Soil mechanics
Lateral earth pressure
References:
1. Budhu, Muni, D. Soil Mechanics & Foundations. New York; John Wiley & Sons,
Inc, 2000.
2. Schroeder, W.L., Dickenson, S.E, Warrington, Don, C. Soils in Construction. Fifth
Edition. Upper Saddle River, New Jersey; Prentice Hall, 2004.
Learning objectives:
1. Lateral Earth Pressure Formula
2. Rankine Analysis
3. Coulomb Method
The Lateral Earth Pressure or Horizontal Pressure(stress):
- Once you find the vertical stress (σ), it is relatively simple to calculate the lateral earth
pressure for the soil.
- The key concept to understand is that the vertical pressure in soil is different than the
horizontal pressure. This is different than water, in which the vertical pressure and
horizontal pressure is the same.
- In soil the lateral Earth pressure is equal to the effective vertical stress (σ’) times a earth
pressure coefficient (K). This coefficient depends on the soil type and where the soil is
allowed to move.
- Lateral Earth Pressure at a distance (σ’H) = K * γ * h = K * σ’v
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Soil mechanics
Lateral earth pressure
Rankine Analysis: Lateral earth pressures are derived from the summation of all
individual pressure areas behind the retaining wall. These pressure area are
triangular in shape with the base of the triangle at the base of the wall for the soil
component and pore water component. Pressure areas for surcharges are
rectangular in shape. For the Rankine analysis the major assumption is that the
retaining wall is smooth wall (no friction).
Height (H)
Pq
Ps
Pw
The resultant lateral earth pressure (F), is the summation of all lateral
earth pressure components.
F = Earth Pressure due to soil (Ps) +Pore Pressure (Pw) + Surcharge (Pq)
Earth Pressure due to soil (Ps) = ½ K γ H2 (lb/ft)(kN/m)
Earth Pressure due to pore water (Pw) = ½ K γw H2 (lb/ft)(kN/m)
Earth Pressure due to surcharge (Pq) = qKH (lbs/ft)(kN/m)
Where: γ = effective unit weight (lb/cf) ; γw = density of water = 62.4pcf;
H = height of soil ; q = surcharge pressure (psf);
K = Earth pressure coefficient
The Earth Pressure has three different Coefficients, the active conditions (Ka),
passive conditions (Kp), at rest conditions (Ko).
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Soil mechanics
Lateral earth pressure
-The three different types of coefficients are below; for the Rankine’s analysis
below are the equations to determine the coefficients. When the backface of
the retaining wall is vertical and the backfill is horizontal below are the
equations.
Ka = tan2(45 – φ/2) or (1 – sin φ) / (1 + sin φ)
 Ka is known as active earth pressure coefficient, which means that the
retaining wall is moving away from the soil
Kp = tan2(45 + φ/2) or (1 – sin φ) / (1 - sin φ)
 Kp is known as passive earth pressure coefficient, which means that the
retaining wall is moving into the soil.
KO = 1 – sin φ (for sand)
KO = .44 + .42 (PI%/100)  (for clay)
 KO is known as at rest earth pressure coefficient, which means no
movement of the retaining wall
Ka < KO < Kp  Kp = 1 / Ka
φ = angle of internal friction
12 ft
γsat = 140 lbs / cf
φ = 30o
F
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The angle of internal
friction φ, and the
density of soil will
allows be given in
problems
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Example:
Find the active lateral earth pressure on the frictionless wall shown in the below figure.
Strategy: The lateral earth pressure coefficients can only be applied to the effective stresses.
You need to calculate the vertical effective stress, apply Ka, and then add the pore water
pressure.
Sand
γsat = 140 lbs / cf
φ = 30o
12 ft
Step 1: Calculate Ka.
- Ka = tan2 (45 – φ/2) = (1 – sin φ) / (1 + sin φ) = 1/3
Step 2: Calculate the vertical effective stress.
