## 1 Introduction  
Unlike shallow foundations, the load transfer mechanism through the pile to the soil layers is more sophisticated [1]. Generally, the end (point) bearing and skin friction are the two mechanisms that transfer the normal loads on piles into the soil. In some cases, such as weak soil deposits, the pile foundation system may be the only reliable system for transmitting the structural load to the ground. When there is no rock or stiff soil layer at a reasonable depth, the pile's end (point) bearing capacity is comparatively small and much of the resistance is caused by lateral friction or adhesion between the materials of the pile and the surrounding soil around the pile shaft. Estimating the frictional capacity of the pile shaft is difficult and has been a matter of long discussions because there is great uncertainty about parameters related to soil and soil-pile interaction. Usually, there are two common methods in practice for prediction the shaft resistance of axially loaded piles: (1) estimating the pile-soil load transfer mechanism with empirical relations [2; 3; 4; 5; 6] and (2) using finite element method analysis [7; 8; 9; 10; 11; 12; 13]. In practice, the ultimate pile shaft friction resistance in cohesionless soil can be calculated from the following equation, which was proposed by [PERSON] [14]. \[Q_{f}=f_{\rm s}.\rm P.L=K.\sigma^{\prime}_{v}.\;tan\delta^{\prime}.\rm P.L \tag{1}\]  
where \(f_{\rm s}\) is unit skin friction, K is lateral earth pressure coefficient, \(\sigma^{\prime}_{v}\) is an average vertical effective overburden pressure, \(\delta^{\prime}\) is effective skin friction angle between soil and pile materials, \(\rm P\) is the perimeter of the pile, and \(\rm L\) is the embedded length of the pile. The (K) value is considered the most difficult factor to estimate in Equation (1) [15]. Several researchers have attempted to estimate the K value from a retrospective analysis of field pile load tests, small-scale model pile load tests in the laboratory, and finite element analysis. Due to the large number of suggested K values equations, the estimated pile shaft resistance may differ by several hundred percent. [PERSON] and [PERSON] [16] reported that the K value is a function of the internal friction angle of soil. The American Petroleum Institute's RP2A code [17] recommended that the K value of 0.8 to 1 be used for estimating shaft friction resistance of driven piles with open and closed ends in sand. However, it does not take into account whether the pile is in compression or tension. In addition, some studies have uncovered the restrictions of the API code design instructions for tension piles, and new design methods have been proposed [18; 19; 20]. [PERSON] [5] stated that the value of K ranges widely between Rankine's active earth pressure coefficient (\(\rm K_{a}\)) and passive earth pressure coefficient (\(\rm K_{p}\)), in some cases, maybe greater than \(\rm K_{p}\). He also asserted that the maximum K value is a function of the relative density of sand, pile diameter, and initial stress condition. [PERSON] and [PERSON] [21], using finite element methods, indicated that the K value is approximately equal to earth pressure coefficient at rest (\(\rm K_{0}\)) for non-displacement piles. When estimating the friction capacity of piles, the interface friction angle (\(\delta\)) between the soil and the pile materials is crucial. The interface friction between sand and other construction materials has been the subject of several investigations in the past. To this end, In the literature, a wide range of geotechnical laboratory experiments have been employed, such as a direct shear test, large-scale shear box test, and ring torsion apparatus.