## 3 Results and Discussion
### Lining Damage Under Different Initial Fault Displacements  
In order to study the damage response of the tunnel under the superimposed seismic action with different initial fault displacements, the initial fault displacements in Case 4 and Case 5 are set to be 5 cm and 10 cm, and the same seismic excitation is applied to the model.  
As depicted in Figure 12, with escalating initial fault displacement, the compression damage intensified, predominantly clustering around the tunnel vault and the tunnel invert at the fault plane, with a proclivity to extend toward the side walls. Concurrently, severe tensile damage afflicted the tunnel lining across all three cases, exhibiting a consistent distribution pattern: a heightened damage concentration adjacent to the fault plane, gradually diminishing as distance from the fault plane increased.  
In order to further evaluate the damage level of the tunnel under the action of different initial fault displacements, the distribution patterns of OLDC and OLDT along  
Figure 11: Overall lining damage under different cases. (**a**) OLDC. (**b**) OLDT.  
Figure 12: Lining damage under different initial fault displacements. (**a**) Compression damage. (**b**) Tensile damage.  
the axial direction of the tunnel for different working cases are obtained according to Equations (8) and (9), as shown in Figure 13.  
Figure 13a illustrates the distribution of OLDC along the axial direction of the tunnel under different initial fault displacements. It is evident from the figure that fault dislocation remains the primary determinant of lining compression damage. As the distance of fault dislocation increases, the degree of damage to the lining escalates. When the initial displacement is 5 cm, the maximum OLDC value is 0.50, reaching moderate damage (DS2). Notably, the lining experiences the most severe damage at the fault surface, with damage gradually diminishing with distance from the fault plane. The damage zone is primarily concentrated within a 14 m radius on both sides of the fault plane.  
Figure 13b depicts the OLDT distribution along the tunnel's axial direction with different initial fault displacements. As can be seen from the figure, with initial displacement, some tunnel areas reach the full damage level (DS4) in all cases. Furthermore, when fault dislocation and seismic forces are combined, the OLDT distribution within the lining is consistent, showing the most severe tensile damage from \(-20\) m to 30 m adjacent to the fault plane. Notably, the maximum OLDT value escalates with the initial fault displacement.  
The analysis of tunnel damage with different initial fault displacements reveals that the initial displacement of the fault has a significant impact on the damage characteristics of the tunnel. Under identical seismic excitations, the compressive damage to the tunnel becomes more severe as the initial fault displacement increases. By comparing the OLDC distribution patterns of the tunnel in Case 1, it is evident that the initial damage caused by fault dislocation amplifies the effects of seismic action, leading to a noticeable increase in the maximum compressive damage of the tunnel.  
On the other hand, the increase in initial fault displacement does not expand the distribution range of the primary compressive damage to the tunnel. This damage remains concentrated within a 14 m radius on both sides of the fault plane, providing valuable reference information for determining the seismic fortification range of this project.  
The tensile damage to the tunnel is significantly more severe, with a distribution pattern primarily characterized by circumferential tensile damage. This indicates that the tunnel is subjected to strong axial tensile stresses. Therefore, in the design of tunnels crossing active fault zones, particular attention should be paid to effectively improving the axial stress state of the tunnel.  
Figure 13: Overall lining damage under different initial fault displacements. (**a**) Compression damage. (**b**) Tensile damage.