## 2 Model Test of the XCC Pile-Raft Foundation for a Ballastless High-Speed Railway
### Dynamic Loading System  
A train's structure is composed of the carriage, bogie, train wheel, and other structures. The dynamic inertial effects are mainly associated with the train speed and train geometry, such as the dominant axle distance, bogie distance, and carriage length. The dynamic responses of substructures are caused by a series of trains' moving loads. Adjacent trains \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Filling Materials** & \(\gamma\) **(kN/m\({}^{3}\))** & \(w\) **(\%)** & \(Dr\) **(\%)** & **Liquid Limit (\%)** & **Plastic Limit (\%)** & **Plastic Index** & \(Cu\) & \(Cc\) \\ \hline Silty soil & 18.6 & 27.8 & — & 31.0 & 24.1 & 6.9 & 2.76 & 1.71 \\ AB Granular & 21.9 & 5.35 & 63 & — & — & — & 6 & 2.89 \\ Gravel & 23.51 & 7.07 & 92 & — & — & — & 19.23 & 1.34 \\ Supporting layer & 19.43 & 25.5 & 69.5 & — & — & — & 2.42 & 0.93 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical properties and parameters of the geomaterials used. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & \(K\) & \(K_{30}\)**/(MPa/m)** & \(E_{vd}\) **(MPa/m)** \\ \hline Value & 0.97 & 243 & 98 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Tested results of the embankment. Figure 3: Grain-size distributions of geomaterials in ballastless railways. have a cyclic superposition effect. The experiments simulated the cyclic passages of trains by applying loads on the railway with a certain shift in time. Every cycle of the applied loads corresponded to the passage of one carriage. The geometry configuration of the standard carriage of a China Railways high-speed train is shown in Figure 4a. The carriage length \(l_{c}\) is 25 m, axle distance \(l_{wb}\) is 2.5 m, and bogie distance \(l_{ab}\) is 7.5 m.  
Theory and empirical evidence from reported studies ([PERSON] et al. [26], [PERSON] et al. [31], [PERSON] et al. [32], [PERSON] et al. [33]) show that the waveform of the dynamic load of a high-speed strain is similar to the shape of M, with each peak corresponding to the load exerted by an axle of the bogie. These two dominant frequencies of the dynamic response correspond to the length of a carriage (25 m) and the distance of two bogies in the adjacent carriages (7.5 m) ([PERSON] et al. [32]). The experimental results from the full-scale model testing show good agreement with the field measurements, indicating that the applied equivalent loadings at the fastener positions generated by the proposed sequential loading system can replace the train's moving loads at the rail surface in a realistic high-speed railway. The M-shaped dynamic loads generated by the electric hydraulic actuator were applied on the concrete slab track, to simulate the dynamic loading process of high-speed moving train loads. The cyclic load \(F\) in the full-scale model can be described by the following three Fourier equation:  
\[ \begin{array}{l}F(t)=a_{0}+a_{1}\cos(\omega t)+b_{1}\sin(\omega t)+a_{2}\cos (2\omega t)+b_{2}\sin(2\omega t)\\ \hskip 14.226378 pt+a_{3}\cos(3\omega t)+b_{3}\sin(3\omega t)\end{array}  \tag{1}\]  
where \(F(t)\) is the axle load. When the axle load is 157 kN, the train speed \(v\) is 225 km/h and the simulated coefficients in the Fourier series are: \(a_{0}\) = 85.52, \(a_{1}\) = \(-\)71.84, \(a_{2}\) = \(-\)32.29, \(a_{3}\) = 12.89, \(b_{1}\) = 15.24, \(b_{2}\) = 14.97, \(b_{3}\) = \(-\)10.45. The relationship between \(\omega\) and different train speeds is shown in Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Train Speed \(V\) (km/h)** & **45** & **90** & **135** & **180** & **225** & **270** \\ \hline \(\omega\) (rad/s) & 10.06 & 20.128 & 30.192 & 40.25 & 50.321 & 60.387 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Relationship between \(\omega\) and different train speeds. Figure 4: Geometry configuration of the high-speed trains and M-shaped wave. (**a**) Geometry configuration of the high-speed trains, (**b**) M-shaped wave.