4.2.2  Analysis of wedge failure with cohesion and friction angle

The most commonly employed analytical technique for this purpose is a rigid-block analysis, in which failure is assumed as linear sliding along one of the discontinuities or along the line of intersection formed by the discontinuities. This analysis takes into account the dimensions and shape of the wedge, different cohesion and friction angles on each slide plane, water pressure and a number of external forces. The stability of the wedge can be evaluated by resolving the forces acting along normal to the discontinuities and in the direction parallel to the line of intersection. These forces include the weight of the wedge, external forces such as foundation loads, seismic accelerations, tensioned reinforcing elements, water pressure acting on the surfaces, and the shear strength developed along the sliding plane or planes. A complete explanation related to different features of the slope involved in this analysis is explained by Hoek (et.al).

Geometry of the wedge is defined by the location and the orientation of as many as five bounding surfaces (figure 10). This includes two intersecting discontinuities, the slope face, the upper slope surface, and the plane of representing a tension crack, if present. Because the calculation is extended, this general analysis is best adapted to computer solution. However, in most cases, assumptions can be made that significantly simplifies the controlling stability equations so that a simplified solution could be obtained. Figure 10 defines the calculation of wedge stability under various assumptions. The formulas contain dimensionless factors (X, Y, A, and B), which depend on the geometry of the wedge. The calculation of these factors requires correct numbering of intersections of the planes (Hoek and Bray, 1981) as given below:

1 – Intersection of plane a with the slope face

2 - Intersection of plane b with the slope face

3 - Intersection of plane a with the upper slope surface

4 - Intersection of plane b with the upper slope surface

5 - Intersection of planes a and b

It is assumed that sliding of the wedge always takes place along the line of intersection numbered 5.

Figure 10: Pictorial View of wedge showing the numbering of intersection lines and planes

Where, Ca and Cb are the cohesive strength of plane a and b, фa and фb are the angle of friction along plane a and b,  QUOTE   is the unit weight of the rock, and H is the total height of the wedge. X, Y, A and B are dimensionless factors, which depend upon the geometry of the wedge, Ψa and Ψb are the dips of planes a and b, whereas, Ψi is the plunge of the line of their intersection.

Under fully drained slope condition, the water pressure is zero. Therefore, factor of safety of the wedge against failure is given by:

In case of frictional strength only, the cohesion is zero. Therefore, the factor of safety becomes