Geo Technical Engineering (lateral earth pressure)

Lateral Earth PressuresLateral Earth Pressures

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
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

Retaining WallsRetaining Walls - Applications- Applications
4
Road
Train

5
highway

6
basement wall
High-rise
building

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

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

Sheet PileSheet Pile
9
Sheet piles marked for driving

10
Sheet pile wall

11
During installation Sheet pile wall

12
Reinforced earth wallsReinforced earth walls are increasingly becoming
popular.
geosynthetics

13
Crib wallsCrib walls have been used in
Queensland.
Interlocking
stretchers and
headers
filled with
soil
Good drainage & allow plant
growth.Looks good.

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

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.

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
=

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

 For over-consolidated clays
where
pc is pre-consolidation pressure
22
OCRKK NCoOCo )()( =
o
cP
OCR
'σ
=

 Distribution of earth pressure at rest is
shown below
Total force per unit length, P0
2
00
2
1
HKP γ=
23
H

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

25

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

Earth Pressure StatesEarth Pressure States
- retaining wallsretaining walls
Active Passive
“At rest” – an intermediate state
Both are failure states

Normal stress
Shear stress
σ′1o
Active
state
Passive state
The 3 States:
At Rest

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

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.

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
- in granular soils
τ
σ
failure envelope
σv’
decreasing
σh’
Initially (K0 state)
Failure (Active
state)
soil,
active
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
- 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
- in granular soils
soil, σh’ decreases till failure occurs.
wall movement
σh’

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.

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

(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

(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′
+±

(Cont.)(Cont.)
 The variation of
with depth:
'
aσ
40
 The slip planes:

ACTIVE EARTH PRESSUREACTIVE EARTH PRESSURE
COEFFICIENTCOEFFICIENT

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

 ÷

 
= − − − ÷
 

( ) ( )
( )
( )
( )
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

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
- in granular soils
τ
σ
failure envelope
σv’
Initially (K0 state)
Failure (Active
state)
increasing
σh’
passive earth
pressure

46
- in granular soils
σv’ [σh’]passive
τ
σ
failure envelope
φ
']'[ vPpassiveh K σσ =
)2/45(tan
sin1
sin1 2
φ
φ
φ
+=
−
+
=PK
Rankine’s coefficient of
passive 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
- in granular soils
B
σv’
σh’
σh’ increases till failure
occurs.
wall movement
σh’

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.

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

Wall movement
(not to scale)
σh’
Passive state
Active state
K0 state

Rankine’s Earth PressureRankine’s Earth Pressure
TheoryTheory
52
 Assumes smooth
wall
 Applicable only on vertical
walls
PvPpassiveh KcK 2']'[ += σσ
AvAactiveh KcK 2']'[ −= σσ

PASSIVE EARTH PRESSUREPASSIVE EARTH PRESSURE
COEFFICIENTCOEFFICIENT

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
÷

 
= + + + ÷
 

( ) ( )
( )
( )
( )
( )
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
=
 
= + − + 
 
 = + + +
  

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

(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

(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:

(Cont.)(Cont.)
 The variation of
with depth:
'
pσ
59
 The slip planes:

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

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

zKpp γσ =
2
2
1
HKP pp γ=
62
Horizontal backfill with Cohesionless soil
2. Passive Case

)]('[ 11
'
HzHqKaa −++= γγσ
63
Horizontal backfill with Cohesionless, partially
submerged soil
1. Active Case

)]('[ 11
'
HzHqKpp −++= γγσ
64
Horizontal backfill with Cohesionless, partially submerged
1. Passive Case

aaa KczK '
2−= γσ
65
Horizontal backfill with Cohesive soil
1. Active Case

aK
c
z
γ
'
0
2
=
γ
uc
z
2
0 =
66
 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

γ
γ
2'
'2 2
2
2
1 c
HcKHKP aaa +−=
γ
γ
2
2 2
2
2
1 u
ua
c
HcHP +−=
67
 Active total pressure force will be
 Active total pressure force when φ = 0

2. Passive Case
 Pressure
 Passive force
Passive force when φ = 0
ppp KczK '
2+= γσ
HcKHKP ppp
'2
2
2
1
+= γ
HcHP up 2
2
1 2
+= γ
68

Sloping backfill, cohesionless soil
2. Passive case (c’=0)
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′))

Sloping backfill, cohesive soil (Mazindrani &
Ganjali, 1997)
1. Active case
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

Sloping backfill, cohesive soil
2. Passive case
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

Geo Technical Engineering (lateral earth pressure)

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