Lateral Earth

PressureMUHAMMAD AZRIL BIN HEZMI

2015

Lateral Earth Pressure

� Analysis and determination of lateral earth pressure are

necessary to design retaining walls or retaining

structures.

� There are three categories of earth pressure:

� Earth pressure at rest

� Elastic equilibrium with no lateral strain taking place

� Active earth pressure

� Plastic equilibrium with lateral expansion taking place

� Passive earth pressure

� Plastic equilibrium with lateral compression taking place

Lateral Earth Pressure

Lateral Pressure (At Rest)

Lateral Pressure (At Rest)

�The horizontal effective stress is :

vh σσ ′=′ oK

φsin1Ko −=

where,

K0 = coefficient of earth pressure

at rest

vσ ′ = coefficient of earth pressure at rest

Lateral Pressure (At Rest)

- The wall is very rigid, soil behind

wall ‘at rest’ condition.

i. Sketch Lateral pressure diagram on

the Figure.

ii. Determine the lateral earth

pressure ‘at rest’ Po

iii. Determine the hydrostatic force

Pw

3.5 m

2.5 m

2/18,0,25 mkNc bo === γφ

Fill sand

2/20,0,25 mkNc sato === γφ

Silt

Lateral Pressure (At Rest)

� For both layer, Ko = 1 – sin 25o = 0.577

3.5 m

2.5 m

18 x 3.5 x 0.577 = 36.35

36.35 + (20 – 9.81) x 2.5

x 0.577 = 51.069.81 x 2.5 = 24.5

Lateral Pressure (At Rest)

� For both layer, Ko = 1 – sin 25o = 0.577

3.5 m

2.5 m

18 x 3.5 x 0.577 = 36.35

36.35 + (20 – 9.81) x 2.5

x 0.577 = 51.069.81 x 2.5 = 24.5

a

bc d

Lateral Pressure (At Rest)

� Lateral earth pressure

� Po = ½ x 36.35 x 3.5 + 36.35 x 2.5 + ½ x 14.71 x 2.5

� = 63.61 + 90.88 + 18.39

� = 172.88 kN

� Hydrostatic pressure

� Pw = ½ x 24.5 x 2.5

� = 30.63 kN

� Total horizontal pressure = 172.88 + 30.63 = 203.51kN

Active and Passive pressure

Rankine Theory

� No adhesion or friction between wall and soil.(wall is smooth)

� Failure is assumed to occur in the form of a sliding wedge along a

failure plane (see figure)

� The direction of resultants pressure is parallel to the backfill and act to

1/3 from the wall base.

� Could be used for cohesionless and cohesion material.

Rankine’s (Cohesionless soils)

Rankine’s (Cohesionless soils)

� If the backfill surface is level, (β = 0)

� or

�

φφ

sin1sin1

K a +−=

−=2

45tanK 02aφ

+=2

45tanK 2pφ

Rankine’s (Cohesionless soils)

� Active earth pressure

�

� Where;

�

�Passive earth pressure

�

� Where;

�

aa KHP2

2

1 γ=

φββ

φβββ2

22

a

coscoscos

coscoscoscosK

2

−+

−−=

pp KHP2

2

1 γ=

ap K

1K =

Rankine’s (Active)

- The wall has moved sufficiently to

develop ‘active’ condition.

i. Sketch Lateral pressure diagram on

the Figure.

ii. Determine the lateral earth

pressure ‘active’ Po

iii. Determine the hydrostatic force

Pw

2.5 m

2.8 m

0,30 == coφSand

3/4.20 mkNb =γ

3/22 mkNsat =γ

Smooth

Rankine’s (Active)

� For both layer, Ka = tan2(45 – 30/2) = 0.333

2.5 m

2.8 m

20.4 x 2.5 x 0.333 = 17

17 + (22 – 9.81) x 2.8 x

0.333 = 28.389.81 x 2.8 = 27.45

Rankine’s (Active)

� For both layer, Ka = tan2(45 – 30/2) = 0.333

2.5 m

2.8 m

20.4 x 2.5 x 0.333 = 17

17 + (22 – 9.81) x 2.8 x

0.333 = 28.389.81 x 2.8 = 27.45

a

b c d

Rankine’s (Active)

� Lateral earth pressure

� Po = ½ x 17 x 2.5 + 17 x 2.8 + ½ x 11.38 x 2.8

� = 21.25 + 47.6 + 15.93

� = 84.78 kN/m

� Hydrostatic pressure

� Pw = ½ x 27.45 x 2.8

� = 38.43 kN/m

� Total horizontal pressure = 84.78 + 38.43 = 123.2kN/m

Rankine’s (Active)

