Shrinkage Limit Formula

The shrinkage limit ($w_s$) is the moisture content at which a soil sample, upon drying, ceases to decrease in volume. At this limit, the soil is saturated.

The relationship is defined by:

$w_s \times G_s = e_s$

where:

• $w_s$ is the shrinkage limit (in decimal form)
• $G_s$ is the specific gravity of soil solids
• $e_s$ is the void ratio at the shrinkage limit

Void Ratio and Specific Gravity Values

From the question details:

• Specific gravity, $G_s = 2.5$.
• Void ratio, $e = 0.5$. This is interpreted as the void ratio at the shrinkage limit ($e_s = 0.5$), representing the void volume relative to the solid volume when shrinkage ceases.

Calculating the Shrinkage Limit

To find the shrinkage limit ($w_s$), rearrange the formula:

$w_s = \frac{e_s}{G_s}$

Substitute the known values:

$w_s = \frac{0.5}{2.5}$

Perform the division:

$w_s = 0.20$

Convert this decimal value to a percentage:

$w_s = 0.20 \times 100\% = 20\%$

Thus, the shrinkage limit of the clay sample is 20%.