ABSTRACT
This chapter is concerned with the resistance of soil to failure in shear, a knowledge of which is required in the analysis of the stability of soil masses and, therefore, for the design of geotechnical structures. Many problems can be treated by analysis in two dimensions, i.e. where only the stresses and displacements in a single plane need to be considered. This simplification will be used initially in this chapter while the framework for the constitutive behaviour of soil is described. An element of soil in the field will typically by subjected to total normal stresses in the vertical (z) and horizontal (x) directions due to the self-weight of the soil and any applied external loading (e.g. from a foundation). The latter may also induce an applied shear stress which additionally acts on the element. The total normal stresses and shear stresses in the x-and z-directions on an element of soil are shown in Figure 5.1(a), the stresses being positive in magnitude as shown; the stresses vary across the element. The rates of change of the normal stresses in the x-and z-directions are ∂σx/∂x and ∂σz/∂z respectively;
the rates of change of the shear stresses are ∂τxz/∂x and ∂τxz/∂z. Every such element within a soil mass must be in static equilibrium. By equating moments about the centre point of the element, and neglecting higher-order differentials, it is apparent that τxz = τzx. By equating forces in the x-and z-directions the following equations are obtained:
(5.1a)
(5.1b)
These are the equations of equilibrium in two dimensions in terms of total stresses; for dry soils, the body force (or unit weight) γ = γdry, while for saturated soil, γ = γsat. Equation 5.1 can also be written in terms of effective stress. From Terzaghi’s Principle (Equation 3.1) the effective body forces will be 0 and γ′ = γ – γw in the x-and z-directions respectively. Furthermore, if seepage is taking place with hydraulic gradients of ix and iz in the x-and z-directions, then there will be additional body forces due to seepage (see Section 3.6) of ixγw and izγw in the x-and z-directions, i.e.:
(5.2a)
(5.2b)
The effective stress components are shown in Figure 5.1(b). Due to the applied loading, points within the soil mass will be displaced relative to the axes and to one another, as shown in Figure 5.2. If the components of displacement in the x-and z-directions are denoted by u and w, respectively, then the normal strains in these directions (εx and εz, respectively) are given by
Figure 5.1 Two-dimensional state of stress in an element of soil: (a) total stresses, (b) effective stresses.
