Chapter (7) Lateral Earth Pressure, Summaries of Engineering

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Chapter ( 7 )

Lateral Earth

Pressure

Page ( 158 )

Foundation Engineering

Introduction

Vertical or near vertical slopes of soil are supported by retaining walls,

cantilever sheet-pile walls, sheet-pile bulkheads, braced cuts, and other

similar structures. The proper design of those structures required estimation

of lateral earth pressure , which is a function of several factors, such as (a)

type and amount of wall movement, (b) shear strength parameters of the

soil, (c) unit weight of the soil, and (d) drainage conditions in the backfill.

The following figures shows a retaining wall of height H. For similar types

of backfill.

Page ( 160 )

Foundation Engineering

At Rest Lateral Earth Pressure:

As stated above, the soil in this case is static and can’t pushes the wall with

any movement, the transformation factor of vertical pressure to horizontal

pressure in this case is "K

୭

" and the lateral earth force is termed by "P

୭

Calculation of at rest lateral earth force "۾

ܗ

" for different cases:

Firstly the value of K

୭

can be calculated as following:

K

୭

= 1 − sinϕ

Always, (at rest) lateral earth pressure at any depth (z) may be calculated as

following:

σ

୦,୭

= vertical effective pressure

σ

୴

ᇱ

× K

୭

• Pore water pressure (u)

σ

୴

ᇱ

= q + γ

ᇱ

× z , u = γ

୵

× z

→ σ

୦,୭

= σ

୴

ᇱ

× K

୭

• u

Note that the value of (u) doesn’t multiplied by any factor since the

horizontal pressure of water is the same as vertical pressure.

Page ( 161 )

Foundation Engineering

As shown there is no water table.

Firstly we calculate the vertical stress at each depth (each change):

At depth z = 0. 0 :

σ

୴

ᇱ

= q

At depth z = H:

σ

୴

ᇱ

= q + γ × H

So, lateral at rest pressure at each depth now can be calculated:

At depth z = 0. 0 :

σ

୦,୭

= qK

୭

At depth z = H:

σ

୦,୭

= (q + γ × H) × K

୭

Now calculate the lateral forces P

ଵ

and P

ଶ

and then calculate P

୭

P

ଵ

= Area of rectangle

= qK

୭

× H (per unit lenght)

P

ଶ

= Area of triangle ( 2 ) =

× (γ × H × K

୭

) × H =

× γ × H

ଶ

× K

୭

P

୭

= P

ଵ

+ P

ଶ

To find the location of P

୭

, take summation moments at point A (for ex.):

P

ଵ

×

H

+ P

ଶ

×

H

= P

୭

× zത →→ zത = ✓.

qK

୭

(q + γH)K

୭

P

୭

P

ଵ

P

ଶ

zത

γ, ϕ

Page ( 163 )

Foundation Engineering

Lateral Earth PressureTheories

Two theories are used to calculate lateral earth pressure (active and passive):

Rankine Earth Pressure theory and Coulomb’s Earth Pressure theory.

Firstly we will learn rankine earth pressure theory (the most important) and

then coulomb earth pressure theory.

Rankine Active Lateral Earth Pressure

This theory is based mainly on the assumption of neglecting friction between

the soil and the wall, so no shear forces are developed on soil particles.

As previously introduced, the soil in this case pushes the wall far away.

The transformation factor of vertical pressure to horizontal pressure in this

case is "K

ୟ

" and the lateral earth force is termed by "P

ୟ

Firstly the value of K

ୟ

can be calculated as following:

K

ୟ

= tan

ଶ

ϕ

There are different cases:

In case of granular soil (pure sand):

Exactly as the case of at rest LEP but here the transformation factor is K

ୟ

P

ୟ

= P

ଵ

+ P

ଶ

If the soil is ۱ − ૖ ܔܑܗܛ :

The clay exerts a lateral earth pressure with value of 2c √

K (in general).

