## 3 Pseudo-Dynamic Tests and Numerical Analysis of Model Structure
### Experimental and Numerical Results
#### 3.3.3 Changing Rules of Dynamic Characteristic Parameters and Lateral Stiffness  
With the increase of PGA, the structural damage generated and accumulated, resulting in the elongation of fundamental period, the increase of the damping ratio and the decrease of dynamic magnification factor. Meanwhile, the lateral stiffness gradually deteriorated with the accumulation of structural damage.  
Table 5 shows test results of the fundamental period, damping ratio, dynamic magnification factor and lateral stiffness of the specimen under PDTs. The dynamic magnification factor was defined as the ratio between the maximum acceleration response and input PGA. The lateral stiffness \(K_{i}\) was calculated by the following formula [16]:  
\[K_{i}=\frac{|+F_{i}|+|-F_{i}|}{|+X_{i}|+|-X_{i}|} \tag{14}\]  
where \(+F_{i}\) and \(-F_{i}\) are respectively the peak load under the \(i\)th loading case of the PDTs in two opposite directions; \(+X_{i}\) and \(-X_{i}\) are respectively the displacement corresponding to the peak load under the \(i\)th loading case of the PDTs in two opposite directions.  
In order to investigate the changing rules of dynamic characteristic parameters and lateral stiffness of the specimen with the increase of PGA, the relative ratio \(\gamma_{\text{d}}\) was defined as the ratio between the dynamic characteristic parameter, i.e., fundamental period, damping ratio and dynamic magnification, under one specific loading case of the PDTs to that under the first loading case with PGA of 0.05 g. Specially, when PGA is 0.05 g, the relative ratios of the dynamic characteristics parameters are 1. Similarly, the relative stiffness ratio \(\gamma_{\text{s}}\) was defined as the ratio between the lateral stiffness under one  
Figure 11: Comparison of time-history lateral displacement responses between the FE simulation and test results: (**a**) Top of A-shaped frame under 0.40 g; (**b**) Middle part of truss under 0.40 g; (**c**) Middle part of column under 0.40 g; (**d**) Top of A-shaped frame under 0.20 g, 0.40 g and 0.60 g.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**PGA (g)** & **Fundamental** & **Damping Ratio** & **Dynamic** & **Lateral Stiffness** \\  & **Period (s)** & **(\%)** & **Magnification Factor** & **(kN/mm)** \\ \hline
0.05 & 0.31 & 4.78 & 1.60 & 9.63 \\
0.10 (Level 1) & 0.35 & 4.93 & 1.59 & 9.04 \\
0.20 (Level 2) & 0.46 & 6.95 & 1.52 & 6.47 \\
0.30 & 0.48 & 6.96 & 1.50 & 6.18 \\
0.40 (Level 3) & 0.49 & 6.97 & 1.44 & 5.78 \\
0.60 (Level 4) & 0.53 & 8.57 & 1.29 & 4.98 \\
0.80 & 0.59 & 12.19 & 1.18 & 3.95 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Dynamic characteristic parameters and lateral stiffness of the specimen subjected to different PGAs.  
specific loading case of the PDTs to that under the first loading case with PGA of 0.05 g. It is 100% when PGA equals to 0.05 g.  
Figure 12 presents the relative stiffness ratio \(\gamma_{\mathrm{s}}\) and relative ratios \(\gamma_{\mathrm{d}}\) of dynamic characteristic parameters of the specimen subjected to different PGAs. It can be found that the specimen had obvious stiffness deterioration characteristics, due to the cracks occurring and propagation on RC tubular columns. When PGA was less than 0.20 g, the stiffness deteriorated sharply due to the initial cracking of concrete. Then, the speed of stiffness deterioration slowed after PGA came up to 0.20 g. When PGA equaled to 0.40 g which referred to the seismic hazard level 3 as introduced in Section 2, the stiffness was about 60% of its initial value, indicating that the lateral stiffness performance of the specimen was good enough to meet the design requirement of \"not collapse under rare earthquake\" in the China seismic design code [22]. Correspondingly, with the increase of PGA, the fundamental period and damping ratio increased, while the dynamic magnification factor decreased.