## 1 Introduction  
With the growth of population and industrial activities, the shortage of available space directly restricts the rapid development of big cities. To solve this problem, many countries have begun to increase their development and utilization of urban underground space. Earthquake damage to underground structures is thus mitigated, as the confining pressure exerted by the surrounding soil can improve the level of structural safety in the event of an earthquake. Hence, seismic calculations and seismic measures are not applied to the underground structures associated with subway systems. However, earthquake disasters in recent decades have affected thinking around this traditional concept, especially the Hanshen earthquake which damaged the 3-km long subway tunnel and five subway stations, indicating the possibility of subway underground structural damage and secondary disasters remains significant [1; 2; 3]. Subway stations are a service-oriented public facility and host a concentrated population (many service facilities), and require a long time to expedite emergency evacuation. At present, the seismic design research of underground structures in loess areas is lagging compared with the engineering requirements, for example, the selection of design parameters related to the characteristics of loess soil lacks a clear set of rules, thus hindering the safe and efficient seismic design of underground structures in loess area. Currently, the relevant codes for the design of underground structures include: the _Code for Seismic Design of Buildings_ (GB 50011-2010) [4] and the _Code for Design of Civil Air Defense Basements_ (GB 50038-2005) [5]. However, these do not provide relevant seismic calculation methods. The inertial force method for underground structures in soft soil has been given in the _Code for Seismic Calculation of Subway Building Structures_ (DG/TJ08-2008) [6], but it is not necessarily suitable for those loess areas. The response acceleration method and response displacement method provided in the _Code for Seismic Calculation of Urban Rail Transit Structures_ (GB 500909-2014) [7] have no specific provisions on the horizontal relative displacement of the stratum and the foundation spring stiffness parameters. The differences of ground motion characteristics among different soil bodies are not considered in the _Code for Seismic Design of Underground Structures_ (GB/T 51336-2018) and other international codes [8; 9; 10; 11; 12; 13; 14; 15; 16; 17]. In recent years, many studies about seismic design and analysis of underground structures have been published, but most of them do not consider the dynamic characteristics of soil or the methods are too complex to be suitable for engineering design [18; 19; 20; 21; 22]. Seismic design and analysis methods mainly include coupled and decoupled approaches, numerical dynamic analyses, and quasi-static calculation methods. A soil-structure interaction analysis of underground tunnels was performed by [PERSON] in1983 [23]. The magnitude and distribution of static normal soil stresses against underground structural cylinders were studied by [PERSON] and [PERSON] in 1998 [24]. New analytical solutions for a deep tunnel in a saturated poro-elastic ground were explored by [PERSON] [25]. The analytical solutions for the thrust and moment in the lining of a circular tunnel due to seismic-induced ovaling deformation were studied by [PERSON] et al. [26]. An analytical solution for a rectangular opening in an infinite elastic medium subjected to far-field shear stresses was proposed for drained and undrained loading conditions [27]. The main limitations of the decoupled approaches were investigated and discussed through a large set of numerical simulations [28]. The main results of several numerical dynamic analyses of propped embedded retaining structures in the time domain were demonstrated by [PERSON] and [PERSON] [29]. The seismic behavior of a multi-propped retaining structure was evaluated considering soil-structure interaction effects [30].