Numerical Problem: Soil Properties
Given:
- Porosity, \( n = 40\% = 0.4 \)
- Specific Gravity of Solids, \( G = 2.70 \)
- \( \gamma_w = 9.81 \; \text{kN/m}^3 \)
(a) Void Ratio
The void ratio is calculated as:
\( e = \frac{n}{1 – n} \)
Substituting the values:
\( e = \frac{0.4}{1 – 0.4} = 0.667 \)
Physical Meaning: The void ratio \( e \) represents the ratio of the volume of voids to the volume of solids in the soil. A higher void ratio indicates a looser soil structure.
(b) Dry Density
The dry density is calculated as:
\( \gamma_d = \frac{G \cdot \gamma_w}{1 + e} \)
Substituting the values:
\( \gamma_d = \frac{2.7 \cdot 9.81}{1 + 0.667} = 15.89 \; \text{kN/m}^3 \)
Physical Meaning: Dry density \( \gamma_d \) is the weight of soil solids per unit volume. It is a key parameter in soil compaction and stability.
(c) Unit Weight at 50% Saturation
The water content is calculated as:
\( w = \frac{e \cdot S}{G} \)
Substituting the values:
\( w = \frac{0.667 \cdot 0.5}{2.7} = 0.124 \)
The unit weight is then calculated as:
\( \gamma = \gamma_d (1 + w) \)
Substituting:
\( \gamma = 15.89 \cdot (1 + 0.124) = 17.86 \; \text{kN/m}^3 \)
Physical Meaning: Unit weight at partial saturation reflects the combined weight of soil solids and water in the voids, providing insights into the soil’s state under various moisture conditions.
(d) Unit Weight at Full Saturation
The saturated water content is calculated as:
\( w_{sat} = \frac{e}{G} \)
Substituting the values:
\( w_{sat} = \frac{0.667}{2.7} = 0.247 \)
The saturated unit weight is then calculated as:
\( \gamma_{sat} = \gamma_d (1 + w_{sat}) \)
Substituting:
\( \gamma_{sat} = 15.89 \cdot (1 + 0.247) = 19.81 \; \text{kN/m}^3 \)
Physical Meaning: Saturated unit weight \( \gamma_{sat} \) is the maximum possible unit weight of soil, representing the weight of soil solids and water completely filling the voids.
Summary of Results
- Void Ratio: \( e = 0.667 \)
- Dry Density: \( \gamma_d = 15.89 \; \text{kN/m}^3 \)
- Unit Weight at 50% Saturation: \( \gamma = 17.86 \; \text{kN/m}^3 \)
- Unit Weight at Full Saturation: \( \gamma_{sat} = 19.81 \; \text{kN/m}^3 \)
Explanation:
This problem illustrates fundamental soil mechanics concepts such as void ratio, dry density, and unit weight under varying moisture conditions. These parameters help determine the soil’s structural behavior and suitability for engineering applications like foundation design, slope stability, and compaction quality assessment.
