# Stress Distribution in the Soil

 Question 1
In the given figure, Point $\mathrm{O}$ indicates the stress point of a soil element at initial non-hydrostatic stress condition. For the stress path (OP), which of the following loading conditions is correct ? A $\sigma_{v}$ is increasing and $\sigma_{h}$ is constant. B $\sigma_{v}$ is constant and $\sigma_{h}$ is increasing. C $\sigma_{v}$ is increasing and $\sigma_{h}$ is decreasing. D $\sigma_{v}$ is decreasing and $\sigma_{h}$ is increasing.
GATE CE 2023 SET-2   Geotechnical Engineering
Question 1 Explanation: For $1: 1$ slope, $\frac{d q}{d P}=1$
$\Rightarrow \frac{d \sigma_{V}-d \sigma_{H}}{d \sigma_{V}+d \sigma_{H}}=1$
$\Rightarrow d \sigma_{V}-d \sigma_{H}=d \sigma_{V}+d \sigma_{H}$
$\Rightarrow \quad-\mathrm{d} \sigma_{\mathrm{H}}=\mathrm{d} \sigma_{\mathrm{H}}$
$\Rightarrow \quad \mathrm{d} \sigma_{\mathrm{H}}=0$
Hence, if $\sigma_{H}$ is constant then increasing $\sigma_{V}$ or decreasing $\sigma_{\mathrm{V}}$ will lead to $1: 1$ slope.
 Question 2
An unconfined compression strength test was conducted on a cohesive soil. The test specimen failed at an axial stress of $76 \mathrm{kPa}$. The undrained cohesion (in $\mathrm{kPa}$, in integer) of the soil is ___
 A 18 B 38 C 28 D 54
GATE CE 2023 SET-2   Geotechnical Engineering
Question 2 Explanation:
UCS Test :
$\sigma_{3}=0$ [Cell pressure]
$\therefore$ Axial stress $=$ Deviatoric stress $=\sigma_{1}-\sigma_{3}=\sigma_{1}$
Given, $\sigma_{1}=76 \mathrm{kPa}$
We know,
$\sigma_{1}=\sigma_{3} \tan ^{2}\left(45^{\circ}+\frac{\phi}{2}\right)+2 C \tan \left(45^{\circ}+\frac{\phi}{2}\right)$
$[\mathrm{C}=$ Undrained cohesion, $\phi=0$ (Cohesive soil)]
$\Rightarrow \quad \sigma_{1}=2 \mathrm{C}$
$\Rightarrow \quad 76=2 \mathrm{C}$
$\Rightarrow \quad C=38 \mathrm{kPa}$

 Question 3
Consider that a force $P$ is acting on the surface of a half-space (Boussingesq's problem). The expression for the vertical stress $\left(\sigma_{z}\right)$ at any point $(r, z)$ with the half-space is given as,

$\sigma_{z}=\frac{3 P}{2 \pi} \frac{z^{3}}{\left(r^{2}+z^{2}\right)^{5 / 2}}$

where, $r$ is the radial distance, and $z$ is the depth with downward direction taken as positive. At any given $r$, there is a variation of $\sigma_{z}$ along $z$, and at a specific $z$, the value of $\sigma_{z}$ will be maximum. What is the locus of the maximum $\sigma_{z}$ ?
 A $z^2=\frac{3}{2}r^2$ B $z^3=\frac{3}{2}r^2$ C $z^2=\frac{5}{2}r^2$ D $z^3=\frac{5}{2}r^2$
GATE CE 2023 SET-1   Geotechnical Engineering
Question 3 Explanation:
The expression for the vertical stress $\left(\sigma_{2}\right)$ at any point $(r, z)$ with the half-space is given as
$\sigma_{z}=\frac{3 P}{2 \pi} \frac{z^{3}}{\left(r^{2}+z^{2}\right)^{5 / 2}}$
For $\sigma_{\mathrm{z}}$ to be maximum
\begin{aligned} & \frac{\sigma_{z}}{\partial z}=0 \Rightarrow \frac{3 P}{2 \pi} \frac{\partial}{\partial z}\left[\frac{z^{3}}{\left(r^{2}+z^{2}\right)^{5 / 2}}\right] \\ &=\frac{3 P}{2 \pi} \frac{\left[\left(r^{2}+z^{2}\right)^{5 / 2}-z^{3} \times \frac{5}{2}\left(r^{2}+z^{2}\right)^{3 / 2} \times 2 z\right]}{\left(r^{2}+z^{2}\right)^{5}}=0 \\ &=\left[\left(r^{2}+z^{2}\right)\left(3 z^{2}\right)-5 z^{4}\right]=0 \\ &=3 r^{2}+3 z^{2}=5 z^{2} \\ & z^{2}=\frac{3 r^{2}}{2} \end{aligned}
 Question 4
A concentrically loaded isolated square footing of size 2 m x 2 m carries a concentrated vertical load of 1000 kN. Considering Boussinesq's theory of stress distribution, the maximum depth (in m) of the pressure bulb corresponding to 10% of the vertical load intensity will be ________. (round off to two decimal places)
 A 1.25 B 4.35 C 5.36 D 6.25
GATE CE 2022 SET-2   Geotechnical Engineering
Question 4 Explanation:
Considering Boussinesq's theory of stress distribution
$\sigma _x=\frac{3Q}{2\pi z^2}\left [ \frac{1}{1+\left ( \frac{r}{z} \right )^2} \right ]^{5/2}$
For $r=0, \sigma _z=\frac{3Q}{2\pi z^2}$
$\sigma _z=0.1q=0.1 \times \frac{Q}{B^2}=\frac{0.1}{2^2}Q$
$\frac{1}{40}Q=\frac{3Q}{2 \pi z^2}$
$z^2=\frac{3 \times 40}{2 \pi}\Rightarrow z=4.37m$
 Question 5
A 5 m high vertical wall has a saturated clay backfill. The saturation unit weight and cohesion of clay are $18kN/m^3$ and 20 kPa, respectively. The angle of internal friction of clay is zero. In order to prevent development of tension zone, the height of the wall is required to be increased. Dry sand is used as backfill above the clay for the increased portion of the wall. The unit weight and angle of internal friction of sand are $16kN/m^3$ and $30^{\circ}$, respectively. Assume that the back of the wall is smooth and top of the backfill is horizontal. To prevent the development of tension zone, the minimum height (in m, round off to one decimal place) by which the wall has to be raised, is __________.
 A 1.5 B 2.5 C 6.2 D 5.2
GATE CE 2020 SET-2   Geotechnical Engineering
Question 5 Explanation: To prevent tension crack,
\begin{aligned} q&=\frac{2c}{\sqrt{k_a}}=\frac{2 \times 20}{1}=40\\ q&=\gamma _d x=40\\ x&=\frac{40}{16}=2.5m \end{aligned}

There are 5 questions to complete.

### 4 thoughts on “Stress Distribution in the Soil”

1. Some of the questions are from another topic. So they have been placed here mistakenly.
I request you to please see into this.

• Can you please write the question no

2. 3. 