## 2 Proposed \"Hybrid BEM-p-y Curve\" Method
### Influence of Suction on Pile Lateral Response  
Suction is an important aspect in pile foundation subjected to lateral loads because the response of this foundation system is mainly affected by the shallower soil layers. The proposed method uses the \"MK-Model\" (Modified-Kovacs Model) described by [PERSON] et al. [35]. This model makes use of a parameter defined as the equivalent capillary rise \(h_{\text{c0}}\) in the porous medium. The role of this parameter is the same as the average capillary rise in the original model developed by [PERSON] and is calculated using the expression for the rise of water in a capillary tube (\(h_{\text{c}}\)) with a diameter \(d\).  
For the sake of convenience, the expression to estimate the equivalent capillary rise in granular soils (Equation (8)) and in cohesive/plastic soils is reported below (Equation (9)).  
\[h_{\text{c0}}(cm)=\frac{0.75}{e\cdot D_{10}[1+1.17\cdot\log C_{\text{U}}]} \tag{8}\]  
where \(D_{10}\) (in cm) is the diameter corresponding to 10% passing on the grain-size distribution curve, C\({}_{\text{U}}\) is the coefficient of uniformity (=\(D_{60}/D_{10}\)), and \(e\) is the void ratio.  
\[h_{\text{c0}}=\frac{0.15\cdot\rho_{s}}{e}w_{L}^{1.45} \tag{9}\]  
where \(w_{L}\) is the liquid limit, and \(\rho_{s}\) is the solid grain density (kg/m\({}^{3}\)).  
The MK-Model uses the equivalent capillary rise as a reference parameter to define the relationship between the degree of saturation \(S_{r}\) (or volumetric water content \(\theta\)) and the matric suction \(\Psi\). The model considers that water is held by capillary forces, responsible for capillary saturation \(S_{c}\), and by adhesive forces, causing saturation by adhesion \(S_{a}\). The \(S_{c}\) component is more important at relatively low suction values, while the \(S_{a}\) component becomes dominant at a higher suction when most capillary water has been withdrawn. The relationship proposed in the MK-Model is written as in Equation (10) for the degree of saturation:  
\[S_{r}=\frac{\theta}{n}=S_{c}+S_{a}^{*}\cdot(1-S_{c}) \tag{10}\]  
In this equation, to ensure that this component does not exceed unity at low suction a truncated value of the adhesion component \(S_{a}{}^{*}\) is introduced in place of \(S_{a}\) used in the original model. The contribution of the capillary and the adhesion components to the total degree of saturation is defined as a function of \(h_{\text{c0}}\) and \(\Psi\) using the equations reported in [PERSON] et al. [35].  
Once the suction value \(\Psi\) has been obtained, the relationship (Equation (11)) proposed by [PERSON] [36] can be used to estimate the effective stress state.  
\[\sigma^{\prime}{}_{\text{c0}}=(\sigma_{\text{c0}}-u_{a})+\chi\cdot(u_{a}-u_{w}) \tag{11}\]  
where \(u_{a}\) = air pressure; \(u_{w}\) = water pressure; \(\chi\) = effective stress parameter that is expressed as a function of the degree of saturation (\(S_{r}\)); and suction \(\Psi\) is equal to \(u_{a}-u_{w}\).  
In order to use Equation (11) correctly, the value of \(\chi\) should be assessed on the basis of the actual degree of saturation of the soil layers above the water table. The relationship between \(\chi\) and \(S_{r}\) is available in the literature for different soil types (for example in: [PERSON] [37], [PERSON] [38] and [PERSON] et al. [39]).  
Implementing the \"MK-Model\" in the \"hybrid BEM-py curve\" method takes suction into account and increases the effective stress state of the upper soil layers. This thus increases both the stiffness and the resistance of the non-linear springs located close to the ground surface, which are expressed as a function of the soil stress state.