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Lateral Earth Pressures and
Retaining Walls
Lateral SupportIn geotechnical engineering, it is often necessary to
prevent lateral soil movements.
Cantilever
retaining wallBraced excavation Anchored sheet pile
Tie rod
Sheet pile
Anchor
Lateral Support
We have to estimate the lateral soil pressures acting on
these structures, to be able to design them.
Gravity Retaining
wall
Soil nailingReinforced earth wall
Soil Nailing
Sheet Pile
Sheet piles marked for driving
Sheet Pile
Sheet pile wall
Sheet Pile
During installation Sheet pile wall
Lateral Support
Reinforced earth walls are increasingly becoming popular.
geosynthetics
Lateral Support
Crib walls have been used in Queensland.
Interlocking
stretchers
and headers
filled with
soil
Good drainage & allow plant growth.
Looks good.
Retaining Walls
A retaining wall is a wall, often made of concrete, built for the purpose of retaining, or holding back, a soil mass (or other material).
Types of retaining walls
gravity wall: A simple retaining wall depending on its weight to achieve its stability
cantilever wall: a taller wall with extended toe and heel to offset the large lateral pressure tending to overturn the wall. A cantilever wall has part of the base extending underneath the backfill, and the weight of the soil above this part of the base helps prevent overturning.
Other types: retaining wall with anchor;
retaining wall with stepped back
Retaining Walls
Structure of retaining walls
Base
heel and toe
Stem
batter: the outer face of the wall which is built inward to prevent the
wall tipping over.
backfill: The material placed behind a retaining wall.
Selection of backfill soils
It is highly desirable that backfill be a select, free-draining,
granular material, such as clean sand, gravel, or broken stones.
Clayey soils make extremely objectionable backfill material
because they create excessive lateral pressure.
Retaining Walls
Earth pressure undertaken by retaining walls
(1) Three categories of earth pressure:
a. earth pressure at rest (lateral pressure cause
by earth that is prevented from lateral
movement by an unyielding wall)
b. active earth pressure (the earth pressure
exerted on the wall, when the wall moves
away from the backfill);
c. passive earth pressure (the earth pressure
exerted on the wall when the wall moves
toward the soil).
Active/Passive Earth Pressures- in granular soils
smooth wall
Wall moves
away from soil
Wall moves
towards soil
A
B
Let’s look at the soil elements A and B during the
wall movement.
A retaining wall must
a. be able to resist sliding along the base,
b. be able to resist overturning, and
c. not introduce a contact pressure on the
foundation soil beneath the base of the
wall that exceeds the allowable bearing
pressure of the foundation soil.
Types of backfill materials for retaining walls
• coarse-grained soil without admixture of fine soil particles,
very free-draining (clean sand, gravel, or broken stone);
• coarse-grained soil of low permeability, owing to
admixture of particles of silt size;
• fine silty sand; granular materials with conspicuous clay
content; or residual soil with stone;
• soft or very soft clay; organic silt; or soft silty clay;
• medium of stiff clay that may be placed in such a way that
a negligible amount of water will enter the spaces between
the chunks during floods or heavy rains.
Stability Analysis
1. Movement of retaining walls: horizontally (by sliding),
vertically (by excessive settlement and /or bearing capacity
failure of the foundation soil), and by rotation (by
overturning).
2. The checks for sliding and for overturning , to the basic laws of
statics.
3. The checks for settlement and bearing capacity of foundation
soil are done by settlement analysis and bearing capacity
analysis.
4. The factor of safety against overturning is determined by
dividing the total righting moment by the total overturning
moment. Since overturning tends to occur about the front base
of the wall (at the toe), the righting moments and the
overturning moments are computed about the toe of the wall.
5. The factor of safety against bearing capacity failure is
determined by dividing the ultimate bearing capacity by
the actual maximum contact (base) pressure.
6. Some common minimum factors of safety for sufficient
stability are as follows:
Factor of safety against sliding = 1.5 (if the passive earth
pressure of the soil at the toe in front of the wall is
neglected)
= 2.0 (if the passive earth pressure of the soil at the toe in
front of the wall is included)
Factor of safety against overturning = 1.5 (granular backfill
soil)
= 2.0 (cohesive backfill soil)
Factor of safety against bearing capacity failure =3.0
Retaining Walls
Backfill Drainage
1. The necessity to drain backfill
2. Methods of drainage
Selection of backfill soil (sand, gravel or crushedstones are highly desirable);
Placement of weep holes: 4- to 6-in. in diameter,extending through the wall for every 5 to 10 ft alongthe wall;
Placement of a perforated drain pipe longitudinallyalong the back of the wall: surrounded by filtermaterial and water drains through the filter materialinto the pipe and then through the pipe to one end ofthe wall.
