# Effect of Subsequent Subgrade on Seismic Response of the High-Speed Railway Track-Bridge System
## 1 Introduction  
Due to the extensive use of continuous welded rail technology in the construction of high-speed railway bridges, the track structure is gradually extended from the bridge structure to the subgrade structure [1, 2]. Although the continuous track structure allows for smoother train travel on high-speed railway bridges, during major natural disasters such as earthquakes, the continuity of the track structure can be disrupted due to variations in structural integrity. During an earthquake, the seismic displacement response of the track structure above the bridge is more pronounced than that of the track structure above the subgrade. Consequently, as the only longitudinal connecting element in the high-speed railway continuous beam bridge, the track structure partially constrains the seismic displacement response of the bridge structure. This interaction results in a certain degree of coupling between the seismic responses of the bridge structure and the subgrade structure [3]. Moreover, this extension of the track structure to the subgrade-bridge transition section results in substantial internal force and deformation. This phenomenon demonstrates that a portion of the seismic inertia force of the bridge structure can be transferred to the flexible subgrade structure through the longitudinally continuous track structure, thereby achieving energy dissipation in the subgrade-bridge transition section, i.e., the longitudinal constraint effect of the subsequent subgrade track structure (SSTS) [4]. This effect helps to balance the differences in seismic response not only between bridge and bridge, but also between bridges and the subgrade. Nevertheless, the design of continuous beam bridges recommended by the current Chinese seismic design code does not consider the longitudinal confinement effect of the SSTS, leading to a potential underutilization of this effect and possible adverse impacts on bridge design. In order to obtain more accurate results from the seismic response analysis of bridge structures, many scholars have analyzed the railway track-bridge system considering the subgrade [5]. [PERSON] et al. created a coupling system encompassing a train, track, and subgrade, revealing that the bridge-subgrade transition section experiences peak dynamic stress and displacement [6]. [PERSON] et al. examine the seismic vulnerability of HSR bridges with track constraints, emphasizing the importance of track-bridge interaction in seismic response analysis [7]. [PERSON] et al. employed a 100 m finite element model for the subgrade track outside the bridge, aiming to minimize boundary condition impacts [8]. [PERSON] et al. modeled track extensions at viaduct ends for accurate transition zone representation [9]. [PERSON] extended the subgrade length to 150 m as per DS899 and UIC774-3 codes to negate boundary condition influences [10]. [PERSON] et al. discovered that there is a critical track length at which the subgrade structure significantly impacts the dynamics of high-speed railway simply supported beam bridges [11]. [PERSON] et al., considering longitudinal constraint and beam-track interaction, found that the range of subgrade constraint on the bridge is only partially related to the subgrade itself [12]. [PERSON] et al. developed a high-speed railway simply supported beam bridge-subgrade model and conducted an in-depth study and analysis of the damage patterns of key bridge components [13]. Based on prior investigations, the subgrade-track constraint effect influences the seismic response of bridge structure during seismic events [14; 15; 16]. Nevertheless, the focus of existing studies predominantly centers on multi-span, simply supported beam bridges, with comparatively scant attention given to continuous beam bridges. In the finite element analysis of multi-span, simply supported beam bridges, these structures exhibit characteristics of repetition and symmetry.