# Proposal of a Design Procedure for Steel Frames with Viscoelastic Dampers
## 6 Nonlinear Dynamic Analyses  
The seismic response of the structures designed through the proposed procedure is evaluated by nonlinear dynamic analyses considering P-\(\Delta\) effects. Numerical analyses are performed by means of the OpenSees v. 3.6.0 computer program [58]. The seismic input consists of two sets of 10 artificial accelerograms generated for a PGA equal to 0.35 g with the computer program SIMQKE [59] to match the elastic response spectrum with 5% equivalent viscous damping provided by Eurocode 8 for soil type A or C.  
The two-dimensional structural model of the MRF has the same geometrical characteristics and boundary conditions as the model used for design analyses. Under the same assumptions formulated in Section 5.1, only two frames of the structure are modeled (i.e., a perimetral frame and an inner frame). The presence of rigid slabs is simulated by means of constraints applied at each floor between a master node belonging to the perimetral frame and the other nodes belonging to the same floor to force the constrained nodes to have the same horizontal displacements. All members are represented by a single force-based element, except for the beams of the braced bay that are modeled with two elements to have a node at midspan for the connection with the VEDs. Members of the MRF that are expected to develop plastic hinges (i.e., C1-, C2-columns and beams) are modeled as forceBeamColumn elements with fiber sections and elastic \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Floor** & \(\mathbf{T_{1}}\)**[s]** & \(\mathbf{G^{\prime}}\)**[N/mm\({}^{2}\)]** & \(\mathbf{t}\)**[mm]** & \(\mathbf{n_{\text{l}}}\) & \(\mathbf{A_{\text{l},\text{l}}}\)**[cm\({}^{2}\)]** & \(\mathbf{b_{\text{l}}}\times\mathbf{h_{\text{l}}}\)**[cm]** \\ \hline
1 & & & & & 838.80 & 42 \(\times\) 20 \\
2 & & & & & 692.19 & 35 \(\times\) 20 \\
3 & 1.548 & 0.268 & 25.7 & 4 & 495.79 & 25 \(\times\) 20 \\
4 & & & & & 457.38 & 23 \(\times\) 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Areas and dimensions of the elastomeric layers and relevant quantities for the calculations. interior. The length of the plastic hinges is assumed to be equal to the height of the member cross-section. The integration method for the plastic hinge is \"HingeRadau\". Members that are expected to remain elastic at the attainment of the NC limit state (i.e., braces and C0-columns) are modeled as elasticBeamColumn elements. Each VED is modeled using six zeroLength elements connected in series between a node belonging to the braces and the midspan node of the beam of the braced bay. A uniaxialMaterialElastic, uniaxialMaterialViscous, and uniaxialMaterialViscousDamper are assigned to the zeroLength elements representing the spring, the dashpot, and the Maxwell elements of the GMM4 model (see Figure 5), respectively. The parameters that characterize the response of the single springs and dashpots of the GMM4 are calculated using the following relations  
\[k_{\text{E,in}}=\frac{A_{\text{i,i}}\ n_{\text{i}}\ G_{\text{E,n}}}{t_{\text{i}}} \tag{53}\]  
\[c_{\text{C,m}}=\frac{A_{\text{i,i}}\ n_{\text{i}}\ G_{\text{C,m}}}{t_{\text{i}}} \tag{54}\]  
where \(k_{\text{E,n}}\) and \(c_{\text{c,m}}\) are the stiffness and viscous coefficients of the n-nth spring and m-nth dashpot of the model (\(n\), \(m\) = 0, 1,..., 4) at the \(i\)-th story; \(G_{\text{E,n}}\) and \(G_{\text{C,m}}\) are the constants of the model as determined by the optimization on the experimental data of the elastomer ISD-111H (see Table 2). Ai, \(n_{\text{i}}\), and \(t_{\text{i}}\) are the geometric characteristics of the VED. The calculated values of \(k_{\text{E,in}}\) and \(c_{\text{c,m}}\) are shown in Table 11.  
The Rayleigh formulation is used to introduce the inherent damping of the frame (mass and stiffness coefficients are defined so that the first and second modes of vibration are characterized by an equivalent viscous damping ratio equal to 0.03). No stiffness proportional damping is considered for the braces.