The paper presents the limit equilibrium equations for the analysis of toppling blocks on a rock slope. The safety conditions are assessed by a numeric procedure implemented in an Excel worksheet. The UDEC numerical program was exploited to validate the procedure. A probabilistic model considering the spacing variability of the discontinuities was developed to define the most relevant parameters in this kind of problems.

La communication presente les equations d'equilibre limite pour l'analyse de blocs rocheux sur une pente. Les conditions de securite sont evaluees par une procedure numerique en Excel. Le logiciel UDEC a ete exploite pour valider cette procedure. Un modèle probabilistique a ete developpe pour definir les plus importants paramètres dans ces problèmes.

Das Referat stellt die Grenzgleichgewichtsgleichungen fuer die Analyse von umstruerzenden Blök-ken auf eines Felshang vor. Die Sicherheitsbedingungen werden durch ein numerisches Verfahren mit Excel beurteilt. Um die Gueltigkeit des Verfahrens festzustellen, wurde das Programm UDEC benutzt. Ein die Klufabstandsvariabilitat beruecksichtigendes probabilistisches Modell wurde entwickelt, um die wichtigsten Parameter dieser Probleme zu bestimmen.

Stability of a rigid block on an inclined plane

In rock slopes, the simplest failure modes are sliding and toppling. Sliding occurs due to a translation movement of a rock block or a rock mass along a plane failure surface. Toppling involves rotation of rock blocks or columns around a fixed point on its base. The common starting point for the study of rock slopes is to establish the statics limit equilibrium equations for a single rigid block on an inclined plane. Considering a parallelepiped block with dimensions b(base length) × h(height) on α dipping plane and a f friction angle, the static equilibrium equations are (no sliding condition)

(Equation in full paper)

The first equation arises from the condition of a null resultant force and is intimately related to the definition of friction angle. Equation (2) is derived from the equivalent condition for the moments and means that the resultant forces (in this case, just the dead weight of the block) falls inside the block's base. Both equations can be plotted on a b/hvstg α graph (Figure 1).

This figure allows to establish the regions where the block is stable, where it slides or tilts. However, it can be seen that there is a zone where the block can slide and tilt at the same time. Inside this region, the correct failure mode cannot be deduced from the limit equilibrium equations (1) and (2) because they are not formally valid. In fact, the zone where both sliding and tilting may occur does not correspond to the intersection of both equations, since it is required to the force acting at the toe to be large enough to provide block fixity (Bray and Goodman).

(Figure in full paper)

As Sagaseta (1986) states, to define correctly the boundaries between the regions where sliding, tilting and sliding plus tilting occur it is necessary to consider a dynamics approach.

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