## 3 Vehicle-Bridge Coupling Analysis for Corrugated Steel Web Box
### Simulation Conditions and Calculation Results  
Based on the three-dimensional contact constraint method, the dynamic response of the vehicle-bridge coupling system of the corrugated-web box girder bridge is analyzed. To investigate the effect of vehicle speed, pavement grade, and the dimension of pavement roughness on the impact factor of the corrugated steel web bridge, a total of 36 simulation conditions were considered. The vehicle speed ranges from 20 km/h to 120 km/h, covering the typical driving speeds of vehicles on bridges. The ABC-level pavement roughness was designed according to the Chinese standard GB/T 7031-2005 [37]. Additionally, two-dimensional and three-dimensional pavement roughness were specifically considered to investigate the effect of pavement roughness dimension on the impact factor. All simulation conditions are summarized in Table 5. The vertical displacement time history data of the bridge deck node at the mid-span of the bridge's central axis is output, as shown in Figure 10. Due to space limitations, only a partial set of response results is presented.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model Parameter** & **Symbol** & **Unit** & **Parameter Value** \\ \hline Vehicle body mass & M\({}_{1}\) & t & 24.81 \\ Pitching moment of inertia & I\({}_{\text{zx}}\) & kg\(\cdot\)m\({}^{2}\) & 172,160 \\ Rolling moment of inertia & I\({}_{\text{zy}}\) & kg\(\cdot\)m\({}^{2}\) & 31,496 \\ Suspension and wheel mass & m\({}_{\text{i}}\) & t & 0.73 \\ Upper spring stiffness & K\({}_{\text{u}}\) & kN/m & 727.81 \\ Upper damping coefficient & C\({}_{\text{u}}\) & kN\(\cdot\)s/m & 2.19 \\ Lower spring stiffness & K\({}_{\text{d}}\) & kN/m & 1972.9 \\ Lower damping coefficient & C\({}_{\text{d}}\) & kN\(\cdot\)s/m & 0 \\ Distance from the front axle to the & L\({}_{1}\) & m & 4.56 \\ vehicle center of mass & & & \\ Distance from the rear axle to the & L\({}_{2}\) & m & 1.69 \\ vehicle center of mass & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Vehicle model parameters.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Order** & **Frequency** & **Mode Characteristics** \\ \hline
1 & 4.84 Hz & First-order symmetric vertical bending \\
2 & 12.92 Hz & First-order antisymmetric vertical bending \\
3 & 12.94 Hz & Bending-torsion coupling \\ \hline \hline \end{tabular} This paper selects the standard two-axle spatial vehicle model commonly used by scholars [19; 32] in the study of vehicle–bridge coupling vibration. The model considers the pitch, roll, and the floatation and sinking degrees of freedom between the vehicle body and the wheelset. The detailed parameters are shown in Table 4.  
\end{table}
Table 3: Dynamic characteristics of the bridge model.  
In the mid-span node vertical displacement time history curve, three stages can be identified: vehicle entry onto the bridge, crossing the bridge, and exit from the bridge. The entry stage appears as a horizontal segment in the time history curve, indicating that the vehicle has not yet exerted force on the bridge. During the crossing stage, the bridge is in a forced vibration phase, where the time history curve of the bridge nodes includes both the static load effect from the vehicle on the bridge and the bridge vibration induced by the vehicle. Since bridge damping is not considered, after the vehicle exits the bridge and the excitation ends, the bridge enters a free vibration phase.