## 1 Introduction  
With the rapid development of megarojects such as supertall buildings and cross-sea bridges, the application of long and ultra-long pile foundations has become increasingly widespread. Under complex geological and environmental conditions, such as seismic events and scouring, the lateral load-bearing performance of these piles has emerged as a critical challenge in engineering design [1, 2, 3, 4, 5]. Due to the diversity of site conditions and loading forms, pile foundations are required to exhibit high lateral load-bearing capacity in addition to supporting axial loads [6, 7, 8, 9]. To more accurately describe the mechanical characteristics of laterally loaded long piles, significant progress has been made in modeling theories and computational methods by scholars both domestically and internationally [10, 11, 12]. The mechanical behavior analysis of laterally loaded long piles is a core issue in soil-structure interaction research, closely related to the selection and applicability of the foundation models [13; 14; 15]. An accurate foundation model is not only a key tool for comprehensively understanding the mechanical behavior of pile foundations under complex geological conditions but also serves as an important basis for guiding practical engineering design. [PERSON] et al. conducted a systematic study on the selection of an appropriate constitutive model for deep excavation optimization, emphasizing the importance and rationality of model selection [16]. Existing research has demonstrated that the Vlasov foundation model, with its clear physical significance and precise characterization of pile-soil interaction, has significant advantages in capturing the impact of the foundation on the lateral load-bearing performance of piles [17; 18]. Compared to traditional Winkler and Pasternak models, the Vlasov model is better suited for simulating the complexity of layered foundations and the continuity of soil, particularly in describing the deformation and internal force distribution of pile foundations under lateral loads [19]. From the perspective of practical engineering needs, recent years have witnessed increasing attention to the static and dynamic mechanical characteristics of laterally loaded piles in layered foundations, providing a reliable theoretical basis for pile foundation design under complex conditions [20; 21; 22; 23; 24]. The study by [PERSON] et al. focuses on the dynamic response of a single pile under long-term, repeated loading and proposes a new method for calculating bearing capacity [25]. [PERSON] et al. demonstrated that pile head stiffness significantly influences lateral displacement and bending moment distribution [26]; [PERSON] et al., using the Laplace transform method, proposed a new lateral vibration impedance model that not only improves computational efficiency but also reveals the key influence of soil layer stiffness on pile dynamic responses [27]. Further, [PERSON] et al. found that flexible constraints at the pile head during horizontal vibration significantly alter the dynamic characteristics of pile foundations, offering new insights for optimizing pile foundation design under complex conditions [28]. [PERSON] et al. investigated the resonance compaction effect of laterally loaded piles in layered soils and proposed an improved foundation reinforcement method [29]. [PERSON] et al. reviewed the efficiency of laterally loaded pile groups and suggested correction factors for bearing capacity under different soil layering conditions [30]. [PERSON] et al. utilized a three-dimensional finite element method to analyze soil resistance around piles in clay, revealing the impact of boundary effects on pile foundation performance [31]. These studies collectively indicate that accurately analyzing the impact of varying soil layer properties and structural conditions on pile mechanical performance is fundamental to ensuring the safety and reliability of pile foundation design.