###### Abstract  
The mutual interaction between base isolation (BI) and soil-structure interaction (SSI) needs to be considered in design applications. In the literature, the key parameters that may affect such interaction have been rarely considered. Closed formulations for the assessment of the mutual role of BI and SSI have not been proposed yet. The paper aims to cover this gap by considering a 3 DOF model to propose closed relationships with the most significant parameters (structural period, isolation ratio and shear wave velocity). The outcomes show that there are two different interactions. When soil deformability and structural stiffness are not significant, the isolation ratio may be used as a main parameter. However, when the foundation soil is particularly deformable and structures are significantly rigid, a more complex relationship needs to be considered. Parametric case studies have been performed with different structures, three isolation ratios and several soil conditions (soil A, B, C and D). base isolation; soil-structure interaction; structural period; isolation ratio; shear wave velocity 0  
deformability may significantly reduce the beneficial effects of BI. In particular, in ref. [11], seismically isolated bridge piers on shallow soil stratum with soil-structure interaction were considered. The seismic response of isolated bridges with soil-structure interaction were also investigated by ref. [12]. In ref. [13], two bridge systems were compared in order to verify that SSI effects may even become non-conservative when compared with the fixed-pier configurations. Recent applications have been proposed [15; 16]. In ref. [15], a hybrid system of a BI system and a shape memory alloy inerter (SMAI) has been applied by considering the nonlinear hysteretic relationships of both the isolation layer and SMA. The authors demonstrated that the inerter may increase the energy dissipation capacity, and thus the effectiveness, of a BIS and SMAI system over the more common BIS and SMA. Another study [16] proposes an identification method for structures equipped with TMD, including the effects of soil deformability. The approach applied the APSO-SSR method that uses the frequency response function and transmissibility functions, and was verified via numerical simulations. In addition, numerical simulations were also applied in order to study the mutual role of SSI and BI. In particular, mid-rise buildings were investigated in ref. [17] by performing a shear-building model with linear elastic isolators on a flexible foundation that was subjected to the great Michoacan earthquake of 1985, which occurred in Mexico. In ref. [18], the authors investigated the role of radiation damping on the performance of one EPS-isolated building. In addition, ref. [19] identified the overall SSI effects by considering the elongation of a fundamental period of the system and decreased composite damping, especially for the high-rise and slender isolated structures. The existing literature proposed several applications without a specific focus on investigating the most important parameters that need to be considered for the design procedure of base-isolated structures on deformable soil conditions. Such an issue may be considered a hole in the literature that the present paper is aiming to cover. In particular, the analytical relationship proposed in [20] is herein considered in order to perform several parametric case studies to investigate the three parameters that influence the mutual effects of SSI and BI: the fundamental period of the structure, the ratio between the isolated period and the structural period and the shear wave velocity of the soil layer. In particular, the 3-DOF system proposed in [21] was applied herein to model the entire system (structure, isolation system and soil layer) and three parametric studies were performed.