# Structural Health Monitoring Using Machine Learning and Cumulative Absolute Velocity Features
## 4 Applications
### MDOF Systems
#### 4.1.1 Model Description
The acceleration was computed at each node along with the force and displacement for each spring element. Damage state was evaluated using Equation (1) for each story. The worst damage state among the five stories was assigned as the damage state of the entire building and its location was assigned as the worst damage location. For the cases when the worst damage state occurred simultaneously at several locations, the lowest story was identified as the damage location. For the MDOF-US model, out of the 1710 cases, 1376 (80%), 150 (9%), 133 (8%), and 51 (3%) cases were respectively \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Story** & **Code Minimum** & **MDOF-NS Design** & **MDOF-US Design** \\ \hline
1 & 1.00 & 1.00 & 1.00 \\
2 & 0.93 & 0.94 & 1.00 \\
3 & 0.81 & 0.82 & 1.00 \\
4 & 0.61 & 0.67 & 1.00 \\
5 & 0.34 & 0.49 & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Story shear to base shear ratios. Figure 6: (**a**) MDOF models used in the study, (**b**) Story shear distribution along the height, (**c**) Story force-displacement relationship [34]. undamaged, minor damage, moderate damage, and major damage (Figure 7). For the MDOF-NS model, 1382 (81%), 55 (3%), 86 (5%), and 187 (11%) cases were respectively undamaged, minor damage, moderate damage, and major damage (Figure 7). The higher percentage of major damage in MDOF-NS was due to the non-uniform distribution of its shear capacity. The location of the worst damage for MDOF-US was the first story for all the 334 damaging events. In the case of MDOF-NS, damage locations of the 328 damaging events were distributed among all the bottom four stories (no damage was observed in the fifth story) but the majority of the damage locations were in the first and third stories. As previously discussed, the MDOF-US had a conservative design of uniform strength throughout the height. Therefore, the damage in this system occurred sequentially from bottom to top, i.e., the damage initiated at the first story and then moved up. For this reason, the extent of damage decreased with height for MDOF-US. However, for MDOF-NS, the strength variation resulted in concurrent damage at different stories, i.e., in many cases, the damage initiated at the second or third stories. The features (\(CAV,R_{CAV}\)) and the ML approach (OLR algorithm) identified in the previous section were utilized to detect the location and severity of damage of the MDOF-US and MDOF-NS models. The \(CAV\) of the first story was chosen where the earthquake force demand is expected to be the highest and the \(R_{CAV}\) of the top story was selected as it is expected to capture the impact of damage at all lower stories. Moreover, in practice, the ground floor and roof of buildings are usually instrumented. Thus, using features associated with these two stories would be useful for direct applications to instrumented buildings. Using these features, the OLR was trained with TR-1 and tested against TE-1 and TE-2 for both MDOF-US and MDOF-NS with damage categories defined similar to the SDOF system. Damage location was determined based on the assigned damaged state, i.e., the story that experienced the worst damage state was identified as the damage location. Using the damage state and location information obtained from the training set, the probability of the location of damage for a given damage state \(P(n|y)\) was calculated where \(n\) is the story number, i.e., \(n=1,\ldots,5\) and \(y\) is the damage state, i.e., \(y=1,2,3\) (\(y=0\) corresponds to the undamaged state). The probabilities of the worst damage location were subsequently  
Figure 7: Worst damage state and location distribution of the MDOF-US and MDOF-NS models. determined by Equation (21) for all five stories. The story with the highest \(P(n)\) was considered as the damage location. Here, \(P(y=i)\) was obtained using Equations (11)-(14). \[P(n)=\sum_{i=1}^{3}P(n|y=i)\times P(y=i) \tag{21}\]