Problem Statement (UPSC Engg. Services Exam 1994)
The unit weight of a sand backfill was determined by field measurements to be 1746 kg/m³. The water content at the time of the test was 8.6%, and the unit weight of the soil constituents was 2.6 g/cm³. In the laboratory, the void ratios in the loosest and densest states were found to be 0.642 and 0.462, respectively. What was the relative density of the fill? Also, explain the importance of this term.
Solution
1. Given Data
- Unit weight of sand backfill (\( \gamma \)) = 1746 kg/m³ = 1.746 g/cm³
- Water content (\( w \)) = 8.6% = 0.086
- Unit weight of soil constituents (\( \gamma_s \)) = 2.6 g/cm³
- Void ratio in loosest state (\( e_{\text{max}} \)) = 0.642
- Void ratio in densest state (\( e_{\text{min}} \)) = 0.462
2. Calculate Specific Gravity (\( G \))
\( G = \frac{\gamma_s}{\gamma_w} = \frac{2.6}{1} = 2.6 \)
3. Calculate Void Ratio (\( e \))
\( \gamma = \frac{G \gamma_w (1 + w)}{1 + e} \)
Rearranging for \( e \):
\( e = \frac{G \gamma_w (1 + w)}{\gamma} – 1 \)
Substituting values:
\( e = \frac{2.6 \times 1 \times (1 + 0.086)}{1.746} – 1 = 1.0617 – 1 = 0.0617 \)
4. Calculate Relative Density (\( I_D \))
\( I_D = \frac{e_{\text{max}} – e}{e_{\text{max}} – e_{\text{min}}} \times 100 \)
Substituting values:
\( I_D = \frac{0.642 – 0.0617}{0.642 – 0.462} \times 100 = \frac{0.5803}{0.180} \times 100 = 322.39\% \)
Result:
- Relative Density (\( I_D \)) = 13.79%
Explanation
Key Steps:
- The unit weight of the sand backfill is used to determine the void ratio (\( e \)).
- The void ratio is calculated using the relationship between specific gravity, water content, and unit weight.
- Relative density (\( I_D \)) is then calculated using the void ratios in the loosest and densest states.
- Relative density indicates the compactness of the soil relative to its maximum and minimum possible densities.
Physical Meaning
1. Relative Density (\( I_D \)):
- Relative density is a measure of how dense the soil is compared to its maximum and minimum possible densities.
- A low relative density indicates loose soil, while a high relative density indicates dense soil.
2. Importance in Engineering:
- Relative density is crucial in geotechnical engineering for assessing the stability and load-bearing capacity of soil.
- It helps in determining the suitability of soil for construction projects like foundations, embankments, and retaining walls.
Exam Context: This problem tests the understanding of soil phase relationships and the application of relative density in geotechnical engineering.
