# Nonlinear Solid-Fluid Coupled Seismic Response Analysis of Layered Liquefiable Deposit
## 2 Numerical Model of Liquefiable Site
### Input Ground Motions  
In this study, two acceleration time histories were used as input motions in the horizontal direction of the elastic bedrock for the numerical analyses. The two motions were recorded during the 1940 Imperial Valley earthquake at the El-Centro station (termed as El-Centro motion in this study) and the 1995 Kobe earthquake at the KJMA station (termed as Kobe motion in this study), respectively. The El-Centro motion, with an amplitude of 0.348 g and the main shaking duration being about 26 s, has the obvious characteristics of the middle- to far-field ground motions. The frequency content of the El-Centro motion concentrates in a range of 0-10 Hz. The Kobe motion, with an amplitude of 0.83 g and the main shaking duration being about 10 s, has typical characteristics of near-fault ground motion with a velocity pulse and the concentration range of the excitation frequency is 0-5 Hz. Figure 3 presents the acceleration time histories, Arias intensity time histories, Fourier spectra, and the energy flux time histories of the two input motions after being scaled to a peak bedrock acceleration of 0.30 g. It is worth noting that the energy flux for the scaled El-Centro motion is 2218 J/m\({}^{2}\)/s, which is more than twice that of the scaled Kobe motion (991 J/m\({}^{2}\)/s). The more detailed information of the two selected input motions is introduced, as shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Parameters** \\ \end{tabular} } & \multicolumn{1}{c}{\(D_{r}\)} & Relative density & 35\% & 75\% \\ \cline{2-4}  & \(G_{o}\) & Shear modulus coefficient & 476.0 & 890.0 \\ \cline{2-4}  & \(h_{po}\) & Contraction rate parameter & 0.53 & 0.63 \\ \cline{2-4}  & \(\rho\) & Mass density (ton/m\({}^{3}\)) & 1.7 & 2.1 \\ \hline \multirow{6}{*}{\
\begin{tabular}{c} **Secondary** \\ **parameter** \\ \end{tabular} } & \(c_{max}\) & Maximum void ratio & 0.8 & 0.8 \\ \cline{2-4}  & \(\epsilon_{min}\) & Minimum void ratio & 0.5 & 0.5 \\ \cline{2-4}  & \(\varphi_{cv}\) & Critical state effective friction angle & 33 & 40 \\ \cline{2-4}  & \(c_{z}\) & Controls the strain level at which & 200.0 & 200.0 \\ \cline{2-4}  & \(h_{0}\) & Variable that adjusts the ratio of & 0.4 & 0.4 \\ \cline{2-4}  & \(h_{0}\) &
\begin{tabular}{c} Variable that adjusts the ratio of \\ plastic modulus to elastic modulus \\ \end{tabular} & 2 & 0.97 \\ \cline{2-4}  & \(n_{d}\) & Dilatancy surface parameter & 0.1 & 0.1 \\ \cline{2-4}  & \(n_{u}\) & Poisson’s ratio & 0.3 & 0.3 \\ \cline{2-4}  & \(R\) & Critical state line parameter 1 & 1.5 & 1.5 \\ \cline{2-4}  & \(m\) & Yield surface constant & 0.01 & 0.01 \\ \cline{2-4}  & \(Q\) & Critical state line parameter 2 & 10.0 & 10.0 \\ \hline \multicolumn{4}{c}{Permeability coefficient/(m/s)} & \(6.88\times 10^{-4}\) & \(1\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: PM4S and input parameters of the loose and dense sand layers.  
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Parameters**} & \multicolumn{1}{c}{**Crust Clay**} \\ \hline \multicolumn{1}{c}{Saturated soil mass density, \(\rho\)/(ton/m\({}^{3}\))} & 1.60 \\ \multicolumn{1}{c}{Reference shear modulus, \(Gr\)/kPa} & 50,000 \\ \multicolumn{1}{c}{Reference bulk modulus, \(Br\)/kPa} & 250,000 \\ \multicolumn{1}{c}{Cohesion, \(c\)/kPa} & 33 \\ \multicolumn{1}{c}{Octahedral peak shear strain, \(\gamma_{max}\)} & 0.1 \\ \multicolumn{1}{c}{Friction angle, \(\varphi\)/\({}^{\circ}\)} & 0.0 \\ \multicolumn{1}{c}{Reference mean effective confining} & 100 \\ \multicolumn{1}{c}{pressure/kPa} & 0.0 \\ \multicolumn{1}{c}{Pressure coefficient} & 20 \\ \multicolumn{1}{c}{Number of yield surfaces} & 20 \\ \multicolumn{1}{c}{Permeability coefficient/(m/s)} & \(1.2\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Constitutive model parameters of the crust clay.  
Figure 3: _Cont._