# Structural Health Monitoring Using Machine Learning and Cumulative Absolute Velocity Features
## 4 Applications
### MDOF Systems  
#### 4.1.1 Model Description  
Two MDOF models representing two five story buildings were used in this case study (Figure 6a). The models were developed using _Steel01_ material in OpenSees [32] with the force-displacement behavior shown in Figure 6c. The mass, stiffness, and damping of each story of the MDOF systems were based on pushover and eigenvalue analysis conducted in previous studies [33; 34]. One of the MDOF systems, MDOF-US, had uniform shear capacity along the height of the building which is equal to the calculated base shear (\(Vy_{base}\)) at yield. The other MDOF system, MDOF-NS, was designed to have non-uniform shear capacity distribution along the height of the building. The shear capacity of MDOF-NS varied as follows,  
\[Vy_{n}=\left[1-(1-\delta)\left(\frac{h_{n}}{H}\right)^{\lambda}\right]\times Vy _{base}, \tag{19}\]  
where \(Vy_{n}\) is the yield shear at story \(n\), \(h_{n}\) is the height of the \(n\)-th story from the base of the building, \(\delta\) is the ratio of the lateral stiffness at the top of the building to that at the base of the building [45], and \(H\) is the total building height. Equation (19) was proposed in [45] to take into account nonuniform stiffness of multistory buildings. The values of \(\delta\) and exponent \(\lambda\) were taken as 0.3 and 1.5, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Input Feature** & **OLR** & **LR** & **ANN\({}_{10}\)** & **ANN\({}_{100}\)** & **SVM** \\ \hline \(CAV\) & 80.54 & 82.88 & 80.54 & 81.71 & 79.38 \\ \(R_{CAV}\) & 87.16 & 86.72 & 88.72 & 89.49 & 88.33 \\ \(\Delta_{CAV}\) & 75.10 & 75.10 & 75.10 & 77.04 & 75.10 \\ \(CAV,R_{CAV}\) & 90.27 & 89.44 & 88.72 & 90.66 & 91.05 \\ \(R_{CAV},\Delta_{CAV}\) & 86.77 & 84.72 & 89.11 & 87.94 & 87.94 \\ \(CAV,\Delta_{CAV}\) & 80.54 & 83.27 & 80.54 & 81.32 & 79.38 \\ \(CAV,R_{CAV},\Delta_{CAV}\) & 90.27 & 89.05 & 90.27 & 90.66 & 89.88 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Accuracy (%) of ML models for different feature combinations (Set-1: training & Set-2: test). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Input Feature** & **OLR** & **LR** & **ANN\({}_{10}\)** & **ANN\({}_{100}\)** & **SVM** \\ \hline \(CAV\) & 36.67 & 12.50 & 18.33 & 15.83 & 8.33 \\ \(R_{CAV}\) & 60.00 & 42.50 & 30.83 & 37.50 & 20.83 \\ \(\Delta_{CAV}\) & 61.67 & 45.00 & 42.50 & 40.00 & 21.67 \\ \(CAV,R_{CAV}\) & 74.14 & 61.67 & 18.33 & 40.00 & 25.00 \\ \(R_{CAV},\Delta_{CAV}\) & 65.83 & 45.00 & 60.00 & 40.00 & 22.50 \\ \(CAV,\Delta_{CAV}\) & 70.00 & 60.00 & 51.67 & 36.67 & 24.17 \\ \(CAV,R_{CAV},\Delta_{CAV}\) & 70.00 & 61.67 & 38.33 & 54.17 & 25.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Accuracy (%) of ML models for different feature combinations (Set-1: training & Set-2: test). According to ASCE 7 [46], the induced shear force at story \(z\), \(V_{z}\), is determined as follows,  
\[V_{z}=\sum_{i=z}^{n}C_{viz}Vy_{base},\ \ \ \ \ C_{viz}=\frac{w_{z}h_{z}^{p}}{\sum_{i= 1}^{n}w_{i}h_{i}^{p}}, \tag{20}\]  
where \(C_{viz}\) is the vertical distribution factor, \(w_{z}\) is the portion of the total effective seismic weight at story \(z\), \(h_{z}\) is the height of the story \(z\) from the base of the building, and \(p\) is an exponent related to the building natural period, \(T_{n}\). For \(T_{n}<0.5\) s, \(p=1.0\), for \(T_{n}>2.5\) s, \(p=2.0\), and for \(0.5\leq T_{n}\leq 2.5\), \(p\) is linearly interpolated. Thus, for the MDOF system at hand, \(p=1.075\). Table 3 shows the story values of the shear to base shear ratios according to code [46] and for the two MDOF systems where the design of both systems was code conforming. However, MDOF-NS marginally met the code whereas MDOF-US had significantly higher values throughout the height and is a more conservative design which is often the case for low-to-medium rise buildings. For each model, a corresponding linear system was also developed using the _Elastic_ material from OpenSees [32]. NTHA is performed on the models using both Set-1 and Set-2 records described in Section 2.1.2.