# Numerical Study on Track-Bridge Interaction of Integral Railway Rigid-Frame Bridge
## 1 Introduction  
Track-bridge interaction (TBI) is an important phenomenon occurring between continuous welded rails (CWRs) and bridges that is a vital factor to take into account during the design, construction, and maintenance of high-speed railways (HSRs). Under the effects of temperature change, vehicle load, trains braking or accelerating, concrete shrinkage and creep, strong earthquakes, wind and other load conditions, the track-bridge system produces various deformations. Due to the nonlinear constraints between the track and the bridge, relative longitudinal displacement can arise and result in additional longitudinal stresses in the track system [1; 2; 3]. When the additional stress of the track is too large, it may lead to instability in summer or even fracture in winter [4; 5], which affects safety and comfort when high-speed trains (HSTs) pass over the bridge structures. In past decades, scholars both nationally and internationally have carried out numerous computational and experimental studies on the interactions between beam and track. In 1974, [PERSON] [6] established the differential equations of a track-bridge systemand set up a quasistatic method to calculate the distribution of train braking and traction forces in rails on the bridges. In 1985, the Bridge Institute of the Railway Bridge Authority in Germany issued special procedures DS899/59 for the Shinkansen Railway Bridge [7], which proposed that a bilinear model should be used to simulate longitudinal resistance. The International Union of Railways (UIC) recommendation 774-3R was issued in 1995 and revised in 2001 [8]. [PERSON] et al. [9; 10] addressed the influence of load history on the analysis of track-bridge interaction and researched the longitudinal forces in CWRs due to nonlinear track-bridge interaction. In 2008, [PERSON] et al. [11] published the book _Track-Bridge Interaction on High-Speed Railways_, which comparatively systematically introduced the finite element analysis (FEA) method to calculate the TBI effect. [PERSON] and [PERSON] et al. [12] proposed a nonlinear TBI analysis framework that accounted for loading history effects and changes in the longitudinal resistance of rail fasteners in various load cases. [PERSON] and [PERSON] [13] conducted longitudinal resistance tests on track fastening systems and proposed a numerical model using interface elements adaptive to various loading cases for the sequential analysis of nonlinear track-bridge interaction. Furthermore, increasing emphasis has been laid on research about the track-bridge interaction in the last decade, especially with the rapid development and wide application of new track types and special bridge systems for HSRs. [PERSON] and [PERSON] et al. [14] carried out an interaction analysis of continuous slab tracks in long-span continuous HSR bridges and compared the force distribution rules of ballasted tracks and continuous slab tracks. [PERSON] et al. [15] analyzed the track-bridge interaction of a continuous bridge with a sliding slab track, considering the behavior of end-supporting anchors. [PERSON] et al. [16] presented a model to analyze the interaction between quick-hardening tracks and a bridge considering the interlayer friction. [PERSON] and [PERSON] et al. [17] conducted comparative research on the track-bridge longitudinal interaction of CWRs on three types of arch bridges, including deck, half-through and through arch bridges. [PERSON] et al. [18] developed an integrated model for the TBI analysis of long-span steel truss arch bridges, including rails, deck system, stringers, cross beams, suspenders, main trusses, piers and foundations. [PERSON] et al. [19] established a track-beam-cable-tower spatial model for analyzing the TBI of CWRs in a long-span cable-stayed HSR bridge and put forward an optimization method for choosing the installation location of rail expansion joints (REJs). [PERSON] et al.