## 5 Dissipative Bracing Retrofit Solution
### Numerical Verification of the Retrofit Solution  
A perspective view of the model including the dissipative bracing system is displayed in Figure 12. The modal analysis carried out in retrofitted conditions confirms the sequence of modes computed in current state, with differences on periods and EMMs lower than 10%, as a consequence of the small stiffening effect of this technology. The periods and EMMs of the two first horizontal translational modes along \(X\) and \(Y\) pass to 0.83 s (\(Y\)) and 0.32 s (\(X\)), and to 79.5% (\(Y\)) and 88.1% (\(X\)); the periods and EMMs of the fourth and fifth mode, translational along \(X\) and \(Y\), to 0.25 s (\(Y\)) and 0.105 s (\(X\)), and to 20.3% (\(Y\)) and 11.7% (\(X\)). The relevant summed modal masses are nearly equal to 100% along both axes, in this case too. The third and sixth mode, rotational around \(Z\), have periods of 0.33 s and 0.037 s, and EMM equal to 84.9% and 13.8%, giving a summed modal mass of 98.7%.  
The results of the time-history verification analyses in rehabilitated configuration are synthesized in Figures 13-16, all referred to the response induced by the most demanding of the seven groups of input ground motions scaled at the MCE level. The \(M_{Xc}\)-\(M_{Yc}\) interaction curves of the first and second level base sections of column C17, plotted in Figure 9c,d above for the original structure, are duplicated in Figure 13 in retrofitted conditions. The two graphs show that the dissipative action of the protective system allows confining the interaction curves within the biaxial moment safe domain, reducing the maximum \(M_{Yc}\) (Figure 13a) and \(M_{Xc}\) (Figure 13b) moments nearly by the targeted \(\alpha_{MY}\) and \(\alpha_{MX}\) factors of 1.77 and 2.07, as given by (24) and (29).  
The response cycles of the pairs of spring-dampers situated along the opposite vertical alignments _Al. X1, Al. X3_, and _Al. Y2, Al. Y4_ are visualized in Figure 14. The cycles exhibit peak displacements equal to 6.3 mm (_Al. X1, Al. X3_) and 8.8 mm (_Al. Y2, Al. Y4_), far below the available stroke limits of \(\pm 30\) mm (in \(X\)) and \(\pm 40\) mm (in \(Y\)) mentioned above. Furthermore, the response of the devices placed in _Al. X1_ and _Al. X3_ are nearly coincident, and the same occurs for the devices situated in _Al. Y2_ and _Al. Y4_, highlighting that torsion effects in plan are virtually null.  
Figure 12: View of the finite element model incorporating the dissipative bracing system.  
The energy time-histories graphed in Figure 15, where \(E_{I}\) and \(E_{D}\) denote the total input and dissipated energies, and \(E_{I,X}\), \(E_{D,X}\), \(E_{I,Y}\), \(E_{D,Y}\) the relevant portions in \(X\) and \(Y\), show \(E_{D,X}\), \(E_{D,Y}\) values of 66.9 kJ and 101.9 kJ, which differ only by 2% and 1% from the corresponding \(E_{D,\alpha F}\) estimates (27) and (33), respectively.  
Furthermore, practically identical values of the \(E_{D}/E_{I}\) ratio are found in \(X\) and \(Y\), namely: \(E_{D,X}/E_{I,X}\) = 0.855; \(E_{D,Y}/E_{I,Y}\) = 0.85, identifying a well-balanced energy dissipation demand in the two directions.  
The roof top displacement time-histories illustrated in Figure 16 show a reduction factor on the peak values equal to about 2.3 (\(X\)) and 2.2 (\(Y\)) when passing from current (CS) to retrofitted (RS) conditions. In the most deformable direction \(Y\), this corresponds to a drop of the second-level drift from 72.7 mm to 33.8 mm, i.e., below the corresponding elastic limit value \(ILD_{e,2L,Y}\) = 36.8 mm, as targeted in the design.  
Figure 14: Retrofitted structure (RS). Response cycles of the spring–damper pairs installed in _Al. X1_ (**a**), _Al. X3_ (**b**), _Al. Y2_ (**c**) and _Al. Y4_ (**d**) vertical alignments obtained from the most demanding MCE-scaled group of accelerometers.  
Figure 13: Retrofitted structure (RS). \(M_{X,c}\)–\(M_{Y,c}\) biaxial moment interaction curves at the base section of column C17 on the first level (**a**) and second level (**b**) obtained from the most demanding MCE-scaled group of accelerometers.