# Estimation of Cyclic Demand in Metallic Yielding Dampers Installed on Frame Structures
## 3 Results  
The results for the two sets of analyses are presented separately. The first set corresponds to the design earthquake. At this hazard level, seismic damage and inelastic energy dissipation is sustained by the dampers and the results correlate to the efficacy \({}_{s}\eta\)/\({}_{s}\eta\) of the dampers with significant structural and seismological parameters. The second sets of analyses correspond to the ultimate capacity of the mixed structural system under the action of very rare earthquakes. In this case, the results are upper bound values of the efficacy of the two parts of the structural system. Estimation of Average \({}_{s}\eta\)/\({}_{s}\mu\) of Dampers under Design Earthquake Hazard Level  
The results at the end of the design earthquake hazard level showed that the inelastic demand concentrated at the dampers while the structure remained elastic, in both the with and without capacity design cases. Therefore, just the plastic response of the dampers is of interest herein. For each nonlinear time history analysis, the following variables were collected in a dataset: the ratio \(\tau\) of the fundamental period \(T_{1}\) to the initial period of medium \(T_{g}\), the reduction factor \(R\), the seismological parameter \(I_{d}\), and the average cyclic demand parameter \({}_{s}\eta\)/\({}_{s}\eta\) along the height of the building. Regular statistical procedures were used to analyse the results and find significant relationships between the variables. A preliminary principal component analysis was performed to determine the minimum number of variables that account for the most variation in the data, and found that the first two components (\(I_{d}\) and \(R\)) cover 92% of the variability of the data. On the other hand, a poor correlation between the  
Figure 3: Schematic of the structural model. parameter \(\tau\) and the average value of \({}_{\rm{s}}\eta\)/\({}_{\rm{s}}\mu\) was found. Figure 4 shows a scatter of these principal variables (\(R\)-1) and \(I_{d}\) versus the average cyclic demand parameter \({}_{\rm{s}}\eta\)/\({}_{\rm{s}}\mu\) for each analysis in the dataset. Data is arranged around two clusters: near-fault seismicity in red, and far-field seismicity in blue. It can be observed that far-field seismicity produces significantly higher values of \({}_{\rm{s}}\eta\)/\({}_{\rm{s}}\mu\). In Figure 4, the results from building models incorporating (or not) capacity design rules are plotted with triangles pointing downward or upwards respectively, and show very little variation one to each other. Using a non-linear least squares fitting, the average cyclic demand parameters of the dampers in a structure can be predicted with Equation (10) as a function of the principal variables, the strength reduction factor \(R\) and the seismological parameter \(I_{d}\). The prediction with this equation is depicted in Figure 4 by the grey surface and black dots. Equation (10) correlates moderately with a regression score \(R^{2}=0.59\). \[{}_{\rm{s}}\eta/{}_{\rm{s}}\mu=1+1.18\cdot(R-1)^{0.24}.\ I_{d}^{0.44} \tag{10}\]  
Behaviour of \({}_{\rm{s}}\eta\)/\({}_{\rm{s}}\mu\) of Dampers and \(\eta\)/\(\mu\) of Frames under the Maximum Considered Earthquake  
In this study, the maximum considered earthquake that the structures were able to withstand was defined as that in which one of the dampers in the building reaches its ultimate energy dissipation capacity. The ultimate energy dissipation capacity was assessed using the damage model proposed by [30] and calibrated with the experimental results from a previous study [28]. At this earthquake level, plastic excursions are likely to occur on the reinforced concrete frame as well. For each nonlinear time history analysis under the maximum earthquake, the following variables were collected in a dataset: the parameter \(I_{d}\) of the ground motion, the number of equivalent inelastic cycles in the dampers \({}_{\rm{s}}\eta\)/\({}_{\rm{s}}\mu\), and the average \({}_{\rm{t}}\eta\)/\({}_{\rm{t}}\mu\) in the reinforced concrete frame.