## 4 Overview of Pseudo-Dynamic Testing and Result Analysis
### Pseudo-Dynamic Testing System and Test Methods  
Figure 9 is a conceptual illustration of the pseudo-dynamic test system developed in the present study; the figure also shows how specimens are set and tested. As shown in Figure 9, it is a two-degree-of-freedom (TDF) system and composed of the numerical analysis unit, in which the input seismic ground motion is determined by the control computer and the loading test unit in which specimens are actually subjected to testing. The displacement response that has been calculated during the test is actually applied to the specimen using two hydraulic actuators that are installed in the horizontal direction. The restoring force is experimentally measured during the test, and this measurement is fed to the control computer to calculate the corresponding displacement response. Data conversion is performed by an analog-to-digital/digital-to-analog converter (DA-16A) [42], and the seismic response during pseudo-dynamic testing is calculated by the closed-loop control system. Control computer-based numerical analysis was conducted using the Pseudo-dynamic Testing Program [43]. Based on the restoring force of the specimen against the applied deformation measured with LVDT during the loading test, along with the input seismic acceleration and the measured response in the current step, the amount of response in the next step is calculated through numerical integration using the motion equation, as shown in Equation (6). \[M\ddot{y}(t)+\mathrm{C}\dot{y}(t)+r(t)[=Ky(t)]=-M\ddot{y_{0}} \tag{6}\]  
Here, \(M\), \(C\), and \(K\) refer to the mass, damping, and stiffness matrix of the structure, respectively. \(\mathrm{y}\) refers to the relevant displacement vector of each layer weight for the foundation. \(r\) is the restoring force vector, while \(\ddot{y_{0}}\) is the input ground acceleration. The numerical integration of the motion equation was conducted using the \(\alpha\)-method [44], and the numerical integration algorithm for pseudo-dynamic testing is shown in Equations (7)-(9). \[Ma_{i+1}+(1+\alpha)Cv_{i+1}-\alpha Cv_{i}+(1+\alpha)-\alpha r_{i}=(1+\alpha)f _{i+1}-\alpha f_{i} \tag{7}\]  
\[y_{i+1}=y_{i}+\Delta tr_{i}+\Delta t^{2}\bigg{[}\bigg{(}\frac{1}{2}-\beta \bigg{)}a_{i}+\beta a_{i+1}\bigg{]} \tag{8}\]  
\[v_{i+1}=v_{i}+\Delta t[(1-\gamma)a_{i}+\gamma a_{i+1}] \tag{9}\]  
Here, \(y_{i}\), \(v_{i}\), and \(a_{i}\) refer to the joint displacement at the time corresponding to \(i\Delta t\), velocity, and acceleration, respectively. \(\Delta t\) is the time interval for integration, \(r_{i}\) is the restoring force vector at the joint displacement, and \(f_{i}\) is the external load vector \((-M\ddot{y_{0}})\). In an elastic structure, \(r_{i}=Ky_{i}\) applies (\(K\) is the elastic stiffness matrix of the structure). \(\alpha,\ \beta\), and \(\gamma\) are parameters that control the numerical characteristics of the algorithm. The conditions \(-5\leq\alpha\leq 0\), \(\beta=\frac{(1-\alpha)^{2}}{4}\), and \(\gamma=\frac{1}{2}-\alpha\) indicate that the system is in a state of unconditional stability. In the next step, the displacement response is calculated based on the stiffness (\(K\)), the mass (\(M\)), and the coefficient of damping proportional to stiffness (\(C\)) using Equations (6)-(9). The damping factor (\(\xi\)) was set to 0.03, which amounts to 3% of the critical damping. As shown in Figure 9, the horizontal seismic response deformation is applied on the specimen using two 2000-kN hydraulic MTS actuators on the first and second floors. The horizontal displacement used to calculate the displacement response is measured using a 300-mm linear variable differential transformer (LVDT) installed on each floor. The axial force was constantly applied to each column of the specimen using 1000-kN oil jacks installed on each side of the specimen by properly distributing the axial load applied to the actual existing frame.