## 5 Results: Seismic Capacity of a WFP Structure Upgraded with Hysteretic Dampers
### Energy Dissipated by Plastic Deformations for SPL NC  
The energy dissipated by plastic deformations, i.e., the damage obtained in the analyses for SPL NC in both the WFP system and the hysteretic dampers, is studied here. The damage at the \(i\) story, \(W_{p,i}\), is obtained by adding the damage in the flexible part (WFP system), \(fW_{p,i}\), and that of the stiff part (hysteretic dampers), \(sW_{p,i}\), i.e., \(W_{p,i}\ =\ fW_{p,i}+sW_{p,i}\). Further, \(W_{p,i}\), \(W_{p,i}\) and \(sW_{p,i}\) can be expressed from their X and Y components as \(W_{p,i}\ =\ W_{p,X,i}+W_{pY,i}\), \(w_{p,i}\ =\ fW_{pX,i}+fW_{pY,i}\) and \(sW_{p,i}\ =\ sW_{pX,i}+sW_{pY,i}\). Then, the following relationships are deduced: \(W_{pX,i}\ =\ fW_{pX,i}+sW_{pX,i}\) and \(W_{pY,i}\ =\ _{p}\ fW_{pY,i}+sW_{pY,i}\). The dissipated energy of the main structure at the \(i\) story along X and  
Figure 12: Mean and error bars for \(\mu\) along X and Y directions: (**a**) main structure; (**b**) hysteretic dampers. Y directions under a given seismic record is obtained as follows: (i) first, the contribution of the \(j\) column to the dissipated energy along the X and Y directions is obtained through the expressions \({}_{f}W_{pX,ij}~{}=~{}\int_{f}Q\chi_{x,ij}d\delta x_{,ij}\) and \({}_{f}W_{pY,ij}~{}=~{}\int_{f}Q\chi_{x,ij}d\delta\chi_{,ij}\), respectively, where \(\
ot{\sim}Q\chi_{x,ij}\) and \({}_{f}\
ot{\sim}Y\chi_{x,ij}\) are the shear force histories of the \(j\) column along X and Y directions, and \(\delta\chi_{x,ij}\) and \(\delta\chi_{x,ij}\) are the counterpart interstory drift histories; (ii) next, \(\
ot{W}_{pX,i}\) and \(\
ot{W}_{pY,i}\) are obtained by adding the contribution of the dissipated energy of all the columns of the story along X and Y directions, i.e., \({}_{f}W_{pX,i}~{}=~{}\sum\limits_{j~{}=~{}1}^{Nc}{}_{f}W_{pX,ij}\) and \({}_{f}W_{pY,i}~{}=~{}\sum\limits_{j~{}=~{}1}^{Nc}{}_{f}W_{pY,ij}\), where \(N_{c}\) is the number of columns at the \(i\) story. The dissipated energy of the dampers at the \(i\) story is obtained by adding the contribution of all of them along the X and Y directions. The WFP system considered in this research was upgraded with dampers located only at the first story, arranged as indicated in Figure 8. The energy dissipated by a \(k\) damper under a seismic record, \({}_{s}W_{p,ik}\), is obtained through the expression \({}_{s}W_{p,ik}~{}=~{}\int_{s}N_{k}d(_{s}\Delta u_{k})\), where \({}_{s}N_{k}\) and \({}_{s}\Delta u_{k}\) are respectively the axial force history and the relative axial displacement history of the damper. Damper 3 is aligned in Y direction, hence its contribution to the dissipated energy for the first story is entirely accounted for in \({}_{s}W_{pY,1}\). Dampers 1 and 2 contribute to dissipating energy in both X and Y directions (Figure 8); their contributions \({}_{s}W_{pX,1}\) and \({}_{s}W_{pY,1}\) are \({}_{s}W_{p,1k}(\cos(\alpha))^{2}\) and \({}_{s}W_{p,1k}(\sin(\alpha))^{2}\), respectively. Figure 13 shows that the energy absorbed and dissipated by the dampers with respect to the total one at the first story, \({}_{s}W_{p,1}/W_{p,1}\) ranges, on average, from 80% to 90%. The highest values are obtained under Sets X, 2, and 3--higher input energy in X component--and the lowest under set Y. This means that most of the energy is absorbed by the dampers. Additionally shown in Figure 13 is the former ratio, but expressed from the X and Y components, \({}_{s}W_{pX,1}/W_{pX,1}\) and \({}_{s}W_{pY,1}/W_{pY,1}\), respectively. \({}_{s}W_{pX,1}/W_{pX,1}\) ranges, on average, from 83% to 97% for bidirectional loadings (Sets 1 through 5) and 100% under set Y--without a seismic component in X direction. \({}_{s}W_{pY,1}/W_{pY,1}\) ranges, on average, from 77% to 88% for bidirectional loadings and 100% under set X--without a seismic component in Y direction--for which reason the torsional effects led to displacements and damage entirely absorbed by the dampers.