Understanding Upward Hydraulic Gradient and Factor of Safety in Sand

The question asks about the maximum permissible upward hydraulic gradient in a previous sand layer, considering its properties and a factor of safety. The upward hydraulic gradient is related to the concept of effective stress and quicksand conditions in soil mechanics. When water flows upwards through a soil, it exerts a seepage force that opposes the weight of the soil particles. If this upward seepage force becomes equal to the submerged weight of the soil, the effective stress becomes zero, and the soil loses its shear strength, behaving like a viscous fluid. This phenomenon is known as quicksand or boiling condition.

Critical Hydraulic Gradient Calculation

The hydraulic gradient at which the quicksand condition occurs is called the critical hydraulic gradient (\(i_c\)). The formula for the critical hydraulic gradient is given by:

\( i_c = \frac{G_s - 1}{1 + e}

Where:

• \(G_s\) is the specific gravity of soil solids.
• \(e\) is the void ratio of the soil.

We are given the porosity (\(n\)) and specific gravity (\(G_s\)). We first need to calculate the void ratio (\(e\)) from the given porosity (\(n\)). The relationship between void ratio and porosity is:

\( e = \frac{n}{1 - n}

Given porosity \(n = 45\%\) or \(n = 0.45\). Specific gravity \(G_s = 2.65\).

Calculating the void ratio \(e\):

\( e = \frac{0.45}{1 - 0.45} = \frac{0.45}{0.55}

Now, substitute the value of \(e\) into the formula for the critical hydraulic gradient (\(i_c\)):

\( i_c = \frac{2.65 - 1}{1 + \frac{0.45}{0.55}} = \frac{1.65}{\frac{0.55 + 0.45}{0.55}} = \frac{1.65}{\frac{1.00}{0.55}} = 1.65 \times 0.55

Calculating the value of \(i_c\):

\( i_c = 1.65 \times 0.55 = 0.9075

So, the critical hydraulic gradient for this sand is approximately 0.9075.

Maximum Permissible Upward Gradient

The question asks for the maximum permissible upward gradient, which is the critical hydraulic gradient divided by the factor of safety (FS). The factor of safety is used to ensure that the actual gradient is well below the critical gradient, preventing the quicksand condition.

\( i_{\text{permissible}} = \frac{i_c}{\text{FS}}

Given factor of safety \(FS = 4\).

Calculating the maximum permissible upward gradient:

\( i_{\text{permissible}} = \frac{0.9075}{4}

\( i_{\text{permissible}} = 0.226875

Comparing this value to the given options, the closest value is 0.225.

Step-by-Step Solution Summary

1. Identify the given soil properties: porosity \(n = 0.45\), specific gravity \(G_s = 2.65\).
2. Identify the required factor of safety \(FS = 4\).
3. Calculate the void ratio \(e\) using the porosity \(n\): \(e = \frac{n}{1-n}\).
4. Calculate the critical hydraulic gradient \(i_c\) using the void ratio \(e\) and specific gravity \(G_s\): \(i_c = \frac{G_s - 1}{1 + e}\).
5. Calculate the maximum permissible upward gradient by dividing the critical hydraulic gradient \(i_c\) by the factor of safety \(FS\): \(i_{\text{permissible}} = \frac{i_c}{FS}\).

Calculation Steps

• Void ratio \(e = \frac{0.45}{1-0.45} = \frac{0.45}{0.55} \approx 0.8182\)
• Critical hydraulic gradient \(i_c = \frac{2.65 - 1}{1 + 0.8182} = \frac{1.65}{1.8182} \approx 0.9075\)
• Maximum permissible upward gradient \(i_{\text{permissible}} = \frac{0.9075}{4} \approx 0.226875\)

The calculated value is approximately 0.226875, which rounds to 0.227. The closest option is 0.225.

Conclusion on Maximum Permissible Upward Gradient

Based on the calculations, the maximum permissible upward gradient for the given previous sand with a factor of safety of 4 is approximately 0.2269. Among the provided options, 0.225 is the closest value.

Revision Table: Key Concepts in Upward Gradient

Additional Information: Quicksand Prevention and Effects

The quicksand phenomenon is particularly dangerous in excavations, especially below the water table. Upward seepage can cause the bottom of the excavation to heave or 'boil', leading to instability and collapse. Preventing quicksand requires managing the upward hydraulic gradient.

Methods to prevent or mitigate quicksand include:

• Lowering the groundwater table outside the excavation using dewatering techniques (e.g., wellpoints, deep wells).
• Increasing the downward pressure by excavating in stages or using berms.
• Increasing the flow path length for upward seepage using cutoff walls or sheet piling.
• Using surcharge load at the bottom of the excavation.
• Injections of stabilizing agents (e.g., grout).

Understanding the critical hydraulic gradient and applying an adequate factor of safety are crucial steps in geotechnical design to ensure stability against quicksand conditions in permeable soils like sand.