# Proposal of a Design Procedure for Steel Frames with Viscoelastic Dampers
## 4 Proposed Design Procedure
### Design of the MRF

Table 3: Drift limits provided by FEMA 365 for MRFs for the considered limit states, and drift limits assumed for viscoelastically damped MRFs. where \(P_{\rm L}\) is the probability of exceedance of the seismic action (2% and 63% in the case of NC and DL, respectively) in TL years (i.e., 50 years), \(P_{\rm LR}\) is the reference probability of exceedance (10% in the case of LS) over the same TL years, and \(k=3\) is a constant. The design base shear is calculated by means of the following expression, as reported in Eurocode 8  
\[V_{\rm base}=S_{\rm e}(T_{1},\,\xi_{0})_{a_{\rm g,LS}}m\,\lambda \tag{36}\]  
where \(S_{\rm e}(T_{1},\,\xi_{0})a_{\rm g,LS}\) is the spectral acceleration obtained at the period of vibration \(T_{1}\) for the equivalent viscous damping ratio \(\xi_{0}\) of the bare frame, m is the seismic mass of the building, and \(\lambda\) is a correction factor (=0.85) accounting for the difference between the base shear force predicted by the lateral force method of analysis and the response spectrum method of analysis. The base shear obtained by Equation (36) is reduced based on the MSE method, as shown in the following steps. According to the MSE method, the equivalent viscous damping ratio of the damped structure can be expressed as  
\[\xi_{\rm add}=\frac{\eta_{\rm v-b}}{2}\frac{K_{\rm v-d}}{K_{\rm s}}=\frac{ \eta_{\rm v-b}}{2}\left(\frac{K_{\rm v-d}}{K_{0}+K_{\rm v-d}}\right) \tag{37}\]  
where \(\eta_{\rm v-b}\) is the loss factor of the damper-brace subassembly, \(K_{0}\), \(K_{\rm s}\), and \(K_{\rm v-d}\) are the lateral stiffness of the MRF, the lateral stiffness of the damped structure, and the lateral stiffness of the damper-brace subassembly, respectively. Assuming that the base shear is proportional to the lateral stiffness of the structure, Equation (37) can be rewritten as [36]  
\[V_{\rm base}=V_{\rm v-d}+V_{0}=2\frac{\xi_{\rm add}}{\eta_{\rm v-b}-2\xi_{\rm add }}V_{0}+V_{0} \tag{38}\]  
where \(V_{\rm v-d}\) is the fraction of \(V_{\rm base}\) to be sustained by the dissipative system and \(V_{0}\) is the fraction to be resisted by the MRF. The value of the seismic base shear to be used to design the MRF is then derived from Equation (38) as  
\[V_{0}=B\frac{\eta_{\rm v-b}-2\xi_{\rm add}}{\eta_{\rm v-b}}V_{\rm base} \tag{39}\]  
where \(B\) is a corrective coefficient (equal to \(\sqrt{10/(5+\xi_{\rm add})}\geq 0.55\)) of the equivalent viscous damping ratio given in Eurocode 8 Part 1-1 (SS3.2.2.2). The loss factor of the VED-brace subassembly, \(\eta_{\rm v-b}\), is given by the subsequent expression [6; 36]  
\[\eta_{\rm v-b}=\frac{\frac{k_{\rm b}}{K_{\rm v}}}{\eta_{\rm v}^{2}+\frac{k_{ \rm b}}{K_{\rm v}}+1}\eta_{\rm v} \tag{40}\]  
where \(k_{\rm b}\) is the lateral stiffness of the brace and \(k_{\rm v}\) is the lateral stiffness of the VED. The loss factor of the elastomer (i.e., the ratio of the loss modulus to the storage modulus) is a stable quantity with respect to both frequency and temperature and for the elastomer ISD-111H, it can be assumed equal to unity. The ratio \(k_{\rm b}/k_{\rm v}^{{}^{\prime}}\) is a design parameter that has to be established a priori and denotes how stiff the brace is with respect to the lateral stiffness of the VED. Clearly, the stiffer the brace, the higher the deformation of the VED. [PERSON] et al. [36] found that the efficiency of the dissipative system is sensitive to the ratio \(k_{\rm b}/k_{\rm v}^{{}^{\prime}}\) up to a value of 40. Further increases in the lateral stiffness of the brace do not improve the performance of the VEDs significantly. The base shear \(V_{0}\) is used to perform a linear static analysis on the initial MRF and obtain the internal forces due to seismic action. In this phase, the behavior factor \(q\) and the importance factor \(\gamma_{\rm NC}\) are applied. The resulting internal forces are combined with the internal forces due to gravity loads in the seismic combination and the resistance and lateral stability checks are conducted on members of the MRF and on the dissipative systems.