## 4 Discussion of the Results
\begin{table}
\begin{tabular}{c c c c c} \hline
**DoF** & **Pushover P-1 x** & **Pushover P-2 x** & **Pushover P-1 x** & **Pushover P-2 x** \\ \hline \(u_{1}\) & 0.77 & 0.68 & 0.76 & 0.68 \\ \(u_{2}\) & 0.71 & 0.78 & 0.70 & 0.79 \\ \(u_{3}\) & 0.58 & 0.63 & 0.58 & 0.63 \\ \(u_{4}\) & 0.53 & 0.58 & 0.58 & 0.57 \\ \(u_{5}\) & 0.39 & 0.55 & 0.45 & 0.54 \\ \(u_{6}\) & 0.27 & 0.65 & 0.27 & 0.64 \\ \hline \end{tabular}
\end{table} 
Table 4: Main diagonal deviation (%) of the damage stiffness matrix terms of the six-story steel-braced frame at the inelastic \(i\)-step where the roof displacement \(u_{\text{top},i}=0.18\) m is observed in pushover analyses P-1 and P-2 with positive and negative forces. except the first floor. Specifically, this degradation is approximately 40% and 140% greater on the two highest floors compared with the P-1 case. In contrast, on the first floor, stiffness degradation is 12% lower in the P-2 case compared with P-1. Figure 21: Deviation (%) of the damage stiffness matrix terms of the six-story steel-breaced frame at the inelastic \(i\)-step where the roof displacement \(u_{\text{top},i}=0.18\) m is observed in pushover analysis P-2 with positive forces. Figure 20: Deviation (%) of the damage stiffness matrix terms of the six-story steel-breaced frame at the inelastic \(i\)-step where the roof displacement \(u_{\text{top},i}=0.18\) m is observed in pushover analysis P-1 with positive forces. These observations align with the plastic mechanisms illustrated in Figure 15 (P-2) compared with Figure 14 (P-1). In general, the severity of damage in elevation in the P-2 pushover is greater in the upper half and less in the lower half of the steel frame compared with the P-1 case. Specifically, at the fifth floor in the P-2 case, the brace in compression fails, while the brace in tension experiences greater deformation. It is noted that in the P-1 pushover, along the negative direction, the brace in compression (Link 14) has only marginally failed, as shown in Figure 19. At the highest floor (sixth floor), both braces in the P-2 case undergo significant deformation. Additionally, in the P-2 case, all the end sections of beams with negative bending on the fifth-floor experience moderate inelastic  
Figure 23: Deviation (%) of the damage stiffness matrix terms of the six-story steel-braced frame at the inelastic \(i\)-step where the roof displacement \(u_{\text{top},i}=0.18\) m is observed in pushover analysis P-2 with negative forces. Figure 22: Deviation (%) of the damage stiffness matrix terms of the six-story steel-braced frame at the inelastic \(i\)-step where the roof displacement \(u_{\text{top},i}=0.18\) m is observed in pushover analysis P-1 with negative forces. deformation, unlike in the P-1 case, where no plastic hinges develop. On the other hand, in the P-2 case, the braces experience lower levels of deformation at the first two floors compared with the P-1 case. Specifically, the two braces in compression on the second floor do not fail, and one of the braces in tension on the first floor remains elastic. Additionally, the end sections of the beams on the first floor do not develop plastic hinges in the P-2 case, unlike in the P-1 case. Therefore, it is demonstrated that for a specific seismic damage image in an existing multi-story steel frame with an X- or V-bracing system, the stiffness matrix at the health (undamaged) state of the structure degrades, leading to a change in the eigenfrequencies of the structure. These changes are experimentally detected in the field through an instrumental monitoring system. Subsequently, using the eigenfrequency diagram, which is the \"key diagram\" of the present methodology, the seismic displacement at the top of the frame is revealed. This displacement corresponds to the observed damage and ensures the same eigenfrequency values as those detected in the field. All parameters used in the hybrid \"\(M\) and \(P\)\" technique are interrelated, making it a self-proven and accurate damage identification process.