## 2 Design Procedure
### Passage from member to storey (\(\alpha_{s}{\rightarrow}\alpha_{F}\)). The A\({\rightarrow}\)B transition in the most critical member, governed by factor \(\alpha_{Vj}\) or \(\alpha_{Mj}\), implies a similar shift in the response of the frame structure storey to which the member belongs. This is visualized in the graph of Figure 1b, where the storey shear, \(F\), and the interstorey drift, \(ID\), are assumed as response parameters, and the member response points A and B are replaced by the corresponding storey response points A\({}^{\prime}\), with coordinates (\(F_{A^{\prime}}=F_{max}\), \(ID_{A^{\prime}}=ID_{max}\)), and B\({}^{\prime}\), with coordinates (\(F_{B^{\prime}}=F_{e}\), \(ID_{B^{\prime}}=ID_{c}\)), where the indexes \"max\" and \"e\" denote the maximum response value and the corresponding elastic limit in this case too. Based on this correlation, the storey response points A\({}^{\prime}\), B\({}^{\prime}\) in Figure 1b are reached when the critical member attains points A, B in Figure 1a. Therefore, the reduction factor at the storey level, \(\alpha_{F}\):  
\[\alpha_{F}=\frac{F_{A^{\prime}}}{F_{B^{\prime}}} \tag{4}\]  
coincides with \(\alpha_{Vj}\) or \(\alpha_{Mj}\), depending on the lack of strength assessed in the most critical member. 3. Correlation of \(\alpha_{F}\) to the dissipated energy \(E_{D}\) and the equivalent viscous damping ratio \(\xi_{eq}\) of the spring-dampers. The correlation is established consistently with the design objective of reducing the storey response from point A\({}^{\prime}\) to point B\({}^{\prime}\), thanks to the incorporation of the protective system. The sizing of the devices is performed first by evaluating their equivalent viscous damping ratio \(\xi_{eq}\), using the general expression:  
\[\xi_{eq}=\frac{E_{D}}{4\pi E_{e}} \tag{5}\]  
where \(E_{D}\) is the energy dissipated by the set of FV spring-dampers installed on the storey containing the critical member, and \(E_{e}\) is the strain energy of the system estimated by referring to the global response cycle of the set of devices, schematically drawn in Figure 2. Therein, \(d_{dmax}\) is the maximum displacement, \(F_{D}\) is the damping force component, \(F_{e}\) is the elastic reaction force, and \(F_{ed}=F_{e}+F_{D}\) is the total reaction force. Figure 2: Schematic response cycle of the set of fluid viscous (FV) spring–dampers installed on a storey, and parameters for evaluating \(\xi_{eq}\). In order to meet the design objective of passing from point A\({}^{\prime}\) to point B\({}^{\prime}\), \(F_{D}\) is set as equal to the difference between \(F_{A^{\prime}}\) and \(F_{B^{\prime}}\):  
\[F_{D}=F_{A^{\prime}}-F_{B^{\prime}} \tag{6}\]  
In doing so, the reduction of storey response is totally assigned to the dissipative action of the protective system, as planned by this retrofit strategy. When substituting (4) in (6), the following \(F_{D}\) expression is obtained:  
\[F_{D}=F_{A^{\prime}}-F_{B^{\prime}}=\alpha_{F}F_{B^{\prime}}-F_{B^{\prime}}=F_ {B^{\prime}}(\alpha_{F}-1) \tag{7}\]  
from which it follows:  
\[F_{B^{\prime}}=\frac{F_{D}}{(\alpha_{F}-1)} \tag{8}\]  
The \(F_{e}\) elastic force component is set as equal to the elastic limit value of the storey shear, \(F_{B^{\prime}}\).