# Seismic Design of Steel Moment-Resisting Frames with Damping Systems in Accordance with KBC 2016
## 4 Design Example
### Three-Story Steel Moment-Resisting Frames with Metallic Yielding Dampers (3F-OMRF-MD)
#### 4.2.2 Design of Displacement-Dependent Damping Devices  
In order to achieve target reduction factors for base shear and maximum story drift, those two response values are recorded from nonlinear response history analyses repeated with incremental damping of 0.05. Thus, the nonlinear response history analysis was performed only five times. Response reduction factors obtained from the nonlinear response history analysis are plotted in Figure 10. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Member** & **Story** & **Section** & **Material** \\ \hline \multirow{3}{*}{**3F-OMRF-MD**} & Beam & 1-3 & H-506 \(\times\) 201 \(\times\) 11/19 & \\ \cline{2-5}  & Column & 1-3 & H-394 \(\times\) 405 \(\times\) 18/18 & \\ \cline{2-5}  & Brace & 1-3 & H-200 \(\times\) 200 \(\times\) 8/12 & SM490 \\ \hline \multirow{3}{*}{**6F-SMRF-VED**} & \multirow{3}{*}{Beam} & 5-6 & H-354 \(\times\) 176 \(\times\) 8/13 & \(f_{y}=315\) MPa \\ \cline{3-5}  & & 3-4 & H-450 \(\times\) 200 \(\times\) 9/14 & \(f_{u}=490\) MPa \\ \cline{3-5}  & & 1-2 & H-496 \(\times\) 199 \(\times\) 9/14 & \\ \cline{2-5}  & \multirow{2}{*}{Column} & 4-6 & H-414 \(\times\) 405 \(\times\) 18/28 & \\ \cline{3-5}  & & 1-3 & H-428 \(\times\) 407 \(\times\) 20/35 & \\ \cline{2-5}  & \multirow{2}{*}{Brace} & 1-6 & H-244 \(\times\) 252 \(\times\) 11/11 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Properties of moment-resisting frames. Figure 10: Base shear and peak story drift reduction factors. Target damping ratio corresponding to the base shear reduction factor 0.79 is interpolated to be 19.1%. The damping devices were designed to achieve an added damping of 0.141 except 0.05 inherent damping of the moment-resisting frame. To design each damping device, the total dissipated energy \(\sum E_{dj}\) was calculated from Equation (12) in combination with Equation (13). Then \(\sum E_{dj}\) was distributed to each story in proportion to the story shear force. It is taken into account that yield strength or friction force of displacement-dependent damping devices are distributed based on the distribution of story shear force to maximize energy dissipation [34; 35]. First estimation of target \(\beta_{v}\) and corresponding \(\beta_{t}\) were 0.141 and 0.191. Characteristics of damping devices determined to achieve the target \(\beta_{v}\) on the basis of Equation (14) are given in Table 4. The post-yield stiffness ratio of the damping device was assumed to be 0.02 in the calculation of dissipated energy. The fundamental period of the structure without damping devices was 1.04 sec. The fundamental period of the moment-resisting frame with damping devices substituted by secant stiffness thereof at respective peak deformations of them was reduced to 0.92 sec. Considering the change of the fundamental period, the story drift reduction factor was decreased, as shown in Figure 10. Then, \(\beta_{v}\) and corresponding \(\beta_{t}\) were modified into 0.12 and 0.17, respectively, using updated damping device deformations corresponding to the adjusted story drift reduction factor. As a result, the corresponding maximum story drift response is reduced to 64% compared to the structure with 0.05 damping ratio. Using those updated damping device deformations, the fundamental period based on the secant stiffness of damping devices was calculated to be 0.91 second. Then, the base shear reduction factor was elevated corresponding to 0.91 second period as shown in Figure 10. Finally, damping device yield strengths are modified in order to compensate reduced deformations due to period change and to achieve an added damping ratio \(\beta_{v}\) of 0.12. The final damping device properties are listed in Table 4 for each story. Based on the modified base shear reduction factor represented by the thicker solid line in Figure 10, the expected damping correction factor is 0.93/1.14 = 0.82, which is slightly higher than the target \(\eta\) = 0.79.