## 4 Pseudo-Dynamic Test Overview and Result Analysis
### Pseudo-Dynamic Test System and Method  
Figure 12 shows the concept of the pseudo-dynamic test system constructed in this study, as well as the specimen setting. As shown in the figure, the system can be expressed in two degrees of freedom (TDF), and it consists of a numerical calculation based on the ground motion input by the control computer and an experiment in which the specimen is loaded. During the test, the calculated displacement response is applied to the specimen using two hydraulic actuators installed in the horizontal direction. The actual restoring force is physically measured during the test and then used in the control computer to calculate the displacement response. Data conversion is performed using an analog-to-digital/digital-to-analog converter (DA-16A, Tokyo Soki Kenkyujo Company [41]. In the pseudo-dynamic test, the seismic response is calculated by a closed-loop control system. MTS's Pseudodynamic Testing Program [42] was used in the numerical calculation by the control computer. In the experiment, the response of the next step was calculatedusing numerical integration and the equation of motion shown in Equation (5) based on the restoring force of the specimen after deformation, the input seismic acceleration, and the response of the current step measured using LVDTs. \[M\ddot{y}(t)+\mathrm{C}\dot{y}(t)+r(t)[=Ky(t)]=-M\ddot{y_{0}} \tag{5}\]  
Here, \(M\), \(C\), and \(K\) represent the mass, damping, and stiffness matrix of the structure, respectively; \(y\) is the relative displacement vector of each layer mass for the foundation; \(r\) is the restoring force vector; \(\ddot{y_{0}}\) is the input ground acceleration. The \(\alpha\)-method [43] was used for the numerical integration of the equation of motion. The algorithm for the numerical integration of the pseudo-dynamic test is given by Equation (6), as follows:  
\[Ma_{i+1}+(1+\alpha)\mathrm{C}v_{i+1}-\alpha\mathrm{C}v_{i}+(1+\alpha)-\alpha r _{i}=(1+\alpha)f_{i+1}-\alpha f_{i} \tag{6}\]  
\[y_{i+1}=y_{i}+\Delta tv_{i}+\Delta t^{2}\bigg{[}\bigg{(}\frac{1}{2}-\beta \bigg{)}a_{i}+\beta a_{i+1}\bigg{]} \tag{7}\]  
\[v_{i+1}=v_{i}+\Delta t[(1-\gamma)a_{i}+\gamma a_{i+1}] \tag{8}\]  
Here, \(y_{i}\), \(v_{i}\), and \(a_{i}\) are the nodal displacement, velocity, and acceleration at the same time as \(i\Delta t\), respectively; \(\Delta t\) is the integral time interval; \(r_{i}\) is the restoring force vector at the node; \(f_{i}\) is the external load vector (\(-M\ddot{y_{0}}\)). For elastic structures, \(r_{i}=Ky_{i}\) (\(K\): the elastic stiffness matrix of the structure) holds. Here, \(\alpha,\ \beta\), and \(\gamma\) are variables used to control the numerical characteristics of the algorithm, where \(-5\leq\alpha\leq 0\), \(\beta=\frac{(1-\alpha)^{2}}{4}\), and \(\gamma=\frac{1}{2}-\alpha\) indicate unconditional stability. The displacement response in the next step was calculated using Equations (5)-(8) based on the stiffness (\(K\)), mass (\(M\)), and stiffness proportional damping coefficient I of the structure. The damping factor was assumed to be 0.03, i.e., 3% of critical damping. The horizontal seismic response deformation was applied to the specimen using a 2000 kN hydraulic MTS actuator on the first and second floors, as shown in Figure 12. The horizontal displacement used for the displacement response calculation was measured using 300-mm LVDTs installed on each floor. As for the axial force, the axial load applied to the actual existing frame was distributed, with the load applied to each column using 1000-kN oil jacks installed on both sides of the specimen. Hachinohe (EW) [44], which exhibited the largest seismic response displacement among the seismic response displacement characteristics (ductility ratio) of existing historical seismic waves, was selected for the ground motion. The test was conducted using the pseudo-dynamic test system with acceleration magnitudes of 200, 300, 400, and 500 cm/s\({}^{2}\).