## 4 Results and Analysis
### Damage Assessment of Tunnel Lining  
The tensile damage contours in the longitudinal view of the tunnel lining under right-side-incident SV-waves for the different incidence angles considered are shown in Figure 14, where the selected time intervals are (i) the moment when the damage first appeared and (ii) the end time of the total duration of the input wave. As it can be seen in the longitudinal dimension, the position in which the damage first showed up is located in the tunnel interval of 60\(\sim\)150 m. This is because the existence of a local rock mountain brings about the reflection of waves (see Figures 12 and 13), and the superposition of the reflected wave and the incident wave aggravates the deformation of the lining and then damages it, whereas, in a plain site, the above-mentioned reflection and superposition of waves are absent. In the circumferential dimension, the positions are different among the cases: as the incidence angle decreases, the position in which the damage first showed up tends to move from the vault towards the hance. Since the damage of the tunnel's lining is highly related to deformation, the deformation of the lining under right-side-incident SV-waves with different incidence angles at the moment when the damage first appeared was observed, as shown in Figure 15, where the scaling factor is set to 300 so that the deformation of the lining can be investigated more clearly and two profiles are selected for each case therein: (i) the middle of the plain site with a longitudinal distance = 30 m; and (ii) the middle of the plain-mountain site with a longitudinal distance = 105 m. As it can be seen, thanks to the 300 scaling factor used, the ovaling deformation of the lining seems to be significant, and the compressive deformation is approximately perpendicular to the incidence direction of the SV-wave. To better observe the relation between the ovaling deformation of the tunnel lining and the incidence angles, the major axis (dash line) of the ovaling lining in the profile for the middle of the plain-mountain site with a longitudinal distance = 105 m is given for each case. Clearly, the inclined angle of the major axis tends to be greater with a decreasing incidence angle, meaning that the part of lining which is initially subjected to severe tension, i.e., the position where tensile damage first appears, gradually changes from the vault to the hance with a decreasing incidence angle. In addition, for each case with right-side incidence, the final damage area reduces as the incidence angle decreases. As shown in Figure 13b, when the incidence angle = 33.69\({}^{\circ}\), the tunnel lining is subjected to a directly incident SV-wave and reflected SV-waves from the soil-rock interface and the site's surface, and the difference in the angle between the incident and reflected waves essentially induces the tunnel lining to experience compressive deformation in two directions. The fact that tensile damage shows up at the arch vault, arch bottom, left arch, and right arch of the 60\(\sim\)150 m tunnel interval can well verify the above-mentioned statement. By contrast, when the incidence angle = 11.31\({}^{\circ}\) (see Figure 12b), the difference in the angle between the incident and reflected waves is much smaller, and, thus, the tunnel lining is approximately subjected to a unidirectional deformation, and, thus, tensile damage only appears in a single, diagonal direction. In summary, as the incidence angle decreases, the mode of compressive deformation changes from multi-directional compression to unidirectional compression.