Fuzzy Randomness -Towards a new Modeling of Uncertainty

International Journal for Numerical Methods in Engineering, 2007

The probabilistic and the possibilistic methods of safety evaluation of structure under uncertain parameters have been developed independently. When the structural system is defined with some of the input parameters as possibilistic and others are sufficient enough to model as probabilistic, available literatures normally start with either probabilistic or possibilistic description of all the variables. This may pose restriction on necessary flexibility to the designer at early stage of modelling of the structural system. The primary objective of the present work is to critically examine various emerging methods of transformation of the possibilistic variables to equivalent probabilistic variables so that probabilistic safety evaluation approach becomes compatible with the nature and quality of the input data. Relying on the fundamental concept of equivalent transformations, i.e. the entropy based transformation and the scaling of fuzzy membership function, the reliability analysis is proposed in the framework of second moment format. In doing so, the bounds on the reliability indices based on the evidence theory are also obtained encompassing the first-order reliability analysis for consistent comparison among alternative transformations. Finally, the reliability computation under hybrid uncertainty is elucidated numerically with examples for comparative study on the suitability of the transformation alternatives.

There are several types of uncertainty in a material characterization arisen from different sources of measurement errors, such as methodological, instrumental, and personal. As a reason of the uncertainty in material models, it is plausible to consider model parameters in an interval instead of a singleton. The probability theory is widely known method used for the consideration of uncertainties by means of a certain distribution function and confidence level concept. In this study, fuzzy logic is considered within a material characterization model to deal with the uncertainty coming from random measurement errors. Data points are treated using fuzzy numbers instead of single values to cover random measurement errors. In this context, an illustrative example, prepared with core strength-rebound hammer data obtained from a concrete structure, is solved and evaluated in detail. Results revealed that there is a potential for fuzzy logic to characterize the uncertainty in a material model arisen from measurement errors.

International Journal for Numerical and Analytical Methods in Geomechanics, 1992

This paper addresses the issue of uncertainty treatment in geotechnical engineering. Emphasis is placed on modelling and analysis of non-random uncertainties using fuzzy sets. Although uncertainties were modelled with fuzzy sets in this study, subsequent analysis or processing of the uncertain information was performed using traditional, non-fuzzy techniques. These techniques, including the vertex method and Monte Carlo simulation, are discussed in detail. An example application using soil liquefaction susceptibility is presented. The paper concludes that non-random uncertainties can be successfully modelled and processed using fuzzy sets.

IEEE Transactions on Instrumentation and Measurement, 2004

The good measurement practice requires that the measurement uncertainty is estimated and provided together with the measurement result. The practice today, which is reflected in the reference standard provided by the IEC-ISO "Guide to the expression of uncertainty in measurement," adopts a statistical approach for the expression and estimation of the uncertainty, since the probability theory is the most known and used mathematical tool to deal with distributions of values. However, the probability theory is not the only tool to deal with distributions of values and is not the most suitable one when the values do not distribute in a totally random way. In this case, a more general theory, the theory of the evidence, should be considered. This paper recalls the fundamentals of the theory of the evidence and frames the random-fuzzy variables within this theory, showing how they can usefully be employed to represent the result of a measurement together with its associated uncertainty. The mathematics is defined on the random-fuzzy variables, so that the uncertainty can be processed, and simple examples are given.

Computers & Structures, 2012

Fuzzy structural analysis is oriented to estimate the membership function of an output structural variable given the membership functions of the input ones, which are discretized in so-called a-cut levels. Such an estimation requires the solution of two optimization problems for each of them. In this paper an alternative approach is presented. It consists in solving a single optimization problem for the entire problem, which is that of FORM (first-order reliability analysis). It is shown that FORM solution has a valuable property, namely that it allows organizing the order statistics of the output variable along the design point vector. This property is exploited by a nonlinear projection of a large amount of standard Gaussian numbers, truncated according to the membership functions, onto a bi-dimensional space. From this mapping the samples necessary for directly solving the optimization problems are easily drawn using two curves, whose equations are derived from a similar transformation of the second-order approximation of the limit state function. A detailed structural example shows that the desired membership function of the response can be accurately estimated by the proposed method.

Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing

Uncertainty analysis of any physical model is always an essential task from the point of decision making analysis. Two kinds of uncertainties exist: (1) aleatory uncertainty which is due to randomness of the parameters of models of interest and (2) the epistemic uncertainty which is due to fuzziness of the parameters of the same models. So far both these uncertainties are addressed independently; however since in any practical problem both the types of uncertain variables present, it is required to address them jointly. In order to solve practical problems on uncertainty modeling, it is required to replace the abstract definition of hybrid set by fuzzy random set. Since uncertainty modeling using fuzzy random set has not been carried out so far, the present chapter will address the utility of fuzzy random set for uncertainty modeling on geotechnical and hydrological applications. This chapter will present the fundamentals of fuzzy random set and their application in uncertainty anal...

Computers & Structures, 2004

Vietnam Journal of Mechanics, 2011

This article presents an approach to assess safety levels of structures. A new formula for determining the fuzzy reliability of structures is proposed for the case where the set of loading effect and set of structural durability are general fuzzy sets. Illustration example concerning the bending strength evaluation of a simple-beam structure, is presented with the choice of triangular fuzzy sets for loading effect and structural durability.

Computational Mechanics, 2010

A new two-factor method based on the probability and the fuzzy sets theory is used for the analyses of the dynamic response and reliability of fuzzy-random truss systems under the stationary stochastic excitation. Considering the fuzzy-randomness of the structural physical parameters and geometric dimensions simultaneously, the fuzzy-random correlation function matrix of structural displacement response in time domain and the fuzzy-random mean square values of structural dynamic response in frequency domain are developed by using the two-factor method, and the fuzzy numerical characteristics of dynamic responses are then derived. Based on numerical characteristics of structural fuzzy-random dynamic responses, the structural fuzzy-random dynamic reliability and its fuzzy numerical characteristic are obtained from the Poisson equation. The effects of the uncertainty of the structural parameters on structural dynamic response and reliability are