# Dynamic Lateral Response of the Partially-Embedded Single Piles in Layered Soil
## 1 Introduction  
As a common form of deep foundation, pile foundations are usually used in bridges, pile foundations of the abutment, and hydraulic structures. They are often subjected to horizontal loads such as wind, waves, traffic, and earthquakes. Existing studies have shown that scouring reduces the strength and stiffness of the pile-soil interaction system and becomes one of the significant causes of structural failure [1, 2, 3]. [PERSON] et al. proposed a simplified method to evaluate the effect of scour-hole dimensions on the response of laterally-loaded piles in the sand [4]. [PERSON] et al. studied the effects of stress history, scour depth, scour width, and scour-hole slope angle on the response of laterally-loaded piles in soft clay [5]. Due to the requirement of lateral load-bearing of pile foundations, the dynamic research of laterally-loaded piles has become a hot issue [6]. Scholars at home and abroad have conducted extensive studies on the dynamic characteristics of single piles under horizontal loads. Their theoretical models can be classified into the Continuum model, finite element or boundary element model, and Winkler foundation beam model. Using thecontinuum model, [PERSON] analyzed the deflection of piles in clay under a quasi-static cyclic load [7]. Based on the coupling of finite element method and boundary element method, [PERSON] et al. studied the dynamic response of pile support structure under harmonic excitation [8]. Using the linear elastic Winkler foundation model, [PERSON] analyzed the dynamic response of a single pile in the frequency domain [9]. For the study of the dynamic response of the partially-embedded single pile, it is critical to simulate the effect of the soil-structure interaction accurately. The Vlasov foundation model not only overcomes the shortcomings of the Winkler model that cannot show the shear characteristics of the foundation, but is also developed from the elastic half-space model, reducing the complexity of the half-space model, and has a complete theoretical basis [10]. [PERSON] correlated the elastic foundation parameters with the attenuation parameters and proposed a modified Vlasov foundation model [11]. Based on the modified Vlasov foundation model, [PERSON] and [PERSON] obtained the equation of displacement of laterally-loaded piles under dynamic load [12]. Since most of the soil fields in the project are layered, the physical properties of the soil fields that change with the buried depth have a significant impact on the mechanical properties of the pile foundation. Using the Mindlin integral equation, [PERSON] numerically solved the horizontally-loaded pile in the two-layer homogeneous isotropic elastic foundation [13]. [PERSON] et al. analyzed the elastic solution of horizontally-loaded piles in layered soil [14]. The numerical method Is the simplest way to solve the dynamic response of pile foundations in complex sites, and the results are accurate [15; 16]. For the evaluation of the seismic behavior of the foundation system, [PERSON] et al. have conducted different types of non-linear numerical analysis, taking into account the soil-structure interaction effects [17]. Using the finite difference method, [PERSON] et al. analyzed the effect of the soil-structure interaction on the natural frequencies of partially-embedded single piles [18]. Compared with fully-embedded single piles, the boundary conditions of partially-embedded single piles in layered soil are more complicated. It is more difficult to use the method of undetermined coefficients to solve the function of dynamic response. The above-mentioned finite element method or boundary element method usually requires much work and often requires professional software and complex models. [PERSON]'s function method has a clear concept and can obtain an accurate solution of the closed-form of the system, which is especially important for determining the dynamic response of the pile foundation.