## 4 Results and Discussion
### Natural Frequency Characteristics  
Cracks cause a reduction in the stiffness of the structure. Decreasing stiffness can reduce the natural frequency of the structural system. This section shows and discusses the effects of crack depth and position on the natural frequencies of the shield tunnel lining. The natural frequencies of the cracked lining in the results are smaller than those of the ideal lining. Moreover, the shield tunnel lining is a symmetrical structure. Hence, locating the cracks only in the left or right half of the lining is a reasonable method. Figure 6 illustrates the relationship between the first three orders of natural frequency and crack relative depth for a partly circumferential crack. The vertical coordinate is the difference between the natural frequencies of the ideal and cracked linings. The crack's three relative positions, 0.3, 0.4 and 0.5, correspond to the three curves in the figure. In Figure 6a, as the crack's relative depth increases, the tunnel lining's first-order natural frequency shows a downward trend. The natural frequency with a relative position of 0.5 decreases the fastest among them. When the crack position moves from the middle to both ends of the lining, the trend of the first-order natural frequency changing with the relative depth slows down. Figure 6b presents the results for second-order natural frequency. In contrast to the first-order one, as the crack position approaches either end, the trend of the second-order natural frequency accelerates concerning the crack's relative depth. Notably, the second-order natural frequency of the tunnel lining, when the crack is at a relative position of 0.5, is almost unchanged as the relative depth increases. In  
Figure 5: Vibration response validation of the shield tunnel lining with different crack depths based on radial displacement amplitudes of circumferential monitoring points. the third-order mode, as illustrated in Figure 6c, all such frequencies decline as the crack depth increases. The natural frequency experiences the most significant reduction when the relative crack position is at 0.5. Conversely, the natural frequency gradually decreases as the crack position approaches either end of the tunnel lining. Additionally, the third-order natural frequency curves for relative positions of 0.3 and 0.4 closely resemble each other. Figure 7 demonstrates the correlation between the initial three orders of natural frequency and the relative position of a partly circumferential crack. The four curves in the graph represent the relative depths of 0.2, 0.4, 0.6, and 0.8. In Figure 7a, the curves of first-order natural frequency all exhibit a parabolic shape. A reduced relative depth of the crack leads to a gentle curve and decreases effects on first-order natural frequency. Meanwhile, all four curves reach their maximum values when the crack is positioned in the central section of the tunnel lining. As the crack approaches either end of the lining, the rate of decrease in the first-order natural frequency becomes gradual. In Figure 7b, the middle position of the tunnel lining is a second-order mode node. Consequently, the second-order natural frequency remains consistent at a relative position of 0.5, regardless of varying crack depths. Furthermore, there is a maximum value on either side of the tunnel lining. This placement of the crack results in the most pronounced variation in the second-order natural frequency. Figure 7c reveals two nodes corresponding to the third-order mode, exhibiting similar characteristics to the second-order mode node. Moreover, there are maximum values on either side of each node. These peak values pinpoint the most sensitive crack  
Figure 6: Influence of multiple crack relative depths on natural frequency characteristics: (**a**) first order, (**b**) second order, (**c**) third order. positions for the third-order natural frequency. Also, as the crack depth increases, the sensitivity of the natural frequency intensifies.