## 4 Numerical Analyses of a WFP Structure Upgraded with Slit-Plate Dampers (SPDs)
### Design of the Hysteretic Dampers  
The dampers were designed in view of two conditions. The first was for them to have the maximum strength established in seismic codes. The second was to use a single type of damper in each story, i.e., all dampers installed in a given story would have the same stiffness and strength. The second condition is common in practice, simplifying the construction process and helping to avoid execution errors. Figure 8 shows a detail of the layout of the dampers at the first story in the WFP system. They were placed as diagonal members connecting the ends of columns. This arrangement of the dampers was intended to decrease the eccentricity in Y direction between the center of masses and the center of stiffness. For the dampers located at the first story, the yield-strength was defined according to the criteria proposed by standard ASCE-SEI/41-13 [35], which establishes a base shear for the dampers equal, at most, to 50% of the main structure. The base shear strength of the WFP specimen tested in laboratory in X and Y directions, \(\mathcal{Q}_{y,X,1}\) and \(\mathcal{Q}_{y,Y,1}\), were 57.5 kN and 62.5 kN, respectively [7]. These results were corroborated against those provided by the numerical model through a pushover analysis. Therefore, the base shear of the dampers in X and Y directions, \({}_{s}\mathcal{Q}_{y,X,1}\) and \({}_{s}\mathcal{Q}_{y,Y,1}\), was limited to 28.75 kN (=57.5/2) and 31.25 kN (=62.5/2), respectively. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline
**Uniaxial** & \({}_{s}\mathcal{N}_{y}\) & \({}_{s}\mathcal{k}\) & **b** & **R0** & **cR1** & **cR2** & **a1** & **a2** & **a3** & **a4** & **sigInit** \\
**Material** & **[kN]** & **[kN/mm]** & & & & & & & & & \\ \hline Steel02 & 16 & 100 & 0.005 & 22.0 & 0.925 & 0.15 & 0.13 & 1.0 & 0.13 & 1.0 & 0.0 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the constitutive model of a slit-plate damper (SPD) calibrated with an experimental test. Figure 7: Force displacement curves obtained from the test and the calibrated model. Using these \({}_{s}\)Q\({}_{y,X,1}\) and \({}_{s}\)Q\({}_{y,Y,1}\), the axial yield strength of the dampers at the first story for X and Y directions, \({}_{s}\)\(N\)\({}_{y,X,1}\) and \({}_{s}\)\(N\)\({}_{y,Y,1}\), were obtained as follows. For X direction, \({}_{s}\)Q\({}_{y,X,1}\)\(=\)\(2_{s}\)\(N\)\({}_{y,X,1}\)\(\cos\alpha\)\(\cos\gamma\), from which \({}_{s}\)\(N\)\({}_{y,X,1}\)\(=\) 17.72 kN. For Y direction, \({}_{s}\)Q\({}_{y,Y,1}\)\(=\)\({}_{s}\)\(N\)\({}_{y,Y,1}\)\((\cos\theta+2\cos\beta\cos\gamma)\), thus \(N\)\({}_{y,Y,1}\)\(=\) 20.93 kN. The meaning of angles \(\alpha\), \(\beta\), \(\gamma\), \(\theta\) is shown in Figure 8. Finally, a common value for the axial yield strength of the dampers installed in the first-story, \({}_{s}\)\(N\)\({}_{y,1}\), was determined as \({}_{s}\)\(N\)\({}_{y,1}\)\(=\) min \(\{}_{s}\)\(N\)\({}_{y,X,1}\), \({}_{s}\)\(N\)\({}_{y,Y,1}\)\(=\) 17.72 kN, in order to fulfill the criteria explained above. Furthermore, the yield interstory drift for the dampers, \({}_{s}\)\(\delta\)\({}_{y,1}\), was limited to 0.40\(\delta\)\({}_{y,1}\) on the basis of the results obtained by [PERSON] et al. [9], where \(\delta\)\({}_{y,1}\) is the yield interstory drift of the WFP system (without dampers) at the first floor. The results of the test [7] and the pushover analyses conducted with the numerical model indicate that \(\delta\)\({}_{y,1}\)\(\approx\) 0.01\(h\)\({}_{p,1}\)\(=\) 14 mm for both X and Y directions, where \(h\)\({}_{p,1}\) is the height of the first story; hence, the value adopted for \({}_{s}\)\(\delta\)\({}_{y,1}\) is \(\delta\)\({}_{y,1}\)\(=\) 0.40 \(\times\) 0.01\(h\)\({}_{p,1}\)\(=\) 5.6 mm.