# Seismic Design of Steel Moment-Resisting Frames with Damping Systems in Accordance with KBC 2016
## 3 Damping System Design Procedure
### Damping Device Design
#### 3.3.3 Design of Damping Devices Considering Change of the Natural Frequency  
Once damping device properties are determined, it is necessary to update the effective damping ratio \(\beta_{v}\) because the effective damping ratio \(\beta_{v}\) given by Equations (12) or (17) is dependent on the stiffness and/or strength of the damping device. A rational method to update the effective damping ratio considering response reduction and stiffening effects of damping devices is described in this section and Figure 2 illustrates the procedure for updating the effective damping ratio conceptually.  
(Step 1) Base shear and maximum story drift reduction factors are plotted with respect to \(\beta_{t}\), broken lines in Figure 2.  
(Step 2) The target damping ratio \(\beta_{t1}\) corresponding to the initial base shear reduction factor \((V/V_{o})_{1}\) is interpolated from the plot 'Initial \(V/V_{o}\)' (thick broken line) in Figure 2.  
(Step 3) The maximum story drift reduction factor \((D/D_{o})_{1}\) is interpolated from the plot 'Initial \(D/D_{o}\)' (thin broken line) in Figure 2.  
(Step 4) Damping devices are designed to achieve the target damping ratio Equation (12) or (17) subjected to deformations corresponding to \((D/D_{o})_{1}\). Those deformations can be approximated by multiplying \((D/D_{o})_{1}\) to the initial response for only \(\beta_{tH}\).  
(Step 5) Effective stiffness of damping devices is calculated and added to the structure without damping devices. Then, the fundamental frequencies of the structure are updated.  
(Step 6) Response reduction factors are updated corresponding to change of the fundamental frequencies. The peak story drift reduction factor is reduced on the basis of displacement design  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Temperature** & **Frequency** & **Strain** & **Shear Storage** & **Shear loss** & **Loss** \\
**(\({}^{\circ}\)C)** & **(Hz)** & **(\%)** & **modulus, G** & **modulus, G** & **Factor** \\
**(MPa)** & **(MPa)** & **(MPa)** & **(MPa)** & **(MPa)** & **(\(\eta_{d}\))** \\ \hline
24 & 1.0 & 20 & 0.958 & 1.151 & 1.20 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Properties of the viscoelastic damping device [20].  
Figure 4: Kelvin model of viscoelastic damping devices.  
spectrum, as shown in Figure 5a due to fundamental period shortening. The modified peak story drift reduction factor is plotted as 'Updated \(D/D_{o}\)' (thin solid line) in Figure 2.  
(Step 7) The effective damping ratio is updated to \(\beta_{t2}\) considering the decrease of deformations in (Step 6), and corresponding maximum story drift reduction factor is interpolated from the plot 'Updated \(D/D_{o}\)' in Figure 2.  
(Step 8) The base shear reduction factor is determined on the basis of pseudo-acceleration design spectrum and increased due to shortening of the fundamental period, as shown in Figure 5b. The modified base shear reduction factor is represented as 'Updated \(V/V_{o}\)' in Figure 2 (thick solid line).  
(Step 9) Damping correction factor \(\eta\) for \(\beta_{t2}\) is calculated as a ratio between \(V/V_{o}\)'s at \(\beta_{t2}\) and \(\beta_{tH}\) on 'Updated \(V/V_{o}\)' in Figure 2.  
(Step 10) Adjust (\(V/V_{o}\))\({}_{1}\) and repeat (Step 2) to (Step 9) until \(\eta\) becomes sufficiently close to the target.  
(Step 11) If \(\eta\) converges to the target, nonlinear response history analysis is performed in order to confirm whether the actual response reduction factor satisfies design requirements or not.  
The design procedure proposed above utilizes nonlinear response history analysis only at Step 1. Additional response prediction is performed using the elastic design spectrum. Thus, the proposed design procedure is computationally efficient compared to the trial-and-error method based on fully nonlinear response history analysis.