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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:
- Lateral Earth Pressure Formula
- Rankine Analysis
- 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

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).
- 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).

Pq

Height (H)

Ps

Pw

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

γsat = 140 lbs / cf

φ = 30o

12 ft

The angle ofinternal friction φ, and the density of soil will allows be given in problems

F

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.

γsat = 140 lbs / cf

φ = 30o

12 ft

749 lbs/sf

Hydrostatic

1680 lbs/sf

From soil

Lateral earth pressure

- Rankine also developed a formula for earth retaining structures with a backfill which is not horizontal.
- Ka = cosbcos b - SQRT (cos 2b - cos 2φ ) cos b + SQRT (cos 2b - cos 2φ )

- Kp = 1 / Ka = cosbcos b + SQRT (cos 2b - cos 2φ ) cos b - SQRT (cos 2b - cos 2φ )
- Where:
- φ = angle of internal friction
- b = angle of inclined backfill

b

γsat = 140 lbs / cf

φ = 30o

b = 10o

12 ft

F

b

The angle ofinternal friction φ, the angle of backfill b and the density of soil will allows be given in problems

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

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 (SQRTKa) = -351 psf

Step 3: Find lateral pressure at h = 15 ft

σ’H = Ka γ h- 2 c (SQRTKa)

= (.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

z

Clay

γ = 100 pcf

φ = 24o

c = 270 psf

15 ft

279 psf

Lateral earth pressure

- The Coulomb Method:
- 1. Allows for friction between the retaining wall and soil
- 2. May be used for non-vertical walls
- 3. Allows for non-horizontal backfill (inclined), but must be planar
- 4. Backfill must be cohesionless for inclined backfill
- Assumes a planar slip surface, similar to Rankine
- Is used for Active and Passive conditions only
- 7. Assumes a homogeneous backfill
- 8. 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 = sin2 ( + ) cos sin (sin - d)[1 + SQRT[(sin ( + d) sin ( - b))/(sin (q - d) sin (q + b))]]2
- KP = 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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