## 4 Pseudo-Dynamic Test System
### Pseudo-Dynamic Test System  
Figure 8 shows the method and pseudo-dynamic specimen setting of the pseudo-dynamic test system used to evaluate the seismic performance of the CCSS method developed in this study. The pseudo-dynamic test conducted in this study can be modeled as a two-degree-of-freedom system, as shown in Figure 8. It consists of a numerical analysis part that is based on the input seismic wave from the control computer and the loading part of the structural specimen. The seismic response displacement calculated in the pseudo-dynamic test was applied to the specimen by two lateral actuators. The restoration force of the framework is measured mechanically during structural testing and used by the control computer to determine the seismic response displacement.  
Data conversion was performed using analog to digital or digital to analog converters [19]. In the pseudo-dynamic test, the seismic response was numerically calculated using the closed-loop control system. For the analysis by the control computer and numerical integration, the pseudo-dynamic testing program developed by MTS [20] was used. In the loading test, the seismic response in the next step was calculated using the equation of motion shown in Equation (1), and based on the numerical integration method, the restoration force of the framework specimen owing to the displacement measured using a displacement meter (LVDT) and the seismic response value in the current step, including the input seismic acceleration.  
\[M\ddot{y}(t)+\mathrm{C}\dot{y}(t)+r(t)[=Ky(t)]=-M\ddot{y}_{0} \tag{1}\]  
Figure 8: Proposed pseudo-dynamic testing system.  
where \(M\), \(C\), and \(K\) are the mass, damping, and stiffness matrix of the structure, respectively, y is the relative displacement vector of each foundation layer mass, \(r\) is the restoration force vector, and \(\ddot{y_{0}}\) is the input ground acceleration.  
The \(\alpha\)-method [21] was used for the numerical integration of the equation of motion. Equations (2)-(4) show the algorithms for the numerical integration of the pseudo-dynamic test.  
\[Ma_{i+1}+(1+\alpha)\mathrm{C}v_{i+1}-\alpha\mathrm{C}v_{i}+(1+\alpha)-\alpha r _{i}=(1+\alpha)f_{i+1}-\alpha f_{i} \tag{2}\]  
\[y_{i+1}=y_{i}+\Delta tv_{i}+\Delta t^{2}\bigg{[}\bigg{(}\frac{1}{2}-\beta \bigg{)}a_{i}+\beta a_{i+1}\bigg{]} \tag{3}\]  
\[v_{i+1}=v_{i}+\Delta t[(1-\gamma)a_{i}+\gamma a_{i+1}] \tag{4}\]  
where \(y_{i}\), \(v_{i}\), and \(a_{i}\) are the joint displacement, velocity, and acceleration at the same time as \(i\Delta t\), \(\Delta t\) is the integration time interval, \(r_{i}\) is the restoration force vector at the joint, and \(f_{i}\) is the external load vector (\(-M\ddot{y_{0}}\)).  
For elastic structures, \(r_{i}=Ky_{i}\) (\(K\): elastic stiffness matrix of the structure) holds, and \(\alpha\), \(\beta\), and \(\gamma\) are variables for controlling the numerical characteristics of the algorithms; \(-5\leq\alpha\leq 0\), \(\beta=\frac{(1-\alpha)^{2}}{4}\), and \(\gamma=\frac{1}{2}-\alpha\) indicates unconditional stability. The displacement response in the next step was then calculated using Equations (1)-(4) based on the stiffness (\(K\)), mass (\(M\)), and stiffness proportional damping coefficient (\(C\)) of the structure. The damping ratio (\(\xi\)) was set to 0.03.  
Meanwhile, the seismic displacement value was applied to the structural specimen using the 1000 kN actuators installed on the first and second floors. The horizontal displacement value used to calculate the seismic response displacement value was then measured using the 300 mm displacement meters (LVDTs) installed on each floor (see Figure 8).  
As described in Section 3.2, during the pseudo-dynamic test, the 1000 kN oil jack set was used to apply an axial load (1000 kN) of 500 kN to each of the two actual columns (framework). Figure 9 shows time-acceleration relations of the normalized ground motion accelerations used in the pseudo-dynamic test, respectively, together with their acceleration response spectrum.