## 2 Design Procedure
### Structures with Excessive InterStorey Drifts  
The attainment of an elastic structural response up to the MCE is assumed as the basic design objective also when the poorest response capacity is in terms of storey drifts. This is more likely verified in structures with a translational period \(T_{1}\) next to the 0.8 s value approximately fixed as the upper limit for the application of the procedure. The design objective is reached by reducing the maximum interstorey drift computed in current conditions, identified by \(ID_{A^{\prime}}=ID_{max}\) in Figure 1b, to the corresponding elastic limit \(ID_{B^{\prime}}=ID_{e}\), i.e., by scaling the drift response by a deformation-related reduction factor \(\alpha_{d}\):  
\[\alpha_{d}=\frac{ID_{A^{\prime}}}{ID_{B^{\prime}}}=\frac{ID_{max}}{ID_{e}} \tag{19}\]  
When an elastic finite element assessment analysis is carried out on the original structure, \(\alpha_{d}\) is proportional to \(\alpha_{F}\). For cases where an inelastic evaluation analysis is developed, assuming a typical degrading strength and degrading stiffness post-elastic behaviour--like the one qualitatively schematized by curve B\({}^{\prime}\)-K\({}^{\prime}\) in Figure 1b--\(\alpha_{d}\) significantly differs from \(\alpha_{F}\). Therefore, the expressions of \(E_{D}\) and \(\xi_{\alpha q}\) must be reformulated as a function of \(\alpha_{d}\) to allow quickly estimating both quantities also for the structures where the interstorey drift is the critical response parameter. Starting from relation (13), the two changes introduced as compared to Section 2.1 consist in assuming \(d_{dmax}=ID_{A^{\prime}}-ID_{B^{\prime}}=ID_{max}-ID_{e}\), and \(F_{cd}=F_{K^{\prime}}=F_{D}\). The first assumption corresponds to assigning to the dampers the capacity of absorbing the post-elastic displacement demand of the structure computed in original conditions, so as to meet the design objective of limiting its response to the elastic field after retrofit. The second assumption derives from the fact that the displacement performance enhancement must be achieved essentially by means of the dissipative capacity of the FV devices, by neglecting at the sizing stage the slight stiffening effects related to their elastic spring function. Then, according to (19), the following \(\xi_{\alpha q}\) expression is obtained:  
\[\xi_{\alpha q,\alpha d}=\frac{4F_{D}(ID_{A^{\prime}}-ID_{B^{\prime}})}{4\pi( F_{cd}ID_{e}/2)}=\frac{4F_{D}(ID_{max}-ID_{e})}{4\pi(F_{cd}ID_{e}/2)}=\frac{2}{ \pi}(\alpha_{d}-1) \tag{20}\]  
By referring to (13), the corresponding energy dissipation capacity to be assigned to the dampers on the considered storey results as follows:  
\[E_{D,\alpha d}=2\pi F_{e}\xi_{\alpha q,\alpha d}ID_{e} \tag{21}\]Similarly to the case of poor strength of the members, the tentative design choice of the device is carried out by referring to the spring-damper with the nearest energy dissipation capacity to \(E_{D}\), estimated by (20), and a stroke not less than (\(ID_{max}-ID_{e}\)). **3. Geometrical and Structural Characteristics of the Case-Study Building**  
The case study examined for a first demonstrative application of the design criterion is the gym in a school in Florence, built in 1971, two external views and an internal view of which are displayed in Figure 3. Figures 4 and 5 show the structural plan and the transversal cross section, respectively. The reference Cartesian coordinate system assumed in the analyses is indicated in Figures 4 and 5, too. As highlighted in Figure 4, the plan is rectangular, with sides of 14.4 m in the transversal direction, parallel to \(X\), and 24.25 m in the longitudinal direction, parallel to \(Y\).