## 2 Retrofitting Method Using Viscous Dampers
### Design Method
#### 2.2.2 Step 2: Optimum Placements of the Dampers at the Same Retrofitting Cost  
The initial design of dampers will be further optimized in this section to increase the structural collapse-resistant capacity at the same retrofitting cost. It is assumed that the cost of retrofit is proportional to the total damping coefficient [14; 34]. Therefore, in the optimization, the total damping coefficient obtained in Step 1 is kept unchanged. The following procedure can be followed:  
1. By referring to initial scheme 1, perform IDA analysis on the structure to obtain its collapse-resistant capacity _CC_. Note that the selected GM for initial scheme 1 corresponds to GM 1. 2. Scale GM to _CC_ and perform a further dynamic analysis on the initial retrofitted structure. Obtain the inter-story drift ratio (\(\delta_{i}\)) of each story. 3. If \(\delta_{i,\,\text{max}}\) is close enough to the pre-defined collapse inter-story drift ratio \(\delta_{collapse}\), no iteration is necessary; otherwise, modify the distribution of the story damping coefficient \(C_{i}\). The story damping coefficient \(C_{i}\) is reduced in stories with \(\delta_{i}\) lower than \(\delta_{collapse}\) and increased in stories with \(\delta_{i}\) higher than \(\delta_{collapse}\) (using Equation (2) for the adjustment and Equation (3) to keep the total damping coefficient unchanged). The adjustment is repeated until all the inter-story drift ratios satisfy Equation (4). \[C_{i}^{j+1}=(\frac{\delta_{i}^{j}}{\delta_{average}})^{\alpha}\cdot C_{i}^{j}\] (2) \[\sum_{i=1}^{6}C_{i}=C_{total}\] (3) \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
** Retrofitting Scheme** & \(C_{1}\) & \(C_{2}\) & \(C_{3}\) & \(C_{4}\) & \(C_{5}\) & \(C_{6}\) & \(\sum\limits_{i=1}^{6}C_{i}\) \\ \hline Initial scheme 1 & 300,000 & 20,000 & 250,000 & 250,000 & 20,000 & 20,000 & 860,000 \\ Initial scheme 2 & 150,000 & 150,000 & 50,000 & 50,000 & 20,000 & 20,000 & 440,000 \\ Initial scheme 3 & 1,300,000 & 1,300,000 & 1,000,000 & 500,000 & 50,000 & 20,000 & 4,170,000 \\ Initial scheme 4 & 1,000,000 & 1,000,000 & 800,000 & 250,000 & 20,000 & 20,000 & 3,090,000 \\ Initial scheme 5 & 1,200,000 & 1,300,000 & 750,000 & 200,000 & 50,000 & 50,000 & 3,550,000 \\ Initial scheme 6 & 800,000 & 750,000 & 600,000 & 100,000 & 20,000 & 20,000 & 2,290,000 \\ Initial scheme 7 & 20,000 & 20,000 & 500,000 & 50,000 & 20,000 & 20,000 & 630,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Initial placements of dampers to satisfy code-specific requirements (unit: N-(s/m)). \[\delta_{i}^{j+1}<\delta_{collapse} \tag{4}\]  
In the above equations, \(C_{i}^{j+1}\) and \(C_{i}^{j}\) are the damping coefficients of dampers placed at story \(I\) after \(j\)+1 th and \(j\)th adjustments; \(\delta_{i}^{j+1}\) and \(\delta_{i}^{j}\) are the inter-story drift ratios at story \(I\) after \(j\)+1 th and \(j\)th adjustments under the scaled GM; \(\delta_{average}\) is the average value of inter-story drift ratios of all stories; and \(\alpha\) is a coefficient between 0 and 1 that controls the convergence gradient. In the study, \(\alpha\) is set to 0.3; and \(C_{total}\) is the total damping coefficient of the initial retrofitted structure. Please note that 6, appearing in Equation (3), denotes the number of stories of the example frame. \(\delta_{collapse}\) is the inter-story drift ratio to determine the collapse state of the optimum structure. According to FEMA 356 [44] and to experimental studies [45; 46; 47], stating that the inter-story drift ratio corresponding to \"Collapse Prevention\" state is in the range 2-4%, \(\delta_{collapse}\) is set to 4% in this study. Note that 10% is used previously in the IDA analyses to check a real collapse state, whereas 4% is used here to assess a \"Collapse Prevention\" state which is actually a nominal collapse suitable to practical design. * Perform IDA analysis on the structure to obtain its new collapse-resistant capacity \(CC\)'. Scale the PGA of GM to \(CC\)' and perform (3) again.