## 6 Conclusions  
The energy-based design criterion formulated in this study for the seismic retrofit of frame buildings by means of dissipative bracing technologies does not require any preliminary evaluation of the input energy demand on the original structure. At the same time, the most critical response parameters in current conditions--the reduction of which within the boundary of relevant safe domains (in case of lack of strength), or below the limits preventing damage to structural and non-structural elements (in case of excessive lateral displacements)--are evaluated by a conventional elastic finite element analysis. Both aspects of the initializing step of the sizing procedure allow simplifying the design of supplemental damping-based retrofit solutions, which can be useful especially for professional engineers not familiar with seismic energy computation and the development of non-linear time-history analyses.  
The criterion was detailed here for relatively stiff structures, i.e., with a fundamental translational vibration period in original conditions below 0.8 s, where the retrofit design objectives can be met by an added damping, with a marginal role of supplemental stiffness. This prompted to select the bracing systems that incorporate FV spring-dampers as protective devices, because they provide a moderate contribution to the lateral stiffness of the retrofitted structures. However, the procedure can be extended with little modifications to dissipative bracing technologies that significantly increase the translational stiffness too, like the systems including metallic dampers, which will be the subject of a further step of the study.  
The demonstrative application to the considered case-study structure allowed checking the quick sizing characteristics of the design criterion, even when stress state-related and drift-related deficiencies are both found in the original structure (as occurs in \(Y\) direction of the gym building). Furthermore, the values of the equivalent damping coefficient ratio calculated as a function of the reduction factors \(\alpha_{F}\), \(\alpha_{d}\) relevant to stress states and drifts, using formulas (14) and (20), respectively, resulted to be  
Figure 16: Rooftop displacement time-histories in X and Y direction obtained from the most demanding MCE-scaled group of accelerograms.  
Figure 15: Retrofitted structure (RS). Energy time-histories obtained from the most demanding MCE-scaled group of accelerograms.  
notably different. On the other hand, a slight difference was found between the corresponding energy dissipation measures, \(E_{D,\alpha F}\) and \(E_{D,\alpha d}\). This identified \(E_{D}\) as a more stable and reliable design parameter, as compared to \(\xi_{eq}\), consistently with the fact that \(E_{D}\) is only determined by the area covered by the response cycles of the dissipaters. This was also confirmed by the fact that the \(E_{D}\) values in \(X\) and \(Y\) computed from the time-history verification analysis were very similar to the \(E_{D,\alpha F}\) estimates.  
As targeted in the retrofit design, the incorporation of the protective system in the gym building allows reaching an elastic and safe response of all members, as well as constraining the interlevel drifts below the Immediate Occupancy drift limit, up to the MCE, starting from a rather poor performance in original conditions.  
The study reported in this paper was sponsored by the Italian Department of Civil Protection within the ReLUIS-DPC Project 2014/2018, Research Line 6: Isolation and Dissipation. The author gratefully acknowledges this financial support.  
The author declares no conflict of interest.