## 6 Damage Detection of IASC-ASCE Structural Health Monitoring Benchmarks
### Feature Extraction and Training Phase for Damage Cases 4 and 5  
Since the damage cases 4 and 5 are more inclusive and are also more complex damage scenarios, they will be used to assess the merits of the proposed algorithm. Training data is generated by simulating damage cases 4 and 5 on a three-dimensional structure with 12-DOF. Damage is induced by reducing the stiffness parameter of each story. The mathematical expression of which is as follows:  
\[k_{ij}^{pd}=\theta_{ij}k_{ij}^{u}. \tag{19}\]  
Figure 5: Location of all structure’s sensors and the sensors used in this research.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Description**} & \multicolumn{5}{c}{**Case**} \\ \cline{2-6}  & **1** & **2** & **3** & **4** & **5** \\ \hline
12 DOF model & O & & O & O & O \\
120 DOF model & O & O & O & O & O \\ Symmetric Mass & O & O & O & O & O \\ Asymmetric Mass & & O & O & O & O \\ Ambient Excitation & & O & O & O & O \\ Shaker on Root & & & & & \\ Damage Patterns: Remove Followings & & & & & \\ (1) All Braces in the 1 st Story & & O & O & O & O \\ (2) All Braces in the 1 st and 3 rd Story & O & O & O & O & O \\ (3) One Brace in the 1 st Story & O & O & O & O & O \\ (4) One Brace in the 1 st and 3 rd Story & & & & O & O \\ (5) 4 and Loosen Floor Beam at 1 st Level & & & & & O \\ (6) 2/3 Stiffness in One Brace at 1 st Story & & & & & O \\ \hline \hline \end{tabular}
\end{table}
Table 2: Damage Cases of the Benchmark Model [45].  
where \(\theta_{ij}\) is the stiffness loss parameters for the \(i^{th}\) story (\(i=1,\ldots 4\)) and face \(j\) (\(j=+x,-x,+y,-y\)) (see Figure 6). Furthermore, \(k^{u}_{ij}\) are stiffnesses of the undamaged shear building model, e.g., \(k^{u}_{i,+x}=k^{u}_{i,-x}=34.0\ MN/m\) and \(k^{u}_{i,+y}=k^{u}_{i,-y}=53.5\ MN/m\)[45]. The damage severities are expressed as stiffness reduction factor (SRF) \(\theta_{ij}\).  
After imposing damage, recorded acceleration response of the structure from each of 4 sensors (Figure 5) are decomposed using the procedure described in Section 3. The Battle-Lemarie [32] wavelet is used as a basic function to analyze the output acceleration response of the sensors. According to the WPE process, up to 7 levels of decomposition are conducted. Based on this approach, 128 frequency bands with a width of 3.91 Hz will be created. The resulting energy components are arranged according to their magnitudes 95% of the WPE is mostly distributed below 100 Hz frequency bands, which are both important in value and sensitive to the damage occurred in the structure. Therefore, WPT's first 8 component energies are selected as damage features.  
Extracted features from different sensors are fused to construct input vectors of the surrogate model as expressed in Equation (8). Training output vectors for the first and second surrogate model are damage severities which is expressed in Equation (19) as stiffness reduction factor \(\theta_{ij}\). In the first phase, to construct the damage signatures for the purpose of comparing the performance of PoDE and PDE [13], 50% stiffness reduction is imposed on the stiffness in the X and Y directions (single damage level). When the damage is assigned, the stiffness of both elements facing each other, e.g., elements in +Y and \(-\)Y faces, are reduced together. In the second phase, to evaluate the damage severity index (DMI), two levels of damage, i.e., 30 and 70% of stiffness losses, are imposed on the structural elements in order to create the training dataset.