## 1 Introduction  
The use of reliable behavioral models for bridges and viaducts is the fulcrum for both the design of new infrastructures and safety assessment of existing ones. Very often, the viaduct piers are founded on deep foundations, such as piles or caissons, to transmit the structural loads to deeper soil layers characterized by enhanced mechanical properties. As found by previous studies [1; 2; 3; 4; 5], a concrete caisson is characterized by a significant stiffness contrast with respect to the surrounding soil, and its massive volume constitutes a major portion of the weight of the entire viaduct structure. Consequently, the seismic behavior of viaducts can be strongly influenced by the response of the foundation system, and in a seismic analysis, proper caisson modeling is of pivotal importance to evaluate its effect on the superstructure. The influence of the Soil-foundation Structure Interaction (SSI) phenomenon on the dynamic response of relatively flexible structures characterized by isolated foundations has been extensively studied in the last decades [6; 7; 8; 9; 10; 11; 12]. A proper schematization of the SSI includes: (i) the interpretation of the modal properties of the bridges as identified through operational modal analysis [13; 14], (ii) their estimated state of health (field of Structural Health Monitoring (SHM)) [15], and (iii) the prediction of the dynamic response to the occurrence of medium/high-intensity seismic phenomena [16; 17; 18; 19; 20]. In general, the SSI problem shows a high level of complexity: by definition, it is a coupled problem in which the action strongly depends on the reaction of the structural system [21]. Although EN 1998-5 seismic code states that the effects of SSI under dynamic conditions have to be considered for structures with massive and deep foundations, most of the structural design and assessment procedures adopt the assumption that the structural elements are fixed to the ground preventing translation and rotation, as highlighted in the introduction of several recent studies [1; 22; 23; 24] that show the improved modeled response of structures when SSI is modeled. In general, the static and, especially, dynamic structural responses are strongly influenced by the stiffness relationships between the superstructure, the foundation, and the ground. Thus, ignoring the effects of the SSI phenomena can lead to relevant errors in the evaluation of the structural response under dynamic loads. A clear understanding of the relevance of this error still deserves more investigations for different foundation types and boundary conditions, despite it being well known that it depends on several parameters, such as stiffness ratio between soil and structure, structure slenderness, and seismic input characteristics [2; 3; 5; 10; 11]. This paper aims to provide a further contribution to knowledge by studying a specific case of a stocky foundation (caisson) resting on a relatively stiff soil layer, and by exploring the importance of the SSI effect when the structure is loaded with different seismic inputs. The approaches for the SSI analysis are classified into direct and indirect methods, as reported, for example, by [25; 26]. Among the indirect methods, the most widespread one addresses the SSI problem by an uncoupled approach (\"substructure method\") in which the superstructure analysis is solved independently from the soil-caisson system. The SSI problem is determined independently by separating the effects caused by the kinematic interaction from those caused by inertial interaction. The response of the overall system is then obtained through the application of the principle of superposition of effects [25]. The inertial interaction is solved by a series of (visco-)elastic elements--one for each degree of freedom of the system--whose stiffness and damping coefficients are defined through equivalent springs and dashpots [10; 11; 12].