# Proposal of a Design Procedure for Steel Frames with Viscoelastic Dampers
## 6 Nonlinear Dynamic Analyses
### Seismic Response of the Case Study  
The achievement of yielding at the ends of the beams and/or at the base of the C1- and C2-columns (i.e., members designed considering \(q\) = 1.5) is expected for values of the PGA equal to or larger than 0.2 5 g/1.5 = 0.167 g. The members of the braced bay are expected not to develop internal forces associated with the plasticization of the \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Floor**} & \multicolumn{4}{c}{**Spring Stiffness [kN/mm]**} & \multicolumn{4}{c}{**Damper Viscous Factors [kN/mm]**} \\ \cline{2-10}  & \(k_{\text{E,0}}\) & \(k_{\text{E,1}}\) & \(k_{\text{E,2}}\) & \(k_{\text{E,3}}\) & \(k_{\text{E,4}}\) & \(\epsilon_{\text{C,0}}\) & \(c_{\text{C,1}}\) & \(c_{\text{C,2}}\) & \(\epsilon_{\text{C,3}}\) & \(c_{\text{C,4}}\) \\ \hline
1 & 0.376 & 3.512 & 1.128 & 0.034 & 0.905 & 0.335 & 0.434 & 0.787 & 0.739 & 4.954 \\
2 & 0.310 & 2.898 & 0.931 & 0.028 & 0.747 & 0.277 & 0.358 & 0.649 & 0.610 & 4.088 \\
3 & 0.222 & 2.076 & 0.667 & 0.020 & 0.535 & 0.198 & 0.256 & 0.465 & 0.437 & 2.928 \\
4 & 0.205 & 1.915 & 0.615 & 0.018 & 0.493 & 0.183 & 0.237 & 0.429 & 0.403 & 2.702 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Values of the stiffness and of the coefficient of viscosity of the springs and the dashpots of the GMM4 model. cross-section or the buckling of the member until the value of the PGA associated with the NC limit state is reached. The elements of the braced bays, instead, are expected to remain elastic until the value of the PGA associated with the NC limit state has been reached, i.e., 0.25 g \(\times\) 1.71 = 0.429 g. VEDs are expected to reach the maximum allowed deformation at the NC limit state (i.e., 200%) at the attainment of a story drift equal to 51.4 mm. The numerical analysis shows that the damper located at the first story reaches the NC limit state first at a PGA equal to 0.443 g, which is slightly higher than the expected value (i.e., 0.429 g). Braces remain elastic and do not buckle even when the maximum expected value of the PGA (0.429 g) is reached. C0-columns remain elastic until the NC limit state of the structure has been reached. The force-displacement responses of the VEDs recorded at a PGA equal to 0.443 g for one of the accelerograms are shown in Figure 10. The effects of the higher modes of vibrations are particularly apparent in the hysteretic cycles of the dampers located at the first and second story whose shape is irregular and not perfectly elliptical. However, the hysteretic cycles of the VEDs are well enveloped by the response of the GMM4 model calibrated based on the maximum forces and maximum displacements recorded in the dampers and considering the frequency associated with the first period of the structure (red dashed line in the figure). The values of the maximum forces recorded in the dampers and the ratios of the maximum allowed displacement to the maximum displacement experienced are shown in Table 12. The latter data show that the dissipative system is efficient as almost all the dampers approach their maximum stroke during the time history. Figure 10: Hysteretic cycles of the VEDs located at the first (**a**), second (**b**), third (**c**), and fourth (**d**) story recorded at the PGA of 0.437 g and enveloped by the GMM4 model for \(\omega\)1 = 4.077 rad/s (i.e., T1 = 1.541 s). As shown in Figure 11a, the average values of the axial force of C0-columns recorded with OpenSees are well predicted by the response spectrum analysis with \(\xi=9.62\%\). The average values of the shear force and bending moment are slightly overestimated, as shown in Figure 11b,c. The average values of the axial force in braces are satisfactorily predicted by the SRSS analysis, even though the axial force is slightly underestimated at the first floor, as shown in Figure 11d.