## 3 Numerical Strategies
### Nonlinear Static Analyses  
Concerning the nonlinear static analyses, after the application of the vertical loads, two vertical distributions of the lateral loads are applied, respectively: (i) a \"uniform\" pattern, based on lateral forces that are proportional to mass and (ii) a \"modal\" pattern, proportional to lateral forces consistent with first mode pattern obtained by the performed modal analysis. The structural performances of the existing and of the retrofitted building are discussed and compared in terms of capacity curves, i.e., the relations between base shear force and control node (top) displacement. In detail, peak shear force, ductility, and mode of failure are thoroughly evaluated and some considerations concerning the improvement obtained by retrofitting intervention are provided.  
The numerical models are realized using a lumped plasticity approach with finite length of plastic hinges. Beams and columns are modelled as force-based nonlinear elements adopting the iterative flexibility formulation with the integration method proposed in [16], i.e., a modified two-point Gauss-Radau integration over each hinge region, so guaranteeing that localized deformations are integrated over the specified plastic hinge length. The length of the plastic hinge is evaluated according to the relation A.9 of EN1998-3 [2]:  
\[L_{pl}=\frac{L_{v}}{30}+0.2h+0.11\frac{d_{bL}f_{y}}{\sqrt{f_{c}}} \tag{1}\]  
where \(L_{v}\) is the shear span assumed equal to half the length of the element, \(h\) is the cross-section depth, \(d_{bL}\) is the mean diameter of the tension reinforcement, and \(f_{c}\) and \(f_{y}\) are the concrete compressive strength (MPa) and the steel yield strength (MPa), respectively. The P-Delta effect is considered for the columns in all the performed nonlinear static analyses.  
The constitutive model adopted for concrete is the one proposed by [PERSON] et al. [17] (based on the equation proposed by [PERSON] [18]) for both existing RC building and RC-framed skin, while the model proposed by Menegotto-Pinto model, as modified by [PERSON] et al. [19], is adopted for steel. In the concrete model, a zero tensile strength is assumed and the compressive stress is set equal to zero when the crushing strain is achieved.  
The concrete model adopted in the column cross section's core accounts for the effects of passive confinement. To this aim, the formulation proposed in [PERSON] et al. [17] is considered in this work both for existing columns (with rectangular stirrups) and for the columns of the reinforcing RC-framed skin (with circular spirals).  
Concerning the implementation phase, for the columns of the reinforcing RC-framed skin, an ad-hoc GUI (Graphical User Interface) for STKO [15] has been specifically developed providing a utility for users to input easily the geometric and material characteristics and computing automatically the effect of confinement on core material. In particular, the plugin recognizes the class of constitutive model adopted for cover concrete in OpenSees (e.g., Concrete01, Concrete02, Concrete04, etc.) and computes the effect of confinement according to several formulations, including the one developed by [PERSON] et al. [17], adopted in this work as recommended in EN1998-3 [2] and Italian National Standard [20]. Finally, the plugin creates on the fly (when the model runs) a new material for core concrete (confined) of the same class of the unconfined concrete. Of course, the user always has the possibility to specify a customized confined material law when he does not want to use the plugin automatism. The window of the GUI for the reinforcing column of the RC-framed skin is shown in Figure 2, where both the constitutive laws adopted in this work for confined and unconfined concrete are depicted in terms of stress-strain curve.