## 1 Introduction  
The structural integrity of straddling oil and gas pipelines is a critical aspect for ensuring the safety of long-distance transportation, with seismic activity posing one of the most significant threats. Extensive engineering practice highlights the importance of studying the failure mechanisms caused by earthquakes in these pipeline structures. When investigating damage mechanisms, it is essential to consider the influence of oil and gas mediums on pipeline behavior. In the case of cable-stayed pipeline bridges used for oil and gas transportation, the load within the pipes consists of various materials, such as oil and natural gas, which can affect the pipeline's performance during transport operations. While this effect may not be prominent under normal circumstances, strong earthquakes can induce vibrations in both the pipeline structure and its contents, leading to mutual interactions that impact the overall system stability. Therefore, theoretical analysis and numerical simulations analyzing earthquake failure mechanisms in straddling oil and gas pipelines must incorporate considerations for medium-pipeline interactions [1, 2]. Fluid-structure coupling (FSC) is a multi-physical field coupling problem that involves the interaction between fluid dynamics and structural mechanics. Since the 1970s, foreign scholars have been extensively studying fluid-structure coupling vibration. [PERSON] [3] conducted a comparative analysis of the dynamic characteristics of an experimental structural system under both coupled and uncoupled conditions to investigate the effect of fluid-structure coupling. [PERSON] et al. [4] analyzed the phenomenon of pressure attenuation in three-dimensional pipeline systems and attributed it to the influence of fluid-structure coupling, which was further supported by theoretical verification. The establishment of a nonlinear dynamic model for pipeline vibration considering fluid-structure coupling requires assumptions such as non-viscous fluids, incompressible pipelines, neglecting shear deformation and moment of inertia in section vibrations, etc. Building upon these assumptions, numerous scholars have focused on investigating other influencing factors. [PERSON] et al. [5] examined gravity effects, damping properties of pipe materials, and viscous damping while proposing a general equation to describe nonlinear dynamics in structures. Based on this work, [PERSON] et al. [6] developed a nonlinear model capturing the impact of fluid-structure coupling by disregarding Poisson's ratio effects. In China, research on the seismic response considering fluid-structure coupling vibrations in pipelines emerged relatively late. [PERSON] et al. [7] proposed a comprehensive mathematical model for fluid-structure coupling in pipeline systems with complex boundary conditions, employing two distinct modeling methods. [PERSON] et al. [8] developed a mathematical model based on an actual suspended oil pipeline, considering the fluid-structure coupling effect during earthquakes. They established the vibration equation of the pipeline using the finite element method and the Hamilton principle and solved it directly to obtain the seismic dynamic response under various flow rates and boundary conditions using the Newmark step-by-step integration method. [PERSON] et al. [9] constructed a finite element model incorporating fluid-structure coupling to analyze the seismic performance of pipelines. By varying parameters such as the medium density and velocity within the pipeline, they identified their close relationship with pipeline failure. [PERSON] et al. [10] employed a fourteen-equation model along with the transfer matrix method (TMM) to describe fluid-structure coupling in liquid-filled pipelines, presenting a general solution applicable to multi-branch pipelines, while also providing a means for predicting the frequency response in complex pipeline systems. [PERSON] et al.