# Influence of Suction on Capacity of Pile Foundation
## 2 Theoretical Background  
The load-carrying capacity for single piles is commonly calculated using conventional \(\alpha\), \(\beta\), and \(\lambda\) methods. In a study conducted by [PERSON] and [PERSON] [7], the contribution of matric suction to single-model pile foundations was investigated. Based on experimental data obtained in a laboratory, modified \(\alpha\), \(\beta\), and \(\lambda\) methods were developed for estimating the shaft capacity of single piles in unsaturated soils. Because in most cases, particularly in arid and semi-arid regions, the groundwater table is located at a great depth, consideration of the soil's unsaturated condition becomes no less important in soil mechanics [7]. The empirical methods of the determination of soil-structure interaction are proposed to estimate skin friction, \(f_{s}\) (Equation (1)), as a function of vertical effective stress, \(\sigma_{v}^{\prime}\); effective friction angle, \(\phi^{\prime}\); undrained shear strength, \(c_{u}\):  
\[f_{s}=f\big{(}\sigma_{v}^{\prime},\ \phi^{\prime},\ c_{u}\big{)} \tag{1}\]  
The \(\beta\) method [24]  
If a pile is loaded at a relatively slow rate, the formula for skin friction is as follows:  
\[f_{s}=c^{\prime}+K_{0}\sigma_{v}^{\prime}tan\phi^{\prime} \tag{2}\]  
where \(K_{0}\)--mean lateral earth coefficient at rest; \(c^{\prime}\)--effective cohesion; \(\phi^{\prime}\)--internal friction angle of soil; \(\sigma_{v}^{\prime}\)--effective vertical stress along the pile length. According to the \(\beta\) method, a new coefficient, \(\beta\), is introduced, incorporating Equation (2) in Equation (3):  
\[Q_{f}=f_{s}A_{s}=\beta\sigma_{v}^{\prime}\pi dL \tag{3}\]  
where \(\beta=K_{0}tan\phi^{\prime}\)--Burland-Bjerrum coefficient [7]; \(A_{s}\)--the surface area of the pile; \(L\)--length of a pile; \(d\)--diameter of a pile. The \(\alpha\) method [25]  
Undrained loading conditions can be assumed when a pile is loaded at a relatively fast rate in saturated fine-grained soils being dependent on the undrained shear strength, \(c_{u}\)[7]. Thus, the \(f_{s}\) can be expressed as Equation (4). \[f_{s}=\alpha c_{u} \tag{4}\]  
where \(\alpha\)--adhesion factor between soil and pile. Hence, the ultimate shaft capacity for cylindrical piles using the \(\alpha\) method can be calculated as:  
\[Q_{f}=f_{s}A_{s}=\alpha c_{u}\pi dL \tag{5}\]  
The adhesion factor, \(\alpha\), can be found in correlation charts or, alternatively, using Equation (6). \[\alpha=0.5\psi^{-0.5}\ if\ \psi\leq 1 \tag{6}\]  
\[\alpha=0.5\psi^{-0.25}\ if\ \psi>1 \tag{7}\]  
where  
\[\psi=\frac{c_{u}}{\sigma_{v}^{\prime}}\]  
The \(\lambda\) method [24]  
\[Q_{f}=\lambda\big{(}\sigma_{v}^{\prime}+2c_{u}\big{)}\pi dL \tag{8}\]  
where \(\lambda\)--frictional capacity coefficient. The modified \(\beta\) method [25]  
Previous research has proposed a model to predict the effect of matric suction on shear strength, \(\tau_{us}\). Thus, the contribution of matric suction on ultimate shaft capacity was estimated as follows:  
\[Q_{(u_{a}-u_{w})}=\tau_{us}A_{s}=\Big{[}(u_{a}-u_{w})S^{k}tan\delta^{\prime} \Big{]}\pi dL \tag{9}\]  
where \((u_{a}-u_{w})\)--metric suction; \(S\)--degree of saturation; \(k\)--fitting parameter used for shear strength; \(\delta^{\prime}\)--effective interface friction angle. A more generalized expression of ultimate shaft capacity for unsaturated soils is given below:  
\[f_{s}=c^{\prime}+K_{0}\sigma_{v}^{\prime}tan\phi^{\prime} \tag{2}\]0The modified \(\alpha\) method (Equation (10)) was derived to represent undrained shear strength with respect to matric suction [7]. \[f_{s}=c^{\prime}+K_{0}\sigma_{v}^{\prime}tan\phi^{\prime} \tag{2}\]1  
where \(c_{u(us)}\)--undrained shear strength under unsaturated conditions; \(c_{u(sat)}\)--undrained shear strength under saturated conditions; \(P_{a}\)--atmospheric pressure; \(v\) and \(\mu\)--fitting parameters.