# Dynamic Response of Railway Subgrade Under Train Load and Freeze-Thaw Action
## 2 Three-Dimensional Finite Element Model of Railway Subgrade
### Train Load  
The dynamic response of subgrade under the action of high-speed train and the long-term performance of railway subgrade soil are two major problems in engineering. Considering the actual engineering situation, the simulation parameters of train load in this paper are high-speed train CRH3 with axle weight of 14 tons. When a carriage with the same set of bogies two wheels in the first wheel through a section, a wave peak is generated, the latter wheel again through the section, another wave peak is generated, it will produce a M-shaped wave, the train load as shown in Figure 1.  
The ABAQUS/Explicit module is dedicated to the simulation of transient dynamic problems, such as explosions and impacts, it uses explicit integral solutions, computed in small time incremental steps for the solution of dynamic problems. The unit finite element equations for transient dynamics calculations are [26]:  
\[[M]\{\ddot{u}\}+[C]\{\dot{u}\}+[K]\{u\}=\{P\} \tag{1}\]  
where \([M]\), \([C]\), and \([K]\) are the unit mass matrix, damping matrix, and stiffness matrix, respectively; \(\{\ddot{u}\}\), \(\{\dot{u}\}\), \(\{u\}\), and \(\{P\}\) are the acceleration, velocity, displacement vectors of the unit and the nodal force loads, respectively.  
The damping coefficient of the roadbed plays a crucial role in influencing its vibration characteristics, where the damping matrix is determined by the Rayleigh linear combination method [27]:  
\[[C]=\alpha[M]+\beta[K] \tag{2}\]  
where coefficients \(\alpha\), \(\beta\) are Rayleigh damping parameters with a following relationship to the commonly used viscous proportional damping coefficient \(\xi\):  
\[\alpha=\frac{2\omega_{j}\omega_{j}}{\omega_{i}+\omega_{j}}\xi \tag{3}\]  
\[\beta=\frac{2\xi}{\omega_{i}+\omega_{j}} \tag{4}\]  
where \(\omega_{i}\) is fundamental frequency of the model, \(\omega_{j}\) is frequency that contributes most to the dynamic response [27].  
The Rayleigh damping parameters are determined as follows. The model analysis of the finite element model is carried out, and the first 40 intrinsic angular frequencies and the vertical response participation coefficients under different frequency excitations are extracted,  
Figure 1: Train axle load distribution (unit: m).  
where the vertical response participation coefficients are larger. It indicates that the vibration mode contributes more to the dynamic response in the vertical direction of the model, the fundamental frequency \(\omega_{i}\) is determined. The frequency with the largest contribution to the dynamic response from the higher-order vibration modes as \(\omega_{j}\) is selected.  
The angular frequency \(\omega_{j}\) denotes the maximum absolute value of the vertical response participation coefficient. Specifically, the values of \(\omega_{i}\) and \(\omega_{j}\) are established as 11.66 rad/s and 14.68 rad/s, respectively. The damping ratio that characterizes the interaction between the roadbed and the underlying ground is conventionally selected within a range of 0.03 to 0.05. For the empirical analysis conducted in this study, the damping ratios are chosen as 0.045 for the roadbed surface layer, 0.039 for the roadbed subgrade, and 0.035 for the ground surface, reflecting the distinct material and structural properties of each layer.