4.3.1 Kinematics of block toppling failure
The potential for toppling failure can be assessed from three kinematic tests (Goodman and Bray, 1976). These tests examine shape of the block potential to failure, relationship between dip of the plane forming the slabs and face angle and alignment of the block with respect to the slope face.
(I) Test for delineation of Block shape
The basic mechanics of stability of a block against toppling on an inclined plane are illustrated in Figure 11. The governing conditions for toppling failure are height (h) and width (t) and dip angle ( QUOTE ). There are four cases to occur in similar conditions
Figure 11: Basic model for toppling failure
FIGURE 12: Conditions for sliding and toppling of a block on an inclined plane (Hoek and Bray, 1977)
Therefore, if the block thickness-to height ratio of the block is less than tan then the resultant force due to its weight will exist out side the toe of the block. Thus an overturning moment will develop about the pivot point (Figure12).
(II) Inter-layer slip test
One of the essential prerequisites for toppling failure is shear displacement along the face to face contacts on the top and bottom faces of the blocks (figure 13). 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 ф with the normal to the layers, where ф is the friction angle of the sides of the blocks. If is the dip of slope face and α is the dip of the planes forming the sides of the blocks, then the condition for interlayer slip is given by:
(180 − − α) ≥ (90 − ф)
α≥ (90 − ) + ф
Figure 13: Block alignment test in toppling failure
(III) Block alignment test
The planes forming the blocks should strike approximately parallel to the slope face so that each layer is free to topple. Field observations shows that instability is possible where the dip direction of the planes forming sides of the blocks, αd is within about 100 of the dip direction of the slope face αf, i.e.
|(αf− αd)| <10◦
Figure 14: Stereographic projection for block alignment test