- σ’ = σ – u
- σ = γsat h = 140 lbs/cf x 12 ft = 1680 lbs/sf
- u = γw h = 62.4 lbs/cf x 12ft = 749 lbs /sf
- σ’ = 1680 – 749 = 931 lbs/sf
Step 3: Calculate the lateral effective stress.
- σ’H = Ka x σ’ \ = .333 x 931 lbs/sf = 310 lbs/sf
Step 4: Sketch the lateral earth pressure distributions.
12 ft
γsat = 140 lbs / cf
φ = 30o
1680 lbs/sf
From soil
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749 lbs/sf
Hydrostatic
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Soil mechanics
Lateral earth pressure
- Rankine also developed a formula for earth retaining structures with a
backfill which is not horizontal.
Ka = cos b cos b - SQRT (cos 2 b - cos 2 φ )
cos b + SQRT (cos 2 b - cos 2 φ )
Kp = 1 / Ka = cos b cos b + SQRT (cos 2 b - cos 2 φ )
cos b - SQRT (cos 2 b - cos 2 φ )
Where:
φ = angle of internal friction
b = angle of inclined backfill
b
γsat = 140 lbs / cf
φ = 30o
b = 10o
12 ft
b
F
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The angle of internal
friction φ, the angle of
backfill b and the
density of soil will
allows be given in
problems
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Soil mechanics
Lateral earth pressure
- The Vertical earth pressure in clay soils is different from the earth pressure in
sandy soils. The vertical earth pressure is given by the following equation.
Vertical earth pressure = γ x h – 2c
So to make that into Lateral earth pressure
Lateral Earth pressure (σ’H ) = K γ h – 2c SQRT (K)
Where:
Ka = earth pressure coefficient
c = cohesion
γ = density of the clay
h = height of the clay
- When clay is used as the backfill, it is expected that a portion of the clay layer will crack.
The thickness of the cracked zone is usually estimated by 2c/γ. In the cracked layer there
is no active pressure generated. However, usually it is estimated that the cracked layer gets
filled with water resulting in a stress due to water.
-351 psf
z
15 ft
Clay
γ = 100 pcf
φ = 24o
c = 270 psf
279 psf
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Example: What is the active horizontal
earth pressure at the surface and base after
cracks occur in the clay soil. Also what is
distance that the pressure becomes 0 (z).
Solution: Find Ka
Ka = tan2(45 – φ/2) = .42
Step 2: Find lateral pressure at h = 0 ft
σ’H = - 2 c (SQRT Ka) = -351 psf
Step 3: Find lateral pressure at h = 15 ft
σ’H = Ka γ h - 2 c (SQRT Ka)
= (.42) (100pcf) (15ft) – 351 psf
= 630 psf-351 psf = 279 psf
Step 4: Find z
use the above equation set σ’H = 0
so z = 2 c / γ SQRT(Ka)
= 2(270)/100 SQRT(.42) = 8.3 ft
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Soil mechanics
Lateral earth pressure
The Coulomb Method:
1.
2.
3.
4.
5.
6.
7.
8.
Allows for friction between the retaining wall and soil
May be used for non-vertical walls
Allows for non-horizontal backfill (inclined), but must be planar
Backfill must be cohesionless for inclined backfill
Assumes a planar slip surface, similar to Rankine
Is used for Active and Passive conditions only
Assumes a homogeneous backfill
Any surcharge must be uniform and cover entire surface of driving wedge
Earth Pressure due to soil (Ps) = ½ K (1/ sin  * cos ) γ H2 (lb/ft)(kN/m)
The Earth Pressure different Coefficients are active conditions (Ka), passive
conditions (Kp).
Ka =
KP =
sin2 ( + ) cos 
sin  (sin  - )[1 + SQRT[(sin ( + ) sin ( - b))/(sin ( - ) sin ( + b))]]2
cos2 
[1 - SQRT[(sin  sin ( - b))/(cos b)]]2
Where:  = angle of wall face from horizontal (90 degrees for vertical
 = angle of wall friction
 = angle of internal friction
b = angle of backfill (0 degrees for horizontal backfill)
b

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