# Estimation of Cyclic Demand in Metallic Yielding Dampers Installed on Frame Structures
## 4 Discussion  
The number of equivalent inelastic cycles \(N\) can be formulated as the ratio \(\eta/\mu\) between the normalised dissipated energy \(\eta\) and the displacement ductility \(\mu\). This ratio is crucial to the design of hysteric dampers within the framework of performance-based design, as it quantifies the capacity to dissipate seismic energy for a given displacement. The ratio \(\eta/\mu\) is also referred to as the efficacy of the dampers. The literature shows that the number of equivalent inelastic cycles varies for each specific lateral carrying system, and that it is strongly influenced by seismological parameters, especially by the intensity and the impulsive effects of the ground motion [16, 17, 25]. This study focused on mixed flexible-stiff structures, made of reinforced concrete frames with supplemental hysteric dampers under two levels of seismic hazard: the design earthquake and the maximum considered earthquake. One interesting result is that the capacity design provisions in the gravity load resisting system, i.e., reinforced concrete frames has a negligible effect on the number of equivalent inelastic cycles of the dampers for both hazard levels. Because the efficacy of the dampers is independent of the lateral collapse pattern, the frame design can be neglected when designing the dampers. This is important in the retrofitting of pre-code frames that have undefined lateral collapse mechanisms. In light of these results, retrofitting strategies that focus on increasing the lateral deformability of the frames or preventing brittle failures (such as FRP wrapping) will have a greater impact on the performance of the dampers compared to strategies that focus on strengthening or reallocating plastic hinges. Under the design earthquake, the ratio \({}_{s}\eta/s\mu\) can be predicted as a function of the structural parameter \(R\) (the seismic force reduction factor, which relates the intensity of the ground motion and the lateral capacity of the structure), and the seismological parameter \(I_{d}\), which has been shown to be an indicator of the impulsive effects of the ground motion. The tendency in the results is for larger ratios of \({}_{s}\eta/s\mu\) as the parameters \(R\) and \(I_{d}\) increase. This latter parameter \(I_{d}\) accounts for most of the variability in the results, therefore it must drive the design of hysteretic dampers. Other parameters such as the ratio \(\tau\) between the fundamental period and ground motion corner period show a poor correlation with \({}_{s}\eta/s\mu\) in this type of structure and were neglected in the regression. The maximum considered earthquake takes the structure to the onset of strength degradation in one of the dampers along the building, and allows for some structural damage to the reinforced concrete frames. Because some dampers reach their ultimate dissipation capacity in this scenario, the results of the ratio \({}_{s}\eta/s\mu\) in the dampers should be taken as an upper-bound limit in the damper design. Our study confirmed that the efficacy of the dampers is significantly larger and more stable in the far-field. However, the ratio \(\eta/\mu\) in the frame remains in the range of 1-2 for all analyses, which agrees with [PERSON]'s [19] approximate design value of \(\eta/\mu\) = 2 for bare elastic-plastic frames. Interestingly, the ratio \(\eta/\mu\) is unchanged despite the impulsive effects of the ground motions, as it shows non-correlation to \(I_{d}\). Because the efficacy of the frames is similar for far-field and near-fault seismicity, the matter of impulsivity can be neglected in the overall design and assessment of frames equipped with dampers. Conceptualisation, L.M. ; Funding acquisition, L.M. ; Methodology, L.M. and D.E.-M.; Visualisation, L.M. ; Writing--original draft, L.M. and D.E.-M.; Writing--review & editing, L.M. and D.E.-M. All authors have read and agreed to the published version of the manuscript.