Engineering Rock Mechanics

Stress as a point property 33

Figure 3.1 (a) Normal forces and shear forces. (b) Normal stresses and shear

stresses.

different notations are in use and we encourage the reader not to be
disturbed by such differences but to establish which notation is being used
and then use it. There is no ‘best’ notation for all purposes: some types of
notation have advantages in specific applications.
We are now in a position to obtain an initial idea of the crucial difference
between forces and stresses. As shown in Fig. 3.2(a), when the force
component, F,, is found in a direction 8 from F, the value is F cos 8.
However, and as shown in Fig. 3.2(b), when the component of the normal
stress is found in the same direction, the value is crcos2 8.
The reason for this is that it is only the force that is resolved in the first
case, whereas, it is both the force and the area that are resolved in the
second case-as shown in Fig. 3.2(b). This is the key to understanding stress
components and the various transformation equations that result. In fact,
the strict definition of a second-order tensor is a quantity that obeys certain
transformation laws as the planes in question are rotated. This is why our
conceptual explanation of the tensor utilized the idea of the magnitude,
direction and ’the plane in question’.

3.4 Stress as a point property

We now consider the stress components on a surface at an arbitrary orienta-

tion through a body loaded by external forces. In Fig. 3.3(a) a generalized

(a) F,= FsinO

F

A, =A/cose

Figure 3.2 (a) Resolution of a normal force. (b) Resolution of a normal stress

component.