## 1 Introduction
The results have enriched and improved the design theory for NT-CEP piles and complemented the practical engineering applications, playing a positive role in promoting the research and application of NT-CEP piles. However, at present, the theoretical research on the horizontal bearing characteristics of squeezed branch piles is not sufficient, so it is necessary to establish a calculation method for the bearing capacity of piles under horizontal loads. The theoretical calculation methods for the bearing characteristics of straight piles under horizontal loads include the elastic-plastic analysis method, ultimate foundation reaction method, and elastic foundation reaction method. In the elastoplastic analysis method, the range of the piles and soil is divided into the elastic zone and plastic zone. In the elastic zone, the elastic foundation reaction method is adopted, while in the plastic zone, the ultimate foundation reaction method is adopted. According to the continuous condition of the boundary of the elastoplastic zone, the lateral resistance of the pile is obtained using the elastic foundation reaction method. The common methods include [PERSON] method [17; 18] and the p-y curve method [19; 20]. The ultimate foundation reaction method assumes that the pile is rigid and the lateral resistance of the pile is calculated according to the static equilibrium under the limit state without considering the foundation deformation. The commonly used methods include the [PERSON] [21] method and the Snitko [22] method. In the elastic foundation reaction method, by using the Winkler foundation model [23], the soil around the piles is simulated as a group of independent springs, and then the displacement and internal force of the piles are solved according to the deflection differential equation of the beam on the elastic foundation [24]. The foundation reaction methods mainly include the constant method, k-method, m-method, and c-method, and the current industry norms [25; 26] clearly stipulate that the m-method is to be used to calculate pile foundations under horizontal loads. Based on the m-method, [PERSON] and [PERSON] [27] discussed the dynamic and static conversion relationship of the m value during testing and proposed a new lateral dynamic load testing method for offshore piles, which was used to analyze the pile-soil interaction and evaluate the horizontal bearing capacity of the piles. Using mathematical statistics, [PERSON] [28] established the correlation formula between the soil's m value and the pile's ground surface displacement under different soil conditions, which was used to determine the mechanical characteristics of the pile under horizontal loads. The m-method is an effective tool to solve the horizontal load pile because it is clear in concept, conforms to the actual situation and the calculation result is more safe. Because of the existence of the bearing plate, the diameter of the pile body becomes larger, which leads to an increase in flexural rigidity and calculated width. Under the action of horizontal loads, the bearing plate will rotate together with the pile body, resulting in the generation of resisting moments, which makes the loading situation of the pile at the bearing plate complicated. [PERSON] [29] gave the analytical solution of the squeezed branch pile according to [PERSON]'s method [30], which is based on the analytical solution of straight piles under horizontal loads. Furthermore, she also combined it with the deformation coordination conditions at the node, initial conditions, and boundary conditions to obtain the numerical solution.