## 1 Introduction  
The rapid urbanization and expansion of underground rail transit have become essential for alleviating surface transportation pressures and promoting urban development. As China is geographically situated between the circum-Pacific seismic belt and the Eurasian seismic belt, many of its cities with subway systems are in seismically active regions. Seismic damage data of underground structures reveal that these structures, like surface structures, are significantly affected during strong earthquakes, with catastrophic consequences in some cases. For instance, the 1995 Osaka Kobe earthquake in Japan completely destroyed over 30 columns at the Dakai subway station, leading to its collapse [1, 2, 3, 4, 5]. Similarly, the 2001 Chi-Chi earthquake in Taiwan and the 2008 Wenchuan earthquake in China caused extensive damage to urban subway systems [6, 7]. At present, substantial progress has been made by Chinese scholars in the seismic research of underground structures [8, 9, 10, 11, 12, 13]. However, most studies have primarily focused on ordinary layered sites. As underground construction scales expand, site conditionsbecome increasingly complex, with weak interlayer sites being a notable example. Weak interlayer sites are characterized by layered or banded weak thin layers formed by soft soil. Due to high moisture content, low strength, and low shear strength, weak interlayer soil is prone to deformation and failure under seismic action, thereby exacerbating damage to underground structures. Consequently, it is imperative to investigate methods for mitigating structural damage at weak interlayer sites. [PERSON] et al. [14] examined the seismic response of underground structures by varying the location and thickness of the weak interlayer. Their results indicated that the presence of weak interlayers significantly impacts seismic response, with the strongest effects occurring when the weak interlayer is located at the center of the structure. [PERSON] et al. [15] analyzed the influence of seismic wave intensity on weak interlayers, demonstrating that the amplification effect is primarily governed by seismic wave velocity. [PERSON] et al. [16] investigated the seismic response of subway station structures in weak saturated soil interlayer sites using finite element analysis, finding that the weak interlayer amplifies seismic effects. The amplification is most pronounced when the interlayer is situated in the middle of the station. [PERSON] et al. [17] explored the dynamic response of a two-story three-span subway station structure by considering various seismic waveforms and relative positions of the station and weak interlayer. Their findings underscored the significant influence of the station's location within the weak interlayer on structural stress and deformation, with the dynamic response peaking when the station is entirely within the weak interlayer. In order to reduce the seismic damage to underground structures, two main methods are used to achieve seismic mitigation control: one method involves adding isolation materials between the structure and the soil, while the other involves applying an auxiliary seismic isolation device to the key components of the underground structure (such as the middle column). In 1987, the friction pendulum bearing (FPB) was introduced by [PERSON] et al. [18] at the University of California, Berkeley. The bearing operates based on the principle of a pendulum, with the structural natural period extended through the swaying motion of the upper support and slider. This motion effectively dissipates seismic energy. In practical engineering applications, FPBs have become widely employed. The idea of applying an isolation layer to underground structures was proposed by [PERSON] et al. [19] and [PERSON] et al. [20], who carried out a series of theoretical analyses and experimental studies. The experiments showed that the seismic mitigation effect becomes more significant as the elastic modulus ratio of the seismic mitigation layer material to the surrounding rock decreases.