4) In both cases (weep holes and drain pipes) a filter
material must be placed adjacent to the pipe to
prevent clogging, and the pipes must be kept clear
of debris.
5) Placement of a wedge of pervious material adjacent
to the wall or a “drainage blanket” of pervious
material If a less pervious material (silt, granular
soil containing clay, etc.) has to be used as backfill.
6) A highly impervious soil (clay) is very undesirable
because, in addition to the excessive lateral earth
pressure, it also is difficult to drain and may be
subject to frost action, and swelling and shrinking.
Settlement and Tilting
1. Settlement by retaining walls is inevitable, just as byany other structure resting on footings or piles.
2. Retaining walls on granular soils: Most of the expectedsettlement will occur by the time the construction of thewall and placement of backfill have been completed.
3. Retaining walls on cohesive soils: Settlement will occurslowly and for a long period of time after constructionhas been completed.
4. Retaining walls on spread footings: The amount ofsettlement can be determined using the principles ofsettlement analysis for footings.
5. Retaining walls on pile foundations: The amount ofsettlement can be determined using the principles ofsettlement analysis for pile foundations.
Non-uniformity of settlement (differential settlement)resulting in vertical cracks in walls
(1) It occurs when the bearing capacity of thefoundation soil along the wall is not uniform andthe wall itself fails to bridge across poor material.
(2) Possible remedies: a. improving the foundationsoil (e.g., by replacement, compaction, orstabilization of the soil); b. changing the width ofthe footing.
2. Tilting is commonly caused by eccentric pressureon the base of the wall. Tilting can be reduced bykeeping the resultant force near the middle of thebase. The amount of tilting may be expected to bein the order of magnitude one-tenth of 1% of theheight of wall or less.
Earth Pressure at Rest
• Coefficient of earth pressure at rest, Ko
where’o = z
’h = Ko( z)
Note:Ko for most soils ranges between 0.5 and 1.0
o
h
oK'
'
Earth Pressure at Rest (Cont.)
• For coarse-grained soils
where ’ - drained friction angle
(Jaky, 1944)
• For fine-grained, normally consolidated soils
(Massarch, 1979)100
(%)42.044.0
PIKo
sin1oK
Earth Pressure at Rest (Cont.)
• For over-consolidated clays
where
pc is pre-consolidation pressure
OCRKK NCoOCo )()(
o
cPOCR
'
Earth Pressure at Rest (Cont.)
• Distribution of earth pressure at rest is shown
below
Total force per unit length, P0
2
002
1HKP
Earth Pressure at Rest (Cont.)
Rankine’s Active Earth Pressure
'
o
LB
'
BA
'
Az
'
a
Frictionless wall
Before the wall moves the stress condition is given by circle “a”
State of Plastic equilibrium represented by circle “b”. This is the
“Rankine’s active state”
Rankine’s active earth pressure is given by'
a
'
o
L
B' B
A' A
z
'
a
• With geometrical manipulations we get:
22
2 45tan245tan
sin1
cos2
sin1
sin1
c'γzσ
c'σσ
'
a
'
o
'
a
For cohesionless soil, c’=0
)2
45(tan'
2'
0
'
a
Rankine’s Active Earth Pressure
Rankine’s Active Pressure Coefficient, Ka
The Rankine’s active pressure coefficient is
given by:
The angle between the failure planes /slip
planes and major principal plane (horizontal)
is:
2
2
'
'
45tano
aaK
245
Rankine’s Passive Earth Pressure
'
o
L
B B’
A A’
z'
p
Frictionless wall
Circle “a” gives initial state stress
condition
“Rankine’s passive state” is represented
by circle “b”
Rankine’s passive earth pressure is given
by '
p
Rankine’s Passive Earth Pressure
(Cont.)
