Fuzzy Randomness. Bernd Moller Michael Beer

Transcription

1 Fuzzy Randomness Bernd Moller Michael Beer

2 Springer-Verlag Berlin Heidelberg GmbH Engineering ONLINE LIBRARY springeronline.com

3 Bernd Moller Michael Beer Fuzzy Randomne Uncertainty in Civil Engineering and Computational Mechanics With 234 Figures Springer

4 Univ.-Prof. Dr. -Ing. habil. Bernd Moller Dr. -Ing. Michael Beer Institute of Structural Analysis (Lehrstuhl ffir Statik) Dresden University of Technology MommsenstraBe Dresden Germany ISBN ISBN (ebook) DOI / Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at < This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under German Copyright Law. springeronline.com Springer-Verlag Berlin Heidelberg 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Softcover reprint of the hardcover 1st edition 2004 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by author Cover-design: Struve & Partner, Heidelberg Printed on acid-free paper 02/3020

5 Everything is vague to a degree you do not realize till you have tried to make it precise. Bertrand Russell Nature has never attended a course in probability theory. This poignant philosophical statement underlines the fact that the phenomenon of uncertainty cannot be described by means of probability theory alone. Data and models encountered in the natural sciences and engineering are more or less characterized by uncertainty. With the inclusion of interval variables and random variables classical mathematical models are available for the treatment of uncertainty. In recent years work on the formulation of new mathematical models for describing uncertainty has intensified. This work is based on chaos theory, convex modeling, fuzzy set theory, and fuzzy randomness. The aim of these uncertainty models is to handle data exhibiting, in particular, nonstochastic properties and informal uncertainty. Fuzzy randomness is a generalized uncertainty model that permits the simultaneous consideration of stochastic, lexical, and informal uncertainty. Initial drafts for the uncertainty model "fuzzy randomness" were developed about 25 years ago. This book considers these developments and enhances fuzzy randomness in such a way that it may be applied in the fields of civil engineering and computational mechanics. These further scientific developments are based on the results of research work carried out under the supervision of the leading author at the Institute of Structural Analysis of Dresden University of Technology with the much-appreciated financial support of the "Deutsche Forschungsgemeinschaft (DFG)" (German Research Foundation). Uncertainty models are the topic of controversial discussion in scientific circles. For this reason the phenomenon of uncertainty is introduced in the first chapter. By way of examples an attempt is made to promote an understanding of uncertainty and to guide readers from a deterministic way of thinking towards the acceptance of uncertainty. The mathematical fundamentals of fuzziness and fuzzy randomness are then introduced in Chap. 2. Particularly with regard to applications, the

6 VI Preface 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). Fuzzy randomness may also be combined with a variety of methods based on reliability theory. This is demonstrated in Chap. 6 by considering the further development of the First Order Reliability Method (FORM) to obtain the Fuzzy First Order Reliability Method (FFORM). As a result of the safety assessment the existing uncertainty is evident in the fuzzy failure probability, which now becomes an uncertain parameter. A general design concept for uncertain structural parameters is finally developed in Chap. 7. Writing a book always requires the assistance of colleagues. The authors wish to thank Professor Wolfgang Graf for many fruitful discussions, and the research coworkers at the Institute of Structural Analysis, Dr-Ing. Andreas Hoffmann, Dr Ing. Nguyen Song Ha, Dipl.-Ing. Martin Liebscher, and Dipl.-Ing. Jan-Uwe Sickert for their valuable support in the preparation of the manuscript. We are also grateful to Dr Ian Westwood (PhD, Civil Engineering) for translating Chaps. 1, 3, and 4, and for his assistance in the formulation of additional parts of the manuscript in English. Finally, we wish to thank the publishers "Springer-Verlag" for their muchappreciated editorial work and printing of the book. Dresden, September 2003 Bernd Moller Michael Beer

