Toppling failure - Rock Slope Engineering: Civil and Mining - page 210

the value the same procedure. It may be found that a

relatively short block that does not satisfy
equation (9.2) for toppling, may still topple
if the moment applied by the thrust force on
the upper face is great enough to satisfy the
condition stated in (v) above. If the condi-
tion P n − 1, t >P n − 1, s is met for all blocks,
then toppling extends down to block 1 and
sliding does not occur.

W n ( cos ψ p tan φ p − sin ψ p )

P n − 1, s = P n −

( 1 − tan φ p tan φ d )

(9.25)

9.4.3 Calculation procedure for toppling
stability of a system of blocks

The calculation procedure for examining toppling
stability of a slope comprising a system of blocks
dipping steeply into the faces is as follows:

(vii) Eventually a block may be reached for

which P n − 1, s >P n − 1, t . This establishes
block n 2 , and for this and all lower blocks,
the critical state is one of sliding. The
stability of the sliding blocks is checked
using equation (9.24), with the block being
unstable if (S n

(i) The dimensions of each block and the

number of blocks are defined using

equations (9.7)-(9.12).

R n tan φ b ) . If block 1 is
stable against both sliding and toppling (i.e.
P 0 < 0), then the overall slope is considered
to be stable. If block 1 either topples or
slides (i.e. P 0 > 0), then the overall slope is
considered to be unstable.

=

(ii) Values for the friction angles on the sides

and base of the blocks (φ d and φ p ) are
assigned based on laboratory testing, or
inspection. The friction angle on the base
should be greater than the dip of the base
to prevent sliding (i.e. φ p >ψ p ).

(iii) Starting with the top block, equation (9.2)

is used to identify if toppling will occur,
that is, when y/x > cot ψ p . For the upper
toppling block, equations (9.23) and (9.25)
are used to calculate the lateral forces
required to prevent toppling and sliding,
respectively.

9.4.4 Cable force required to stabilize a slope

If the calculation process described in
Section 9.4.3 shows that block 1 is unstable, then
a tensioned cable can be installed through this
block and anchored in stable rock beneath the
zone of toppling to prevent movement. The design
parameters for anchoring are the bolt tension, the
plunge of the anchor and its position on block 1
(Figure 9.9(c)).
Suppose that an anchor is installed at a plunge
angle ψ T through block 1 at a distance L 1 above
its base. The anchor tension required to prevent
toppling of block 1 is

(iv) Let n 1 be the uppermost block of the

toppling set.

(v) Starting with block n 1 , determine the lat-

eral forces P n − 1, t required to prevent top-
pling, and P n − 1, s to prevent sliding. If
P n − 1, t >P n − 1, s , the block is on the point of
toppling and P n − 1 is set equal to P n − 1, t ,or
if P n − 1, s >P n − 1, t , the block is on the point
of sliding and P n − 1 is set equal to P n − 1, s .
In addition, a check is made that there is
a normal force R on the base of the block,
and that sliding does not occur on the base,
that is

W 1 / 2 (y 1 sin ψ p − x cos ψ p ) + P 1 (y 1 − x tan φ d )
L 1 cos (ψ p + ψ T )

T t =

(9.26)

while the required anchor tension to prevent
sliding of block 1 is

R n > 0 and ( | S n | >R n tan φ p )

P 1 ( 1 − tan φ p tan φ d ) − W 1 ( tan φ p cos ψ p − sin ψ p )

T s =

(vi) The next lower block (n 1 − 1 ) and all the

lower blocks are treated in succession using

tan φ p sin (ψ p + ψ T ) + cos (ψ p + ψ T )

(9.27)