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Lateral
Earth
Pressure
References:
1) Principles of Geotechnical Engineering
(Fifth Edition)– Braja M. DAS – Chapter 12
2) Soil Mechanics and Foundations (Third
Edition), Muni BUDHU, Chapter 15
Earth Pressure at Rest
• Coefficient of earth pressure at rest, Ko
Wheres’o = gz
s’h = Ko(gz)
Note:Ko for most soils ranges between 0.5 and 1.0
o
hoK
'
'
s
s
Earth Pressure at Rest (Cont.)
• For coarse-grained soils
K0 = 1 – sin f’ (f’ - drained friction angle)
(Jaky, 1944)
• For fine-grained, normally consolidated soils
(Massarch, 1979)
100
(%)42.044.0
PIKo
Earth Pressure at Rest (Cont.)
• For overconsolidated clays
Where
pc is pre-consolidation pressure
OCRKK NCoOCo )()(
o
cPOCR
's
Earth Pressure at Rest (Cont.)
• Distribution of earth pressure at rest is shown
below
Total force per unit length, P0
2
002
1HkP g
Earth Pressure at Rest (Cont.)
Earth Pressure at Rest (Cont.)
Partially submerged soil
• Pressure on the wall can be found from
effective stress & pore water pressure
components
z ≤ H1:
zkh gs 0
' - Variation of σ’h with depth is
shown by triangle ACE
- No pore water pressure component
since water table is below z
Earth Pressure at Rest (Cont.)
Earth Pressure at Rest (Cont.)
z ≥ H1:
Lateral pressure from water
- Variation of σh’ with depth is shown by CEGB
- Variation of U with depth is shown by IJK
Total Lateral pressure is
)}('{ 110
' HzHkh ggs
)( 1Hzu w g
uhh 'ss
Earth Pressure at Rest (Cont.)
Active earth pressure occurs when the wall tilts away from the soil
(a typical free standing retaining wall)
Active earth pressure occurs when the wall tilts away from the soil
(a typical free standing retaining wall)
Active earth pressure occurs when the wall tilts away from the soil
(a typical free standing retaining wall)
Inclination of Failure Plane
q = 45 + f/2
Mohr’s Circle and Failure
Envelope
qss 2sin2
131 f
qsssss 2cos2
1
2
13131 f
State of Failure q = 45 + f/2
Pole
Mohr’s Circle and Failure
Envelope
s’1= s’3 tan2(45+f/2) + 2c tan(45+f/2)
s’3= s’1 tan2(45-f/2) - 2c tan(45-f/2)
Rankine’s Active Earth Pressure
'
os
L
B
'
B
A
'
A
z'
as
Rankine’s Theory
Assumptions
The lateral earth pressure coefficients
are valid for;
Vertical earth retaining wall
Smooth wall in which the interface between the
wall and soil is frictionless
The supported soil is homogeneous and isotropic
The soil is loose and originally in an at-rest state
Lateral earth pressures must be applied to
effective stress
Critical slip planes are oriented at 45o ± f’/2
Rankine’s Active Earth Pressure
'
os
LB
'
BA
'
Az
'
as
Frictionless wall
Before the wall move 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'
as
'
os
L
B
'
B
A
'
A
z'
as
Rankine’s Active Earth Pressure
(Cont.)
• With geometrical manipulations we get:
22
2'
''
45tan'245tan
sin1
cos'2
sin1
sin1
ffgs
f
f
f
fss
cz
c
a
oa
• For cohesionless soil, c’=0
)2
45(tan'
2'
0
' fss a
Rankine’s Active Earth Pressure
(Cont.)
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
'
'
45tanf
s
s
o
aaK
2
45f
Rankine’s Active Earth Pressure
'
os
LB
'
BA
'
Az
'
as
Frictionless wall
Before the wall move 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'
as
'
os
L
B
'
B
A
'
A
z'
as 2
45f
Pole
Basic Concepts on Earth
Pressures
a's
o's)(' ahs
)(' phs
Rankine’s Active Earth Pressure
(Cont.)
The variation of s’awith depth:
'
as
The slip planes:
Rankine’s Passive Earth Pressure
'
os
L
B B’
A A’
z'
ps
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 '
ps
2
45f
Pole
Rankine’s Passive Earth Pressure
(Cont.)
• Rankine’s passive pressure is given by:
22
2'
''
45tan'245tan
sin1
cos'2
sin1
sin1
ffgs
f
f
f
fss
cz
c
p
op
• For cohesionless soil, c’=0
)2
45(tan'
2'
0
' fss p
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:
Rankine’s Passive Earth Pressure
(Cont.)
