Chapter 4-Stress distribution in soil.docx

stress distribution in soil

1. Stress distribution in soil
Q: State the formula for stress in soil mass due to a point load at a point below ground level as given
by Boussinesq’s and gives the meaning of all the terms in it.
Q: Derive expression for Boussinesq’s equatioin for vertical stress due to point load
Soils that effect of external load are subjected to stress. The vertical stress increases in soil due to
various type of loading. At any point in soil the stress applied from own weight of soil which called
effective stress, and from external load which called net stress, the net stress which applied must be
determined.
Boussinesq (1885) has given the solution for the stresses caused by the application of the point load
at the surface of a elastic medium with the aid of the mathematical theory of elasticity.
When a point load Q acting on the surface of a semi infinite solid, a vertical stress σz produces at any
point in addition to lateral and shear stress.
Basic Assumptions of Boussinesq theory:
For soil, the soil mass is elastic, isotropic, homogeneous and semi-infinite.
The soil is weightless.
For load, the load is vertical, concentrated acting on the surface.
Hook’s Low Applied, it is mean that the constant ratio between stress and strain.
From figure,
R=√r
2
+Z
2
Where,
r
2
=x
2
+y
2
Chapter 4 GT I notes Page 1 of 15

2. sinβ=
r
R
∧cosβ=
Z
R
Now, the vertical stress at a point P is given by
σ
z=σ
Rcos
2
β
σ
z=
3Q
2πZ
2
(
1
1+(
r
Z)
2
)
5
2
Hence, σ
z
=
I
B
Q
Z
2
Where,
I
B=
3
2π
(
1
1+(
r
Z)
2
)
5
2
σz      : Vertical stress at point P as shown in figure
Z       : Vertical dimension or depth of point P from GL
IB       : Influence factor depend on (r/Z)
Q       : Point load
Polar stress:
σ
R=
3Q
2π
cosβ
R
2
States assumptions made in Boussinesq theory
Assumptions made by Boussinesq.
a)The soil medium is an elastic, homogeneous, isotropic and semi infinite medium, which extends
infinitely in all directions from a level surface.
b)The medium obeys Hooke’s law.
c)The self weight of the soil is ignored.
d)The soil is initially unstressed
e)The change in volume of the soil upon application of the loads on to it is neglected.
f)The top surface of the medium is free of shear stress and is subjected to only the point load at a
specified location.
Chapter 4 GT I notes Page 2 of 15

3. g)Continuity of stress is considered to exist in the medium.
h)The stresses are distributed symmetrically with respect to z axis.
Q: With neat sketches, explain the use of Newmark’s chart to find stress at a given point under a
loaded area and its applications
The foregoing (Boussinesq's or Westergaard’s)
equations and tables can be used conveniently if
the shape of the loaded area is geometrically
simple.
In some problems the surface loading may cover an
area with irregular boundaries, or be
discontinuous, or involve variable contact
pressures, in which cases application of the
foregoing procedures can lead to unnecessarily
long calculations.
In some of these cases, the most efficient method
for estimating the stresses may be a graphical
procedure developed by Newmark (1942, 1947).
Newmark's charts are based on the vertical
stress at a point P below the centre of a
circular uniformly loaded area.
Consider a circle of radius R1, divided into 20
equal sectors. The vertical stress at a point
below the center of the circle at depth z due
to a uniform load on one sector will be equal
to 1/20th of that due to the load on the entire
circle.
Procedure to determine the value of the Stress at a given point P at a death z
Measure the depth z and precisely locate the point P where the stress is to be determined. 
Plot the plan of the loaded area with a scale of z equal to unit length of the chart (AB).
Chapter 4 GT I notes Page 3 of 15

4. Place the plan on the chart in such a manner that the point P, at which the stress is to be determined is
placed directly below the center of the chart. 
Count the number of elements (N) of the chart enclosed by the plan of the loaded area.
The formula which can be used to determine the stress at depth z, due to the given loaded area can be
given as,    Stress = σz = (IF).q.N
  Here IF = Influence Factor, which we have taken equal to 0.005
          q = pressure intensity at top.
          N = Number of elements of the chart covered by the prepared plan.
Q: States assumptions made in Westergaard theory
Westergaard’s solution assumes that,
a)The soil medium is an elastic, homogeneous, isotropic and semi infinite medium, which extends
infinitely in all directions from a level surface.
b)The medium obeys Hookes law.
c)There are thin sheets of rigid materials sandwiched in a homogeneous soil mass. These thin
sheets are closely spaced
d)Sheets are of infinite rigidity
e)Sheets prevent the medium from undergoing lateral strain.
f)These permit only downward displacement of the soil mass as a whole without any lateral
displacement
Q: Explain pressure bulb and its significance
Chapter 4 GT I notes Page 4 of 15

