This set of Geotechnical Engineering Questions and Answers focuses on “Active Earth Pressure: Rankine’s Theory – 2”.

1. For a dry backfill with no surcharge, the active earth pressure intensity is _________

a) p_{a}=K_{a} γH

b) p_{a}=γH

c) p_{a}=K_{a} H

d) p_{a}=K_{a} γ

View Answer

Explanation: The active earth pressure intensity is given by,

p

_{a}=K

_{a}γH,

where, \(\frac{σ_1}{σ_3} = \frac{σ_v}{σ_h} = K_a=\frac{1}{tan^2(45°+\frac{φ}{2})},\)

γ=unit weight of the back fill

H= height of the retaining wall.

2. The resultant active pressure per unit length of wall for dry backfill with no surcharge is _______

a) \(P_a=\frac{1}{2}K_aγH^2\)

b) P_{a}=γH^{2}

c) P_{a}=K_{a} γH^{2}

d) P_{a}=K_{a} H^{2}

View Answer

Explanation: The pressure intensity p

_{0}is given by,

p

_{a}=K

_{a}γz,

The total earth pressure P

_{0}at rest per unit length is,

\(P_a=\int_0^HK_a γz.dz\)

∴ \(P_a=\frac{1}{2}K_aγH^2\)

3. The resultant active pressure per unit length of wall for dry backfill with no surcharge acting at _________ above the base of wall.

a) H/2

b) H

c) H/6

d) H/3

View Answer

Explanation: The pressure distribution of the stresses in the retaining wall due to the backfill is triangular one. Since the pressure distribution is triangular, the resultant active pressure per unit length of wall will act at the centroid of the wall, which is at a distance of H/3 from the base of the wall.

4. For a submerged backfill, the active earth pressure is given by _________

a) p_{a}=K_{a}γ’z

b) p_{a}=K_{a}γ’z-γ_{w}z

c) p_{a}=K_{a}γ’z+γ_{w}z

d) p_{a}=K_{a}γ’z*γ_{w}z

View Answer

Explanation: For a submerged backfill, the lateral pressure is made up of two components,

- lateral pressure due to submerged weight of backfill
- lateral pressure due to water

∴ p_{a}=K_{a}γ’z+γ_{w}z.

5. If free water stands on both side of a retaining wall, the lateral earth pressure is given by ____

a) p_{a}=K_{a} γ’ z

b) p_{a}=K_{a} γ’ z-γ_{w} z

c) p_{a}=K_{a} γ’ z+γ_{w} z

d) p_{a}=K_{a} γ’ z*γ_{w} z

View Answer

Explanation: When the free water stands on both side of a retaining wall, the water pressure is not considered and the net pressure is given by,

p

_{a}=K

_{a}γ’ z.

6. If the angle of internal friction decreases, then K_{a} ___________

a) decreases

b) increases

c) equal to zero

d) does not change

View Answer

Explanation: Since the coefficient of earth pressure for active state of plastic equilibrium is given by,

\(K_a=\frac{1-sinφ}{1+sinφ}.\) Therefore, from the equation it is clear that as the angle of internal friction decreases, then K

_{a}increases.

7. For the same value of φ, the backfill is partly submerged to height H_{2} and the backfill is moist to a depth H_{1}. Find the lateral pressure intensity at the base of wall.

a) p_{a}=K_{a} γH_{1}-K_{a} γ’ H_{2}+γ_{w} H_{2}

b) p_{a}=K_{a} γH_{1}+K_{a} γ’ H_{2}+γ_{w} H_{2}

c) p_{a}=K_{a} γH_{1}+γ_{w} H_{2}

d) p_{a}=K_{a} γH_{1}+K_{a} γ’ H_{2}

View Answer

Explanation: When the backfill is partly submerged to height H

_{2}and the backfill is moist to a depth H

_{1}, lateral pressure intensity is due to:

- lateral pressure due to moist weight of backfill
- lateral pressure due to saturated weight of backfill
- lateral pressure due to water

∴ p_{a}=K_{a} γH_{1}+K_{a} γ’ H_{2}+γ_{w} H_{2}.

