## 1 Introduction  
Several earthquake events have caused heavy human tolls and substantial property damages [1; 2; 3] in recent years, and, thus, the seismic response characteristics of tunnels have attracted extensive attention. Many researchers have studied the dynamic response of underground tunnels using different methods [4], and, due to the complexity of wave propagation and scattering in soil media, the dynamic analysis of underground structures is always carried out economically and under realistic conditions with the aid of the numerical method [5; 6; 7]. The angle of incidence is a critical factor that determines the seismic tunnel response [8; 9; 10; 11; 12; 13], and the influence of the incidence direction and the angles of the seismic wave on the response of a tunnel was first noticed in the anti-seismic studies of underground tunnels. Through a series of studies and analyses, [PERSON] et al. [7] stated that the incidence direction of earthquake waves has significance on the seismic responses of long lined tunnels, and it is important to include the incidence angles of earthquake waves when mapping the seismic risk of tunnels. [PERSON] and [PERSON] [14] developed a novel numerical method for analyzing the longitudinal seismic response of tunnels under an asynchronous earthquake wave input, based on which the influence of the incidence angle and incidence direction on the longitudinal seismic response of tunnels was discussed. Considering obliquely incident seismic waves, [PERSON] et al. [15; 16] carried out a study on the dynamic response of tunnels using a 2.5D coupled finite-element--boundary-element model. [PERSON] et al. [17] conducted a 3D seismic response analysis of an immersed tunnel in a horizontally layered site subjected to obliquely incident waves based on the precise dynamic stiffness matrix of the soil layer and half-space via combined viscous-spring boundary and equivalent node stress methods. [PERSON] et al. [18] investigated the effects of pulse-like ground motions on tunnels in saturated poroelastic soil for obliquely incident seismic waves using a 2.5D modeling technique. Considering 3D obliquely incident P- and SV-waves, [PERSON] et al. [19] calculated the 3D seismic response of a lined tunnel embedded in a half-space using the 2.5D finite/infinite-element method. [PERSON] et al. [20] investigated the dynamic performance of prefabricated utility tunnels considering the oblique incidence of P-waves. [PERSON] et al. [21] developed a viscous-slip interface model to reflect the contact between the surrounding rock and tunnels, and the dynamic stress concentrations and surface displacements of two shallowly buried lining tunnels were examined using IBIEM, during which the effects of the angle on the medium's responses were considered. [PERSON] et al. [22] evaluated the seismic response of an immersed tunnel in an ocean space under obliquely incident earthquake waves. In addition, the responses of a tunnel in the longitudinal direction should not be neglected [23], especially when considering the tunnel lining damage which has been caused by recent earthquakes, and the longitudinal seismic response is another important perspective in the field of anti-seismic analyses of tunnels. Considering the longitudinal variation in ground properties, [PERSON] et al. [24] developed a longitudinal displacement profile-based procedure for simulating the dynamic response of tunnels and performed a series of pseudo-static 3D finite-element analyses of tunnels under spatially varying ground motion. Based on the generalized response displacement method, [PERSON] et al. [25; 26] studied the longitudinal seismic response of a metro tunnel in a site with non-uniform soil properties and found that the inter-segment opening of a tunnel in soft soil is larger than that in hard rock. [PERSON] et al.