Toppling Failure.docx

Chapter 10

Toppling failure

10.1 INTRODUCTION The failure modes discussed in the three previous chapters all relate to sliding of a rock or soil mass along an existing or induced sliding surface. This chapter discusses a different failure mode – that of toppling, which involves rotation of columns or blocks of rock about a fixed base. Similarly to plane and wedge failure, the stability analysis of toppling failures involves first carrying out a kinematic analysis of the structural geology to identify potential toppling conditions, and then if this condition exists, performing a stability analysis specific to toppling failures. Figure 10.1 shows typical toppling blocks in columnar basalt. One of the earliest references to toppling failures is by Muller (1968) who suggested that block rotation or toppling may have been a contributory factor in the failure of the north face of the Vajont slide (Figure 10.2). Hofmann (1972) carried out a number of model studies under Muller’s direction to investigate block rotation. Similar model studies were carried out by Ashby (1971), Soto (1974) and Whyte (1973), while Cundall (1971), Byrne (1974) and Hammett (1974) incorporated rotational failure modes into the computer analysis of rock mass behaviour. Figure 10.3 shows a computer model of a toppling failure in which the solid blocks are fixed and the open blocks are free to move. When the fixed blocks at the face are removed, the tallest columns of blocks topple because their centre of gravity lies outside the base. The model illustrates a typical feature of toppling failures in which the tension cracks are wider at the top than at the base. This condition, which can best be observed when looking along the strike, is useful in the field identification of topples. Papers concerning field studies of toppling failures include de Freitas and Watters (1973) who discuss slopes in Britain, and Wyllie (1980) who demonstrates stabilisation measures for toppling failures related to railway operations. Most of the discussion that follows in this chapter is based on a paper by Goodman and Bray (1976) in which a formal mathematical solution to a simple toppling problem is shown. This solution, which is reproduced here, represents a basis for designing rock slopes in which toppling is present, and has been further developed into a more general design tool (Zanbak, 1983; Adhikary, Dyskin, Jewell et al., 1997; Bobet, 1999; Sagaseta, Sánchez and Cañizal, 2001).

10.2 TYPE FAILURES

OF

TOPPLING

Goodman and Bray (1976) have described a number of different types of toppling failures that may be encountered in the field, and each is discussed briefly in the following pages. The importance of distinguishing between types of toppling is that two distinct 269

270

Rock Slope Engineering

Toppling failure 270

Figure 10.1 Toppling blocks in columnar basalt showing typical tension cracks with widths greater at the top and narrowing near the base (Sea to Sky Highway near Whistler, British Columbia, Canada).

Figure 10.2 Suggested toppling mechanism of the north face of Vajont slide (Müller, 1968).

Figure 10.3 Computer-generated model of toppling failure; solid blocks are fixed in space while open blocks are free to move (Cundall, 1971).

methods of stability analysis for toppling failures can occur as described in the following pages – block and flexural toppling – and it is necessary to use the appropriate analysis in design.

10.2.1 toppling

Block

As illustrated in Figure 10.4a, block toppling occurs when, in a strong rock, individual columns are formed by a set of discontinuities dipping steeply into the face, and a second set of widely spaced orthogonal joints defines the column height. The short columns forming the toe of the slope are pushed forward by the loads from the longer overturning columns behind, and this sliding of the toe allows further toppling to develop higher up on the slope. The base of the failure generally consists of a stepped surface rising from one cross-joint to the next. Typical geological conditions in which this type of failure may occur are bedded sandstone and columnar basalt in which orthogonal jointing is well developed.

10.2.2 toppling

Flexural

The process of flexural toppling is illustrated in Figure 10.4b that shows continuous columns of rock, separated by well-developed, steeply dipping discontinuities, breaking in flexure as they bend forward. Typical geological conditions in which this type of failure may occur are thinly bedded shale and slate in which orthogonal jointing is not well developed. Generally, the basal plane of a flexural topple is not as well defined as a block topple. Sliding, excavation or erosion of the toe of the slope allows the toppling process to start and it retrogresses back into the rock mass with the formation of deep tension cracks that

Figure 10.4 Common classes of toppling failures: (a) block toppling of columns of rock containing widely spaced orthogonal joints; (b) flexural toppling of

slabs of rock dipping steeply into face; (c) block flexure toppling characterised by pseudo-continuous flexure of long columns through accumulated motions along numerous cross-joints (Goodman and Bray, 1976).

become narrower with depth. The lower portion of the slope is covered with disordered fallen blocks and it is sometimes difficult to recognise a toppling failure from the bottom of the slope. Detailed examination of toppling slopes shows that the outward movement of each cantilevered column produces an interlayer slip and a portion of the upper sur- face of each plane is exposed in a series of back-facing, or obsequent scarps, such as those illustrated in Figure 10.4a.

10.2.3 toppling

Block–flexure

As illustrated in Figure 10.4c, block–flexure toppling is characterised by pseudo-continuous flexure along long columns that are divided by numerous cross-joints. Instead of the flexural failure of continuous columns resulting in flexural toppling, toppling of columns in this case results from accumulated displacements on the cross-joints. Because of the large number of small movements in this type of topple, fewer tension cracks develop under these conditions than in flexural toppling, and fewer edge-to-face contacts and voids than in block toppling.

