Engineering Rock Mechanics

Understanding stress 29

0, acts over this area, A normal stress = force/area, on = F/A

noma1 stress = Fcos@,(AkosB)=(F/A) cos2@ =Gnc0s26

Resultant acts over this area

AkosB

Figure 3.2 Resolution of a stress component, from the heavier arrow to the lighter arrow

(for a prism of unit depth).

Because the shear stress on the plane can be resolved into two perpen-
dicular components, there will be a total of three orthogonal stress com-
ponents acting on the plane: the normal stress and two shear stresses.
For example, on a vertical plane in the E-W direction, there could be a
normal stress of 10 MPa acting due north, a shear stress of 5 MPa acting
due west, and a shear stress of 7 MPa acting vertically downwards.
Thus, to specify a tensor in two dimensions requires three pieces of
information: either (a) two normal stresses acting in the specified x, and
y directions, plus the shear stress; or (b) the two principal stresses (see
Section 3.2, Q3.5) and their orientation. To specify a tensor in three di-
mensions requires six pieces of information: (a) either the three normal
stresses and the three shear stresses acting on three specified orthogonal
planes; or (b) the three principal stresses and their three directions.
We noted above that when a force, F, is resolved through an angle 8,
the result is F cos 8. However, when a stress component, say a normal
component a,,, is resolved through an angle 8, the resultant is a, cos2 8.
The cos2, term arises because a double resolution is required: i.e. a
resolution of the force component and a resolution of the area on which
it is acting. This is illustrated in Fig. 3.2, where the original normal
stress component (represented by the heavy arrow) is transformed to
the new stress component (represented by the Iighter arrow) using the
cos2 8 term. The second key to understanding stress is understanding
this double resolution.
When all the stress components are transformed in this way in two
dimensions, the resulting equations for the stresses on a plane give the
locus of a circle in normal stress-shear stress space. The third key to
understanding stress is realizing that the cos2 8 resolution of one normal
stress and the sin28 resolution of the other normal stress enables a
graphical resolution of the stress components using Mohr 's circle (see
Q3.4). The circle occurs due to the cos28 resolution of the first normal
stress, the sin2 0 resolution of the second normal stress (sin2 because the
second normal stress is perpendicular to the first normal stress), and the
fact that a circle with radius r is represented in (r, e) space as

r2 cos2 e + r2 sin2 0 = r2.
These stress principles apply in all materials from chalk to cheese,
and hence in all rock types. Moreover, in engineering rock mechanics,
a knowledge of the values of the in situ natural stress components is
required to understand the pre-engineering stress conditions. This is