## 6 Results of Non-Linear Analyses  
The results of the non-linear analyses are evaluated in terms of maximum displacement at the roof level (\(d\)), selected as control point, maximum inter-story drift ratio (\(\Delta\)), and maximum shear force at the base (\(V\)), which are engineering demand parameters (EDPs) usually considered to investigate the performance of the structural system and the effectiveness of the retrofit design [1; 5; 6; 15; 17; 18; 41; 43; 44]. As an example, Figure 11 shows the results at LLS for the six-story building in Tramutola retrofitted with damped braces with ductility \(\mu_{DB}=9\). The damped spectrum depicted in red corresponds to the damping reduction factor \(\eta\) resulting from the actual \(\xi_{eq}\) evaluated as \(\sqrt{10/\left(5+\xi_{eq}\right)}\) which in this case amounts to 0.487. The yellow curve refers to the case in which the damping reduction factor is evaluated in accordance with the IBC [30] as \(\eta_{IBC}=\max\left(0.55;\sqrt{10/\left(5+\xi_{eq}\right)}\right)\). The actual \(\eta\) results in a \(-22.4\%\) in the displacement at roof level and \(-4.1\%\) in the global shear force in comparison to the values calculated by assuming \(\eta_{IBC}\). For the same case study structure, Figure 12 shows the comparison between the maximum displacements at each floor obtained from static and dynamic non-linear analyses in X and Z directions. In the graph, NLTHA,N and NLTHA,A stand for Non-Linear Time-History analyses with either natural (N) or artificial (A) accelerograms, while NLSA,\(\eta\) and NLSA,\(\eta_{IBC}\) refer to non-linear static analyses considering either \(\eta\) or \(\eta_{IBC}\), respectively. NLTHA,N calculates the largest displacements at each floor, but the last one where its results are similar (or even smaller than) to NLSA,\(\eta_{IBC}\). For the first three floors, NLTHA,A, NLSA,\(\eta\), and NLSA,\(\eta_{IBC}\) provide consistent results, smaller than those calculated from NLTTHA,N, but their estimates diverge at the upper floors. Regarding the displacement of the control point, curiously, in the X direction, the maximum and minimum values are calculated by the two time-history analyses, with a deviation of about 27% in the X direction and 29% in the Z direction; in the Z direction, the largest displacement is calculated by NLSA,\(\eta_{IBC}\). There is a deviation of approximately 19% in both directions between NLSA,\(\eta\) and NLTTHA,N, while NLSA,\(\eta_{IBC}\) yields results quite similar to those of NLTHA,N, with a difference of about 5% at the top floor in the Z direction. For each analysis, results similar to those shown in Figure 12 are obtained for displacements, inter-story drift ratio and shear forces in the columns at the base floors. Hereinafter, the results of the non-linear analyses are compared to the values obtained from NLTHAs  
Figure 11: Comparison between the SDOF bilinear capacity curves of the 6-story structure retrofitted with damped braces with \(\mu_{DB}=9\) and the damped demand spectra. with natural accelerograms, assumed as benchmark by means of some simple indexes. Specifically, the three dimensionless ratios, \(d/d_{TH,N}\), \(\Delta/\Delta_{TH,N}\), and \(V/V_{TH,N}\) are introduced. In these ratios, \(d\), \(\Delta\) and \(V\) represent the maximum displacement at the roof level, the maximum inter-story drift ratio, and the maximum shear force at the base, respectively. Each response parameter (\(d\), \(\Delta\) and \(V\)) is calculated in a non-linear analysis (NLSA with actual \(\eta\), NLSA with \(\eta\)\(I\)\(B\), and NLTHA,A) and then compared to the corresponding quantity determined from the NLTHAs with natural accelerograms (\(d_{TH,N}\), \(A_{TH,N}\) and \(V_{TH,N}\), respectively). Figures 13-15 show the dimensionless ratios for the two buildings associated to the examined \(\mu_{DBs}\) (for each index, the largest between the values for the X and Z directions is reported).