Stress analysis is often necessary in the design of foundations of all types of structures, particularly buildings, retaining structures, dams, highway pavements, and embankments. In this chapter, the mathematical definitions of stress and strain and the elasticity of an isotropic material are first treated. This is followed by the classical theory of Boussinesq for the stress in a semi-infinite, elastic, isotropic, and homogeneous continuum loaded normally on its upper plane surface by a concentrated load. The Boussinesq solution is later extended to analyze the stresses produced by a uniformly distributed load over a flexible circular foundation, rectangular loading, strip loading, line loading, triangular loading, and embankment loading. The case of irregular loading using the Newmark's Chart is also considered. The settlement of a foundation under external loadings by the use of both the Boussinesq theory and the semi-empirical strain influence factor method proposed by Schmertmann et al. (1978) are considered.
Top4.1 Definition Of Stress At A Point
The stress at a point is uniquely defined by specifying the magnitude and direction of a single vector having the units of force per unit elemental area, Figure 1(a). For engineering purposes, it is more useful to define stress at a point in terms of the direct and shear components of stress acting normally and tangentially to the six faces of an elemental prism as shown in Figure 1 (b). The stress at a point is defined when the stress components, σxx, σxy, σxz, ... etc. are known and the gradient of stress is defined when the values of the 
, 
, 
, etc. are known.
It is often convenient to choose a prism so oriented that the shear components are zero, Figure 1(c). The direct stresses on such an elemental prism are called principal stresses. The stress at a point can therefore be specified either by the magnitudes and directions of the principal stresses or by the magnitudes of the corresponding conjugate stresses. Similarly, deformations may be defined in terms of direct and shear components (angular distortions) of strain denoted by εx, εy, εz and εxy, εxz, εyz respectively.
4.1.1 Sign Convention
In soil mechanics, stresses which tend to cause compression and compressive strains are usually taken as positive while those that cause extension or tensile strains are taken as negative. This is the opposite of the sign convention used in structural mechanics.
4.1.2 Hooke’s Law and Poisson’s Ratio
In 1678 Robert Hooke on the basis of experiments on a steel wire deduced that stress is proportional to strain if the elastic limit is not exceeded. Later Poisson showed that as the length of a specimen increased under load, the diameter or lateral dimension decreased, and that for many materials the lateral and longitudinal strains are always in a fixed ratio. These two discoveries expressed as Hooke’s law and Poisson’s Ratio form the basis of the classical theory of elasticity, Timoshenko and Goodier (1951). They can be expressed as:
(4.1)(4.2) where
σ = stress
ε = strain
E = Modulus of Elasticity
ν = Poisson’s ratio
Following conventional definition of stress, the normal stress is defined as:
(4.3)Similarly the shear stress is given by
(4.4)The simplest form of a continuum is given by considering an elementary unit of material that is so small that one does not need to involve local moments to establish equilibrium. In this class of continua, the stress tensor can be written as:
(4.5) which has 6 independent components.
For one set of axes (x, y, z), Figure 1b, the shear stresses vanish giving the principal stresses σ1, σ2, σ3, Figure 1c. It is helpful to concentrate on principal stresses because
- 1.
Most test data are from tests in which only principal stresses are applied.
- 2.
We do not know much about the effects of rotation of principal axes on the stress-strain behaviour. It is therefore most often assumed that rotation of the axes, as such, has no effect on the stress-strain behaviour.