2
2
'
'
45tanf
s
s
o
p
pK
2
45f
Rankine’s Passive Earth Pressure
'
os
L
B B’
A A’
z'
ps
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 '
ps
Basic Concepts on Earth
Pressures
a's
o's)(' ahs
)(' phs
Rankine’s Passive Earth Pressure
(Cont.) The variation of s’p
with depth: The slip planes:
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 gs
Horizontal backfill with Cohesionless soil
1. Active Case
2
2
1Hkp aa g
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
zk pp gs
Horizontal backfill with Cohesionless soil
2. Passive Case
2
2
1Hkp pp g
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
Horizontal backfill with Cohesionless, partially submerged soil
1. Active Case
)}('{ 11
' HzHqkaa ggs
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
Horizontal backfill with Cohesionless, partially submerged soil
1. Passive Case
)}('{ 11
' HzHqk pp ggs
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
aaa kczk '2 gs
Horizontal backfill with Cohesive soil
1. Active Case
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
ak
cz
g
'
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, f = 0, then ka becomes 1 (tan245 = 1)
and c=cu . Therefore,
guc
z2
0
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.)
HckHkP aaa
'2 22
1 g
Horizontal backfill with Cohesive soil
Active total pressure force will be
Active total pressure force when f = 0
HcHP ua 22
1 2 g
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
gg
2''2 2
22
1 cHckHkP aaa
Horizontal backfill with Cohesive soil (by considering the
tensile cracks into account).
Active total pressure force will be
Active total pressure force when f = 0
gg
22 2
22
1 uua
cHcHP
)2
)(2(2
1 2''2
a
aaak
cHckHkP
gg
Horizontal backfill with Cohesive soil
2. Passive Case
Pressure
Passive force
Passive force when f = 0
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
ppp kczk '2 gs
HckHkP ppp
'2 22
1 g
HcHP up 22
1 2 g
Working Example(M.Budhu - Pg 454)
Ex (cont.)
1 - Lateral earth pressure coefficients:
Layer Ka Kp
0 – 2m
2 – 6m
41.02
2545tan2
3
1
2
3045tan2
3
2
3045tan2
Ex (cont.)
2 - Lateral earth pressures distribution with depth
)219(41.0'11
azaK s)20(41.01 sa qK
)20(3
12 sa qK
)219(3
1'
12 azaK s
1
2 )4()81.920(3
1'
22 azaK s
481.92 Hwg481.92 Hwg
)4()81.920(3'22
azpK s
Ex (cont.)
3 – Calculate lateral forces and their locations
Ex (cont.)
• Location of resultant active lateral earth
pressure,
• Location of passive lateral force,
• Ratio of Moments
mforceslateralActive
Momentsza 09.2
2.215
9.450
mz p 33.13
4
96.09.450
9.430
momentsActive
momentsPassivemomentsofRatio
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 gs '
2
2
1Hkp aa g
This force acts H/3 from bottom and inclines to the horizontal
f
f
22
22
coscoscos
coscoscoscosaK
Sloping backfill, cohesionless soil
2. Passive case (c’=0)
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
zk pp gs '
2
2
1Hkp pp g
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 f)
f
f
22
22
coscoscos
coscoscoscospK
Sloping backfill, cohesive soil
1. Active case
Lateral Earth Pressure Distribution
Against Retaining Walls (Cont.)
ggs cos"'
aaa zkzk
'sin1
'sin1'20
f
f
g
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.)
ggs cos"'
ppp zkzk
cos
" p
p
kk
(Table 11.4 in page 361 gives variation of and with α, and Φ’)"
ak"
pkz
c
g
'
'cos'sincos
'8'cos
'4'coscoscos4'sin'cos2cos2*
'cos
1, 22
2
222'
2
2
"" ffg
fg
fffg
f z
c
z
c
z
ckk pa
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 ffq
aPW
Coulomb’s Earth Pressure Theory
(Cont.)
''90sinsincos
'sincoscos
2
12
2
fqq
fqqgHPa
WPa''90sin
'sin
fq
f
or
γ, H, θ, α, Φ’, 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 g
2
2
2
)cos()'cos(
)'sin()''sin(1'coscos
)'(cos
ffqq
qfak
After solving
Ka – Coulomb’s active earth pressure coefficient and given by
Note: α=0, θ=0, δ=0 then
'sin1
'sin1
f
f
ak 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 Φ’ and δ’ = Φ’/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 g
Kp – Coulomb’s passive earth pressure coefficient and given by
2
2
2
)cos()'cos(
)'sin()''sin(1'coscos
)'(cos
ffqq
qfpk
Note: α=0, θ=0, δ=0 then
'sin1
'sin1
f
f
pk
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)
Basic Concepts on Earth
Pressures
a's
o's)(' ahs
)(' phs
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