This document discusses lateral earth pressures and methods for estimating soil pressures on retaining structures like retaining walls. It introduces the concepts of active and passive earth pressures, which depend on whether the wall is moving towards or away from the soil. Rankine's theory is described for calculating the active and passive earth pressure coefficients (Ka and Kp) in terms of the soil friction angle. The pressure distribution behind retaining walls is illustrated, showing the higher passive pressures and lower active pressures. Formulas are provided for determining the active and passive earth pressures based on soil and design parameters.
2. Lateral SupportLateral Support
2
In geotechnical engineering, it is often
necessary to prevent lateral soil
movements.
Cantilever
retaining wall
Braced excavation Anchored sheet pile
Tie rod
Sheet pile
Anchor
3. Lateral SupportLateral Support
3
We have to estimate the lateral soil pressureslateral soil pressures
acting on these structures, to be able to design
them.
Gravity Retaining
wall
Soil nailing
Reinforced earth wall
7. Gravity Retaining WallsGravity Retaining Walls
7
cobbles
cement mortar
plain concrete or
stone masonry
They rely on their self weight to
support the backfill
They rely on their self weight to
support the backfill
8. Cantilever Retaining WallsCantilever Retaining Walls
8
They act like vertical cantilever,
fixed to the ground
They act like vertical cantilever,
fixed to the ground
Reinforced;
smaller
section than
gravity walls
13. Lateral SupportLateral Support
13
Crib wallsCrib walls have been used in
Queensland.
Interlocking
stretchers and
headers
filled with
soil
Good drainage & allow plant
growth.Looks good.
14.
15.
16.
17. Lateral Earth PressureLateral Earth Pressure
TheoriesTheories
Outline:
• Earth pressure at rest
• Rankine’s theory for active and
passive earth pressures
• Coulomb’s theory for active and
passive earth pressures
17
18.
19. Earth Pressure at RestEarth Pressure at Rest
19
In a homogeneous natural soil
deposit,
X
σh’
σv’
the ratio σh’/σv’ is a constant known as
coefficient of earth pressure at rest (Kcoefficient of earth pressure at rest (K00).).
Importantly, at K0 state, there are no
lateral strains.
Importantly, at K0 state, there are no
lateral strains.
20. Earth Pressure at RestEarth Pressure at Rest
Coefficient of earth pressure at rest, Ko
where
σ’o = γz
σ’h = Ko(γz)
Note:
Ko for most soils ranges between 0.5 and 1.0
20
o
h
oK
'
'
σ
σ
=
21. Earth Pressure at Rest (Cont.)Earth Pressure at Rest (Cont.)
For coarse-grained soils
where φ’ - drained friction angle
(Jaky, 1944)
For fine-grained, normally consolidated soils
(Massarch, 1979)
21
+=
100
(%)
42.044.0
PI
Ko
φ′−= sin1oK
22. Earth Pressure at Rest (Cont.)Earth Pressure at Rest (Cont.)
For over-consolidated clays
where
pc is pre-consolidation pressure
22
OCRKK NCoOCo )()( =
o
cP
OCR
'σ
=
23. Earth Pressure at Rest (Cont.)Earth Pressure at Rest (Cont.)
Distribution of earth pressure at rest is
shown below
Total force per unit length, P0
2
00
2
1
HKP γ=
23
H
24. 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 γσ 0
'
=
24
- Variation of σ’h with depth is
shown by triangle ACE
- No pore water pressure component
since water table is below z
26. 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 −+= γγσ
26
)( 1Hzu w −= γ
uhh += '
σσ
27. Earth Pressure StatesEarth Pressure States
- retaining walls- retaining walls
Active Passive
“At rest” – an intermediate state
Both are failure states
29. The 3 States – consider a verticalThe 3 States – consider a vertical
retaining wallretaining wall
σ′H/σ′z
Wall movement
Kp
Ka
NB: Passive needs LARGE strains
KO
30. Active/Passive Earth PressuresActive/Passive Earth Pressures
30
- 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.
