## 3 Numerical Example
Figure 11: Diagram of stepping frequencies (Hz) of the steel-breaced frame in the nonlinear area. Figure 12: Diagram of stepping frequencies (Hz) in the nonlinear area closely linked with the capacity curve of the existing six-story, steel-breaced frame (pushover analyses P-1 and P-2 with target displacement \(u_{\rm top}=0.18\) m). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Damage State** & \(T_{1}\) **(s)** & \(T_{2}\) **(s)** & \(T_{3}\) **(s)** & \(T_{4}\) **(s)** & \(T_{5}\) **(s)** & \(T_{6}\) **(s)** \\ \hline Health & 0.34 & 0.13 & 0.08 & 0.06 & 0.04 & 0.03 \\ \hline First plastic hinge (\(u_{\rm top}=0.02\) m) & 0.35 & 0.13 & 0.08 & 0.06 & 0.04 & 0.03 \\ \hline DL (\(u_{\rm top}=0.05\) m) & 0.59 & 0.17 & 0.11 & 0.07 & 0.05 & 0.04 \\ \hline At detected frequency (\(u_{\rm top}=0.18\) m) & 1.67 & 0.36 & 0.16 & 0.09 & 0.07 & 0.05 \\ \hline SD (\(u_{\rm top}=0.24\) m) & 1.83 & 0.39 & 0.18 & 0.10 & 0.07 & 0.05 \\ \hline NC (\(u_{\rm top}=0.32\) m) & 2.00 & 0.42 & 0.19 & 0.11 & 0.07 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Increase in the period values (s) between the health state and the NC state of the existing steel-breaced frame. (a) The fundamental eigenfrequency \(f_{1}\) (Hz) of the existing steel-braced frame, detected through structural health monitoring in the field, is inserted into the frequencies diagram (Figure 2). This reveals the corresponding roof displacement \(u_{\text{top},i}\) (or \(\theta_{\text{pr},i}\), respectively). (b) With the roof displacement \(u_{\text{top},i}\) known, we move upward to the capacity curve of the existing steel-braced frame. Here, the seismic damage state is visually identified based on the achieved performance level. For clarity, only the capacity curves of pushovers P-1 and P-2 with positive floor forces are shown in Figure 12. The eigenfrequency diagram of the steel-braced frame is plotted in Figure 12, with the best-fit lines determined in Figure 10. The best-fit line for the fundamental eigenfrequency provides an excellent approximation of its mean values across all pushover cases. According to phase (d) of the methodology, the fundamental eigenfrequency \(f_{1,i}=0.60\) Hz of the frame detected through structural health monitoring is inserted into the eigenfrequency diagram of Figure 12. This reveals the target displacement at the roof of the frame, \(u_{\text{top},i}=0.18\) m (or equivalently, a \(\theta_{\text{pr},i}\) value approximately equal to 0.01 rad), which corresponds to the seismic damage state of the existing steel frame. All other, higher eigenfrequencies (\(f_{2,i}\)to \(f_{6,i}\)) can also be determined within Figure 12 at the same inelastic \(i\)-step. Therefore, we focus on the part of the capacity curves up to \(u_{\text{top},i}=0.18\) m, while the rest of the curves up to the NC state (dashed line) are not considered. This roof displacement is closely linked to the fundamental frequency, \(f_{1,i}=0.60\) Hz, of the existing steel-braced frame, which was previously detected through the structural health monitoring in the field. Table 3 presents the eigenfrequencies of the existing steel-braced frame at a target displacement of \(u_{\text{top}}=0.18\) m, based on four pushover analyses (P-1 and P-2 in both the positive and negative directions). This table highlights the minor differences between the mean values of the eigenfrequencies from these pushovers and the corresponding predictions of the fit line, as seen in Table 1 and Figure 12, for \(u_{\text{top}}=0.18\) m.  
Figure 13 illustrates the first three stepping mode shapes (\(\varphi_{1,i}\) to \(\varphi_{3,i}\)) of the six-story, steel-braced frame at the same inelastic \(i\)-step where \(u_{\text{top}}=0.18\) m is shown, in the context of pushover analyses P-1 and P-2 along the positive direction. The mode shapes for the negative direction are similar.