## 1 Introduction
[18] analyzed the response of cast iron connecting pipe and ductile iron connecting pipe to single tunnel excavation by using a laboratory model, and found that cast iron connecting pipe was a pipe type with great influence and easy damage from tunnel excavation. Using the indoor shrinkage model, [PERSON] et al. [19] studied the development process of underground pipeline deformation and surface settlement in dry sand. Centrifugal model test can compensate for the loss of self-weight of structures caused by the reduction of model size through the acceleration field formed by the centrifugal testing machine. [PERSON] et al. [20] used a centrifuge model test to study the response of pipelines when the tunnel crosses obliquely. [PERSON] et al. [21] studied the relationship between soil strain and pipe bending behavior by centrifugal model test, and obtained the relationship between tunnel volume loss, soil strain and pipe bending deformation. [PERSON] et al. [22] conducted a large number of centrifugal simulation tests and studied the relationship between pipeline stress and deformation and soil stress. With the development of computer technology, finite element numerical simulation has become the main method to study the crossing problem. Using the finite-difference continuous elastic analysis method, [PERSON] et al. [23] simulated the response of continuous pipe and jointed pipe under tunnel-induced multi-layer soil motion. Based on finite element software, [PERSON] et al. [24] studied the influence of the position between pipeline and tunnel on pipeline deformation. [PERSON] et al. [25] studied the stress and displacement changes of the upper underground pipeline under the conditions of earthquake and tunnel exavation by using FLAC\({}^{\text{3D}}\) software. The three-dimensional finite element analysis model was established by using the finite-difference software, and [PERSON] et al. [26] simulated the influence of double pipe parallel tunnel excavation with different buried depth, material properties and diameter on underground pipelines. Based on Plaxis 3D, [PERSON] et al. [27] conducted a simulation analysis on surface settlement and pipeline deformation by using the soil hardening small strain constitutive model, and the results showed that considering the small strain stiffness of soil could predict the actual deformation of the project more reasonably. [PERSON] et al. [28] performed numerical simulations, the results show that when the position of the municipal pipeline and subway tunnel was vertical, the displacement of the pipeline was less than the horizontal displacement, while the stress of the pipeline was opposite. [PERSON] et al. [29] discussed the main influencing factors of the existing tunnel on the seismic performance of adjacent underground pipelines, including the joint type of the pipeline, the angle and distance between the pipeline and the tunnel. The numerical results show that the flexible joint was an important factor to reduce the settlement difference and axial stress of the pipeline. Using the formula of finite element lower limit analysis and second-order cone programming, [PERSON] et al. [30] estimated the support pressure required to stabilize a circular tunnel in two layers of clay under undrained conditions, and evaluated the stability of a long tunnel excavated from normally consolidated and overconsolidated saturated clay under undrained conditions [31]. Additionally, the calculation method of the minimum support pressure for the circular tunnel lining was given considering the anisotropic undrained shear strength parameter [32]. The existing research on the response of adjacent underground pipelines under tunnel excavation had considered the geometrical shape of a tunnel, the buried depth of a pipeline, the material, the size of a pipeline and the relative position with tunnel.