## 5 Results: Seismic Capacity of a WFP Structure Upgraded with Hysteretic Dampers
### Input Energy and Dissipated Energy
The highest values for the latter are found under Set X, Set 1 and especially Set 2 (\(V_{E,NC}/V_{E,FO}\) = 12.23). The structure under Sets 4, 5, and Y showed the highest values for the elastic input energy \(V_{E,FO}\), while the trend was reversed for \(V_{E,NC}\) (i.e., when the structure undergoes plastic deformations), giving the highest values under Sets X, 1, and 2. Therefore, a reduction of the seismic capacity of the structure under plastic deformations is observed when the Y component of the seismic action is the strongest. It is worth noting that the structure analyzed in this study presents eccentricities of the center of mass with respect to the center of stiffness in the Y direction. Figure 10: Energy capacity of the structure: input energy for seismic performance levels (SPL) FO (**a**), SPL OP (**b**) and SPL NC (**d**); hysteretic energy for SPL OP(**c**) and SPL NC (**e**). Figure 11 shows the mean and error bars of \(\theta_{VE}\) defined as \(\theta_{VE}~{}=~{}\text{atan}(V_{EY}/V_{EX})\), obtained for each SPL under the different set of bidirectional ground motions (Set 1 to Set 5). In the ground motions of Set 1, the energy input in the X direction is markedly larger than in the Y direction (see Section 4.3). As seen in Figure 11, for Set 1, as the level of plastic deformations increases (i.e., from the SPL FO to SPL NC) the value of \(\theta_{VE}\) decreases; this means that the energy input in the X direction keeps increasing in relation to the energy input in the Y direction. A similar trend but for the Y component of the ground motion is observed in Sets 3, 4, and 5. In these sets, the energy input in the Y direction is initially larger than in the X direction and keeps increasing with the level of plastic deformation (i.e., the value of \(\theta_{VE}\) increases as the SPL goes from FO to NC). Set 2 presented an intermediate behavior. Accordingly, the structure investigated does not show a tendency to balance the energy input in the two orthogonal horizontal directions as the structure enters in the non-linear range. This is opposite to what has been reported in previous studies [37] for conventional structures, where the torsional response makes the energy input in the X and Y directions tend to grow near as the structure undergoes larger plastic deformation. This finding is most probably due to the presence of the dampers, effectively controlling (i.e., reducing) the torsional response. The EDDs installed in the WFP system successfully reduced the torsional response because: (i) they reduced the initial eccentricity, (ii) the EDDs did not present stiffness degradation, and (iii) overstrength was provided by the EDDs due to strain hardening. Moreover, Figure 10c,e show the maximum energy dissipated through plastic deformations for SPLs OP and NC in terms of equivalent velocity, i.e., \(V_{D,OP}\) and \(V_{D,NC}\), respectively. The same representation criteria as in Figure 10a were used. The average values shown in the Figures with red dashed lines are \(\overline{V}_{D,OP}~{}=~{}77\) cm/s and \(\overline{V}_{D,NC}~{}=~{}241\) cm/s, with standard deviations equal to 22 cm/s