## 2 Proposed \"Hybrid BEM-p-y Curve\" Method
### Pile Modelling
For the analysis of steel piles, the flexural rigidity \(E_{p}I_{p}\) is assumed to be constant (which means hypothesizing a linear-elastic behaviour of the section until the ultimate bending moment occurs). For reinforced concrete sections, the development of cracks, even at low values of the bending moment, requires a different modeling for the pile response. For this material, in fact, it is also necessary to know the mechanical properties of both the concrete and steel, the number of longitudinal bars, and the spacing of the transverse reinforcement. These data provide the basis for the moment-curvature relationship (for each axial load value) for the reinforced concrete cross-section. The \"moment-curvature-axial load\" relationship is obtained by imposing the equilibrium equations to the translation and rotation at the geometric center of gravity of the section, varying the curvature and the deformation at the most compressed fiber in the section. This model has the additional feature of taking the influence of tension stiffening into account, as presented in [PERSON] et al. (2013). The main assumptions of the tension stiffening model are:  
* the conservation of planar sections;
* concrete tension strength equal to zero for cracked sections;
* perfect bonding between steel bars and the surrounding concrete;
* the constitutive model proposed by [PERSON] et al. (2014) for confined concrete in compression;
* the constitutive model proposed by [PERSON] (2015) for unconfined concrete in compression;
* the constitutive model proposed by CEB-FIP (2016) for concrete in tension;
* a simple strain-hardening model or a bilinear model for steel reinforcement;
* a bond-slip law as suggested by [PERSON] (2017);
* the tension stiffening is considered using a variable elastic modulus for the concrete in tension along a pile-block between two consecutive cracks;
* the crack spacing can be estimated using the expression suggested by the CEB-FIP;
* no other secondary cracks develop between two consecutive cracks. Using these assumptions, the \"average moment-curvature\" relationship for a representative pile-section can be estimated as the weighted average of the moment-curvature relationships computed in the middle, at a quarter, at three quarters and at the cracked section of a block between two cracks. Details on this model are presented in [PERSON] et al. (2013), however this model represents an extension to the circular section of another model, which considers the tension stiffening influence for rectangular reinforced concrete sections, developed by [PERSON] et al. (2018). Once the average moment-curvature relationship has been obtained, the coefficients of the flexibility matrix need to be defined using Equation (3) for the reinforced concrete pile, which is modeled as a beam with a variable flexural rigidity, \(E_{p}I_{p}\), along the pile shaft. In Equation (3), the variation of both \(E_{p}\) and \(I_{p}\) along the shaft is fully considered by changing \(I_{p}\) of the section, while \(E_{p}\) is kept constant. Consequently, in an incremental analysis, the pile flexibility matrix needs to be updated at each load increment.