# Validation and Parametric Study of a Soil-Structure Interaction Nonlinear Simplified Model
## 1 Introduction  
The majority of national standards (e.g., [1; 2; 3]) currently assume a rigid foundation for the analysis of seismic performance in the superstructure, with the soil-structure interaction (SSI) being largely overlooked. However, due to the flexible nature of foundation soil, the dynamic response and failure modes of structures under seismic effects differ from those assumed with rigid foundations. Numerous studies (e.g., [4; 5; 6; 7; 8; 9]) have demonstrated that the damage to structures constructed on soft soil foundations is significantly exacerbated during earthquakes due to the influence of SSI. Currently, research on SSI is predominantly conducted through experimental studies and theoretical analyses. Experimental studies include prototype tests (e.g., [10; 11]), centrifuge methods (e.g., [12; 13; 14]), forced vibration tests (e.g., [15; 16; 17]), and shaking table tests (e.g., [18; 19; 20; 21]). Theoretical analysis methods (e.g., [22; 23; 24; 25]) primarily employ analytical formulations and numerical analysis techniques. Due to the complexity of SSI research and the uncertainty of foundation soil parameters, the consideration of SSI effects increases the complexity of structural numerical models and computational redundancy. Consequently, in engineering applications, a simplified model calculation method that is both convenient and simple, yet accurately reflects structure responses, is required. In recent years, numerous scholars have proposed various analytical models and corresponding research methods for the study of the soil-structure interaction. [PERSON] and [PERSON] [26] proposed the SR model (Swing-Rocking Model), which simplifies the superstructure into a centralized mass, and set horizontal and rotating springs at the base position to simulate the horizontal movement and rotation of the foundation. The free field system adopted the shear particle system, with the acceleration of the top mass of the free field system serving as the input seismic motion for the foundation. This model isrelatively simple and straightforward to employ, yet it is unable to simulate the nonlinear soil-foundation interaction. [PERSON] and [PERSON] (2017) proposed the Winkler model, which treats piles as beams in the soil. The site soil is divided into two regions: the near field and the far field. The near field region employs nonlinear springs and dampers to simulate the nonlinear soil-structure interaction. The far-field region employs linear springs and dampers to simulate the free field. This model considers the nonlinear soil-structure interaction, enabling the analysis of the vertical vibration effects of pile groups. However, as a single-parameter model, the Winkler model is unable to simulate the continuous behavior of the ground and is therefore only capable of predicting vertical displacement at specific points. In addition, the determination of spring parameters is difficult, prompting numerous scholars to improve the Winkler model. [PERSON] et al. (2018) used an approximate frequency-independent parameter model as the far-field unit to simulate the linear elastic free field and solve the non-dynamic soil-pile response. [PERSON] (2019) used frequency-dependent springs to simulate the foundation soil impedance. [PERSON] et al. (2019) developed a simplified model to calculate the lateral nonlinear dynamic soil-pile interaction. All the above models effectively simulate the soil-structure interaction mechanism. [PERSON] and [PERSON] (2019) proposed the Penzien model, which consists of three parts: the superstructure, the pile and the equivalent soil. The superstructure is simplified to a lumped-mass model and the site soil is divided into equivalent soil layers according to certain principles. The pile group is combined into a single pile, which is divided into different units according to the soil layers, and the equivalent soil around the pile is simplified into a centralized mass. The soil-structure interaction is represented by springs and dampers.