LECTURE 28. Module 8 : Rock slope stability 8.3 WEDGE FAILURE

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LECTURE 28 8.3 WEDGE FAILURE When two or more weak planes in the slope intersect to from a wedge, the slope may fail as wedge failure.

1 LECTURE WEDGE FAILURE When two or more weak planes in the slope intersect to from a wedge, the slope may fail as wedge failure. The basic condition at which wedge mode of slope failure happens are as follows, 1. Two planes will always intersect in a line 2. Plunge of the line of intersection must be flatter than the dip of the face and steeper that the average friction angle of the two slide planes, i.e (ψ fi > ψ i > ϕ) 3. The line of intersection must dip in a direction out of the face for sliding to be feasible. (a) (b) Figure 8.7: Wedge failure in rock a) actual field b) line diagram with tension crack 235

2 ξ/2 ξ/2 β R A R B ξ = included angle W cos ψ i β = angle of tilt Figure 8.8: Analysis of wedge failure in rock Factor of safety of the wedge is defined by assuming sliding is resisted only by friction and the friction angle ϕ is same for both the surface. Factor of safety (FS) = (RR AA+ RR BB ) tan φφ WW tan ψψ ii cos ψψ ii R A and R B are the normal reactions provided by plane A and B. The component of weight acting down the line of intersection is w sin ψ i. The forces R A and R B are found by reducing them into components normal and parallel to the direction along the line of intersection. RR AA sin (β ξ 2 ) = RR BBsin (β + ξ 2 ) RR AA cos (β ξ 2 ) + RR BBcos (β + ξ 2 ) = W cos ψ i RR AA + RR BB = W cos ψψ ii ssssssβ sin ξ 2 Hence, FS = ssssssβ sin ξ 2 tan φφ. tan ψ i In other words, FS w = K. FS P where FS w is the factor of safety of a wedge supported by friction only. FS p is the factor of a plane failure in which the slide plane with friction angle ϕ dips at the same angle as the line of intersection ψ i. K is the wedge factor depends upon the included angle of the wedge ξ and the angle of tilt β of the wedge. 236

3 Figure 8.9: Wedge factor K as a function of wedge geometry (Duncan and Wiley, 1999), 237

4 8.4 CIRCULAR FAILURE It is understood that, the slope failure in rocks largely controlled by the weak planes. But when the fractures are too many and closely spaced, the slope automatically finds the least resistance path to failure. The failure surface in such highly fractured cased is mostly circular. The conditions under which circular failure (Duncan and Wiley, 1999), Occurs when the individual particles in soil or rock mass are very small compared with the size of the slope. Broken rock in a fill will tend to behave as soil and fail in a circular mode when the slope dimensions are substantially greater than the dimensions of the rock fragments. When soil consisting of sand, silt and smaller particle sizes will exhibit circular slide surfaces, even in small slopes. Highly altered and weathered rocks, as well as rock with closely spaced, randomly oriented discontinuities such as some rapidly cooled basalts, will also tend to fail in this manner. Face failure Deep seated / base failure Toe failure Figure 8.10: Different modes of circular slope failure 238

5 (a) (b) Figure 8.11: The shape of typical sliding surfaces: (a) large radius circular surface in homogeneous, weak material, (b) non-circular surface in weak, surfacial material with stronger rock at base. 239

6 8.4.1 Limit equilibrium stability analysis The factor of safety of the circular failure is performed based on limit equilibrium analysis. Procedure involves comparing the available shear strength along the sliding surface with the force required to maintain the slope in equilibrium. The application of this procedure to circular failures involves division of the slope into a series of slices that are usually vertical, but may be inclined to coincide with certain geological features. The base of each slice is inclined at angle ψ b and has an area A. In the simplest case, the forces acting on the base of each slice are the shear resistance S due to the shear strength of the rock (cohesion c; friction angle φ), and forces E (dip angle ψ; height h above base) acting on the sides of the slice (see detail in figure XXX). The analysis procedure is to consider equilibrium conditions slice by slice, and if a condition of equilibrium is satisfied for each slice, then it is also satisfied for the entire sliding mass. Figure 8.12: Forces acting on a slice in the limit equilibrium analysis 240

7 The analyses are statically indeterminate and assumptions are required to make up the imbalance between equations and unknowns (Duncan, 1996). The various limit equilibrium analysis procedures either make assumptions to make up the balance between known and unknowns, or they do not satisfy all the conditions of equilibrium. For example, the Spencer Method assumes that the inclination of the side forces is the same for every slice, while the Fellenius and Bishop methods do not satisfy all conditions of equilibrium. Later, computers made it possible to more readily handle the iterative procedures inherent in the limit equilibrium method, and this lead to mathematically more rigorous formulations which include all interslice forces and satisfy all equations of statics. Table 8.1: Equations of statics to be satisfied Figure 8.13: Software generated analysis with method of slices 241

(Refer Slide Time: 01:15)

(Refer Slide Time: 01:15) Soil Mechanics Prof. B.V.S. Viswanathan Department of Civil Engineering Indian Institute of Technology, Bombay Lecture 56 Stability analysis of slopes II Welcome to lecture two on stability analysis of