## 3 Numerical Example
As a result, the diagram showing the stepping eigenfrequencies (Hz) of the steel-braced frame as a function of the top displacement \(u_{\text{top},i}=u_{6,i}\) can be plotted up to a roof displacement \(u_{\text{top},i}=0.32\) m, where the NC state is attained. Figure 11 illustrates this diagram for the two pushover analyses, P-1 and P-2, with horizontal floor forces applied in the positive and negative directions. The sudden peaks shown in this figure in each eigenfrequency line correspond to unbalanced situations caused by the sudden failure of braces in compression. This pattern becomes more pronounced near the NC state, where immediately before it, the braces in compression on the first floor fail, and immediately after it, the braces in tension on the third and fourth floors also fail (Figure 10). As the flexibility of the frame should be higher with increasing seismic damage (e.g., at the NC state relative to the SD state), best-fit mean lines of the eigenfrequencies are plotted as smooth, wide, colored lines in Figure 4:0 These best-fit lines can be used with good approximation, especially for the fundamental eigenfrequency under consideration. To further estimate the dynamic characteristics of the existing steel-braced frame at the inelastic \(i\)-step, the mean values of the stepping eigenfrequencies for each pattern, P-1 and P-2, should be calculated. These mean eigenfrequency values \(f_{1}\) to \(f_{6}\) (Hz), derived from the best-fit lines, are presented in Table 1 for characteristic values of \(\theta_{\text{pr}}\). As shown in Table 1, the stepping periods \(T_{1}\) to \(T_{6}\) (s) of the existing steel-braced frame (where period \(T\) is the inverse of the frequency \(f\), i.e., \(T(s)=1/f\)) increase with the increasing roof displacement \(u_{\text{top}}\). Consequently, the flexibility of the six-story steel frame progressively increases due to the evolution of seismic damage. This increase in the periods \(T_{1}\) to \(T_{6}\) (s) of the existing steel-braced frame between the health state and the NC state is clearly presented in Figure 4:1 As is evident in this table,the fundamental period of the steel-braced frame is about six times higher at the NC state compared with the health state. In Figure 12, the diagram of stepping eigenfrequencies (Hz) is combined with the capacity curve of the existing steel-braced frame for a target roof displacement equal to \(u_{\text{top},i}\). The two diagrams are plotted up to a roof displacement of 0.32 m, corresponding to the NC state of the frame. These interconnected diagrams form the \"key diagram\" of the hybrid \"\(M\) and \(P\)\" technique, illustrating its application as follows:  
Figure 4:2 Sequence of yielding and failure of the X-braces of the steel frame for the (**a**) P-1 and (**b**) P-2 pushover along the positive direction. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(u_{\text{top}}\) (m) & \(\theta_{\text{pr}}\) (rad) & \(f_{1}\) (Hz) & \(f_{2}\) (Hz) & \(f_{3}\) (Hz) & \(f_{4}\) (Hz) & \(f_{5}\) (Hz) & \(f_{6}\) (Hz) \\ \hline
0 & 0 & 2.97 & 7.74 & 12.20 & 17.75 & 23.51 & 30.67 \\ \hline
0.02 & 0.0011 & 2.82 & 7.57 & 11.91 & 17.36 & 22.94 & 30.03 \\ \hline
0.05 & 0.003 & 1.70 & 5.83 & 9.00 & 13.50 & 18.65 & 23.90 \\ \hline
0.08 & 0.004 & 1.14 & 4.10 & 8.25 & 12.60 & 16.61 & 20.23 \\ \hline
0.13 & 0.007 & 0.70 & 3.35 & 7.00 & 11.40 & 15.42 & 19.43 \\ \hline
0.17 & 0.009 & 0.61 & 2.90 & 6.31 & 10.70 & 14.78 & 18.96 \\ \hline
0.18 & 0.010 & 0.60 & 2.80 & 6.14 & 10.53 & 14.64 & 18.84 \\ \hline
0.2 & 0.011 & 0.58 & 2.60 & 5.80 & 10.20 & 14.35 & 18.60 \\ \hline
0.23 & 0.012 & 0.55 & 2.55 & 5.70 & 9.95 & 14.07 & 18.50 \\ \hline
0.26 & 0.014 & 0.54 & 2.50 & 5.60 & 9.70 & 13.88 & 18.40 \\ \hline
0.28 & 0.015 & 0.52 & 2.47 & 5.53 & 9.53 & 13.75 & 18.33 \\ \hline
0.3 & 0.016 & 0.51 & 2.43 & 5.47 & 9.37 & 13.63 & 18.27 \\ \hline \end{tabular}
\end{table} 
Table 1: Mean values (best-fitted lines) of stepping eigenfrequencies (Hz) of the existing six-story, steel-breaced frame, for the two pushover sets P-1 \(\kappa\alpha\) P-2.