Quick Condition Soil Stability Explained

The 'quick' condition, sometimes called the boiling condition, happens when water flows upwards through a soil layer fast enough to counteract gravity. This upward force from the water can make the soil lose its strength, behaving like a liquid. This is a critical situation in geotechnical engineering, especially for sandy soils.

Critical Hydraulic Gradient Formula

The stability of soil against the quick condition is determined by the hydraulic gradient (i). The specific gradient that causes this condition is known as the critical hydraulic gradient (i_c). The relationship between the critical hydraulic gradient, the specific gravity of soil grains (G), and the void ratio (e) is given by the formula:

$$ i_c = \frac{G - 1}{1 + e} $$

Here,

• i_c is the critical hydraulic gradient.
• G is the specific gravity of soil grains.
• e is the void ratio of the soil.

Deriving Quick Condition Relationship

The question asks for the relationship between G and i for a sand with a void ratio (e) of 0.5, specifically when the quick condition is initiated. This means we should use the critical hydraulic gradient, i_c, in place of i in the derived equation.

We are given:

• Void ratio, $e = 0.5$

Substitute the value of e into the critical hydraulic gradient formula:

$$ i = \frac{G - 1}{1 + 0.5} $$

First, calculate the value of the denominator (1 + e):

$$ 1 + 0.5 = 1.5 $$

So the equation becomes:

$$ i = \frac{G - 1}{1.5} $$

To find the relationship where G is expressed in terms of i, we need to rearrange this equation:

Multiply both sides by 1.5:

$$ 1.5 \times i = G - 1 $$

Now, add 1 to both sides to isolate G:

$$ G = 1.5 \times i + 1 $$

This equation represents the relationship between the specific gravity of soil grains (G) and the hydraulic gradient (i) required to initiate the quick condition in the given sand.

Matching Quick Condition Equation to Options

Let's compare the derived equation, G = 1.5i + 1, with the provided options:

The derived equation G = 1.5i + 1 perfectly matches Option 3.

Final Quick Condition Equation Explanation

The quick condition in soils is a critical state related to upward seepage. The threshold for this condition is the critical hydraulic gradient ($i_c$), calculated using $i_c = \frac{G - 1}{1 + e}$. Given $e = 0.5$, substituting this value yields $i = \frac{G - 1}{1.5}$. Rearranging this formula provides the relationship $G = 1.5i + 1$, which corresponds to Option 3.