## 1 Introduction
Both procedures are based on the Capacity Spectrum Method (CSM), as defined in ATC 1996 [12]: the CSM compares the capacity curve of the structure, represented by a multilinear force-displacement relationship, with a damped response spectrum representing the seismic demand; the intersection of the two curves identifies the performance point, which defines the maximum global response expected during the ground motion. In the design force approach, the goal is to limit the maximum force transmitted by the isolation systems to an assigned percentage of the pier's yield strength, while in the design displacement approach, the goal is to limit the maximum displacement of the isolation systems to a percentage of its ultimate displacement capacity. The iterative process involves adjusting the maximum displacement or maximum force of the isolation systems, which in turn affects the effective period and the overall damping of the system, until convergence is achieved. [PERSON] et al. [13] developed a design procedure which takes into account five different cyclic force-displacement behaviors of the isolation system, namely, viscoelastic, elastic-plastic with hardening, rigid-plastic with hardening, double-flag-shaped, linear, or nonlinear viscous behavior. At the beginning of the procedure, the designer assigns the displacement of the isolation system in each direction, considering the characteristics of the selected isolator type. During the design phase, the maximum displacements of the piers and of the joints are verified to ensure that they are lower than the relevant yield displacements and the available clearances, respectively. The choice of the isolation system is linked to the amount of damping that the designer wants to obtain. For this purpose, the Authors developed a graphical procedure, valid for both existing and new bridges. The principle consists in plotting the yield point of the equivalent SDOF bridge on the ADRS plane, along with damped demand spectra for various damping values associated with the different types of isolators. Once the damping is chosen, an iterative procedure is then performed to define the optimal stiffness of the isolator that achieves the target displacement in both directions. [PERSON] and [PERSON] [14] developed a system-level repair cost ratio design surface for various base-isolated bridges, facilitating performance-based design and device optimization. The methodology is based on the derivation, for bridges with different isolation systems, of component fragility surfaces as functions of multiple engineering demand parameters. The system-level repair cost ratio surface is derived by incorporating the damage probabilities, damage ratios, and replacement costs of bridge components. Subsequently, the uniform repair cost ratio function is used as a performance index, and a genetic optimization procedure is implemented to obtain the mechanical properties of isolation systems that lead to the lowest repair cost ratio. [PERSON] and [PERSON] (2015) presented a simplified procedure for the isolation of a continuous reinforced concrete (RC) bridge girder. The goal is to simplify the determination of isolator response parameters and the definition of an optimal layout based on predefined structural performance criteria. [PERSON] et al. (2016) proposed an equivalent energy-based design procedure to improve the seismic performance of high-speed railway (HSR) bridges by using friction pendulum bearings (FPBs). The procedure balances the total input seismic energy with energy dissipation and deformation capacity, providing a holistic approach to the structural response. The method offers simpler calculations compared to traditional methods, but it is strictly related to the characteristics of the Chinese HSR, for which it was developed. [PERSON] and [PERSON] (2017) proposed a procedure based on the Deformation-Based Design method.