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Über dieses Buch

sections dealing with fuzzy functions and fuzzy random functions are certain to be of special interest. The reader is expected to be in command of the knowledge gained in a basic university mathematics course, with the inclusion of stochastic elements. A specification of uncertainty in any particular case is often difficult. For this reason Chaps. 3 and 4 are devoted solely to this problem. The derivation of fuzzy variables for representing informal and lexical uncertainty reflects the subjective assessment of objective conditions in the form of a membership function. Techniques for modeling fuzzy random variables are presented for data that simultaneously exhibit stochastic and nonstochastic properties. The application of fuzzy randomness is demonstrated in three fields of civil engineering and computational mechanics: structural analysis, safety assessment, and design. The methods of fuzzy structural analysis and fuzzy probabilistic structural analysis developed in Chap. 5 are applicable without restriction to arbitrary geometrically and physically nonlinear problems. The most important forms of the latter are the Fuzzy Finite Element Method (FFEM) and the Fuzzy Stochastic Finite Element Method (FSFEM).

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
For quantifying physical parameters such as geometry, material, or loading parameters, real numbers or integers are mainly applied, i.e., a deterministic data model is applied. In order to describe randomness resulting from imprecise readings or fluctuating ambient conditions, measurements are repeated and lumped together in a concrete data sample. Mathematical statistics offers methods for describing data samples with the aid of random variables. A common approach for this purpose is to specify a probability distribution function in order to obtain a stochastic data model.
Bernd Möller, Michael Beer

2. Mathematical Basics for the Formal Description of Uncertainty

Abstract
In this chapter some selected theoretical basics and enhancements concerning the definition and the treatment of uncertainty are stated and explained. These explanations do not represent an overall and detailed mathematical overview but only summarize fundamentals that may be helpful for understanding the developed methods and engineering applications considered subsequently.
Bernd Möller, Michael Beer

3. Description of Uncertain Structural Parameters as Fuzzy Variables

Abstract
The specification of the membership function μA(x) of a fuzzy set à is referred to as fuzzification.
Bernd Möller, Michael Beer

4. Description of Uncertain Structural Parameters as Fuzzy Random Variables

Abstract
Structural parameters that possess the uncertainty characteristic fuzzy randomness are modeled as fuzzy random variables [119, 124]. For each fuzzy random variable it is necessary to generate the fuzzy probability distribution function, the fuzzy probability density function, and the associated fuzzy parameters on the basis of limited statistical data material. The data material possesses stochastic and nonstochastic properties, the latter resulting in particular from informal uncertainty or uncertainty due to fluctuating reproduction conditions.
Bernd Möller, Michael Beer

5. Fuzzy and Fuzzy Stochastic Structural Analysis

Abstract
In deterministic structural analysis crisp input vectors xX representing load, geometry, and material parameters are known.
Bernd Möller, Michael Beer

6. Fuzzy Probabilistic Safety Assessment

Abstract
In fuzzy probabilistic safety assessment of structures fuzzy random vectors are introduced for the mathematical description of the uncertainty characteristic fuzzy randomness. The failure and survival of a structure may be assessed with the aid of the probability measure for fuzzy random vectors, which has been introduced in Sect. 2.3.1.2; the fuzzy failure probability and fuzzy survival probability are to be determined.
Bernd Möller, Michael Beer

7. Structural Design Based on Clustering

Abstract
In fuzzy structural analysis according to Eq. (5.2) fuzzy input vectors x are mapped onto fuzzy result vectors z, i.e.
Bernd Möller, Michael Beer

Backmatter

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