## 3 Materials, Methods, and Equipment
### Classical Bearing Capacity Methods
#### 3.6.4. [PERSON] (1970) Method  
The [PERSON] method (1970) provides a comprehensive framework for analyzing pile performance in sandy soils, where load-bearing capacity primarily stems from frictional resistance. The approach integrates empirical data with theoretical principles to address the complex interplay between soil properties, pile geometry, and loading conditions [30]. Key to this method are the bearing capacity factors \(N_{q}\) and \(N_{\gamma}\), which represent contributions from surcharge and skin friction. These factors vary with the angle of internal friction (\(\phi\)) and soil density. In [PERSON]'s formulation, \(N_{q}\) and \(N_{\gamma}\) increase with higher friction angles, reflecting the enhancement of soil resistance as \(\phi\) rises. This relationship emphasizes the critical importance of accurately determining soil frictional properties [31]. [PERSON]'s method distinguishes itself by incorporating additional refinement factors: shape factors adjust for the pile's cross-sectional geometry; depth factors account for increased resistance from deeper embedment; and inclination factors consider the effects of pile tilt on overall capacity. When applying this method, engineers must first determine the soil's angle of internal friction and sand density, and then use [PERSON]'s equations or graphs to establish appropriate \(N_{q}\) and \(N_{\gamma}\) values. For cohesionless soils (\(c=0\)), the end-bearing component is calculated using Equation (9), providing a refined estimate of pile bearing capacity in sandy conditions.  
\[Q_{b}=A(q^{\prime}N_{q^{\prime}}s_{q}d_{i}q_{b}+0.5\gamma^{\prime}B^{\prime}N _{\gamma}s_{q}d_{\gamma}i_{\gamma}b_{\gamma}) \tag{9}\]  
where  
\(A=\) pile base area (\(\pi B^{2}/4\) for circular piles);  
\(q^{\prime}=\) effective overburden pressure at the pile toe;  
\(B^{\prime}=\) pile diameter.  
[PERSON] proposed Equations (10) and (11) to assess bearing capacity factors.  
\[N_{q}=e^{\pi tan\phi^{\prime}}tan^{2}(45^{\circ}+\frac{\phi^{\prime}}{2}) \tag{10}\]  
\[N_{\gamma}=1.5\big{(}N_{q}-1\big{)}tan\phi^{\prime} \tag{11}\]  
Shape factors for circular piles are expressed as follows:  
\[s_{q}=1+\frac{B^{\prime}}{L^{\prime}}\sin\phi^{\prime}\]\[s_{\gamma}=1-0.4\frac{B^{\prime}}{L^{\prime}}\geq 0.6\]  
Depth factors for deep foundations (\(D/B>1\)):  
\[d_{q}=1+2\tan\phi^{\prime}(1-\sin\phi^{\prime})^{2}\tan^{-1}(\frac{D}{B})\]  
\[d_{\gamma}=1.0\]  
Inclination factors for vertical loading:  
\[i_{q}=i_{\gamma}=1.0\]  
Base factors for horizontal ground surface:  
\[b_{q}=b_{\gamma}=1.0\]  
[PERSON] (1970) maintained the complete theoretical bearing capacity equation structure, even for deep foundations, including both the overburden term (\(q^{\prime}N_{q}\)) and the self-weight term (\(0.5\;\gamma^{\prime}B^{\prime}N_{\gamma}\)). This follows his general approach to foundation design, where he applied modification factors to the complete Terzaghi equation. In contrast, [PERSON] (1976) simplified his pile capacity equation for deep foundations. He recognized that the self-weight term becomes negligible at significant depths compared to the overburden pressure term. [PERSON] empirically determined that the base capacity could be only adequately represented for typical pile depths by applying the modified \(N_{q}{}^{*}\) factor to the overburden pressure. This difference reflects their distinct approaches. [PERSON] maintained theoretical completeness with comprehensive modification factors, and [PERSON] developed a more practice-oriented, simplified approach based on field observations. This is why the \(N_{\gamma}\) term appeared in the [PERSON] formulation but was intentionally omitted by [PERSON] for deep foundations. [PERSON]'s shaft resistance formula for total shaft resistance along the pile is the same as Equation (5). This formulation accounts for the varying effective stress with depth and allows for different soil parameters in each segment.