1. GEOTECHNICAL ENGINEERING - II
Engr. Nauman Ijaz
LATERAL EARTH PRESSURE
Chapter # 04
UNIVERSITY OF SOUTH ASIA

2. LATERAL EARTH
PRESSURE
Lateral Earth pressure is an important
parameter for the design of bridge abutment,
different types of retaining walls (Such as
gravity retaining walls, cantilever walls,
counterforts or buttresses), sheet piles and
other retaining structures.

3. GRAVITY RETAINING
WALL

4. CANTILEVER WALL

5. COUNTERFORT OR BUTTRESS
RETAINING WALL

6. SHEET PILES

7. BRIDGE ABUTMENT

8. LATERAL EARTH PRESSURE
AND WALL MOVEMENT
Lateral earth pressure are the direct
result of horizontal stresses in the soil.
In order to understand the lateral earth
pressure we have to define the
Coefficient of lateral earth pressure, K.

9. COEFFICIENT OF LATERAL
EARTH PRESSURE “K”
It is defined as the;
“Ratio of the horizontal effective stress to the
vertical effective stress at any point in a soil.”
K = σ’x / σ’z
K
= Coefficient of lateral earth pressure.
σ’x = Horizontal effective stresses.
σ’z = Vertical effective stresses.

10. 1.
2.
3.
K is important because it is an indicator
of the lateral earth pressures acting on
retaining wall.
For purpose of describing lateral earth
pressures, geotechnical engineers have
defined three important soil conditions;
At – rest Condition
The Active Condition
Passive condition

11. Two classic Earth pressure theories has
been put forward in the eighteen and
nineteen centuries by Coulomb and Rankine
respectively.
1) Rankine (1857) Earth Pressure Theory
2) Coulomb’s(1776) Earth Pressure Theory
These two theories are still in use in their original
form and in some modified forms to calculate the
earth pressure.

12. Consider an element of soil at depth z
below the ground surface level (GSL) as
shown in the figure.
The vertical stress due to the self weight of
soil, σ’z (also known as overburden pressure
or gravitational stress) is given by;
σ’z = γz
Where;
γ = unit weight of in-situ soil

13. Figure # (a)

14. Figure # (b)

15. When confined (as in general case
below GSL due to the pressure of
surrounding soil), this vertical
stress,(σz) will tend to cause the
expansion of soil element and in doing
so a secondary lateral pressure is
generated.
These vertical (σz) and horizontal (σx)
stresses are the major and minor
principal stresses in this particular
case respectively.

16. The ratio of σx to σz is termed as the co-efficient
of earth pressure at rest and denoted by Ko.
Thus;
Ko = σx / σz = σ3 / σ1………..(a)
Ko value in general is variable and depends upon t
soil type and its history of deposition
Numerous relations have been derived for its
evaluation, but the following relationships given by
Jaky (1948) is commonly used;
Ko = 1 – SinΦ’ under root(OCR)……(b)

17. Φ’
= Effective angle of internal friction.
OCR = Over-consolidation Ratio.
For normally consolidated soils, the equation(b) is
reduced to;
Ko = 1 – SinΦ’

18. TYPICAL VALUES OF Ko
SOIL TYPE
Ko
LOOSE SAND
0.59
DENSE ASND
0.36
NORMALLY CONSOLIDATED CLAY
0.56 – 0.80
PRECONSOLIDATED CLAY
>1

19. ACTIVE CASE
Consider the figure (b), if the wall moves
away from the backfill the soil expands and
the confining stress, σx gradually decreases.
If the movement is sufficiently large the σx
will decrease to minimum value and the state of
equilibrium will then attained.
As σz > σx in this case, σz is the major principal
stress and σx is the minor principal stress.

20. This condition of wall movement is
said to generate an active stress
condition and the ratio σx / σz is
defined as the active earth pressure
co-efficient, Ka.
Thus:
Ka = σx / σz = σ3 / σ1

21. PASSIVE CASE
When the wall moves towards the backfill,
and against the soil mass, the soil will be
subjected to lateral compression.
Under this condition, σx is the principal
stress and σz becomes the minor principal
stress.
this condition is known as Passive Earth
Pressure condition and ratio is given by;
Kp = σx / σz = σ1 / σ3

22. where,
Kp = the coefficient of Passive Earth
pressure.
Thus soil can exist in any condition ranging
from the active, through the at rest to the
passive state.

23. DIAGRAMTIC RELATIONSHIP
BETWEEN THE LATERAL STRAIN AND
LATERAL EARTH PRESSURE

24. RANKINE THEORY (1857)
In original form the theory was
developed for purely non-cohesive
soils (i.e. c = 0), but subsequently Bell
(1915) extended this theory to c-Φ soil
as well.

25. ASSUMPTIONS
1.
2.
3.
4.
5.
Soil is non-cohesive (c = 0) dry, isotropic
and homogenous.
Backfill is horizontal.
Wall is vertical,
Wall friction is neglected.
Failure is a plain strain problem.
Consider a unit length of an infinitely long
wall.

26. Soil Element : Rankine Theory

27. Vertical Stress = σ’z = γz
Major Principal stress.
At failure Horizontal stress , σ’x
Minor Principal Stress.

28. From ∆OAB,
SinΦ = AB/OB = ½(σz – σx)
½(σz + σx)
SinΦ(σz + σx) = (σz – σx)
σz (1 – SinΦ) = σx (1 + SinΦ)
Ka = σx / σz = (1 – SinΦ) = tan² (45 – Φ/2)
(1 + SinΦ)
and σx = active earth pressure = σz Ka
σa
= γz
tan² (45 – Φ/2)

29. Similarly, passive earth pressure,
Kp = tan² (45 – Φ/2) and,
σp = γz tan² (45 + Φ/2)
m.irfan B-15952