## 3 Materials, Methods, and Equipment
### Direct Shear Test (DST)  
The direct shear test determined sand shear strength parameters using a standard 6 cm \(\times\) 6 cm \(\times\) 2 cm apparatus. Prior to testing, the equipment was cleaned, inspected, and assembled with alignment screws, and we applied silicon grease to inner walls to minimize friction. Sand specimens were prepared at 10%, 35%, 50%, 75%, and 95% relative densities, leveled with a straight edge, and topped with a cap before transfer to the apparatus. Tests employed three normal stresses (54.5 kPa, 109 kPa, and 163.5 kPa) for each density to establish shear strength envelopes. Consolidation preceded shearing, with testing commencing after vertical displacement stabilization. Figure 5 illustrates the apparatus for measuring soil strength parameters through controlled horizontal displacement under normal stress. The strain-controlled procedure (0.5 mm/min) ensured proper drainage while documenting shear stress, horizontal displacement, and vertical displacement until reaching peak stress or constant stress at approximately 10% of specimen length. Fifteen tests were conducted across all density-stress combinations, with moisture content determined after each test. Analysis involved plotting peak shear stress against normal stress to establish Mohr-Coulomb failure envelopes, yielding internal friction angles (\(\varphi\)) that typically increased with relative density due to enhanced particle interlocking, providing critical insights for geotechnical applications. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Relative Density, D\({}_{\text{r}}\) (\%)** & **10** & **35** & **50** & **75** & **95** \\ \hline Sand Description & Very Loose & Loose & Medium & Dense & Very Dense \\ \hline Material Model & Mohr-Coulomb & Mohr-Coulomb & Mohr–Coulomb & Mohr–Coulomb & Mohr–Coulomb & Mohr–Coulomb \\ \hline Drainage Type & Drained & Drained & Drained & Drained & Drained \\ \hline Poisson’s Ratio, \(\
u\) & 0.3 \(\pm\) 0.02 & 0.3 \(\pm\) 0.02 & 0.3 \(\pm\) 0.02 & 0.3 \(\pm\) 0.02 & 0.3 \(\pm\) 0.02 \\ \hline Modulus of Elasticity, E (kN/m\({}^{2}\)) & 12,000 \(\pm\) 600 & 26,500 \(\pm\) 1325 & 32,300 \(\pm\) 1615 & 58,800 \(\pm\) 2940 & 70,100 \(\pm\) 3505 \\ \hline Dry Unit Weight, \(\gamma_{\text{d}}\) (kN/m\({}^{3}\)) & 14.1 \(\pm\) 0.3 & 15.2 \(\pm\) 0.3 & 15.9 \(\pm\) 0.3 & 17.0 \(\pm\) 0.3 & 18.0 \(\pm\) 0.4 \\ \hline Saturated Unit Weight, \(\gamma_{\text{sat}}\) (kN/m\({}^{3}\)) & 22.60 \(\pm\) 0.45 & 22.60 \(\pm\) 0.45 & 22.60 \(\pm\) 0.45 & 22.60 \(\pm\) 0.45 & 22.60 \(\pm\) 0.45 \\ \hline Internal Friction Angle, \(\phi\) (\({}^{\circ}\)) & 28 \(\pm\) 0.5 & 30.9 \(\pm\) 0.6 & 33.7 \(\pm\) 0.6 & 36.6 \(\pm\) 0.7 & 40 \(\pm\) 0.8 \\ \hline Dilatancy Angle, \(\psi\) (\({}^{\circ}\)) & 0 \(\pm\) 0.2 & 5.9 \(\pm\) 0.3 & 9.4 \(\pm\) 0.4 & 15.3 \(\pm\) 0.5 & 20 \(\pm\) 0.6 \\ \hline Cohesion, c (kN/m\({}^{2}\)) & 0 \(\pm\) 0.05 kN/m\({}^{2}\) & 0.115 \(\pm\) 0.01 & 0.15 \(\pm\) 0.01 & 0.445 \(\pm\) 0.03 & 0.706 \(\pm\) 0.04 \\ \hline Interaction Factor, R\({}_{\text{int}}\) & 0.8 \(\pm\) 0.05 & 0.8 \(\pm\) 0.05 & 0.8 \(\pm\) 0.05 & 0.8 \(\pm\) 0.05 & 0.8 \(\pm\) 0.05 \\ \hline \hline \end{tabular}  
* Notes: 1. Uncertainties in E and c reflect \(\pm\)5% variability in PLAXIS 3D calibration. 2. Dilatancy angles derived from triaxial test correlations (ASTM D7181).  
\end{table} 
Table 6: Parametric analysis of sand properties for numerical simulation of hybrid pile analysis.