# Influence of Degree of Saturation on Soil-Pile Interactions for Piles in Expansive Soils
## 2 Total Uplift Force on Pile in Expansive Soils  
In the effective stress method, [PERSON] and [PERSON] (1980) [10] proposed a simple equation for pile design based on the adhesion between the soil and the pile. The equation calculates the total tension force, \(T\), used in pile design by integrating the adhesion, \(\tau_{a}\), along the pile shaft:  
\[T=\int_{0}^{h}\pi d\tau_{a}dz \tag{1}\]  
where \(\tau_{a}\) represents the adhesion between the soil and the pile. To evaluate the load that would occur if adhesion were mobilized along the entire pile shaft (\(P_{FS}\)), the equation integrates the adhesion along the length of the pile:  
\[P_{FS}=\int_{0}^{L}\pi d\tau_{a}dz \tag{2}\]For saturated soil under drained conditions, the adhesion, \(\tau_{a}\), can be determined using the following equation, which involves effective cohesion, \(C_{a}^{\prime}\), and the normal effective stress acting on the pile shaft, \(K_{s}\sigma_{v}^{\prime}\), along with the friction angle, \(\phi_{a}^{\prime}\):  
\[\tau_{a}=C_{a}^{\prime}+K_{s}\sigma_{v}^{\prime}lam\phi_{a}^{\prime} \tag{3}\]  
Thus,  
\[P_{FS}=\pi\int_{0}^{L}d\big{(}C_{a}^{\prime}+K_{s}\sigma_{v}^{\prime}lam\phi_{ a}^{\prime}\big{)}dz \tag{4}\]  
However, for unsaturated conditions, the shear strength of the soil is defined by the constitutive surface of unsaturated soil, which depends on matric suction (\(u_{af}-u_{mf}\)) and net normal stress (\(\sigma_{mf}-\sigma_{af}\)). The widely used [PERSON]'s (1993) [20] equation expresses the shear strength of unsaturated soil using the effective cohesion, \(C^{\prime}\), net normal stress, \(\sigma_{nf}\), matric suction, (\(u_{af}-u_{mf}\)), and the friction angle contributed by soil suction, \(\phi^{\rm b}\):  
\[\tau_{f}=C^{\prime}+(\sigma_{nf}-u_{af})tan\phi^{\prime}+(u_{af}-u_{mf})tan \phi^{b} \tag{5}\]  
[PERSON] and [PERSON] (2009) [21] proposed an equation to determine the shear strength of non-expansive unsaturated soil, which considers the interface friction angle, \(\phi^{\prime}\), and volumetric water content, \(\theta\):  
\[\tau_{f}=C_{a}^{\prime}+(\sigma_{nf}-u_{af})tan\phi^{\prime}+(u_{af}-u_{mf})tan \bigg{[}\phi^{\prime}\bigg{(}\frac{\theta-\theta_{r}}{\theta_{s}-\theta_{r}} \bigg{)}\bigg{]} \tag{6}\]  
where  
\(C_{a}^{\prime}\) is the effective cohesion of soil;  
\((\sigma_{nf}-\sigma_{af})\) is the net normal stress;  
\(\phi^{\prime}\) is the interface friction angle between soil and pile;  
\((u_{af}-u_{mf})\) is the matric suction of soil;  
\(\theta\) is the current volumetric water content;  
\(\theta_{r}\) is the residual volumetric water content (in this case, \(\theta_{r}=0\));  
\(\theta_{s}\) is the saturated volumetric water content. The migration of water into expansive soils involves complex changes in soil stress state parameters [22; 23]. Additionally, [PERSON] et al. (2015) [5] emphasized that the infiltration of water into expansive soil leads to the development of lateral swelling pressure, thereby increasing the lateral earth pressure in the horizontal direction. The normal stresses acting on the pile shaft were found to be influenced by two main components: the earth pressure exerted by the non-expansive soil, denoted as (\(K\sigma_{v}^{\prime}\)), and the lateral constant-volume swelling pressure, \(\big{(}\sigma_{cv}^{\prime\prime}\big{)}_{h}\). \[\tau_{f}=C_{a}^{\prime}+\Big{[}K\sigma_{v}^{\prime}+\big{(}\sigma_{cv}^{\prime \prime}\big{)}_{h}\big{]}tan\phi^{\prime}+(u_{af}-u_{af})tan\bigg{[}\phi^{ \prime}\bigg{(}\frac{\theta-\theta_{r}}{\theta_{s}-\theta_{r}}\bigg{)}\bigg{]} \tag{7}\]  
The lateral constant-volume swelling pressure, \(\big{(}\sigma_{cv}^{\prime\prime}\big{)}_{h}\), is not commonly measured in practicing geotechnical engineering. Instead, the vertical constant-volume swelling pressure, \(\big{(}\sigma_{cv}^{\prime\prime}\big{)}_{v}\), can be measured using an oedometer machine or determined from the consolidation-swell swelling pressure, \(\sigma_{cs}^{\prime\prime}\)[5; 24; 25].