Lateral earth pressure
Contents:
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
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.
Total force of active earth pressure:
Pa = gH2Ka/2-2CÖKa
where C is soil cohesion.
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
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.
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 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)
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