**
Limit equilibrium analysis for toppling failure**

**
**Toppling failure of rock blocks along the slope occurs when there are formed by
closely spaced and steeply inclined discontinuity system dipping into the
excavation. For analysis of slope stability against toppling failure, a
kinematic analysis of the structural geology is first carried out to identify
potential toppling conditions. If failure condition exists, a stability analysis
specific to toppling failure is conducted.

Toppling failure of an individual block or column can exist only in the following conditions

· There is an inclined surface upon which the block rests,

· There is a joint set approximately perpendicular to the inclined surface,

·

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**

Stability analysis involves an iterative process in which the dimensions of all the blocks and the forces acting on them are calculated, and stability of each block is examined, starting with

· 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 and the height is limited so the centre of gravity lies inside the base.

· An intermediate set of toppling blocks where the centre of gravity lies outside the base.

· A set of blocks in the toe region, which are pushed by the toppling blocks above.

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**

The factor of safety can be calculated as the ratio of resisting moments to driving moments

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**

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**

Figure 15 shows a typical block (*n*) with the normal force (*R _{n}*)
and shear forces (

When sliding and toppling occurs, frictional forces are generated on the bases
and sides of the blocks. If friction angles are different on the two surfaces,
then the friction angle of the sides of the blocks (ф_{d}) will be lower
than friction angle on the bases (ф_{p}). For limiting friction on the
sides of the block:

Q_{n} = P_{n} tan ф_{d}

Q_{n−1 }= P_{n−1 }tan ф_{d}

By resolving perpendicular and parallel to the base of a block with weight W_{n},
the normal and the shear forces acting on the base of block *n* are given
by:

R_{n} =
W_{n }cosθ + (P_{n} − P_{n−1}) tan
ф_{d}

S_{n} =
W_{n}sinθ_{ }+ (P_{n} − P_{n−1})

**
∑ **
F_{Y} = 0, **
∑ **F_{X} = 0

S_{n} =
R_{n} tan
ф_{p}

R_{n}= W_{n}
cosθ

Where, S_{n} is the shear force along the column-base contact and R_{n}
is the normal force across the column base contact.
Thus

Considering rotational equilibrium, it is found that the force P_{n−1}
that is just sufficient to prevent toppling has the value

P_{n−1},t = [P_{n}(M_{n} − ∆x tanф_{d})
+ (W_{n}/2) (y_{n} sinθ − ∆x cosθ)]/Ln)

When the block under consideration is one of the sliding set

S_{n} = R_{n} tan
ф_{p}

To detemined the magnitude of P_{n} we set the momnets about the pivot
point, O,

If the friction angle is same for both the surfaces,

Figure 15: Model for limiting equilibrium analysis of toppling on a stepped base (Goodman and Bray, 1976).

Figure 16: Forces acting on the n^{th} column sitting on a stepped base

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**

Figure 17:
Limiting equilibrium conditions for toppling and sliding of *
n*th
block: (a) forces acting on *
n*th
block; (b) toppling of *
n*th
block;

**
**

*
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:

(i) The dimensions of each block and the number of blocks are defined using equations.

(ii) Values for the friction angles on the sides and base of the blocks (ф_{d}
and ф_{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. ф_{p}>θ_{p}).

(iii) Starting with the top block, equation is used to identify if toppling will
occur, that is, when *y/x >*cot ф_{p}. For the upper toppling
block, equations are used to calculate the lateral forces required to prevent
toppling and sliding, respectively.

(iv) Let*n*_{1} be the uppermost block of the toppling set.

(v) Starting with block *n*_{1}, determine the lateral forces *P _{n}*

(vi) The next lower block *(n*_{1} − 1*) *and all the lower
blocks are treated in succession using 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 condition *P _{n}*

(vii) Eventually a block may be reached for which *P _{n}*

In the toppling simulation where, the slope is composed of rock columns (blocks)
on a flat base, an unstable rock column (n+1) will exert a force of P magnitude
P_{n} on the down slope adjacent column (n). Therefore, the n^{th}
column is no longer be under gravitational forces only. The forces acting on
the n^{th }column for limiting equilibrium conditions are shown in
figure 17. When the base plane of the columns is flat, overturning of the
columns is not kinematically possible without displacement of the pivot points
(O) of the columns due to required dilation of the columnar structure. Such
dilation is possible if the magnitude of the overturning forces exveeds the
shear resistance of the rotation points of the columns.

When the slope is composed of rock columns on a steeped base, rotation of the n^{th
}block about its pivot will cause shear (S_{n}) and normal forces
(N_{n}) to develop on the block. While overturning about the pivot
point, O, the n^{th} block will maintain contact with block n and n-1 as
shown in Figure 17 because the locations of the shear and normal forces are
known, the stability of the column against rotation and sliding is determinate.