# Proposal of a Design Procedure for Steel Frames with Viscoelastic Dampers
## 4 Proposed Design Procedure
### Design of the Braces  
In accordance with Equation (38), the design value of the lateral stiffness of the VED-brace subassembly at the \(i\)-th floor, \(k_{\text{v-b,i}}\) is given as follows:  
\[k_{\text{v-b,i}}=\alpha_{\text{MSE}}k_{0,\text{i}} \tag{41}\]  
where  
\[\alpha_{\text{MSE}}=\frac{2\xi_{\text{add}}}{n_{\text{v-b}}-2\xi_{\text{add}}} \tag{42}\]  
where \(k_{0,\text{i}}\) is the lateral stiffness of the MRF at the \(i\)-th floor. The stiffness of the VED at the \(i\)-th floor is given by the subsequent expression [36]  
\[k^{\prime}_{\text{v,i}}=\frac{\left[\left(1+\eta_{\text{V}}^{2}\right)+\frac{ k_{\text{h}}}{k_{\text{v}}}\right]\left[1+\eta_{\text{v-b}}^{2}\right]}{\frac{ k_{\text{h}}}{k_{\text{v}}}\left(1+\eta_{\text{v}}^{2}\right)}k_{\text{v-b,i}} \tag{43}\]  
Recall that the response of the VEDs is sensitive to the changes in the load frequency. Thus, the force that arises in VEDs and that is transferred to the braces is affected by the higher modes of vibration. Further, due to the phase shift of the VE material, the force associated with the maximum displacement through the stiffness \(k^{\prime}_{\text{v,i}}\) is not the maximum force that occurs in the device (see Figure 2b). To obtain an accurate prediction of the maximum axial force in braces due to seismic actions, the following steps are required in the design process. First, an initial value of the lateral stiffness \(k_{\text{b,i}}\) of the braces at the \(i\)-th floor is estimated from the prefixed ratio \(k_{\text{b}}/k_{\text{v}}\) considering the value of \(k^{\prime}_{\text{v,i}}\) calculated by Equation (43). The calculation of the lateral stiffness and the cross-sectional area of the equivalent brace by Equations (29) and (33) is straightforward. The structural model is then updated, including the equivalent braces modeled as hinged rods with area \(A_{\text{eq}}\). Secondly, the period of vibration \(T_{\text{j}}\) and the seismic lateral forces \(F_{\text{i}}\)[(0)] associated with the \(j\)-th mode are determined by modal analysis of the MRF with equivalent braces. Then, the in-phase stiffness \(G^{\prime}\) of the VE material is calculated for each mode of vibration by Equation (23) considering the respective circular frequency. If \(k^{\text{v,i}}_{r}\) are the stiffnesses of the VEDs in the first mode of vibration, the stiffnesses of the VEDs \(k^{\prime}_{\text{v,i}}\) corresponding to the \(j\)-th mode (with \(j>1\)) are calculated through the following expression  
\[k^{\prime}_{\text{v,i}}=k^{\prime}_{\text{v,i}}\,\frac{G^{\prime}(\text{j})}{ G^{\prime}(\text{1})} \tag{44}\]  
The stiffnesses \(k^{(\text{j})}_{\text{eq,i}}\) and the areas \(A^{(\text{j})}_{\text{eq,i}}\) of the equivalent braces are evaluated by Equations (29) and (33) for each mode of vibration. For the \(j\)-th mode of vibration, the axial forces in braces are obtained by linear static analysis performed on the structure with equivalent braces with area \(A^{(\text{j})}_{\text{eq,i}}\) using the equivalent seismic forces \(F_{\text{i}}\)[(0)]. Third, the axial forces obtained in braces for the \(j\)-th mode are multiplied by a corrective factor \(\gamma_{\rm k}^{\rm(j)}\) that accounts for the shift between the occurrence of the maximum deformation and maximum force in VEDs  
\[\gamma_{\rm k}^{\rm(j)}=\frac{\left|G^{*(j)}\right|}{G^{\prime(j)}} \tag{45}\]  
The maximum axial forces in the braces \(N_{\rm max,i}^{\rm SIS}\) due to seismic action are finally obtained, combining the effects of the modes of vibration using the SRSS rule. As braces are expected to remain elastic at the NC limit state, the importance factor \(\gamma_{\rm NC}\) is applied, i.e.,  
\[N_{\rm max,i}^{\rm SIS}=\gamma_{\rm NC}\sqrt{\sum_{j=1}^{n}\left(N_{\rm i}^{ \rm(j)}\,\gamma_{\rm k}^{\rm(j)}\right)^{2}} \tag{46}\]  
The axial forces in braces due to gravity loads (\(N_{\rm max,i}^{\rm GL}\)) in the seismic combination are computed through a separated analysis performed on a different structural model.