# Foundation Engineering Questions and Answers – Passive Earth Pressure: Rankine’s Theory

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This set of Foundation Engineering Multiple Choice Questions & Answers focuses on “Passive Earth Pressure: Rankine’s Theory”.

1. In case of passive state of plastic state of equilibrium, the lateral pressure is the _________
a) minor principal stress
b) major principal stress
c) major principal plane
d) minor principal plane

Explanation: In case of passive state of plastic state of equilibrium, the lateral pressure is the major principal stress while the vertical pressure is the minor principal stress.

2. The passive earth pressure intensity is given by _______
a) pp=γz cot2 α
b) pp=γz tan2 α
c) pp=γz sin2 α
d) pp=γz cos2 α

Explanation: In case of passive state of plastic state of equilibrium,
pph1, and σ3v=γz,
∴ for non-cohesive soils, σ13 tan2 α,
∴ pp=γz tan2 α.

3. The passive earth pressure intensity in terms of coefficient of passive earth pressure is _________
a) pp=Kp γH
b) pp=γH
c) pp=Kp H
d) pp=Kp γ

Explanation: The passive earth pressure intensity is given by,
pp=γz tan2 α,
since, Kp= tan2 α,
∴ pp=Kp γH.
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4. The ratio between coefficients of passive and active earth pressure is __________
a) cot2 $$(45°+\frac{φ}{2})$$
b) cos2 $$(45°+\frac{φ}{2})$$
c) tan4 $$(45°+\frac{φ}{2})$$
d) sin2 $$(45°+\frac{φ}{2})$$

Explanation: The coefficient of passive earth pressure is Kp= tan2 α and that of the coefficient of active earth pressure is $$K_a=\frac{1}{tan^2(45°+\frac{φ}{2})},$$
∴ $$\frac{K_p}{K_a}$$ = tan2 α* tan2 α
$$\frac{K_p}{K_a} =tan^4 (45°+\frac{φ}{2}).$$

5. For angle of internal friction of 250, the ratio of coefficient of passive and active earth pressure is __________
a) 2
b) 4
c) 6
d) 8

Explanation: Given,
angle of internal friction φ=250,
∴ $$\frac{K_p}{K_a} =tan^4 (45°+\frac{φ}{2}) = tan^4 (45°+\frac{25}{2}),$$
Kp/Ka =6.

6. The relation between Kp and Ka is _____________
a) Kp=Ka
b) Kp=1+Ka
c) Kp=1-Ka
d) Kp=1/Ka

Explanation: The coefficient of passive earth pressure is,
Kp= tan2 α,
the coefficient of active earth pressure is,
$$K_a=\frac{1}{tan^2(45°+\frac{φ}{2})},$$
∴ Kp=1/Ka.

7. For a cohesion less soil, the total passive pressure Pp at a depth H is ___________
a) Pp=$$\frac{1}{2}$$ Kp γH2
b) Pp=γH2
c) Pp=Kp γH2
d) Pp=Kp H2

Explanation: The pressure intensity pp is given by,
pp=Kp γz,
The total earth pressure P0 at rest per unit length is,
Pp=$$\int_0^H$$Kp γz.dz
∴ Pp=$$\frac{1}{2}$$ Kp γH2.

8. For a cohesion less soil with surcharge Q, the passive pressure intensity is ___________
a) pp=Kp γz+Kp q
b) pp=Kp γz-Kp q
c) pp=Kp γz*Kp q
d) pp=Kp γz/Kp q

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 Kp q. Hence, lateral pressure at the depth of wall H is pp=Kp γz+Kp q.

9. For a surcharge angle β, the passive earth pressure is __________
a) $$p_p=γz \frac{\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}}$$
b) $$p_p=γz \frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}}$$
c) $$p_p=γz \frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}}$$
d) $$p_p=γzcosβ \frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}}$$

Explanation: If the backfill is having its top surface inclined to the horizontal at an angle β, then the passive earth pressure is given by,
$$p_p=γzcosβ \frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}},$$
where, β= surcharge angle
γ=unit weight of the soil
φ= angle of internal friction.

10. For a surcharge angle β the coefficient of passive earth pressure is _______
a) Kp=$$\frac{\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}}$$
b) Kp=$$cosβ \frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}}$$
c) Kp=$$\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}}$$
d) Kp=$$cosβ \frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}}$$

Explanation: The vertical pressure in case of backfill with surcharge,
σ= γzcosβ,
Therefore, from the assumption of stresses being conjugate,
Kp=$$cosβ \frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}}$$

11. When the surcharge angle β is reduced to zero, then the coefficient of passive earth pressure is ______
a) 1
b) $$\frac{1-sinφ}{1+sinφ}$$
c) $$\frac{1+sinφ}{1-sinφ}$$
d) 0

Explanation: For backfill with sloping surface, the coefficient of passive earth pressure is given by,
Kp=$$cosβ \frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}},$$
When the surcharge angle reduces to zero, β=0,
substituting this in the equation, we get,
Ka=$$\frac{1+sinφ}{1-sinφ}.$$

12. In case of cohesive backfill, the passive pressure intensity is given by _______
a) pp=γz tan2 α-2c tan⁡α
b) pp=γz tan2 α+2c tan⁡α
c) pp=-2c tan⁡α
d) pp=γz tan2 α/2c tan⁡α

Explanation: Since the principal stress relationship on a failure plane is given by,
σ1=2c tan⁡α+σ3 tan2 α,
σ1=pp and σ3=γz,
∴ pp=2c tan⁡α+γztan2 α
∴ pp=γz tan2 α+2c tan⁡α.

13. In case of cohesive backfill, the total passive pressure is _________
a) Pp=$$\frac{1}{2}$$ Kp γH2+2cH tanα
b) Pp=γH2+2c tanα
c) Pp=Kp γH2-2c tanα
d) Pp=Kp H2-2c tanα

Explanation: The pressure intensity pp for a cohesive soil is given by,
pp=Kp γz+2c tanα,
The total earth pressure P0 at rest per unit length is,
Pp=$$\int_0^H$$Kp γz+2c tanα.dz
∴ Pp=$$\frac{1}{2}$$ Kp γH2+2cH tanα.

14. With respect to the flow value, the total passive pressure for cohesive backfill is _______
a) Pp=$$\frac{1}{2}$$ Nφ γH2+2cH √Nφ
b) Pp=γH2+2c √Nφ
c) Pp=Nφ γH2-2c √Nφ
d) Pp=Nφ H2-2c √Nφ

Explanation: The flow value is given by,
Nφ= tan2 α=Kp,
since the the total passive pressure for cohesive backfill is given by,
Pp=$$\frac{1}{2}$$ Kp γH2+2cH tanα,
therefore on substituting the value of Nφ in the equation, we get,
Pp=$$\frac{1}{2}$$ Nφ γH2+2cH √Nφ.

15. At the ground surface, there will be no passive pressure in the case of cohesive backfill.
a) True
b) False

Explanation: The passive pressure intensity is given by,
pp=Kp γz+2c tanα,
At the ground surface, z=0,
∴ pp=2c tanα.

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