8. For the different value of φ, the backfill is partly submerged to height H_{2} and the backfill is moist to a depth H_{1}. Find the lateral pressure intensity at the base of wall for φ_{1}>φ_{2}.

a) p_{a}=K_{a2} γH_{1}-K_{a2} γ’ H_{2}+γ_{w} H_{2}

b) p_{a}=K_{a2} γH_{1}+K_{a2} γ’ H_{2}+γ_{w} H_{2}

c) p_{a}=K_{a2} γH_{1}+γ_{w} H_{2}

d) p_{a}=K_{a2} γH_{1}+K_{a2} γ’ H_{2}

View Answer

Explanation: For the different value of φ, the coefficient of active earth pressure is different. Since, the lateral pressure intensity is due to:

- lateral pressure due to moist weight of backfill
- lateral pressure due to saturated weight of backfill
- lateral pressure due to water

∴ p_{a}=K_{a2} γH_{1}+K_{a2} γ’ H_{2}+γ_{w} H_{2}. We have to consider K_{a2} over K_{a1} as the φ_{1}>φ_{2}.

9. If the backfill carries a uniform surcharge q, then the lateral pressure at the depth of wall H is ____________

a) p_{a}=K_{a} γz+K_{a} q

b) p_{a}=K_{a} γz-K_{a} q

c) p_{a}=K_{a} γz*K_{a} q

d) p_{a}=K_{a} γz/K_{a} q

View Answer

Explanation: When the backfill is horizontal and carries a surcharge q, then the vertical pressure increment will be by q. Due to this, the lateral pressure will increase by K

_{a}q. Hence, lateral pressure at the depth of wall H is p

_{a}=K

_{a}γz+K

_{a}q.

10. The height of fill Z_{e}, equivalent to uniform surcharge intensity is __________

a) q/γ

b) q-γ

c) q+γ

d) q*γ

View Answer

Explanation: The height of fill Z

_{e}, equivalent to uniform surcharge intensity is given by,

K

_{a}γz

_{e}=K

_{a}q,

∴ \(z_e=\frac{q}{γ}.\)

11. For finding out the active earth pressure for a backfill with sloping surface, the Rankine’s theory makes as additional assumption of ________

a) vertical and lateral stresses are normal to surcharge

b) vertical and lateral stresses are tangential to surcharge

c) vertical and lateral stresses are conjugate

d) vertical and lateral stresses are negligible

View Answer

Explanation: The additional assumption of vertical and lateral stresses are conjugate is made as, it can be shown that stresses on a given plane at a given point is parallel to another plane, the stresses on the latter plane at the same point must be parallel to the first plane.

12. The vertical and the lateral pressures have the same angle of obliquity β.

a) True

b) False

View Answer

Explanation: In Rankine’s theory, the additional assumption of the vertical and lateral stresses are conjugate is made. Being conjugate, both the vertical and the lateral pressures have the same angle of obliquity β.

13. The Rankine’s lateral pressure ratio is given by ________

a) \(K=\frac{\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)

b) \(K=\frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}} \)

c) \(K=\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)

d) \(K=cosβ\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)

View Answer

Explanation: The ratio K is the conjugate ratio or the Rankine’s lateral pressure ratio, which is given by,

\(K=\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}}, \)

where, β=surcharge angle

φ=angle of internal friction.

14. For backfill with sloping surface, the coefficient of active earth pressure is given by ______

a) \(K_a=\frac{\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)

b) \(K_a=\frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}} \)

c) \(K_a=\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)

d) \(K_a=cosβ\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)

View Answer

Explanation: The vertical pressure in case of backfill with surcharge,

σ= γzcosβ,

Therefore, from the assumption of stresses being conjugate,

\(K_a=cosβ\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}}. \)

15. When the surcharge angle reduces to zero, the coefficient of active earth pressure is given by

a) K_{a}=1

b) \(K_a=\frac{1-sinφ}{1+sinφ}\)

c) \(K_a=\frac{1+sinφ}{1-sinφ}\)

d) K_{a}=0

View Answer

Explanation: For backfill with sloping surface, the coefficient of active earth pressure is given by,

\(K_a=cosβ\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}}, \)

When the surcharge angle reduces to zero, β=0,

substituting this in the equation, we get,

\(K_a=\frac{1-sinφ}{1+sinφ}.\)

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