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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}=(1sinΦ)/(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)/(1sinΦ)=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)
(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.
where, Z_{C} = depth of tension crack.
where, H_{c} = Cirtical depth = 27º = max. Possible unsupported depth.
PASSIVE EARTH PRESSURE FOR COHESIVE SOIL
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
acts at centroid of trapezoidal.
Acts at centroid of trapezoidal
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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