## 3 Modeling Approach  
The Establishment of Finite Element Model of Bridge and the Calculation and Analysis of Its Natural Vibration Characteristics  
In the process of modeling in this paper, space beam elements were used to simulate all components, and the interaction between piles and soil was simulated using the m method. The elastic modulus E, density \(\rho\), Poisson's ratio \(\upmu\), and other parameters of all components were set according to the current norms. The finite element analysis model of the 10 \(\times\) 25 m simply supported beam bridge selected in this paper is shown in Figure 5.  
The natural vibration characteristics of a bridge are among the most important parameters for reflecting its dynamic characteristics, and serve as the basis for vehicle-bridge coupled vibration analysis. These characteristics include natural frequencies and mode shapes, which are closely related to the structural configuration, mass distribution, stiffness distribution, and material properties of the bridge. They can be used to evaluate the bridge stiffness and validate the finite element model. Based on the 10 \(\times\) 25 m simply supported girder bridge model established previously, we calculated and analyzed its natural vibration characteristics. The first three natural frequencies corresponding to lateral bending, vertical bending, and longitudinal vibration modes are summarized in Table 1, while the mode shapes for the first-order lateral bending, vertical bending, and longitudinal vibration are shown in Figures 6-8.  
Figure 4: Section of pier top and pier body (unit: cm).  
Figure 3: Elevation of pier (unit: cm).  
Figure 5: Finite element analysis model of 10 \(\times\) 25 m simply supported beam bridge.  
Figure 6: Transverse bending mode of the first step of the 10 \(\times\) 25 m simply supported beam bridge.  
As shown in the frequency analysis results, the first vertical bending mode frequency of the 10 \(\times\) 25 m simply supported beam is 5.319 Hz, which exceeds the minimum requirement of 3.600 Hz (i.e., 90/L, where L = 25 m) specified in the Design Code for Medium and Low Speed Maglev Transportation (2018) (hereinafter referred to as the Code). The initial vibration modes are dominated by longitudinal floating displacements, indicating relatively weak longitudinal constraints, which is consistent with the structural characteristics of a simply supported beam bridge. Furthermore, the transverse bending mode appears before the vertical bending mode, suggesting that the bridge's transverse stiffness is lower than its vertical stiffness.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Structural Mode Shape** & \begin{tabular}{c} **The Order of the Mode** \\ **Shape** \\ \end{tabular} & \
\begin{tabular}{c} **Natural Frequency** \\ **(Hz)** \\ \end{tabular} &
\begin{tabular}{c} **Characteristics of the Mode** \\ **Shapes of a Structure** \\ \end{tabular} \\ \hline \multirow{3}{*}{Lateral bending of the girder} & 9 & 1.139 & Lateral flexural behavior of \\  & 11 & 1.268 & pier-girder system \\  & 12 & 1.403 & \\ \hline \multirow{3}{*}{Vertical flexure of the girder} & 19 & 5.319 & Vertical flexural vibration of \\  & 20 & 5.340 & pier-girder assembly \\  & 21 & 5.387 & \\ \hline \multirow{3}{*}{Longitudinal deflection} & 1 & 0.916 & Longitudinal floating behavior of \\  & 2 & 0.918 & pier-girder system \\  & 3 & 0.940 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of natural vibration frequency of 10 \(\times\) 25 m simply supported beam bridge.  
Figure 7: Vertical bending mode of the first order of 10 \(\times\) 25 m simply supported beam bridge.