Retaining Wall/Earth Pressure Theories

Retaining Wall/Earth Pressure Theories

Contrasting Points of Active and Passive Pressures

Earth Pressure at Rest

for C-soil

where, σ_h = Earth pressure at rest

K₀ = Coefficient of earth pressure at rest

μ = Poissons ratio of soil ∼ 0.4

K₀ = 1 – sin ∅ → for ∅ soil.

where, ∅ = Angle of internal friction.

(K₀)_{over consolidation} = (K₀)_{normally consolidation} √OCR

where, OCR = Over Consolidation Ratio.

Type of soil - K₀

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 consolidation clay - 1.0 to 4.0

Active Earth Pressure

• Length of Failure block 
• ΔH = 0.2% of H for dense sand
  ΔH = 0.5% of H for loose sand
  ΔH = 0.4% of H for clay's
• 

where k_a = Coefficient of active earth pressure.

Passive Earth Pressure

• Length of Failure block 
• ΔH = 0.2% of H for dense sand

ΔH = 15% of H for loss sand

 or 

where k_P = Coefficient of passive earth pressure.

where, P_a = Active earth pressure.

P₀ = Earth pressure at rest.

P_P = Passive earth pressure.

Active Earth pressure by Rankine Theory

(i)  acts at H/3 from base.

where, P_a = Active earth pressure force on unit length of wall.

(ii)  acts at H/3 from base

Where γ = Submerged unit weight of soil.

(iii)

 --- acts of  from base 

 --- acts of  from base 

 --- acts at  from base 

 --- acts of  from base 

• Total Earth Pressure (p_a)

 acts at  from base

where, 

(iv)

 acts at  from base.

 acts at  from base.

 acts at  from base.

(v)

•  acts at  from base.
•  acts at  from base.
•  acts at H₂ from base.

(vi)

acts at H/3 from base but line of action 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

∝ = Angle of inclination of resultant P_a with the horizontal.

(viii)

Active Earth Pressure for Cohesive Soil

Assuming no tension crack development

 where  = Influence Factor.

• Active Earth Pressure of Any Depth z
  
• Active Earth Pressure of Surface. i.e., at z = 0 
• At 
• 

Where, z_c = depth of tension crack.

where, H_C = Critical depth = 2Z_c

= Max. possible unsupported depth.

• When Tension Cracks are not Developed

acts at centroid of trapezoidal.

• When Tension Cracks are Developed

or  acts at 

Passive Earth Pressure for Cohesive Soil

• Passive Earth Pressure at any depth 'z',

• Total P_p on Unit Length

Coulombs Wedge Theory

Assumption

1. Backfill-Dry, Cohesionless, Isotropic
2. Back of wall can be inclined
3. Backfill can be inclined
4. Friction exist between wall + soil
5. Failure plane assumed planer
6. Sliding wedge assumed rigid body
7. Position and line of action of earth pressure in advance.

For C = 0

where, ∝ = Angle of back of wall with the horizontal

β = Angle of sloping ground.

δ = Angle of roughness between soil and wall = 0° for smooth walls

Special points:

• Retaining wall are designed for active earth P.
• Ranking theory
• Overstimate → Active earth pressure
  Underestimates → Passive earth pressure