## 1 Introduction
This study aims to fill these gaps by investigating how soft soil interactions influence seismic performance using a simplified dynamic model. Developing and validating simple mechanical models that accurately capture SSI behavior under seismic conditions represents a key advancement in SSI research. Additionally, this study seeks to clarify how various factors--such as foundation block dimensions, variations in superstructure height, and different soil types--affect structural behavior under seismic loads, providing insights to improve reliability and uniformity in engineering applications. This work focuses on quantitatively investigating how the response of soil-interacting structures differs from that of fixed-base (non-interacting) structures. Another unique aspect of this research is its ability to systematically examine the time-domain response of the dynamic SFSI system. The increased flexibility and natural vibration period of the dynamic system in the presence of soil are presented in detail through examples. Dynamic SFSI analysis is performed by modeling the behavior of unbounded soil extending to infinity using a discrete-element approach. For a rigid basemat, the degrees of freedom at the structure-soil interface are considered as those of a rigid body. Specifically, for this configuration, physical approximations, such as truncated, semi-infinite one-dimensional cones, are used to represent unbounded soil. The dynamic system is analyzed by considering the coupling between rocking and horizontal motions. In the context of SSI and dynamic analysis, coupled rocking and horizontal motions provide a more realistic representation, as soil behavior typically involves both vertical and horizontal movements simultaneously. This approach more accurately reflects the physical interactions that occur during seismic or dynamic loading events, where rocking and horizontal motions occur simultaneously due to the complex nature of the soil response. This analysis is conducted using the 1940 El Centro earthquake (Mw = 6.9), focusing solely on the inertial interaction while excluding the effects of seismic input modification on structural response. Dynamic SFSI analysis is performed using the central Finite Difference (FD) method in the time domain. This method is advantageous for its simplicity and efficiency, as it bypasses the need for computational complexities, saving both time and effort. Unlike traditional frequency-domain studies, which require computationally intensive Fourier transformations, the proposed time-domain approach is faster. The proposed time-domain computational technique is both precise and well suited for SSI calculations. Many SSI problems involve dynamic, time-dependent behavior, especially under seismic loading or other transient events. The FD method is well suited for explicit time-stepping schemes [22]. When implemented with small time steps, the FD method can accurately capture high-frequency wave components, which are often critical in SSI problems, particularly during seismic events. This enables the precise resolution of small, rapid changes in stress or strain, providing accurate predictions of wave propagation and soil-structure interactions [23].