# Proposal of a Design Procedure for Steel Frames with Viscoelastic Dampers
## 4 Proposed Design Procedure
### Design of the MRF  
In this study, the initial MRF is designed to sustain gravity loads only, according to prescriptions of Eurocode 3. Then, the modal analysis is performed, and the first period \(T_{1}\), the first mode shape \(\Phi_{1}\), and the modal participation factor \(\Gamma^{(1)}\) are obtained. Limits on inter-story drifts \(\delta_{\text{max}}\) are established for Near Collapse (NC), Significant Damage (SD), and Damage Limitation (DL) limit states using the limits reported in FEMA 365 [57] as reference values for the corresponding limit states (i.e., Collapse Prevention, Life Safety, and Immediate Occupancy, respectively). These latter limits, reported in the first row of Table 3, are stipulated for the rehabilitation of existing MRFs. Since one of the objectives of the examined structural typology is to limit damage, it is reasonable to consider drift limits in the design of new buildings stricter than those reported in FEMA 365. Owing to this, the limit drift values adopted in this study are reported in the second row of Table 3.  
For the considered limit states, the maximum accepted values of the top displacement \(u_{\text{NC}}^{\text{top}}\), \(u_{\text{LS}}^{\text{top}}\), \(u_{\text{DL}}^{\text{top}}\) are calculated assuming that the drift limit is attained at all floors simultaneously. Then, the top displacements are converted into the displacements of equivalent SDOF systems \(u_{\text{NC}}^{\text{SDOF}}\), \(u_{\text{LS}}^{\text{SDOF}}\) and \(u_{\text{DL}}^{\text{SDOF}}\) using the following expression  
\[u^{\text{SDOF}}=\frac{u^{\text{top}}}{\Gamma^{(1)}\,\phi_{\text{top}}^{(1)}} \tag{34}\]  
where \(\phi_{\text{top}}^{(1)}\) is the component of the first mode shape at the top floor. For the considered limit states, the minimum demanded values of the equivalent viscous damping ratio, \(\xi_{\text{d,NC}}\), \(\xi_{\text{d,LS}}\), and \(\xi_{\text{d,DL}}\), are evaluated by means of the elastic response spectrum scaled at the peak ground acceleration (PGA) corresponding to earthquakes with probability of exceedance equal to 2% (NC), 10%(SD), and 63% (DL) in 50 years, respectively. The values of \(\xi_{\text{d,NC}}\), \(\xi_{\text{d,LS}}\), and \(\xi_{\text{d,DL}}\) are found by minimization of the difference between the spectral ordinates \(S_{\text{d}}\)(\(T_{1}\), \(\xi_{\text{d,NC}}\))\({}_{\text{g,NC}}\), \(S_{\text{d}}\)(\(T_{1}\), \(\xi_{\text{d,LG}}\))\({}_{\text{g,LG}}\), and \(S_{\text{d}}\)(\(T_{1}\), \(\xi_{\text{d,DL}}\))\({}_{\text{g,DL}}\), and the maximum displacements \(u_{\text{NC}}^{\text{SDOF}}\), \(u_{\text{LS}}^{\text{SDOF}}\), and \(u_{\text{DL}}^{\text{SDOF}}\) of the equivalent SDOF system. The value of the equivalent viscous damping ratio to be considered in the next steps of the procedure is the maximum among the values determined for the considered limit states, i.e., \(\xi_{\text{d}}=\max\{\xi_{\text{d,NC}},\,\xi_{\text{d,LS}},\,\xi_{\text{d,DL}}\}\). The equivalent viscous damping \(\xi_{\text{add}}\) that shall be provided by the VEDs is equal to \(\xi_{\text{d}}\) minus the inherent damping \(\xi_{0}\) of the bare frame, assumed equal to 3% for steel MRFs. For the sake of simplicity, the seismic base shear and the internal forces are obtained by means of the elastic response spectrum scaled at the PGA associated with the LS limit state. The behavior factor \(q\) and the importance factor \(\gamma_{\text{NC}}\) will be applied to the relevant members when internal forces are combined. The importance factor is calculated for the relevant limit state using the following expression, as suggested in Eurocode 8 Part 1 (SS2.1 (4))  
\[\gamma=\left(\frac{P_{\text{L}}}{P_{\text{LR}}}\right)^{-\frac{1}{L}} \tag{35}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\delta_{\text{max,NC}}\) & \(\delta_{\text{max,LS}}\) & \(\delta_{\text{max,DL}}\) \\ \hline FEMA 365 (Table C1-3) & 5\% & 2.5\% & 0.7\% \\ Adopted limits & 2.5\% & 1\% & 0.5\% \\ \hline \hline \end{tabular}
\end{table}