# Hybrid Simulation of Seismic Responses of a Typical Station with a Reinforced Concrete Column
## 3 Numerical Substructure
### Structure Modeling  
The finite element model of the stratum-station is shown in Figure 1. The subway station was a one-story two-bay RC frame with section dimensions given in Figure 3. The cross-section of the subway station measured 11.5 m \(\times\) 7.17 m. The buried depth of the station structure was 4.8 m, and the column spacing was 3.5 m. The thickness of the station roof was 0.8 m, and the thickness of the bottom plate was 0.85 m, while the thickness of the sidewalls was 0.7 m. The cross-section of the center column measured 0.4 m \(\times\) 0.4 m. The structure elements and the surrounding soil elements were fixed together, and the sliding effect of soil and structure was neglected in the analysis. The subway station was modeled with distributed plasticity Euler-Bernoulli beam elements, which had four Gauss-Legendre integration points. The displacement based element was adopted to simulate the slabs, lateral walls, and central column. The element assumed the displacement field derived the formulation through the virtual displacement principle. For a given element displacement vector \(q\), the section deformation vector \(d(x)\) at location \(x\) is approximated as below [22],  
\[d(x)=N_{d}(x)q \tag{3}\]  
where \(N_{d}(x)\) is the displacement interpolation function, and the Hermite polynomials can be used here. Then, the section force vector \(D(x)\) can be obtained through the fiber section model,  
\[D(x)=K_{s}(x)d(x) \tag{4}\]  
where \(K_{s}(x)\) is the section stiffness matrix. According to the virtual displacement principle, the element resisting force \(Q\) and stiffness \(K_{e}\) are expressed by Equation (5),  
\[ \begin{array}{l}Q=\int_{t}N_{d}^{T}(x)D(x)dx\\ K_{e}=\int_{L}N_{d}^{T}(x)K_{s}(x)N_{d}(x)dx\end{array}  \tag{5}\]  
where the integral can be computed through Gauss quadrature. Figure 3: Section of the subway station (units in mm). The Concrete 01 constitutive model as used to construct a uniaxial Kent-Scott-Park concrete material object with degraded linear unloading/reloading stiffness. In addition, no tensile strength was considered for such concrete material. The Concrete 01 model was utilized for the concrete, whilst the Steel 02 model was utilized for the reinforcements of the central column specimen. The whole structure had a density of 25 kN/m\({}^{3}\), a Poisson's ratio of 0.2, and a Young's modulus of 32,500 MPa. The main parameters of the Concrete 01 model were the peak compressive yielding stress 36.4 MPa and peak strain 0.002, while the extreme compressive yielding stress 7.3 MPa and peak strain 0.004. The longitudinal reinforcements in the central column had a density of 7800 kN/m\({}^{3}\), a Poisson's ratio of 0.3, a Young's modulus of 200 GPa, and a yielding stress of 431.5 MPa. The hardening ratio was set to be 0.01. During the hybrid simulation process, the appropriate experimental substructure should be adopted to satisfy the boundary conditions. Figure 4 shows that the experimental specimen was selected based on the inflection point of the central column. It was assumed that the inflection point of the central column was located in the center. Therefore, the half bottom central column was adopted as the experimental substructure with a height of 3.56 m.  
Two analysis steps should be involved. Firstly, geostatic analysis procedure should be carried out to balance the gravity load. Then, the dynamic analysis procedure should be carried out with the designated accelerations. Actually, only the dynamic process was conducted in this study because thrust was not applied on the top of the column specimen. The end point of the Euler-Bernoulli beam element shared three degrees of freedom, corresponding to the axial direction, tangential direction, and rotational direction. Therefore, axial force, shear force, and bending moment should be taken into consideration in the traditional substructure hybrid simulation. In the hypothetical model, the top point of the column was the inflection point, and the position of inflection point did not change in the hybrid simulation test.