ABSTRACT
Toppling is a mode of failure in rock slopes which has not been discussed until recently in the literature. A large degree of understanding of the process is due to Goodman and Bray (1976) who have suggested three toppling failure classes, together with a kinematic test and an analytical model of limit equilibrium to assess the factor of safety of toppling slopes. The purpose of this paper is to present nomograms that have been constructed according to the limit equilibrium model and a procedure for assessment of toppling slopes which accounts for the degree of cross-fracturation of the toppling rock columns. The general character of the nomograms is discussed, together with the possibility of plane failure associated with toppling under certain conditions. The nomograms are then used to discuss the validity of the simple kinematic test intended for use on a stereonet. Its validity is enforced while a modification in its formulation is suggested.
1 INTRODUCTION
Toppling in rock slopes is a mode of failure which can occur in formations presenting well developed and narrow-spaced to moderately narrow-spaced sub-vertical fractures dipping opposite to a slope face. Although not only restricted to them, most failures which have been described in the litterature by MUller (1968), De Freitas and Watters (1973), Heslop (1974), Bukovansky et al. (1974), Goodman and Bray (1976) and Piteau et al. (1981), have occured in sedimentary or metamorphosed sedimentary rock types. In these particular cases, the fractures involved were the bedding planes of limestones and sand- stones, or the schistosity planes of shales and slates. A second set of cross-fracturation of the toppling rock columns can also be involved in the failure process. Goodman and Bray (1976) have classified toppling failures into three classes (fig. 1), one involving the flexure of columns with no pre-existing cross-fracturation, a second one, the toppling of blocks already delineated by a widely spaced cross-fracturation, and thirdly, a combination of these two modes into a so called block flexure toppling class for columns containing numerous cross-fractures. The authors have also recognized the existence of secondary toppling modes whereby the process is initialed by external forces such as the weight of material overlying the columns. In order to understand the phenomenae involved, many authors including Ashby (1971), Hoffman (1972), Whyte (1973) and Soto (1974) have studied the overturning of blocks on base friction models, which enable to represent fairly accurately the real situations, especially for the primary modes of block toppling and block flexure toppling. Analytical aspects have been studied by Cundall (1971), Burman (1971), Byrne (1974), Hammer (1974) and Hitringer (1978). Even though finite element analysis techniques have been used, the overall tendency coming out of these many contributions is that a limit equilibrium analysis such as the one of Goodman and Bray, described in a further section, is best suited to account for the field and experimental observations. Moreover, finite element analyses have not allowed until now the assessment of instability for a particular toppling slope. On the other hand, Cundall's rigid block method, even though not discussed in this paper, is believed to be also very promising in the future for the study of toppling.