5. An ‘isobar’ is a stress contour or a curve which connects all points below the ground surface at
which the vertical pressure is the same.
In fact, an isobar is a spatial curved surface and resembles a bulb in shape; this is because the
vertical pressure at all points in a horizontal plane at equal radial distances from the load is the
same.
Thus, the stress isobar is also called the ‘bulb of pressure’ or simply the ‘pressure bulb’. The vertical
pressure at each point on the pressure bulb is the same.
Pressures at points inside the bulb are greater than that at a point on the surface of the bulb; and
pressures at points outside the bulb are smaller than that value.
Any number of pressure bulbs may be drawn for any applied load, since each one corresponds to an
arbitrarily chosen value of stress.
A system of isobars indicates the decrease in stress intensity from the inner to the outer ones and
reminds one of an ‘Onion bulb’. Hence the term ‘pressure bulb’. An isobar diagram, consisting of a
system of isobars appears somewhat as shown in Fig
Procedure for plotting pressure bulb
The procedure for plotting an isobar is as follows:
1. Let it be required to plot an isobar for which σ
z
=0.1Q  per unit area (10% isobar).
Stress distribution is given by formula:
σ
z=
K
B
Q
z
2
…… equation (1)
where,
K
B
=
3
2π
[
1
(
1+(
r
z)
2
)]
5
2
…… equation (2)
Thus, K
B=0.1z
2
…… equation (3)
2. Assuming various values for z, the corresponding KB-values are computed.
3. For the computed values of KB, the corresponding r/z-values are obtained.
4. For the assumed values of z, r-values can be calculated.
It is obvious that, for the same value of r on any side of the z-axis, or line of action of the point load, the
value of σ
z is the same; hence the isobar is symmetrical with respect to this axis. So, the other half can
be drawn from symmetry.
When r= 0, KB= 0.4775; the isobar crosses the line of action of the load at a depth of:
The calculations are best performed in the form of a table as given below:
Chapter 4 GT I notes Page 5 of 15

6. Data for isobar of i , σ
z
=0.1Q  per unit area.
Depth z (units)
Influence
coefficients KB
r/z r (units)
0.5 0.0250 1.501 0.750 0.1Q
1.0 0.1000 0.932 0.932 0.1Q
1.5 0.2550 0.593 0.890 0.1Q
2.0 0.4000 0.271 0.542 0.1Q
2.185 0.4775 0 0 0.1Q
Short notes: Pressure Distribution on horizontal and vertical plane
It is possible to calculate the following pressure distributions by equation of Boussinesq and present
them graphically.
(i)Vertical stress distribution on a horizontal plane, at a depth z below the ground surface.
(ii)Vertical stress distribution along a vertical line, at a distance r from the line of action of the
single concentrated load.
Chapter 4 GT I notes Page 6 of 15

7. Prove that maximum vertical stress on a plane at a distance ‘r’ from concerned load Q acting at ground
will be at a depth z=1.225r
Chapter 4 GT I notes Page 7 of 15

8. Boussinesq equation for point load is,
σ
z=
3Q
2πZ
2
(
1
1+(
r
Z)
2
)
5
2
It is clear that vertical stress decrease with the
depth z. on any vertical line distant r from the axis,
of the load the variation of vertical stress can be
plotted from relation
σ
z=K
B
Q
Z
2
In above expression r is associated with KB is constant. Hence various values of Z and r/Z can be selected
and KB can be found out as below.
Let say r=1 unit
Z r/Z KB KB/Z
2
σ
z
0 ∞ - - -
0.2 5 0.0001 0.0025 0.0025Q
0.5 2 0.0085 0.034 0.0340Q
1 1 0.0844 0.0844 0.0844Q
1.225 0.817 0.1332 0.0888 0.0888Q
2 0.5 0.2733 0.0683 0.0683Q
4 0.25 0.4103 0.0256 0.0256Q
5 0.2 0.4329 0.0172 0.0172Q
From above table of calculations it is clear that, maximum vertical stress on a plane at a distance ‘r’ from
concerned load Q acting at ground will be at a depth z=1.225r
A concentrated load of 1000kN acts at the ground surface. Construct an isobar for
σ
3
=40 kN/sqm making use of Boussinesq equation.
We have Boussinesq equation for point load is,
σ
z=
3Q
2πZ
2
(
1
1+(
r
Z)
2
)
5
2
It is clear that vertical stress decrease with the depth z. on any vertical line distant r from the axis, of the
load the variation of vertical stress can be plotted from relation
σ
z=K
B
Q
Z
2
Chapter 4 GT I notes Page 8 of 15

