## 5 Results: Seismic Capacity of a WFP Structure Upgraded with Hysteretic Dampers
### Input Energy and Dissipated Energy
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Set**} & **mean** & \multirow{2}{*}{**COV**} & **mean** & \multirow{2}{*}{**COV**} & **mean** & \multirow{2}{*}{**COV**} & **mean** & **COV** & **mean** & **COV** \\  & & & & & & & & & \\  & **(cm/s)** & & & & & & & & \\ \hline X & 30.36 & 0.25 & 98.58 & 0.33 & 294.89 & 0.28 & 3.28 & 0.21 & 10.15 & 0.31 \\
1 & 32.15 & 0.15 & 104.14 & 0.17 & 289.98 & 0.30 & 3.29 & 0.18 & 9.19 & 0.31 \\
2 & 32.94 & 0.29 & 127.21 & 0.29 & 391.20 & 0.24 & 4.00 & 0.29 & 12.23 & 0.23 \\
3 & 38.64 & 0.20 & 136.04 & 0.16 & 301.72 & 0.12 & 3.64 & 0.24 & 8.04 & 0.19 \\
4 & 40.87 & 0.35 & 113.43 & 0.40 & 255.92 & 0.24 & 2.92 & 0.32 & 6.69 & 0.26 \\
5 & 40.85 & 0.36 & 92.30 & 0.26 & 278.61 & 0.16 & 2.44 & 0.24 & 7.55 & 0.30 \\ Y & 36.77 & 0.25 & 105.36 & 0.27 & 293.24 & 0.31 & 2.93 & 0.25 & 8.02 & 0.19 \\ \hline
**All** & \multirow{2}{*}{**36.31**} & \multirow{2}{*}{**0.31**} & \multirow{2}{*}{**111.81**} & \multirow{2}{*}{**0.31**} & \multirow{2}{*}{**301.39**} & \multirow{2}{*}{**0.28**} & \multirow{2}{*}{**3.22**} & \multirow{2}{*}{**0.30**} & \multirow{2}{*}{**8.82**} & \multirow{2}{*}{**0.33**} \\
**records** & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Analysis of the input energy obtained from non-linear time history analyses (NLTHAs). Figure 11: Mean and error bars of \(\theta_{VE}\) for the different SPLs. and 69 cm/s, respectively. A more detailed analysis of results regarding the energy that contributed to damage is given in Table 3. As seen, under the sets of ground motions X, 1 and 2, the structure shows the highest values for the capacity to dissipate energy by plastic deformation, as with input energy. Moreover, this capacity is on average more than threefold, in terms of equivalent velocity, for SPL NC than for SPL OP, exhibiting the highest differences under unidirectional seismic loadings (Sets X and Y) and the bidirectional seismic loadings with Sets 1 and 2 for which the X component of the seismic action is the strongest. Table 3 furthermore gives the mean of the ratios \(V_{D,OP}/V_{E,OP}\) and \(V_{D,NC}/V_{E,NC}\) obtained for the different set of records for SPL OP and NC, respectively. The mean of \(V_{D,OP}/V_{E,OP}\) under each set of records was in a range between 0.66 and 0.72, the absolute mean--taking into account all the records--being equal to 0.70. For \(V_{D,NC}/V_{E,NC}\), a range between 0.76 and 0.82 was obtained and the absolute mean was 0.80. Two aspects of these results deserve special attention. The first one is the low dispersion achieved for both cases, with COV equal to 0.11 at the most for \(V_{D,OP}/V_{E,OP}\) and 0.04 for \(V_{D,NC}/V_{E,NC}\). The second noteworthy finding is: the stronger the X component of the seismic action applied along a symmetrical direction, the higher the aforementioned relationships, and thus the capacity to dissipate the energy through plastic deformations. As indicated above, the mean of \(V_{D,OP}/V_{E,OP}\) was 0.70 (COV = 0.09) and the mean of \(V_{D,NC}/V_{E,NC}\) was 0.80 (COV = 0.04). Numerous expressions are proposed in the literature to estimate \(V_{D}/V_{E}\). Among them, the one proposed by [PERSON] [15] is expressed as follows:  
\[\frac{V_{D}}{V_{E}}\ =\ \frac{1}{1+3\xi+1.2\sqrt{\xi}} \tag{2}\]  
Substituting the damping ratio 0.038 established in Section 4.2 in Equation (2) gives \(V_{D}/V_{E}\) = 0.74, which is very close to the value obtained, on average, in the analyses for \(V_{D,OP}/V_{E,OP}\) (=0.70). Nevertheless, for SPL NC both the WFP system and the dampers undergo plastic deformations. This means that higher values are expected for the modified frequency and damping ratio. Substituting the damping ratio 0.09 considered in Section 4.2 for the WFP system with dampers when it is near collapse, Equation (2) gives \(V_{D}/V_{E}\) = 0.61, far from the value obtained from the analysis, (\(V_{D,NC}/V_{E,NC}\) = 0.80).