# Development of Innovative Lateral Resistance Systems
## 2 Guideline Design Procedures for Butterfly-Shaped Links
### Drift Ratio Control Requirement, Based on the Stiffness  
Figure 2 depicts the geometric properties of a plate consisting of butterfly-shaped shear links. Taking into account various factors contributing to the total stiffness of a plate with butterfly-shaped links [1,30], the total stiffness of the innovative damper is estimated by Equation (9). \[K_{T}=\frac{K_{b}K_{v}K_{c}K_{d}}{K_{b}K_{v}K_{c}+K_{b}K_{v}K_{d}+K_{b}K_{c}K_{d}+ K_{v}K_{c}K_{d}} \tag{9}\]  
Therefore, the stiffness partial terms of \(K_{b}\), \(K_{v}\), \(K_{c}\), and \(K_{d}\) are derived from Equations (10)-(13):  
\[K_{b}=Eb^{3}t\left(\frac{2\left(\frac{a}{b}-1\right)^{3}}{3\left[2L_{n}\left( \frac{a}{b}\right)+\left(\frac{a}{b}-1\right)\left(\frac{a}{b}-3\right)\right] L^{3}}\right) \tag{10}\]  
\[K_{v}=\frac{\frac{5 mGt}{6L}(b-a)}{\text{Ln}\left(\frac{b}{a}\right)} \tag{11}\]  
\[K_{c}=n\frac{P}{\delta_{c}}=\frac{Et{L_{b}}^{3}}{6C{(L)}^{2}} \tag{12}\]  
\[K_{d}=\frac{5}{6}\frac{GtL_{b}}{L} \tag{13}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Shear Fuse** & & \(b/L\) & \(\Omega\) \\ \hline \multirow{6}{*}{_a/b_} & & 0.1 & 4.1 \\  & & 0.2 & 3.3 \\  & & 0.3 & 2.8 \\  & & 0.4 & 1.8 \\ \hline \multirow{6}{*}{_a/b_} & & 0.1 & 2.3 \\  & & 0.2 & 1.65 \\  & & 0.3 & 1.35 \\  & & 0.4 & 1.3 \\ \hline \multirow{6}{*}{_a/b_} & & 0.1 & 4.13 \\  & & 0.2 & 3.18 \\  & & 0.3 & 2.51 \\  & & 0.4 & 1.95 \\ \hline \multirow{6}{*}{_a/b_} & & 0.1 & 4.35 \\  & & 0.2 & 3.35 \\ \cline{1-1}  & & 0.3 & 2.75 \\ \cline{1-1}  & & 0.4 & 2.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The over strength factor for various geometrical properties of steel dampers. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Normalized PEEQ at 5\%** & & \multicolumn{3}{c}{**Mid Width (a)/End Width (b)**} \\ \hline \(b/L\) & 0.1 & 0.33 & 0.75 & 1 \\
0.1 & 5.82 & 2.35 & 2.59 & 5.18 \\
0.2 & 8.41 & 6.18 & 5.47 & 15.94 \\
0.3 & 16.35 & 1.03 & 1.00 & 16.76 \\
0.4 & 20.71 & 1.29 & 1.24 & 21.76 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The over-strength factors for various geometrical properties of steel dampers. Equations (14) and (15) determine the calculations of the total flexure stiffness (\(K_{\text{flexurefactor}}\)) and total shear stiffness (\(K_{\text{shearfactor}}\)), respectively. These two factors could be interpolated based on data provided in Tables 3 and 4 as well, which summarize the factors for estimating the flexural and shear stiffness values. \[K_{b}=\text{n}\times\text{E}\times\text{t}\times\text{K}_{\text{ flexurefactor}} \tag{14}\]  
\[K_{v}=\text{n}\times\text{G}\times\text{t}\times\text{K}_{\text{shearfactor}} \tag{15}\]  
Slit shear link stiffness could be similarly established by Equation (16):  
\[K_{T}=\frac{K_{b}K_{v}K_{c}K_{d}}{K_{b}K_{c}K_{c}+K_{b}K_{v}K_{d}+K_{b}K_{c}K_{ d}+K_{v}K_{c}K_{d}} \tag{16}\]  
Therefore, the stiffness partial terms of \(K_{b}\), \(K_{v}\), \(K_{c}\), and \(K_{d}\) are derived from Equations (17)-(20):  
\[K_{b}=\frac{nEtb^{3}}{L^{3}} \tag{17}\]  
\[K_{v}=K_{v}=\frac{5n}{6}\frac{Gtb}{L} \tag{18}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & & \multicolumn{3}{c}{_a/b_} \\ \hline  & & 0.1 & 0.33 & 0.66 & 1 \\  & & 0.2 & 0.065 & 0.100 & 0.136 & 0.2 \\
**K shear factor** & \(b\)/\(L\) & 0.4 & 0.130 & 0.201 & 0.272 & 0.4 \\  & & 0.6 & 0.195 & 0.302 & 0.409 & 0.6 \\  & & 0.8 & 0.260 & 0.402 & 0.545 & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: \(\text{K}_{\text{shearfactor}}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & & \multicolumn{3}{c}{_a/b_} \\ \hline  & & 0.1 & 0.33 & 0.66 & 1 \\  & & 0.2 & 0.065 & 0.100 & 0.136 & 0.2 \\
**K shear factor** & \(b\)/\(L\) & 0.4 & 0.130 & 0.201 & 0.272 & 0.4 \\  & & 0.6 & 0.195 & 0.302 & 0.409 & 0.6 \\  & & 0.8 & 0.260 & 0.402 & 0.545 & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(\text{K}_{\text{flexurefactor}}\). Figure 2: The structural shear damper modes of behavior: (**a**) butterfly-shaped geometry; (**b**) moment deformation; (**c**) shear deformation.