9. K
B=
3
2π
(
1
1+(
r
Z)
2
)
5
2
…..(1)
Given Q = 1000, σ
z
=40
K
B/Z
2

Z
(assu
me)
KB
r
from
equation
(1)
40/1
000
=
0.04
0.1
0.0
004 0.25
0.25
0.0
025 0.39
0.5
0.0
1 0.49
0.75
0.0
225 0.48
1
0.0
4 0.35
1.1
0.0
484 0.21
1.15
0.0
529 0.04
Q: Prove that stresses below the point load as calculated by Westergaad’s and Boussinesq’s theory
are in the ration of 1:1.5 approximately
We have Boussinesq equation for point load,
σ
zB=
3Q
2πZ
2
(
1
1+(
r
Z)
2
)
5
2
……..(1)
Westergaad’s equation for point load ( for μ=0),
σ
zw=
Q
πZ
2
(
1
1+2(
r
Z)
2
)
3
2
……..(2)
Dividing equation (1) by (2) we get,
Chapter 4 GT I notes Page 9 of 15

10. σ
zB
σ
zw
=
3Q
2πZ
2
(
1
1+(
r
Z)
2
)
5
2
Q
πZ
2
(
1
1+2(
r
Z)
2
)
3
2
σ
zB
σ
zw
=
3
2
(
1
1+(
r
Z)
2
)
5
2
(
1
1+2(
r
Z)
2
)
3
2
For below point load, r=0
σ
zB
σ
zw
=
3
2(
1
1+0)
5
2
(
1
1+2×0)
3
2
σ
zB
σ
zw
=
3
2
(1)
5
2
(1)
3
2
σ
zB
σ
zw
=
3
2
=1.5
Chapter 4 GT I notes Page 10 of 15

11. Vertical stress due to a line load
Vertical stress due to a Strip load
Chapter 4 GT I notes Page 11 of 15

12. Equivalent point- load method
•The vertical stress at a point under a loaded area of any of any shape can be determined by
dividing the loaded area into small area and replacing the distributed load on each on small area
by an equivalent point load acting at the centroid of the small area.
•The principle of superposition is then applied and the required stress at a specified point is
obtained by summing up the contributions of the individual.
•Point loads from each of the units by applying the approximate point load formula, such as that
of Boussinesq’s or Westergaard’s.
•As show in figure, if aarea of Size B is acted on by a uniform load q, the same area can be divided
in to four small area.
•And the load on each area can be converted into an equivalent point load assumed to act at its
centroid.
•Then the vertical stress at any point below or outside the loaded area is equal to the sum of the
vertical stresses due to these equivalent point loads
•Then
σ
z=
Q
1
I
B1
+Q
2
I
B2
+Q
3
I
B3
…….
Z
2
σ
z
=
1
Z
2∑
i=1
n
Q
i
I
Bi
Chapter 4 GT I notes Page 12 of 15

13. Contact pressure
1.The upward pressure due to soil on the underside of the footing or foundation is termed contact
pressure.
2.In the derivations of vertical stress below the loaded areas using Boussinesq’s theory or
Westergaard’s theory, it has been assumed that the footing is flexible and the contact pressure
distribution is uniform and equal to ‘q’.
3.Actual footings are not flexible as assumed.
4.The actual distribution of the contact pressure depends on a number of factors.
5.Factors affecting contact pressure distribution: Flexural rigidity of base of footing, type of soil,
confinement
Sandy soil
Clayey soil
C-Ø soil
Chapter 4 GT I notes Page 13 of 15

14. Sample problems
Q:- Find intensity of vertical pressure at a point 3 m
directly below 25 kN point load acting on a horizontal
ground surface. What will be the vertical pressure at a
point 2m horizontally away from the axis of loading and at
same depth of 3 m? Use Boussinesq’s equation.
A line load of 100 kN/m run extends to a
long distance. Determine the intensity of
vertical stress at a point, 2m below the
surface for the following two cases:
i) Directly under the line load, and
ii) At a distance of 2 m perpendicular to
the line load.
Use Boussinesq’s theory
Chapter 4 GT I notes Page 14 of 15

15. A strip footing 2 m wide is loaded on the ground surface with a pressure of 150 kN/m 2 . A 4 m thick
soft clay layer exists at a depth of 10 m below the foundation. Find the average increase in vertical
stress at the centre of clay layer below the centre line and at the edge of footing. Adopt Boussinesq’s
theory for strip load
Case (i): Point at the centre of clay layer below the centre line of strip load
Case (ii): Point at the centre of clay layer below the edge of footing (strip load)
Chapter 4 GT I notes Page 15 of 15