## 6 Results of Non-Linear Analyses
The curves are labelled as follows: curve (a) shows the results from NLSAs where the actual damping reduction factor \(\eta=\sqrt{10/\left(5+\xi_{eq}\right)}\) is considered; curve (b) reports the results from NLSAs in which \(\eta_{IBC}=\max\left(0.55;\sqrt{10/\left(5+\xi_{eq}\right)}\right)\) is used; curve (c) shows the results from NLTHAs with artificial accelerograms. The deviation on the top building displacement is examined in Figure 13. For the three-story building, NLSAs overestimate the top building displacement calculated by NLTHA,N except for Benevento (CLS) where \(d/d_{TH,N}\) is approximately equal to unity for NLSA,\(\eta_{IBC}\), and about 0.8 for NLSA,\(\eta\). Also, in the case of Benevento (LLS), for \(\mu_{DB}\) = 3, NLSA,\(\eta_{IBC}\) underestimates the results from NLTHA,N. Considering the site of Tramutola, for both limit states, \(d/d_{TH,N}\) ranges from 1.05 to 1.40 for NLSA,\(\eta\) and from 1.20 to about 2.0 for NLSA,\(\eta_{IBC}\). The opposite trend is revealed for the six-story building: NLSA,\(\eta\) underestimates the benchmark displacement for both sites; \(d/d_{TH,N}\) is approximately constant at LLS, while it tends to increase with \(\mu_{DB}\) at CLS and approaches unity for the case of Tramutola when \(\mu_{DB}\geq 9\). In general, NLSA,\(\eta_{IBC}\) yields conservative results at CLS when \(\mu_{DB}\geq 5\) (Tramutola) or \(\mu_{DB}\geq 7\) (Benevento), while at LLS it underestimates the displacement of the top floor for small \(\mu_{DB}\). NLTHA,A always estimates lower values than NLSAs. More specifically, when referring to \(d/d_{TH,N}\) (Figure 13), NLTHA,A underestimates the top displacement calculated from NLTHA,N in either building. Only in the particular case of the three-story building in Benevento do artificial accelerograms provide larger (for \(\mu_{DB}\) = 3) or similar displacements than natural ones. As a general trend, for the three-story building, ratio \(d/d_{TH,N}\) decreases as \(\mu_{DB}\) increases, while for the six-story building the ratio is virtually independent of the properties of the damped brace system, at least when \(\mu_{DB}\geq 5\). Additionally, for the six-story building at  
Figure 12: Comparison of maximum displacements at each floor of the 6-story building retrofitted with damped braces with \(\mu_{DB}\) = 9 obtained from the different non-linear analyses. both sites, similar values of \(d\)/\(d\)/\(d\)\({}_{TH,N}\) are obtained at LLS and CLS, showing that the accuracy is little affected by the intensity of the earthquake. Disregarding the case of \(\mu_{DB}\) = 3, for the three-story building, the maximum deviation from the benchmark is about \(-20\%\) in Benevento and \(-22\%\) in Tramutola, while for the six-story buildings the corresponding figures are \(-28\%\) and \(-25\%\), respectively. It is worth noticing that for the three-story building in Benevento, NLTHA,A and NLSA,\(\eta\) provide results remarkably close to each other, and this similarity increases with the increase in the ductility of the damping system. Figure 13: Comparison of \(d\)/\(d\)/\(d\)\({}_{TH,N}\) for the case study structures; the curves refer to (a) results from NLSAs with actual \(\eta\); (b) results from NLSAs with \(\eta_{BC}\); (c) results from NLTHA,A. By looking at Figure 14 relevant to \(\Delta/\Delta_{TH,N}\), non-linear static analyses provide rough estimates of the maximum inter-story drift, with overestimates that increase almost proportionally with \(\mu_{DB}\), with the only exception of Benevento CLS where \(\Delta/\Delta_{TH,N}\) is close to unity. Notably, for low damper ductility (\(\mu_{DB}=3\)), \(\Delta/\Delta_{TH,N}\) is always close to one. The ratio between inter-story drifts calculated by NLSA,\(\eta\) and by NLSA,\(\eta_{IBC}\) is as large as \(\eta_{IBC}\). Figure 14: Comparison of \(\Delta/\Delta_{TH,N}\) for the case study structures; the curves refer to (a) results from NLSAs with actual \(\eta\); (b) results from NLSAs with \(\eta_{IBC}\); (c) results from NLTHA,A.