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2012 | Buch

Nondeterministic Mechanics

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Table of contents: Stochastic methods in nonlinear structural dynamics.- Stochastic models of uncertainties in computational structural dynamics and structural acoustics.- The tale of stochastic linearization techniques: over half a century of progress.- Comprehensive modeling of uncertain systems using fuzzy set theory.- Bounding uncertainty in civil engineering: theoretical background and applications.- Combined methods in nondeterministic mechanics.

In this book the current state of the art of nondeterministic mechanics in its various forms is presented. The topics range from stochastic problems to fuzzy sets; from linear to nonlinear problems; from specific methodologies to combinations of various techniques; from theoretical considerations to practical applications. It is specially designed to illuminate the various aspects of the three methodologies (probabilistic or stochastic modelling, fuzzy sets based analysis, antioptimization of structures) to deal with various uncertainties and deepen the discussion of their pros and cons.

Inhaltsverzeichnis

Frontmatter

Stochastic Models

Frontmatter
Stochastic Methods in Nonlinear Structural Dynamics
Abstract
The uncertainties are inherent in any structural problem. Here attention is focused only on the uncertain nature of the dynamic actions and its consequences on the structural response. In the framework of stochastic dynamics, only three methods are the most used: the Moment Equation Method (MEM), the Stochastic Linearization (SL) and the Monte Carlo Simulation (MCS). The MEM in conjuction with a closure method (CM) allows to obtain the response statistical moments, but it increases in complexity as the problem dimension increases. The SL is easily applied to a large variety of engineering problems. Providing information limited to the first two moments of the system response, unfortunately it suffers of accuracy in the case of strongly nonlinear behavior. MCS is able to give additional information on the structural response, yielding estimates for the probability density function of the nonlinear response, but it is computationally expensive. In this paper some improvements of these three methods are presented, which allow to overcome the aforementioned drawbacks.
Umberto Alibrandi, Giuseppe Ricciardi
Stochastic Models of Uncertainties in Computational Structural Dynamics and Structural Acoustics
Abstract
We present an overview concerning the main concepts, formulations and advances for the stochastic modeling of uncertainties in computational structural dynamics and structural acoustics. The parametric probabilistic approach, the nonparametric probabilistic approach and the generalized probabilistic approach of uncertainties are presented in the context of structural dynamics and in structural acoustics and vibration, including not only the construction of prior probability models but also the identification of posterior probability models.
Christian Soize
The Tale of Stochastic Linearization Technique: Over Half a Century of Progress
Abstract
Developments in stochastic linearization since its inception in 1953 are summarized in this review paper along with the new, statistical orthogonality based derivation of the method. The developments that are described in this review mostly took place after extensive accounts on the classical version of the stochastic linearization technique, such as the monographs by Roberts and Spanos (1990), and by Socha (2008), and the review articles by Socha (2005) and Crandall (2006), have been published the recent decade. This essay is an updated version of our previous reviews (1995, 2000) along with the new derivation of the formulas of the classical stochastic linearization technique.
Isaac Elishakoff, Lova Andriamasy

Non-Stochastic Models

Frontmatter
Comprehensive Modeling of Uncertain Systems Using Fuzzy Set Theory
Abstract
Non-determinism in structural mechanics is ubiquitious. Whenever a mathematical model of a real world engineering system is developed, simplification techniques are employed which lead to systematic errors in the modeling procedure. The fuzzy arithmetical approach which is presented in this chapter is designed to handle epistemic uncertainties, as those systematic errors are also termed, in numerical simulations by the use of fuzzy numbers. The approach consists of the transformation method which is designed to compute fuzzy-valued output quantities and an inverse fuzzy arithmetical method which additionally determines the fuzzy-valued model parameters on the basis of measurement data in a second step. Additionally, a kind of a sensitivity analysis is provided along with a criterion to assess the quality of different competing mathematical models.
Thomas Haag, Michael Hanss
Bounding Uncertainty in Civil Engineering: Theoretical Background and Applications
Abstract
The design of civil engineering constructions frequently involves a great uncertainty about loading conditions, material properties and their degradation in time, human errors in modeling, construction and successive management. These uncertainties seldom can be described by mapping probabilistic input variables through a deterministic model, to obtain the precise expectation of output parameters. In the last years researchers referred to the more general idea of imprecise probabilities to derive lower/upper bounds of output expectations. In this field, the theory of random sets appears as the most appropriate and relatively simple approach for many typical engineering problems, containing probabilistic methods, interval analyses and fuzzy sets as particular cases. The theoretical background and its connection to the more general theory of imprecise probabilities are briefly summarized. Finally the results of real world field applications of the theory to large scale urban building constructions in order to evaluate their seismic vulnerability are presented1.
Alberto Bernardini, Fulvio Tonon

Combined Methods

Frontmatter
Combined Methods in Nondeterministic Mechanics
Abstract
The goal of the lectures on Combined Methods is to discuss various (mathematical and conceptual) approaches that have been put forth as tools for modeling uncertainty in engineering, among them probability, interval arithmetic, random sets, fuzzy sets, sets of probability measures, and previsions. After recalling the definitions, we stress their interpretations (semantics), axioms, interrelations as well as numerical procedures and demonstrate how the concepts are applied in practice. As an accompanying example we use the dimensioning of an elastically bedded beam. Further applications of combined methods in aerospace engineering, to vibrations of belltowers, in queueing theory, and to tuned massed dampers will be sketched.
Michael Oberguggenberger
Metadaten
Titel
Nondeterministic Mechanics
herausgegeben von
Isaac Elishakoff
Christian Soize
Copyright-Jahr
2012
Verlag
Springer Vienna
Electronic ISBN
978-3-7091-1306-6
Print ISBN
978-3-7091-1305-9
DOI
https://doi.org/10.1007/978-3-7091-1306-6