## 2 Numerical Modeling
### Modeling of the Tested Structures
The material parameters of the concrete and steel reinforcement in the structural models are listed in Table 2. These parameters are requested in Concrete02 and Steel02 material models in the OpenSees. In this research, there are uniaxial eccentric structural models and bidirectional eccentric structural models in the dynamic time history analyses. The eccentricities of both types of models are set as 0%, 5%, 10%, 15%, 20%, 25%, and 30% (0% eccentricity corresponds to the symmetric structural model). In the bidirectional eccentric structural models, the eccentricities in the X and Y directions are the same. Figures 3 and 4 show the layouts of the uniaxial eccentric structure and the biaxial eccentric structure. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & \begin{tabular}{c} **Sectional Dimension** \\ **(mm \(\times\) mm)** \\ \end{tabular} & \
\begin{tabular}{c} **Longitudinal Bar** \\ **(mm2)** \\ \end{tabular} &
\begin{tabular}{c} **Hooping Bar** \\ **(mm)** \\ \end{tabular} \\ \hline Beam (X direction) & 250 \(\times\) 500 & Top 1140/bottom 1140 & \(\Phi\)8@100 \\ Grider (Y direction) & 250 \(\times\) 500 & Top 1140/bottom 1520 & \(\Phi\)8@100 \\ Column & 500 \(\times\) 500 & Top 942/bottom 942 & \(\Phi\)10@100 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Component dimensions and reinforcement information. Figure 2: Reinforcement monotonic envelope curves. Figure 4: Plan and vertical layout of the biaxial eccentric structure: (**a**) Plan layout; (**b**) vertical layout of the biaxial eccentric structure in the X direction; (**c**) vertical layout of the biaxial eccentric structure in the Y direction. Note: CM is the center of mass and CR is the rigidity center. Figure 3: Plan and layout of the uniaxial eccentric structure. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Material** & \begin{tabular}{c} **Maximum** \\ **Stress/\({}_{e}\)** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Strain at** \\ **Maximum** \\ **Stress \(\epsilon_{a}\)** \\ **Stress \(\epsilon_{b}\)** \\ \end{tabular} & \begin{tabular}{c} **Ultimate Stress** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Strain at** \\ **Ultimate Stress** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Ratio between** \\ **Unloading** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Tensile Strength** \\ **Softening** \\ **Stiffness** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Tension** \\ **Softening** \\ **E\({}_{e}\) **(MPa)** \\ \end{tabular} \\ \hline \begin{tabular}{c} Unconfined \\ concrete \\ \end{tabular} & \(-27.4\) & \(-0.003\) & \(-5.5\) & \(-0.01\) & \(0.1\) & \(2.7\) & \(1.4\times 10^{3}\) \\ \begin{tabular}{c} Confined \\ concrete \\ \end{tabular} & \(-35.6\) & \(-0.006\) & \(-7.1\) & \(-0.012\) & \(0.1\) & \(3.6\) & \(1.8\times 10^{3}\) \\ \hline \begin{tabular}{c} **Yield stress \(\mathbf{Y_{y}}\)** \\ **(MPa)** \\ \end{tabular} & \
\begin{tabular}{c} **Modulus of** \\ **Stress \(\mathbf{E_{u}}\)** \\ **(MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Strain-** \\ **hardening ratio** \\ **(MPa)** \\ \end{tabular} & **Parameter R0** & **Parameter cR1** & **Parameter cR2** \\ \hline Steel & 457 & \(2\times 10^{5}\) & \(0.01\) & \(18\) & \(0.925\) & \(0.15\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material parameters of the concrete and steel reinforcement in the structural models. The eccentricity is obtained by using Equation (2):  
\[\omega_{\rm i}=e_{\rm i}/L_{\rm i} \tag{2}\]  
where \(\omega_{\rm i}\) denotes the eccentricity of the structural model and \(e_{\rm i}\) denotes the eccentric distance, which is the distance between the mass center and rigidity center. \(L_{\rm i}\) denotes the building outline length in eccentric direction. For mass eccentric structural models, the basic assumptions are as follows:  
1. Each floor is a rigid floor. 2. The total mass and moment of inertia of each floor are appointed at the geometric center of the floor. 3.