# Effect of Subsequent Subgrade on Seismic Response of the High-Speed Railway Track-Bridge System
## 3 Finite Element Model  
In this study, two bridge models were established using the SAP2000 finite element software, as shown in Figure 2. Figure 2a shows the bridge model considering the longitudinal constraint effect of the SSTS in the 130 m length section, referred to as BMCS. Figure 2b shows the bridge model without considering the longitudinal constraint effect of SSTS, referred to as BMWS. The piers and foundations of bridges are critical components of bridge substructures, playing a vital role in ensuring the safety and stability of the entire structure [19; 20]. Their primary function is to support the bridge span structure and transfer the deck loads to the foundation [21]. The substructures, including rectangular piers and circular piles, are simulated using nonlinear beam-column elements with discrete fiber sections that encompass cover concrete, core concrete, and reinforcement steel. (Figure 3) The side piers of a continuous beam bridge and the piers of a simply supported beam bridge are simulated using beam units at 0.8 m intervals. In contrast, the central pier of a continuous girder bridge is simulated using beam units at 1 m intervals. Additionally, this study takes into  
Figure 2: Finite element model: (**a**) the bridge model considering the longitudinal constraint effect of the SSTS(BMCS); (**b**) the bridge model without considering the longitudinal constraint effect of SSTS(BMWS). account the influence of the pile-soil effect. The interaction between the piles and soil at the base of the foundation is simulated using zero-length units (Figure 3), which are essentially fully elastic soil-spring-connected units with six degrees of freedom [22; 23]. In general, the superstructure of a bridge tends to maintain their elasticity under seismic activity. In the finite element software, we use the elastic frame unit to simulate the box beam, base plate, track plate, and rail [24; 25]. The sliding layer, CA mortar layer, fasteners, shear tooth grooves, shear reinforcement, and horizontal blocks play pivotal roles in the transmission of internal forces within essential elements of the track structure [26]. Consequently, in the finite element software, elastoplastic connection units are utilized for simulation purposes, with a Rayleigh damping ratio set at 5%. In this simulation, the unit is divided according to the actual size of the standard track slab (6.5 m). The spacing between the box girder, the bottom plate, the track slab, and the rail is set at 0.65 m. The distance between the sliding layer, the CA mortar layer, and the fastener is also 0.65 m. The horizontal blocks units are positioned at intervals of 6.5 m, while the shear reinforcement units are located at the beam ends [27]. The bearing is simulated using a nonlinear plastic unit, featuring a dynamic hysteresis curve that mirrors the stress-strain relationship typical of ideal elastoplastic materials. The continuous beam bridge model displays the longitudinal force-displacement curves of interlayer components as shown in Figure 3b. With the unit length set at 0.65 m, the stiffness of the connecting components between layers is detailed in Table 1. The materials and material properties of key components in the HERTBS are shown in Table 2.  
Figure 3: (**a**) Finite element simulation of critical components of HERTBS; (**b**) the materials of critical components of HERTBS and their stress\(-\)strain relationship. The track structure exhibits longitudinal continuity in both the subgrade and bridge sections, particularly when considering the longitudinal constraint effects of the SSTS. As a result, the modeling of the track structure in the upper part of the subgrade section is consistent with that of the bridge structure. Additionally, the subgrade section also includes friction plates, end spurs, and water-hardened support layers. According to [PERSON]'s research, most components of the subgrade structure maintain an elastic state under longitudinal seismic excitation.