- The wall has moved sufficiently to

develop ‘active’ condition. Ground

surface behind the wall is inclined at

a slope of 3H:1V.

ii. Determine the normal and shear

forces acting on the back of this wall.6.0 m

0,30 == coφSand

3/2.19 mkNb =γ

Smooth

13

Rankine’s (Active)

� 3H:1V slope – β = tan-1(1/3) = 18o

� = 0.415

� = ½(19.2)(6)2(0.415) = 143.3kN/m

� PaH = Pacos18o = 136 kN/m

� PaV = Pasin18o = 44 kN/m

φββ

φβββ2

22

a

coscoscos

coscoscoscosK

2

−+

−−=

aa KHP2

2

1 γ=

Rankine’s (Cohesive soils)

Rankine’s (Cohesive soils)

Rankine’s (Cohesive soils)

Rankine’s (Cohesive soils)

� Active earth pressure

�

�

� Passive earth pressure

�

)2

)(2(2

1

a

aaaK

cHcKHKP

γγ −−=

γγ

22 22

2

1 ccHKHK aa +−=

cHKHKP ppp 22

1 2 += γ

Rankine’s (Cohesive soils)

� Active pressure existed behind the

wall.

� Fill of sand (1 m) was placed in

front of the wall to reduce the

movement.

� a) Determine Pa working on the

wall

� b) Determine Pp from the sand

� c) Does the sand fill sufficient to

retain the movement of the wall

5 m

1 m

0,30 == coφ

Sand

3/5.18 mkNb =γ

Clay

10,0 == cφ

3/20 mkNb =γ

Rankine’s (Cohesive soils)

� = For a clay, Ka = 1

� For sand, Kp = 2.46

� a) Pa = 250 kN/m

� b) Pp = 22.75 kN/m

� c) Pa > Pp

−=2

45tanK 02aφ

Coulomb’s Theory

�Developed nearly a century before Rankine theory.

�Assumes that failure occurs in the form of wedge and that

friction occurs between wall and soil.

Coulomb’s Theory

@

2

2

2

sinsinsinsin

1sinsin

sin

++++++

−=

β)(θ)(

β)()()(θ

)(K p

δθφδφδθ

φθ

ap K

1K =

aa KHP2

2

1 γ= pp KHP2

2

1 γ=

2

2

2

sinsinsinsin

1sinsin

sin

+−−++−

+=

β)(θ)(

β)()()(θ

)(Ka

δθφδφδθ

φθ

Coulomb’s Theory

� Calculate the active pressure on

the wall using Coulomb’s theory.

9 m

0,30 == coφ2/6.17 mkNb =γ

δ = 25o

Coulomb’s Theory

2

2

2

sinsinsinsin

1sinsin

sin

+−−++−

+=

β)(θ)(

β)()()(θ

)(Ka

δθφδφδθ

φθ

296.0

090sin2590sin030sin2530sin

12590sin90sin

3090sin2

2

2

=

+−−++−

+=

)()(

)()()(

)(Ka

mkNKHP aa /211296.096.172

1

2

1 22 =×××== γ

Types of Retaining Wall

�Gravity walls

�Embedded walls

Gravity wall

� Depend largely upon their own weight for stability, wide base and rigid

construction

� Example of gravity walls:

�Masonry walls

�Gabion walls

�Crib walls

�RC walls

Embedded walls

�Consist of vertical sheets or piles that anchored, tied or simple cantilevers

and flexible retaining structures

�Example of embedded walls:

�Driven sheet pile walls

�Anchored walls

�Secant bored-pile walls

Design Considerations

(Gravity Wall)

� Check for overturning

� Check for sliding along the base

� Check for bearing capacity failure

Check for overturning

5.2M

MFS

O

R ≥=∑∑

=∑ RM

=∑ oM

sum of the moments of forces tending to resist overturning about point C

sum of the moments of forces tending to overturn about point C

Check for sliding along the

base

0.2P

PBcVtanφ

F

FFS

h

p22

d

Rsliding ≥

++==

∑∑

∑∑

=∑ RF

=∑ dF

where,

sum of the horizontal resisting forces

sum of the horizontal driving forces

=2φ angle of friction between base and the soil (normally taken as ½ to 2/3 of friction angle of soil 2)

=2c cohesion of soil 2 (normally reduced to ½ to 2/3)

Check for bearing capacity failure

±= ∑− B6e

1B

Vq minmax

where,

6

B

2Xe ≤

−= B

∑∑ ∑

∑−

==V

MM

V

MX ORnet