## 2 Methodology Steps
### Phases of the \"M and P\" Technique
Pushover analysis is conducted using the following elevation patterns for the horizontal floor forces (for a unit base shear, \(V_{\text{o}}=1\)): (i) The triangular pattern or that according to the fundamental mode shape is used in frames up to four floors. (ii) For taller frames, an additional pattern should be used, where an additional floor force equal to \(0.20\cdot V_{\text{o}}\) is applied at the top of the frame while the remainder of the unit base shear (i.e., \(0.80\cdot V_{\text{o}}\)) is distributed to the storeys according to the triangular pattern or that according to the fundamental mode shape. This second pattern was proposed for RC structures in [36], but it was found to be effective for steel buildings as well. The aim of the second additional pattern is to consider the influence of the higher modes on the seismic response of the frame. This is particularly important in the nonlinear region, where even tall steel-frame buildings braced with vertical links can become flexible at high levels of roof target displacement. It is noted that a modified pattern for floor forces in pushover analysis of steel moment-resisting frames is also presented in equation (B.5) of Informational Annex B of Eurocode EN 1998-3 [37]. In this pattern, floor forces are distributed in elevation relative to the frame period; specifically, in more flexible frames, higher floor forces are applied as one moves toward the roof. It is emphasized that P-\(\Delta\) effects should always be considered in pushover analysis, especially in more flexible frame structures. (c) Following the final step of each pushover analysis with increasing target displacement \(u_{\text{top},i}\) from phase (b) of the methodology, a stepping modal analysis of the existing steel-braced frame is performed. Specifically, the modal analysis corresponding to a roof displacement \(u_{\text{top},i}\) is conducted using the stiffness matrix of the damaged frame obtained at the final step of the corresponding pushover analysis with the same target displacement. Care must be taken when performing sequential modal analyses. These analyses should correspond to target roof displacements on the upward (positive) slope of the pushover curve. Conversely, if the target displacement corresponds to a point on a downward (negative) slope of the pushover curve, negative eigenvalues will result, which lack physical significance. Consequently, a diagram of the stepping eigenfrequencies \(f_{N,i}\) (Hz) of the damaged frame is plotted as a function of the roof displacement \(u_{\text{top},i}\) (m). The general form of such a diagram is illustrated in Figure 2. In this key aspect of the methodology, the inelastic roof displacement \(u_{\text{top},i}\) is plotted on the \(x\)-axis (abscissa), while the eigenfrequency \(f_{N,i}\) of the damaged frame is plotted on the \(y\)-axis (ordinate). By incorporating the fundamental natural frequency \(f_{1}\), detected through the structural health monitoring in step (a), into this diagram, the seismic inelastic roof displacement of the existing damaged steel-braced frame can be determined. This displacement is directly correlated with the seismic damage state of the frame. The next phase of the methodology involves identifying this seismic damage state. It is emphasized that, unlike reinforced concrete structures that crack, the moment of inertia of the steel cross-sections in steel structures remains constant from the unloaded state to yielding and is equal to the geometric moment of inertia. Hence, the nonlinear model of the steel-braced frame in the sequential pushover and stepping modal analyses, which target increasing roof displacements, includes steel sections with a stable geometric moment of inertia. Figure 1: Pushover capacity curve of a multistory steel-braced frame. Target displacement at NC.