## 2 Design Procedure
### Structures with Poor Shear or Bending Moment Strength of Constituting Members  
The basic design objective of supplemental damping-based retrofit interventions is always a significant enhancement of the seismic performance as compared to the original structural conditions. When high-damping capacity dissipaters are adopted, like pressurized FV devices, the attainment of the Immediate Occupancy (IO) seismic performance level can be planned up to the maximum considered earthquake (MCE), with reasonable costs and architectural impact. This objective corresponds to keeping the response of the structural members within the elastic field, thus replacing the plastic demand in current conditions with the dissipative action of the protective system. The design procedure is based on this target and articulated in the four steps described below.  
1. Development of an elastic finite element verification analysis and relevant stress state checks in current conditions.  
The shear and bending moment-related stress states in columns and beams are calculated by means of an elastic finite element (either modal superposition with response spectrum or time-history) analysis of the structure in current conditions. Then, the maximum stress states are checked by comparison with the corresponding elastic limit values for the two axes in plan of the reference Cartesian coordinate system, \(X\) and \(Y\) (\(Z\) being the vertical axis). The shear and bending moment values in the member that proves to be subjected to the highest unsafe stress conditions are named \(V_{j,c}^{a}\) and \(M_{j,c}^{a}\), where index \(j\) refers to the axis (\(j=X\),\(Y\)), \(c\) denotes the \"critical\" member(s), and \(a\) means that the values are derived from the elastic assessment analysis.  
Evaluation of the target stress reduction factors \(\alpha_{s}\).  
This evaluation is carried out by referring to the scheme in Figure 1a, where \(d,s\) represent the general deformation-related and stress-related parameters for the critical member(s). The objective of the retrofit design consists in reducing the seismic demand in current state calculated by the elastic analysis in step 1--denoted by point A with coordinates (\(d_{A}=d_{max}\), \(s_{A}=s_{max}\)), where \(d_{max}\), \(s_{max}\) are the maximum \(d\) and \(s\) computed values--below point B with coordinates (\(d_{B}=d_{e}\), \(s_{B}=s_{e}\)), where \(d_{e}\), \(s_{e}\) are the elastic limit deformation-related and stress-related parameters for the critical member(s), respectively. On the contrary, in a traditional design approach, the reduction of the force-related parameter is pursued by exploiting the ductile plastic response of the structural members, symbolically represented by the dashed line K-B, by moving the maximum response point from A to K, with coordinates (\(d_{A}\), \(s_{B}\)).  
Hence, the force reduction factor \(\alpha_{s}\), targeted to reach an elastic response of the critical member (and thus of all remaining members) when passing from original to retrofitted conditions, is given by:  
\[\alpha_{s}=\frac{s_{A}}{s_{B}} \tag{1}\]  
The \(s_{A}\) and \(s_{B}\) parameters in Figure 1 are detailed below according to the specific lack of strength affecting the structural members in original conditions.