Calculating Relative Density of Sand Deposit

The question asks us to determine the relative density of a sand deposit given its void ratios at different states: the densest possible state, the loosest possible state, and its natural state.

Relative density ($\boldsymbol{I_D}$) is a key parameter in soil mechanics used to describe the degree of packing of a cohesionless soil, such as sand or gravel. It indicates how dense the soil is relative to its minimum and maximum possible densities.

Given Void Ratios

We are provided with the following void ratios:

• Void ratio at the densest state ($\boldsymbol{e_{min}}$): 0.2
• Void ratio at the loosest state ($\boldsymbol{e_{max}}$): 0.6
• Void ratio at the natural state ($\boldsymbol{e}$): 0.4

Formula for Relative Density

The relative density ($\boldsymbol{I_D}$) is calculated using the following formula:

$\boldsymbol{I_D} = \frac{e_{max} - e}{e_{max} - e_{min}} \times 100\%$

Where:

• $\boldsymbol{e_{max}}$ is the maximum void ratio (loosest state).
• $\boldsymbol{e}$ is the natural void ratio.
• $\boldsymbol{e_{min}}$ is the minimum void ratio (densest state).

Step-by-Step Calculation of Relative Density

Now, we substitute the given values into the relative density formula:

$\boldsymbol{I_D} = \frac{0.6 - 0.4}{0.6 - 0.2} \times 100\%$

First, calculate the numerator and the denominator:

• Numerator ($\boldsymbol{e_{max} - e}$): $\boldsymbol{0.6 - 0.4 = 0.2}$
• Denominator ($\boldsymbol{e_{max} - e_{min}}$): $\boldsymbol{0.6 - 0.2 = 0.4}$

Now, divide the numerator by the denominator and multiply by 100%:

$\boldsymbol{I_D} = \frac{0.2}{0.4} \times 100\%$

$\boldsymbol{I_D} = 0.5 \times 100\%$

$\boldsymbol{I_D} = 50\%$

Thus, the relative density of the sand deposit is 50%.

Comparing with Options

Let's compare our calculated value with the given options:

• Option 1: 100%
• Option 2: 75%
• Option 3: 50%
• Option 4: 25%

Our calculated relative density of 50% matches Option 3.

Revision Table: Void Ratio and Relative Density

Additional Information: Understanding Relative Density

Relative density is particularly important for granular soils like sands because their behavior (strength, stiffness, compressibility) is strongly influenced by how tightly the particles are packed. It is expressed as a percentage.

• An $\boldsymbol{I_D}$ of 0% corresponds to the loosest state ($\boldsymbol{e} = e_{max}$).
• An $\boldsymbol{I_D}$ of 100% corresponds to the densest state ($\boldsymbol{e} = e_{min}$).
• An $\boldsymbol{I_D}$ between 0% and 100% indicates an intermediate density state.

Based on relative density, sands are often described qualitatively:

In this problem, a relative density of 50% falls exactly at the boundary between "Loose" and "Medium Dense," often categorized as "Medium Dense."