# A Vehicle-Bridge Interaction Element: Implementation in ABAQUS and Verification
## 5 Conclusions and Discussions  
In this paper, we have implemented the vehicle-bridge interaction (VBI) element theory proposed by [PERSON] [43] in ABAQUS using a user-element subroutine (UEL). To verify the accuracy and reliability of the implementation, we conducted verification studies involving simple VBI problems. These problems include a moving sprung mass acting on a simply supported beam and a two-wheel suspended rigid beam moving on a simply supported beam system. The obtained results for the displacement and velocity at the midpoint of the beam, using the VBI element theory, exhibit a close agreement with the analytical solution based on the first-mode approximation obtained through MATLAB. This successful match between the analytical solution and the results obtained from the implementation in ABAQUS demonstrates the effectiveness and validity of the implemented VBI element theory in ABAQUS. Furthermore, we showcased a practical application example to demonstrate the implementation's real-world relevance. For this purpose, we developed a detailed 3D model to perform an FE analysis of the vehicle-bridge interaction on a multi-girder high-speed railway bridge. In the model, solid elements were used to represent the bridge deck and piers, ensuring a comprehensive representation of their structural behavior. On the other hand, beam elements were employed to model the rails, capturing their dynamic response under vehicle loads effectively. To simulate the connection between the rail and rail pad, we implemented spring and dashpot systems, allowing for an accurate representation of the dynamic interaction between the rail and the rest of the structure during train movement. A good correlation is demonstrated between the Eigen analysis and FE simulation for the train-left time window. Partial validation is illustrated in comparison to the natural frequency of the bridge. An acceptable discrepancy is interpreted by the vehicle load effect and road roughness contribution. Based on [PERSON]'s theory, the interaction between wheels and rail is derived through equations of motion for the vehicle and rail parts. Consequently, an iteration process within a single time step is not required. Thus, the implemented procedure for the VBI element theory in ABAQUS using the user element (UEL) is found to be both applicable and convenient. This successful implementation opens the door for future studies on vehicle-bridge interactions, allowing for more in-depth and comprehensive analyses of such dynamic interactions in high-speed railway systems and other relevant scenarios. The versatility and reliability of the implemented approach make it a valuable tool for exploring and understanding the complexities of  
Figure 15: FFT of the node A2 for free decay. vehicle-bridge interactions, paving the way for further advancements, which enables researchers to conduct broader research on:  
* The effects of culvert properties (its depth from track, stiffness, and geometry) on high-speed train dynamics;
* The impacts of embankment on high-speed train dynamics;
* The investigation of Rayleigh wave propagation in soil under the trainload;
* The exploration of the bridge abutment transition zones and their optimal design;
* The investigation of the soil boundary conditions such as perfect-matched layer;
* The analysis of derailment risk under moderate-to-large earthquake excitation on both regular track and long-span bridges;
* The investigation of the influence of sub-grade and soil nonlinearities on high-speed train dynamics and critical speed analyses. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app13158812/s1](https://www.mdpi.com/article/10.3390/app13158812/s1)]([https://www.mdpi.com/article/10.3390/app13158812/s1](https://www.mdpi.com/article/10.3390/app13158812/s1)), File S1: SM2023.m; File S2:uel_VBIBPS3D.for; File S3: MRB2023.m; File S4: UEL3 Dbeam8w.for; File S5:uel_VBIBPS3D201911116w.for. Conceptualization, Y.D. and L.S. ; methodology, L.S. ; software, Y.D. ; validation, Y.D.