## 3 Materials, Methods, and Equipment
### Classical Bearing Capacity Methods
#### 3.6.6 [PERSON] (1976) Method  
The [PERSON] method (1976) offers a comprehensive approach to this challenge, providing a robust framework that accounts for various factors influencing pile performance. This method addresses the complexities of sandy soils, which primarily derive their load-bearing capacity from frictional resistance [34]. [PERSON]'s methodology is grounded in empirical data and theoretical principles, considering the interplay between soil properties, pile geometry, and loading conditions. The relationship between these factors and the parameters above is meticulously analyzed through graphical (Supplementary Material S1) and analytical methods. In [PERSON]'s formulation, \(N_{q}\) functions increase with the angle of internal friction, reflecting the enhanced resistance the soil offers as \(\phi\) rises. This relationship underscores the importance of understanding the soil's frictional properties when applying [PERSON]'s method to sandy conditions. The base resistance is calculated as shown in Equation (14).  
\[Q_{b}=Aq^{\prime}N_{q}^{*} \tag{14}\]  
where  
\(Q_{b}=\) pile base area;  
\(q^{\prime}=\) effective overburden pressure at pile tip;  
\(N_{q}^{*}=\) [PERSON]'s modified bearing capacity factor.  
[PERSON]'s bearing capacity factor incorporates the surface failure angle, which is calculated as shown in Equation (15).  
\[N_{q}^{*}=\left[\tan\phi+\sqrt{1+\tan^{2}\phi}\right]^{2}e^{2\psi\tan\phi} \tag{15}\]  
where \(\psi=\) angle parameter.  
The angle parameter (\(\psi\)) in [PERSON]'s bearing capacity equation correlates systematically with soil relative density (Dr), reflecting fundamental soil mechanics principles regarding particle arrangement and failure mechanisms. Based on the literature review and field observations, specific values are established for different density conditions [35]. For very loose sands (Dr = 10%), \(\psi=60^{\circ}\) reflects minimal particle interlocking and wide failure zones as particles readily rearrange during shearing. In loose sands (Dr = 35%), \(\psi\) increases to 70\({}^{\circ}\) due to enhanced particle interaction, creating steeper failure surfaces, although they are still relatively wide compared to denser states. Medium-density sands (Dr = 50%) represent a transition point where \(\psi=75^{\circ}\), capturing the balance between particle sliding and interlocking mechanisms with moderate dilation behavior. For dense sands (Dr = 75%), \(\psi\) increases substantially to 85\({}^{\circ}\) due to significant particle interlocking, resulting in more confined failure zones and steeper surfaces. In very dense sands (Dr = 95%), \(\psi\) reaches its practical maximum of 90\({}^{\circ}\), corresponding to the highest degree of particle interlocking and the most direct failure path. This progression aligns with the influence of void ratio and particle arrangement on strength characteristics. These values should be considered approximate, with potential adjustments based on site-specific conditions.  
The total shaft resistance calculation follows [PERSON]'s equations, accounting for soil conditions both above and below the water table and considering effective stress changes due to buoyancy effects.