# Proposal of a Design Procedure for Steel Frames with Viscoelastic Dampers
## 5 Application to a Case Study
### Design of the MRF  
The initial MRF is sized to sustain gravity loads only in the non-seismic situation. Beam and column cross-sections obtained from gravity loads analysis are shown in Table 4. Then, modal analysis is performed on the initial MRF and the period \(T_{1}\), the mode shape  
Figure 8: Floor plan of the case studies.  
Figure 9: Schematic representation of the plane model.  
\(\Phi_{1}\) and the modal participation factor \(\Gamma^{(1)}\) associated with the first mode of vibration are obtained. The minimum values of the required equivalent viscous damping ratio are determined for NC, LS, and DL limit states based on the prefixed drift limits established in Section 4.1 (see Table 3). These values are shown in Table 5.  
The damping ratio that should be provided by the dampers is equal to  
\[\xi_{\text{add}}=\xi_{\text{d}}-\xi_{0}=9.62\%-3\%=6.62\% \tag{50}\]  
The ratio \(k_{\text{b}}/k_{\text{v}}\) is assumed to be equal to 40, and the loss factor \(\eta_{\text{v}}\) of the elastomer ISD-111H is taken equal to 1. The reduced value of the seismic base shear \(V_{0}\) is evaluated by Equation (39) considering the spectral acceleration \(\text{S}_{\text{e}}(T_{1},\,k_{\text{d}})a_{\text{gL}S}\). Internal forces due to seismic actions are determined by the lateral force method of analysis. The value of the reduced seismic base shear \(V_{0}\) and the other quantities involved in the calculations are shown in Table 6 for the first and the last design iteration. It should be noted that \(V_{0}\) is about 80% \(V_{\text{base}}\).  
Capacity design rules are applied to size the elements of the frame. Beam and column cross-sections obtained at the last iteration are shown in Table 7.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(T_{1}\) [s] & \(\xi_{\text{add}}\) [\%] & \(k_{\text{b}}/k_{\text{v}}\) [-] & \(\eta_{\text{v}}\) [-] & \(\eta_{\text{v-b}}\) [-] & \(\beta_{\text{MSE}}\) & \(V_{\text{base}}\) [kN] & \(V_{0}\) [kN] \\ \hline
2.096 & 6.62 & 40 & 1.00 & 0.952 & 0.799 & 733.14 & 585.42 \\
1.668 & 6.62 & 40 & 1.00 & 0.952 & 0.928 & 965.5 & 770.9 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Values of the design base shear at the first and at the last iteration on the procedure and relevant quantities for the calculations.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Columns**} & \multicolumn{2}{c}{**Beams**} \\ \cline{2-5}
**Floor** & **C0** & **C1** & **C2** & **Braced Bay** & **MRF** \\ \hline
1 & HEB200 & HEB140 & HEB260 & IPE 300 * & IPE 360 * \\
2 & HEB200 & HEB140 & HEB260 & IPE 300 * & IPE 360 * \\
3 & HEB160 & HEB120 & HEB180 & IPE 300 * & IPE 360 * \\
4 & HEB160 & HEB120 & HEB180 & IPE 300 * & IPE 360 * \\ \hline \hline \end{tabular} Note: all profiles are made of S355 steel grade except the ones marked with * S275.  
\end{table}
Table 4: Cross-sections of beams and columns obtained through gravity load analysis.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Limit State** & \(\Phi^{(1)}\)**4 [-]** & \(\Gamma^{(1)}\) [-] & \(T_{1}\) [s] & \(u_{\text{maxMDOF}}\) [m] & \(\eta_{\text{SDOF}}\) [mm] & \(\xi_{\text{d}}\) [\%] \\ \hline DL & & & & 0.06 & 46.271 & 8.77 \\ LS & 0.0592 & 22.004 & 2.096 & 0.12 & 92.542 & 9.62 \\ NC & & & & 0.30 & 231.355 & 3.48 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Minimum values of the required equivalent viscous damping ratio for the considered limit states.