# Proposal of a Design Procedure for Steel Frames with Viscoelastic Dampers
## 3 Numerical Model of Viscoelastic Dampers
### Equivalent Brace Model  
Since the proposed design procedure is based on linear elastic analyses (lateral force method of analysis and response spectrum analysis), only the contribution of the VED-brace subassembly to the lateral stiffness of the frame needs to be represented in the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(\gamma_{0}=100\%\)} & \multicolumn{4}{c}{\(f\) [Hz]} \\ \cline{2-5}  & **0.1** & **0.3** & **1.0** & **2.0** \\ \hline \(G^{\prime}\) [MPa] & 0.123 & 0.191 & 0.327 & 0.446 \\ \(G^{\prime\prime}\) [MPa] & 0.101 & 0.183 & 0.366 & 0.517 \\ \(\eta\) [-] & 0.82 & 0.96 & 1.12 & 1.16 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Experimental values of the storage and loss moduli obtained for the elastomer ISD-111 HH at 20 \({}^{\circ}\)C [52].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**T = 20 \({}^{\circ}\)C** & **Spring 0** & **Spring 1** & **Spring 2** & **Spring 3** & **Spring 4** \\ \hline \(G_{\text{E}}\) [MPa] & 0.033337 & 0.311289 & 0.099964 & 0.003002 & 0.080206 \\ \hline  & Dashpot 0 & Dashpot 1 & Dashpot 2 & Dashpot 3 & Dashpot 4 \\ \hline \(G_{\text{C}}\) [MPa/\(\omega\)] & 0.029715 & 0.038467 & 0.069748 & 0.065506 & 0.439183 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the GMM4 obtained by optimizing the equations of G\({}^{\prime}\) and G\({}^{\prime\prime}\) on experimental values.  
Figure 5: Schematic representation of the Generalized Maxwell Model with four Maxwell elements (GMM4).  
structural model. The VED is modeled as a simple spring whose stiffness corresponds to the in-phase stiffness of the VED at the expected ambient temperature and at the frequency corresponding to the dominant mode of vibration of the structure. The VED-brace subassembly is modeled as a single rod whose lateral stiffness is equal to that of the subassembly. Specifically, the lateral stiffness \(k_{\text{eq}}\) of the equivalent brace is obtained by the in-series sum of the lateral stiffness \(k^{\prime}v\) of the VED and the lateral stiffness \(k_{\text{b}}\) of the brace.  
\[k_{\text{eq}}=\frac{1}{\frac{1}{k^{\prime}_{\text{v}}}+\frac{1}{k_{\text{b}}}} \tag{29}\]  
Referring to the schematic representation of the VED-brace subassembly shown in Figure 6, the lateral displacement \(u_{\text{tot}}\) of the node A due to a horizontal force \(F\) is given by the sum of the lateral deformation of the damper \(u_{\text{v}}\) and the lateral deformation of the brace \(u_{\text{b}}\), i.e.,  
\[u_{\text{tot}}=u_{\text{v}}+u_{\text{b}} \tag{30}\]  
Both the VED and the bracing system are subjected to the force \(F\). Therefore, the stiffnesses \(k^{\prime}v\) and \(k_{\text{b}}\) can be written as  
\[k^{\prime}_{\text{v}}=\frac{F}{u_{\text{v}}} \tag{31}\]  
\[k_{\text{b}}=\frac{2 EA_{\text{b}}\cos^{2}\alpha}{l_{\text{b}}} \tag{32}\]  
where \(\alpha\) is the angle of inclination of the brace with respect to the horizontal line, \(l_{\text{b}}\) is the length of the brace, \(A_{\text{b}}\) is the area of the cross-section of the brace, and E is [PERSON]'s modulus of steel. Once the equivalent stiffness \(k_{\text{eq}}\) is determined by Equation (29), the VED-brace subassembly is represented in the numerical model by means of a fictitious brace with cross-sectional area \(A_{\text{eq}}\) equal to  
\[A_{\text{eq}}=\frac{k_{\text{eq}}\,l_{\text{b}}}{2E\,\cos^{2}\alpha} \tag{33}\]