## 4 Results and Discussions  
### Properties and Behavior of Sand at Various Relative Densities  
Table 4 illustrates the relationship between relative density and sand's physical and mechanical properties, demonstrating how sand's characteristics systematically change across different density states. This comprehensive characterization is crucial for geotechnical engineering applications. The relative density values range from 10% to 95%, corresponding to very loose to very dense sand classifications. This range effectively captures the full spectrum of naturally occurring sand states in geotechnical practice. The void ratio shows an inverse relationship with relative density, decreasing from 0.886 for very loose sand to 0.476 for very dense sand. This reflects the more efficient particle packing achieved at higher densities. The average dry unit weight demonstrates a clear positive correlation with relative density, increasing from 14.1 kN/m\({}^{3}\) for very loose sand to 18 kN/m\({}^{3}\) for very dense sand. This relationship reflects the increased particle contact and reduced void space as the sand is denser, resulting in higher mass per unit volume. The internal friction angle exhibits significant variation across the density range, increasing from 28 degrees in very loose sand to 41 degrees in very dense sand. This substantial change in friction angle has important implications for the sand's shear strength and bearing capacity, with denser states providing considerably higher resistance to deformation and failure.  
Permeability shows a marked decrease with increasing relative density, ranging from 0.01 cm/s for very loose sand to 0.0001 cm/s for very dense sand. This inverse relationship results from the reduction in void spaces and interconnected pore channels as the sand becomes more densely packed, thereby reducing the ease with which water can flow through the soil matrix. These interrelationships among sand properties demonstrate the fundamental importance of relative density in determining the engineering behavior of sandy soils. Also, Table 1 provides essential input parameters for geotechnical design calculations and numerical modeling, enabling engineers to make informed decisions based on site-specific soil conditions. Understanding these relationships is particularly crucial in foundation engineering, slope stability analysis, and soil improvement projects, where the accurate characterization of soil properties is essential for safe and efficient design. The systematic variation in these properties with relative density also provides valuable insights for predicting soil behavior under different loading conditions and environmental factors. The permeability values in Table 4 were derived from constant-head permeability tests (ASTM D2434) for loose to medium-density sands (Dr = 10-50%) and falling-head tests (ASTM D5084) for dense to very dense sands (Dr = 75-95%), reflecting method-specific precision. The apparent variation in decimal places arises from the inherent resolution of each test: constant-head results are reported to align with flow measurement accuracy (\(\pm\)5%) to two decimal places (e.g., 0.01 cm/s), while falling-head tests for low-permeability sands are truncated to four decimal places (e.g., 0.0001 cm/s) due to smaller volumetric increments. To enhance consistency, we revised all permeability values to scientific notation (e.g., 1 \(\times\) 10\({}^{-2}\) cm/s to 1 \(\times\) 10\({}^{-4}\) cm/s) and included uncertainty ranges (\(\pm\)8% for constant-head, \(\pm\)12% for falling-head) in Table 4. The experimental uncertainties, direct shear (DST) and interface shear (IST) test results include \(\pm\) values for friction angles (e.g., \(\varphi\) = 28\({}^{\circ}\)\(\pm\) 0.5\({}^{\circ}\)) and unit weights (e.g., \(\gamma_{\rm d}\) = 14.1 \(\pm\) 0.3 kN/m\({}^{3}\)), calculated from triplicate trials per condition.