Engineering Rock Mechanics

Question and answers: foundation and slope instability mechanisms^303

(c) This analysis is for CHILE materials. By making use of the +j
theory, show qualitatively how the bulb of pressure may be expected
to change when the ground beneath the load consists of a layered
discontinuous rock mass dipping at some arbitrary angle.
(d) On the basis of these results, what are the ramifications this
analysis has for site investigation in layered rocks?

A17.8 (a) This is a problem in which the stress components are defined
in polar co-ordinates, which are expressed in terms of a radius r and an
angle 6. If we examine the expression for radial stress,

2P sin6

nr

or = -,

we see that it is a maximum when 6 is 90". This maximum must also
occur when the direction of action of the radial stress is parallel to the
applied load, and so we conclude that the geometry of the problem is as
shown below.

In this problem, the radial and tangential stresses are principal stresses,
and so the magnitude of tre is zero; also, the magnitude of oe is
zero. These simplifications to the stress state mean that the solution to
problems associated with forces applied to the surface of a semi-infinite
body are usually conducted using polar co-ordinates.

(b) Rearranging the expression for radial stress gives

2p.

r = - sm8.

and so, knowing that P and a, are constant, we can find radius values
for given angles. If we assume that 2P/nor = 1, the following results are
obtained.

n or

eo 15 30 45 60 75 90

r 0.000 0.259 0.500 0.707 0.866 0.966 1.000

These are plotted in the diagram below, and show how a cylindrical

contour of constant radial stress develops (remembering that a line load

is being applied). It is the cylindrical form of this contour that gives rise

to the term 'bulb of pressure'.