## 4 Results and Discussion
### Forced Vibration under Different Load Positions  
This section investigates the influence of varying load locations on the displacement response surrounding the crack. The model parameters and boundary conditions are consistent with the previous section. The locations of the crack and load points are mentioned in Section 2.5. The analysis concentrates on variations in displacement response within the crack's impact region. Accordingly, six monitoring positions are distributed along the axial direction, with each section housing a corresponding monitoring point set at a circumferential angle of 0\({}^{\circ}\). Figure 10a indicates the crack's influence on the radial displacement response of the 0\({}^{\circ}\) circumferential angle monitoring point at a relative load position of 0.45. The curves correspond to the three relative crack depths of 0.3, 0.4, and 0.5, respectively. The ordinate is the displacement amplitude difference between the cracked and undamaged lines. Meanwhile, the horizontal axis is the axial relative position. All curves increase first and then decrease. The shallower the crack depth, the lower the initial point of the respective curve and the more apparent the upward trend. This phenomenon occurs due to the interaction and superposition of the four transmitted waveforms [11]. As the distance between the monitor and the crack increases, the radial displacement amplitude of the cracked lining first increases and then decreases. Meanwhile, the amplitude difference is the amplitude of the undamaged lining minus that of the cracked lining. Therefore, the amplitude difference curve shows an initial rising and then falling trend. In addition, as the distance between the load position and the crack increases, this initially rising and then decreasing trend gradually disappears, transitioning into a trend of complete decline. Figure 10b shows the impact of the crack on the axial displacement response of the 0\({}^{\circ}\) circumferential angle monitoring point at a relative load position of 0.45. This demonstrates that the phenomena observed in the radial displacement response are absent in the axial direction. All curves exhibit a descending trend, which becomes increasingly evident as the relative depth of the crack rises. Furthermore, Figure 10 reveals that monitoring points with axial relative positions of 0.65, 0.7, 0.75, and 0.8 provide regular amplitude difference  
Figure 9: Displacement response at the monitoring points (circumferential angle 0\({}^{\circ}\)) of the shield tunnel lining with a partly circumferential crack: (**a**) radial direction, (**b**) axial direction. outcomes. This section analyzes the monitoring point results with axial relative positions of 0.65 and 0.8. Figure 11 displays the effect of load points at different locations on the displacement response of the monitoring point for the cracked tunnel lining. The monitoring point is at a circumferential angle of 0\({}^{\circ}\) and an axial relative position of 0.65. The vertical coordinate is the difference between the displacement response amplitudes of the ideal and cracked linings. The horizontal coordinate is the axial relative load position. The three relative crack depths, 0.3, 0.4 and 0.5, correspond to the three curves in the figure. In Figure 10(a), as the load point approaches the crack, the amplitude differences show an increasing trend regardless of the crack's relative depth. This upward trend intensifies with the growing relative crack depth. However, the rising trend of amplitude differences weakens when the load moves from an axial relative position of 0.4 to 0.45. Figure 10(b) shows that the amplitude difference in the axial direction follows a similar trend to that observed in the radial direction. The distinction is that the variation range in the cracked lining's amplitude differences in the axial direction is smaller than the variation range in the radial direction. Figure 12 shows the impact of the load position on another monitor's displacement response.