## 2 Methodology Steps
### Phases of the \"M and P\" Technique  
To assess the seismic damage of existing, ductile, planar, steel moment-resisting braced frames (with an X or V configuration) using the \"\(M\) and \(P\)\" technique, the following phases should be performed:  
(a) The fundamental natural frequency \(f_{1}\) (Hz) of the existing damaged steel-braced frame is detected by structural health monitoring. Note that up to the first three eigenfrequencies can be detected with the instrumental monitoring procedure. (b) An appropriate mathematical nonlinear model of the steel-braced, moment-resisting frame with rigid beam-to-column connections and pin brace-to-joint (X-brace) or brace-to-beam (V-brace) connections is created. In this model, the nonlinear bending behavior of the end sections of the frame elements (both beams and columns) is represented by incorporating fiber _P_-\(M_{2}\)-\(M_{3}\) hinges. These hinges are distributed over a distance approximately equal to the depth of the element's section. Additionally, the nonlinear behavior of the vertical concentric braces (with an X or V configuration) is best modeled using links (axial springs) with an axial force-deformation (_P_-\(\delta\)) nonlinear law along their longitudinal direction. This modeling approach is preferred over using plastic hinges in a pinned frame element to represent the nonlinear behavior of the vertical brace, as these nonlinear links can more accurately capture significant drops in strength. The skeleton curve of the nonlinear _P_-\(\delta\) law for the brace links is defined by points that characterize yield, ultimate failure, residual strength, and fracture behavior, with buckling behavior anticipated for the compressed braces. This curve will be illustrated in the numerical example. A series of separate pushover analyses are conducted on the nonlinear model, each time targeting a gradually increasing roof displacement \(u_{\text{top},i}\). For each target displacement, two pushover analyses, with positive and negative floor forces, are conducted. These analyses lead to the drawing of the capacity curve in terms of base shear \(V_{\text{o}}\) and top displacement \(u_{\text{top}}\). Figure 1 shows the general form of the capacity curve of a steel-braced frame, along with its elastoplastic idealization, which defines the idealized yield point \(\left(u_{y},V_{\text{oy}}\right)\) and the effective stiffness at yield \(K_{\text{eff}}\) (secant stiffness at yield) of the frame. Additionally, various performance levels of the structure corresponding to different roof displacements \(u_{\text{top},i}\) are depicted within the figure. The characteristic damage states correspond to different performance levels (or limit states). Starting from the health state (without damage), an increase in roof displacement leads the steel-braced frame to the first yield state (formation of the first hinge). Further increasing the roof displacement leads the steel-braced frame to the damage limitation (DL) state, subsequently to the significant damage (SD) state, and finally, at the ultimate roof displacement \(u_{\text{{\it{ult}}}}\), the near collapse (NC) state appears. This ultimate displacement is defined as the point where a sudden drop in strength exceeding 20% of the maximum observed strength occurs. Intermediate damage states are characterized by their location between the aforementioned characteristic states where they are observed. The general pattern of the capacity curve for ductile steel-braced frames is characterized by high effective stiffness at yield and sequential strength drops in the inelastic range. These drops are smaller when a compressive brace fails and larger when a tension brace fails. In general, the first change in the slope of the elastic branch of the pushover curve is due to the yielding of the braces, and the following changes are due to the yielding of beams/columns [35].