In case of active earth pressure the value of K is K

a

, and when the wall

moves away from soil, the soil particles will disturbed and the cohesion of

soil will decreased, so in case of active earth pressure we subtract the lateral

earth pressure of clay because the cohesion of clay decreased.

qK

ୟ

qK

ୟ

γHK

q + γH

K

ୟ

γ , ϕ

Page ( 164 )

Foundation Engineering

The value of 2c√K is constant along the layer, and differ when the value of

C or ϕ change (i.e. constant for each layer)

As we see, at depth z=0.0, the lateral earth pressure is −2cඥK

ୟ

this negative

value (i.e. the soil will exerts a tensile stress on the wall) and this tensile

stress will causes cracking on the wall from depth z=0.0 to depth = z

c

Calculation of ܢ

܋

The lateral pressure will be zero at depth ܢ

܋

γ z

ୡ

K

ୟ

− 2c ඥ

K

ୟ

= 0. 0 → z

ୡ

2cඥK

ୟ

γK

ୟ

2cඥK

ୟ

γ ඥ

K

ୟ

×

K

ୟ

2c

γ ඥ

K

ୟ

But, we know that soil can’t develop any tension, so in design (in practice)

we modify negative pressure to be zero and design for it (more safe because

we enlarge lateral pressure) as shown:

2c ඥ

K

ୟ

γHK

ୟ

γHK

ୟ

− 2c ඥ

K

ୟ

−2cඥK

ୟ

γHK

ୟ

− 2cඥK

ୟ

γ

C, ϕ

Page ( 166 )

Foundation Engineering

α = inclination of backfill with horizontal

θ = inclination of wall with vertical

β = inclination of P

ୟ

with the normal to the wall

From trigonometry, the angle between the normal to the wall and horizontal

is θ.

Calculation of ۾

܉

P

ୟ

= Vertical force × K

ୟ

Vertical force = area of vertical pressure digram =

γH

ଶ

The value of K

ୟ

in this case is calculated from the following equation:

K

ୟ

cos

α − θ

ඥ 1 + sin

ଶ

ϕ − 2 sinϕ cos ψ

ୟ

cos

ଶ

θ൫cosα + ඥsin

ଶ

ϕ − sin

ଶ

α൯

ψ

ୟ

= sin

ି ଵ

sinα

sinϕ

൰ − α + 2θ

The angle β is:

β = tan

ି ଵ

sinϕ sinψ

ୟ

1 − sinϕ cosψ

ୟ

P

ୟ

γH

ଶ

× K

ୟ

The location of P

ୟ

is

ୌ

ଷ

from base as shown above

If we need the horizontal and vertical components of P

ୟ

P

ୟ,୦

= P

ୟ

cos(β + θ)

P

ୟ,୴

= P

ୟ

sin(β + θ)

Page ( 167 )

Foundation Engineering

Case of vertical wall and inclined backfill:

Two cases:

**1. For pure sand

2. For** ۱ − ૖ ܔܑܗܛ

For Pure sand:

Here P

ୟ

is inclined with angle ߙ with horizontal.

Effective vertical pressure = γH

P

ୟ

γH

ଶ

K

ୟ

K

ୟ

in this case is calculated from the following equation:

K

ୟ

= cos α

cos α − ඥcos

ଶ

α − cos

ଶ

ϕ

cos α + ඥcos

ଶ

α − cos

ଶ

ϕ

Or, by using ( Table 7.1Page 337) easier than the equation above.

Page ( 169 )

Foundation Engineering

P

ୟ,୦

= P

ୟ

cos(α)

P

ୟ,୴

= P

ୟ

sin(α)

Rankine Passive Lateral Earth Pressure

As previously introduced, the wall in this case pushed into the soil.

The transformation factor of vertical pressure to horizontal pressure in this

case is "K

୔

" and the lateral earth force is termed by "P

୔

Firstly the value of K

୔

can be calculated as following:

K

୔

= tan

ଶ

ϕ

The only difference between passive and active is in the formula of

calculating K.

If the soil is ۱ − ૖ ܔܑܗܛ :

In case of passive earth pressure the value of K is K

P

, and when the wall

moves into the soil, the soil particles will converges and the cohesion of soil

will increased, so in case of passive earth pressure we add the lateral earth

pressure of clay because the cohesion of clay increased.

The value of 2c √

K is constant along the layer, and differ when the value of

C or ϕ change (i.e. constant for each layer)

If there exist surcharge, will be added to vertical pressure and the final form

for passive rankine pressure will be:

σ

୦,୔

= (q + γH)K

୔

• 2c ඥ

K

୔

+2c ඥ

K

୔

2c ඥ

K

୔

γHK

୔

γHK

୔

• 2c ඥ

K

୔

Page ( 170 )

Foundation Engineering

Rankine Passive Pressure (Vertical wall and inclined backfill):

The same as rankine active pressure for this case, the only difference is in

the equation of calculating K

P

(negative sign in active transformed to

positive sign in passive). (Page 363).

