Elsevier

Computers and Geotechnics

Volume 27, Issue 2, September 2000, Pages 101-115
Computers and Geotechnics

Reliability analysis of slopes using fuzzy sets theory

https://doi.org/10.1016/S0266-352X(00)00009-4Get rights and content

Abstract

The stability assessment of slopes is difficult because of many uncertainties. Possibilistic approach using fuzzy sets allows for a logical and systematic analysis of the uncertainties. In this paper an attempt has been made to present a new approach for the stability analysis of slopes incorporating fuzzy uncertainty. Uncertain parameters are expressed as fuzzy sets. A methodology has been presented in the study to process the fuzzy uncertainties in a slope reliability analysis. Fuzzy uncertainty is incorporated in the estimated probability of failure. A numerical example of the finite earth slope problem illustrates the methodology. The approach allows assessment of the likelihood that a particular slope section will have a higher failure probability than the failure probability of the ‘critical’ deterministic failure surface.

Introduction

Instability of both natural and man-made slopes constitutes a major share of civil engineering failures. The need to treat uncertainties in the design or assessment of slope stability today induces a lot of concern in order to ensure the best security levels. The value of almost any quantity of interest measured in civil engineering is to a certain extent uncertain, and hence may be considered as an uncertain variable [1]. The presence of uncertainties and their significance in relation to analysis and design of slopes has long been appreciated. The variability of geologic materials cannot be taken into account by the conventional deterministic methods of slope stability analysis and thus subjective decisions have to be taken for the selection of such properties as shear strength parameters, unit weight and pore pressure. Reliability analysis requires information about these uncertainties. In view of the uncertainties in the problem, satisfactory performance cannot be absolutely ensured. Instead, assurance can only be given in terms of the probability of success in satisfying some performance criterion. In engineering terminology, this probabilistic assurance of performance is referred to as reliability. Historically, probability theory has been the primary tool for representing uncertainty in mathematical models. Because of this, all uncertainty was assumed to follow the characteristics of random uncertainty. However, not all uncertainties are random or objectively quantifiable. Some uncertainties, specially those based on incomplete information are due to cognitive sources [2]. These uncertainties cannot be handled satisfactorily in the probability theory and the theory of fuzzy sets is more appropriate.

The data base in most cases is quite limited, and the distribution type of the uncertain variable is not known. This situation makes the application of classical estimation procedure extremely difficult. Under such conditions of limited information it appears reasonable to base the estimation on concepts of fuzzy set theory. There are many uncertainties with regard to the evaluation of slope stability, and even in homogeneous soil, the uncertainties associated with shear strength parameters are significant. The variability of slope material and other uncertainties associated with slope stability led to the development of possibilistic approaches based on conceptual models that parallel those used widely for deterministic studies. If the form of the uncertainty happens to arise because of imprecision, ambiguity or vagueness, then the variable is probably fuzzy and can be represented by a possibility distribution or membership function. The main objective of the paper is to incorporate the fuzzy uncertainty in reliability analysis of slopes.

A new approach, called fuzzy point estimate method (FPEM), has been presented to estimate the reliability of a finite earth slope. The approach incorporates the degree of membership value associated with each of the uncertain input parameters used in the solution model. The FPEM is used to estimate the expected value and variance of the factor of safety.

Section snippets

Concepts of fuzzy set theory

The nature of uncertainty in a problem is a very important point that engineers should ponder prior to the selection of an appropriate method to express the uncertainty. Fuzzy set theory was developed specially to deal with uncertainties that are nonrandom in nature [2].

Possibilistic reliability analysis

It is now well known that the probabilistic techniques alone for reliability evaluation are not adequate to treat imprecise system data. In such situations, the possibilistic approach based on fuzzy set theory provides a powerful approach for solving this type of problem. Fuzzy variables and distributions of possibility can be compared, respectively, to random variables and probability density functions in the probability theory [4].

The usual model of the safety factor is used to determine the

Background

The probability theory has been the most formidable tool for handling uncertainty. Requirements of treating objective and subjective types of uncertainties in a rigorous way might not be practical for routine geotechnical analysis. The probabilistic characteristics of a random variable would be described completely if the form of the distribution function and the associated parameters are specified. In practice, the form of the distribution function may not be known; consequently, approximate

Bishop's simplified method

A general slope stability problem is statically indeterminate. There are various techniques for solving stability problems depending on which assumption is used to make the problem determinate. The greatest uncertainties in stability problems arise in the selection of the pore pressure and strength parameters. The error associated with the rigorous method of analysis, amounting to about 5% difference in computed factor of safety for the relatively better techniques, is small compared to that

Numerical example

A sample cross-section of the slope consisting of two layers, bottom layer overlying a hard rock is chosen (Fig. 2). The soil properties with their coefficients of variation for both the layers is given in Table 1. The effect of pore water pressure is included in the form of pore pressure ratio (ru). The mean value of pore pressure ratio is 0.30 with a coefficient of variation V(ru)=0.40. The most reasonable values of coefficients of variation are taken from the published literature [10]. This

Discussion

In the present study, a simple methodology is proposed for conducting a reliability analysis of slope with uncertain input parameters. The uncertain parameters considered in the analysis are soil strength parameters c1′, φ1′, c2′ and φ2′, and pore pressure ratio ru. These uncertain parameters are considered uncorrelated as well as correlated. For the analysis, these parameters are represented as fuzzy sets and processed with the FPEM. The trapezoidal fuzzy numbers are used to represent the

Summary and conclusions

The evaluation of reliability and safety of a given or proposed system is one of the purposes of slope reliability analysis and is necessary in the development of proper criteria for design. The problem of defining better and more consistent measures of safety of slopes can be approached by a possibilistic approach based on fuzzy sets theory. Using the Bishop's simplified method as an adequate model, the uncertainties in the input variables have been incorporated into a fuzzy reliability

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