# Seismic Response Analysis of Concrete Box-Type Subgrade in High-Speed Railways
## 3 Computational Methods and Models
### Seismic Wave Input  
Earthquakes are caused by violent movements in the earth's crust, and unlike traditional mechanical load characteristics, they propagate and load in the form of acceleration. Due to the small span of the box-type subgrade structure, the effects of wave propagation and non-uniform excitation can be neglected. The direct acceleration method is selected to input the seismic wave excitation at the bottom of the box-type subgrade with a stiffness spring unit. The structure employs Rayleigh damping, with the damping ratio \(\xi\), taken as 0.05, and the damping coefficients \(\alpha\) and \(\beta\) are calculated according to Equation (4) below.  
\[\alpha=2h\frac{\omega_{1}\omega_{2}}{\omega_{1}+\omega_{2}}\quad\beta=2h\frac {1}{\omega_{1}+\omega_{2}} \tag{4}\]  
where \(\omega_{1}\) and \(\omega_{2}\) represent the first-order frequency and the primary frequency contributing to the longitudinal vibration mode of the structure.  
This paper selects the classic Wenchuan Wolong earthquake wave for analysis. To approximate the actual state of seismic waves, the input of multi-dimensional seismic waves is considered [34]. In the 3D track model, the direction along the longitudinal axis of the box-type subgrade is defined as the primary horizontal direction; the direction perpendicular to the longitudinal axis of the box-type subgrade is defined as the secondary horizontal direction, and the direction perpendicular to the track plane is defined as the vertical direction, as shown in Figure 8a. Considering the computational time cost, the seismic wave near the peak value of vibration acceleration is truncated, and values are extracted at certain time intervals. Referring to the provisions of Clause 7.2.4 of the \"Seismic Design  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Beam Structure** & **Vertical Natural Frequency** & **Horizontal Natural Frequency** \\  & **(Hz)** & **(Hz)** \\ \hline
16.99 m Box-Type Subgrade & 23.10 & 9.24 \\
24 m Simply Supported Beam & 11.25 & 21.34 \\
32 m Simply Supported Beam & 8.08 & 15.59 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparative data for conventional single-span supported beams.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Order** & **Frequency (Hz)** & **Box-Type Subgrade Deformation State** \\ \hline
1 & 9.24 & Lateral deformation in the same direction \\
2 & 10.74 & Lateral deformation in opposite directions \\
3 & 13.84 & Lateral deformation in opposite directions \\
4 & 23.10 & Vertical deformation in the upward and downward directions \\
5 & 23.21 & Vertical torsional deformation \\
6 & 24.64 & Vertical torsional deformation \\
7 & 25.00 & Horizontal torsional deformation \\
8 & 26.15 & Horizontal torsional deformation \\
9 & 27.19 & Vertical torsional deformation \\
10 & 27.50 & Vertical torsional deformation \\ \hline \hline \end{tabular}
\end{table}
Table 5: Natural frequency of box-type subgrade.  
Code for Railway Engineering\", the actual seismic wave is adjusted for peak value as per Equation (5), and the adjustment ratio is based on the \"Seismic Design Code for Buildings\" set as the primary horizontal direction: secondary horizontal direction: vertical direction = 1:0.85:0.65. Figure 8b below shows the seismic wave curve adjusted to 0.4 g.  
\[a_{h}^{\prime}(t)=\frac{A_{g}}{A_{\max}}\cdot a_{h}(t) \tag{5}\]  
Here, \(a_{h}^{\prime}(t)\) is the adjusted horizontal ground motion acceleration time-history curve; \(a_{h}(t)\) is the horizontal ground motion acceleration time-history curve before adjustment; \(A_{g}\) is the maximum ground motion value determined according to the seismic fortification intensity in the specification; \(A_{max}\) is the maximum value of the acceleration time-history curve recorded in the actual seismic wave data.