# Seismic Response of a Three-Dimensional Asymmetric Multi-Storey Reinforced Concrete Structure
## 4 Validation of the Finite Element Model
### Seismic Analysis to Low-Intensity Ground Motions  
Alongside the modal analysis result, the seismic response of the SMART 2013 structure to low-intensity ground motions was investigated. A classical Rayleigh damping was introduced to the SMART 2013 RC building with the following equation:  
\[\mathbf{C}=\alpha\mathbf{M}+\beta\mathbf{K} \tag{10}\]  
where \(\mathbf{M}\) and \(\mathbf{K}\) are the mass and stiffness matrices of the structure, respectively. The two coefficients \(\alpha\) and \(\beta\) are determined by solving the following system of equations:  
\[\frac{1}{2}\left[\begin{array}{cc}1/\omega_{i}&\omega_{i}\\ 1/\omega_{j}&\omega_{j}\end{array}\right]\left[\begin{array}{c}\alpha\\ \beta\end{array}\right]=\left[\begin{array}{c}\zeta_{i}\\ \zeta_{j}\end{array}\right] \tag{11}\]  
where \(\omega_{i}\) and \(\omega_{j}\) are the natural frequencies corresponding to modes \(i\) and \(j\), respectively, and \(\zeta_{i}\) and \(\zeta_{j}\) are the modal damping ratios for each mode. Table 6 shows the natural frequencies of the first three modes, and the corresponding modal damping ratios obtained by modal experiments. The modal damping ratios were obtained by analyzing the response of the structure with 28 accelerometers under the low-level (PGA \(<0.1\) g) biaxial random ground motion using a stochastic subspace identification bottom approach [23]. The coefficients \(\alpha\) and \(\beta\) were calculated by solving Equation (11) using the frequencies and damping ratios for the first and second modes. The calculated \(\alpha\) and \(\beta\) are \(-1.649\) and \(0.002\), respectively.  
Figure 18 shows the time history of the low-intensity ground motion. The time signal is the one scaled by 50% from the Northridge earthquake. The PGA is about 0.1 g in both the \(x\) and \(y\) directions. The structural response to the ground motion is calculated without considering material nonlinearity. Figures 19-21 show the seismic response of each floor of the SMART 2013 RC structure to the low-intensity ground motion at sampling point A. The calculated displacement and acceleration showed excellent agreement with the experimental results.  
Table 7 presents the absolute maximum seismic response of the structure at the sampling point A. Both the displacement and acceleration values agreed quite well with the experimental results.  
\begin{table}
\begin{tabular}{c c c} \hline
**Mode** & **Frequency, 1 (Hz)** & **Modal Damping Ratio, 1** \\ \hline
1 & 6.28 & 0.026 \\
2 & 7.86 & 0.042 \\
3 & 16.50 & 0.055 \\ \hline \end{tabular}
\end{table}
Table 6: Modal damping ratios obtained by experiments.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Floor**} & \multicolumn{2}{c}{**Displacement, \(u_{x}\) (mm)**} & \multicolumn{2}{c}{**Acceleration, \(a_{x}\) (m/s\({}^{2}\))**} \\ \cline{2-4}  & **Analysis** & **Experiment** & **Analysis** & **Experiment** \\ \hline
1 st floor & 1.06 & 0.91 & 1.37 (0.14 g) & 1.29 (0.13 g) \\
2 nd floor & 1.08 & 1.00 & 1.88 (0.19 g) & 1.74 (0.18 g) \\
3 rd floor & 1.10 & 1.01 & 2.66 (0.27 g) & 2.49 (0.25 g) \\ \hline \end{tabular}
\end{table}
Table 7: Absolute maximum seismic responses at point A due to the low-intensity ground motions.  
Figure 19: First-floor response in the \(x\) direction due to the low-intensity ground motion; responses sampled at point A.  
Figure 20: Second-floor response in the \(x\) direction due to the low-intensity ground motion; responses sampled at point A.  
Figure 18: Low-intensity ground motions; (**a**) ground displacement; (**b**) ground acceleration.  
Figure 21: Third-floor response in the \(x\) direction due to the low-intensity ground motion; responses sampled at point A.