## 1 Introduction  
Bridge structural deformation is one of the main excitation sources in a system involving train-track-bridge dynamic interactions (TTBDI) [1, 2]. The deformation of a bridge caused by pier settlement and deck variation directly affects the track irregularity and then impact the comfort and safety of train operation [3, 4, 5, 6, 7, 8]. Accurate evaluation and analysis of the effect of the foundation deformation on the safety of vehicle operation are of great significance in railway maintenance. In recent years, many scholars have conducted a series of studies on the characteristics and dynamic response of the vehicle rail system. Some of them have verified the accuracy of the vehicle rail coupling model by comparing the field test with numerical simulation models [9, 10]. Someone else has conducted in-depth research on the dynamic response of a track structure in the presence of a long-span suspension bridge [11], abutment transition area [12], simply supported beam bridge [13, 14], irregularities in front of the turnout [15], through numerical methods. Others optimized the vehicle rail coupling dynamic model and calculation method [16, 17, 18, 19, 20, 21]. Track irregularity, as an important excitation source of the vehicle-rail coupling system, affects the operation safety. Therefore, the measurement and analysis of track irregularitiesare very important. [PERSON] et al. proposed a method to analyze rail profilometric data, detected different states of degradation of the rails and identified a level of deterioration associated with the need for maintenance through rail grinding [22; 23]. [PERSON] et al. discussed the ground vibration level caused by the vertical irregularity of the track under different speeds and track structures [24]. Furthermore, more information about the causes, measurements, and analysis methods of rail irregularities can be found in the review by [PERSON] [25; 26]. In published studies, track irregularities caused by infrastructure disruptions were mostly described by harmonic irregularities [27; 28]; meanwhile, the vehicle-track coupled dynamics model was used to conduct simulations, wherein the vehicle acceleration, reduction rate of wheel load, derailment coefficient, etc., were taken as evaluation indexes to study the effect of infrastructure disruptions on vehicles' vertical safety. [PERSON] et al. [27] [PERSON] et al. [29], [PERSON] et al. [30], and [PERSON] et al. [31; 32] explored the amplitude and wavelength limits of track irregularities at different operation speeds by the train-track interaction system. [PERSON] et al. [33] studied the sensitive wavelengths of track irregularities corresponding to different operating speeds. Despite their great significance for the construction, operation, and maintenance of higher-speed railways, these studies only considered the coupling of trains and tracks and ignored the influence of bridges. In order to explore the impact of pier settlements on driving safety, [PERSON] et al. [3; 4; 5; 34] established a detailed FE model of a bridge to analyze the track deformation caused by pier settlement and then used TTBDI to study its influence on the reduction rate of wheel load, vertical acceleration, and wheel/rail force wheel. The above research studied the running safety of high-speed railways based on deterministic analysis. Some scholars introduced stochastic analysis into TTBIM, making their conclusions more statistically significant. [PERSON] et al. [35; 36; 37], on the basis of the energy principle, derived equations for the train-track-bridge coupled system and investigated the random vibrations of a high-speed train traversing a slab track on a continuous girder bridge subjected to track irregularities and traveling seismic waves by the pseudo excitation method. [PERSON] et al. [38] provided an investigation as to how uncertainty in the parameters influences the dynamic responses of time-varying TTBS, which refers to dynamic sensitivity analysis in the context of a stochastic dynamic system.