# Coulombs Lateral Earth Pressure

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## Introduction on Coulomb's Lateral Earth Pressure :

Lateral earth pressure is the pressure that soil exerts in the horizontal direction. Retaining and sheet-pile walls, both braced and unbraced excavations, grain in silo walls and bins, and earth or rock contacting tunnel walls and other underground structures require a quantitative estimate of the lateral pressure on a structural member for either a design or stability analysis.

## Concepts and Formulas of Coulomb's Lateral Earth Pressure:

### Theory:

Coulomb (1776) first studied the problem of lateral earth pressures on retaining structures. He used limit equilibrium theory, which considers the failing soil block as a free body in order to determine the limiting horizontal earth pressure.

The lateral effective earth pressure is a function of effective vertical pressure

P'h = K P'v

Where

P'h lateral effective earth pressure,

P'v is vertical effective pressure,

The equation above relates EFFECTIVE stresses not TOTAL stresses! Thus, for calculation of horizontal total stress in presence of ground water table, first effective vertical stress must be calculated then based on At-restRankine, or Coulomb theory (whichever is appropriate), horizontal effective stress will be calculated. Summing the effective horizontal stress with pore water pressure at that point will result in total horizontal stress.

K is lateral earth pressure coefficient. There are three type of lateral earth pressure:

1. Active pressure: when retaining wall is moving away from soil, K=Ka.
2. Passive pressure: when retaining wall is moving against soil, K=Kp
3. At rest pressure: when earth is at rest connection such as earth pressure against basement walls, K=Ko

Figure 1. Assumed conditions for failure in Coulomb earth pressure theory

### Assumptions of Coulomb earth pressure theory:

1. Soil is isotropic and homogeneous and has both internal friction and cohesion.
2. The rupture surface is a plane surface and the backfill surface is planar (it may slope but is not irregularly shaped).
3 .The friction resistance is distributed uniformly along the rupture surface and the soil-to-soil friction coefficient f=tanf.
4. The failure wedge is a rigid body undergoing translation.
5. There is wall friction, i.e., as the failure wedge moves with respect to the back face of the wall a friction force develops between soil and wall. This friction angle is usually termed d.
6. Failure is a plane strain problem, that is, consider a unit interior slice from an infinitely long wall.

### Coulomb earth pressure coefficients

There are two commonly uses lateral earth pressure theories: Coulomb (1776) and Rankine (1857).

Coulomb active earth pressure coefficient:

$K_{a} = \frac{sin^{2}(\alpha+\phi)}{sin^{2}\alpha sin(\alpha-\delta)[1+\sqrt{\frac{sin(\phi+\delta)sin(\phi-\beta)}{sin(\alpha-\delta)sin(\alpha+\beta)} }]}$

Coulomb passive earth pressure coefficient:

$K_{p} = \frac{sin^{2}(\alpha-\phi)}{sin^{2}\alpha sin(\alpha+\delta)[1-\sqrt{\frac{sin(\phi+\delta)sin(\phi+\beta)}{sin(\alpha+\delta)sin(\alpha+\beta)} }]}$

Where

f is internal friction angle of the soil,

b is the slope of the backfill

a is the angle of the back of retaining wall

d is friction angle between soil and back of retaining wall

An illustration of differences between at-rest, active and passive states is given in At-Rest State article.

### For cohesive soils (Clays):

Neither the Coulomb nor Rankine method explicitly incorporated cohesion as an equation parameter in lateral earth pressure computations. For soils with cohesion, Bell (1915) developed an analytical solution that uses the square root of the pressure coefficient to predict the cohesion's contribution to the overall resulting pressure. These equations represent the total lateral earth pressure (not effective). The first term represents the non-cohesive contribution and the second term the cohesive contribution. The first equation is for an active situation and the second for passive situations.

