## 1 Introduction  
A \(p\)-\(y\) curve depicts the relationship between soil horizontal resistance \(p\), at a certain depth under the ground and pile deflection \(y\), at the corresponding position under the lateral load (horizontal force and bending moment) of the pile. This curve can help engineers understand the nonlinear state of pile-soil interaction, which is why it is the most used method globally for calculating the large deformation of a horizontally loaded pile. In 1970, [PERSON] [1] established the formula for a \(p\)-\(y\) curve for soft clay under a horizontal load, in the form of a subsection function, based on field tests[PERSON] Then, [PERSON] et al. [2] established the \(p\)-\(y\) curve for hard clay. Both of these papers laid the foundations for the \(p\)-\(y\) curve method, but the method still had limitations. In order to make the \(p\)-\(y\) curve method more broadly applicable, [PERSON] et al. [3] proposed a unified method suitable for arbitrary clay, and [PERSON] et al. [4] proposed a unified method suitable for clay, based on the constitutive relationship of the soil. Along the same lines, [PERSON] et al. [5] proposed a calculation method for a \(p\)-\(y\) curve suitable for soft clay in the middle and lower reaches of the Yangtze River in China, through field pile testing, and [PERSON] et al. [6] expanded upon this, by considering the weakening effect of soil. These authors constructed \(p\)-\(y\) curves of saturated sand under different weakening states. Additionally, based on field tests and a theoretical analysis, [PERSON] et al. [7] and others combined the [PERSON]m method\" with the s hyperbolic \(p\)-\(y\) curve, to modify the calculation of \(p\)-\(y\) curves in layered soil. [PERSON] et al. [8] dealt with pile side soil resistance by using the stress distribution law of the pile side soil, and considered the influence of the longitudinal stress of the soil, in order to establish a \(p\)-\(y\) curve model based on the stress state of the pile side soil. [PERSON] et al. [9] went further, and used the finite difference method to solve the deflection differential equation, based on the combination of load transfer and \(p\)-\(y\) curve methods for layered soil foundations, and validated the model's feasibility with real-world engineering examples. Similarly, [PERSON] et al. [10] adopted a hyperbolic horizontal load transfer model, based on a consideration of the layering of the surrounding soil, and obtained the nonlinear solution of the deformation and internal forces of the pile body under a composite load. Their results show that the large deformation of horizontally loaded piles cannot be ignored. [PERSON] et al. [11] studied the mechanism of composite loads on piles in sandy soil, and indicated that they are caused by the superposition of the soil compaction and second-order effect. [PERSON] et al. [12] also studied a \(p\)-\(y\) curve method of piles in layered sandy soil, based on numerical analysis, and compared it to the iso-depth method proposed by [PERSON] [13], and the results showed little difference. Finally, [PERSON] et al. [14] used a model test that considered the influence of pile top constraints, and constructed the hyperbolic \(p\)-\(y\) curve of an inclined pile in sand, using the two parameters of initial foundation reaction modulus and ultimate soil reaction at the pile side. At present, there are still some problems that are generally ignored by the \(p\)-\(y\) curve method. First, \(C_{u}\) is taken as fixed, ignoring the influence of its change with depth on the \(p\)-\(y\) curve. Second, when the \(p\)-\(y\) curve is used to calculate the forces on a laterally loaded pile, the influence of the vertical load on the pile top is not considered.