## 2 Methodology Steps
### Phases of the \"M and P\" Technique
(d) The known fundamental eigenfrequency \(f_{1}\) (Hz) of the existing damaged steel-braced frame of phase (a) is inserted into the diagram of stepping eigenfrequencies (Figure 2), and thus, the corresponding target inelastic displacement at the top of the frame \(u_{top,i}\) is revealed. All other stepping higher eigenfrequencies of the steel frame \(\left(f_{2,i}\,\ f_{3,i}\,\dots\ f_{N,i}\right)\) are depicted within the diagram along the same vertical line passing through the target displacement \(u_{\text{top},i}\). (e) The seismic damage status of the existing steel-braced frame can now be identified based on the results of two pushover analyses. These analyses are conducted with positive and negative floor forces, applied according to the two-floor force patterns discussed in phase (b) of the methodology. The target displacement in these analyses is equal to the roof displacement \(u_{\text{top},i}\) determined in the previous phase of the methodology. The locations and states of the plastic hinges observed in the final step of these pushover analyses provide an estimate of the damage state of the existing steel-braced frame. An overall assessment of the damage can be derived by considering the envelope of the damage states obtained from both sets of pushover analyses. (f) Following the final step of each pushover analysis in phase (e) with target displacement \(u_{\text{top},i}\), a modal analysis is conducted. In this analysis, the stiffness matrix of the damaged steel frame from the final step of the pushover analysis serves as the initial condition. Through this stepping modal analysis, all cyclic eigenfrequencies \(\omega_{i,g}\) and all mode shapes \(\boldsymbol{\uppsi}_{i,g}\) of the existing damaged steel frame are determined. (g) The stepping lateral (condensed) stiffness matrix \(\mathbf{k}_{i}\) of the damaged steel-braced frame is calculated at the specific inelastic \(i\)-step where the roof displacement \(u_{\text{top},i}\) was observed. To achieve this, it is more straightforward to first compute the flexibility matrix of the frame at the specific inelastic \(i\)-step. This involves conducting separate linear analyses where lateral unit floor forces are successively applied to each floor, aligned with the direction of the horizontal dynamic degrees of freedom of the frame. These analyses are conducted in the final step of each pushover analysis during phase (e), with a target displacement \(u_{\text{top},i}\). The resulting horizontal displacements of the floors are then organized as columns in the flexibility matrix of the damaged steel frame, which is of order \(N\times N\) for an \(N\)-degree frame structure. The inversion of this flexibility matrix results in the \(N\times N\) stepping lateral stiffness matrix \(\mathbf{k}_{i}\) of the existing damaged steel frame at the same inelastic \(i\)-step where the roof displacement \(u_{\text{top},i}\) is observed. Therefore, the \(N\times N\) damage stiffness matrix of the steel frame \(\boldsymbol{\Delta}\mathbf{k}_{i}\) within the same inelastic \(i\)-step can be calculated  
Figure 2: Diagram of stepping eigenfrequencies \(f\) (Hz) of a multistory steel-braced frame in the nonlinear area. from the general relation \(\mathbf{\Delta}\mathbf{k}_{i}=\mathbf{k}_{o}-\mathbf{k}_{i}\), where \(\mathbf{k}_{o}\) represents the initial lateral stiffness matrix of the steel frame in its health (undamaged) state. The calculation of \(\mathbf{k}_{o}\) follows the aforementioned procedure, which is conducted in the final step of a pushover analysis considering only the incremental action of the vertical loads from the seismic combination \(g+\psi_{E}q\). The steps for applying the hybrid \"\(M\) and \(P\)\" technique to assess seismic damage in existing, ductile, planar steel frames braced with X- or V-links are clearly outlined in the flow diagram depicted in Figure 3.  
Figure 3: Flowchart for identification of damage in existing steel-braced frames using the “\(M\) and \(P\)” technique.