Lateral Earth Pressure Theories

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


• For over-consolidated clays

where

pc is pre-consolidation pressure

OCRKK NCoOCo )()(

o

cPOCR

'


• Distribution of earth pressure at rest is shown

below

Total force per unit length, P0

2

002

1HKP


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


Against Retaining Walls (Cont.)

zKaa

Horizontal backfill with Cohesionless soil

1. Active Case

2

2

1HKP aa



zK pp

Horizontal backfill with Cohesionless soil

2. Passive Case

2

2

1HKP pp


Against Retaining Walls (Cont.)Horizontal backfill with Cohesionless, partially submerged soil

1. Active Case

)]('[ 11

' HzHqKaa


Against Retaining Walls (Cont.)Horizontal backfill with Cohesionless, partially submerged soil

1. Passive Case

)]('[ 11

' HzHqKpp



aaa KczK '2

Horizontal backfill with Cohesive soil

1. Active Case



aK

cz

'

0

2


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



2''2 2

22

1 cHcKHKP aaa


Active total pressure force will be

Active total pressure force when = 0

22 2

22

1 uua

cHcHP


2. Passive Case

Pressure

Passive force

Passive force when = 0



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)



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)



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



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



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 )



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

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


(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


(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


(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)


(Cont.)

Passive case


(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


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


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



• 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


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)


Resisting Moment:

A table similar to following need to be prepared (neglecting Pp)


)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)


'' 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


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)


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


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


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

Lateral Earth Pressure Theories

Documents

partially

major principal

ultimate bearing

seated shear

bearing capacity

kp z1 kp

earth pressure

bc curve amp