## 5 Results: Seismic Capacity of a WFP Structure Upgraded with Hysteretic Dampers
### Differences for the Seismic Capacity Under Unidirectional and Bidirectional Loadings  
The seismic capacity of structures determined through independent unidirectional analysis along two orthogonal directions is a matter of controversy [27; 37; 38; 39]. To shed light on this issue, a comparison is made in this section between: (i) the capacity of the WFP system with hysteretic dampers obtained from the analysis under unidirectional seismic loadings using the records included in Set X and Set Y, and (ii) that obtained from the analysis under bidirectional seismic loadings using both components of those same records. In this case, only SPLs OP and NC are considered. The input energy and dissipated energy derived for each SPL in terms of equivalent velocity under unidirectional loading are represented by \(V^{uni}_{EX,SPL}\) and \(V^{uni}_{DX,SPL}\) along X direction, and by \(V^{uni}_{ETY,SPL}\) and \(V^{uni}_{DY,SPL}\) along the Y direction. Moreover, \(V^{uni}_{EX,SPL}\) and \(V^{uni}_{ETY,SPL}\) can be used to obtain an estimation of the capacity of the structure in terms of input energy, \(V^{*}_{E,SPL}\), through the expression \(V^{*}_{E,SPL}=\sqrt{\left(V^{uni}_{EX,SPL}\right)^{2}+\left(V^{uni}_{ETY,SPL} \right)^{2}}\). By applying a similar procedure, an estimation of the total dissipated energy, \(V^{*}_{D,SPL}\), is obtained through the expression \(V^{*}_{D,SPL}=\sqrt{\left(V^{uni}_{DX,SPL}\right)^{2}+\left(V^{uni}_{DY,SPL} \right)^{2}}\).  
Table 4 reports the results, so that for each SPL the seismic capacity of the WFP system with hysteretic dampers under unidirectional loadings can be compared to the values obtained under bidirectional loadings in terms of input energy and dissipated energy. For SPL OP, the capacity seen for most records under unidirectional loadings is smaller than under bidirectional loadings, being on average 79% to 85% for \(V_{E}\) and 70% to 80% for \(V_{D}\). The same trend is observed for SPL NC, but with higher values--86% to 89% for \(V_{E}\), and 78% to 88% for \(V_{D}\). Similar results were obtained by [PERSON] et al. in columns subjected to biaxial cyclic loadings and variable axial loads [27]. Yet, there are exceptions, such as the response under the Y component of Kobe-1115 or the X component of Landers-888 in SPL NC, which show values of \(V_{E}\) and \(V_{D}\) that are higher than those under bidirectional seismic action (about 50% higher for Kobe-115 and about 20% greater for Landers-888). The limited number of records used in the present study impedes the formulation of a general statement about the higher capacity of structures under bidirectional loading versus unidirectional loads. It is worth noting that the columns of the WFP system are prone to undergoing variable axial loads under both unidirectional and bidirectional cyclic loadings due to the scheme of the structure itself--three columns. This could explain the results obtained.  
Furthermore, Table 4 indicates the differences observed between the capacity in terms of energy estimated from the uniaxial analyses under X and Y components of the selected seismic actions and that obtained in the bidirectional analyses under both components of the referred seismic actions. In all cases, \(V^{*}_{E,SPL}\) and \(V^{*}_{D,SPL}\) exceeded the counterpart \(V_{E,SPL}\) and \(V_{D,SPL}\). For SPL OP the differences, on average, are about 18% higher with COV = 0.07. For SPL NC the differences are greater, being about 27% higher with COV = 0.20.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Rec. Name** & \(V