# Parameter Analysis on Seismic Response of Long Lined Tunnel by 2.5D Substructure Method
## 5 P-SV Wave Scattering of Long Lined Tunnel
### Influence of Different Incidence Angle on Structural Seismic Response  
Taking a circular section tunnel as an example, the tunnel section form and lining material parameters remain unchanged along the longitudinal direction. The calculation model is the same as Figure 6b. The buried depth \(H\) of the tunnel is 15 m. The thickness of the soil layer is 80 m. The outer diameter \(r_{0}\) = 5.5 m, and the inner diameter \(r_{1}\) = 5 m. Assuming that the tunnel lining maintains linear elastic deformation, the lining materials and circular tunnel parameters are shown in Table 1. The Kobe wave is selected for the incident seismic wave record, and the acceleration-time curve and spectrum curve are shown in Figure 11. For the incident with the P wave and the SV wave, the incident angle \(\theta_{v}\) ranges from 0\({}^{\circ}\) to 90\({}^{\circ}\), and the conversion angle \(\theta_{t}\) ranges from 0\({}^{\circ}\) to 90\({}^{\circ}\). The increment interval between the incident angle and the conversion angle is 15\({}^{\circ}\). The external surface seismic responses of tunnel lining are calculated under different incident angles and conversion angles.  
Figure 10: Amplification of axial displacement along the external tunnel surface: (**a**) P wave obliquely incident at 30\({}^{\circ}\) and (**b**) SV wave obliquely incident at 30\({}^{\circ}\).  
Figure 9: Amplification of radial displacement along the external tunnel surface: (**a**) P wave obliquely incident at 30\({}^{\circ}\) and (**b**) SV wave obliquely incident at 30\({}^{\circ}\).  
Figure 12 is a schematic diagram of the internal force of the lining; \(N\), \(\tau\) and \(M_{c}\) represent the axial force, shear force and bending moment of the lining in the cross section, \(T\) represents the longitudinal axial force of the lining. \(M_{v}\) and \(M_{h}\) represent the vertical and horizontal bending of the lining in the horizontal plane moment.  
By changing the incident angle of the seismic wave, the law of the maximum bending moment values, \(M_{c}\), \(M_{v}\) and \(M_{h}\) corresponding to the lining changing with the incident angle, are obtained. When the P wave is incident, as shown in Figure 13a, \(M_{c}\) increases with the increase of the incident angle and the conversion angle, and the maximum value is reached when both the incident angle and the conversion angle are \(90^{\circ}\). The \(M_{v}\) and \(M_{h}\) results are shown in Figure 13c,e, the bending moment values are mainly affected by the transition angle. Both of them demonstrate peak value at vertical incidence and the transition angle of \(90^{\circ}\). When the incident seismic wave is an SV wave, as shown in Figure 13b, \(M_{c}\) is also mainly affected by the conversion angle. The maximum lining bending moment first increases and then decreases with the conversion angle. At the same conversion angle, \(M_{c}\) increases with the increase of the incident angle. Figure 13d,f shows that the maximum value of \(M_{v}\) and \(M_{h}\) increase first and then decrease with the increase of the conversion angle under seismic waves' vertical incidence. It can be seen that when the P wave is incident, considering the seismic response, the maximum bending moment of the lining is at the conversion angle of \(90^{\circ}\). When the SV wave is incident, the unfavorable bending moment increases first and then decreases with the same incident angle. The different seismic waves' incident angles also affect the internal force of the tunnel, which provides a way for us to further analyze the seismic response of underground structures.  
Figure 11: Kobe seismic wave (**a**) acceleration\(-\)time curve and (**b**) spectrum.  
Figure 12: Schematic diagram of lining axial force, shear force and bending moment.