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## Lateral Earth Pressure

Contents:

• Theory
• Determine resultant force of lateral earth pressure

• Lateral Earth pressure coefficients

Rankine earth pressure coefficient

## Theory

The lateral earth pressure is a function of vertical pressure

Ph = K Pv

Where

ph lateral earth pressure,

pv is vertical pressure,

K is lateral earth pressure coefficient.

There are three type of lateral earth pressure as shown in Figure 4.1

1.        Active pressure: when retaining wall is moving away from earth, 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

## Determine resultant force of lateral earth pressure

### Lateral earth pressure due to weight of soil for cohesionless soil

Total force of active earth pressure:

Pa = gH2Ka/2

Total force of passive earth pressure:

Pp = gH2Kp/2

Total force of at rest earth pressure:

Po = gH2Ko/2

Where:

Pa: total active earth pressure.

Pp: total passive earth pressure.

Po: total at rest earth pressure.

g: unit weight (density) of soil

H: height of retaining wall

Ka: active earth pressure coefficient.

Kp: passive earth pressure coefficient.

Ko: at rest earth pressure coefficient.

# Lateral earth pressure due to weight of soil for cohesive soil

Total force of active earth pressure:

Pa = gH2Ka/2-2CÖKa

where C is soil cohesion.

### Lateral earth pressure due to surcharge:

Retaining wall is often subjected to surcharge such as addition depth of soil, weight of parking vehicle or traffic, etc. Lateral earth pressure due to surcharge is normally assumed distributed uniformly along the depth. Lateral earth pressure due to surcharge are calculated as follows:

Total force of active earth pressure from surcharge:

P'a = qHKa

Total force of passive earth pressure:

P'p = qHKp

Total force of at rest earth pressure:

P'o = qHKo

Where q is the weight of surcharge

## Lateral Earth pressure coefficients

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

## Rankine earth pressure coefficient

### Retaining wall with horizontal backfill

Rankine’s Active earth pressure coefficient

Ka = tan2(45-f/2)2

Rankine’s Passive earth pressure coefficient

Kp = tan2(45+f/2)2

Where f is internal friction angle (degree) of soil.

### Retaining wall with slope backfill

Rankine’s active earth pressure coefficient Rankine’s passive earth pressure coefficient Where b is the slope of the backfill from horizontal surface.

Assumptions for Rankine’s theory:

1.        Soil is isotropic and homogeneous and is cohesionless.

2.        Soil rapture in a plane and earth pressure applies to the wall is from movement of soil wedge

3.        Back of retaining wall is a vertical plane with no friction between soil and wall.

4.        The retaining wall is long and soil is in plain strain condition.

### Example 4.1: Rankine's lateral earth pressure with horizontal backfill

Given:

Height of earth at heel, H = 12 ft

Height  of earth at toe , h = 2 ft

Friction angle of soil: f = 30 degree

Horizontal backfill,

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

Requirement:

Using Rankine's lateral earth pressure

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

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

Solution:

Active earth pressure coefficient:

Ka = tan2(45-f/2)2 = 0.333

Total active force:

Pa = gH2Ka/2 = 2760 lb/ft               (per one ft width of wall)

Passive earth pressure coefficient:

Kp = tan2(45+f/2)2 = 3

Total passive force:

Pp = gH2Kp/2 = 690 lb/ft               (per one ft width of wall)

### Example 4.2: Rankine's earth pressure with slope backfill

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: 30 degree

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

Requirement:

Using Rankine'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:  Total active force:

Pa = gH2Ka/2 = 3430 lb/ft               (per one ft width of wall)

Passive earth pressure coefficient:  Total passive force:

Pp = gH2Kp/2 = 490 lb/ft               (per one ft width of wall)

## Coulomb earth pressure coefficients

Coulomb active earth pressure coefficient: Coulomb passive earth pressure coefficient: 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

Assumptions for Coulomb’s theory:

1.        Soil is isotropic, homogeneous, and cohesionless.

2.        Soil rapture in a plane and earth pressure applies to the wall is from movement of soil wedge

3.        Back of retaining wall is a plane with friction between soil and wall distributes uniformly

4.        The retaining wall is long and soil is in plain strain condition.

### Example 4.3: Coulomb's lateral earth pressure with horizontal backfill on smooth vertical back face

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:  Total active force:

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

Passive earth pressure coefficient:  Total passive force:

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

### Example 4.4: Coulomb's earth pressure with slope backfill on smooth vertical back face

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: 30 degree

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:  Total active force:

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

Passive earth pressure coefficient:  Total passive force:

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

### Example 4.5: Coulomb's earth pressure with slope backfill on rough slope back face

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: 30 degree

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:  Total active force:

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

Passive earth pressure coefficient:  Total passive force:

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

### Related topics

Soil parameters

Lateral earth pressure