# Seismic Upgrade of Steel Frame Buildings by Using Damped Braces
## 6 Seismic Response Assessment
On the other hand, the procedure was shown to provide conservative results, as the averaged roof displacement of all retrofitted buildings was smaller than the target displacement \(d_{p}\): considering the three MRFs, the gap in the X-direction ranged between 17 and 25% with \(\mu_{DB}\) = 4 and between 35 and 45% with \(\mu_{DB}\) = 10, while in the Z-direction (8-story MRF only) the gap was about 30% with \(\mu_{DB}\) = 4 and 40% with \(\mu_{DB}\) = 10. The tendency of the method to provide a conservative design of the damped brace system when checked against the NLDAs has been already highlighted and discussed in previous works [11; 94]. This behavior can be explained by considering that the results of the NLSAs used to evaluate the capacity curve of the main frame at the beginning of the design procedure depend on the assumed distribution of lateral forces and neglect the dynamic  
Figure 14: Buckling checks in the most stressed column in (**a**) two-story MRF and (**b**) four-story MRF. Figure 13: Peak Floor Acceleration of the case-study MRFs with and w/o damped braces: (**a**) two-story, (**b**) four-story, and (**c**) eight-story structures. effects. Moreover, the design procedure accounts for the damping introduced from both the inelastic deformation of the main frame and the engagement of the dissipation system by modifying the response spectrum by the damping correction factor \(\eta=\sqrt{10/(5+\xi_{F+DB})}\). However, owing to the large damping capability of the damped braces investigated in the study, the ensuing value of \(\eta\) is less than 55%, which is the threshold over which the reduced shapes of the pseudo-acceleration and displacement response spectra are no longer valid [53]. The maximum bidirectional peak floor acceleration (PFA) at each story is plotted in Figure 13. The two-story and four-story frames do not show a significant increase in the PFA moving from the bare to the upgraded configurations. In the eight-story structure, the PFA has a non-negligible increase, especially from the fifth up to the last floor, where the PFA passes from 6.5 m/s\({}^{2}\) in the bare configuration to 8.3 m/s\({}^{2}\) and 9.1 m/s\({}^{2}\) in the retrofitted configurations for \(\mu_{DB}=10\) and \(\mu_{DB}=4\), respectively. The introduction of the damped braces in a frame causes an increment of the internal forces in the structural members adjacent to the upgraded bay. Therefore, the maximum compressive axial forces of the columns are evaluated at each floor and divided by the corresponding buckling load (N\({}_{\text{buckling}}\)) calculated according to IBC [51], to check if the buckling capacity is exceeded. Figure 14 shows the check results for the most stressed element of the two-story and four-story MRFs (column P2 according to the layout in Figure 10a), highlighting that, for these structures, the buckling limitation requirement is satisfied. On the contrary, the most stressed elements of the eight-story MRF (columns P2 and P5 according to Figure 10b) do not respect the buckling limit at the first floor in the retrofitted configuration with \(\mu_{DB}=10\) and at both the first and second floors in the retrofitted configuration with \(\mu_{DB}=4\) (Figure 15a). In \(Z\)-direction, the buckling threshold is not respected only at the ground floor by column P5 in the retrofitted configuration with \(\mu_{DB}=4\), where N/N\({}_{\text{buckling}}\) is equal to 1.025 (Figure 15b). In order to fulfill the buckling limitations, the design of the chevron braces with the hysteretic devices is repeated for the eight-story structure by inserting two braces per each perimetral frame in the X-direction, as shown in Figure 16a; in the Z-direction, only one chevron brace per frame is kept. The properties of the devices in the new configuration with two dampers per frame are half of those calculated in the previous configuration.