• Rankine’s passive pressure is given by:
22
2'
''
45tan'245tan
sin1
cos'2
sin1
sin1
cz
c
p
op
For cohesionless soil, c’=0
)2
45(tan'
2'
0
'
p
2
2
'
'
45tano
p
pK
245
Rankine’s Passive Pressure Coefficient Kp
The Rankine’s passive pressure coefficient is
given by:
The angle between the failure planes /slip planes
and major principal plane (horizontal) is:
Lateral Earth Pressure Distribution
Against Retaining Walls
• There are three different cases considered:
– Horizontal backfill• Cohesionless soil
• Partially submerged cohesionless soil with surcharge
• Cohesive soil
– Sloping backfill• Cohesionless soil
• Cohesive soil
– Walls with Friction
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
zKaa
Horizontal backfill with Cohesionless soil
1. Active Case
2
2
1HKP aa
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
zK pp
Horizontal backfill with Cohesionless soil
2. Passive Case
2
2
1HKP pp
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Horizontal backfill with Cohesionless, partially submerged soil
1. Active Case
)]('[ 11
' HzHqKaa
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Horizontal backfill with Cohesionless, partially submerged soil
1. Passive Case
)]('[ 11
' HzHqKpp
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
aaa KczK '2
Horizontal backfill with Cohesive soil
1. Active Case
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
aK
cz
'
0
2
Horizontal backfill with Cohesive soil
The depth at which the active pressure becomes equal to zero (depth
of tension crack) is
For the undrained condition, = 0, then Ka becomes 1
(tan245 = 1) and c=cu . Therefore,
ucz
20
Tensile crack is taken into account when finding the total active force.
i.e., consider only the pressure distribution below the crack
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
2''2 2
22
1 cHcKHKP aaa
Horizontal backfill with Cohesive soil
Active total pressure force will be
Active total pressure force when = 0
22 2
22
1 uua
cHcHP
Horizontal backfill with Cohesive soil
2. Passive Case
Pressure
Passive force
Passive force when = 0
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
ppp KczK '2
HcKHKP ppp
'2 22
1
HcHP up 22
1 2
Sloping backfill, cohesionless soil
Earth pressure acts an angle of to
the horizontal
1. Active case (c’=0)
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
zKaa
'
2
2
1HKP aa
This force acts H/3 from bottom and inclines to the horizontal
22
22
coscoscos
coscoscoscosaK
(Table 11.2 in page 359 gives ka values for various combinations of and ’)
Sloping backfill, cohesionless soil
2. Passive case (c’=0)
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
zK pp
'
2
2
1HKP pp
This force acts H/3 from bottom and inclines to the horizontal
(Table 11.3 in page 360 gives kp values for various combinations of and )
22
22
coscoscos
coscoscoscospK
Sloping backfill, cohesive soil (Mazindrani & Ganjali, 1997)
1. Active case
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
cos"'
aaa zKzK
'sin1
'sin1'20
cz
Depth to the tensile crack is given by
cos
" aa
KK
Sloping backfill, cohesive soil
2. Passive case
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
cos"'
ppp zKzKcos
" p
p
KK
(Table 11.4 in page 361 gives variation of and with α, and Φ’)"
aK"
pK
z
c'
'sin'cos2cos2*'cos
1,
'2
2
""
z
cKK pa
'cos'sincos'
8'cos'
4'coscoscos4cos
1 22
2
222
2 z
c
z
c
Friction walls
Rough retaining walls with granular backfill. Angle of friction between the
wall and the backfill is δ'
1. Active case
Case 1: Positive wall friction in the active case (+δ )
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
Friction walls
Downward motion of soil
Wall AB A’B causes a downward motion of soil relative to
wall. Causes downward shear on the wall (fig. b)
Pa will be inclined δ’ to the normal drawn to the back face of the
retaining wall
Failure surface is BCD (advanced studies): BC curve & CD straight
Rankine’s active state exists in the zone ACD
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
Friction walls
Case 2: Negative wall friction in the active case (-δ’)
- Wall is forced to a downward motion relative to the backfill
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
Friction walls
2. Passive case
Case 1: Positive wall friction in the passive case (+δ’)
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
Friction walls
Downward motion of wall
Wall AB A’B causes a upward motion of soil relative to wall.
Causes upward shear on the wall (fig. e)
Pp will be inclined δ’ to the normal drawn to the back face of the
retaining wall
Failure surface is BCD: BC curve & CD straight
Rankine’s passive state exists in the zone ACD
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
Friction walls
Case 2: Negative wall friction in the passive case (-δ’)
- The wall is forced to a upward motion relative to the backfill
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
Coulomb’s Earth Pressure Theory Failure surface is assumed to be plane. Also, wall friction is taken
into account
Active case
Coulomb’s Earth Pressure Theory
(Cont.)
