## 1 Background  
The fundamental period and the damping are properties that significantly affect the dynamic behaviour of structural systems in terms of base shear, deformations and thus seismic forces. In this regard, soil-structure interaction (SSI) effects are particularly connected with the mutual relationship between the fundamental periods of the soil and of the structure. In particular, the comparison between the fixed-based structures (SSI neglected) and the flexible-based ones (where the effects of soil deformability are considered) shows that the fundamental period of the latter configuration elongates due to SSI effects. In addition, the overall damping of the system (soil + structure) also increases. These aspects have been extensively considered in literature both theoretically [1] and numerically [2; 3]. In particular, the first studies (such as [4]) demonstrated that period elongation results from inertial interaction that affects the displacement demand of the structure. In particular, [PERSON]. [5; 6; 7] proposed the so-called \"structure to soil stiffness ratio\" (that depends on the height and the shear velocity of the flexible layer) and it defines the elongation of the structural period due to SSI. Later, [PERSON]. [8] extensively investigated the formulas in ASCE 7-05 for several buildings by considering over 800 fundamental periods. [PERSON]. [9] proposed an empirical formula for assessing the fundamental period of reinforced concrete structures by performing 3D numerical simulations of SSI effects. Other approaches were applied, such as [4; 10] that used system identification to evaluate the effects of SSI on 57 buildings subjected to strong motions. Moreover, [PERSON]. [11; 12; 13; 14] proposed analytical surveys on the effects of SSI in terms of fundamental period and damping ratio. Experiments were also conducted, such as [15] that studied the effects of SSI on an instrumented 16-story, reinforced concrete building in Moldova, and [PERSON]. [16] examined the experimental results of forced-vibration tests on a school building in Taiwan. In addition, [PERSON]. [17] studied the response of a 14-story reinforced concrete building in SrSpka to 20 recorded earthquakes, while [18] assessed the frequency and damping variation of low-rise masonry buildings. Moreover, [PERSON]. [19] proposed to use [PERSON]'s formula to assess the variation of natural frequencies demonstrating high accuracy from experimental and numerical results. In addition, [PERSON]. [20] investigated the effects of SSI with series of free-vibration experiments on a 1/4-scale steel-frame structure by reproducing 34 scenarios under fixed-base and flexible-base conditions. Moreover, Ref. [21] investigated the effects of SSI on several residential RC buildings with different capacity design principles founded on four different soil conditions and applying 20 acceleration records. Furthermore, Ref. [22] proposed a comparison between FEM simulations and dynamic tests with large scale soil-foundation-structure models, Ref. [23] studied the effects of soil-foundation interaction on the seismic response of a cooling tower by applying 3D-FEM analysis and [24] proposed a theoretical lumped-mass model calibrated by experimental results to perform an in-depth parametric analysis. The main novelties of the paper are: (1) it proposes a methodology that calculates the period elongation and the damping increase with an equivalent 2-DOF (degree of freedom) model. Then, (2) several equivalent MDOF (multi-degree of freedom) systems fixed at the base (named EQ) were produced to represent more complex MDOF systems that account for SSI effects. In particular, EQ models are built up with the two parameters that are calculated in point 1. (3) Finally, in the paper, the models were validated with numerical case studies performed with Opensees PL to consider realistic response of the soil in terms of plastic and hysteretic mechanisms. Section 2 describes the adopted 2-DOF model, the formulations for the period elongation and damping increase.