28 Stress
Fcose
Figure 3.1 Resolution of a force.are not consciously encountered in daily life. This means that we do not
have intuitive feelings about quantities such as stress and strain, except
in reduced one-dimensional forms (where the stress or strain is acting
in only one direction) or when the values of the normal components
are equal (so that stress becomes pressure and strain becomes a uniform
contraction or expansion). An extra effort is required to develop an un-
derstanding of a tensor quantity and hence the nature of stress in a solid.
Readers who have already been exposed to some stress analysis
and who may be somewhat sceptical about the lack of their intuitive
understanding of stress might like to try Q3.10 in Part B of the book. In
Part B, the questions can be tackled without the answers being visible. If
you can solve problem Q3.10 without recourse to mathematics, you do
have a feeling for stress!
Force is a vector quantity: it has magnitude and direction. For ex-
ample, we might say that ’a force of 5 MN is acting horizontally in a
northeasterly direction’. To specify a vector in two dimensions requires
two pieces of information: either the magnitude and direction or the x
and y components of the vector. In three dimensions, three pieces of
information are required: either the magnitude and two directions or the
x, y and z components of the vector. When a force, F, is resolved through
an angle 8, the resultant is F cos 8, as the diagram in Fig. 3.1 illustrates.
The units of force are the newton (N) or the pound force (lbf) with
dimensions LMT-2.
Stress is a tensor quantity developed from the idea of normal forces
and shear forces acting within the rock. Pushing your hand down on a
table generates a normal force. Pushing your hand along the surface of
the table generates a shear force. It is the fact that a solid can sustain
a shear force which causes the stress field: otherwise, there would just
be a scalar pressure, with the same value in all directions, and rock
tunnels would float to the surface. The first key to understanding stress
is understanding the existence of the shear force. The normal and shear
forces are scaled by the areas on which they act, giving normal stresses
and shear stresses’. The units of stress are newtons per metre squared,
N m-’, known as pascals, Pa, or pounds force per inch squared, lbf/in2,
with dimensions L-lMT-*.
When a plane is specified on which the stresses are acting, the normal
and shear stress components have magnitude and direction. The normal
stress acts normal to the plane; the shear stress acts along the plane.* In fact the stress components are defined as the limiting values at a point when the
area on which the forces act is reduced to zero.