# Nonlinear Solid-Fluid Coupled Seismic Response Analysis of Layered Liquefiable Deposit
## 2 Numerical Model of Liquefiable Site  
### Liquefiable Site Model  
The Open System for Earthquake Engineering Simulation (OpenSees) [12] is an open-source software framework that can perform the pseudo-static and dynamic analyses with the finite element method and add additional constitutive models and algorithms based on user needs. Moreover, OpenSees has good capability for a nonlinear analysis of soil and structure, and can perform a comprehensive analysis from the material stress-strain level to the section level to the element level, and finally, the structure level. This computational platform, OpenSees, has been employed to simulate the nonlinear behaviors of soil-structure interaction systems under static and dynamic loading by many researchers [13; 14; 15; 16].  
Figure 1 shows the schematic diagram of a generic site consisting of three soil layers, following existing studies [17; 18] without the initial static shear stress. The soil domain was discretized using four-node quadrilateral elements with _u-p_ formulations [19] such that water pressure and soil deformation could be computed simultaneously. The overall depth and width of the soil deposit are 21 m and 50 m, respectively. The soil deposit mainly consists of sand with relative density, \(D_{r}=35\%\) and 75% and crust stayed soil. The water table is assumed to be coincident with the ground surface level. Therefore, the ground surface is supposed to be a free drainage surface, while the lateral and base boundaries are impermeable. Horizontal kinematic constraints are introduced to the nodes on two side boundaries to ensure the same horizontal movement of the two nodes at the same burial depth and effectively simulate the shear deformations of the soil layers under upward propagation of in-plane waves. The analyses consist of two steps: a gravity application step and a dynamic excitation step. Gravity is firstly applied to the model with the nodes fixed at the base to generate the distributions of the initial effective confinement and pore pressure in the soil model for the site before the application of the seismic excitation. The permeabilities for all of the elements in the gravity analysis were initially set to 1.0 m/s to facilitate the development of hydrostatic conditions in the model. Subsequently, the seismic input motion is applied to the base of the soil model as a force-time history at the node, which shares equal degrees-of-freedom with the Lysmer-Kuhlemeyer [20] dashpot. This dashpot depicts the finite rigidity of the underlying bedrock by using a bedrock shear wave velocity of 760 m/s for the soil profiles and a mass density of 2200 kg/m\({}^{3}\). Meanwhile, the nodes at the base of the soil model are only constrained in the vertical direction and are left free to move in the horizontal direction.  
Figure 1: Configuration of numerical model of generic liquefiable site.  
In the dynamic analysis, the constant average acceleration Newmark integrator method with the parameters \(\gamma\) = 0.6 and \(\beta\) = 0.3025 is used in order to resolve the integration in time. The energy increment test with a tolerance of 1.0 \(\times\) 10\({}^{-6}\) was utilized to control convergence checking. For each time step, the solution is obtained using the Krylov-Newton algorithm with Krylov subspace acceleration [21]. Rayleigh damping with an initial damping ratio of 3% is applied for the purpose of stabilizing the numerical results in dynamic numerical analyses. For the soil deposit, the first- and second-order natural frequencies of the site are 0.46 Hz and 1.42 Hz, which were obtained through model analysis. The damping coefficients for the mass matrix and stiffness matrix are calculated by selecting the damping ratio of 3% and the ground natural frequencies are equal to 0.124 and 0.0008 by numerical model analysis, respectively.