Geotechnical Engineering

DHARM

STRESS DISTRIBUTION IN SOIL 353

where σz = vertical stress in soil at depth z below the surface due to its self-weight,

and γ = unit weight of soil.

If there are imposed structural loadings also on the soil, the resultant stress may be

obtained by adding algebraically the stress due to self-weight and stress transmitted due to

structural loadings.

10 .2 Point Load

Although a point load or a concentrated load is, strictly speaking, hypothetical in nature, con-

sideration of it serves a useful purpose in arriving at the solutions for more complex loadings

in practice.

The most fundamental of the solutions of stress distribution in soil is that for a point

load applied at the surface. Boussinesq and Wastergaard have given the solution with differ-

ent assumptions regarding the soil medium. These solutions which form the basis for further

work in this regard and other pertinent topics will be dealt with in the following sub-sections.

10.2.1Boussinesq’s Solution
Boussinesq (1885) has given the solution for the stresses caused by the application of a point
load at the surface of a homogeneous, elastic, isotropic and semi-infinite medium, with the aid
of the mathematical theory of elasticity. (A semi-infinite medium is one bounded by a horizon-
tal boundary plane, which is the ground surface for soil medium).
The following is an exhaustive list of assumptions made by Boussinesq in the derivation
of his theory:
(i) The soil medium is an elastic, homogeneous, isotropic, and semi-infinite medium,
which extends infinitely in all directions from a level surface. (Homogeneity indicates identi-
cal properties at all points in identical directions, while isotropy indicates identical elastic
properties in all directions at a point).

(ii) The medium obeys Hooke’s law.
(iii) The self-weight of the soil is ignored.
(iv) The soil is initially unstressed.
(v) The change in volume of the soil upon application of the loads on to it is neglected.
(vi) The top surface of the medium is free of shear stress and is subjected to only the
point load at a specified location.

(vii) Continuity of stress is considered to exist in the medium.
(viii) The stresses are distributed symmetrically with respect to Z-axis.
The notation with regard to the stress components and the co-ordinate system is as
shown in Fig. 10.1.
In Fig. 10.1 (a), the origin of co-ordinates is taken as the point of application of the load
Q and the location of any point A in the soil mass is specified by the co-ordinates x, y, and z. The
stresses acting at point A on planes normal to the co-ordinate axes are shown in Fig. 10.1 (b).
σ’s are the normal stresses on the planes normal to the co-ordinate axes; τ’s are the shearing
stresses. The first subscript of τ denotes the axis normal to which the plane containing the
shear stress is, and the second subscript indicates direction of the axis parallel to which the