## 3 Numerical Example
Additionally, in the figure, it is evident that for the P-1 pattern in the positive or negative direction, the NC state was reached at a \(\theta_{\text{pr}}\) value approximately equal to \(0.32/18.5=0.0173\) rad. As mentioned in phase (b) of the methodology, the ultimate roof displacement \(u_{\text{ult}}=0.32\)m is defined as the point where a sudden drop in strength, exceeding 20% of the maximum observed strength, occurs. It is also noteworthy that the capacity curves for the second pattern, P-2, exhibit similar values of \(\theta_{\text{pr}}\) at failure and a slightly lower effective (secant) stiffness, \(K_{\text{eff}}\), at yield. Figure 8: Capacity curves of the steel-breaced frame resulted from pushover analyses along the positive and negative directions using the two floor force patterns P-1 and P-2 (target displacement to NC state). Figure 9: The chord rotation profile angle \(\theta_{\text{pr}}=u_{6}/H_{\text{tot}}\) of the steel-breaced frame in pushover analysis. The sequence of yielding and failure of all vertical X-braces in the developed plastic mechanism of the steel-braced frame for the pushover analysis P-1 and P-2 along the positive direction is illustrated clearly in Figure 10a,b, respectively, in terms of axial force and roof displacement. The behavior is similar for both pushovers along the negative direction. These graphs are not exactly linear because the axial force of the braces and the roof displacement of the frame do not align in the same direction. In the same figures, the capacity curve of the steel-braced frame in terms of base shear and roof displacement, along with its counterpart due to base columns and base X-braces, is also shown. It is noted that the contribution of the base X-braces to the base shear is obtained by multiplication of their axial force with \(\cos(\alpha)\), where \(\alpha\) is the angle of the base X-braces. This contribution accounts for approximately 80% of the total base shear, similar to that observed in dual RC frames, as mentioned in the Introduction. As evident in Figure 10, at the NC state, the braces in compression have failed on all floors except the top floor in the P-1 pushover and the first floor in P-2. The NC state occurs with the failure of the braces in tension on the third and fourth floors in the P-1 pushover, and on the fourth and third floors in the P-2 pushover, following this sequence. The first plastic hinge forms in the braces under compression very early in both pushovers P-1 and P-2 at a roof displacement of approximately 0.02 m. The DL state is reached at a roof displacement of 0.05 m in both pushovers P-1 and P-2, when the braces in tension begin to yield. According to phases (b) and (c) of the methodology, a series of pushover analyses are conducted on the existing steel-braced frame, each targeting a successively increased roof displacement until the NC state is reached. These analyses are performed along the positive and negative directions using the two loading patterns P-1 and P-2. The objective of these analyses is to establish the natural eigenfrequency curves of the existing damaged steel-braced frame. The first pushover analysis in the series focuses on zero target displacement and considers only the gradual action of vertical loads from the seismic combination \(g+\psi_{E}q\), representing the health state of the structure. In the final step of each separate pushover analysis in the series, a stepping modal analysis is conducted, starting with the observed damage state from this last step. Specifically, each modal analysis considers the stiffness matrix of the damaged steel frame as determined in the final step of the respective pushover analysis. In each of these stepping modal analyses, the six stepping eigenfrequencies \(f_{1,i}\) to \(f_{6,i}\) (in Hz) of the steel-braced frame are recorded, corresponding to the number of dynamic degrees of freedom of the frame.