# Effective Stress and Permeability

 Question 1
A vertical trench is excavated in a clayey soil deposit having a surcharge load of $30 \mathrm{kPa}$. A fluid of unit weight $12 \mathrm{kN} / \mathrm{m}^{3}$ is poured in the trench to prevent collapse as the excavation proceeds. Assume that the fluid is not seeping through the soil deposit. If the undrained cohesion of the clay deposit is $20 \mathrm{kPa}$ and saturated unit weight is $18 \mathrm{kN} / \mathrm{m}^{3}$, what is the maximum depth of unsupported excavation (in $\mathrm{m}$, rounded off to two decimal places) ? ____
 A 3.33 B 2.25 C 1.45 D 5.65
GATE CE 2023 SET-1   Geotechnical Engineering
Question 1 Explanation: We know, $\alpha=45+\frac{\mathrm{Q}}{2}$ here $\mathrm{Q}=0$
$\therefore$ Failure plane is at $45^{\circ}$
It is said that there is no seepage hence no flow of fluid. But considering the fluid is there in the pores.
Also, considering given $\gamma_{\text {sat }}$ is the value for given fluid.
\begin{aligned} \therefore \quad \gamma_{\text {sub }} & =\gamma_{\text {sat }}-\gamma_{\text {fluid }} \\ & =18-12 \\ & =6 \mathrm{kN} / \mathrm{m}^{3} \end{aligned}
Now, equating the forces on the considered failure plane. As it is clay $\mathrm{Q}=0, \mathrm{~K}_{\mathrm{a}}=1$.
$\therefore \quad \frac{1}{2} \times x \times x \times 1 \times \gamma_{\text {sub }} \times \sin 45^{\circ}+30 \times x \sin 45^{\circ}$
$=C \times x \sqrt{2}$
$3 x=10$
$\therefore \quad x=3.33 \mathrm{~m}$ Solving with lateral earth pressure approach.
$\mathrm{K}_{\mathrm{a}}=1$ \begin{aligned} & P=\sigma_{v} K_{a}-2 C \sqrt{K_{a}} \\ & P=\left(30+\gamma_{\text {sub }} z\right) \times 1-2 C \times \sqrt{1} \end{aligned}
When $\mathrm{P}=0$,
$30+6 z-2 \times 20=0$
$\therefore \quad 6 z=10$
$z=\frac{10}{6}$
$\therefore \quad$ Unsupported excavation $=2 \times \frac{10}{6}=3.33 \mathrm{~m}$
 Question 2
A soil sample is underlying a water column of height $h_1$, as shown in the figure. The vertical effective stresses at points A, B, and C are $\sigma '_A$, $\sigma '_B$, and $\sigma '_C$ respectively. Let $\gamma _{sat}$ and $\gamma '$ be the saturated and submerged unit weights of the soil sample, respectively, and $\gamma _{w}$ be the unit weight of water. Which one of the following expressions correctly represents the sum ($\sigma '_A+ \sigma '_B+\sigma '_C$ )? A $(2h_2+h_3 )\gamma \;'$ B $(h_1+h_2+h_3 )\gamma \;'$ C $(h_2+h_3 )(\gamma _{sat}-\gamma _w )$ D $(h_1+h_2+h_3 )\gamma _{sat}$
GATE CE 2022 SET-2   Geotechnical Engineering
Question 2 Explanation:

\begin{aligned} \sigma _A'&=0\\ \sigma _B'&=h_2\gamma '\\ \sigma _C'&=(h_2+h_3)\gamma '\\ \sigma _A'+\sigma _B'+\sigma _C'&=(2h_2+h_3)\gamma '\\ \end{aligned}

 Question 3
In an Oedometer apparatus, a specimen of fully saturated clay has been consolidated under a vertical pressure of $50 \mathrm{kN} / \mathrm{m}^{2}$ and is presently at equilibrium. The effective stress and pore water pressure immediately on increasing the vertical stress to $150 \mathrm{kN} / \mathrm{m}^{2}$, respectively are
 A $150 \mathrm{kN} / \mathrm{m}^{2}$ and 0 B $100 \mathrm{kN} / \mathrm{m}^{2}$ and $50 \mathrm{kN} / \mathrm{m}^{2}$ C $50 \mathrm{kN} / \mathrm{m}^{2}$ and $100 \mathrm{kN} / \mathrm{m}^{2}$ D 0 and $150 \mathrm{kN} / \mathrm{m}^{2}$
GATE CE 2021 SET-1   Geotechnical Engineering
Question 3 Explanation:
Stress is increased suddenly, hence entire change will be taken by water $\Delta \bar{\sigma}=\Delta U=100 \mathrm{kPa}$.
There will be no change in effective stress
$\therefore \qquad \qquad\bar{\sigma}=50 \mathrm{kPa}$
 Question 4
A constant head permeability test was conducted on a soil specimen under a hydraulic gradient of 2.5. The soil specimen has specific gravity of 2.65 and saturated water content of 20%. If the coefficient of permeability of the soil is 0.1 cm/s, the seepage velocity (in cm/s, round off to two decimal places) through the soil specimen is ________.
 A 0.34 B 0.46 C 0.72 D 0.96
GATE CE 2020 SET-2   Geotechnical Engineering
Question 4 Explanation:
\begin{aligned} \text{Void ratio,}\; e&=\frac{wG}{s}\\ &=\frac{0.2 \times 2.65}{1}=0.53\\ \text{Porosity,}\; n&=\frac{e}{1+e}=0.3464\\ \text{Seepage velocity,}\; V_s&=\frac{v}{n}=\frac{k_i}{n}\\ &=\frac{0.1 \times 2.5}{0.3464}=0.72 cm/sec \end{aligned}
 Question 5
Permeability tests were carried out on the samples collected from two different layers as shown in the figure (not drawn to the scale). The relevant horizontal ($k_h$) and vertical ($k_v$) coefficients of permeability are indicated for each layer. The ratio of the equivalent horizontal to vertical coefficients of permeability, is
 A 37.29 B 80.2 C 68.25 D 0.03
GATE CE 2020 SET-2   Geotechnical Engineering
Question 5 Explanation:
\begin{aligned} \frac{k_{eq.H}}{k_{eq.V}}&=\frac{\frac{\Sigma K_iZ_i}{\Sigma Z_i}}{\frac{\Sigma Z_i}{\Sigma \frac{Z_i}{K_i}}}\\ &=\frac{\frac{4.4 \times 10^{-3} \times 3 +6 \times 10^{-1} \times 4}{7}}{\frac{7}{\frac{3}{4 \times 10^{-3}}+\frac{4}{5.5 \times 10^{-1}}}}\\ &=\frac{0.3447}{9.24 \times 10{-3}}=37.29 \end{aligned}

There are 5 questions to complete.

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