1. DR. S & S. S. GHANDHY GOVERNMENT ENGINEERING COLLEGE , SURAT
DR. S & S.S. GHANDHY GOVERNMENT ENGINEERING COLLEGE , SURAT. Stress Distribution of soils Subject name: Soil mechanics Subject code:

2. NAME
EN. NO.
Parmar Bhavesh V.
Parmar Kartikey M.
Patel Bhargav A.
Patel Deep A.
Patel Gaurang B.
Patel Hiren C.
Patel Kiran S.
Patel kush V.
Patel Reepal K.
Patel Virat P.

3. CONTENT INTRODUCTION GEOSTATIC STRESSES BOUSSINESQ’S SOLUTION
ISOBAR DIAGRAM
VERTICAL STREES DISTRIBUTION ON A HORIZONTAL PLANE
VERTICAL STREES DISTRIBUTION ON A VERTICAL PLANE
VERTICAL STREES DUE TO LINE LOAD
VERTICAL STREES DUE TO STRIP LOAD
VERTICAL STREES UNDER A CIRCULAR AREA
VERTICAL STREEES UNDER A RECTANGULAR AREA
EQUIVALENT POINT LOAD METHOD
NEWMARK’S INFLUENCE CHART
WESTERGURD’S SOLUTION
STREES UNDER TRIANGLE LOADS

4. INTRODUCTION
Stresses are induced in a soil mass due to soft weight of soil and due to applied structural loads.

5. GEOSTATIC STRESSES
Vertical stress in soil due to self weight is called geostatic stress.
Due to self weight of soils stresses are large.
When the ground surface is horizontal the stresses due to self weight of soil are normal to the horizontal and vertical planes
No shearing stresses on principal planes.

6. Vertical Stresses The vertical stress at depth z below ground surface
where,
= vertical stress
z = depth below ground surface
γ = unit weight of soil
Horizontal Stresses
Depends upon vertical stresses, type of soil and conditions weather the soil is stretched or compressed.
Horizontal stress,
k0 =coefficient of lateral earth pressure at rest =
(μ= Poison’s ratio)

7. BOUSSINESQ’S SOLUTION
Boussinesq (1885) gave the theoretical solution for the stress distribution in an elastic medium subjected to a concentrated load on its surface.
Assumptions:
The soil mass is an elastic medium (elasticity is constant)
The soil is homogeneous.
The soil is isotropic.
The soil mass is semi-infinite.
Self weight of soil is neglected.
The soil is initially stress free.
The change in volume of soil is neglected.
Top surface of medium is shear stress.
Community of stress is considered to exist in the medium.

8. By Boussinesq’s solution polar radial stress at P(x, y, z) is
Let P (x, y, z) be the point in the soil mass where vertical stresses are to be determined due to applied load Q on the ground surface.
By Boussinesq’s solution polar radial stress
at P(x, y, z) is
(i)
Where,
R=polar distance between the origin O and point P
Β=angle which line PQ makes with vertical
Stresses due to concentrated load
With,
and

9. The vertical stress at point P, With substituting equation (i)OR
Where,
= Boussinesq influence coefficient
for the vertical stress.

10. PRESSURE ISOBARS-Pressure Bulb
An isobar is a line which connects all points of equal stress below the ground surface. In other words, an isobar is a stress contour. We may draw any number of isobars as shown in Fig. for any given load system.
Each isobar represents a fraction of the load applied at the surface. Since these isobars form closed figures and resemble the form of a bulb, they are also termed bulb of pressure or simply the pressure bulb.
Normally isobars are drawn for vertical, horizontal and shear stresses. The one that is most important in the calculation of settlements of footings is the vertical pressure isobar.

11. we may draw any number of isobars for any given load system, but the one that is of practical significance is the one which encloses a soil mass which is responsible for the settlement of the structure.
The depth of this stressed zone may be termed as the significant depth Ds which is responsible for the settlement of the structure. Terzaghi recommended that for all practical purposes one can take a stress contour which represents 20 per cent of the foundation contact pressure q, i.e, equal to 0.2q.
Terzaghi's recommendation was based on his observation that direct stresses are considered of negligible magnitude when they are smaller than 20 per cent of the intensity of the applied stress from structural loading, and that most of the settlement, approximately 80 per cent of the total, takes place at a depth less than Ds.
The depth Ds is approximately equal to 1.5 times the width of square or circular footings

12. Example of Pressure bulb.

14. Vertical stress distribution on a horizontal plane

15. The vertical stress at various points on horizontal plane at a particular depth z, below the ground surface due to concentrate load Q is given by Boussinesq equation, in which horizontal distance r change but depth z remains constant.

