This paper presents a new approach to assessing the stability of rock slope. The approach is developed by mathematically incorporating judgement and experience of practising engineers, with conventional limit equilibrium analysis. Fuzzy Set Theory is used for evaluating design parameters, and for assessing slope stability. Some examples are presented to illustrate the procedure for this proposed method.
La determination de la stabilite des versants rocheux est abordee de façon nouvelle. La methode combine le traitement mathematique du jugement et de l'experience de l'ingenieur praticien avec l'analyse traditionnelle par equilibre limite. La theorie des ensembles flous est utilisee pour l'evaluation des, paramètres de dimensionnement et pour le calcul de la stabilite de la pente. Quelques exemples sont presentes pour illustrer la demarche de la methode proposee.
Die vorliegende Arbeit beschreibt ein neues Verfahren zur Bestirnrnung der Stabilitat von Felsboeschungen. Das Verfahren beruht auf der Kombination von Ingenieurerfahrung und -verstandnis mit der Grenzgleichgewichtsmethode. Die Fuzzy Set Theorie dient zur Bestirnrnung der Entwurfsparameter und der Stabilitaet von Boeschungen. Einige Beispiele illustrieren die Anwendung des vorgeschlagenen Verfahrens.
Mechanical characteristics of rock masses contain various types of uncertainties. These uncertainties cause great difficulty in the determination of mechanical constants as definite values, even for the same grade classification of rock masses. In order to overcome this difficulty, the Fuzzy Set Theory (introduced by Zadeh 1965) is a useful tool, which allows us to consider the mechanical constants to be uncertain values. Thus, when analyzing rock engineering problems, the Fuzzy Set Theory must be adopted. It is particularly important for the assessment of rock slope stability, because in rock slope problems, the results of analyses are greatly influenced by the uncertainties of mechanical characteristics. Of course, one can use an ordinary probabilistic approach to solve rock slope problems (Marek & Savely 1978, Chowdhury 1986). However, compared with materials such as steel and concrete, the determination of a probability density function for the mechanical constants of rock masses is extremely difficult. In other words, there is no reliable way to determine the input data for the probabilistic approach. This means that the probabilistic approach may be less applicable to practical engineering problems. On the other hand, the Fuzzy Set Theory can easily provide all the input data necessary for the analyses on the basis of engineers' subjective judgements. Therefore, the Fuzzy Set Theory is recommended. Recently the Fuzzy Set Theory has begun to win recognition as a potential tool for solving rock mechanics problems (Fairhurst & Lin 1985, Nguyen & Ashworth 1985a, Nguyen 1985b, c, Shimizu & Sakurai 1986). In this paper, a method based on the Fuzzy Set Theory for determining the strength of rock masses with uncertainties, is described. This method determines the strength parameters by considering their applicability to actual engineering practices. An evaluation method of the stability of rock slopes, using the strength parameters obtained on the basis of the Fuzzy Set Theory, is therefore proposed. According to the Fuzzy Set Theory, the uncertainties in mechanical constants of rock masses are considered as "fuzziness" instead of "randomness" defined in probability theory. The fuzziness is defined as the uncertainties appearing due to either the complexity of the characteristics, or to the lack of knowledge in understanding the characteristics (Yao & Furuta 1986).
The authors have proposed a rock classification method based on the Fuzzy Set Theory (Shimizu & Sakurai 1986). This classification consists of two major steps;
The ambiguous description for the judgement of parameters of rock classifications, for instance, "strength is high", "spacing of discontinuties is very close", are firstly represented by a fuzzy set.
The process of giving an evaluation of the parameters of rock classification is mathematically formulated by fuzzy integral and other fuzzy operations. The proposed rock classification gives the distribution of "fuzzy expected value". This distribution assigns a grade signifying to what degree the rock mass belongs to each rock mass class. The classification parameters and the results of the evaluation are shown in Table 1 and Fig. 1.
For the strength parameters of rock masses, internal friction angle and cohesion, values are given for each class of rock classification, although they have uncertainties (Bieniawski 1979). In this study, we express these ambiguous values by fuzzy numbers. A fuzzy number can model an ill-known quantity whose value is "approximately M" (denoted by M) (Dubois & Prade 1980).
