## 1 Introduction
The results of these previous physical model tests and three-dimensional simulations, however, are only applicable to the specific cases researched in the above-mentioned studies. When the construction conditions change, the previous research results cannot be directly used to quantitatively evaluate the response of the existing tunnel to ground surcharge. Therefore, a simplified analytical method that can be used to directly predict the tunnel response to ground surcharge is urgently required. The theoretical method has the advantages of a clear concept, convenient implementation, and time effectiveness, and it is often used to quickly predict the deformation response of a tunnel subjected to ground surcharge in the early stage of analysis and research. [PERSON] et al. [14] derived a longitudinal deformation balance equation of an existing tunnel subjected to ground surcharge based on a beam model with a Winkler elastic foundation, and they obtained the theoretical analytical solution of the longitudinal displacement of the existing tunnel through the finite difference method. The maximum displacement diagram of the existing tunnel for different ground surcharge distances was presented. The safe horizontal distance between the center of the existing tunnel and that of the surcharge was determined to be greater than 32 m. [PERSON] et al. [19], treating the soil as an elastic layered foundation in half-space, considered the pipeline to be an elastic long beam based on the Winkler foundation model. They derived and solved the differential equation of pipeline deformation under high-fill conditions. By comparing the solution with the outputs of finite element simulations, the reliability of the proposed theoretical method was verified, and the theoretical prediction scheme could be applied to the same type of construction project. [PERSON] et al. [20] analyzed the accompanying shear deformation of the existing tunnel in the bending process and simplified the tunnel to a Timoshenko beam. An ordinary differential equation for the longitudinal deformation of an existing tunnel subjected to ground surcharge was established, and the analytical solution of the longitudinal deformation was obtained. The applicability of the proposed method was verified by comparing the results with field monitoring data and finite element software simulation analysis. In addition, the influence of ordinary differential equation parameters on tunnel longitudinal deformation was analyzed. The theoretical analytical method typically uses the Winkler foundation model for the tunnel. However, as shown in Figure 1, the drawback of this approach is the inherent discontinuity of adjacent springs, which does not reflect the mechanical properties of foundation soil materials. This assumption greatly simplifies the calculation and affects the prediction results. In this paper, a new simplified analytical method is proposed to estimate the longitudinal displacement of a shield tunnel subjected to ground surcharge. A two-parameter foundation model, namely the Pasternak foundation model, was established. This model is capable of capturing continuity. In the proposed method, the existing shield tunnel is treated as a continuous Euler-Bernoulli long beam with reduced bending stiffness, resting on a Pasternak foundation. An analytical solution for a shield tunnel subjected to ground surcharge is derived. On the basis of existing research, assuming the continuity of the existing tunnel in the soil and based on an analysis of the tunnel and its interaction with the soil, the Pasternak model in Figure 2 [21] uses the Boussinesq elastic solution to obtain the ground-heaped load distribution in the additional load matrix at the tunnel location. The adjusted coefficient of the bed pressure is used to establish the differential equation of tunnel deformation, and the finite difference method is used to obtain the tunnel longitudinal displacement matrix.