlement at the bottom of the arch is 2.44 mm, with an increase ratio of about 317.5%. To sum up, the spatial intersection angle \(\beta\) has a great influence on the settlement at the bottom of the existing tunnel arch and at the top of the arch. Figure 20 shows the maximum settlement and height change rates of the existing tunnel at different spatial intersection angles. From the figure, it is clear that, as the spatial intersection angle \(\beta\) increases, the maximum settlement at the vault and arch bottom of the existing tunnel also goes through four stages: rapid decrease (0\({}^{\circ}\)-15\({}^{\circ}\)), slow decrease (15\({}^{\circ}\)-30\({}^{\circ}\)), steady (30\({}^{\circ}\)-75\({}^{\circ}\)) and slow decrease (75\({}^{\circ}\)-90\({}^{\circ}\)). Since the change in the maximum  
Figure 19: Settlement in an existing tunnel at different spatial intersection angles: (**a**) vault; (**b**) arch bottom. Figure 18: Numerical simulation model diagram. settlement at the bottom of the arch is consistent with the pattern of change at the top of the arch, the rate of change in the height of the existing tunnel also goes through the same four stages. For a more convenient subsequent presentation, the different spatial intersection angles can be divided into four phases: Phase 1 (0\({}^{\circ}\)\(\sim\)15\({}^{\circ}\)), Phase 2 (15\({}^{\circ}\)\(\sim\)30\({}^{\circ}\)), Phase 3 (30\({}^{\circ}\)-75\({}^{\circ}\)), and Phase 4 (75\({}^{\circ}\)\(\sim\)90\({}^{\circ}\)). Figure 21 shows the maximum settlement increment ratio at the bottom of the existing tunnel arch under different shield construction stages, where S1 represents RT construction to half, S2 represents RT penetration, S3 represents LT construction to half with RT penetration, and S4 represents double penetration. The settlement increment ratio \(N\) is shown in the following equation:  
\[N=\frac{S_{\text{max}\beta}-S_{\text{max}90^{\circ}}}{S_{\text{max}90^{\circ}}} \tag{2}\]  
where \(S_{\text{max}\beta}\) represents the maximum settlement at the bottom of the existing tunnel arch at any spatial intersection angle, and \(S_{\text{max}90^{\circ}}\) represents the maximum settlement at the bottom of the existing tunnel arch at spatial intersection angle \(\beta\) = 90\({}^{\circ}\). The incremental settlement ratio \(N\) increases from 46.29% to 73.66% from shield construction phases S1 to S2, as shown in Figure 21, which indicates that the greater the volume loss induced by the shield tunnel excavation, the greater the impact of the spatial intersection angle on the maximum settlement at the bottom of the existing tunnel is. It is worth noting that as the shield construction phase progresses, the trend of the different phases in this process is the same as the trend of the rate of change in height shown in Figure 20. When the construction phase moves from S2 to S3, the secondary disturbance of the existing tunnel soil due to the shield construction will cause the maximum settlement increment ratio to increase cumulatively from 69.36% to 272.52%.