7 Abbreviations... XK 1 Kntroduction The Phenomenon of Uncertainty Definition and Classification of Uncertainty Examples of Data and Model Uncertainty On the State of Development of Uncertainty Models MathematicallBasics for the Formal IDiescription of Uncertainty Fuzziness - Definitions and Arithmetic Crisp Set and Fuzzy Set Fuzzy Number a-level Set Linguistic Variables Set Theoretical Operations Cartesian Product Extension Principle Interaction between Fuzzy Variables a-discretization of Fuzzy Sets Defuzzification of Fuzzy Variables Fuzzy Functions Definition of Fuzzy Functions, Fuzzy Processes, and Fuzzy Fields Point and Time Discretization of Fuzzy Functions Approximative Description of Fuzzy Functions Elements of Measure Theory Measure and Uncertain Measure of Crisp Sets Probability Other Uncertain Measures Measures of Uncertainty Fuzzy Randomness Fuzzy Random Vectors Definition of Fuzzy Random Vectors Probability Measure for Fuzzy Random Vectors.. 68

8 VIII Contents Fuzzy Probability Distributions Parameters of Fuzzy Random Vectors Fuzzy Random Functions Definition of Fuzzy Random Functions Parameters and Properties of Fuzzy Random Functions Fuzzy Random Field and Fuzzy Random Process as Special Cases of Fuzzy Random Functions ]])escri.ptiolill of UlIllcertailIll StmctlllraR Parameters as lfuzzy Variables Data Uncertainty - Specification of Membership Functions Model Uncertainty - Construction of Fuzzy Models g ]])escriptiolill of UlIllcertai.1Ill StmcturaR Parameters as lfuzzy lramllom Variables ll Techniques for Modeling Fuzzy Random Variables Fuzzy Probability Distribution Function for Known Fuzzy Parameters Fuzzy Probability Distribution Function with a Fuzzy Distribution Type Fuzzy Random Variables and Typical Data Situations Small Sample Size Samples with Unknown, Nonconstant Reproduction Conditions Samples with Known, Nonconstant Reproduction Conditions lfuzzy amllfuzzy Stoch.astic Structural Analysis ll Uncertain Structural Analysis and Deterministic Fundamental Solution Fuzzy Structural Analysis Fuzzy Structural Analysis with the Aid of the Extension Principle Fuzzy Structural Analysis with the Aid of a-level Optimization Developing a-level Optimization from the Extension Principle Properties of the Mapping Model Solution Techniques for a-level Optimization Specification of the Optimization Problem Monte Carlo Simulation and Mesh Search Techniques Gradient Method and Evolution Strategy Modified Evolution Strategy

9 Contents IX Fuzzy Finite Element Method (FFEM) Application of Fuzzy Structural Analysis Multistorey Frame, Linear Dynamic Analysis - Initial Value Problem Steel Frame, Geometrically Nonlinear Statical Analysis Reinforced-Concrete Frame, Nonlinear Statical Analysis Prestressed Reinforced-Concrete Frame, Nonlinear Dynamic Analysis Reinforced-Concrete Folded-Plate Structure, Fuzzy Finite Element Method, Physically Nonlinear Analysis Fuzzy Stochastic Structural Analysis Fuzzy Stochastic Finite Element Method Application of the Fuzzy Stochastic Finite Element Method Numerical Simulation of the Structural Behavior of a Textile-Reinforced Specimen Reinforced-Concrete Plate, Physically Nonlinear Analysis Reinforced-Concrete Folded-Plate Structure, Physically Nonlinear Analysis ]Fuzzy Probabilistic Safety Assessment Conceptual Idea of the Fuzzy First Order Reliability Method (FFORM) Original Space of the Fuzzy Probabilistic Basic Variables Fuzzy Probabilistic Basic Variables and Joint Fuzzy Probability Density Function Fuzzy Limit State Surface Fuzzy Design Point Transformation of Fuzzy Random Variables Standard Normal Space of the Normalized Basic Variables Standard Normal Joint Probability Density Function and Fuzzy Limit State Surface Fuzzy Design Point and Fuzzy Reliability Index Safety Verification Numerical Realization Application offform Steel Girder, Physically Nonlinear Statical System Behavior Reinforced-Concrete Frame, Nonlinear Statical System Behavior

10 X Contents 7 Structural Design Based on Clustering Conceptual Idea On the Application of Cluster Analysis Methods Introductory Remarks k-medoid Cluster Method Fuzzy Cluster Method Composition of Fuzzy Cluster Design Mapping Spaces Algorithmic Procedure Assessment of the Results from Fuzzy Cluster Design Examples Steel Girder, Structural Design for Time-dependent Load Reinforced-Concrete Frame, Structural Design with FFORM References 308 Index

