Stress distribution in soil
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 equation 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= r2+ Z2
Where,
r2= x2+ y2
sin = rR andcos = ZR
Now, the vertical stress at a point P is given by
z= R cos2 β
z= 3 Q2 π Z2 11+ rZ252
Hence, z= IB Q Z2
Where,
IB= 3 2 π 11+ rZ252
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:
R= 3 Q2 π cos R2
States assumptions made in Boussinesq theory
Assumptions made by Boussinesq.
The soil medium is an elastic, homogeneous, isotropic and semi infinite medium, which extends infinitely in all directions from a level surface.
The medium obeys Hooke’s law.
The self weight of the soil is ignored.
The soil is initially unstressed
The change in volume of the soil upon application of the loads on to it is neglected.
The top surface of the medium is free of shear stress and is subjected to only the point load at a specified location.
Continuity of stress is considered to exist in the medium.
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
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).
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,
The soil medium is an elastic, homogeneous, isotropic and semi infinite medium, which extends infinitely in all directions from a level surface.
The medium obeys Hookes law.
There are thin sheets of rigid materials sandwiched in a homogeneous soil mass. These thin sheets are closely spaced
Sheets are of infinite rigidity
Sheets prevent the medium from undergoing lateral strain.
These permit only downward displacement of the soil mass as a whole without any lateral displacement
Q: Explain pressure bulb and its significance
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 z=0.1 Q per unit area (10% isobar).
Stress distribution is given by formula:
z=KB Qz2 …… equation (1)
where, KB= 32π 11+ rz252…… equation (2)
Thus, KB= 0.1 z2 …… 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 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:
Data for isobar of i , z=0.1Q per unit area.
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.
Vertical stress distribution on a horizontal plane, at a depth z below the ground surface.
Vertical stress distribution along a vertical line, at a distance r from the line of action of the single concentrated load.
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
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
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
3=40 kN/sqm making use of Boussinesq equation.
We have Boussinesq equation for point load is,
z= 3 Q2 π Z2 11+ rZ252
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
z= KB QZ2
KB= 3 2 π 11+ rZ252…..(1)
Given Q = 1000, z=40
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, zB= 3 Q2 π Z2 11+ rZ252……..(1)
Westergaad’s equation for point load ( for μ=0), zw= Q π Z2 11+ 2 rZ232……..(2)
Dividing equation (1) by (2) we get,
zBzw= 3 Q2 π Z2 11+ rZ252 Q π Z2 11+ 2 rZ232
zBzw= 3 2 11+ rZ252 11+ 2 rZ232
For below point load, r=0
zBzw= 3 2 11+ 052 11+ 2 × 032
zBzw= 3 2 152 132
zBzw= 32=1.5
Vertical stress due to a line load
Vertical stress due to a Strip load
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
z= Q1IB1+ Q2IB2+ Q3IB3…….Z2
z=1Z2 i=1nQiIBi
Contact pressure
The upward pressure due to soil on the underside of the footing or foundation is termed contact pressure.
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’.
Actual footings are not flexible as assumed.
The actual distribution of the contact pressure depends on a number of factors.
Factors affecting contact pressure distribution: Flexural rigidity of base of footing, type of soil, confinement
Sandy soil
Clayey soil
C-Ø soil
Sample problems
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