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Determine resultant force of lateral earth pressure
Lateral earth pressure due to weight of soil for cohesionless soil
Lateral earth pressure due to weight of soil for cohesive soil
Lateral Earth pressure coefficients
Rankine earth pressure coefficient
Example 4.1: Rankine's lateral earth pressure with horizontal backfill
Coulomb earth pressure coefficients
Example 4.3: Coulomb's lateral earth pressure with horizontal
backfill on smooth vertical back face
Example 4.4: Coulomb's earth pressure with slope backfill on
smooth vertical back face
Example 4.5: Coulomb's earth pressure with slope backfill on
rough slope back face
The lateral earth pressure is a function of vertical pressure
P_{h} = K P_{v}
Where
p_{h} lateral earth pressure,
p_{v} 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
Total force of active earth pressure:
P_{a} = gH^{2}K_{a}/2
Total force of passive earth pressure:
P_{p} = gH^{2}K_{p}/2
Total force of at rest earth pressure:
P_{o} = gH^{2}K_{o}/2
Where:
P_{a}: total active earth pressure.
P_{p}: total passive earth pressure.
P_{o}: total at rest earth pressure.
g: unit weight (density) of soil
H: height of retaining wall
K_{a}: active earth pressure coefficient.
K_{p}: passive earth pressure coefficient.
K_{o}: at rest earth pressure coefficient.
Total force of active earth pressure:
P_{a} = gH^{2}K_{a}/2-2CÖK_{a}
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} = qHK_{a}
Total force of passive earth pressure:
P'_{p} = qHK_{p}
Total force of at rest earth pressure:
P'_{o} = qHK_{o}
Where q is the weight of surcharge
There are two commonly uses lateral earth pressure theories: Coulomb (1776) and Rankine (1857)
Rankine’s Active earth pressure coefficient
K_{a} = tan^{2}(45-f/2)^{2}
Rankine’s Passive earth pressure coefficient
K_{p} = tan^{2}(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.
Given:
Height of earth at heel
Height of
earth at toe
Friction angle of soil:
Horizontal backfill
Unit weight of backfill soil:
Requirement:
Using Rankine's lateral earth pressure
1. determine Rankine total active
force, P_{a}, at heel per foot width of wall
2. determine Rankine's total
passive force, P_{p} at
toe per foot width of wall
Solution:
Active earth pressure coefficient:
K_{a} = tan^{2}(45-f/2)^{2} = 0.333
Total active force:
P_{a} = gH^{2}K_{a}/2 = 2760 lb/ft (per one ft width of wall)
Passive earth pressure
coefficient:
K_{p} = tan^{2}(45+f/2)^{2 }= 3
Total passive force:
P_{p} = gH^{2}K_{p}/2 = 690 lb/ft (per one ft width of wall)
Given:
Height from top of earth to bottom
of footing
Height from top of backfill to
bottom of toe
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:
Requirement:
Using Rankine's lateral earth
pressure theory
1. Determine total force, P_{a},
at heel per foot width of wall
2. Determine total passive force,
P_{p} at toe per
foot width of wall
Solution:
b = 20 deg
Active earth pressure coefficient:
Total active force:
P_{a} = gH^{2}K_{a}/2 = 3430 lb/ft (per one ft width of wall)
Passive earth pressure
coefficient:
Total passive force:
P_{p} = gH^{2}K_{p}/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.
Given:
Height of earth at heel
Height of
earth at toe
Friction angle of soil:
Horizontal backfill
Unit weight of backfill soil:
Angle of back of retaining wall:
Friction angle between soil and
back of retaining wall:
Requirement:
Using Coulomb's lateral earth
pressure theory
1. determine total active force, P_{a},
at heel per foot width of wall
2. determine total passive force,
P_{p} at toe per
foot width of wall
Solution:
b = 20 deg
Active earth pressure coefficient:
Total active force:
P_{a} = gH^{2}K_{a}/2 = 2313 lb/ft (per one ft width of wall)
Passive earth pressure
coefficient:
Total passive force:
P_{p} = gH^{2}K_{p}/2 = 356 lb/ft (per one ft width of wall)
Given:
Height from top of earth to bottom
of footing
Height from top of backfill to
bottom of toe
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:
Angle of back of retaining wall:
Friction angle between soil and
back of retaining wall:
Requirement:
Using coulomb's lateral earth
pressure theory
1. Determine total force, P_{a},
at heel per foot width of wall
2. Determine total passive force,
P_{p} at toe per
foot width of wall
Solution:
b = 20 deg
Active earth pressure coefficient:
Total active force:
P_{a} = gH^{2}K_{a}/2 = 3652 lb/ft (per one ft width of wall)
Passive earth pressure
coefficient:
Total passive force:
P_{p} = gH^{2}K_{p}/2 = 356 lb/ft (per one ft width of wall)
Given:
Height from top of earth to bottom
of footing
Height from top of backfill to
bottom of toe
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:
Angle of back of retaining wall:
Friction angle between soil and
back of retaining wall:
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:
P_{a} = gH^{2}K_{a}/2 = 4474 lb/ft (per one ft width of wall)
Passive earth pressure
coefficient:
Total passive force:
P_{p} = gH^{2}K_{p}/2 = 386 lb/ft (per one ft width of wall)