296 Surface excavation instability mechanisms77.7.4 Toppling
To complete the set of fundamental mechanical modes of structurally
controlled instability, toppling failure is considered. Toppling failure has
traditionally been regarded as occurring in two modes: direct toppling and
flexural toppling. The former occurs when the centre of gravity of a block of
rock lies outside the outline of the base of the block, with the result that a
critical overturning moment develops. The latter occurs under certain cir-
cumstances when a layered rock mass outcrops at a rock slope, and the
principal stress parallel to the slope face induces inter-layer slip which causes
the intact rock to fracture and the resulting blocks to overturn. The distinction
between these two toppling modes of instability is illustrated in Fig. 17.9.
Direct toppling instability. Will a block resting on an inclined rock surface
be stable, or slide, or topple, or simultaneously slide and topple? The nature
of the instability, if any, is determined from considerations of the block
geometry and the angle of friction between the block and the surface on
which it is resting. The four possibilities are shown in Fig. 17.10, as the
various regions in a graph of block aspect ratio versus friction angle.
Sliding will only occur when the dip of a plane exceeds the angle of
friction. This results in the vertical line dividing Fig. 17.10 into regions-
with no sliding on the left and sliding on the right.
To establish the equilibrium due to toppling, consider the location of the
line of action of the force due to gravity. This passes through the centre of
gravity of the block and will coincide with the lower apex of the block if
blh = tan y, which is the limiting equilibrium condition. Thus, toppling will
not occur if blh > tan y, and will occur if blh < tan y.
The resulting four categories of equilibrium are
(a) no sliding and no toppling: y < Q and blh > tan y;
(b) sliding but no toppling: y > Q and b/h > tan y;
(c) no sliding but toppling: y < Q and blh < tan y;
(d) sliding and toppling: y > Q and blh < tan y.
These four fundamental categories represent the basic circumstances of
toppling and related sliding, and enable a rapid initial analysis of whether
direct toppling could take place and hence whether further analysis is
necessary.
Figure 17.9 Direct and flexural modes of toppling instability.