## 4 Energy Dissipation Capacity  
According to an existing procedure [15; 18] developed for the dimensioning of dissipative additional systems to be installed inside buildings (dissipative bracings), the energy dissipated by the \(S\) and ADS can be evaluated at each deformation value (Figure 4), and, according to [PERSON] [29], it can be evaluated by calculating \(\xi_{eq,S}\), that is, the equivalent viscous damping of the structure (as a function of the displacement, \(D\)); it can be expressed as follows:  
\[\xi_{eq,S}=\frac{1}{4\pi}\frac{I_{D,S}}{I_{S,S}} \tag{1}\]  
Equation (1) can be solved by determining all the necessary quantities from the capacity curve, where  
\(D\) is the displacement reached by the control joint;  
\(F_{s}(D)\) is the force (base shear) corresponding to \(D\);  
\(D_{sy}\) is the displacement at yielding;  
\(F_{sy}\) is the yielding force (base shear at yielding);  
\(I_{D,S}\) is the energy that is dissipated (cycle of \(D\) amplitudes);  
\(I_{S,S}\) is the elastic strain energy at \(D\).  
In a simplified approach, an equivalent bilinear capacity curve (BCC) can be easily used. The BCC can be determined (according to one of the methods available in the literature or technical codes) from the \"real\" capacity curve (output of the pushover analysis).  
In this way, the terms of Equation (1), considering an ideal elastoplastic hysteretic cycle, are determined as follows:  
\[I_{D,S}^{id}=\big{(}F_{sy}D-D_{sy}F_{s}(D)\big{)}/0.25 \tag{2}\]  
\[I_{S,S}=DF_{s}(D)/2 \tag{3}\]The hysteretic cycle of a real structure, which differs from the mathematically evaluated ideal cycle, can be evaluated according to specific corrective coefficients \(c_{S}\) (for the structure) and \(c_{A}\) for the ADS (\(c=1\) for ideal elastoplastic behavior).  
Therefore,  
\[I_{D,S}=c_{S}I_{D,S}^{id} \tag{4}\]  
\[I_{D,A}=c_{A}I_{D,A}^{id} \tag{5}\]  
with \(I_{D,A}^{id}\) being the energy dissipated by the ideal hysteretic cycle of the ADS (elastoplastic behavior defined by the elastic stiffness, yielding limit, and hardening ratio).  
Then, \(c_{S}\) can be determined through specific analysis or by simply referring to the provision's technical codes or the scientific literature (e.g., [30]); according to the author's experience, the assumption of \(c_{A}\approx 1\) can be considered as reasonable, and the force-displacement relationship of the \(ADS\) can be idealized as a bilinear curve.  
The equivalent viscous damping (\(\xi_{\alpha,S+T}\)) of the \(S+ADS\), to be added to the inherent damping (\(\xi_{I}\)--usually, \(\xi_{I}=5\%\) for r.c. structures, and \(\xi_{I}=2\%\) for steel structures) can be evaluated using the following expressions:  
\[\xi_{\alpha,S+A}=0.25\left[\frac{c_{S}I_{D,S}^{id}}{I_{S,S+A}}+\frac{c_{A} \sum_{I}I_{D,A,j}^{id}}{I_{S,S+A}}\right]1/\pi \tag{6}\]  
\[\xi_{\alpha,S}=c_{S}0.25\frac{I_{D,S}^{id}}{I_{S,S+A}}\pi_{I};\;\xi_{\alpha,A }=c_{A}0.25\frac{\sum_{I}I_{D,A,j}^{id}}{I_{S,S+A}}\pi \tag{7}\]  
where \(I_{D,A,j}^{id}\) is the energy dissipated by \(j\) ADSs connected to the structure (e.g., in practical applications, one or more dissipative towers can be designed).  
Note that \(\xi_{\alpha,S}\) and \(\xi_{\alpha,A}\) are obtained by dividing the dissipated energy, determined from the capacity curves of the \(S\) and \(ADS\), respectively, by the elastic strain energy of the retrofitted building, which is determined from the curve of the \(S+ADS\).