## 3 Numerical Strategies
### Structural Capacity Assessment  
After the nonlinear static analyses, both ductile and brittle mechanisms (i.e., shear) capacities are evaluated for all elements according to EN1998-3 [2]. In detail, for ductile mechanism, capacity checks are performed at the element level, evaluating the chord rotation demand and the corresponding capacity at the ends of each structural element (both beams and columns). The total chord rotation capacity at ultimate limit state (Near Collapse, NC) is evaluated according to the relation A.1 of EN1993-3 [2]:  
\[\theta_{u,m}=\frac{1}{\gamma_{el}}0.016\cdot(0.3^{v})\left[\frac{max(0.01; \omega^{\prime})}{max(0.01;\omega)}f_{c}\right]^{0.225}\left(min\left(9;\frac{ L_{V}}{h}\right)\right)^{0.35}25^{(a\rho_{xx}\frac{f_{yy}}{f_{c}})}\left(1.25^{100 \rho_{d}}\right) \tag{2}\]  
where \(\gamma_{el}\) is equal to 1.5 for primary seismic elements and to 1.0 for secondary seismic elements; \(\
u\) is the axial load ratio; \(\omega\), \(\omega^{\prime}\) are the mechanical reinforcement ratios of the tension (including the web reinforcement) and compression longitudinal reinforcement; \(\alpha\) is the confinement effectiveness factor; \(\rho_{xx}\), \(\rho_{d}\) are the ratios respectively of transverse steel parallel to the direction x of loading, and of diagonal reinforcement (if any), in each diagonal direction; \(f_{yyw}\) is the stirrup yield strength (MPa). For RC existing frame designed without detailing for earthquake resistance, the value given by (1) must be divided by the factor 1.2. Moreover, all the mean values of material strengths should be divided by the appropriate confidence factors based on the Knowledge Level, EN1998-3 [2]. The assessment with respect to Life Safety (LS) limit state refers to a chord rotation capacity equal to \(3\theta_{u,m}/4\). Figure 2: GUI developed for RC-framed skin column accounting automatically for effect of confinement. For brittle mechanism, the shear demand is compared with the cyclic shear capacity for all elements evaluated according to the relation A.12 of EN1993-3 [2] (with units: MN and meters):  
\[ \begin{split}& V_{R}=\frac{1}{\gamma_{el,s}}\Big{[}\frac{b-x}{2L_{c}} min(N;0.55A_{c}f_{c})+\Big{(}1-0.05 min\Big{(}5;\mu_{\Delta}^{pl}\Big{)}\Big{)}\\ &\cdot\Big{[}0.16 max(0.5;100\rho_{tot})\Big{(}1-0.16 min\Big{(}5; \frac{L_{w}}{h}\Big{)}\Big{)}\sqrt{f_{c}}A_{c}+V_{w}\Big{]}\Big{]}\end{split}  \tag{3}\]  
where \(\gamma_{el,s}\) is equal to 1.15 for primary seismic elements and to 1.0 for secondary seismic elements; \(x\) is the compression zone depth; \(N\) is the compressive axial force (positive, taken as being zero for tension); \(A_{c}=b_{w}\cdot d\) with \(b_{w}\) the width and \(d\) the effective depth of the cross-section; \(\rho_{tot}\) is the total longitudinal reinforcement ratio; \(V_{w}\) is the contribution of transverse reinforcement to shear resistance. According to the Biskinis model [21], adopted in EN1998-3 [2], the decrease of shear strength is assumed as a function of the plastic part of ductility demand \(\mu_{\Delta}^{pl}=\mu_{\Delta}-1\) that can be calculated as the ratio between the plastic part of the chord rotation and the chord rotation at yielding. Note that shear resistance of primary seismic elements should be evaluated by adopting mean values of material strengths divided by the appropriate confidence factors based on the Knowledge Level and also by the partial factors for materials in accordance with EN1998-1 [22]. In the following analyses, a unitary confidence factor is assumed with the aim of evaluating the level of improvement attainable by adopting the proposed retrofitting technology in the limit case of maximum Knowledge Level for the existing building. The attainment of ductile and brittle failures in the first elements, according to Equations (2) and (3), is marked in the capacity curves using for chord rotation capacity a yellow circle and for shear capacity a green triangle.