Coulomb’s Lateral Earth Pressure Theory

The main assumption of this theory is considering the friction between the

wall and the soil, this friction angle between the soil and the wall is (δ).

So there exist shear stresses on the soil particles and the equations for

calculating passive lateral earth coefficient will differ from equations of

active lateral earth coefficient.

This theory deal only with granular soil (pure sand).

Coulomb’s Active Lateral Earth Pressure

General case (inclined wall and inclined backfill):

α = inclination of backfill with horizontal

θ = inclination of wall with vertical

β = inclination of wall with the horizontal

δ = friction angle between soil and wall

From trigonometry, the angle between the normal to the wall and horizontal

is θ.

δ

θ θ

δ

θ

β

β

β

Page ( 172 )

Foundation Engineering

Problems:

For the shown figure below. Plot the pressure diagram and find the resultant

force F and its location under active conditions.

Solution

Note that the value of ϕ is differ for each layer, so the value of K

ୟ

will differ

for each layer, thus the first step is to calculate the value of K

ୟ

for each layer.

K

ୟ

= tan

ଶ

ϕ

For ϕ = 32 → K

ୟ

= tan

ଶ

ଷଶ

ଶ

Calculate K

ୟ

for each layer by the same way.

The values of K

ୟ

are written on each layer on the figure above.

ϕ = 32

ఖ

γ = 110 pcf

ϕ = 30

ఖ

γ = 125 pcf

ϕ = 10

ఖ

γ = 126 pcf

c = 600 pcf

ϕ = 0

ఖ

γ = 120 pcf

c = 800 pcf

ϕ = 20

ఖ

γ = 120 pcf

c = 400 pcf

K

ୟ

K

ୟ

K

ୟ

K

ୟ

K

ୟ

Page ( 173 )

Foundation Engineering

Now, calculate the lateral earth pressure at each depth (each change) from

soil alone (i.e. vertical effective pressure x K

a

), then the water will

considered alone.

Before calculating the pressure, at each change we calculate the lateral earth

pressure just before and just after the layer because the value of K

a

is differ

before and after the layer.

The general formula for active lateral earth pressure at any depth is:

σ

୦,ୟ

q + γH

K

ୟ

− 2c ඥ

K

ୟ

γ

୵

= 62 .4pcf

q = 2ksf = 2000psf

σ

୦,ୟ

= ( 2000 + 0 ) × 0. 307 − 0 = 614 psf

Just before (K

ୟ

= 0. 307 , c = 0 ):

σ

୦,ୟ

2000 + 110 × 6

× 0. 307 − 0 = 816. 62 psf

Just after (K

ୟ

= 0. 333 , c = 0 ):

σ

୦,ୟ

2000 + 110 × 6

× 0. 333 − 0 = 885. 8 psf

Just before (K

ୟ

= 0. 333 , c = 0 ):

σ

୦,ୟ

2000 + 110 × 6 + ( 125 − 62. 4 ) × 2

× 0. 333 − 0 = 927. 5 psf

Just after (K

ୟ

= 0. 704 , c = 600 ):

σ

୦,ୟ

= ( 2000 + 110 × 6 + ( 125 − 62. 4 ) × 2 ) × 0. 704

− 2 × 600 × √ 0. 704 = 953. 9 psf

Just before (K

ୟ

= 0. 704 , c = 600 ):

σ

୦,ୟ

=

( 2000 + 110 × 6 +

( 125 − 62. 4

) × 2 +

( 126 − 62. 4

) × 9

) × 0. 704

− 2 × 600 × √ 0. 704 = 1356. 9 psf

Just after (K

ୟ

= 1 , c = 800 ):

σ

୦,ୟ

2000 + 110 × 6 +

× 2 +

× 9

× 1

− 2 × 800 × √ 1 = 1757. 6 psf

Page ( 175 )

Foundation Engineering

Now calculate the force for each shape (1 to 11) i.e. area of each shape and

then sum all of these forces to get total active lateral force P

a

P

ୟ

≅ 57214 Ib/ft

ᇱ

To calculate the location of P

ୟ

, take summation moments at point A

(Include the moment from water force “don’t forget it”)

The location of P

ୟ

is ≅ 10.7 ft (above point A) ✓.