## 2 Proposed \"Hybrid BEM-p-y Curve\" Method  
### Pile Modelling  
The proposed method was developed to capture the response of a single pile subjected to horizontal load. It consists of a hybrid BEM-p-y curve approach. The analysis is performed using a non-linear incremental tangent method. The pile is modeled as a vertical strip, geometrically defined by the outer diameter \(D\) and length \(L\) of the actual pile. In the proposed method the pile is a vertical floating pile, which cannot settle. The pile is discretized in 60 blocks of variable length with depth. With this kind of discretization, it is possible to minimize the calculation-time. The definition of the discretization criterion was suggested by [PERSON] [22] based on results obtained with a parametric study on a free-to-rotate single pile embedded in a homogeneous elastic half-space. Adopting a homogeneous discretization with depth when the number of blocks gradually increases, the solution becomes increasingly accurate, reaching an \(n^{*}\) value beyond which the solution remains unchanged. The parametric study in the work of [PERSON] [22] showed that by decreasing the relative stiffness \(K=E_{p}/E_{s}\) the dependence of the response on the number of blocks starts to increase. Using an \(n\) value equal to 60 leads to an error that varies as a function of the relative stiffness \(K=E_{p}/E_{s}\): for \(K=10^{1}\), \(10^{2}\), \(10^{3}\), \(10^{4}\) the error is 20%, 10%, 5%, 2% respectively, in the evaluation of the pile-headdeflection and the maximum bending moment. Since the displacements are mainly localized at a depth corresponding to 10-15 pile diameters, a suitable discretization is when the element height is sufficiently small for depths close to the ground surface. On the basis of these observations, [PERSON] [22] introduced the discretization shown in Figure 1, and in this case for \(K=10^{1}\), \(10^{2}\), \(10^{3}\), \(10^{4}\) the error was \(8\%\), \(7\%\), \(4\%\), \(2\%\), respectively, in the prediction of pile-head deflection. For the maximum moment along the pile shaft, the error was \(8\%\), \(5\%\), \(2.5\%\), \(1\%\), respectively. The resulting discretization is shown as follows (Figure 1):  
* 20 blocks with a thickness \(\Delta=D/8\), starting from the ground level up to a depth of \(2.5D\);
* 10 blocks with a thickness \(\Delta=D/4\), starting from a depth of \(2.5D\) up to a depth of \(5D\);
* 10 blocks with a thickness \(\Delta=D/2\), starting from a depth of \(5D\) up to a depth of \(10D\);
* 10 blocks with a thickness \(\Delta=D\), starting from a depth of \(10D\) up to a depth of \(20D\);
* 10 blocks with a thickness \(\Delta=(\mathrm{L}-20D)/10\), starting from a depth of \(20D\) up to the pile base depth. The pile flexibility matrix, in case of linear elastic behaviour, is obtained using the elastic beam theory, and each coefficient of this matrix can be expressed using Equation (1) (Figure 2). \[ \begin{array}{ll}a_{ij}=\frac{z_{i}^{3}}{3E_{p}I_{p}}+\frac{z_{i}^{2}\left( z_{i}-z_{i}\right)}{2E_{p}I_{p}}&\text{if }z_{i}<z_{j}\\ a_{ij}=\frac{z_{i}^{3}}{3E_{p}I_{p}}+\frac{z_{i}^{2}\left(z_{i}-z_{j}\right)}{2E _{p}I_{p}}&\text{if }z_{i}\geq z_{j}\end{array}  \tag{1}\]  
Figure 1: Pile—discretization. Figure 2: Pile flexibility matrix using the elastic beam theory (auxiliary restraint approach). In this way, the horizontal displacement of each pile-block assumes the expression as in Equation (2). \[y_{i}=-\sum_{j=1}^{n}a_{ij}P_{j}+y_{0}+\theta_{0}z_{i} \tag{2}\]  
in which \(P_{j}\) represents the load applied at the generic pile-block \(j\) (located at depth \(z_{j}\)), and \(y_{0}\) and \(\theta_{0}\) are the unknown displacement and rotation at the pile-head. Obviously if the pile-head is fixed, the rotation becomes a known term. Each pile-point displacement is a function of \(n+2\) (or \(n+1\), for fixed condition) unknowns, \(n\) pile-soil interface pressures, \(y_{0}\) and \(\theta_{0}\). The proposed method analyses both steel-pipe and reinforced concrete piles.