BC is a trial failure surface and the probable failure wedge is ABC
Forces acting: W - effective weight of the soil wedge; F – resultant
of the shear and normal force on the surface of failure BC; Pa –
active force per unit length
Angle of friction between soil and wall is δ’
The force triangle for wedge is shown in figure b
From the law of sines,
'sin''90sin
aPW
Coulomb’s Earth Pressure Theory
(Cont.)
''90sinsincos
'sincoscos
2
12
2HPa
WPa''90sin
'sin
or
γ, H, θ, α, ’, and δ' are constants and β is the only variable. To
determine the critical value of β for maximum Pa
0d
dPa
Coulomb’s Earth Pressure Theory
(Cont.)
2
2
1HKP aa
2
2
2
)cos()'cos(
)'sin()''sin(1'coscos
)'(cosaK
After solving
Ka – Coulomb’s active earth pressure coefficient and given by
Note: α=0, θ=0, δ=0 then
'sin1
'sin1aK Same as Rankine’s earth
pressure coefficient
Coulomb’s Earth Pressure Theory
(Cont.)
The variation of Ka for retaining walls with vertical back (θ=0)
and horizontal backfill (α=0) is given in Table 11.5 in page
367
Tables 11.6 (pages 368 & 369) and 11.7 (pages 370 & 371)
give the values of Ka for δ = ⅔ and δ = /2 respectively
(useful in retaining wall design)
Coulomb’s Earth Pressure Theory
(Cont.)
Passive case
Coulomb’s Earth Pressure Theory
(Cont.)
Similarly in the active case
2
2
1HKP pp
Kp – Coulomb’s passive earth pressure coefficient and given by
2
2
2
)cos()'cos(
)'sin()''sin(1'coscos
)'(cospK
Note: α=0, θ=0, δ=0 then
'sin1
'sin1pK
Same as Rankine’s earth
pressure coefficient
Table 11.8 in page 373 gives variation of Kp with Φ’ and δ’ (for θ=0 & α=0)
Conventional Retaining Walls
Types of Retaining Walls
Two phases in the design of a conventional
retaining walls
1.Check for stability as a whole
– Check for Overturning
– Check for Sliding
– Check for Bearing Capacity Failure
2.Check each component for Strength and the
Steel Reinforcement
– (Structural design)
Proportioning Retaining Walls
• Assume initial dimensions and check for stability
– Top of the stem ≥ 12 in (feasibility for concreting)
– Depth, D to the bottom ≥ 24 in
– Counterfort slab thickness ≈ 12 in & spacing 0.3H -0.7H
Proportioning Retaining Walls
• Assume initial dimensions and check for stability
– Top of the stem ≥ 12 in (feasibility for concreting)
– Depth, D to the bottom ≥ 24 in
– Counterfort slab thickness ≈ 12 in & spacing 0.3H -0.7H
Application of Lateral Earth
Pressure Theories
• Cantilever Walls
– Rankine active condition is
assumed along a vertical
plane goes through the
edge of the heel of the
base slab
– Consider the weight of soil
above the heel and the
weight of the concrete of
the structure'
1'
sin
sinsin
2245
Application of Lateral Earth
Pressure Theories
• Gravity Walls
– Similar type of analysis as cantilever walls (Fig.a)
– Also, Coulomb’s active earth pressure theory can be used
(Fig.b)
Figure a Figure b
Application of Lateral Earth
Pressure Theories
• Gravity Walls
– Wall friction angle, ranges are shown below for
masonry of mass concrete for various types of backfills
(If Coulomb’s active earth pressure theory is used, this
is required)
Backfill material Range of (deg.)
