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,

·

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.

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

Figure 15 shows a typical block (n) with the normal force (Rn) and shear forces (Sn) developed on the base. The forces developed at the interfaces with adjacent blocks are Pn, Qn, Pn−1, Qn−1. When the block is one of the toppling set, the points of application of all forces are known, as shown in figure 17. The points of application of the normal forces Pn are Mn and Ln on the upper and lower faces respectively of the block, and are given by the following.  The value of blocks, yn, Ln and Mn can be determined graphically.

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 (фd) will be lower than friction angle on the bases (фp).  For limiting friction on the sides of the block:

Qn = Pn tan фd

Qn−1 = Pn−1 tan фd

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

Rn = Wn cosθ  + (Pn − Pn−1) tan фd

Sn = Wnsinθ + (Pn − Pn−1)

FY = 0, FX = 0

Sn = Rn tan фp

Rn= Wn cosθ

Where, Sn is the shear force along the column-base contact and Rn is the normal force across the column base contact. Thus

Considering rotational equilibrium, it is found that the force Pn−1 that is just sufficient to prevent toppling has the value

Pn−1,t = [Pn(Mn − ∆x tanфd) + (Wn/2) (yn sinθ − ∆x cosθ)]/Ln)

When the block under consideration is one of the sliding set

Sn = Rn tan фp

To detemined the magnitude of Pn 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 nth column sitting on a stepped base

Figure 17: Limiting equilibrium conditions for toppling and sliding of nth block: (a) forces acting on nth block; (b) toppling of nth 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 (фd and фp) 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. фpp).

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

(iv) Letn1 be the uppermost block of the toppling set.

(v) Starting with block n1, determine the lateral forces Pn−1,trequired to prevent toppling, and Pn−1,sto prevent sliding. If  Pn−1,t> Pn−1,s, the block is on the point of toppling and Pn−1 is set equal to Pn−1,t, or if Pn−1,s> Pn−1,t, the block is on the point of sliding and Pn−1 is set equal to Pn−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 Rn>0 and (|Sn| >Rntanфp)

(vi) The next lower block (n1 − 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 Pn−1,t> Pn−1,sis met for all blocks, then toppling extends down to block 1 and sliding does not occur.

(vii) Eventually a block may be reached for which Pn−1,s> Pn−1,t. This establishes block n2, 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 (Sn= Rntanфb). If block 1 is stable against both sliding and toppling (i.e. P0<0), then the overall slope is considered to be stable. If block 1 either topples or slides (i.e. P0>0), then the overall slope is considered to be unstable.

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 Pn on the down slope adjacent column (n). Therefore, the nth column is no longer be under gravitational forces only.  The forces acting on the nth 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 nth block about its pivot will cause shear (Sn) and normal forces (Nn) to develop on the block. While overturning about the pivot point, O, the nth 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.