Foundation instability 303foundations under high loads and where the rock mass is effectively con-
tinuous and weak. Such circumstances will be rare, and so the simplified
analysis presented here is included mainly for completeness.
With the same loading geometry as for the discontinuum analysis illus-
trated in Fig. 17.14, but for a mesh of square elements, consider the stresses
acting on the sides of the elements to determine whether and, if so, where
local plastic failure occurs according to a suitable yield criterion. In the inter-
ests of simplicity, a Mohr-Coulomb criterion with @ = 0 has been used here,
with the added assumption that the sides of the elements have zero cohesion.
Figure 17.17 illustrates the basic problem. The stresses acting on elements
I and I1 can be estimated by considering the stresses resulting from the
overburden and the applied load in conjunction with the yield criterion.
Analysing elements I shows that the overburden stress acting at these
locations-remote from the loaded area-is yz. It follows from inspection
of the yield criterion that the horizontal stress cannot exceed yz + 2c (see
inset Mohr’s circle in Fig. 17.17). By inspection, we see that at element I1
the vertical stress due to the applied load and the overburden is greater
than the horizontal stress. However, the horizontal stress has the same
magnitude throughout, i.e. yz + 2c, and hence the vertical stress acting on
element I1 cannot exceed p + 2c + 2c, that is, p + 4c. But, because we can
approximate the vertical stress acting on element I1 asp + yz, it follows that
p = 4c-which is a lower bound solution and should be compared to the
result of p = 6c as an upper bound solution found earlier.
In the case of a more realistic yield criterion and stress distribution, the
analysis becomes much more complex. Closed form solutions exist for the
simpler cases experienced in soil mechanics but, in general, numerical
methods are required to produce solutions.
7 7.2.2 Stress distributions beneath applied loads
Two of the classic closed form solutions in stress analysis are for normal
and shear line loads applied to the surface of a CHILE half-space. These
are commonly attributed to Boussinesq (1883) and Cerruti (1882), respec-
tively. We illustrate these problem geometries and key aspects of the
solutions in Figs 17.18(a) and (b).
Figure 17.17 A lower bound solution for foundation collapse load with associated
Mohr’s circle.