## 3 Description of the Case Study: An Existing RC Building
### Definition of the Non-Linear Numerical Models for Fixed-Base/Base-Isolated RC Structure  
The numerical model of the RC structure has been realized according to the geometric features described in the previous section. In particular, the columns have been assumed as fully restrained by the stiff inverted beams at the foundation level and the behavior of rigid floors has been accounted for by means of in-plane diaphragm constrains [25]. The interaction in the non-linear dynamic simulations between the structure of the ground-floor and the surrounding soil embankment has been accounted for by means of appropriate restraints to limit the horizontal displacements in X and Y directions. In this way, the seismic response only of the ground-floor of the structure (i.e., story 1 of Figure 3) is restrained due to the presence of the surrounding soil embankment as will be achieved in the following. The cross-sections of the structural members (i.e., beams and columns) have been realized according to the fiber approach [25], as also developed in [28]. In particular, the mechanical properties of the fibers in each cross-section have been defined differentiating between concrete cover, core and longitudinal reinforcements. The concrete behavior in both cover and core fibers has been reproduced by means of [PERSON] et al. [29] constitutive relationships accounting for non-linear response in compression and elastic with linear softening response in tension. The properties of concrete cover fibers (i.e., tensile strength, ultimate deformation in compression) have been derived from both the mean value of cylinder compressive strength \(f_{cm}\) and Young's modulus \(E_{cm}\) according to [30]. The response of concrete core fibers has been defined evaluating the enhanced properties due to the confinement effects of the stirrups evaluated adopting the model of [31]. The properties of the reinforcement have been defined by means of an elastic-plastic law with a limited ultimate strain to reproduce both tensile ad compressive response. The yielding strength has been set equal to the mean value \(f_{ym}\) estimated from the tests with a Young's modulus assumed equal to 200,000 MPa and an ultimate strain set to 7.5%. The mechanical non-linearities have been included in the numerical model adopting a distributed plasticity approach (the so-called \"Fiber Hinge\") implemented in the software platform SAP2000 [25]. The fiber-plastic hinges so far defined are able to account for the interaction between axial load and bending moments in X and Y directions. The moment-curvature law of the fiber-hinge law is determined on the basis of the fiber cross-section properties previously introduced. The mechanical non-linearity is developed in each fiber of the cross-section, but is not extended to the whole element [32], but are included in a portion of the RC member with a predefined length (i.e., plastic hinge length). In particular, the plastic hinge length \(L_{p}\) is evaluated according to a well-validated formulation by [PERSON] et al. [33] formulation:  
\[Lp=0.08\cdot l+0.022\cdot f_{ym}\cdot d^{\prime} \tag{4}\]  
where \(l\) is the length of the member, \(f_{ym}\) is the mean yield strength of steel reinforcement from tests and \(d^{\prime}\) is the diameter of the smallest longitudinal reinforcing bar adopted within the cross-section. The geometric non-linearity (i.e., P-\(\Delta\) effect) has been accounted for within the numerical simulations with the evaluation of the equilibrium equations with reference to the deformed structural configuration. The fundamental vibration period of the fixed-base structures is 0.80 s from the eigenvalue analysis without considering any stiffness reduction in the RC beams and RC columns. This assumption is aimed not to underestimate the effects in terms of both accelerations and seismic forces to the structure. The retrofitting using the single-concave friction pendulum (FP) has been simulated introducing a disconnection between superstructure and substructure at the top of the columns of the groundfloor.