16. IB cam be obtained from table, and hence Oz can be computed.
From the above table, it can be concluded that at a given depth, when horizontal radial distance is equal to twice the depth (r = 2z), the vertical pressure due to single concentrate load is vary small, I.e present of the maximum,and can be neglected.

17. show the vertical stress distribution diagram on a horizontal plane.
the diagram is symmetrical about the vertical axis. the maximum stress occurs just below the load ( r = 0 ).

18. Influence Diagram
If the point load Q is taken as unity, (I.e. Q=1), tha vertical stress distribution diagram is called influence Diagram.

19. Shows three influence Diagram marked I1, I2, I3 due to unit load Q1, Q2, Q3 applied at point A', B', C' respectively on the ground surface.
the stress at and point A on the horizontal plane atdepth z due to three load Q1, Q2, Q3 is below.

20. LINE LOADS
By applying the principle of the above theory, the stresses at any point in the mass due to a line load of infinite extent acting at the surface may be obtained.
The state of stress encountered in this case is that of a plane strain condition. The strain at any point P in the Y- direction parallel to the line load is assumed equal to zero. The stress бy normal to the XZ-plane is the same at all sections and the shear stresses on these sections are zero.
The vertical бz stress at point P may be written in rectangular coordinates as
where, / z is the influence factor equal to at x/z =0.

22. STRIP LOADS
Such conditions are found for structures extended very much in one direction, such as strip and wall foundations, foundations of retaining walls, embankments, dams and the like.

23. Fig. shows a load q per unit area acting on a strip of infinite length and of constant width B. The vertical stress at any arbitrary point P due to a line load of qdx acting at can be written from Eq. as
Applying the principle of superposition, the total stress бz at point P due to a strip load distributed over a width B(= 2b) may be written as
The non-dimensional values can be expressed in a more convenient form as

25. Example
Three parallel strip footings 3 m wide each and 5 m apart center to center transmit contact pressures of 200, 150 and 100 kN/m2 respectively.
Calculate the vertical stress due to the combined loads beneath the centers of each footing at a depth of 3 m below the base. Assume the footings are placed at a depth of 2 m below the ground surface. Use Boussinesq's method for line loads.

26. We know

27. Topic: Vertical Stresses Under A Circular Area

28. Introduction:
The boussinesq’s equation for the vertical stress due to point load can be used to find the vertical stress at any point beneath the center of a uniformly loaded circular area.
This type of study arise in the structures with circular foundation, such as gasoline tank, storage bins and grain elevators.

29. Determining Vertical Stress:
Determine the stress at point p at depth z below the uniformly loaded circular area. q = intensity of load per unit area Figure:

34. NEWMARK’S INFLUENCE CHARTS
The Newmark’s Influence Chart is useful for the determination of vertical stress(σ) at any point below the uniformly loaded area of any shaped.
This method is based on the concept of the vertical stress at point below the centre of uniformly loaded circular area A charts, consisting of number of circles and radiating lines, is so prepared that the influence of each area unit is the same at the centre of the circles, i.e. each area unit causes the equal vertical stress at the centre of the circle.

35. NEWMARK’S INFLUENCE CHART

36. σ/20= q/20 *[1- {1/(1+(R1/z)^2)}^3/2]
Consider a uniformly loaded circular area of radius R1,divided into 20 equal sectors(area units) as shown in fig. If q is the intensity of loading and σ is the vertical stress at point P at depth z below the centre of the loaded area , then due to each area unit ,the vertical stress at point P is given by equation as
σ/20= q/20 *[1- {1/(1+(R1/z)^2)}^3/2]
In order to use the Newmark’s influence chart for determining the vertical stress at point P lying under a uniformly loaded area of any shape.

37. WESTERGAARD’S SOLUTION

38. Westergaard solved the problem of pressure distribution in soil under point load.
He assume that there are thin sheets of rigid material sand-wiched in a homogeneous soil mass.
The thin sheets are closely spaced and are of negligible thickness and infinite rigidity, which permits only downward displacement of the soil mass as a whole without allowing it to undergo any lateral strain.

39. Boussinesq solution assumes that the soil mass is isotropic.
But there are generally thin layer of sand embedded in homogeneous clay strata which accentuates the non isotropic condition.
Therefore Westergaard’s solution represents more closely the actual field condition.

40. Application of Westergaard’s formula for stress
distribution
N 1 N IW
σW = =
ΠZ2 [1 + 2(r/Z)2]3/2 Z2
(1/Π)
IW =
[1 + 2(r/Z)2]3/2

41. Where σW = Westergaard stress coefficient
IW = Westergaard vertical stress
N = Point load from the end bearing pile
Z =Vertical distance from the end point of pile
r = Radial distance from Z (Murthy, 1992).