Foundations: stress distributions beneath variably loaded areas 325The analysis indicates that, were a circular (in plan) quarry to be oper-
ating in this rock mass, then the absence of potential instability mechanisms
for slopes with dip directions in the range 150-165' indicates that they could
be steep, but slopes with dip directions in the ranges 90-120' and 180-240'
would be vulnerable to instability unless cut to shallow dip angles. Is this
acceptable, or is an alternative solution required? One such alternative is
to avoid creating slopes within these ranges of dip directions. A generalized
corollary of the example is that circular excavations can never be optimal
in terms of maximizing slope dip: an elliptical or irregular polygonal
geometry will always be better, in that these allow the flexibility
necessary to harmonize the engineering geometry with the rock structure
geometry. A correctly orientated elliptical floor plan will always be better
than a circular plan for a quarry based on these slope instability
considerations.
The entire analysis is based on the simple criteria established for each of
the instability mechanisms. Further analysis is required to confirm that
the failure mechanisms are likely to be operative. The strength of the tech-
nique lies in its underlying philosophy utilizing primary instability criteria.
With the technique, it is possible to design a stable excavation without
recourse to mathematical analysis and subsequent interpretation of
factors of safety.
18.3 Foundations: stress distributions beneath
variably loaded areas
We have extended the slope mechanisms approach of Chapter 17 to the
study of the kinematic feasibility of four different potential slope
instability mechanisms. By analogy, we now extend the earlier considera-
tion of the stress distributions beneath point loads for foundations to the
stress distributions that occur beneath variably loaded areas, i.e. consider-
ing the more realistic circumstances. In the next section of this chapter, we
consider other factors as they relate to both slopes and foundations.
78.3.7 Cartesian form of the Boussinesq and Cerruti
solutions
In Section 17.2.2, the cylindrical polar form of the solutions was pen for
the stress distributions associated with single normal and shear point loads
on the surface of an infinite CHILE half-space, due respectively to
Boussinesq and Cerruti. In order to give these solutions greater utility in
the case of loaded areas and varying loads, it is helpful to first express them
in Cartesian form so that loaded areas can be discretized as elemental
components, each of a pen magnitude, and then compute the total
solution by integration of the components over the area in question.
Poulos and Davis (1974) provide the solutions for various Cartesian
components of stress and displacement in a form similar to those
tabulated in Fig. 18.12. Given that these are available, and knowing from
the theory of elasticity that the solutions for two or more separate loadings