## 3 Numerical Simulation
### Database Setup  
In this study, an SHM system based on RCMCSE and the BPN was developed and used for damage analysis. RCMCSE was used for damage feature extraction, and the BPN was applied for damage quantification. To reflect the possible damage condition of and location on the structure, the numerical model was simulated to have residual stiffness of 0% (representing a damage case), 35%, 70%, and 100% (representing a healthy case) for each floor. This study defined different damage cases with the removal of bracings. Both bracings on each floor were replaced with different section bracings to simulate stiffness reductions. Accordingly, ETABS was used to create a database of different stiffness reduction cases.  
According to previous research [12] discussing detection accuracy under different parameter combinations, RCMCSE parameters such as the template length m, threshold r, and signal length N were optimized as 4, 0.08 \(\times\) SD of the time series, and 20,000 points, respectively. [PERSON] et al. [14] recommended a data length of between 14 m and 23 m points in the MSE analysis. Therefore, the time scale was calculated as 10 (\(\tau\) = 10). Inspired by [PERSON] et al. [15], this study calculated the statistical value of RCMCSE, namely maximum, minimum, mean, geometric mean, and SD, which served as training data and took as inputs for dimensionality reduction.  
The five RCMCSE statistical values were calculated for each floor and served as the input nodes for the ANN-based SHM system; therefore, the system comprised 35 individual nodes. Seven output nodes representing the seven floors were derived. To enhance training efficiency, all training data were normalized. A hidden layer was adopted for the ANN model, and after a series of trials. The node number was determined to be 80 for the hidden layer in order to achieve optimal performance for predicting the damage condition and location. The structure of the ANN model is detailed in Figure 4. The diagnosis results, expressed as health index (HI) values, are listed in Table 2. To achieve effective training results for the ANN model, the HI was constrained to between 1 and 0. Thus, a positive unit value was assigned if the structure was evaluated as healthy, and 0 was assigned to reflect 100% loss of stiffness.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Damage Group** & **Damage Floors** & **Frequency (Hz)** \\ \hline Undamaged & None & 3.25 \\ \hline \multirow{6}{*}{One-story damage} & 1F & 2.98 \\ \cline{2-3}  & 2F & 2.18 \\ \cline{2-3}  & 3F & 1.9 \\ \cline{2-3}  & 4F & 2.08 \\ \cline{2-3}  & 5F & 2.34 \\ \cline{2-3}  & 6F & 2.76 \\ \cline{2-3}  & 7F & 3.54 \\ \hline \multirow{2}{*}{Two-story damage} & 1k2F & 1.79 \\ \cline{2-3}  & 3k\textasci{}4F & 1.51 \\ \cline{2-3}  & 5k\textasci{}6F & 2.01 \\ \hline \multirow{2}{*}{Three-story damage} & 1k2\textasci{}3F & 1.22 \\ \cline{2-3}  & 4k\textasci{}5k\textasci{}6F & 1.56 \\ \hline \multirow{2}{*}{Multistory damage} & 1k2\textasci{}3k\textasci{}4F & 2.5 \\ \cline{2-3}  & 4k\textasci{}5k\textasci{}6k\textasci{}7F & 1.54 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Damage cases and modal analysis.