Effective Stress and Permeability

Question 1
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 1 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 2
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 2 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 3
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 3 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 4
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 4 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}
Question 5
Water flows in the upward direction in a tank through 2.5 m thick sand layer as shown in the figure. The void ratio and specific gravity of sand are 0.58 and 2.7, respectively. The sand is fully saturated. Unit weight of water is 10 kN/m^3.

The effective stress (in kPa, round off to two decimal places) at point A, located 1 m above the base of tank, is __________.
A
3.14
B
5.36
C
8.94
D
11.24
GATE CE 2020 SET-1   Geotechnical Engineering
Question 5 Explanation: 


\begin{aligned} \gamma _{suv}&=\left ( \frac{G-1}{1+e} \right )\gamma _\omega \\ &=\left ( \frac{2.7-1}{1+0.58} \right ) \times 10\\ &=10.759\\ \bar{\sigma }&=z\gamma _{suv}-iz\gamma _\omega \\ &=1.5(\gamma _{suv})-\left ( \frac{1.2}{2.5} \right )\times 1.5 \times \gamma _\omega \\ &=8.939 kN/m^2 \end{aligned}
Question 6
Constant head permeability tests were performed on two soil specimens, S1 and S2. The ratio of height of the two specimens (L_{S1}:L_{S2}) is 1.5, the ratio of the diameter of specimens (D_{S1}:D_{S2}) is 0.5, and the ratio of the constant head (h_{S1}:h_{S2}) applied on the specimens is 2.0. If the discharge from both the specimens is equal, the ratio of the permeability of the soil specimens (k_{S1}:k_{S2}) is ____
A
1
B
2
C
3
D
4
GATE CE 2019 SET-2   Geotechnical Engineering
Question 6 Explanation: 
\begin{aligned} \frac{L_{s1}}{L_{s2}}&=1.5\\ \frac{D_{s1}}{D_{s2}}&=0.5\\ \frac{h_{s1}}{h_{s2}}&=2\\ \frac{k_{s1}}{k_{s2}}&=? \end{aligned}
Discharge is same.
\begin{aligned} k_1i_1A_1&=k_2i_2A_2\\ k_1\frac{h_{s1}}{L_{s1}} \times \frac{\pi}{4} \times D_{s1}^2&=k_2\frac{h_{s2}}{L_{s2}} \times \frac{\pi}{4} \times D_{s2}^2\\ \frac{k_{s1}}{k_{s2}}&=\frac{L_{s1}}{L_{s2}} \times \frac{h_{s2}}{h_{s1}} \times \frac{D_{s2}^2}{D_{s1}^2}\\ &=1.5 \times \frac{1}{2} \times \left ( \frac{1}{0.5} \right )^2=3 \end{aligned}
Question 7
An anisotropic soil deposit has coefficient of permeability in vertical and horizontal directions as k_z \; and \; k_x, respectively. For constructing a flow net, the horizontal dimension of the problem's geometry is transformed by a multiplying factor of
A
\sqrt{\frac{k_z}{k_x}}
B
\sqrt{\frac{k_x}{k_z}}
C
\frac{k_x}{k_z}
D
\frac{k_z}{k_x}
GATE CE 2019 SET-2   Geotechnical Engineering
Question 7 Explanation: 
X=X_T\sqrt{\frac{k_x}{k_z}}
Transformed horizontal dimension, X_T=X\sqrt{\frac{k_z}{k_x}}
Question 8
In a soil specimen, the total stress, effective stress, hydraulic gradient and critical hydraulic gradient are \sigma ,\sigma ',i, \; and \; i_c, respectively. For initiation of quicksand condition, which one of the following statements is TRUE?
A
\sigma '\neq 0 \; and \;i =i_c
B
\sigma '= 0 \; and \;i =i_c
C
\sigma '\neq 0 \; and \;i \neq i_c
D
\sigma = 0 \; and \;i =i_c
GATE CE 2019 SET-1   Geotechnical Engineering
Question 8 Explanation: 
Quick sand condition occurs if \sigma '=0 and i=i_c
Question 9
Three soil specimens (Soil 1, Soil 2 and Soil 3), each 150 mm long and 100 mm diameter, are placed in series in a constant head flow set-up as shown in the figure. Suitable screens are provided at the boundaries of the specimens to keep them intact. The values of coefficient of permeability of Soil 1, Soil 2 and Soil 3 are 0.01, 0.003 and 0.03 cm/s, respectively.

The value of h in the set-up is
A
0 mm
B
40mm
C
255mm
D
560mm
GATE CE 2018 SET-2   Geotechnical Engineering
Question 9 Explanation: 
In normal to bedding plane flow (series arrangement), Discharge will be same and Head loss and Hydraulic gradient will be different
\begin{aligned} q &=K_{1} i_{1} A=K_{2} i_{2} A \\ &=K_{3} i_{3} A=K_{\mathrm{avg}} \cdot\left(\frac{H_{L}}{L}\right) A \\ K_{\mathrm{avg}_{1}} &=\frac{\Sigma Z_{i}}{\Sigma \frac{Z_{i}}{K_{i}}}=\frac{150+150+150}{\frac{150}{0.01}+\frac{150}{0.003}+\frac{150}{0.03}} \\ &=0.0064\\ \text{Total head loss}\\ &=H_{L}=560 \mathrm{mm} \\ \therefore K_{3} \cdot \frac{h}{150} \times A &=K_{a v g} \cdot \frac{560}{(150+150+150)} A \\ 0.03 \cdot\left(\frac{h}{150}\right) &=0.0064\left(\frac{560}{450}\right) \\ h &=40 \mathrm{mm} \end{aligned}
Question 10
At a construction site, a contractor plans to make an excavation as shown in the figure.

The water level in the adjacent river is at an elevation of +20.0 m. Unit weight of water is 10 kN/m^{3}. The factor of safety (up to two decimal places) against sand boiling for the proposed excavation is ______
A
1
B
2
C
3
D
4
GATE CE 2018 SET-1   Geotechnical Engineering
Question 10 Explanation: 
\mathrm{FOS}=\frac{10 \times \gamma_{\mathrm{sat}}}{20 \times \gamma_{\mathrm{w}}}=\frac{10 \times 20}{20 \times 10}=1
There are 10 questions to complete.

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