## 1 Introduction  
With the growing populations in metropolitan cities around the world, underground urban infrastructure development has become a major focus. If these underground structures are constructed in earthquake areas, their dynamic response to seismic loads must be determined in order to reduce economic damage and threats to human safety. In contrast to overground structures where design is based on the fact that earthquakes manifest as inertial forces, in underground structures such as tunnels, which have a significant degree of containment, design and analysis are based on the assumption that earthquakes primarily deform the soil structure. Previously, engineers believed that earthquakes caused less destruction of underground structures than surface structures, but the earthquakes of the 1990s caused serious damage to tunnels, forcing a re-appraisal of this view. In recent years, there have been numerous examples in different countries of earthquakes causing extensive damage to underground structures (e.g., the Kobe earthquake, 1995; Taiwan, 1999) [1; 2; 3; 4; 5; 6; 7], which indicates that it is crucial for studies to be made of the seismic behavior of tunnels. As earthquake waves are usually amplified by passing through different layers of soil (e.g., clay or sand) [8], the surrounding soil can affect the performance of structures such as tunnels [9; 10]. Previous studies have shown that tunnels located in clay soils are more likely to be damaged by earthquakes than those constructed in sandy soils [11; 12; 13; 14]. However, there are fewer studies on the seismic behavior of tunnels located in soft soils than in sandy soils. Different approaches such as numerical [15; 16; 17] and experimental methods [18; 19] are used to investigate the seismic behavior of tunnels in terms of the interactions between soil and tunnel. Using numerical methods to investigate the dynamic behavior of tunnels requires a suitable constitutive model for the materials involved. In recent years, researchers have used numerical methods with different constitutive models to study the seismic behavior of tunnels. [PERSON] et al. [5] used a modified Cam-clay constitutive model to investigate the dynamic response of tunnels during the 1999 Duzce earthquake in Turkey. Their results showed that this constitutive model was not suitable for simulating the seismic behavior of a tunnel during strong earthquakes. [PERSON] et al. [20] investigated the dynamic response of shallow rectangular tunnels to earthquake loads. Their analysis was conducted mainly with pseudo-static finite element analyses, and it showed that these structures are strongly affected by nonlinear frictional effects at the soil-structure interface. [PERSON] et al. [21] conducted a numerical study on the behavior of tunnels under seismic loads using PLAXIS2D software and a modified Cam-Clay model to model the soil medium. The results of the nonlinear plane strain analysis of the soil-tunnel system showed that the earthquake frequency affected the maximum and residual dynamic Earth pressure around the tunnel. [PERSON] and [PERSON] [13] studied the seismic behavior of tunnel in clays under earthquake loads, with consideration of the hysteresis-hyperbolic constitutive model for soil. Their results suggested that randomness in the small-strain shear modulus of clay can significantly influence the shear force, the seismic responses of bending moments, and the lateral deformation of tunnels. [PERSON] et al. [22] studied the dynamic behavior of a metro subway tunnel under earthquake loads, using a cyclic elastoplastic constitutive model for foundation soils so as to obtain acceptable results, and they also studied various parameters that affect the seismic behavior of a tunnel. Their work showed that the acceleration response decreased at different depths in the model due to the phenomenon of liquefaction. Moreover, the increase in the amplitude of vertical acceleration increased the liquefaction area. Soil response to loads is strain-dependent.