# Seismic Design of Steel Moment-Resisting Frames with Damping Systems in Accordance with KBC 2016
## 4 Design Example
### Six-Story Steel Moment Frames with Viscoelastic Dampers (6F-SMRF-VED)
#### 4.3.2 Design of Velocity-Dependent Damping Devices  
The natural frequency of the first mode was 0.5 Hz from the eigenvalue analysis of the structure with only the stiffness component of the viscoelastic damping devices. Viscoelastic damper characteristics corresponding to an excitation frequency of 1.0 Hz, which is the closest one to 0.5 Hz, was adopted among those dependent on excitation frequencies. The stiffness and damping coefficients of each damping device were calculated using Equation (15) and (16). The average maximum story drift ratios from nonlinear response history analysis for seven ground motion records representing design earthquake are listed in Table 6. The maximum story drift ratio is found to be 1.48% for the third story. Target story drift reduction factor is 1.0%/1.48% = 0.68. The maximum story drift reduction factor was obtained from the nonlinear response history analysis of the moment-resisting frame with an incremental damping ratio of 0.05 that substitute damping devices. Thus, the nonlinear response history analysis was performed only five times and the maximum story drift reduction factor was plotted in Figure 13 and the target damping ratio \(\beta_{t}\) interpolated from the plot is 0.20. To achieve the target damping ratio, \(\beta_{p}\) = 0.15 excluding \(\beta_{IH}\) = 0.05 is necessary to be added by damping devices. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Seismic Risk** \\ **Category** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Importance** \\ **Factor** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Story** \\ \end{tabular} } & \multicolumn{2}{c}{\begin{tabular}{c} **Story-Drift Ratio without Damping** \\ **Devices (\%)** \\ \end{tabular} } & \multicolumn{2}{c}{\begin{tabular}{c} **Allowable** \\ **Story-Drift** \\ **Ratio (\%)** \\ \end{tabular} } \\ \cline{3-5}  & & & \begin{tabular}{c} **Equivalent** \\ **Static Analysis** \\ \end{tabular} & \begin{tabular}{c} **Nonlinear Response** \\ **History Analysis** \\ \end{tabular} & \\ \hline \multirow{4}{*}{\
\begin{tabular}{c} **S** \\ \end{tabular} } & \multirow{4}{*}{
\begin{tabular}{c} **1.5** \\ \end{tabular} } & 6 & 1.32 & 1.24 & \multirow{4}{*}{1.00} \\ \cline{3-5}  & & 5 & 1.59 & 1.28 & & \\ \cline{3-5}  & & 4 & 1.75 & 1.31 & & \\ \cline{3-5}  & & 3 & 1.75 & 1.48 & & \\ \cline{3-5}  & & 2 & 1.54 & 1.39 & & \\ \cline{3-5}  & & 1 & 0.93 & 0.76 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Peak story drifts of six-story steel moment frames with viscoelastic dampers (6F-SMRF-VED). Figure 13: Normalized response vs. incremental damping ratios. The damping ratio added by viscoelastic damping devices is calculated by Equation (17), in which the added damping ratio is dependent on the fundamental frequency of the structure with effective stiffness of damping devices. In order to design damping devices, the stiffness \(K_{d}\) of damping devices represented by Kelvin model are increased until the fundamental frequency becomes the target value corresponding to the target damping ratio. This work is conducted by eigenvalue analysis of the linear elastic model and does not require additional nonlinear response history analysis. The same stiffness was applied to all the damping devices in this design example, but more efficient distribution may be investigated [18]. When \(K_{d}\) is determined, a corresponding \(C_{d}\) can be calculated using Equation (16). However, change of the fundamental frequency due to damping devices affects the maximum story drift reduction factor. As a result, the maximum story drift reduction factor in Figure 13 is updated repeatedly. For each update of the maximum story drift reduction factor, the target damping ratio changes correspondingly. Four times of update were performed and updated parameters including target damping ratios and target frequencies are summarized in Table 7. The final fundamental frequency converged to 1.89 sec and \(K_{a}\) and \(C_{d}\) reached 6800 kN/m and 2718 kN-sec/m, respectively.