31. Active Earth PressureActive Earth Pressure
31
- in granular soils
As the wall moves away from the
soil,
Initially, there is no lateral
movement.
σv’ = γz
∴σh’ = K0 σv’ = K0 γz
σv’ remains the same; and
σh’ decreases till failure
occurs.
Active
state
Active
state
32. Active Earth PressureActive Earth Pressure
32
- in granular soils
τ
σ
failure envelope
σv’
decreasing
σh’
Initially (K0 state)
Failure (Active
state)
As the wall moves away from the
soil,
active
earth
pressure
33. Active Earth PressureActive Earth Pressure
33
- in granular soils
σv’[σh’]activ
e
τ
σ
failure envelope
φ
']'[ vAactiveh K σσ =
)2/45(tan
sin1
sin1 2
φ
φ
φ
−=
+
−
=AK
Rankine’s coefficient of
active earth pressure
WJM Rankine
(1820-1872)
34. Active Earth PressureActive Earth Pressure
34
- in granular soils
σv’[σh’]activ
e
τ
σ
failure envelope
φ
A
σv’
σh’45 +
ϕ/2
90+ϕ
Failure plane is at
45 + φ/2 to
horizontal
35. Active Earth PressureActive Earth Pressure
35
- in granular soils
As the wall moves away from the
soil, σh’ decreases till failure occurs.
wall movement
σh’
36. Active Earth PressureActive Earth Pressure
36
- in cohesive soils
Follow the same steps
as for granular soils.
Only difference is that
c ≠ 0.
AvAactiveh KcK 2']'[ −= σσ
Everything else the same as
for granular soils.
37. Rankine’s Active Earth PressureRankine’s Active Earth Pressure
'
aσ
37
'
oσ
∆
L
B
'
BA
'
Az
'a
σ
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
'
oσ
∆L
B' B
A' A
z
'
aσ
38. Rankine’s Active Earth PressureRankine’s Active Earth Pressure
(Cont.)(Cont.)
With geometrical manipulations we get:
( ) ( )22
2
45tan245tan
sin1
cos
2
sin1
sin1
φφ
φ
φ
φ
φ
′′
−−−=
′+
′
−
′+
′−
=
c'γzσ
c'σσ
'
a
'
o
'
a
)
2
45(tan
'
2'
0
' φ
σσ −=a
38
For cohesionless soil, c’=0
39. Rankine’s Active Earth PressureRankine’s Active Earth Pressure
(Cont.)(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
'
'
45tan φ
σ
σ ′
−==
o
a
aK
39
( )245 φ′
+±
40. Rankine’s Active Earth PressureRankine’s Active Earth Pressure
(Cont.)(Cont.)
The variation of
with depth:
'
aσ
40
The slip planes:
42. 1
2 tan
1
sin
2
1 1
sin sin
2 2 tan
2
1 sin sin cos
2
sin 1 sin cos
2
(1 sin ) 1 sin cos
1 sin 2 cos
1 sin 1 sin
A
A
A A
A A
A A
A
A
K c
R z
K
r z R
K K c
z z
c
K K
z
c
K K
z
c
K
z
c
K
z
γ
φ
γ φ
γ γ φ φ
φ
φ φ φ
γ
φ φ φ
γ
φ φ φ
γ
φ φ
φ γ φ
+
= + ÷
−
= = ÷
− +
= + × ÷ ÷
− = + +
− − = − + +
+ = − −
−
= − ÷
+ +
( ) ( )2 2
tan 45 tan 45
2 2A
c
K
z
φ φ
γ
÷
= − − − ÷
43. ( ) ( )
( )
( )
( )
2
NOTE:
1 sincos
1 sin 1 sin
1 sin 1 sin
1 sin
1 sin
1 sin
tan 45
2
φφ
φ φ
φ φ
φ
φ
φ
φ
−
=
+ +
− +
=
+
−
=
+
= −
( ) ( )tan 45 2 tan 45
2 2AP z cφ φγ = − − −
Thus, the active earth pressure coefficient is as shown on the
previous page and the active earth pressure is
44. Passive Earth PressurePassive Earth Pressure
44
- in granular soils
B
σv’
σh’
Initially, soil is in K0 state.
