## 4 Numerical Analyses of a WFP Structure Upgraded with Slit-Plate Dampers (SPDs)
### Design of the Hysteretic Dampers
Then, the axial yield relative displacement for the dampers, \({}_{s}\)\(\Delta\)\({}_{y,1}\), is obtained from \({}_{s}\)\(\delta\)\({}_{y,1}\) in X direction, for which \({}_{s}\)\(N\)\({}_{y,1}\) was selected as \({}_{s}\)\(\Delta\)\({}_{y,1}\)\(=\)\({}_{s}\)\(\delta\)\({}_{y,1}\)\(\cos\gamma\)\(/\)\(\cos\alpha\), giving that \({}_{s}\)\(\Delta\)\({}_{y,1}\)\(\in\) 5.35 mm. Finally, the stiffness of the dampers is \({}_{s}\)\(k\)\({}_{y,1}\)\(=\)\({}_{s}\)\(N\)\({}_{y,1}\)\(/_{s}\)\(\Delta\)\({}_{y,1}\)\(=\) 17.72/5.35 \(=\) 3.31 kN/mm. The yield-strength for the upper stories, \(Q\)\({}_{y,i}\) for \(i\)\(>\) 1, was determined for the purpose of preventing damage concentration. Here, damage is characterized through the parameter \(\eta_{i}\)\(=\)\(W\)\({}_{p,i}\)/\(\left(Q\)\({}_{y,i}\)\(\delta\)\({}_{y,i}\)\(\right)\), where \(W\)\({}_{p,i}\) is the energy dissipated by plastic deformations at the \(i\) story [15]. \(Q\)\({}_{y,i}\), can be expressed as dimensionless by the yield-force coefficient, \(\alpha\)\({}_{i}\), defined as \(\alpha\)\({}_{i}\)\(=\)\(Q\)\({}_{y,i}\)/\(\sum\limits_{k\)\(=\)\(i\)\(}^{N}\)\(m\)\({}_{k}\)g, where \(m\)\({}_{k}\) is the mass of the \(k\) story, \(N\) is the number of stories and \(g\) is the gravity acceleration. The distribution \(\overline{\alpha}\)\({}_{i}\)\(=\)\(\alpha\)\({}_{i}\)/\(\alpha\)\({}_{1}\) that makes \(\eta\)\({}_{i}\) approximately equal in all stories is called optimum yield-shear strength coefficient distribution [15]. Different expressions for \(\overline{\alpha}\)\({}_{i}\) have been proposed in the literature [15; 16; 17; 18]. In this study, the approach proposed by Japanese Building Code [18], \(\overline{\alpha}\)\({}_{j\)BC,\(i}\) is used, which is expressed as follows:  
\[\overline{\alpha}\)\({}_{j\)BC,\(i}\)\(=\)\(1+\left(\frac{1}{\sqrt{\overline{m}_{i}}}-\overline{m}_{i}\right)\)\(\frac{2T_{1}}{1+3T_{1}}\) (1)  
where \(\overline{m}_{i}\)\(=\)\(\sum\limits_{k\)\(=\)\(s\)\(}^{N}\)\(m\)\({}_{k}\)/\(M\), \(M\) is the total mass and \(T_{1}\) is the fundamental period of the structure without dampers. The eigenvalue analysis carried out in the numerical model of the WFP system showed that the fundamental periods along X and Y directions are \(T\)\({}_{X,1}\)\(=\) 0.32 s and \(T\)\({}_{Y,1}\)\(=\) 0.38 s, respectively. Therefore, by applying \(T\)\({}_{1,X}\)\(=\) 0.32 s and \(\overline{m}_{2}\)\(=\) 0.527 in Equation (1) for the X direction, the value obtained for \(\overline{\alpha}\)\({}_{j\)BC,\(2}\) is equal to 1.278. Further, as the total shear strength at the first story along X direction, \(Q\)\({}_{y,X,1}\), is  
Figure 8: Layout of dampers at the first story: (**a**) elevation of the installation scheme of damper 3; (**b**) plan view; (**c**) elevation of the installation scheme of dampers 1 and 2.  
obtained as \(Q_{yX,1}~{}=~{}_{f}Q_{yX,1}+_{s}Q_{yX,1}~{}=~{}57.5+28.75~{}=~{}86.25\) kN, this implies that \(\alpha_{1}=86.25\cdot 10^{3}\) N/[(6450 kg + 5780 kg)9.8 m/s\({}^{2}\)] = 0.7196. Then, the optimum shear strength coefficient at the second story in X direction is \(\overline{\alpha}_{jBC,2}\alpha_{1}~{}=~{}0.92\) and the required shear strength is \(\overline{\alpha}_{jBC,2}\alpha_{1}~{}\sum\limits_{k~{}=~{}2}^{2}m_{k}\bar{g}~ {}=~{}58.15\) kN. Following a similar procedure for Y direction, \(Q_{yY,1}~{}=~{}_{f}Q_{yY,1}+_{s}Q_{yY,1}~{}=~{}62.5+26.5~{}=~{}89\) kN, \(\alpha_{1}~{}=~{}0.743\), \(\overline{\alpha}_{jBC,2}~{}=~{}1.302\) (for \(T_{Y,1}\) = 0.38 s), \(\overline{\alpha}_{jBC,2}\alpha_{1}~{}=~{}0.97\) and finally \(\overline{\alpha}_{jBC,2}\alpha_{1}~{}\sum\limits_{k~{}=~{}2}^{2}m_{k}\bar{g}~ {}=~{}61.3\) kN. In addition, the yield-shear strength at the second story of the WFP specimen along X and Y direction was obtained through a pushover analysis, obtaining that \({}_{f}Q_{yX,2}\approx{}_{f}Q_{yY,2}~{}=~{}138\) kN.