11 arithmetics a,..., Z.!!,..., ~ A,,Z L IT In ld variables, a,..., Z E R vectors matrices sum product natural logarithm logarithm to base 2 analysis (...,... ) (...,... ) [...,... J ordered n-tuple, elements of a column matrix open interval closed interval I I, II x. II absolute value, norm Cartesian product sup min sup min max lim ()() A d a f supremum (least upper bound, maximum) minimum max-min operator maximum limes, limit infinity difference differentiation partial differentiation integration mapping

12 XU Abbreviations set theory A;,'l!. d,,~ dc,..., ~c fundamental sets crisp sets complement of,1.,..., ~ with regard to the assigned fundamental seta;... ; 'l!. natural numbers real numbers n-dimensional Euclidean space {...,... ) E (f o n u \ \P(... ) m(... ) (... ) Mm(... ) set of..., elements of a set element of not element of empty set intersection union set difference,... without... subset of power set family of sets a-algebra set function measure theory M S m(... ) M U m(... ) V(... ) [...,... J E H Hu (crisp) measure uncertain measure distribution function measure space event Shannon's entropy modified Shannon's entropy, Shannon's uncertainty measure logic /\ V conjunction, logical and alternative, logical or for which the following holds implication; if..., then equivalence; if and only if...

13 Abbreviations XIII universal quantifier, for all existential quantifier, there exists fuzzy set theory fuzziness ~,..., ~ <a, b, c> <a,b,c,d> A,...,Z Ac,...,tC E 0; d,...,z a j (lj x(t), x (1) x(.ft) x('r)!! 1: x(1) = X(~,l) x(!) E x(1) Xa(!) fuzzy vectors fuzzy triangular number with the interval bounds of the support a and c and the mean value b fuzzy trapezoidal interval with the interval bounds of the support a and d and the interval bounds of the mean interval band c membership function of the fuzzy vector K fuzzy sets complement of A;... ; Z fuzzy event a-level for 0; = ~ a-level sets for a = 0; fuzzy function fuzzy field fuzzy process spatial coordinates time coordinate bunch parameter representation of a fuzzy function fuzzy bunch parameter vector trajectory of x(1) a-function set fuzzy structural analysis fuzzy input variables including fuzzy model parameters fuzzy result variables ~, x-space ~, z-space x z x. 0.. Z' o.i space of the fuzzy input variables and fuzzy model parameters space of the fuzzy result variables fuzzy input set fuzzy result set crisp input subset for a = a j crisp result subset for a = a j

14 XN Abbreviations fuzzy probabilistics A,,~ A,...,i Kj EX i, fixi 2-2 D Xi' axi P(E) :Pel) f(x) X(i),X(1) Xef!') XCr) ft X(!) = XU:, 1).s. real-valued random vectors fuzzy random vectors original of the fuzzy random vector X fuzzy expected value of the i-th component of X fuzzy variance of the i-th component of X fuzzy standard deviation of the i-th component of X fuzzy probability of the event E fuzzy probability distribution function of X fuzzy probability density function of X fuzzy random function fuzzy random field fuzzy random process spatial coordinates time coordinate bunch parameter representation of a fuzzy random function fuzzy bunch parameter vector original function random a-function set fuzzy mean of the i-th component of a sample belonging to X fuzzy variance of the i-th component of a sample belonging to X fuzzy standard deviation of the i-th component of a sample belonging to X fuzzy probabilistic safety assessment ~, x-space Y, y-space g(it) h(y) l(y) original space of the basic variables standard normal space limit state surface in! limit state surface in y linearized limit state surface in y permissible required

15 Abbreviations XV probability distribution function of the standard normal distribution probability density function of the standard normal distribution fuzzy design point in x-space fuzzy design point in y-space fuzzy reliability index fuzzy failure probability structural design based! on clustering d l,..., d n d l,..., d n D n f1'.. " r m 1'['..., I'm R m permissible design parameters nonpermissible design parameters space of the design parameters restricted parameters nonrestricted parameters space of the restricted parameters design constraint cluster - [vi - [vi XI,.." Xn - [vi - [vi Zl,.. " Zm modified fuzzy input variables, alternative design variants modified fuzzy result variables