## 1 Introduction  
Wind energy is a clean and sustainable energy source that is gaining popularity in nations around the world. Compared to onshore wind power, offshore wind power enjoys the advantage of abundant resources, leading to great efficiency in power generation, minimal environmental contamination, and no land resources, among other things. The vast ocean makes up 71 percent of the Earth's land, is rich in natural resources, and has huge potential for development. Offshore wind energy resources are abundant off the coast of China, and can largely solve the problem of insufficient electricity supply in the country. However, different to the European countries, the most of China's offshore wind farms are located on claycy or silty seabeds [1]. A monopile is a simple structure with clear force resistance and a high bearing capacity, which makes it the most commonly used foundation for offshore wind turbines. Generally, the offshore wind turbine pile foundations are positioned in a challenging maritime environment. Under the long-term cycle of low-frequency loading, the pile foundation accumulates deformation and permanently destroys the tilt of the superstructure. With the development of deep-sea wind turbines and giants, the production cost of monopiles substantially increases, and traditional large-diameter piles eventually cannot meet the needs of complex loading. To optimize the horizontal bearing capacity of monopile foundations and avoid excessive increases in pile size, relevant researchers proposed a composite monopile foundation [2; 3; 4]. Skirt-pile foundations are an innovative form of offshore wind power foundation that incorporate a skirt to boost the ultimate bearing capacity. The skirt design is similar to that of the bucket foundation, featuring a capped top and an open bottom. The study of skirt-pile foundations builds upon research on traditional bucket foundations [5; 6; 7]. The offshore wind power team of Tianjin University [8; 9; 10] developed a wide and shallow composite foundation for wind conditions and addressed the characteristics of China's offshore wind farms. Subsequently, the team conducted a series of studies on the dynamic response and bearing performance of pile-bucket composite foundations on this basis [11; 12; 13; 14; 15]. Recently, the pile-bucket foundation was successfully used in wind farm practices in southeastern China [16]. [PERSON] et al. [17] employed a numerical model to scrutinize the factors influencing the bearing capacity of a cylindrical foundation. [PERSON] et al. [15] conducted experimental and numerical studies on the bearing mode, bearing capacity, and damage mode of pile-bucket foundations under different loads in saturated clay soils. This study delves into a composite foundation composed of a single pile and a skirt foundation. The skirt foundations are similar to the bucket foundation in that they are connected to the single pile foundation by reserving a round hole at the top with the same diameter as the single pile. Typically, skirt foundations distribute the load over a larger area through the skirt, while pile foundations transmit the load deep into the soil. By combining the bearing capacity of the skirt foundation and the pile foundation, the combined system can effectively support greater loads and provide enhanced stability. Moreover, skirt-pile foundations offer greater resistance to lateral loads. Piles provide resistance to lateral movement, while skirts help to stabilize the foundation by increasing the effective foundation width, thereby preventing overturning or sliding. Theoretical investigations of piles under horizontal cyclic loading primarily focus on understanding the alterations in the stiffness and strength of foundations subjected to cyclic loading. For pile-soil interaction under horizontal loading, [PERSON] et al. [18] conducted two-way cyclic loading experiments in dense sand and showed that pile-soil interaction exhibits strong nonlinearity.