## 5 Conclusions  
This paper explores the dynamic response of a shield tunnel lining with a partly circumferential crack under a harmonic point load. The dynamic model is established based on the Flugge shell theory. The wave propagation method is employed to solve the dynamic model and obtain the free vibration characteristics of the tunnel lining. Subsequently, the residue theorem is applied to acquire the forced vibration response. Additionally, the simulation of a partly circumferential crack utilizes the LSM. The displacement response around the crack is ultimately determined by employing the wave superposition principle. The methodology enables the investigation of the evolution law of vibration responses in shield tunnel linings with various crack morphologies. The conclusion has three parts:  
1. The partly circumferential crack reduces the natural frequency of the shield tunnel lining because it causes local flexibility in the lining. Moreover, the rise in crack depth accelerates the decrease in natural frequency. Meanwhile, when the partly circumferential crack is positioned at a modal node, the crack minimally impacts the natural frequency of this mode. In contrast, when the crack is located in the middle  
Figure 12: Influence of various relative load positions on the displacement response of the monitoring point located at an axial relative position of 0.8 and a circumferential angle of 0°: (**a**) radial direction, (**b**) axial direction. of two nodes, the natural frequency of this mode undergoes the most significant variation. Therefore, the partly circumferential crack intensifies the fluctuation of natural frequency changes in high-order modes. This finding can aid in identifying the axial location of the crack and evaluating its depth;
2. The monitoring point at \(0^{\circ}\) circumferentially experiences the highest amplitudes of displacement response in both radial and axial directions under a time-harmonic normal point load. Hence, the displacement response in both directions can be used to analyze a partly circumferential crack at the \(0^{\circ}\) circumferential position. The radial and axial amplitude differences show a downward trend on the other side of the crack as the distance from the monitoring point to the partly circumferential crack increases. In the meantime, the difference in radial displacement amplitudes exceeds that of axial displacement amplitudes. Furthermore, as the crack deepens, the scope in displacement amplitude difference increases. The variation in displacement response occurs in the region affected by the partly circumferential crack. Consequently, the alteration in radial and axial displacement amplitudes helps identify the crack's circumferential position, and the radial displacement is more effective than the axial one;
3. When the time-harmonic normal point load is near the crack, the radial amplitude difference on the other side of the crack initially increases and subsequently decreases. This phenomenon diminishes progressively as the load position moves away from the crack. Since the amplitude difference of the required monitoring points needs to have evolution laws, the axial relative positions of the monitor should be 0.65, 0.7, 0.75, and 0.8. As the distance between the load point and the crack decreases, the radial and axial amplitude differences of the selected monitoring points show a downward trend. Moreover, as the monitoring point relocates away from the crack, the range of variation in amplitude differences gradually decreases. Meanwhile, the deepening crack exhibits an enlarged variation range in radial and axial displacements. The results contribute to the identification of the crack's circumferential position, providing a basis for selecting the load point and corresponding monitoring point locations. This paper explores the dynamic response of the independent shield tunnel lining. Future research could integrate soil into the model to investigate the dynamic response of the lining structure within the tunnel-soil system.