# Hybrid Simulation of Seismic Responses of a Typical Station with a Reinforced Concrete Column
## 3 Numerical Substructure  
### Soil Modeling  
The FE model was established by the object-oriented and open source software OpenSees, with a width of 200 m (11.7 times of structural width) and a height of 58 m, as shown in Figure 1. The thickness of plane-strain model was 3.5 m because the column spacing was 3.5 m [19]. The central column was located at the center position of the model along the out-of-plane direction. The ground surface was set to be free. The fixed boundary was set at the bottom of the FEA model. The boundaries at the lateral sides of the model were set as the equal displacement boundary in order to reduce the impact of seismic wave reflection. All boundaries were supposed to be undrained.  
The soil was modeled with the pressure-dependent multi-yield-surface J2 plasticity material model [20]. In the multi-yield-surface clay model, it was postulated that the shear behavior was dependent on the soil confining pressure. In this constitutive model, a set of von Mises yield surfaces with different sizes formed a hardening zone, as shown in Figure 2. The outermost surface defined the shear strength (or failure) envelope. Nonlinear kinematic hardening and associative flow rules were employed to reproduce the Masing-type hysteretic behavior. The soil layers and parameters of the site are listed in Table 1.  
The quad elements were applied to simulate the deposit, and such quad elements were four-node plane strain elements. The size of the soil elements should be less than one-eighth of the wavelength corresponding to the maximum frequency of the input seismic wave [21]. The Rayleigh damping  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Soil Layer** & **Depth (m)** & \begin{tabular}{c} **Unit Weight** \\ **(kN/m3)** \\ \end{tabular} & \begin{tabular}{c} **Friction** \\ **Angle (\({}^{\circ}\))** \\ \end{tabular} & \begin{tabular}{c} **Cohesion Yield** \\ **Stress (kPa)** \\ \end{tabular} & \
\begin{tabular}{c} **Young’s Modulus** \\ **(MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} \\ \hline Fill & 0–1 & 19 & 20 & 70 & 101.3 & 0.333 \\ Holocene clay & 1–2 & 19 & 20 & 70 & 100.3 & 0.333 \\ Holocene sand & 2-4.8 & 19 & 20 & - & 147.9 & 0.32 \\ Pleistocene sand & 4.8–8.0 & 19 & 20 & - & 195.9 & 0.40 \\ Pleistocene clay & 8–17 & 19 & 20 & 70 & 290.3 & 0.30 \\ Pleistocene gravel & 17–58 & 20 & 20 & - & 560.0 & 0.26 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Soil material properties.  
Figure 1: The stratum-station system FEA model (units in m).  
Figure 2: Pressure-dependent soil material model [20].  
model was applied in the analysis procedure. To be specific, the target damping ratio was set as 0.02. The stiffness parameter and the mass parameter were determined by Equation (2). The frequencies of first order and third order were derived to determine the damping coefficients.  
\[\left\{\begin{array}{c}\alpha\\ \beta\end{array}\right\}=\frac{2\xi}{\omega_{1}+\omega_{3}}\left\{\begin{array} []{c}\omega_{1}\omega_{3}\\ 1\end{array}\right\} \tag{2}\]  
where \(\alpha\) is the mass parameter, \(\beta\) is the stiffness parameter, \(\xi\) is the damping ratio, \(\omega_{1}\) is the angular frequency of the first mode, and \(\omega_{3}\) is the angular frequency of the third mode.