## 1 Introduction  
With the accelerated urbanization process, underground engineering structures, such as metro systems, utility tunnels, and deeply buried tunnels, serve as critical components of national infrastructure, drawing significant attention to their seismic performance. Under seismic loading, underground structures are prone to severe damage or even progressive collapse due to the complexity of the soil-structure interaction (SSI). Historical seismic events such as the 1995 Great Hanshin earthquake in Japan [1] and the 2008 Wenchuan earthquake in China [2] revealed the substantial seismic risk associated with underground infrastructure. Consequently, the development of efficient and accurate analytical methods for evaluating the seismic responses of underground structures has become an urgent requirement for enhancing engineering safety and advancing disaster prevention and mitigation strategies. Current seismic analysis methods for underground structures can be divided into two main categories: the dynamic time history method and simplified practical methods. The dynamic time history method, which directly solves dynamic differential equations to obtain structural seismic responses [3, 4], is highly accurate in simulating the time-dependentdynamic behavior of underground structures. As a result, it has become the most predominant approach for evaluating the seismic performance of underground structures. However, its computational complexity and inefficiency often limit its applicability in the analysis of large-scale engineering projects. Recent advancements have introduced incremental dynamic analysis (IDA) as an extension of dynamic methodologies. By gradually modulating the amplitude of earthquake motion intensity [5], IDA can be used to systematically evaluate the seismic performance margin of structures, making it particularly suitable for probabilistic risk assessment. Nevertheless, the resource-intensive nature of such approaches remains a barrier for routine engineering applications. To address these limitations, researchers have developed various quasistatic simplified methods. Among these methods, the response displacement method (RDM) has gained widespread adoption in practical engineering because of its theoretical clarity and computational efficiency. Notably, this method has been codified in the Standard for Seismic Design of Underground Structures (GB/T 51336-2018) [6], further solidifying its role as a standard approach for underground structural analysis. The accuracy of the traditional RDM is highly dependent on the rational determination of the equivalent foundation spring stiffness. Focusing on this critical aspect, [PERSON] et al. [7] developed an improved algorithm for calculating foundation spring stiffness tailored to underground structures with rectangular cross-sections, significantly enhancing computational efficiency. [PERSON] et al. [8; 9] systematically evaluated the precision of six distinct approaches for deriving foundation spring coefficients by modifying load application modes. Their findings demonstrated that methods that independently solve for horizontal and vertical spring coefficients yield optimal accuracy. [PERSON] et al. [10; 11] conducted a comparative analysis of foundation spring stiffness coefficients determined through the test method, [PERSON]'s method, the MIDAS method, and the finite element method. Their study recommended prioritizing experimental methods and proposed applying a safety correction factor of 1.10 to structural internal force calculations to enhance structural integrity. Furthermore, [PERSON] et al. [12] derived an analytical solution for the foundation modulus incorporating radial-tangential modulus interactions and deformation effects, allowing for a precise and explicit consideration of soil deformation patterns. To overcome the inherent limitations of traditional methods that rely on foundation spring calculations, researchers have developed multiple enhanced methodologies. [PERSON] et al.