## 6 Conclusions  
This study investigated the seismic behavior of reinforced concrete (RC) waffle-flat-plate (WFP) structures upgraded with hysteretic dampers and subjected to bidirectional seismic loadings. To this end, a 3D numerical model representing a 2/5 scale portion of a WFP prototype structure was built, calibrated and validated with the results of bidirectional shake-table tests. Next, hysteretic dampers were added to the numerical model at the first story. The constitutive model used for the dampers was also validated by means of the results of static cyclic tests. The numerical model of the WFP system with dampers was subjected to five sets of seven ground motion records in order to attain SPLs of fully operational (FO, elastic behavior), operational (OP, damage only in dampers), and near collapse (NC, heavy damage in WFP and dampers). Each record was applied in a sequence of scaled seismic simulations until achieving SPL NC. In addition, five records were selected (one from each set) and the X and Y components were applied separately in only one direction until achieving SPL NC. The following conclusions are reached:  
* The total input energy or the total hysteretic energy--expressed in the form of equivalent velocities \(V_{E}\) or \(V_{D}\)--required to attain SPLs FO, OP, and NC remains basically constant, irrespective of the ground motion considered when the two horizontal components of the ground motion are simultaneously applied. * The \(V_{E}\) that the structure can endure until SPL NC was, on average, about three times the value obtained SPL OP, and eight times the value for OP and FO. Meanwhile, the \(V_{D}\) when the structure reaches SPL NC was, on average, about three times that obtained for SPL OP. * Past studies on conventional structures pointed out that torsion effects tend to redistribute the damage in the structure, which in terms of energy means to balance the energy input by the X and Y components of the ground motion. This was not found for the structure under study, most probably due to the fact that the dampers controlled the torsional movements. * The capacity of the structure under bidirectional loadings is in most cases slightly higher than under unidirectional loadings. Nevertheless, for some ground motions the response is opposite. * The values of \(V_{E}\) and \(V_{D}\) estimated from the energies obtained independently for the X and Y components of the ground motion (unidirectional analysis) are always larger than the actual value obtained applying the two components simultaneously (bidirectional analysis). The values are 18% larger for SPL OP and 27% larger for SPL NC. * The relationship \(V_{D}/V_{E}\) obtained for SPL OP (0.70 with COV = 0.09) was very similar to that predicted with [PERSON]'s Equation (2) (0.74). Nevertheless, for SPL NC, the value obtained (0.80) was markedly higher than that provided by [PERSON]'s equation (0.61). * The maximum ductility level, estimated through the maximum apparent plastic deformation, was 1.6 in the WFP system and 5.5 in SPDs. * Most of the energy (80%) input and dissipated by the structure was absorbed by the dampers. * The equivalent number of cycles obtained for the flexible and stiff parts would rely on the characteristics of the main structure and the layout of dampers in X and Y directions. The average and minimum values obtained are \(\mu_{eq,1}=2\) and \(\mu_{eq,min1}=1.20\), respectively, for the flexible part; \({}_{s}n_{eq,1}\) ranged between 15 and 24, with a minimum value of \({}_{s}n_{eq,min1}=6.36\). The values found here are in accordance with those proposed by [PERSON] [15]. Finally, it is important to note that the results shown in this study have been obtained from the analysis of a portion of a structure. Therefore, further research will be required to apply them to a whole structure.