# A Vehicle-Bridge Interaction Element: Implementation in ABAQUS and Verification
## 4 Case Study
### Model Introduction and Numerical Results  
In our FE model in ABAQUS, we create the simplified railway bridge according to the design document. The train load is modeled as a suspended vehicle carriage (rigid mass supported by a combination of stiffness and damper) resting on a rigid wheel mass. Moreover, the irregularities between the wheel and the road/rail are considered following the theory [45; 46]. The irregularity factors are assigned to external force terms. However, contact stiffness is not included. In other words, the wheels are directly in contact with the road. The decks, bearings, and piers are modeled as solid elements(C3D8). The beam element is adopted for the rail part with the vertical displacement and rotational DOF, while the other four are fixed. The rail pad is modeled using shell elements. We use spring and dashpot systems to connect nodes on the rail and rail pad instances. Since the track lines are over 1000 m in reality, it should be considered that the model constructed in ABAQUS only covers three spans for the convenience of applying the VBI element technique, which \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parts** & **Young’s Modulus (Pa)** & **Poisson’s Ratio** & **Mass Density** \\ \hline Rail & \(2\times 10^{11}\) & 0.26 & 7850 \\ Rail pad & \(3.25\times 10^{10}\) & 0.2 & 2440 \\ Rail pad reduction layer & \(5.18\times 10^{9}\) & 0.48 & 1700 \\ Box girder & \(1.98\times 10^{10}\) & 0.2 & 2500 \\ Bearing & \(2\times 10^{11}\) & 0.26 & 7850 \\ Pier & \(3.15\times 10^{10}\) & 0.2 & 2420 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameters. Figure 11: Top views of the bearings and piers. Figure 10: Top views of the tracks. will be stated in the data interpretation section. Here, we define the x, z, and y axles as the longitudinal, transverse, and vertical directions of the bridge. Only vertical vibrations, i.e., along the y axle, are recorded and interpreted in this paper. The 3D FE model in ABAQUS is illustrated in Figure 12. Table 2 summarizes the input parameters of the VBI element. The soil dynamics are neglected in the simulation, meaning the piles and soil base-ments are not built since this study focuses on implementing the VBI element theory in ABAQUS. We presented the nodal velocity at node A2 for the time history of the train passing through and 10 s free decay after the train left. The results are shown in Figures 13-15. Figure 13 presents the numerical results in the time domain for velocities in node A2. It also demonstrates the data under a high-frequency filter and low-frequency filter. To better evaluate the accuracy of the simulation, we conducted the simulation for ten more seconds after the train load left to capture the structure's character. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Mass density of beam & \(\rho_{b}\) & 7850 kg/m\({}^{3}\) \\ Section area of beam & \(A_{b}\) & 0.00764 m\({}^{2}\) \\ Bending rigidity of beam about y axle & \(EI_{by}\) & \(6.11\times 10^{-8}\) N m\({}^{2}\) \\ Bending rigidity of beam about z axle & \(EI_{bz}\) & \(6.11\times 10^{-8}\) N m\({}^{2}\) \\ Mass of vehicle carriage & \(M_{v}\) & 10,000–11,000 kg \\ Mass of vehicle wheel & \(M_{w}\) & 675 kg \\ Stiffness of vehicle suspension & \(k_{sus}\) & \(1.4\times 10^{6}\) N/m \\ Damping of vehicle suspension & \(c_{sus}\) & \(3\times 10^{4}\) N s/m \\ Initial positions for wheels & \(x_{0}\) & \(-\)77.45–11 m \\ Train velocity & \(v\) & 60 km/h \\ Wheel number & \(n_{c}\) & 16 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Model parameters applied in the UEL subroutine. Figure 12: The 3D FE model of railway bridge in ABAQUS. Figure 14 shows the numerical results for velocities of the same node after the train left around 10 s. It also illustrates FE data under the high-frequency filter and low-frequency filter. A free decay process is simulated in this part.