Question:
Density index or relative density is defined as
Density index or relative density is defined as
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To remember the formula, think logically: Relative Density is about how "dense" the soil is relative to its maximum possible density. The numerator, \(e_{max} - e\), represents the 'amount of compaction achieved'. The denominator, \(e_{max} - e_{min}\), represents the 'total possible compaction'. So, it's (Achieved Compaction) (Total Possible Compaction).
Updated On: Aug 13, 2026
- \( (e_{max} - e) / (e_{min} - e) \)
- \( (e_{max} - e) / (e_{max} - e_{min}) \)
- \( (e_{max} - e_{min}) / (e_{max} - e) \)
- \( (e_{min} - e) / (e_{max} - e_{min}) \)
Where, \(e_{max}\) = void ratio in the loosest state
\(e_{min}\) = voids ratio in the densest state
e = natural void ratio of the deposit
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
This question asks for the definition of the Density Index (ID), also known as Relative Density (Dr). This is a crucial parameter in geotechnical engineering, specifically for characterizing cohesionless soils (like sands and gravels). It indicates the degree of compaction of a soil relative to its loosest and densest possible states.
Step 2: Key Formula or Approach:
The Density Index is a ratio that compares the "room for compaction" of the soil in its natural state to the "total possible room for compaction".
Let's define the terms:
• \(e_{max}\): The maximum possible void ratio (the soil is in its loosest state).
• \(e_{min}\): The minimum possible void ratio (the soil is in its densest state).
• e: The current, natural void ratio of the soil in-situ. The "room for compaction" in the natural state is the difference between the current void ratio and the densest possible void ratio, i.e., \(e - e_{min}\). However, the formula is structured differently.
A more intuitive way to think about it is:
• The range of possible void ratios is from \(e_{min}\) to \(e_{max}\). The total range is \(e_{max} - e_{min}\).
• The amount the soil has compacted from its loosest state is \(e_{max} - e\). The relative density is the ratio of how much the soil has been compacted from its loosest state to the maximum possible compaction.
The standard formula for Density Index (ID) or Relative Density (Dr) is:
\[ I_D = \frac{e_{max} - e}{e_{max} - e_{min}} \]
Step 3: Detailed Explanation:
Let's analyze the formula \( I_D = \frac{e_{max} - e}{e_{max} - e_{min}} \):
• Denominator (\(e_{max} - e_{min}\)): This represents the total possible range of void ratios for the given soil, from its absolute loosest to its absolute densest state. It's the maximum possible change in void ratio.
• Numerator (\(e_{max} - e\)): This represents how far the current void ratio (e) is from the loosest state (\(e_{max}\)). It measures the amount of densification the soil has already undergone. Thus, the ratio expresses the current state of compaction as a fraction (or percentage) of the maximum possible compaction.
• If the soil is in its loosest state, \(e = e_{max}\), then \(I_D = \frac{e_{max} - e_{max}}{e_{max} - e_{min}} = 0\).
• If the soil is in its densest state, \(e = e_{min}\), then \(I_D = \frac{e_{max} - e_{min}}{e_{max} - e_{min}} = 1\). This confirms that the formula correctly represents the concept of relative density, ranging from 0 (loosest) to 1 (densest). Comparing this with the options, option (B) matches the correct formula.
Step 4: Final Answer:
The correct definition for Density Index is \( (e_{max} - e) / (e_{max} - e_{min}) \).
This question asks for the definition of the Density Index (ID), also known as Relative Density (Dr). This is a crucial parameter in geotechnical engineering, specifically for characterizing cohesionless soils (like sands and gravels). It indicates the degree of compaction of a soil relative to its loosest and densest possible states.
Step 2: Key Formula or Approach:
The Density Index is a ratio that compares the "room for compaction" of the soil in its natural state to the "total possible room for compaction".
Let's define the terms:
• \(e_{max}\): The maximum possible void ratio (the soil is in its loosest state).
• \(e_{min}\): The minimum possible void ratio (the soil is in its densest state).
• e: The current, natural void ratio of the soil in-situ. The "room for compaction" in the natural state is the difference between the current void ratio and the densest possible void ratio, i.e., \(e - e_{min}\). However, the formula is structured differently.
A more intuitive way to think about it is:
• The range of possible void ratios is from \(e_{min}\) to \(e_{max}\). The total range is \(e_{max} - e_{min}\).
• The amount the soil has compacted from its loosest state is \(e_{max} - e\). The relative density is the ratio of how much the soil has been compacted from its loosest state to the maximum possible compaction.
The standard formula for Density Index (ID) or Relative Density (Dr) is:
\[ I_D = \frac{e_{max} - e}{e_{max} - e_{min}} \]
Step 3: Detailed Explanation:
Let's analyze the formula \( I_D = \frac{e_{max} - e}{e_{max} - e_{min}} \):
• Denominator (\(e_{max} - e_{min}\)): This represents the total possible range of void ratios for the given soil, from its absolute loosest to its absolute densest state. It's the maximum possible change in void ratio.
• Numerator (\(e_{max} - e\)): This represents how far the current void ratio (e) is from the loosest state (\(e_{max}\)). It measures the amount of densification the soil has already undergone. Thus, the ratio expresses the current state of compaction as a fraction (or percentage) of the maximum possible compaction.
• If the soil is in its loosest state, \(e = e_{max}\), then \(I_D = \frac{e_{max} - e_{max}}{e_{max} - e_{min}} = 0\).
• If the soil is in its densest state, \(e = e_{min}\), then \(I_D = \frac{e_{max} - e_{min}}{e_{max} - e_{min}} = 1\). This confirms that the formula correctly represents the concept of relative density, ranging from 0 (loosest) to 1 (densest). Comparing this with the options, option (B) matches the correct formula.
Step 4: Final Answer:
The correct definition for Density Index is \( (e_{max} - e) / (e_{max} - e_{min}) \).
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