# Retaining Wall-Earth Pressure Theories

 Question 1
The figure shows a vertical retaining wall with backfill consisting of cohesive-frictional soil and a failure plane developed due to passive earth pressure. The forces acting on the failure wedge are: $P$ as the reaction force between the wall and the soil, $R$ as the reaction force on the failure plane, $C$ as the cohesive force along the failure plane and $W$ as the weight of the failure wedge. Assuming that there is no adhesion between the wall and the wedge, identify the most appropriate force polygon for the wedge.

 A A B B C C D D
GATE CE 2023 SET-2   Geotechnical Engineering
Question 1 Explanation:

 Question 2
A vertical sheet pile wall is installed in an anisotropic soil having coefficient of horizontal permeability, $k_{H}$ and coefficient of vertical permeability, $k_{V}$. In order to draw the flow net for the isotropic condition, the embedment depth of the wall should be scaled by a factor of _____, without changing the horizontal scale.
 A $\sqrt{\frac{k_{H}}{k_{V}}}$ B $\sqrt{\frac{k_{V}}{k_{H}}}$ C 1 D $\sqrt{\frac{k_{H}}{k_{V}}}$
GATE CE 2023 SET-2   Geotechnical Engineering
Question 2 Explanation:
We know that, for 2D flow.
$\mathrm{K}=\mathrm{K}_{\mathrm{x}} \frac{\partial^{2} \mathrm{~h}}{\mathrm{dx}}+\mathrm{K}_{\mathrm{z}} \frac{\partial^{2} \mathrm{~h}}{\mathrm{dz}}=0$
When $\mathrm{K}_{1}=\mathrm{K}_{\text {horizontal }}$ and $\mathrm{K}_{2}=\mathrm{K}_{\text {vertical }}$ as shown in figure.
$\frac{\partial^{2} \mathrm{~h}}{\frac{\mathrm{K}_{\mathrm{z}}}{\mathrm{K}_{\mathrm{x}}} \partial \mathrm{x}^{2}}+\frac{\partial^{2} \mathrm{~h}}{\partial \mathrm{z}^{2}}=0$
From the above equation we can note that, when soil is anisotropic with respect to permeability i.e. $k_{x}=k_{z}$, the flow and equipotential line are not necessarily to be orthogonal as because $\frac{\mathrm{K}_{\mathrm{z}}}{\mathrm{K}_{\mathrm{x}}}=1$, and we cannot get the actual form of Laplace equation which is $\frac{\partial^{2} h}{\partial x^{2}}+\frac{\partial^{2} h}{\partial z^{2}}=0$.
But if we replace
$x_{t}=x \sqrt{\frac{K_{z}}{\mathrm{~K}_{x}}}$
Then, $\quad \partial x_{t}^{2}=\partial x^{2} \frac{K_{z}}{\mathrm{~K}_{x}}$
$\frac{\partial^{2} h}{\partial x_{t}^{2}}+\frac{\partial^{2} h}{\partial z^{2}}=0$
Thus, new flow line and equipotential lines are drawn on transformed section with $\mathrm{x}$-distance changed to $x \sqrt{\frac{\mathrm{K}_{\mathrm{z}}}{\mathrm{K}_{\mathrm{x}}}}$ while keeping the vertical dimension constant.
If we scale up the depth by keeping the horizontal distance same the depth will be scaled by factor $\sqrt{\mathrm{K}_{\mathrm{x}} / \mathrm{K}_{\mathrm{z}}}$.

 Question 3
A smooth vertical retaining wall supporting layered soils is shown in figure. According to Rankine's earth pressure theory, the lateral active earth pressure acting at the base of the wall is ____ $\mathrm{kPa}$ (round off to one decimal place)

 A 45.2 B 35.4 C 54.2 D 56.2
GATE CE 2023 SET-1   Geotechnical Engineering
Question 3 Explanation:
$\mathrm{K}_{\mathrm{a} 2}=\frac{1-\sin 25^{\circ}}{1+\sin 25^{\circ}}$
Active earth pressure at the base of the wall.
\begin{aligned} &(\mathrm{Pa})_{\text {Base }}=\mathrm{Ka}_{2}\left(\mathrm{q}+\gamma_{1} \mathrm{z}_{1}+\gamma_{2} \mathrm{z}_{2}\right)-2 \mathrm{C} \sqrt{\mathrm{Ka}_{2}} \\ &=0.405+8(20+18 \times 3+19 \times 4)-2 \times 20 \times \sqrt{0.4058} \\ &= 35.39 \mathrm{kN} / \mathrm{m}^{2} \end{aligned}
 Question 4
As per Rankine's theory of earth pressure, the inclination of failure planes is $(45+\frac{\phi }{2})^{\circ}$ with respect to the direction of the minor principal stress.
The above statement is correct for which one of the following options?
 A Only the active state and not the passive state B Only the passive state and not the active state C Both active as well as passive states D Neither active nor passive state
GATE CE 2022 SET-1   Geotechnical Engineering
Question 4 Explanation:

 Question 5
A concentrated vertical load of 3000 kN is applied on a horizontal ground surface. Points P and Q are at depths 1 m and 2 m below the ground, respectively, along the line of application of the load. Considering the ground to be a linearly elastic, isotropic, semi-infinite medium, the ratio of the increase in vertical stress at P to the increase in vertical stress at Q is ________. (in integer)
 A 12 B 36 C 8 D 4
GATE CE 2022 SET-1   Geotechnical Engineering
Question 5 Explanation:

Increase in vertical stress,
\begin{aligned} \sigma _z&=\frac{KQ}{z^2}\\ \sigma _z&\propto \frac{1}{z^2}\\ \frac{\sigma _P}{\sigma _Q}&=\left ( \frac{z_Q}{z_P} \right )^2\\ &=\left ( \frac{2}{1} \right )^2=4 \end{aligned}

There are 5 questions to complete.

### 6 thoughts on “Retaining Wall-Earth Pressure Theories”

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