## 5 Results: Seismic Capacity of a WFP Structure Upgraded with Hysteretic Dampers  
### Input Energy and Dissipated Energy  
The input energy on a structure subjected to bidirectional (horizontal) seismic loading expressed in terms of equivalent velocity \(V_{E}\) can be simply calculated from the energy input in the X, \(E_{IX}\), and in the Y, \(E_{IX}\), directions by \(V_{E}=(V_{EX}2+V_{EV}2)^{0.5}\), where \(V_{EX}=(2E_{IX}/M)^{0.5}\), \(V_{EY}=(2E_{IX}/M)^{0.5}\). \(V_{E}\) can be interpreted as the modulus of a vector in the \(V_{EX}\)-\(V_{EY}\) plane. The same can be applied to the total energy that contributes to damage, \(E_{D}\), defined [15] as \(E_{I}\) minus the energy dissipated by inherent damping \(E_{\xi}\), i.e., \(E_{D}\)= \(E_{I}-E_{\xi}\). Given the energy balance of the structure [15], it follows that \(E_{D}\) equals the sum of the elastic vibrational energy \(E_{e}\) and the energy dissipated through plastic deformations (hysteretic energy) \(E_{h}\), i.e., \(E_{D}\)=\(E_{e}\)+\(E_{h}\). For high levels of plastic deformations \(E_{e}\) becomes negligible in comparison with \(E_{h}\)[15], which leads to \(E_{D}\approx E_{h}\). The energy that contributes to damage can be calculated independently in the X and Y directions, \(E_{DX}\) and \(E_{DY}\), and is nearly equal to the energy dissipated in the X and Y directions, \(E_{hX}\) and \(E_{hY}\), i.e., \(E_{DX}\approx E_{hX}\) and \(E_{DY}\approx E_{hY}\). The corresponding equivalent velocities are \(V_{DX}=(2E_{DX}/M)^{0.5}\), \(V_{DY}=(2E_{DY}/M)^{0.5}\), and \(V_{D}=(V_{DX}2^{2}+V_{DY}2)^{0.5}\). Based on the above considerations, the values of \(V_{D}\), \(V_{DX}\), and \(V_{DY}\) when the structure is near collapse can be interpreted as the ultimate energy dissipation capacity of the structure in the form of plastic deformations, expressed in terms of equivalent velocities. Figure 10 illustrates the energy capacity of the structure when it reaches the SPL of FO, OP, and NC, respectively. More precisely, Figure 10a,b,d show \(V_{E}\) (defined by its components \(V_{EX}\) and \(V_{EY}\)) obtained through numerical simulations using the model described in Section 4.2, subjected to the five sets of bidirectional ground motion records explained in Section 4.3. The results of each set (Set 1 to Set 5) are identified in the figure with different colors. Additionally plotted are the \(V_{E}\)'s obtained with the numerical model subjected to unidirectional ground motions (referred to as Set X and Set Y); they are identified in the legend with the letter X or Y added to the record name. The mean \(\overline{x}\) (red dash line) and mean plus/minus one standard deviation \(\sigma\) (blue dash line) are also depicted and identified with \(V_{E,FO}\), \(V_{E,OP}\) and \(V_{E,NC}\) in Figure 10.  
The average values shown in Figure 10a,b,d are \(\overline{V}_{E,FO}~{}=~{}36\) cm/s, \(\overline{V}_{E,OP}~{}=~{}112\) cm/s and \(\overline{V}_{E,NC}~{}=~{}301\) cm/s, with standard deviations equal to 11 cm/s, 34 cm/s, and 84 cm/s, respectively. In terms of equivalent velocity this entails that the energy capacity of the structure in terms of input energy for SPL NC is, on average, 2.70 times that for SPL OP. In terms of energy (\(E_{I}\)), the increase is more than sevenfold, i.e., 7.29 (\(=2.70^{2}\)). A more detailed analysis of the results is offered in Table 2, where the mean and the coefficient of variation, COV, of the input energy of the three SPLs are shown for each set of records. Moderate differences are seen among the different groups for \(V_{E,FO}\), which are greater for \(V_{E,OP}\) and especially for \(V_{E,NC}\), with COVs around 0.30. Furthermore, the differences between the input energy achieved for the different SPLs under each set of ground motions were analyzed by means of the ratios \(V_{E,OP}/V_{E,FO}\) and \(V_{E,NC}/V_{E,FO}\). Moderate differences could be seen for \(V_{E,OP}/V_{E,FO}\), with a mean of 3.22. For \(V_{E,NC}/V_{E,FO}\) the mean increases to 8.82.