# Dynamic Response of Railway Subgrade Under Train Load and Freeze-Thaw Action
## 2 Three-Dimensional Finite Element Model of Railway Subgrade
### Parameter Selection and Meshing  
In this paper, ABAQUS 2020 is used to determine the parameters and establish a three-dimensional finite element model according to the code for design on the subgrade of China's high-speed railway [36]. Figure 2 delineates the finite element model of the double-track ballastless track subgrade structure. The railway incorporates CHN60 rails conforming to the standard gauge of 1.435 m, equipped with 300-1a fasteners spaced at 0.625 m. The CRTS-II ballast slab with dimensions of 2.55 m in width and 0.2 m in thickness is supported by a cement-asphal-mortar (CAM) layer that is 0.2 m wide and 0.03 m thick. The trapezoidal support layer with dimensions varying from 2.95 to 3.25 m in width and 0.3 m in thickness is under the ballast slab. The subgrade structure is composed of a surface layer (0.4 m thick), a subgrade layer (1.8 m thick), and an embankment layer (2 m thick), extending 13.6 m in width with a slope gradient of 1:1.5. In adherence to the specified standards, the line spacing for the dual-track railway is established at 5 m. The subgrade's dimensions are critically considered to ensure that the energy transferred from the dynamic loads above is adequately attenuated; hence, the subgrade is dimensioned at 47.7 m in width and 20 m in thickness. The 300-1a fasteners, in conjunction with the rails, are modeled as line-elastic connections, represented by spring-damping elements with a stiffness of 45 kN/mm and a damping coefficient of 45,000. The Mohr-Coulomb model is adopted for soil. The parameters of the track-subgrade model are taken according to the code for design on the subgrade of China's high-speed railway [36]. The parameters are shown in Table 1, where \(E\) is Elastic modulus, \(\mu\) is Poisson's ratio, \(c\) is the Cohesive force of soil, and \(\varphi\) is the internal friction angle of soil. Figure 2: Three-dimensional ballastless track–subgrade model. The existing research shows that the physical and mechanical parameters of unfrozen soil change little. The physical and mechanical parameters of frozen soil below 0 \({}^{\circ}\)C change greatly with temperature. The mechanical and thermophysical parameters of soil are shown in Tables 2 and 3, and the \(f\) and \(u\) represent the frozen state and the unfrozen state, respectively [27]. Within the context of finite element analysis, the maximum allowable mesh size within the finite element model is governed by the minimum wavelength associated with the highest frequency induced by the dynamic train loads. It is established that a Rayleigh wavelength should encompass no fewer than 10 mesh elements [37]. Consequently, the maximum mesh size can be derived from the minimum wavelength utilizing the following calculation formula. \[L_{e}\leq\frac{\lambda_{s}}{6} \tag{8}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(\rho\)**(**kg.m\({}^{-3}\))** & **\(E\)/MPa** & \(\mu\) & **c/MPa** & \(\varphi\)**/\(\circ\) \\ \hline Rail & 7800 & 210,000 & 0.3 & / & / \\ Slab & 3000 & 35,000 & 0.16 & / & / \\ CAM layer & 2000 & 92 & 0.4 & / & / \\ Supporting layer & 2700 & 30,000 & 0.16 & / & / \\ Surface layer of subgrade & 1950 & 21 & 0.24 & 1 & 28 \\ Subgrade fill & 1900 & 21 & 0.24 & 1 & 28 \\ Active layer & 1800 & 34 & 0.45 & 3 & 30 \\ Ice-rich frozen soil & 1600 & 2 & 0.4 & 7 & 12 \\ Ice-poor frozen soil & 1800 & 500 & 0.25 & 32,000 & 14 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters of the ballastless track–subgrade structure.