Problem Statement (Engg. Services Exam 2000)
A sand sample above the water table has the following properties:
- Natural moisture content (\( w \)) = 15%
- Unit weight (\( \gamma \)) = 18.84 kN/m³
- Minimum void ratio (\( e_{\text{min}} \)) = 0.5
- Maximum void ratio (\( e_{\text{max}} \)) = 0.85
- Specific gravity (\( G \)) = 2.65
Calculate:
- Degree of saturation (\( S \))
- Relative density (\( I_D \))
Solution
1. Given Data
- Natural moisture content (\( w \)) = 15% = 0.15
- Unit weight (\( \gamma \)) = 18.84 kN/m³
- Specific gravity (\( G \)) = 2.65
- Unit weight of water (\( \gamma_w \)) = 9.81 kN/m³
- Void ratios: \( e_{\text{min}} = 0.5 \), \( e_{\text{max}} = 0.85 \)
2. Calculate Void Ratio (\( e \))
\( e = \frac{G \cdot \gamma_w \cdot (1 + w)}{\gamma} – 1 \)
Substituting values:
\( e = \frac{2.65 \times 9.81 \times (1 + 0.15)}{18.84} – 1 = \frac{29.93}{18.84} – 1 = 1.588 – 1 = 0.588 \)
Note: The slight discrepancy (0.588 vs. 0.587) is due to rounding.
3. Calculate Degree of Saturation (\( S \))
\( S = \frac{w \cdot G}{e} = \frac{0.15 \times 2.65}{0.588} = \frac{0.3975}{0.588} = 0.6774 \, \text{or} \, 67.74\% \)
4. Calculate Relative Density (\( I_D \))
\( I_D = \frac{e_{\text{max}} – e}{e_{\text{max}} – e_{\text{min}}} \times 100 \)
Substituting values:
\( I_D = \frac{0.85 – 0.588}{0.85 – 0.5} \times 100 = \frac{0.262}{0.35} \times 100 = 75.14\% \)
Results:
- Degree of saturation (\( S \)) = 67.74%
- Relative density (\( I_D \)) = 75.14%
Explanation
Key Steps:
- Void Ratio: Derived from the relationship between unit weight, moisture content, and specific gravity.
- Degree of Saturation: Represents the fraction of voids filled with water.
- Relative Density: Indicates the compactness of the sand relative to its loosest and densest states.
Physical Meaning
1. Degree of Saturation (67.74%):
- About 68% of the voids are filled with water. This partial saturation affects permeability and shear strength.
2. Relative Density (75.14%):
- The sand is in a dense state (above 70% is considered dense). Dense sands have higher bearing capacity and lower settlement.
3. Engineering Significance:
- High relative density reduces liquefaction risk in seismic zones.
- Partial saturation indicates the soil is above the water table but retains some moisture from capillary action or rainfall.
Exam Context: Tests understanding of phase relationships and compaction indices in geotechnical engineering.
