## 5 Results: Seismic Capacity of a WFP Structure Upgraded with Hysteretic Dampers
### Equivalent Number of Cycles for SPL NC  
Sections 5.3 and 5.4 addressed two important aspects of the seismic response of structures. One is the ductility level at each \(i\) story expressed through the variable \(\mu_{i}\); the other is the hysteretic energy dissipated through cumulative cyclic reversals of each story \(i\) expressed through the dimensionless variable \(\eta_{i}\). The ratio \(\eta_{i}/\mu_{i}\) is another key aspect of the seismic response of the structure, characterizing the efficiency of the structure in dissipating energy [15]. The ratio \(\eta_{i}/\mu_{i}\) is referred in the literature as the equivalent number of cycles \(n_{\alpha_{i},i}(=\eta_{i}/\mu_{i})\) and is influenced by the characteristics of the ground motion and the type of structure [43]. In flexible-stiff mixed structures, this ratio can be defined for the main structure, \({}_{\mu}n_{\alpha_{i}j}={}_{f}\eta_{i}/\mu_{i}\), and for the stiff part, \({}_{s}n_{\alpha_{i}j}={}_{s}\eta_{i}/s\mu_{i}\).  
Figure 15 offers the mean of \(\mu_{\alpha_{i},1}\) and \({}_{s}n_{\alpha_{i},1}\) obtained in the analyses for SPL NC at the first story along the X and Y directions, i.e., \({}_{f}n_{\alpha_{i},X}={}_{f}\eta_{X,i}/\mu_{X,i}\) and \({}_{s}n_{\alpha_{i},Y,i}={}_{f}\eta_{Y,i}/\mu_{Y,i}\) for the WFP system (Figure 15a), and \({}_{s}n_{\alpha_{i},X,i}={}_{s}\eta_{X,i}/{}_{s}\mu_{X,i}\) and \({}_{s}n_{\alpha_{i},Y,i}={}_{s}\eta_{Y,i}/s\mu_{Y,i}\) for the hysteretic dampers (Figure 15b). It is important to stress that the components X and Y used to calculate \(\mu_{\alpha_{i},1}\) and \({}_{s}n_{\alpha_{i},1}\) were those for which the maximum ductility levels, \({}_{f}\mu_{max,1}\) and \({}_{s}\mu_{max,1}\), were achieved under each seismic record. Figure 15 also shows \(\mu_{\alpha_{i},min1}\) and \({}_{s}n_{\alpha_{i},min1}\), respectively, the minimum values achieved for \(\mu_{\alpha_{i},1}\) and \({}_{s}n_{\alpha_{i},1}\) in the analysis under each set of records.  
Figure 15: Equivalent number of cycles obtained in the analyses at the first story for WFP system (**a**) and for hysteretic dampers (**b**).  
As seen in Figure 15a, \(\mu_{eq,X,1}\) ranges on average between 1 and 1.5, with a mean of 1.39 and COV = 0.24; in turn \(\mu_{eq,Y,1}\) ranges between 1.5 and 2.5, with a mean of 2.0 and COV = 0.28. Further, it can be observed that \(\mu_{eq,min1}\) ranges between 0.53 (set X) and 1.66 (set Y), with a mean of 1.20 and COV = 0.29. These values obtained for \(\mu_{eq,I}\) are similar to the proposal by [PERSON], \(\mu_{eq}=2\) for flexible-stiff mixed systems whose flexible part exhibits pinching (as is the case of WFP systems) [15].  
On the other hand, Figure 15b shows that \({}_{s}n_{eq,X,1}\) ranges on average between 7 and 35, with a mean about 24 and COV = 0.60. \({}_{s}n_{qY,1}\) ranges between 10 and 20, with a mean about 15 and COV = 0.59. The higher values for both \({}_{s}n_{eq,X,1}\) and \({}_{s}n_{eq,Y,1}\) are concentrated under the sets of records for which the correspondent seismic component (X or Y) is large, i.e., Sets X, 1, and 2 for the former, and Sets 5 and Y for the latter. Furthermore, \({}_{s}n_{eq,min1}\) is found to range between 3.1 (Set 4) and 10.72 (Set X), with a mean of 6.36 and COV = 0.39. For flexible-stiff systems whose stiff parts exhibit an elastic-perfectly plastic behavior, [PERSON] [15] proposed \({}_{s}n_{eq}=8\). This value is slightly higher than \({}_{s}n_{eq,min1}\), but close to the lower limit of the range of both \({}_{s}n_{eqX,1}\) and \({}_{s}n_{eq}\), indicated above.  
It is noteworthy that the results obtained in the analyses for \(\mu_{eq,I}\) and \({}_{s}n_{eq,I}\) come from the successively scaled seismic simulations until achieving SPL NC. Therefore, they can be used to obtain the cumulative damage in WFP structures with hysteretic dampers subjected to shocks and aftershocks.