$\sigma_{h}=K_{a}\sigma_{v}-2c\sqrt{K_{a}}$

$\sigma_{h}=K_{p}\sigma_{v}+2c\sqrt{K_{p}}$

## Solved sample problems of Coulomb's Lateral Earth Pressure:

### Example 1: Coulomb's lateral earth pressure with horizontal backfill on smooth vertical back face (English units)

Given:

Height of earth at heel, H = 12 ft

Height  of earth at toe, h = 2 ft

Friction angle of soil:30 degree

Horizontal backfill

Unit weight of backfill soil:g = 115 lb/ft3

Angle of back of retaining wall:a = 90 deg

Friction angle between soil and back of retaining wall:d = 0 deg

Requirement:

Using Coulomb's lateral earth pressure theory

1. determine total active force, Pa, at heel per foot width of wall

2. determine total passive force, Pp at toe per foot width of wall

Solution:

b = 20 deg

Active earth pressure coefficient:

$K_{a} = \frac{sin^{2}(\alpha+\phi)}{sin^{2}\alpha sin(\alpha-\delta)[1+\sqrt{\frac{sin(\phi+\delta)sin(\phi-\beta)}{sin(\alpha-\delta)sin(\alpha+\beta)} }]}$

Ka=0.279

Total active force:

Pa = gH2Ka/2 = 2313 lb/ft               (per one ft width of wall)

Passive earth pressure coefficient:

$K_{p} = \frac{sin^{2}(\alpha-\phi)}{sin^{2}\alpha sin(\alpha+\delta)[1-\sqrt{\frac{sin(\phi+\delta)sin(\phi+\beta)}{sin(\alpha+\delta)sin(\alpha+\beta)} }]}$

Kp=1.548

Total passive force:

Pp = gH2Kp/2 = 356 lb/ft               (per one ft width of wall)

### Example 2: Coulomb's earth pressure with slope backfill on smooth vertical back face (English units)

Given:

Height from top of earth to bottom of footing, H = 12 ft

Height from top of backfill to bottom of toe, h = 2 ft

Friction angle of soil:

Slope of backfill soil at heel: 20 deg

Slope of backfill soil at toe: -20 deg

Unit weight of backfill soil:g = 115 lb/ft3

Angle of back of retaining wall:a = 90 deg

Friction angle between soil and back of retaining wall:d = 0 deg

Requirement:

Using coulomb's lateral earth pressure theory

1. Determine total force, Pa, at heel per foot width of wall

2. Determine total passive force, Pp at toe per foot width of wall

Solution:

b = 20 deg

Active earth pressure coefficient:

$K_{a} = \frac{sin^{2}(\alpha+\phi)}{sin^{2}\alpha sin(\alpha-\delta)[1+\sqrt{\frac{sin(\phi+\delta)sin(\phi-\beta)}{sin(\alpha-\delta)sin(\alpha+\beta)} }]}$

Ka=0.441

Total active force:

Pa = gH2Ka/2 = 3652 lb/ft               (per one ft width of wall)

Passive earth pressure coefficient:

$K_{p} = \frac{sin^{2}(\alpha-\phi)}{sin^{2}\alpha sin(\alpha+\delta)[1-\sqrt{\frac{sin(\phi+\delta)sin(\phi+\beta)}{sin(\alpha+\delta)sin(\alpha+\beta)} }]}$

Kp=1.548

Total passive force:

Pp = gH2Kp/2 = 356 lb/ft               (per one ft width of wall)

### Example 3: Coulomb's earth pressure with slope backfill on rough slope back face (English units)

Given:

Height from top of earth to bottom of footing, H = 12 ft

Height from top of backfill to bottom of toe, h = 2 ft

Friction angle of soil:

Slope of backfill soil at heel: 20 deg

Slope of backfill soil at toe: -20 deg

Unit weight of backfill soil:g = 115 lb/ft3

Angle of back of retaining wall:a = 80 deg

Friction angle between soil and back of retaining wall:d = 2 0 deg

Requirement:

Using coulomb's lateral earth pressure theory

1. Determine total force, Pa, at heel per foot width of wall

2. Determine total passive force, Pp at toe per foot width of wall

Solution:

b = 20 deg

Active earth pressure coefficient:

$K_{a} = \frac{sin^{2}(\alpha+\phi)}{sin^{2}\alpha sin(\alpha-\delta)[1+\sqrt{\frac{sin(\phi+\delta)sin(\phi-\beta)}{sin(\alpha-\delta)sin(\alpha+\beta)} }]}$

Ka=0.54

Total active force:

Pa = gH2Ka/2 = 4474 lb/ft               (per one ft width of wall)

Passive earth pressure coefficient:

$K_{p} = \frac{sin^{2}(\alpha-\phi)}{sin^{2}\alpha sin(\alpha+\delta)[1-\sqrt{\frac{sin(\phi+\delta)sin(\phi+\beta)}{sin(\alpha+\delta)sin(\alpha+\beta)} }]}$

Kp=1.678

Total passive force:

Pp = gH2Kp/2 = 386 lb/ft               (per one ft width of wall)

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