## 1 Introduction  
In modern society, underground constructions are becoming increasingly prevalent due to rapid population growth and limited urban land resources [1]. From small underground structures such as pipelines to large underground stations, the variety of underground constructions is also increasing. Among them, underground utility tunnels are being used more often in urban construction because of their ability to make full use of underground space and improve the urban environment. However, there are many safety issues with underground structures, especially when they are in liquefiable sites in seismic zones, and there are many cases of damage to underground structures in liquefied soil in previous earthquake damage reports. In 1995, the massive soil liquefaction that occurred in the Hyogo-Ken Nanbu earthquake caused many caisson quay walls along the Kobe coast to move several meters out to sea [2]. During the 2004 Nakagoshi earthquake in Niigata Prefecture, Japan, liquefaction of under-compacted backfill caused a large number of sewer manholes in the area to be lifted more than 1 m above ground level [3, 4]. Liquefaction can also cause damage to large underground structures, and some tunnels in California were damaged by uplift during the 1989 Loma Prieta earthquake [5, 6]. Although underground structures are sheltered and protected by the soil, loosely saturated sandy soil gradually and substantially loses its shear strength during strong ground motion, and the soil changes from a solid to a liquefied state, so that underground structures in the liquefied soils are more vulnerable to strong earthquakes [7]. \"Currently, lots of studies have been conducted on the seismic response of underground structures in liquefied soils and many valuable views have been proposed.\" A large number of centrifuge tests and large-scale shaking table tests have been carried out to study the dynamic deformation characteristics of underground structures such as pipelines,tunnels, and underground stations in liquefiable soils from different perspectives during the shaking [4, 8, 9, 10, 11, 12]. [PERSON] et al. [10] carried out a series of large-scale shaking table tests to investigate the damage mechanism and uplift behavior of a three-arch subway station structure in liquefiable soil during the strong motions. [PERSON] et al. [4] tested 22 centrifuge models to study the response of underground manholes under dynamic loads and explained the mechanism of manhole uplift caused by excess pore water pressure and proposed a solution to reduce the uplift. Numerical calculations of underground structures in liquefiable strata using various complex soil constitutive models are also widely used [6, 13, 14, 15]. Many of these studies are based on homogeneous soil, while in practice, few underground structures are in homogeneous strata, especially linear underground structures such as utility tunnels, which are more commonly located in non-homogeneous strata containing liquefied soils on a site scale of hundreds or more than a dozen kilometers. For example, the utility tunnel of Beijing Daxing International Airport, shown in Figure 1, passes through a non-homogeneous liquefiable stratum. Therefore, it is all the more important to study underground structures partially or fully buried in layered liquefiable soils [16]. In previous research and engineering practice, increasing the burial depth is considered to be an effective method to control the uplift of underground structures in liquefied soils [7, 17, 18, 19, 20]. [PERSON] et al. [9] studied the uplift of tunnels in liquefied soil by numerical analysis and centrifuge tests and analyzed the effect of burial depth and dimensions on the vertical displacement of tunnels with circular cross-sections. [PERSON] et al. [19] established numerical models to investigate the effect of parameters such as the expansion angle and relative density of the soil and the diameter and burial depth of the pipe on the uplift of the pipes in saturated sandy soil.