# Retaining Wall-Earth Pressure Theories

 Question 1
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 1 Explanation: Question 2
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 2 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}
 Question 3
Let $\sigma _v'$ and $\sigma _h'$ denote the effective vertical stress and effective horizontal stress, respectively. Which one of the following conditions must be satisfied for a soil element to reach the failure state under Rankine?s passive earth pressure condition?
 A $\sigma ' _v \lt\sigma ' _h$ B $\sigma ' _v \gt\sigma ' _h$ C $\sigma ' _v = \sigma ' _h$ D $\sigma ' _v + \sigma ' _h =0$
GATE CE 2022 SET-1   Geotechnical Engineering
Question 3 Explanation:
We know, $\sigma _h'=K\sigma _v'$
For passive earth pressure,
\begin{aligned} k&=K_P \gt 1\\ \Rightarrow \frac{\sigma _h'}{\sigma _v'}&=K_P \gt 1\\ \Rightarrow \sigma _h' \gt \sigma _v' \end{aligned}
 Question 4
A retaining wall of height 10 m with clay backfill is shown in the figure (not to scale). Weight of the retaining wall is 5000 kN per m acting at 3.3 m from the toe of the retaining wall. The interface friction angle between base of the retaining wall and the base soil is $20^{\circ}$. The depth of clay in front of the retaining wall is 2.0 m. The properties of the clay backfill and the clay placed in front of the retaining wall are the same. Assume that the tension crack is filled with water. Use Rankine's earth pressure theory. Take unit weight of water, $\gamma_{w}=9.81 \mathrm{kN} / \mathrm{m}^{3}$. The factor of safety (round off to two decimal places) against sliding failure of the retaining wall after ignoring the passive earth pressure will be ________________
 A 2.25 B 3.72 C 4.29 D 1.45
GATE CE 2021 SET-1   Geotechnical Engineering
Question 4 Explanation: Depth of tension crack,
\begin{aligned} Z_{C}&=\frac{2 C}{\gamma } \text { If } \phi=0 \\ &=\frac{2 \times 30}{17.2 }=3.488 \mathrm{~m} \end{aligned}
Tension crack develops only in the back fill (clay)
The tension crack will not develop in the clay which is infront of the wall.
The total active thrust on the wall, due to the backfill, and the water in the tension crack
$F_a=k_a\cdot \frac{\gamma H^2}{2}-2C\sqrt{k_a}\cdot H+\frac{2C^2}{\gamma }+\frac{\gamma _wz_c^2}{2}$
$k_a=1 \text{ since }\phi =0$
$F_a=1 \times \frac{17.2 \times 10^2}{2}-2 \times 30\sqrt{1} \times 10+\frac{2 \times 30^2}{17.2 }+\frac{9.81 \times 3.488^2}{2}=424.326 \; kN$
Factor of safety against sliding neglecting passive earth pressure on front side, $F_s$

$F_s=\frac{\mu \cdot W}{F_a}$
$\mu =$ coefficient of friction at base $= \tan \delta$
$F_s=\frac{\tan \delta \cdot W}{F_a}=\frac{\tan 20^{\circ} \times 5000}{424.326}=4.2888\approx 4.29$
 Question 5
A vertical retaining wall of 5 m height has to support soil having unit weight of 18$kN/m^3$, effective cohesion of 12 $kN/m^2$, and effective friction angle of $30^{\circ}$. As per Rankine's earth pressure theory and assuming that a tension crack has occurred, the lateral active thrust on the wall per meter length (in kN/m, round off to two decimal places), is ______.
 A 18.25 B 21.71 C 28.25 D 32.26
GATE CE 2020 SET-1   Geotechnical Engineering
Question 5 Explanation: After tension crack \begin{aligned} P_a&=\frac{1}{2}\times 16.144(5-2.309)\\ &=21.714kN/m \end{aligned}
 Question 6
An earthen dam of height H is made of cohesive soil whose cohesion and unit weight are c and $\gamma$, respectively. If the factor of safety against cohesionis $F_c$, the Taylor's stability number ($S_n$) is
 A $\frac{\gamma H}{cF_c}$ B $\frac{cF_c}{\gamma H}$ C $\frac{c}{F_c \gamma H}$ D $\frac{F_c \gamma H}{c}$
GATE CE 2019 SET-2   Geotechnical Engineering
Question 6 Explanation:
$S_n=\frac{c}{\gamma H_c}=\frac{c}{\gamma F_cH} \;\;\;\;\left \{ \because \; F_c=\frac{H_c}{H} \right \}$
 Question 7
A retaining wall of height H with smooth vertical backface supports a backfill inclined at an angle $\beta$ with the horizontal. The backfill consists of cohesionless soil having angle of internal friction $\phi$. If the active lateral thrust acting on the wall is $P_a$, which one of the following statements is TRUE?
 A $P_a$ acts at a height H/2 from the base of the wall and at an angle $\beta$ with the horizontal B $P_a$ acts at a height H/2 from the base of the wall and at an angle $\phi$ with the horizontal C $P_a$ acts at a height H/3 from the base of the wall and at an angle $\beta$ with the horizontal D $P_a$ acts at a height H/3 from the base of the wall and at an angle $\phi$ with the horizontal
GATE CE 2019 SET-1   Geotechnical Engineering
Question 7 Explanation: Question 8
A 3 m high vertical earth retaining wall retains a dry granular backfill with angle of internal friction of $30^{\circ}$ and unit weight of 20 $kN/m^{3}$. If the wall is prevented from yielding (no movement), the total horizontal thrust (in kN per unit length) on the wall is
 A 0 B 30 C 45 D 270
GATE CE 2018 SET-2   Geotechnical Engineering
Question 8 Explanation: Soil is dry sand
$\therefore K_{0}=1-\sin \phi=1-\sin 30^{\circ}=0.5$
Total horizontal thrust
\begin{aligned} P_{0} &=\frac{1}{2} K_{0} \gamma H \cdot H=\frac{1}{2} \times \frac{1}{2} \times 20 \times 3^{2} \\ &=45 \mathrm{kN} / \mathrm{m} \end{aligned}
 Question 9
A rigid smooth retaining wall of height 7 m with vertical backface retains saturated clay as backfill. The saturated unit weight and undrained cohesion of the backfill are 17.2 kN/$m^{3}$ and 20 kPa, respectively. The difference in the active lateral forces on the wall (in kN per meter length of wall, up to two decimal places), before and after the occurrence of tension cracks is ______
 A 15.74 B 23.63 C 40.58 D 46.72
GATE CE 2018 SET-1   Geotechnical Engineering
Question 9 Explanation: \begin{aligned} \text{For clay }\phi&=0 \\ \therefore \quad k_{a}&=\frac{1-\sin 0}{1+\sin 0}=1 \\ \end{aligned}
Earth pressure when tension cracks are not developed.
\begin{aligned} P_{a} &=\frac{1}{2}(40+80.4) \times 2.349 \\ &=141.4098 \end{aligned} Earth pressure when tension cracks are developed \begin{aligned} P_{a} &=\frac{1}{2} \times 80.4 \times 4.68=188.136 \\ \text { Difference } &=141.4098-188.136 \\ &=-46.7262 \mathrm{kN} / \mathrm{m}^{2} \end{aligned}
 Question 10
Consider a rigid retaining wall with partially submerged cohesionless backfill with a surcharge. Which one of the following diagrams closely represents the Rankine's active earth pressure distribution against this wall? A A B B C C D D
GATE CE 2017 SET-2   Geotechnical Engineering
Question 10 Explanation: There are 10 questions to complete.

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

1. Truly appreciate ur efforts!
God bless u 🙂

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