Lateral Earth Pressure -...

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


• For overconsolidated clays

Where

pc is pre-consolidation pressure

OCRKK NCoOCo )()(

o

cPOCR

's


• Distribution of earth pressure at rest is shown

below

Total force per unit length, P0

2

002

1HkP g



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



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


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


'

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


(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


(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


'

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


(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


(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:


(Cont.)

2

2

'

'

45tanf

s

s

o

p

pK

2

45f


'

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


Pressures

a's

o's)(' ahs

)(' phs


(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


Against Retaining Walls (Cont.)

zkaa gs

Horizontal backfill with Cohesionless soil

1. Active Case

2

2

1Hkp aa g



zk pp gs

Horizontal backfill with Cohesionless soil

2. Passive Case

2

2

1Hkp pp g



Horizontal backfill with Cohesionless, partially submerged soil

1. Active Case

)}('{ 11

' HzHqkaa ggs



Horizontal backfill with Cohesionless, partially submerged soil

1. Passive Case

)}('{ 11

' HzHqk pp ggs



aaa kczk '2 gs

Horizontal backfill with Cohesive soil

1. Active Case



ak

cz

g

'

0

2


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



HckHkP aaa

'2 22

1 g


Active total pressure force will be

Active total pressure force when f = 0

HcHP ua 22

1 2 g



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


2. Passive Case

Pressure

Passive force

Passive force when f = 0



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)



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)



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



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



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’)



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



Friction walls

Case 2: Negative wall friction in the active case (-δ’)

- Wall is forced to a downward motion relative to the backfill



Friction walls

2. Passive case

Case 1: Positive wall friction in the passive case (+δ’)



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



Friction walls

Case 2: Negative wall friction in the passive case (-δ’)

- The wall is forced to a upward motion relative to the backfill



Coulomb’s Earth Pressure Theory

Failure surface is assumed to be plane. Also, wall friction is taken

into account

Active case


(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


(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


(Cont.)

2

2

1Hkp aa g

2

2

2

)cos()'cos(

)'sin()''sin(1'coscos

)'(cos

qq

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


(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)


(Cont.)

Passive case


(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

qq

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)


Pressures

a's

o's)(' ahs

)(' phs

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Lateral Earth Pressure -...

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