## 3 Materials, Methods, and Equipment
### Classical Bearing Capacity Methods
#### 3.6.7 [PERSON]'s Method (1977)  
[PERSON]'s method (1977) provides a comprehensive approach to analyzing the bearing capacity of circular piles in sand, incorporating soil compressibility and the critical depth concept. The method determines bearing capacity factors by considering both soil friction angle and rigidity index--the ratio of soil shear modulus to the initial mean normal stress times the strength parameter. This inclusion of soil stiffness distinguishes [PERSON]'s method from earlier bearing capacity theories [36]. The approach accounts for the critical depth phenomenon, where the bearing capacity reaches a limiting value rather than increasing linearly with depth beyond a certain point. This critical depth typically ranges between 10 and 20 pile diameters, depending on sand relative density. [PERSON]'s method also differentiates between installation techniques, recognizing that driven piles achieve higher bearing capacities than bored piles due to the surrounding soil's densification during installation [37]. Additionally, the method addresses scale effects in pile foundations, acknowledging that unit base resistance does not increase indefinitely with pile size--a particularly important consideration for large-diameter piles where conventional theories might overestimate capacity [38]. The base resistance is calculated using Equation (16), providing a more nuanced assessment of pile performance in sandy conditions.  
\[Q_{b}=A\times q=A(\overline{\sigma}_{o}N_{q}^{*}) \tag{16}\]  
where \(\overline{\sigma}_{o}=\) mean effective normal stress at the pile tip.  
The mean effective stress is calculated as shown in Equation (17).  
\[\overline{\sigma}_{o}=\bigg{(}\frac{1+2K}{3}\bigg{)}q^{{}^{\prime}} \tag{17}\]  
where \(q^{{}^{\prime}}=\) effective overburden pressure at the pile tip.  
Vesic related the bearing capacity factor to the reduced-rigidity index, which is calculated using Equation (18).  
\[N_{q}^{*}=e^{\pi\tan\phi^{\prime}}\bigg{(}\frac{3}{3-\sin\phi^{\prime}}\bigg{)} e^{[(4\sin\phi^{\prime})/(3-\sin\phi^{\prime})]\ln I_{tr}} \tag{18}\]  
The reduced-rigidity index accounts for soil compressibility:  
\[I_{tr}=\frac{I_{r}}{1+I_{r}\Delta}\]  
where \(I_{r}=\) rigidity index and \(\Delta=\) average volumetric strain in the plastic zone  
\[I_{r}=\frac{E_{s}}{2(1+\mu_{s})(c^{\prime}+q^{\prime}\tan\phi^{\prime})}\]  
where  
\(E_{s}\)= soil elastic modulus;  
\(\mu_{s}=\) [PERSON]'s ratio.  
Vesic recognized a critical depth beyond which effective stress remains approximately constant:  
\[L_{c}\approx 15-20D\]The selection of volumetric strain (\(\Delta\)) values in Vesic's reduced-rigidity index calculation correlates systematically with sand relative density [39; 40; 41]. Very loose sands (10% relative density) exhibit volumetric strains of approximately 0.050, reflecting significant volume change potential due to minimal particle interlocking. Loose sands (35% relative density) demonstrate a moderately high volumetric strain of 0.035, as incipient interlocking begins to limit compression. Medium-density sands (50% relative density) represent a transitional state with volumetric strain values of 0.020, balancing contractive and dilative tendencies. Dense sands (75% relative density) show a decreased volumetric strain of approximately 0.012 due to significant particle interlocking restricting rearrangement, although some compression remains possible under high confining pressures. Very dense sands (95% relative density) exhibit a minimal volumetric strain of approximately 0.006, resulting from tight particle packing and strong dilative tendencies. These correlations derive from theoretical soil mechanical principles, experimental observations, and field performance data, demonstrating the inverse relationship between relative density and compressibility under complex loading conditions. The shaft resistance approach remains fundamentally similar to [PERSON]'s method.