## 4 Numerical Analyses of a WFP Structure Upgraded with Slit-Plate Dampers (SPDs)
### Numerical Analyses  
The numerical model described in the previous section was subjected to a selection of ground motion records to obtain the ultimate capacity of the specimen in terms of energy. The ground motion records were selected from the database prepared by [PERSON] et al. [36], which contains ordinary records (i.e., without recognizable pulses in the ground velocity) with moment magnitude greater or equal to 5. The records were selected on the basis of the following two criteria. First, the scale factor \(SF_{70}\) to be applied to the accelerograms in order to achieve a total input energy--in terms of pseudo velocity \(V_{E}=(2E_{I}/M)^{0.5}\) of \(V_{E}=70\) cm/s--was between 1/3 and 3. \(V_{E}=70\) cm/s is the input energy measured in the WFP specimen tested on the shake-table (Section 2.1) when it was on the brim of yield under uni- and bidirectional loadings [6; 7]; it will be referred to as \(V_{E,70}\) (=70 cm/s) herein. It was expected that by adding the hysteretic dampers, the input energy to produce plastic deformations in WFP structure would be higher than \(V_{E,70}\). The energy input in the X and Y directions at the onset of yielding of the WFP structure without dampers, i.e., when \(V_{E,70}\) (=70 cm/s) is attained, will be respectively referred to as \(E_{IX,70}\) and \(E_{IY,70}\), and the corresponding equivalent velocities are \(V_{EX,70}=(2E_{IX,70}/M)^{0.5}\) and \(V_{EX,70}=(2E_{IX,70}/M)^{0.5}\). In general, \(V_{EX,70}\) and \(V_{EY,70}\) are different for each ground motion. The second criterion is that the PGA of the ground motion after scaling by \(SF_{70}\), be 0.3 g at most. This condition was imposed to avoid excessively large (i.e., unrealistic) values of PGA when the numerical model collapses. The records were classified in five sets, Set 1 to Set 5, according to the angle \(\theta\) formed by the components X and Y of \(V_{E}\), \(\theta=\text{atan}(V_{EY,70}/V_{EX,70})\), as follows: \(22.91^{\circ}<\theta<32.65^{\circ}\) for Set 1; \(32.65^{\circ}<\theta<42.40^{\circ}\) for Set 2; \(42.40^{\circ}<\theta<52.14^{\circ}\) for Set 3; \(52.14^{\circ}<\theta<61.88^{\circ}\) for Set 4; and \(61.88^{\circ}<\theta<71.62^{\circ}\) for Set 5. Seven records corresponding to different earthquakes were chosen within each set. The records are identified by the earthquake name followed by the record sequence number in the database (see Tables A1-A5 in Appendix A for details). The \(X\) and \(Y\) components of each ground motion scaled by \(SF_{70}\) were applied simultaneously to the numerical model in successive seismic simulations using a sequence of scaling factors, 100%, 200%, 300%, etc., until the seismic performance level (SPL) of Near Collapse (NC) was achieved. That is, 100% meant that the scaling factor applied to the original ground motion was \(SF_{70}\), for 200% the scaling factor was \(2\times SF_{70}\), and so on. In addition, within each set, one ground motion was selected, and the \(X\) and \(Y\) components were applied separately to the numerical model until failure, so as to investigate the ultimate capacity under unidirectional seismic loads. In total, 45 NLTHAs were launched in a parallel scheme (OpenseesMP.exe) using a Dell Precision Tower 5810 with 12 cores at 3.60 GHz in order to minimize the computational time. As a reference, the mean computational time required was 40 h per core and record. Three limit states characterized by their SPLs were considered in this study. The first one corresponds to the SPL Fully Operational (FO), in which the structure only undergoes elastic deformations, and therefore no damage occurs. The second is SPL Operational (OP), in which the WFP system remains elastic and only the dampers are damaged. The third SPL is NC, in which the WFP system is severely damaged, and the dampers are also damaged but without failing.