**1.5 VERTICAL STRESSES DUE TO A CONCENTRATED LOAD**

Boussinesq gave the theoretical solutions for the stress distribution in an elastic medium subjected to a concentrated load on its surface. The solutions are commonly used to obtain the stresses in a soil mass due to externally applied loads. The following assumptions are made:

1. The soil mass is an elastic continuum, having a constant value of modulus of elasticity (E), i.e., the ratio between the stress and strain is constant.

2. The soil is homogeneous, i.e., it has identical properties at different points.

3. The soil is isotropic, i.e., it has identical properties in all directions.

4. The soil is semi- infinite, i.e., it extends to infinity in the downward direction and lateral directions. In other wards, it is limited on its top by a horizontal plane and extends to infinity in all other directions.

5. The soil is weightless and is free from residual stresses before the application of the load.

Consider a horizontal surface of the elastic continuum subjected to a point load Q at point O.

The vertical stress (

*σ*) at point P is given by_{z }

**σ**_{z }= I_{B }. Q /z^{ }^{2 …......………………………… (1.3)}

^{}

^{}

The coefficient I

_{B}is known as the Boussinesq influence coefficient for the vertical stress.
The following points are worth nothing when using Eq. 1.3.

1. The vertical stress does not depend upon the modulus of elasticity (E) and the Poisson ratio (ν).But the solution has been derived assuming the soil is linearly elastic. The stress distribution will be the same in all linearly elastic materials.

2. The intensity of vertical stress just below the load point is given by

**σ**

_{z }= 0.4775 Q /z^{ }^{2 ……................………………………(1.5)}

^{}

^{ }

^{}

3. At the surface (z = 0), the vertical stress just below the load is theoretically infinite. However, in an actual case, the soil under the load yields due to very high stresses. The load point spreads over small but finite area and, therefore, only finite stresses develop.

3. The vertical stress (σ

_{z}) decreases rapidly with an increase in r/z ratio. Theoretically, the vertical stress would be zero only at an infinite distance from the load point. Actually, at r/z = 5.0 or more, the vertical stress becomes extremely small and is neglected.
4. In actual practice, foundation loads are not applied directly on the ground surface. However, it has been established that the Boussinesq solution can be applied conservatively to field problems concerning loads at shallow depths, provided the distance z is measured from the point of application of the load.

5. Boussinesq’s solution can even by used for negative (upward) loads. For example, if the vertical stress decrease due to an excavation is required, the load is equal to the weight of the soil removed. However, as the soil is not fully elastic, the stresses determined are necessarily approximate.

6. The field measurements indicate that the actual stresses are generally smaller than the theoretical values given by Boussinesq’s solution, especially at shallow depths. Thus, the Boussinesq’s solution gives conservative values and is commonly used in soil engineering problems.

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