As the wall moves towards the soil,
σv’ remains the same, and
σh’ increases till failure
occurs.
Passive state
45. Passive Earth PressurePassive Earth Pressure
45
- in granular soils
τ
σ
failure envelope
σv’
Initially (K0 state)
Failure (Active
state)
As the wall moves towards the soil,
increasing
σh’
passive earth
pressure
47. Passive Earth PressurePassive Earth Pressure
47
- in granular soils
σv’ [σh’]passive
τ
σ
failure envelope
φ
A
σv’
σh’
90+ϕ
Failure plane is at
45 - φ/2 to
horizontal
45 - ϕ/2
48. Passive Earth PressurePassive Earth Pressure
48
- in granular soils
B
σv’
σh’
As the wall moves towards the soil,
σh’ increases till failure
occurs.
wall movement
σh’
49. Passive Earth PressurePassive Earth Pressure
49
- in cohesive soils
Follow the same steps
as for granular soils.
Only difference is that
c ≠ 0.
PvPpassiveh KcK 2']'[ += σσ
Everything else the
same as for granular
soils.
50. Earth Pressure DistributionEarth Pressure Distribution
50
- in granular soils
[σh’]passive
[σh’]active
H
KAγHKPγh
PA=0.5 KAγH2
PP=0.5 KPγh2
PA and PP are
the resultant
active and
passive thrusts
on the wall
54. 1
2 tan
1
sin
2
1 1
sin sin
2 2 tan
2
1 sin sin cos
2
sin 1 sin cos
2
(1 sin ) 1 sin cos
1 sin 2 cos
1 sin 1 sin
P
P
P P
P P
P P
P
P
K c
R z
K
r z R
K K c
z z
c
K K
z
c
K K
z
c
K
z
c
K
z
γ
φ
γ φ
γ γ φ φ
φ
φ φ φ
γ
φ φ φ
γ
φ φ φ
γ
φ φ
φ γ φ
+
= + ÷
−
= = ÷
− +
= + × ÷ ÷
− = + +
− = + +
− = − −
+
= + ÷
− −
( ) ( )2 2
tan 45 tan 45
2 2P
c
K
z
φ φ
γ
÷
= + + + ÷
55. ( ) ( )
( )
( )
( )
( )
2
2
NOTE:
1 sincos
1 sin 1 sin
1 sin 1 sin
1 sin
1 sin
1 sin
tan 45
2
tan 45
2
φφ
φ φ
φ φ
φ
φ
φ
φ
φ
−
=
− −
+ −
=
−
+
=
−
= +
= +
Thus the passive pressure is,
( ) ( )
( ) ( )
2
tan 45 tan 45
2 2
tan 45 2 tan 45
2 2
P P
P
P K z
c
z
z
P z c
γ
φ φ γ
γ
φ φγ
=
= + − +
= + + +
56. Rankine’s Passive Earth PressureRankine’s Passive Earth Pressure
'
pσ
56
'
oσ
∆L
B B’
A A’
z
'
pσ
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
57. Rankine’s Passive Earth PressureRankine’s Passive Earth Pressure
(Cont.)(Cont.)
Rankine’s passive pressure is given by:
( ) ( )22
2'
''
45tan'245tan
sin1
cos
'2
sin1
sin1
φφ
γσ
φ
φ
φ
φ
σσ
′′
+++=
′−
′
+
′−
′+
=
cz
c
p
op
57
For cohesionless soil, c’=0
)
2
45(tan
'
2'
0
' φ
σσ +=p
58. Rankine’s Passive Earth PressureRankine’s Passive Earth Pressure
(Cont.)(Cont.)
( )2
2
'
'
45tan φ
σ
σ ′
+==
o
p
pK
( )245 φ′
−±
58
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:
59. Rankine’s Passive Earth PressureRankine’s Passive Earth Pressure
(Cont.)(Cont.)
