ABSTRACT: A probabilistic approach for key block analysis was introduced and compared with the current deterministic analysis in key block theory. A case study was made to demonstrate the capability of the probabilistic key block analysis, where three subway tunnels were studied. The probabilistic key block analysis incorporates probability distributions of rock joint orientations, trace lengths, spacings and friction angles to predict the size, shape, frequency of occurrence of key blocks as well as a positional probability of failure, which identifies the parts of the rock mass being most susceptible to key block failure. Deterministic procedures do not take into account these important variables and give a worst case analysis.


Rock joints or discontinuities in a rock mass play an important role in the design and stability analysis of geotechnical structures such as mines, storage chambers, power houses, or nuclear waste disposal sites. Generally a stereographic projection was applied to the kinematic analysis of rock joints for the structural stability (Hoek and Bray 1981). Also, a factor of safety could be obtained for potential failure blocks (John 1968). Recently, Goodman and Shi (1985) developed the key block theory which is a numerical algorithm of joint analysis for structural stability and support design. A common point among the above techniques of joint analysis is that a joint set was represented by a single joint plane whose orientation and geometry is known precisely. However, natural rock joints have significant dispersions and require a cluster analysis to group them into several joint sets. Various joint models were studied and their key parameters were inferred from the field data (Arnold 1941, Grossman 1985). McMahon (1971) realized the influence of joint orientation dispersion over the mean attitude on the slope stability and incorporated the probabilistic distribution of joint orientations into the kinematic slope analysis. Also, a non- parametric geostatistics was introduced to model the joint systems in a rock mass, which predicted not only the global probability distributions of joint parameters, but also their local probability distributions (Young 1987). By applying this model, the superiority of a localized probabilistic approach for kinematic rock slope stability analysis was demonstrated on an open pit mine (Young and Hoerger 1988).

In this paper, the key block theorem (Goodman and Shi 1985) was incorporated with simulation techniques to obtain a probabilistic key block analysis, which takes into account the probabilistic distributions of whole joint parameters; orientations, spacings, and trace lengths. Then, metropolitan subway tunnels were studied to show the advantages of the probabilistic approach over the deterministic key block analysis. It was found that the probabilistic approach yields additional information about the key block failure, which is very important for designing an excavation and supporting systems to prevent key block failure. The probabilistic key block analysis does not require additional field sampling of joints and its whole input data is available from the joint mapping data used for the deterministic key block analysis.


The principle of "block theory" (or key block theorems) is that the potential key blocks are found prior to their failures and their stability is secured, then no block failure will occur anywhere in the entire excavation area.

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