## 4 Results and Discussion
### Forced Vibration with Different Monitoring Positions  
This section explores the displacement response of different monitoring points under a position-fixed harmonic point load. The model parameters and boundary conditions are the same as in Section 4.1. The relative position of the load point is 0.25. The location of the crack and monitoring points are mentioned in Section 2.5. Figure 8 depicts the displacement response of the undamaged tunnel lining subjected to harmonic point excitation. Figure 8a, b, and c represent the radial, circumferential, and axial displacement responses. The vertical axis represents the amplitude of the displacement response. The horizontal axis represents the angular distribution of monitoring points along the circumferential section. Each curve within the figure corresponds to the monitoring points' specific axial relative position.  
Figure 7: Influence of multiple crack relative positions on natural frequency characteristics: (**a**) first order, (**b**) second order, (**c**) third order.  
In Figure 8a, the radial displacement's maximum amplitude at monitoring points is 0\({}^{\circ}\) and 180\({}^{\circ}\) in the circumferential angle. Figure 8c reveals that the maximum amplitude of axial displacement occurs at the same location. Meanwhile, in Figure 8b, the circumferential displacement response differs from the other two directions, exhibiting the minimum value at the corresponding monitoring position. The displacement response of the cracked lining is consistent with the shape and maximum amplitude position in Figure 8. Therefore, this section does not present similar results for cracked tunnel linings. According to the displacement response and the crack's location, the radial and axial displacements are suitable results for further analysis.  
Figure 9 illustrates the impact of cracks on the displacement response at the monitoring point at a circumferential angle of 0\({}^{\circ}\). Figure 9a and b depict the radial and axial displacement responses, respectively. The figure displays three distinct relative crack depths: 0.3, 0.4, and 0.5, each corresponding to a different curve. The vertical axis represents the difference between the displacement response amplitudes of intact and cracked tunnel linings. Notably, the displacement response amplitudes of the cracked tunnel lining in this section are greater than those of the undamaged tunnel lining. Meantime, the horizontal axis denotes the axial relative position. In Figure 9a, the difference in axial displacement amplitudes corresponding to each axial relative position expands with increasing crack depth. Moreover, as the relative position increases, the amplitude difference diminishes, and this diminishing trend becomes more prominent with increasing crack depth. While the trends in amplitude differences between the axial and radial directions are similar, the extent of change in the axial direction is smaller when compared to the radial direction. The  
Figure 8: Displacement response amplitudes of the undamaged tunnel lining subjected to a time-harmonic normal point load: (**a**) radial direction, (**b**) circumferential direction, (**c**) axial direction.  
findings presented in this section reveal that cracks diminish the local stiffness of the tunnel lining, leading to an increase in the displacement response. As a result, differences in the displacement response amplitudes emerge between the undamaged and cracked linings. These discrepancies progressively diminish with increasing distance from the crack. The variance in displacement between the cracked and undamaged regions helps identify the crack's circumferential location, with the circumferential position displacement becoming more pronounced as the monitoring point approaches the crack location.