## 1 Introduction
[15] revealed the mechanical characteristics of slender piles under a lateral monotonic load and a cyclic load, and they deduced the static and cyclic p-y curves of a pile. Through the centrifugal test, [PERSON] et al. [16] revealed the influence of the type and shape of the scour hole on the p-y curve of the rigid pile. Through addressing the Boussinesq point load arising from scour unloading, [PERSON] et al. [17] developed the p-y analysis method under asymmetric scour conditions, developing a simplified method for calculating lateral loads on small-diameter flexible piles. By adapting the Matlock p-y curve and incorporating the Boussinesq solution, [PERSON] et al. [18] introduced the concept of average reduced stress to formulate a comprehensive analytical solution for laterally loaded piles in soft clay. The p-y curve method mentioned above is a relatively common analysis method at home and abroad. However, the p-y curve method should be based on the test results, and the p-y curve calculation method suitable for the test site or similar test site should be obtained via inversion and curve fitting. Therefore, the p-y curve method depends on tests and experience. To accurately evaluate the impact of scouring on the lateral bearing performance of a single pile, further investigation is needed utilizing the achievements from the theory of lateral loaded pile analysis. Introducing the Duncan-Chang model and Mohr-Coulomb model, [PERSON] et al. [19] enhanced the strain wedge and conducted nonlinear analysis of a laterally loaded pile in sand. Leveraging the strain wedge model, [PERSON] et al. [20] incorporated the state-dependent plasticity model to compute the stress-strain relationship of sand within the wedge. Based on a strain wedge, [PERSON] et al. [21] proposed a versatile state-dependent constitutive model to promote its predictive performance across different scenarios. [PERSON] et al. [22] presented a modification of the strain wedge model in sloped pile foundations. Utilizing a modified three-dimensional cone-shaped strain wedge model, [PERSON] et al. [23] proposed a novel technique for calculating the ultimate shaft resistance of piles in sand. Using the modified strain wedge model, [PERSON] et al. [24] derived the p-y criterion for rock-socketed piles while considering Hoek-Brown failure criterion. [PERSON] et al. [25] proposed the conical strain wedge (CSW) model to conduct the nonlinear analysis of a single pile subjected to lateral loading within layered soil. [PERSON] et al. [26] proposed a scour failure wedge (SFW) model to simulate the interaction between laterally loaded piles and sand with local scour. [PERSON] et al. [27] established an approximate cone strain wedge model, considering both free-water-head and fixed-head conditions. In summary, the strain wedge model and its improved model can be derived from the theory of the soil stress-strain relationship, the displacement-strain relationship, and the load-stress relationship, and they are not affected based on the test site or the artificial selection of a curve form. Considering the unique advantages of the strain wedge model, strain wedge theory does not rely on empirical parameters (or field tests), and it can consider the three-dimensional continuity of the soil and the characteristics of the pile. In this paper, based on the improved strain wedge model, the fine size of the scour hole is considered with the Boussinesq solution, and a simplified calculation model of the single pile under the condition of a three-dimensional asymmetric local scour hole is established to reduce the construction cost of the pile foundation and the operation and maintenance cost of an offshore wind farm. At the same time, this paper provides guidance for the scour protection of an offshore pile foundation.