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**Retaining Wall/earth Pressure Theories**

**CONTRASTING POINTS OF ACTIVE AND PASSIVE PRESSURES**

ACTIVE | PASSIVE | ||

1. | Very little movement is required to mobilise the active pressure.(about 0.5% horizontal strain) | 1. | Much higher movement is required to mobilise the pressure (about 2% of horizontal strain) |

2. | Failure plane is inclined at 45º + Φ/2 | 2. | Failure plane is inclined at 45º – Φ/2 |

3. | Width of sliding wedge at the top of wall is H. Cot (45 + Φ/2) | 3. | Width of sliding wedge at the top of wall is H. Cot (45 - Φ/2) |

**EARTH PRESSURE AT REST:**

Earth pressure at rest rest is the horizontal pressure acting on the rigid structure.

Earth pressure is given by

For cohesive soils the Terzaghi formula for computing K_{r} is implemented in the program:

where,

ν=poisson's ratio

For cohesionless soils the Jáky expression is used:

where, Φ=Angle of internal friction of soil

For **overconsolidated soils** the expression proposed by Schmertmann to compute the coefficient of earth pressure at rest K_{r} is used:

where, OCR = Over Consolidation Ratio.**Typical values of K**_{o:}

Type of soil - K _{o}

Dense sand - 0.4 to 0.45

Loose sand - 0.45 to 0.5

Mechanically compacted sand - 0.8 to 1.5

Normally consolidated clay - 0.5 to 0.6

Over consolidated clay - 1.0 to 4.0

** ACTIVE EARTH PRESSURE:**

ΔH = 0.2% to 0.5% of H

-length of failure block = H.cot(45^{0}+Φ/2)

-ΔH= 0.2%of H for dense sand.

- ΔH= 0.5%of H for loss sand.

-ΔH= 0.4% of H clay

K_{a}=(1-sinΦ)/(1+sinΦ)=tan^{2}(45^{0}-Φ/2)**PASSIVE EARTH PRESSURE**

Langth of failure block =H.cot(45^{0}-Φ/2)

- ΔH= 2% of H For dense sand

-ΔH=15% of H For losse sand

-k_{a} . k_{p} =1

where

k_{p} = Coefficient of passive earth pressure.

K_{p}=(1+sinΦ)/(1-sinΦ)=tan^{2}(45^{0}+Φ/2)

-P_{a} < P_{o} < P_{p}

where, P_{a} = Active earth pressure.

P_{o} = Earth pressure at rest.

P_{p} = Passive earth pressure.

**ACTIVE EARTH PRESSURE BY RANKINE THEORY:**

**(i) Cohesionless soil on vertical smooth wall:**

acts at (H/3) from base.

where,P_{a} = Active earth pressure force on unit length of wall.

**(ii) Submerged Cohesionless soil on vertical smooth wall**

acts at (H/3) from base.

where, g ' = Submerged unit weight of soil.

acts at a H_{2}+H_{1}/3 from base =

P_{a2} =K_{a}γ1H_{2}^{2}acts at H_{2}/2 from base =

acts at from base =

acts at from base =

**Total Earth Pressure (P _{a})**

** ** acts at from base

where,

**P _{a1} = k_{a}q.H** acts at

from base

acts at from base

acts at

from base.

**(v)**

- acts at (H
_{2}+H_{1}) from base. - acts at from base.
- acts at H
_{2}from base.

**(vi)**

acts at H/3 from base but line of ation is parallel is parallel to backfill.

acts at H/3 from base but line of action is parallel to backfill.**(vii)**

where, W= Weight of soil over inclined back of wall of unit length.

P_{av} = Active earth pressure on imaginary wall of vertical back.

p_{a} = Resultant active earth pressure on the wall

a = Angle of inclination of resultant P_{a }with the horizontal.

** (viii)**

where, P_{av} = Active earth pressure on Imaginary vertical back of wall. Acts parallel to back fill i.e., of b angle with hori zontal

W = Weigth of soil block of unit length.

**ACTIVE EARTH PRESSURE FOR COHESIVE SOIL**

whereN_{f} = Influence Factor.

- Active Earth Pressure of Any Depth z

- Active Earth Pressure of Surface. i.e., at z = 0

- At Z = Z
_{c}® P_{a}= O

where, Z_{C} = depth of tension crack.

where, H_{c} = Cirtical depth = 27º = max. Possible unsupported depth.

**PASSIVE EARTH PRESSURE FOR COHESIVE SOIL**

- Passive Earth Pressure at any depth ‘Z’,

- Total P
_{p}on Unit Length

- Coulombs Wedge Theory

where, a = Angle of back of wall with the hori zontal

b = Angle of sloping ground.

f = Angle of roughness between soil and wall

= 0º for smooth walls

- When Tension Cracks are not Developed

acts at centroid of trapezoidal.

- When Tension Cracks are Developed

Acts at centroid of trapezoidal

- When tension Cracks are Developed

or

acts at

**Remember:-** Cohesion decreases the active earth pressure while increases the passive earth pressure.

acts at (H/3) from base of an angledwith the normal of wall.

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