## 5 Results: Seismic Capacity of a WFP Structure Upgraded with Hysteretic Dampers
### Energy Dissipated by Plastic Deformations for SPL NC
Therefore, the symmetrical layout of dampers 1 and 2 (Figure 8) leads to higher efficiency in absorbing and dissipating energy along the X direction than along Y direction, where the three dampers contribute to dissipating energy but not in the same proportion. Moreover, according to these results, the WFP system absorbs and dissipates about 20%, at most, of \(W_{p,1}\); this stands as a moderate contribution and means that the dampers are effectively controlling (i.e., limiting) damage to the main structure. Figure 13: Ratio of the dissipated energy in the hysteretic dampers with respect to the total value at the first story, overall (\({}_{s}W_{p,1}/W_{p,1}\)), and by components (\({}_{s}W_{pX,1}/W_{pX,1}\) and \({}_{s}W_{pY,1}/W_{pY,1}\) ). The damage along X and Y directions at the \(i\) story for both the main part (WFP structure) and the stiff part (dampers) can also be expressed through the dimensionless variable \(\eta_{,i}\) defined in Section 4.1. Accordingly, for the WFP system, the damage along X and Y directions is respectively expressed as \({}_{f}\eta X_{,i}={}_{f}W_{p}X_{,i}/\Big{(}QyX_{,1}\hat{f}\phi_{y}X_{,i}\Big{)}\) and \({}_{f}\eta Y_{,i}={}_{f}W_{p}Y_{,i}/\Big{(}QyY_{,1}\hat{f}\phi_{y}Y_{,i}\Big{)}\). For the hysteretic dampers, the counterpart expressions are \({}_{s}\eta X_{,i}={}_{s}W_{p}X_{,i}/\Big{(}sQyX_{,1}\hat{\phi}_{y}X_{,i}\Big{)}\) and \({}_{s}\eta Y_{,i}={}_{s}W_{p}Y_{,i}/\Big{(}sQyY_{,1}\hat{\phi}_{y}Y_{,i}\Big{)}\). Moreover, according to [PERSON] and [PERSON] [38], the damage to the main part and the stiff part at the \(i\) story can be expressed as \({}_{f}\eta_{,i}={}_{f}\eta X_{,i}+{}_{f}\eta Y_{,i}\) and \({}_{s}\eta_{,i}={}_{s}\eta X_{,i}+s_{s}\eta Y_{,i}\), respectively. Figure 14a,b show \({}_{f}\eta X_{,i}\), \({}_{f}\eta Y_{,i}\) and \({}_{f}\eta_{,i}\) for the first (_i_ = 1) and second story (_i_ = 2), respectively. Higher values are seen for \({}_{f}\eta_{,1}\) than for \({}_{f}\eta_{,2}\). This result is a consequence of the greater strength of the WFP system in the second story with respect to the value provided by an optimum distribution (Section 4.1). Figure 14a also shows that the higher the intensity of the X or Y component of the seismic action, the higher the damage \({}_{f}\eta X_{,1}\) or \({}_{f}\eta Y_{,1}\), respectively. The same response is observed for the hysteretic dampers in Figure 14c. Nevertheless, the efficiency exhibited by dampers in the X direction under ground motion records of large intensity for the X component is higher than that of the Y direction under seismic actions having large intensity for the Y component; this is reflected by the comparatively higher values observed for \({}_{s}\eta X_{,1}\) under Sets X, 1, and 2 as opposed to those for \({}_{s}\eta Y_{,  
Figure 14: Dimensionless representation of the damage: for WFP system at the first story (**a**) and at second story (**b**); for hysteretic dampers, damage level, (**c**) and damage index (**d**). It is likewise interesting to note the significant differences observed for damage in the WFP system versus the dampers, respectively shown in Figures 14a and 14c, a finding related to the energy dissipated in them (Figure 13). Despite the values achieved in dampers, none of them failed, as verified by the energy-based damage index _ID_. This damage index is defined in the reference [40] for steel elements and moreover verified by non-destructive evaluation tests [34; 41; 42]. _ID_ varies from 0 to 1, where 0 indicates no damage and 1 failure. Figure 14d shows the damage indices for the three dampers, _ID\({}_{k}\)_ (\(k\) varies from 1 to 3), achieved under each set of records, as well as the maximum damage index achieved for them under each set of records, _ID\({}_{max}\)_. It is seen that _ID\({}_{max}\)_ ranges from 0.07 (Set 3) up to 0.25 (Set Y), indicating that SPL NC is achieved in conjunction with the failure of the WFP system, but not by failure of the dampers.