## 4 Discussion of the Results  
In this section, the location and severity of seismic damage into the existing six-story steel-braced frame is evaluated and compared with its health (undamaged) state. This evaluation is feasible because all the results presented for the stepping eigenfrequencies in Table 3, the target roof displacement in Figure 12, the plastic mechanism in Figures 14 and 15, and the damage stiffness matrices (Equations (12)-(15)) are interconnected. Indeed, by knowing the damage stiffness matrix \(\mathbf{\Delta}\mathbf{k}_{i}\) of the planar steel-braced frame, the final percentage (%) degradation of the stiffness terms within the matrix can be calculated with respect to the stiffness matrix \(\mathbf{k}_{\text{o}}\) in the health (undamaged) state of the frame. This degradation of the main diagonal stiffness terms is presented in Table 4 for all pushover analysis cases, P-1 and P-2, with both positive and negative floor forces. Additionally, visual representations of the stiffness degradation matrices is shown in Figures 20-23, corresponding to the P-1 and P-2 pushover cases with positive and negative floor forces, respectively. The percentage (%) deviation observed in the terms of the main diagonal, within the damage stiffness matrix \(\mathbf{\Delta}\mathbf{k}_{i}\) with respect to the corresponding terms of the stiffness matrix \(\mathbf{k}_{\text{o}}\), demonstrates the magnitude of the seismic damage developed within the floors of the existing six-story steel-braced frame at the final step of the pushovers analyses P-1 and P-2 with target displacement \(u_{\text{top}}=0.18\) m. This is fully compatible with the damage images observed in Figures 14 and 15. For an aggregated assessment of the seismic damage of the existing six-story steel-braced frame in terms of stiffness degradation rates, the mean values of the respective matrices of Figures 20-23 should be considered. This study examines the seismic damage of ductile, planar, multistory steel frames with a concentric X- or V-bracing system. In such structures, the developed plastic mechanism is characterized by severe damage or failure of the braces in compression as well as moderate to severe damage of the braces in tension. Additionally, plastic hinges form at the end sections of the beams, indicating a beam sway mechanism. However, plastic hinges can also develop in some columns, especially those with high axial forces. This latter issue can be mitigated by employing a staggered configuration for the concentric bracing system, as discussed in the Introduction. Such steel frames with a concentric X- or V-bracing structural system, designed according to EN 1998-1 [38], exhibit significant ductile behavior, as demonstrated by the six-story steel-braced frame considered in this study. The terms on the main diagonal of the stiffness matrix correspond to the horizontal dynamic degrees of freedom of the floors of the steel frame (Figure 4). Therefore, the deviation of these terms in the matrix \(\mathbf{\Delta}\mathbf{k}_{i}\) from the corresponding ones in the matrix \(\mathbf{k}_{\text{o}}\) demonstrates the overall damage occurring at the braces and the end cross-sections of the beams of each floor. It is noted that each dynamic degree of freedom (DoF) accounts for the behavior of the columns and braces both above and below the respective floor, in addition to the behavior of the beams. From Table 4, it is observed that the stiffness degradation in the main diagonal due to seismic damage is similar for the P-1 pushover analysis in both the positive and negative directions. The same observation applies to the P-2 pushover analysis in both directions. Additionally, in the P-2 pushover analysis, where an additional top force is applied, the stiffness degradation in the main diagonal is higher on all floors