## 1 Introduction
The line spring model (LSM) can describe the problem of internal and surface cracks in plates and shells. [PERSON] and [PERSON] [23] first used distributed line springs to simulate cracks. The study obtained local flexibility at the crack position according to fracture mechanics' strain energy release rate. [PERSON] et al. [24; 25] improved LSM in a flat plate. The research also used the model to describe cylindrical shells' circumferential or axial partial penetration cracks. After that, [PERSON] et al. [26] optimized LSM and brought it into the finite element models to explore the elastic-plastic response. [PERSON] [27] modeled surface cracks in pipes and elbows by LSM. The validation of the model applied finite elements and experimental results. [PERSON] [28] investigated the effect of a circumferential surface crack on the axisymmetric vibration of an anisotropic cylindrical shell. The crack modelled by LSM increases the flexibility of the cylindrical shell. [PERSON] and [PERSON] et al. [29; 30] combined LSM and the wave propagation method to analyze the vibration power flow of infinitely long cylindrical shells. Moreover, the study evaluated the effect of crack location and depth on the vibration characteristics of the shell. [PERSON] et al. [31] simulated the cylindrical shell's semi-elliptical surface cracks by LSM. The results indicated the effect of crack parameters on a cylindrical shell's natural frequency. In addition, the length of the cracks was smaller than the radius of the cylindrical shell, and the distribution was in any direction. [PERSON] et al. [11] adopted the wave propagation method and LSM to analyze the nonlinear vibration response of the thin cylindrical shell with a straight crack. This illustrates that the cooperation of these two methods is reliable. Overall, LSM effectively presents the stiffness reduction caused by cracks, thereby reflecting displacement changes in their vicinity. This model can integrate with the wave propagation method. The achievements above support the results of the dynamic analysis of a thin cylindrical shell with a crack. This facilitates the exploration of the dynamic response of the shield tunnel lining with a partly circumferential crack subjected to a normal point load. This paper introduces four innovative aspects. First, it focuses on displacement as the primary parameter for analysis, integrating methods for solving free and forced vibrations mentioned in the literature and elucidating the response mechanism in the crack-affected lining region using the LSM model. Second, it uses harmonic point loading, which differs from the commonly used loading methods in the current dynamic response research of thin cylindrical shells. Third, it addresses the crack morphology by establishing an analytical model for a partly circumferential crack located inside shield tunnels, aligning with the characteristics of tunnel damage. Lastly, it innovates in the setup of loading and monitoring positions, introducing varying loading and monitoring positions and analyzing their effects on the dynamic response monitoring of tunnel structures. In addition, this paper presents finite element models to validate the dynamic model considered with and without a partly circumferential crack. The remaining part of the paper proceeds as follows. Section 2 describes the methodology for the dynamic simulation of the shield tunnel lining with a partly circumferential crack. Then, Section 3 uses the literature and the finite element model (FEM) to validate the proposed analytical approach. After that, Section 4 shows the vibration response of the shield tunnel lining without or with a partly circumferential crack. The analysis reveals the evolution of the natural frequency and forced vibration displacement response under different crack morphologies. Section 5 proposes the main conclusion.