# Retaining Wall-Earth Pressure Theories

 Question 1
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 1 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 2
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 2 Explanation:

After tension crack

\begin{aligned} P_a&=\frac{1}{2}\times 16.144(5-2.309)\\ &=21.714kN/m \end{aligned}
 Question 3
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 3 Explanation:
$S_n=\frac{c}{\gamma H_c}=\frac{c}{\gamma F_cH} \;\;\;\;\left \{ \because \; F_c=\frac{H_c}{H} \right \}$
 Question 4
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 4 Explanation:

 Question 5
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 5 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 6
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 6 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 7
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 7 Explanation:

 Question 8
The soil profile at a site consists of a 5 m thick sand layer underlain by a $c-\phi$ soil as shown in figure. The water table is found 1 m below the ground level. The entire soil mass is retained by a concrete retaining wall and is in the active state. The back of the wall is smooth and vertical. The total active earth pressure (expressed in $kN/m^{2}$ ) at point A as per Rankine's theory is _________.
 A 102.15 B 32.5 C 69.65 D 134.65
GATE CE 2016 SET-2   Geotechnical Engineering
Question 8 Explanation:
In $\mathrm{c}-\phi$ soil
Earth pressure
$\begin{array}{l} p_{\mathrm{a}}=k_{\mathrm{a}} \sigma_{\mathrm{v}}-2 \mathrm{c} \sqrt{\mathrm{k}_{\mathrm{a}}} \\ k_{\mathrm{a}}=\frac{1-\sin \phi}{1+\sin \phi}=\frac{1-\sin 24}{1+\sin 24}=0.4217 \end{array}$
Note: Below water table, water pressure will not be multiplied by $k_{\mathrm{a}}$
at point A
$\begin{array}{l} \sigma_{v}=1 \times \gamma_{b}+4 \gamma_{\text {sat }}+3 \gamma_{\text {sat }} \\ \sigma_{v}=1 \times \gamma_{b}+\left(4 \times \gamma^{\prime}+4 \times \gamma_{w}\right)+\left(3 \gamma^{\prime}+3 \gamma_{w}\right) \\ \sigma_{v}=\left(1 \times \gamma_{b}+4 \gamma^{\prime}+3 \gamma^{\prime}\right)+\left(4 \gamma_{w}+3 \gamma_{w}\right) \end{array}$
Earth Pressure at 'A'
$\begin{array}{l} p_{\mathrm{a}}=k_{\mathrm{a}} \sigma_{v}-2 c \sqrt{k_{\mathrm{a}}} \\ =k_{a}\left[\gamma_{b}+4 \gamma^{\prime}+3 \gamma^{\prime}\right]+4 \gamma_{w}+3 \gamma_{w}-2 \times c^{\prime} \sqrt{k_{a}} \\ =0.4217[16.5+4(19-9.81)+3(18.5-9.81)] \\ +7 \times 9.81-2 \times 25 \times \sqrt{0.4217} \\ p_{\mathrm{a}}=69.65 \mathrm{kN} / \mathrm{m}^{2} \end{array}$
 Question 9
A homogeneous gravity retaining wall supporting a cohesionless backfill is shown in the figure. The lateral active earth pressure at the bottom of the wall is 40 kPa.

The minimum weight of the wall (expressed in kN per m length) required to prevent it from overturning about its toe (Point P) is
 A 120 B 180 C 240 D 360
GATE CE 2016 SET-1   Geotechnical Engineering
Question 9 Explanation:

W = weight of wall
$R_{v}=$ Reaction from cohesion less backfill
\begin{aligned} &=\frac{1}{2} \times 40 \times H \\ &=\frac{1}{2} \times 40 \times 6 \\ &=120 \mathrm{kN} \end{aligned}
To prevent overturning about point P.
\begin{aligned} M_{0} &\leq M_{R} \\ R_{v} \times \frac{H}{3} &\leq W \times 2 \\ 120 \times \frac{6}{3} &\leq W \times 2\\ \Rightarrow \quad W &\geq 120 \mathrm{kN} \\ \end{aligned}
$\therefore$ Minimum weight of wall $=120kN$
 Question 10
A vertical cut is to be made in a soil mass having cohesion c, angle of internal friction $\phi$, and unit weight $\gamma$. Considering $K_{a}$ and $K_{p}$ as the coefficients of active and passive earth pressures, respectively, the maximum depth of unsupported excavation is
 A $\frac{4c}{ \gamma \sqrt{K_{p}}}$ B $\frac{2c\sqrt{K_{p}}}{ \gamma }$ C $\frac{4c\sqrt{K_{a}}}{ \gamma }$ D $\frac{4c}{ \gamma\sqrt{K_{a}} }$
GATE CE 2016 SET-1   Geotechnical Engineering
There are 10 questions to complete.

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