The variation of
with depth:
'
pσ
59
The slip planes:
60. Lateral Earth Pressure DistributionLateral Earth Pressure Distribution
Against Retaining WallsAgainst 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
60
61. Lateral Earth Pressure DistributionLateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Against Retaining Walls (Cont.)
zKaa γσ =
2
2
1
HKP aa γ=
61
Horizontal backfill with Cohesionless soil
1. Active Case
62. Lateral Earth Pressure DistributionLateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Against Retaining Walls (Cont.)
zKpp γσ =
2
2
1
HKP pp γ=
62
Horizontal backfill with Cohesionless soil
2. Passive Case
63. Lateral Earth Pressure DistributionLateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Against Retaining Walls (Cont.)
)]('[ 11
'
HzHqKaa −++= γγσ
63
Horizontal backfill with Cohesionless, partially
submerged soil
1. Active Case
64. Lateral Earth Pressure DistributionLateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Against Retaining Walls (Cont.)
)]('[ 11
'
HzHqKpp −++= γγσ
64
Horizontal backfill with Cohesionless, partially submerged
1. Passive Case
65. Lateral Earth Pressure DistributionLateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Against Retaining Walls (Cont.)
aaa KczK '
2−= γσ
65
Horizontal backfill with Cohesive soil
1. Active Case
66. Lateral Earth Pressure DistributionLateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Against Retaining Walls (Cont.)
aK
c
z
γ
'
0
2
=
γ
uc
z
2
0 =
66
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, φ = 0, then Ka becomes 1
(tan2
45° = 1) and c=cu . Therefore,
Tensile crack is taken into account when finding the total
active force. i.e., consider only the pressure distribution
below the crack
67. Lateral Earth Pressure DistributionLateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Against Retaining Walls (Cont.)
γ
γ
2'
'2 2
2
2
1 c
HcKHKP aaa +−=
γ
γ
2
2 2
2
2
1 u
ua
c
HcHP +−=
67
Horizontal backfill with Cohesive soil
Active total pressure force will be
Active total pressure force when φ = 0
68. Horizontal backfill with Cohesive soil
2. Passive Case
Pressure
Passive force
Passive force when φ = 0
Lateral Earth Pressure DistributionLateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Against Retaining Walls (Cont.)
ppp KczK '
2+= γσ
HcKHKP ppp
'2
2
2
1
+= γ
HcHP up 2
2
1 2
+= γ
68
69. Sloping backfill, cohesionless soil
2. Passive case (c’=0)
Lateral Earth Pressure DistributionLateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Against Retaining Walls (Cont.)
zKpp γσ ='
2
2
1
HKP pp γ=
φαα
φαα
α
′−−
′−+
⋅=
22
22
coscoscos
coscoscos
cospK
69
This force acts H/3 from bottom and inclines α to the horizontal
(Table 11.3 in page 360 gives kpp values for various combinations ofvalues for various combinations of αα andand φ′φ′))
70. Sloping backfill, cohesive soil (Mazindrani &
Ganjali, 1997)
1. Active case
Lateral Earth Pressure DistributionLateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Against Retaining Walls (Cont.)
αγγσ cos"'
aaa zKzK ==
'sin1
'sin1'2
0
φ
φ
γ −
+
=
c
z
αcos
" a
a
K
K =
70
Depth to the tensile crack is given by
71. Sloping backfill, cohesive soil
2. Passive case
Lateral Earth Pressure DistributionLateral Earth Pressure Distribution
Against Retaining Walls (Cont.)Against Retaining Walls (Cont.)
αγγσ cos"'
ppp zKzK ==
αcos
" p
p
K
K =
71
(Table 11.4 in page 361 gives variation of and withwith αα, and, and ΦΦ’)’)
"
pK
z
c
γ
'
+= 'sin'cos2cos2*
'cos
1
,
'
2
2
""
φφ
γ
α
φ z
c
KK pa
( )
+
+−
′
± 'cos'sincos
'
8'cos
'
4'coscoscos4
cos
1 22
2
222
2
φφα
γ
φ
γ
φαα
φ z
c
z
c