# Seismic Response Analysis of Concrete Box-Type Subgrade in High-Speed Railways
## 3 Computational Methods and Models
### Finite Element Model  
Based on the theory, a finite element model is established using the nonlinear finite element software ABAQUS (v.2018). The model utilizes the domestically developed CRTS \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Seismic Ground Motion** & **Frequent Earthquakes** & **Design Earthquakes** & **Rare Earthquakes** \\ \hline Structures & Bridges & Subgrades, retaining walls, tunnels, abutments, and connections between upper and lower bridge structures & Bridges with reinforced concrete \\ Seismic Design Objectives & Meet Seismic Performance Requirement I & Meet Seismic Performance Requirement II & Meet Seismic Performance Requirement III \\ \hline Analysis Methods & General Bridges: Response Spectrum Method & Static method & Simplified method for ductility design of reinforced concrete piers \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Seismic fortification goals and analysis methods of railway engineering structures with different ground motion levels. Figure 2: Model of ballastless track and box subgrade. Type-III slab track system in China, which consists of rails, fasteners, track slabs, self-compacting concrete, and base plates from top to bottom. This paper conducts a refined modeling of the ballastless track, where each track component is meshed with 8-node solid elements (C3D8R), reinforcement is simulated with T3D2 elements, and fasteners are of the Type WJ-8 with a spacing of 0.63 m. Considering the significant displacements that structures may undergo under seismic actions, nonlinear fasteners are employed for simulation. The force-displacement relationships of the fasteners in various directions are as follows [26]:  
The longitudinal force-displacement relationship can be expressed as follows:  
\[r_{L}=\left\{ \begin{aligned} 12x&|x|\leq 2\;\text{mm}\\ 24\text{sign}x&|x|>2\;\text{mm}\end{aligned} \right. \tag{2}\]  
where \(r_{\text{L}}\) is the longitudinal resistance of the fastener, kN; \(x\) is the relative displacement between the rail and the bearing plate, mm. The lateral resistance of the fastener is represented by the test-fitted results as follows:  
\[r_{\text{H}}=\left\{ \begin{aligned} 4.5x&|x|\leq 2\;\text{mm}\\ 9\text{sign}x&|x|>2\;\text{mm}\end{aligned} \right. \tag{3}\]  
In the above equation, \(r_{\text{H}}\) is the lateral resistance of the fastener, kN; \(x\) is the relative displacement between the rail and the sleeper, mm. The vertical stiffness of the fastener depends on the loading condition of the fastener, as well as the vertical stiffness of the pad and the clip. The force-deformation curve of the fastener in the vertical direction is presented in Figure 3.  
In practical engineering, a door-type reinforcement bar is utilized between the track slab and the self-compacting concrete layer to ensure a rigid connection; thus, a tied constraint is applied in the simulation, and surface-to-surface contact is employed between the self-compacting concrete and the base plate. The contact between the reinforcements and the track structure uses the embedding feature in the software. The specific parameters derived from the dimensions and arrangement of the CRTS Type-III slabs on bridges are delineated in Table 4 below, and the model is constructed in Figure 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Density (kg/m3)** & **Elastic Modulus (Pa)** & **Poisson’s Ratio** & **Dimensions (m)** \\ \hline Rebar & 7800 & \(1.9\times 10^{9}\) & 0.3 & HRB400 \\ Rail & 7850 & \(2.1\times 10^{11}\) & 0.3 & 60 kg/m \\ Track Slab & 2500 & \(3.6\times 10^{10}\) & 0.167 & \(2.5\times 2.0\times 5.6\) \\ Self-Compacting Concrete & 2000 & \(3\times 10^{9}\) & 0.167 & \(2.5\times 0.9\times 5.6\) \\ Base Plate & 2500 & \(3.25\times 10^{10}\) & 0.167 & \(2.95\times 2.0\times 5.65\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Ballastless track parameters. Figure 3: Vertical resistance–deformation curve of fastener.