## 1 Introduction  
The rehabilitation of existing constructions has been gaining increasing interest in recent years for its huge impact on the socio-economic system. The subject has been deeply investigated for buildings, leading to the formulation of rehabilitation strategies that have now been incorporated into design codes [1; 2; 3], but comparable development for bridges and viaducts is still lacking. In particular, although deck isolation, achieved by creating an isolation layer between the deck and the substructures, has proven to be a viable method for reducing the seismic vulnerability of bridges [4; 5], a survey of the state of the art shows that, to date, only a few codes of practice have been published regarding the application of seismic isolation for bridge retrofitting, despite the widespread use of this technique over the last three decades [6; 7; 8]. Some approaches for the conceptual design of seismically isolated bridges (which can be extended to the retrofit of bridges by simply replacing the existing bearings with isolation systems), have been formulated in the past by different Authors. In 1997, [PERSON] and [PERSON] [9] proposed a displacement-based design procedure that uses a single-degree-of-freedom equivalent linear model of the bridge. The design process begins by defining the maximum displacement allowed during the design earthquake, which is then applied uniformly to all piers and abutments (assuming a perfectly regular structure). The yield displacement of each pier is calculated by integrating the curvature along the height of the column and assuming proportionality between the moment and curvature. The actual displacement of each isolator is determined by requiring that the force at the target displacement does not exceed 85% of the yield force of the pier, and the isolator's yield displacement is calculated by dividing the actual displacement by the isolator ductility, chosen by the designer in order to prevent collapse during extreme events. The effective ductility of the equivalent single-degree-of-freedom (SDOF) isolated bridge is determined as the ratio between the maximum allowed displacement and the sum of the yield displacements of the isolators and the piers. Subsequently, the equivalent viscous damping ratios of the isolation system and the structure are defined. Using the displacement spectrum for the appropriate equivalent damping ratio, the period of the first vibration mode is calculated, thus obtaining the equivalent stiffness of the isolated bridge. Then, the procedure ends with the determination of the stiffness of the isolation system. After presenting the procedure for the design of new bridges, the Authors briefly discuss its application to existing bridges, highlighting two main problems, namely, the different response of the bridge in the two directions and the possible presence of weaker structural elements. This procedure was subsequently extended by [PERSON] and [PERSON] [10] to the case of bridges supported by isolators with a bilinear hysteretic loop, such as lead rubber bearings, taking into account the contribution of the bearings to the equivalent viscous damping of the system. An iterative procedure in nine steps was presented, but its application was conceived for new bridges only. [PERSON] et al. [11] addressed the design of isolation systems with bilinear force-displacement behavior, represented by flat sliding bearings combined with recentering devices and considering three different devices: rubber springs, steel springs, and shape memory alloys, respectively. Two iterative procedures were developed to design the isolation system, the design displacement approach, and the design force approach.