10.2.4 Secondary toppling modes Figure 10.5 illustrates a number of possible secondary toppling mechanisms suggested by Goodman and Bray (1976). In general, these failures are initiated by undercutting of the toe of the slope, either by natural agencies such as scour or weathering, or by human activities. In all cases, the primary failure mode involves sliding or physical breakdown of the rock, (a)

(d) 2050 Elevation (m)

(c)

(b)

Tension cracks

2000

Circular sliding surfaces Toppling at pit crest

1950 1900 1850 1800 1750

Talus

Sandstone Fault

Siltstone

Conglomerate

Figure 10.5 Secondary toppling modes: (a) Toppling at head of slide. (b) Toppling at toe of slide with shear movement of upper slope (Goodman and Bray, 1976). (c) Toppling of columns in strong upper material due to weathering of underlying weak material. (d) Toppling at pit crest resulting in circular failure of upper slope (Wyllie and Munn, 1979).

and toppling is induced in the upper part of the slope as a result of this primary failure (Figure 10.5a and b). Figure 10.5c illustrates a common occurrence of toppling failure in horizontally bedded sandstone and shale formations. The shale is usually significantly weaker and more susceptible to weathering than the sandstone, while the sandstone often contains vertical stress relief joints. As the shale weathers, support for the sandstone is undermined and columns of sandstone, with their dimensions defined by the spacing of the vertical joints, topple from the face. At some locations, the overhangs can be as wide as 5 m (16 ft), and failures of substantial volumes of rock can occur with little warning. An example of a combined toppling and circular slide is shown in the failure of a pit slope in a coal mine where the beds at the crest of the pit dipped at 70° into the face, and their strike was parallel to the face (Figure 10.5d). Mining of the pit slope at an angle of 50° initiated a toppling failure at the crest of the pit where movement monitoring showed that the columns of sandstone initially moved upwards and towards the pit. This movement resulted in a circular failure that extended to a height of 230 m (750 ft) above the base of the topple. Detailed monitoring of the slope showed that a total movement of about 30 m (100 ft) occurred on the slope above the pit, resulting in cracks opening in the crest of the mountain that were several metres wide and up to 9 m (30 ft) deep. As described in Section 15.7.1, continuous movement monitoring was used to allow mining to proceed under the moving slope, and finally the slope was stabilised by back-filling the pit (Wyllie and Munn, 1979). A further example of the toppling mechanism is illustrated in Figure 10.6 (Sjöberg, 2000). In open pit mines where the depth of the slope progressively increases, minor toppling movement may eventually develop into a substantial failure. Careful monitoring of

I

II

Crest

III

Elastic rebound

Joint slip fully developed (exaggerated displacements)

Joint slip

Joints

Stress redistribution

Toe

New mining step VI

V IV

Compression and bending of columns

Tensile bending failure at base of rotation

Movement on slide surface

Tensile bending failure propagated to crest

Displacements starting from toe

Figure 10.6 Failure stages for large-scale toppling failure in a slope (Sjöberg, 2000).

the movement, and recognition of the toppling mechanism, can be used to anticipate when hazardous conditions are developing.

10.3 KINEMATICS FAILURE

OF

BLOCK

TOPPLING

The potential for toppling can be assessed from two kinematic tests described in this section. These tests examine first the shape of the block, and second the relationship between the dip of the planes forming the slabs and the face angle. It is emphasised that these two tests are useful for identifying potential toppling conditions, but the tests cannot be used alone as a method of stability analysis.

10.3.1 Block shape test The basic mechanics of the stability of a block on a plane are illustrated in Figure 10.7a (see also Figure 1.11). This diagram shows the conditions that differentiate stable, sliding or toppling blocks with height y and width Δx on a plane dipping at an angle ψ p. If the

(a)

(c)

x

f

y

(180 –

W

(b)

d

d

p

(d)

f

f

–

d)

20°

d

(90 –

f

f)

d d

Figure 10.7 Kinematic conditions for flexural slip preceding toppling: (a) block height/width test for toppling; (b) directions of stress and slip direction in rock slope; (c) conditions for interlayer slip; (d) kinematic test defined on lower hemisphere stereographic projection.

friction angle between the base of the block and the plane is ϕ p, then the block will be stable against sliding when the dip of the base plane is less than the friction angle, that is, when ψ p < φ p – Stable

(10.1)

but will topple when the centre of gravity of the block lies outside the base, that is, when ∆x < tan ψ p − Topple y

(10.2)

For example, for a 3 m (10 ft) wide block on a base plane dipping at 10°, toppling will occur if the height exceeds 17 m (56 ft).

10.3.2 Interlayer slip test A requirement for toppling to occur in the mechanisms shown in Figures 10.4 and 10.6 is shear displacement on the face-to-face contacts on the front and back faces of the blocks. Sliding on these faces will occur if the following conditions are met (Figure 10.7b). The state of stress close to the slope face is uniaxial with the direction of the normal stress σ aligned parallel to the slope face. When the layers slip past each other, σ must be inclined at an angle ϕd with the normal to the layers, where ϕd is the friction angle of the sides of the blocks. If ψf is the dip of slope face and ψd is the dip of the planes forming the sides of the blocks, then the condition for interlayer slip is given by (Figure 10.7c) (180 − ψ f − ψ d ) ≥ (90 − φ d (10.3) ) or ψ d ≥ (90 − ψ f ) + φd

10.3.3 alignment test

(10.4)

Block

The other kinematic condition for toppling is that the strike of the planes forming the blocks is approximately parallel to the slope face so that each layer is free to topple with little constraint from adjacent layers. Observations of topples in the field show that instability is possible where the dip direction of the planes forming the sides of the blocks αd is within about 10° of the dip direction of the slope face αf, or (α f − α d ) < 10° (10.5)

The two conditions defining kinematic stability of topples given by Equations 10.4 and 10.5 can be depicted on the stereonet (Figure 10.7d). On the stereonet, toppling is possible for planes for which the poles lie within the shaded area, provided also that the base friction properties and shape of the blocks meet the conditions given by Equations 10.1 and 10.2, respectively.