## 1 Introduction
This approach does not consider the possible coupling between the different degrees of freedom and the inelastic response induced by plastic deformations which might develop in the foundation soil even under relatively low loading conditions [27; 28]. Surrogate implementations in the time domain for inertial SSI can also include masses, leading to the well-known monkey tail approach [29]. Other pioneering works on the topic were conducted by [30; 31]. Among the simplified approaches, a possible alternative to correctly consider the effects of nonlinearities in SSI problems is based on the concept of _macro-element_ where the entire foundation-soil system is schematized through a single point at the base of the superstructure (_lumped mass_ model) with three or six degrees of freedom (respectively, for 2D or 3D problems). The first application of the macro element concept in geotechnical engineering is due to the contribution of [PERSON] and [PERSON] [32], who presented the formulation of a method to assess the settlements of a strip foundation on cohesionless soils subjected to quasi-static monotonic loading. Successive developments and applications of the concept of macro-elements applied to bridges founded on shallow foundations are due to [9; 26; 32; 33; 34; 35], among others. Many applications demonstrated that this approach can be very effective in predicting the structural behavior under dynamic loading conditions [32; 33; 34; 35; 36; 37], yet it requires a very detailed characterization of the soil parameters to properly calibrate all the variables included in the mathematical formulations. In the direct approaches [25], the soil volume, the foundation, and the superstructure are part of the same model, analyzed together using numerical methods based on a continuum discretization of the domain (typically, by either the Finite Element Method, FEM, or the Finite Difference Method, FDM). Dynamic numerical analyses performed through FEM approaches can be very effective to analyze the seismic behavior of structures due to their ability to simulate complex geometry, boundary conditions, soil and structure dynamic behavior [38; 39; 40; 41]. The crucial aspects to obtaining reliable predictions rely on the correct selection of the seismic input, proper adoption of domain discretization, and suitable choice of boundary conditions. Following the direct approach described above, in this paper we present the main results of a full 3D FE modeling to assess the dynamic response of a bridge pier founded on a caisson subjected to seismic loads. These results are compared with those derived by a simplified model, where the bridge pier is embedded at the base, neglecting the SSI effects. The analysis has been limited to a single pier considered isolated; this is motivated by the goal of providing evidence of the importance of the phenomena, without aiming at giving an exact and faithful evaluation of the structural safety of the case study. The structure of the paper is the following: after the brief literature introduction of Section 1, the examined viaduct is presented in Section 2, describing its structural features and the main geotechnical characteristics of the foundation soil. In Section 3, the seismic inputs used in the dynamic analyses are presented, while in Section 4, details on the numerical models are given. In Section 5, the main outcomes of the model are discussed and, finally, in Section 6, some concluding remarks and perspectives are listed.