## 1 Introduction
Additionally, compared with one-way cyclic loading, two-way cyclic loading induces more substantial changes in the sand surrounding the piles. [PERSON] et al. [19] analyzed the deformation characteristics of monopiles and group piles under horizontal cyclic loading by centrifugal model tests of pile foundations in sandy soil, and the friction between the pile caps and the earth greatly enhanced the horizontal bearing capacity of pile foundations, according to test data. [PERSON] et al. [20] used computational, analytical, and field model test methodologies to undertake an in-depth analysis of monopile foundations subjected to long-term cyclic loads. Based on the findings of a monopile field test in soft clay soil, [PERSON] et al. [21] developed a modified hyperbolic p-y curve. A cyclic degradation factor (t) was developed and estimated based on the resultant cyclic p-y curve. In the same year, [PERSON] et al. [22] performed centrifugal model tests on semi-rigid piles in soft clay soils and investigated the cumulative deformation characteristics of monopiles under horizontal cyclic loading, concluding that the variation in the horizontal displacement of the pile top with the load amplitude and the number of cycles was primarily caused by the cumulative deformation of the soil surrounding the pile induced by horizontal cyclic loading. Recently, numerous scholars have conducted studies on the effects of the pile diameter and bucket diameter on the horizontal bearing capacity of pile-bucket composite foundations [23; 24; 25]. [PERSON] et al. [26] consider that the bucket diameter has a greater effect on the p-y curve and horizontal bearing capacity of composite piles compared with the bucket height, while [PERSON] et al. [27] consider that the ratio of pile and bucket diameters has a significant effect on the horizontal bearing characteristics. In the aspect of stiffness degradation, the stress-strain relationship and the dynamic modulus of the soil are widely analyzed. The empirical connection between the normalized nonlinear stiffness and strain regarding the elastic modulus is often described using stiffness degradation curves. [PERSON] et al. [28] developed a stress-strain model based on the results of cyclic experiments, which considered the nonlinear and degradation characteristics of soft clay soils under cyclic loading, and first introduced the concept of the degradation index. To examine the cumulative pile-side displacement and pile-side bearing capacity attenuation under axial cyclic loading, a simplified cyclic weakening t-z model was devised by [PERSON] and [PERSON] [29]. [PERSON] and [PERSON] [30] combined the semi-empirical formulation of the long-term cumulative plasticity of soils with the stiffness softening coefficient of sandy soils [28; 31] to propose a new computational model of stiffness attenuation, which can simulate the development of softening and cumulative plastic deformation of sandy soils under cyclic loading. [PERSON] [31] developed a stress-strain model by monotonic shear tests and cyclic triaxial tests; it was concluded that there is a close relationship between the stiffness degradation properties of clay and the axial or shear strain corresponding to a single amplitude during cyclic loading, and the soil softening coefficient tends to vary logarithmically with the number of cycles. [PERSON] et al. [32] examined the horizontal cyclic loading characteristics of monopiles based on field tests and found that the lateral stiffness of the piles degraded significantly during the first five cyclic loading cycles when the cyclic load increased to 35-70% of the ultimate monopile load capacity. However, the lateral stiffness stopped degrading when the number of cyclic loadings was larger and showed the stability of the pile-soil system. An in-depth study of the shear strength, critical state, and cyclic response of highly compressible marine-phase soft clays was conducted by [PERSON] et al. [33].