# Seismic Upgrade of Steel Frame Buildings by Using Damped Braces
## 6 Seismic Response Assessment  
In order to investigate the effectiveness and reliability of the method, the NLSAs and NLDAs were performed on both as-built and retrofitted structures. Figure 11 compares in the ADRS space the capacity curves in the X-direction of the three as-built MRFs (indicated as _Frame_ in the Figure) to the curves of the upgraded structures with \(\mu_{DB}\) = 4 (red line) and \(\mu_{DB}\) = 10 (blue line). The target displacement is matched by the three upgraded frames for both values of \(\mu_{DB}\). As shown in the Figure, for the two-story and the four-story frames, the post-yield branches of the curves relevant to either \(\mu_{DB}\) are practically overlapped; in contrast, though the design target is matched in both configurations, a non-negligible difference exists between the capacity curves of the eight-story structure. The bidirectional NLDAs were performed according to the Italian [51] and European [20] codes, considering seven pairs of artificial ground motions generated by the computer code SIMQKE [93] and characterized by a duration of 25 s each [51]. The performance of the retrofitted frames was evaluated in terms of the maximum displacement (d\({}_{\text{max}}\)) and the maximum acceleration (PFA) at each floor; moreover, the compressive axial force (N) in the columns due to the seismic loads was calculated to check if the buckling limitation defined by the IBC [51] was respected. The mean values of the maxima of these parameters for the seven pairs of bidirectional accelerograms are reported in Figures 12-14. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**X-Direction**} & \multicolumn{4}{c}{**Z-Direction**} \\ \hline
**Story** & \multicolumn{2}{c}{\(\mu_{DB}\) = 4} & \multicolumn{2}{c}{\(\mu_{DB}\) = 10} & \multicolumn{2}{c}{\(\mu_{DB}\) = 4} & \multicolumn{2}{c}{\(\mu_{DB}\) = 10} \\ \cline{2-9}  & \(\text{K}^{D}_{yi}\) & \(\text{N}^{D}_{yi}\) & \(\text{K}^{D}_{yi}\) & \(\text{N}^{D}_{yi}\) & \(\text{K}^{D}_{yi}\) & \(\text{N}^{D}_{yi}\) & \(\text{K}^{D}_{yi}\) & \(\text{N}^{D}_{yi}\) \\  & [kN/mm] & [kN] & [kN/mm] & [kN] & [kN/mm] & [kN] & [kN/mm] & [kN] \\ \hline
8 th & 282.2 & 370.0 & 602.3 & 315.9 & 139.8 & 290.2 & 298.0 & 247.4 \\
7 th & 300.5 & 756.3 & 641.3 & 645.6 & 168.9 & 588.3 & 360.1 & 501.6 \\
6 th & 302.6 & 1111.6 & 645.8 & 949.0 & 173.5 & 857.0 & 369.9 & 730.8 \\
5 th & 303.7 & 1421.7 & 648.1 & 1213.7 & 175.2 & 1084.1 & 373.4 & 924.4 \\
4 th & 325.3 & 1675.0 & 694.2 & 1429.9 & 193.7 & 1259.6 & 413.0 & 1074.1 \\
3 rd & 327.8 & 1865.3 & 699.6 & 1592.4 & 213.5 & 1380.5 & 455.1 & 1177.2 \\
2 nd & 349.0 & 1985.3 & 744.9 & 1694.9 & 274.1 & 1446.7 & 584.2 & 1233.6 \\
1 st & 510.8 & 2034.9 & 1090.2 & 1737.2 & 585.5 & 1468.1 & 1248.2 & 1251.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Eight-story MRF: elastic stiffness \(K^{D}_{yi}\) and yield force \(N^{D}_{yi}\) of the dampers at each story in X- and Z-directions for \(\mu_{DB}\) = 4 and \(\mu_{DB}\) = 10. Figure 11: Results of NSLAs in X-direction of (**a**) two-story, (**b**) four-story, and (**c**) eight-story structures. Figure 12: Maximum displacement at each floor of case-study MRFs with and w/o damped braces: (**a**) two-story, (**b**) four-story, and (**c**) eight-story structures. Figure 12 shows the results in terms of the maximum displacement in the X- and Z-directions, indicated as \(d_{x,max}\) and \(d_{z,max}\), respectively. The introduction of the damped braces produced a significant decrease in the lateral displacement, below the design target. It is also worth noting that the upgraded MRFs exhibited similar stiffness in the two main horizontal directions, respecting the relevant requirement provided in EC8 [20].