## 3 Numerical Example
These mode shapes are determined through modal analysis following the last step of the P-1 and P-2 pushovers with target displacement equal to \(u_{\text{top}}=0.18\) m.  
Additionally, as described in phase (e) of the methodology, Figures 14 and 15 present the seismic damage images of the steel-braced frame at the same inelastic \(i\)-step where \(u_{\text{top}}=0.18\) m is shown in the pushover analyses P-1 and P-2 along both the positive and negative directions, respectively. The developing plastic hinges in these figures are depicted by the following: (i) filled black semi-circles representing the damage state at the upper or lower fiber of the end cross-sections of the steel beam/column elements, indicating a condition between the DL and SD states, (ii) filled colored circles (green for DL, blue for SD, and red for NC) representing the damage state at the X-bracing elements, and (iii) X-signed black circles indicating the failure of various braces in compression. The yield and failure  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mode** & **Pushover P-1 x** & **Pushover P-1 –x** & **Pushover P-2 x** & **Pushover P-2 –x** & **Mean Values** \\ \hline
1 & 0.64 & 0.51 & 0.66 & 0.66 & 0.62 \\ \hline
2 & 3.16 & 2.60 & 2.98 & 3.01 & 2.94 \\ \hline
3 & 6.68 & 6.34 & 6.00 & 5.99 & 6.25 \\ \hline
4 & 10.86 & 10.85 & 10.37 & 10.41 & 10.62 \\ \hline
5 & 15.07 & 14.77 & 14.46 & 14.43  
Figure 16: Sequence of yielding and failure of the X-brraces of the steel frame at roof displacements \(u_{\text{top},i}=0.18\) m in pushover analyses (**a**) P-1 and (**b**) P-2 along the positive direction. Figure 15: Plastic mechanism of the existing six-story, steel-breaced frame for the pushover analysis P-2 with (**a**) positive and (**b**) negative floor forces, at a target roof displacement \(u_{\text{top}}=0.18\) m.  
It is noted that in both pushover analyses P-1 and P-2, along both the positive and negative directions, severe damage occurs primarily at the X-braces, with several of the compressed braces failing. Meanwhile, the plastic mechanism that develops in the existing steel frame, particularly concerning the beams and columns, is a beam sway mechanism, with moderate damage observed only on the end sections of the beams. No plastic hinge develops at the end sections of the columns on any floor or at the base of the frame. Additionally, it is observed that the severity of seismic damage is slightly greater in the upper half of the existing steel frame and slightly lesser in the lower half for the P-2  
Figure 17: Axial force–axial deformation behavior of X-braces of the steel frame at roof displacement \(u_{\text{top},i}=0.18\) m in pushover analyses (**a**) P-1 and (**b**) P-2 along the positive direction. Figure 18: Sequence of yielding and failure of the X-braces of the steel frame at roof displacement \(u_{\text{top},i}=0.18\) m in pushover analyses (**a**) P-1 and (**b**) P-2 along the negative direction. pushover analysis. As previously noted, for a conservative assessment of the damage profile of the existing steel-braced frame, it is advisable to consider the envelope of results from both the pushover analyses, P-1 and P-2. It is important to note that during a seismic event, the actual seismic load on the structure varies at each time step. Therefore, the seismic damage observed in the six-story steel-braced frame may differ from that predicted by the pushover analysis. However, the critical parameter in the proposed methodology is the fundamental natural frequency \(f_{1}\) (Hz), detected in the field through structural health monitoring. By knowing the fundamental natural frequency of the existing planar steel frame, the corresponding seismic displacement at the top of the frame can be estimated using Figure 12. Additionally, the capacity curve linked in the figure allows for the identification and assessment of seismic damage.