stress distribution in soils

1. CHAPTER 2 : STRESS DISTRIBUTION,
COMPRESSIBILITY AND
SETTLEMENT OF SOILS

2. At the end of this lecture/week, the students
will be able to :
LEARNING OUTCOMES
Learning Outcomes :
1. Identify and discuss all parameters required
to determine the increase in vertical stress
below a foundation subjected to different
types of loading.
2. Formulate and evaluate the relevant
increase in vertical stress due to various
types of loading and footing shapes.
Coverage : Stress distribution; Increase in vertical
stress due to different loadings.

3. At the end of this lecture/week, the students
will be able to :
LEARNING OUTCOMES
Learning Outcomes :
1. Formulate and evaluate vertical stress due to
rectangular laoding and use Bulbs of Pressure
Coverage : Stress distribution due to rectangular footing;
Bulbs of pressure chart

4. 2.1 Introduction
2.2 Contact Pressures
2.3 Methods of Estimating Stress
Distribution
2.4 Types of Loading system
2.5 Stress Distribution due to
Different Loads
2.6 Bulbs of Pressure Chart
Stress Distribution, Compressibility and
Settlement of Soils
OUTLINE OF PRESENTATION
2A : Stress Distribution in Soils

5. Introduction
Stress Distribution in Soils
Imposing load on the surface of the soil
will create stresses within the mass. The
loading transferred to the soil mass will
be spread laterally with increasing depth
from the point or area of application.
With increasing depth, the area
over which new stresses develop
will increase but magnitude will
decrease.

6. How the stress is perceived to be
distributed from the surface a point
in the soil mass.
Stress Distribution in Soils

7. Factors affecting stress distribution
:
 Size and shape of footing
 Load distribution
 Contact pressure – depends on the rigidity
of footing and stiffness of foundation soil
 Modulus of Elasticity and Poisson’s ratio
 Position of rigid boundary
Stress Distribution in Soils

8. Effect of Soil Type on Contact Pressure
Contact pressure varies with the rigidity of foundation
and the stiffness of soil beneath the foundation.
Pressure
Distribution
Diagram
Description
Footing on hard soil or rock
Due to high stiffness modulus,
the load is distributed to a
relatively small area since a high
intensity of stress can develop.
Stress Distribution in Soils

9. Effect of Soil Type on Contact Pressure
Pressure
Distribution
Diagram
Description
Footing on stiff soil
Load is distributed laterally which
produces lower values of contact pressure
Footing on soft soil
The contact pressure on soil
beneath the foundation is
distributed almost uniformly.
Stress Distribution in Soils

10. Effect of Footing Rigidity on Contact Pressure
The distribution of pressure depending on footing
rigidity
Pressure Distribution
Diagram
Description
Flexible footing
Uniformly loaded footings
of perfect flexibility will
theoretically distribute a
uniform contact pressure
in compressible soil.
Stress Distribution in Soils

11. Effect of Footing Rigidity on Contact Pressure
Pressure Distribution
Diagram Description
Rigid footing on cohesive soil
A higher contact pressures will be
transmitted while settling uniformly.
However, extremely high edge stresses
cannot occur since the soil passes some
of its load inwards and produces the
arc-like distribution.
Rigid footing on cohesionless soil
Less contact pressure at the edges
of footing but higher at mid-footing
due to higher confining pressure.
Uniform settlement will occur in this
case.
Stress Distribution in Soils

12. Boussinesq Stress Distribution
 19 th.- century French mathematician
 assumed soil as homogeneous, isotropic
(same properties in all directions) and
elastic.
 publish solutions (1885) for stresses beneath
a point load applied at the surface
Stress Distribution in Soils

13. Following the footsteps of Boussinesq, other
solutions were developed for both stresses and
displacements relating to different types of
loading, layers of thickness, multi-layered
masses and internally loaded masses:
 Ahlvin and Ulery, 1962
 Giroud, 1970
 Newmark, 1942
 Poulos and Davis, 1974 and others……
Stress Distribution in Soils

14. Westergaard Stress Distribution
 more suitable for thin layers of
stratified deposits
 assumed that thin layers of a homogeneous
and anisotropic material sandwiched
between closely spaced, infinitely thin
sheets of rigid material.
 permit compression but no lateral
deformation
 formula different from Boussinesq.
Stress Distribution in Soils

15. Methods of Estimating Stress
Distribution
 Boussinesq’s method
 Westergaard’s method
 Newmark’s chart
 Bulbs of Pressure chart
Stress Distribution in Soils

16. Types of Loading
 Point load
 based on Boussinesq
 based on Westergaard
 Line load
 Triangular load
 Strip load
 Uniformly loaded rectangular area
 Uniformly loaded circular area
Stress Distribution in Soils

