# Proposal of a Design Procedure for Steel Frames with Viscoelastic Dampers
## 7 Conclusions  
In this paper, a design procedure for steel moment-resisting frames with viscoelastic dampers has been proposed. The parameters that control the response of viscoelastic materials, namely the storage modulus, the loss modulus, and the loss factor, have been discussed, and their dependency on frequency and ambient temperature has been pointed out. A simple and frequency-dependent numerical model, the Generalized Maxwell Model with four Maxwell elements (GMM4), has been proposed to represent the dynamic response of viscoelastic materials. Mathematical expressions for the storage and the loss moduli of the GMM4 have been derived analytically and calibrated based on experimental data of a commercial elastomer. The proposed design procedure is based on the Modal Strain Energy (MSE) method, and its application requires simple analytical tools, such as the modal response spectrum analysis and the lateral force method of analysis. This approach has been improved in order to accurately predict the internal forces that are transmitted from each damper to the braces due to the sensitivity of the VE material to the effects of the higher modes of vibration. The design procedure has been applied to 4-, 6-, and 8-story buildings equipped with viscoelastic dampers sustained by braces in the chevron configuration. The buildings are founded on different soils (Soil A and C) and are designed based on seismic actions of different intensities (design PGA equal to 0.25 g and 0.35 g). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & \(\mathbf{PGA_{SD}}\) & \(\mathbf{PGA_{NC}}\) & \(\mathbf{PGA_{beams}}\) & \(\mathbf{PGA_{colMRF}}\) & \(\mathbf{PGA_{VED}}\) \\ \hline \multirow{4}{*}{4-story} & Soil A & 0.250 & 0.429 & 0.241 & - & 0.443 \\  & 0.350 & 0.600 & 0.309 & - & 0.603 \\ \cline{2-6}  & Soil C & 0.250 & 0.429 & 0.273 & - & 0.437 \\  & 0.350 & 0.600 & 0.340 & - & 0.541 \\ \hline \multirow{4}{*}{6-story} & Soil A & 0.250 & 0.429 & 0.330 & 0.312 & 0.544 \\  & 0.350 & 0.600 & 0.341 & 0.386 & 0.572 \\ \cline{2-6}  & Soil C & 0.250 & 0.429 & 0.222 & 0.273 & 0.376 \\  & 0.350 & 0.600 & 0.292 & - & 0.513 \\ \hline \multirow{4}{*}{8-story} & Soil A & 0.250 & 0.429 & 0.304 & 0.361 & 0.505 \\  & 0.350 & 0.600 & 0.353 & 0.432 & 0.560 \\ \cline{2-6}  & Soil C & 0.250 & 0.429 & 0.223 & 0.291 & 0.357 \\ \cline{2-6}  & 0.350 & 0.600 & 0.281 & 0.372 & 0.489 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Values of the PGA for which the SD limit state and the NC limit state are attained in members of the MRF and the VEDs. To validate the proposed design procedure, the response of the designed buildings was investigated by nonlinear dynamic analyses. Based on the results of the NLDA, the following conclusions can be drawn:  
* The fulfillment of the main design objective, i.e., the control of the story drifts below prefixed limits, is achieved for all cases. At the same time, the internal forces of the braces and of the other members of the MRF are well predicted. This proves the effectiveness of the procedure based on the MSE method. * The GMM4 model is effective in reproducing the hysteretic behavior of the VEDs as a function of frequency and for a given ambient temperature. * VEDs are effective in conferring the demanded damping to the structure, although in the cases in which high damping is demanded, at least one of the dampers of the dissipative system exceeds the maximum allowed displacement for a value of the PGA that is lower than the expected value. Future developments of this research will concern:  
* Optimal arrangement of the dampers in the structure;
* Comparison of the seismic performance and construction costs of the frames designed by means of the proposed procedure with those of the frames designed by means of other procedures available in the literature;
* Study of the response of viscoelastically damped structures in the case of near-fault design scenarios. Conceptualization, M.B., A.F. and P.P.R. ; methodology, M.B., A.F. and P.P.R. ; software, M.B. and A.F. ; validation, M.B., A.F.