## 5 Three-Dimensional Dynamic Finite Element Analyses  
_Model Development_  
The three-dimensional elastic constitutive finite element model illustrated in Figure 15 was used to calculate the vertical vibration velocities of the track-embankment-pile raft foundation. The elastic dynamic responses for different structure layers were calculated by using dynamic simulations at a reduced scale under train loads. The total length, width, and height of the model was 5, 4, and 5.3 m, respectively. The translation in the z-direction  
Figure 14: Distribution of the vibration velocity amplitude along the vertical direction from the roadbed. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Train Speed** & \(v\) = **90 km/h** & \(v\) = **180 km/h** & \(v\) = **225 km/h** & \(v\) = **270 km/h** \\ \hline
**Depth/m** & **Rate** & **Rate** & **Rate** & **Rate** \\ \hline
0.15 (V5) & 62.03\% & 30.74\% & 33.53\% & 0.47\% \\
0.5 (V6) & 63.29\% & 33.11\% & 35.61\% & 39.57\% \\
0.7 (V7) & 69.62\% & 57.43\% & 60.24\% & 62.56\% \\
1.7 (V8) & 75.95\% & 63.51\% & 66.17\% & 64.45\% \\
2.9 (V9) & 84.18\% & 74.66\% & 67.95\% & 68.01\% \\
4 (V10) & 89.87\% & 78.04\% & 79.82\% & 73.22\% \\
5 (V11) & 94.94\% & 85.84\% & 87.33\% & 89.34\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Vibration attenuation rates of velocity along depth. was restrained on the symmetry boundary. A fixed boundary was applied to the bottom of the model. Infinite elements based on the previous work by [PERSON] [34] and [PERSON] et al. [35] were used on the \(X\) and \(Z\) direction boundaries to represent the infinite boundary condition to absorb \(S\) and \(P\) waves. As for the other four boundaries, infinite elements were applied to represent the infinite boundary condition. The material parameters applied to the infinite elements were the same as their adjacent solid elements. The material parameters for the different structures layers used in the finite element model are listed in Table 9. The railway track, track slab, concrete base, roadbed, concrete raft, supporting layer, and pile were assumed to be liner elastic based on their high stiffness. The embankment, gravel, and silty soil were assumed to be viscoelastic. Rayleigh damping was used to represent the energy-dissipating mechanisms for the viscoelastic material in the simulation model, in which the mass and stiffness proportional damping coefficients were selected to provide slight damping ratios of 2-4% in the frequency range of 3-50 Hz [36; 37; 38]. The mass and stiffness proportional damping coefficients of embankment, gravel, and subsoil were selected and are shown in Table 9.  
In Figure 16, the equivalent moving M-shaped loading wave was used to simulate the train moving load by applying the load on the unit point in the numerical simulation, of which the time and frequency domain features were consistent with the loading waves in the model test. The loading wave includes the passing frequency of the train carriage geometry (\(l_{c}=25\) m), bogie (\(l_{ab}=7.5\) m), and axle distance (\(l_{wb}=2.5\) m). The numerical  
Figure 15: Finite element model of the pile–raft composite foundation. (**a**) finite element mesh, (**b**) wireframe of the model.