## 1 Introduction
The MPMR technique was found to be fast, while FEM analysis required considerable computational time. By considering the dynamic stiffness of soil, ref. [14] examined the SSI effects on structural dynamic responses using a substructure approach. The unbounded soil region was analyzed in the time domain using a coupled model, based on FE and scaled boundary FE methods, and the results from two-dimensional frameworks were compared with those from the cone model. The cone model offers practical and efficient approximation for SSI analysis. In this model, the soil underlying the structure is idealized as a truncated, semi-infinite elastic cone, which is represented by frequency-dependent dashpots and springs for each degree of freedom. Both translational and rotational cone models are available. The cone is treated as a rod (bar), with its cross-sectional displacement pattern determined by the corresponding value on the axis of the cone, where plane cross-sections remain planar. The soil half-space outside the cone is disregarded. The cone model performs well in low-to-intermediate frequency ranges, such as those associated with earthquakes and machine vibrations [15]. According to [16], a semi-infinite truncated cone can approximately represent the soil in SSI analysis. The cone model has been used to study torsional rotation, rocking, and vertical and horizontal translations. Using the approximate cone model method, ref. [17] investigated the dynamic analysis of a foundation resting on saturated porous media under torsional and horizontal vibrations. This study analyzed the response of foundations to torsional and shear vibrations, considering parameters such as density, porosity, shear modulus, and Poisson's ratio. The study by [18] modeled foundations on deformable soil with a simple spring-dashpot-mass system using frequency-independent coefficients. For both homogeneous half-space sites and homogeneous layers fixed at their base, the provided tables list coefficients for various parameters, including dimensional ratios and Poisson's ratio. The study considered both embedded and surface rigid foundations for translational and rotational motions. Using a cone model in the time domain, ref. [19] investigated SSI effects on rigid blocks and single-degree-of-freedom superstructures, comparing results from time and frequency domain analyses. The study also examined the impact of the reflection coefficient under two different soil layering configurations. To estimate the dynamic behavior of simple structures resting on an elastic half-space, ref. [20] developed and applied discrete models. The study derived precise analytical expressions for the fundamental natural period and corresponding damping coefficients as functions of geometry and frequency-dependent foundation impedances. It also introduced the concept of statically and geometrically equivalent SSI systems and compared the radiation damping produced by a single pile and a footing under harmonic loading. An overview of the available modeling techniques for addressing SSI issues and the impact of dynamic SSI on building structures is provided in [21]. While there has been extensive research on soil-structure interaction (SSI), several key gaps remain. (i) Many studies acknowledge that SSI effects are significant in soft soils, yet there is no clear consensus on how these effects influence seismic performance across various types of structures. (ii) Some studies suggest that SSI can reduce structural demand, while others indicate it may amplify forces, particularly in high-rise or flexible structures. (iii) Furthermore, accurately modeling the interaction between soil and structure remains challenging. (iv) Although most studies rely on specific soil models, soil behavior is inherently complex and variable under dynamic conditions. These gaps limit researchers' ability to verify the accuracy of numerical solutions and the underlying assumptions in SSI analyses.