Engineering Rock Mechanics

34 Stress

\FI

AN

(a) (b)

Figure 3.3 (a) Arbitrary loading of any rock shape. @) The normal force, m, and

the shear force, AS, acting on a small area, AA, anywhere on the surface of an

arbitrary cut through the loaded rock

diagram of a body is shown, in this context a piece of intact rock loaded

by the forces F1, Fa, ..., F,. This is a generic illustration of any rock loaded

in any static way. Consider now, as shown in Fig. 3.3(b), the forces that are

required to act in order to maintain equilibrium on a small area of a surface

created by cutting through the rock. On any small area AA, equilibrium can

be maintained by the normal force AN and the shear force AS. Because

these forces will vary according to the orientation of AA within the slice, it

is most useful to consider the normal stress (AN/AA) and the shear stress

(AS/AA) as the area AA becomes very small, eventually approaching zero.

In this way, we develop the normal stress CT and the shear stress z as

properties at a point within the body.

The normal stress and shear stress can now be formally defined as:

AN

normal stress, 0, = lim ~

M+O AA

AS

M+O &4

shear stress, z = lim -.

There are obvious practical limitations in reducing the size of the small area

to zero, but it is important to realize that formally the stress components

are defined in this way as mathematical quantities, with the result that

stress is a point property.

3.5 The stress components on a small cube

within the rock

It is more convenient to consider the normal and shear components

with reference to a given set of axes, usually a rectangular Cartesian x-y-z

system. In this case, the body can be considered to be cut at three

orientations corresponding to the visible faces of the cube shown in Fig.

3.4. To determine all the stress components, we consider the normal and

shear stresses on the three planes of this infinitesimal cube.

The normal stresses, as defined in Section 3.4, are directly evident as