# A Vehicle-Bridge Interaction Element: Implementation in ABAQUS and Verification
## 1 Introduction  
The study of railway-bridge systems under train loads has evolved from simple to complex models [1; 2; 3; 4; 5]. With the advancements in train-track coupled dynamic systems worldwide, particular attention has been directed towards the analysis of vehicle-bridge interaction (VBI), especially when the railway is subjected to moving loads from multiple wheels [6; 7]. The earliest VBI models focused on a simple support beam subjected to a moving point load. [PERSON] [8; 9] and [PERSON] et al. [10] first conducted dynamic analyses of beams resting on elastic foundations subjected to moving loads. [PERSON] and [PERSON] [11] developed the analytical solution of track vibration using an assumed continuous support system, taking into account the ballast mass effect. [PERSON] [12] analyzed the infinite periodic Euler-Bernoulli beam system under moving vehicle loads, regarded as concentrated forces with uniform traversal speeds. With the application of modal superposition, [PERSON] et al. [13] solved the equations of motion for simply supported beams subjected to moving loads in the time domain. [PERSON] [14] solved this equation analytically and established that the deflection of beams would reach a maximum under moving loads at a certain speed. Further research has focused on the interaction between the vehicle and the beam. [PERSON] and [PERSON] [15; 16; 17] explored track dynamics and train-track interactionin time and frequency domains. [PERSON] and [PERSON] [18] studied the dynamic instabilities of train-track coupled systems. [PERSON] et al. [19] analyzed the near-fault velocity pulse effect on a vehicle-bridge structure induced by seismic motions. Extensive studies conducted by [PERSON] et al. and [PERSON] et al. [20,21] provide a nonlinear wheel-track contact force scheme. [PERSON] et al. [22] explored the dynamic behavior of the pantograph-catenary system under vehicle-track excitation and emphasized the importance of spatial effects to better understand and improve the reliability and performance of high-speed railway systems. The authors introduced a novel approach to consider the spatial vibration of the carbody in the pantograph-catenary interaction by developing a spatial contact model between the contact wire and the pantograph strip. [PERSON] et al. [23] presented an advanced pantograph-catenary-vehicle-track model, which analyzes the vertical dynamics of the complete system. The model developed by the authors is able to evaluate the displacements and the contact force generated in the catenary-pantograph as well as the wheel-track interactions. In addition, the study also focused on the possible influence of track irregularities on the catenary-pantograph dynamic interaction. [PERSON] et al. analyzed poroelastic ground vibration based on [PERSON]'s theory [24, 25, 26], while [PERSON] conducted systematic studies on track dynamics [27,28]. [PERSON] introduced a simplified vehicle model with a primary suspension system on a single wheel and a complete car model with primary and secondary suspension systems on four wheels, involving 10 degrees of freedom. [PERSON] [29] subsequently developed the equations of motion for the vehicle-track coupling system, considering the effects of track irregularities and abrupt changes in track stiffness. Moreover, the critical velocity is also widely studied in the vehicle-bridge interaction field. [PERSON] and [PERSON] [30] conducted a dynamic analysis of nanotube structures under the influence of a moving nanoparticle, utilizing the nonlocal continuum theory. They modeled the nanotube structure as an equivalent continuum structure (ECS) based on nonlocal Euler-Bernoulli, Timoshenko, and higher-order beam theories. The study established nondimensional equations of motion for the nonlocal beams, leading to analytical solutions and explicit derivation of critical velocities.