Gravel
Coarse sand
Fine sand
Stiff clay
Silty clay
27 - 30
20 - 28
15 - 25
15 - 20
12 - 16
Stability of Retaining Walls
• Modes of failures
– Overturn about its toe (Fig. a)
– Slide along its base (Fig. b)
– Loss of bearing capacity (Fig. c)
– Deep-seated shear failure (Fig. d)
– Excessive settlement
• Checks for stability against Overturning,
Sliding and Bearing Capacity will be
discussed only
Stability of Retaining Walls
Stability of Retaining Walls
• Modes of failures
– Overturn about its toe (Fig. a)
– Slide along its base (Fig. b)
– Loss of bearing capacity (Fig. c)
– Deep-seated shear failure (Fig. d)
– Excessive settlement
• Checks for stability against Overturning,
Sliding and Bearing Capacity will be
discussed only
Stability of Retaining Walls
Checking for Overturning
Refer to the cantilever retaining wall shown on right
(assume Rankine active pressure is acting)
Checking for Overturning
Refer to the gravity retaining wall shown on right
(assume Rankine active pressure is acting)
Check for Overturning
O
Rgoverturnin
M
MFS )(
cosah PP
where
Where
ΣMO = sum of the moments of forces that tends to overturn about point C
ΣMR = sum of the moments of forces that tends to resist overturning about C
3
'HPM hO
Overturning Moment:
(2 to 3)
Check for Overturning
Resisting Moment:
A table similar to following need to be prepared (neglecting Pp)
Check for Overturning
)3/(cos '
654321)(
HP
MMMMMMMFS
a
vgoverturnin
Factor of safety can be calculated from
Some prefer the following formula
va
goverturninMHP
MMMMMMFS
)3/(cos '
654321)(
■ Desirable value for FOS with respect to overturning
is 2 to 3
Stability Analysis - Sliding
resist
table)(seeFactor Friction Interface where
tan'
vresist
base on the Weight where
tan
or
V
VH resist
5.1
driving
resist
H
H
sF
FFS
Check for Sliding along the Base
d
Rsliding
F
FFS
'
)(
where
ΣFR’ = sum of the
horizontal resisting forces
ΣFd = sum of the
horizontal driving forces
(min. 1.5)
Check for Sliding along the Base
'' tan acs
Shear strength of the soil immediately below the base is
where angle of friction between the soil and the base slab
adhesion between the soil and the base slab'
ac
Resisting force from the bottom of the base:''' tan1 aBcBBsR
)(' tablepreviousreferVforceverticaltheofsumB
'' tan)( aBcVsRSo,
But,
Check for Sliding along the
Base
paRPBcVF 'tan' and cosad PF
cos
tan
a
pa
slidingP
PBcVFS
Minimum FOS 1.5 is required
cos
)tan( '
22
'
21
a
p
slidingP
PcBkkVFS
'
22
''
21 & ckck a
K1 and K2 are in the range of 1/2 to 2/3
Check for Sliding along the Base
Increase the width of the
base
■ Use a key to the base
slab
■ Use a deadman
anchor at the stem of
the wall
If desired FOS is not achieved
following alternatives may be
investigated
Stability Analysis - Bearing
Capacity
• Bearing Capacity
Soft Clay
Fill
Stiff Clay
Check for Bearing Capacity
Failure
maxq
qFS u
capacitybearing
uq
maxq
ultimate bearing capacity
maximum bearing pressure
Where:
(generally 3)
Net moment of forces about point C
ORnet MMM
(ΣMR and ΣMO were previously determined in Table 8.2)
Check for Bearing Capacity
Failure
If the line of action of the resultant R intersects the base slab at E
V
MXCE net
Eccentricity of the resultant R can be expressed as: CEB
e2
Pressure distribution under the base slab can be determined by
I
yM
A
Vq net where: Mnet = (ΣV)e
I = (1/12)(1)(B3)
For maximum and minimum values, y = B/2 and by substituting this
B
e
B
Vqq toe
61max B
e
B
Vqq heel
61min
and
Check for Bearing Capacity
Failure
Ultimate bearing capacity is given by
idqiqdqcicdcu FFNBFFqNFFNcq '
2
'
22
1
Where:
1
)sin1(tan21
4.01
2
'
2'
2
'
2
'
'
2
d
qd
cd
F
B
DF
B
DF
eBB
Dq
V
P
F
FF
a
i
qici
costan
1
901
1
2
'
2
2
Shape factors Fcs, Fqs & Fγs are all equal to one. Therefore not included in the above equation
Check for Bearing Capacity
Failure
maxq
qFS u
capacitybearing
Generally FOS 3 is required
Drainage Considerations
• Recall that water pressure
was added to the soil
pressure at full vertical
component ratio
• Use clean, sandy backfill
to permit liquids to drain
• Use wall drain
– drain pipe
– geonet
Drain Pipe
„Clean‟
Backfill
Gravel or
Geonet