Toppling failure - Rock Slope Engineering: Civil and Mining - page 208

the crest of the slope is P n are M n and L n on the upper and lower faces

respectively of the block, and are given by the
following.
If the n th block is below the slope crest, then

y n = n(a 1 − b) (9.8)

while above the crest

M n = y n (9.13)

y n = y n − 1 − a 2 − b (9.9)

L n = y n − a 1 (9.14)

The three constants a 1 , a 2 and b that are defined
by the block and slope geometry and are given by

If the n th block is the crest block, then

a 1 = x tan (ψ f − ψ p ) (9.10)

M n = y n − a 2 (9.15)

a 2 = x tan (ψ p − ψ s ) (9.11)

L n = y n − a 1 (9.16)

b = x tan (ψ b − ψ p ) (9.12)

If the n th block is above the slope crest, then

9.4.2 Block stability

Figure 9.7 shows the stability of a system of
blocks subject to toppling, in which it is possible
to distinguish three separate groups of blocks
according to their mode of behavior:

M n = y n − a 2 (9.17)

L n = y n (9.18)

For an irregular array of blocks, y n , L n and M n

can be determined graphically.
When sliding and toppling occurs, frictional
forces are generated on the bases and sides of
the blocks. In many geological environments, the
friction angles on these two surfaces are likely to
be different. For example, in a steeply dipping
sedimentary sequence comprising sandstone beds
separated by thin seams of shale, the shale will
form the sides of the blocks, while joints in the
sandstone will form the bases of the blocks. For
these conditions, the friction angle of the sides of
the blocks ( φ d ) will be lower than friction angle
on the bases ( φ p ). These two friction angles can be
incorporated into the limit equilibrium analysis as
follows.
For limiting friction on the sides of the block:

(a) A set of stable blocks in the upper part of the
slope, where the friction angle of the base
of the blocks is greater than the dip of this
plane (i.e. φ p >ψ p ), and the height is limited
so the center of gravity lies inside the base
( y/x < cot ψ p ).
(b) An intermediate set of toppling blocks where
the center of gravity lies outside the base.

(c) A set of blocks in the toe region, which

are pushed by the toppling blocks above.
Depending on the slope and block geometries,
the toe blocks may be stable, topple or slide.

Figure 9.9 demonstrates the terms used to define
the dimensions of the blocks, and the posi-
tion and direction of all the forces acting on
the blocks during both toppling and sliding.
Figure 9.9(a) shows a typical block ( n ) with the
normal and shear forces developed on the base
(R n , S n ) , and on the interfaces with adjacent
blocks (P n , Q n , P n − 1 , Q n − 1 ) . When the block is
one of the toppling set, the points of application
of all forces are known, as shown in Figure 9.9(b).
The points of application of the normal forces

Q n = P n tan φ d (9.19)

Q n − 1 = P n − 1 tan φ d (9.20)

By resolving perpendicular and parallel to the
base of a block with weight W n , the normal and
shear forces acting on the base of block n are,