## 1 Introduction  
The response to the horizontal loading of pile foundations has been the focus of many studies. However, as noted by [PERSON] and [PERSON] [1] and [PERSON] and [PERSON] [2], additional tests are needed to better understand the interactions between the soil, piles and superstructures. For the single pile case, it is well known that the key factors that influence the response include the restraint condition at the pile-head and the pile-soil relative stiffness. The installation technique seems to be less important, probably because of the large volume of soil involved in the response mechanism with respect to that related to vertical loads. The most common analysis methods are continuum-based or Winkler-based approaches (non-linear transfer curves). Methods based on p-y curves are limited to the use of a subgrade soil reaction modulus, which does not represent an actual property of the soil, and the soil is schematized with a series of independent springs. Some of the most common p-y curves (and implemented in software such as LPILE [3]) include those recommended by the American Petroleum Institute. These were obtained from experimental tests on steel pipe piles with an outer diameter of about 30 cm, and which are not affected by the non-linearity of the pile material. More recently new p-y curves were proposed by [PERSON] et al. [5] developed a nonlinear analysis method for laterally loaded single reinforced concrete piles based on the beam-on-nonlinear-Winkler-foundation approach. Continuum-based approaches are usually solved with boundary element methods (BEM) and finite element methods (FEM). Despite their great potential in geotechnical engineering applications, FEM suffers from the complexity of the domain discretization phase and the difficulty in input parameters definition. Furthermore, as evidenced by [PERSON] et al. [6], FEM results are affected by the pile modeling. Often, therefore, they are used as a benchmark to validate other simplified approaches or as a tool to determine p-y curves for comparison with those obtained experimentally in situ [7; 8]. In the work carried out by [PERSON] et al. [9], instead, the influence of both the initial lateral stress state and interface parameters was investigated in depth by finite element method analysis. The results shown in [9] highlight the importance of selecting the most appropriate pile-soil interface parameters. BEM approaches, however, describe the soil as a homogeneous elastic half-space, characterized by a Young modulus and Poisson ratio, and enable pile-soil-pile interactions to be directly evaluated, and group effects can be considered. The computer codes that make use of this approach include DEFPIG [10], PIGLET [11], and PGROUPN [12]. With PGROUPN, the non-linear soil behaviour is modeled in an approximate way using a hyperbolic stress-strain law. These methods provide a complete solution at the interfaces of the problem domain but entail numerical approximations when the analysis involves heterogeneous soils (different layers with different stiffnesses). To evaluate the displacement induced at one point of the subsoil by a load acting in another point, the elastic Mindlin solution is generally used [13]. The most important works and parametric studies using BEM approaches have been carried out by [14; 15; 16; 17; 18; 19]. To account for the non-linear behaviour of the soil, [PERSON] and [PERSON] [15] proposed an elastic-perfectly plastic model, in which pile-soil relative displacements are allowed once the limit pressure at the interfaces is reached. The same authors, however, explained that such a procedure tends to become less accurate as the number of plasticized boundary elements becomes bigger. Another interesting and more rigorous method is the Strain Wedge Model [20; 21] for the analysis of single piles and pile-groups.