17. Load Distribution
Stress Distribution in Soils

18. 2.1 Stresses due to a point load
2/5
2P
2
)/(1
1
2
3
I
z
P









zr
where
IPz



19. Variation of stress due to a point load
(a)Variation with depth

20. Variation of stress due to a point load
(b) Variation with radial offset (r)

21. Table 2.1: Influence factors (Ip) for
vertical stress due to a point load (P)

22. Ex. 1 : Stresses due to a Vertical Point Load
Four column loads of 980 kN, 800 kN, 550 kN and 700 kN
respectively are located at the corners of a square of 4 m side on the
surface of a soil mass. A culvert passes diagonally across the square,
directly under the 980kN and 550kN load, and a depth ( to its top) of
4 m. Calculate the vertical stress imposed on the culvert under the
980 kN load by using
i) formula for the influence factor is IP = 3 1 5/2
2 1 + (r/z)2
ii) influence factor (IP) table.
550kN800kN
700kN
980kN
4m
4m

23. 2.2 Stresses due to the line
load
222
2
Q
)(
2z
I
z
Q
zx
where
IQz






24. Horizontal thrust on a rigid structure due to a line load

25. Table 2.2: Influence factors (IL) for
vertical stress due to a line load (Q)

26. Stresses due to a Long Line Load
Ex. 2 : Figure below shows two line loads and a point
load acting at the ground surface. Determine the
increase in vertical stress at point A, which is
located at a depth of 1.5 m.
Q2 = 10 kN/m
Q1 = 5 kN/m
P = 30 kN
2 m
A
z = 1.5 m
3 m
2 m

27. Stresses due to a Long Line Load
Solution:
     
     
   
 
   
 
  
     
2
mkN0.902






























012006508250
5143
51
2
303
514
51102
512
51152
2
5
222
3
222
3
222
3
...
.
.
.
.
.
.
zr
z
2π
3P
zxπ
z2Q
zxπ
z2Q
ΔσΔσΔσΔσ
2
5
22
3
222
2
3
2
222
1
3
1
3z2z1zz


28. 2.3 Stresses due to a uniform strip load
z = qIs

29. Influence factors (Is) for vertical stress due to
a strip load

30. Stresses Due to a Strip Load
Three parallel strip foundation, each 1.8 m wide and 3.6 m
apart centre to centre are founded at 1.2 m depth
transmit contact pressures of 240 kPa, 180 kPa and 200
kPa respectively. Using the table of influence factor, Is ,
calculate the intensity of vertical stresses due to combined
load beneath the centre of each footing at a depth of 3.0 m
from the ground surface.
240kPa 180kPa 200kPa
3.6m 3.6m
3m
1.2m
A CB
Example 3

31. Stresses Due to a Strip Load
Solution:

32. 2.4 Stresses due to a triangular load
z = qIs

33. Influence factors (IT) for vertical stress due to
a triangular strip load

34. Influence factors (IR) for vertical stress under
one corner of a uniformly loaded rectangular area.

35. Triangular Strip Load Distribution
Example 4
z
CL
A
3 m 10 m 3 m
5 m
B AC
60 kN/m

36. SOLUTION:
z = 5 m
Vertical start point A.
Is (kN/m2
)
Center
(B)
0.48 28.8
Right
slope
(A)
0.172 10.32
Left
slope
(C)
0 0
TOTAL 39.12
b
z
b
x qI sz  
1
5
5  1
5
5 
c
z
c
x
67.1
3
5  1
3
3 
67.1
3
5  33.4
3
13 

37. • For a uniformly loaded circular areas, e.g.
raft foundations, tank bases, etc., the
basic Boussinesq expressions are
integrated over the area.
• An exact solution can be found for the
increase in vertical stress under the
centre, but for points offset from the
centre an approximate method has to be
used.
2.5 Stresses due to a uniformly
loaded circular area

38. Stresses Due to a Uniformly Loaded
Circular Area
a) Stress beneath centre of circle
b) General vertical stress case
Example 5 : See worked example 6.4
--- Page 199 (Text book)
z = qIc =q (A+B)
Parameters = r/a & z/a
r
r
z
R
z
r
aa

39. Influence factors (A and B) for vertical stress
due to a uniformly loaded circular area.

40. • Most widely used
in soil engineering
design.
• Component stress
can be obtained by
integrating the
Boussinesq
expressions
L
B
z
∆z = q IR
Ex. 6 : See worked example 6.5 --- Page 200 (Text book)
2.6 Stresses due to a uniformly
loaded rectangular area

41. Fadum’s Chart
Take note that the values
of m ( =L/z) and n ( =B/z)
are interchangeable !!

42. 2.7 Pressure bulbs
for vertical stress
(a) Circular foundation
(b) Strip foundation

43. Pressure bulbs indicating depth to which soil is
significantly stressed

44. References
1. Roy Whitlow, “Basic Soil Mechanics”, 4th Edition 2001, Prentice
Hall
1 David F. McCarthy, “Essentials of Soil Mechanics and
Foundations”, 5th Edition, 1998, Prentice Hall
2. Braja M. Das, “Principles of Geotechnical Engineering”, 4th
Edition, 1998, PWS Publishing Company
3. G. N. Smith and Ian G. N. Smith, “Basic Soil Mechanics”, 7th
Edition, 2000, Blackwell Science
Stress Distribution in Soils
Text