Boussinesq Equation and Westergaard’s Equation for point load in soil

Application of Boussinesq Equation and Westergaard’s Equation

Boussinesq Equation and Westergaard’s Equation are used to analyze the verticle stress on the soil with respect to the depth of the soil due to the point load of any structure.

Whenever a load is applied to the soil, the soil gets stressed in a different direction which we will discuss below with the help of two equations. After that, we will also see the numerical calculation for the verticle stress distribution pattern. Let us discuss both equations one by one.

1) Boussinesq’s Equation

Boussinesq gave the solution to the problem of vertical stress distribution in soil due to point load (single concentrated vertical load) acting at the soil surface, with the aid of the theory of elasticity.

The following assumptions are made by Boussinesq in obtaining the solutions:

Boussinesq theory assumptions

i) The soil mass is an elastic medium for which elasticity E is constant.

ii) The soil mass is homogenous i.e.; it has identical properties at different points or the same soil properties with depth.

iii) The soil mass is isotropic i.e.; it has identical properties in all directions.

iv) The soil mass is semi-infinite i.e.; it extends infinitely in all directions below a level surface.

v) The self-weight of soil is ignored.

vi) The soil is initially unstressed.

Boussinesq Equation and Westergaard's Equation for point load in soil

Let a vertical point load (single a concentrated vertical load) Q be acting at the soil surface at point O which is taken as the origin of the x, y, and z axes as shown in figure 8.2.

Let, P be the point in the soil mass having coordinates (x, y, z) or having radial horizontal distances r and vertical distance z from the point O.

Using logarithmic stress function for the solution of elasticity problem, Boussinesq showed that the polar radial stress or at point P(x, y, z) is expressed as,

Boussinesq Equation and Westergaard's Equation for point load in soil

where, R = Polar distance between the origin and point P (or polar radial coordinate of point P) 

β = Angle of line OP makes with vertical.

and, the vertical normal stress σv at point P is given by;

σv = σR cos2 β

From the figure; we have,

Boussinesq Equation and Westergaard's Equation for point load in soil
Boussinesq Equation and Westergaard's Equation for point load in soil
Boussinesq Equation and Westergaard's Equation for point load in soil

Now, 

Boussinesq Equation and Westergaard's Equation for point load in soil
Boussinesq Equation and Westergaard's Equation for point load in soil
Boussinesq Equation and Westergaard's Equation for point load in soil
Boussinesq Equation and Westergaard's Equation for point load in soil

where, IB = Boussinesq’s influence coefficient for vertical stress.

Boussinesq Equation and Westergaard's Equation for point load in soil

Thus vertical stress is;

Boussinesq Equation and Westergaard's Equation for point load in soil

2) Westergaard’s Equation for Point Load

Westergaard gave the solution to the problem of stress distribution in a soil mass due to point load acting at a soil surface with the following assumptions:

Westergaard theory assumptions

i) Soil mass is elastic

ii) Soil mass is homogenous

iii) Soil mass is semi-infinite

iv) Soil mass is anisotropic

According to Westergaard’s the vertical stress at a point P at a depth z below the concentrated load Q is given by;

Boussinesq Equation and Westergaard's Equation for point load in soil

where, C depends upon the Poisson ratio (v) and is given as;

Boussinesq Equation and Westergaard's Equation for point load in soil

For elastic materials, the value of Poisson’s ratio (v) ranges from 0 to 0.5. If v is taken as zero for all practical purposes; then,

Boussinesq Equation and Westergaard's Equation for point load in soil

Then, the above equation reduces as;

Boussinesq Equation and Westergaard's Equation for point load in soil

where, Iw is known as Westergaard’s influence coefficient.

Boussinesq Equation and Westergaard's Equation for point load in soil

The value of Iw is considerably smaller than the Boussinesq influence factor (IB) .

Comparison of IB and Iw

From figure 8.3 it is clear that the Westergaard equation gives values consistently less than the Boussinesq for the same point load up to a ratio r/z equal to 1.5. When the ratio exceeds 1.5, the Westergaard formula gives greater stress. For all ratios or r/z less than about 0.8, the vertical stress as per Westergaard’s formula is approximately equal to two-thirds of the values given by Boussinesq’s equation.

Boussinesq Equation and Westergaard's Equation for point load in soil

Derive an expression for the vertical stress at a point due to a point load using Boussinesq’s theory. What are the limitation of Boussinesq’s theory? 

Solution:

For the first part: See above

For the second part: We have from Boussinesq’s solution:

Boussinesq Equation and Westergaard's Equation for point load in soil

The value of vertical normal stress at a point directed below the point load on its axis of leading (where r = 0) is given by; 

Boussinesq Equation and Westergaard's Equation for point load in soil

Equation (2) shows that the vertical normal stress directly below the point load decreases with the square of the depth and at z = 0 the vertical normal stress below the point load is infinite.

Theoretically, the point load has a zero-contact area and hence the resulting stress is infinite. However, according to Boussinesq in actual practice the material near the surface under the point load yields and hence it experiences only finite stresses. 

Again equation (2) shows that the vertical normal stress directly below that point load will be zero only at z = ∞ (i.e., at infinite, depth below the load), but for practical purposes, the vertical normal stress may be considered to be zero at a relatively small finite depth below the load.

Stress distribution in soil solved problems

Q) A point load of 1000 kN acts as a ground surface. Show the variation of vertical stresses on a horizontal plane at a depth of 5m below the surface.

Solution:

The given values are:

Q = 1000 kN 

and, z=5m 

The vertical stress distribution in soil on a horizontal plane at a given depth z = 5 m can be determined by using the Boussinesq equation as given below. 

Boussinesq Equation and Westergaard's Equation for point load in soil

For this the several assumed values of r and constant values of z, σv is calculated from the equation (1).

Now, 

Boussinesq Equation and Westergaard's Equation for point load in soil
Boussinesq Equation and Westergaard's Equation for point load in soil

Using equation (2)  σv  is calculated for different values of r and plotted as shown below.

r(m)0123456
σv (kN/m2)19.117.3113.188.855.543.382.05
Boussinesq Equation and Westergaard's Equation for point load in soil
Vertical stress distribution in soil

Happy Learning – Civil Concept

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