## 3 Numerical Example
According to phase (a) of the methodology, a multi-channel system of uniaxial accelerometers is installed in the existing six-story steel-braced frame, and the horizontal accel  
Figure 5: Stress–strain diagram of structural steel S235. Figure 6: Steel sections in steel-braced frame. Columns/beams are modeled as frame elements and braces as links. eration response due to ambient vibrations is recorded at each floor when the frame is at rest. From the analysis of these recordings, the fundamental natural frequency \(f_{1,i}=0.60\) Hz of the existing damaged steel-braced frame is detected. This frequency will subsequently be associated with the \(i\)-step of the pushover analysis. In the nonlinear model of the steel-braced frame, fiber \(P\)-\(M_{2}\)-\(M_{3}\)-type plastic hinges (FHs) are incorporated into the end cross-sections of the beams and columns (frame elements) to simulate their nonlinear behavior. Five fiber hinges (FHs) are inserted at each beam/column end section, with a plastic hinge length of 0.1 m each, to model distributed plasticity. These plastic hinges extend over a total distance approximately equal to the depth of the cross-section of the beams/columns. The constitutive law governing the behavior of plastic hinges in nonlinear analysis is according to the steel kinematic model [39; 40]. The double-channel X-braces are modeled as links, with their axial constitutive relationships represented by a skeleton curve in terms of axial force (\(N\)) and axial deformation (\(u\)). Figure 7 depicts the nonlinear behavior of these links. The middle diagram in the first row presents the characteristic points of the skeleton curve with their values. As shown in this figure, the behavior of the links in compression is characterized by half the yield axial force and deformation compared with their tensile counterparts to account for buckling. At the ultimate state, this reduction in axial force and deformation of the links in compression exceeds the half relative values in tension. Indeed, the axial deformation is reduced significantly at the ultimate state in compression to account for limited ductility under negative axial forces. A skew drop in axial strength is included in the constitutive law in tension and compression beyond the ultimate deformation to prevent unbalanced situations during analysis. A residual axial strength of 30% of the yield value is included in the skeleton curve following the ultimate state, remaining constant until fracture. To plot the capacity curve of the planar steel-braced frame up to the NC state, two pushover analyses should be conducted in both the positive and negative directions. Since the multistory frame has six floors (\(>\)4), the pushover analyses utilize the two loading elevation patterns described in phase (b) of the methodology: the triangular pattern and the triangular pattern with an additional top force, both corresponding to a unit base shear. These patterns are henceforth referred to as P-1 and P-2 and are depicted in Figure 4. The four capacity curves of the six-story steel-braced frame resulting from the two sets of pushover analyses with floor force patterns P-1 and P-2 are depicted in Figure 8. The capacity curve includes two ordinate axes, the first representing the roof displacement \(u_{6}\) and the  
Figure 7: Constitutive relationships of link elements simulating the axial behavior of the vertical X-braces. second representing the chord rotation profile angle \(\theta_{\text{pr}}=u_{6}/H_{\text{tot}}\), where \(H_{\text{tot}}=18.5\) m is the total height of the frame. This angle represents the mean drift ratio of the steel-breaced frame, as shown in Figure 9. Also shown within Figure 8 is the elastoplastic idealization of these curves, which identifies the idealized yield point of the frame (DL state) at a \(\theta_{\text{pr}}\) approximately equal to \(0.05/18.5=0.0027\) rad.