By Univ.-Prof. Dr.-Ing. habil. Bernd Möller, Dr.-Ing. Michael Beer (auth.)
This publication, for the 1st time, presents a coherent, total inspiration for taking account of uncertainty within the research, the security evaluation, and the layout of buildings. The reader is brought to the matter of uncertainty modeling and familiarized with specific uncertainty versions. For at the same time contemplating stochastic and non-stochastic uncertainty the superordinated uncertainty version fuzzy randomness, which incorporates genuine valued random variables in addition to fuzzy variables as detailed circumstances, is gifted. For this goal uncomplicated mathematical wisdom in regards to the fuzzy set concept and the speculation of fuzzy random variables is imparted. The physique of the publication contains the correct quantification of doubtful structural parameters, the bushy and fuzzy probabilistic structural research, the bushy probabilistic safeguard evaluate, and the bushy cluster structural layout. The thoroughly new algorithms are defined intimately and illustrated when it comes to demonstrative examples.
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Extra resources for Fuzzy Randomness: Uncertainty in Civil Engineering and Computational Mechanics
Example text
The extension principle specifies the mapping of the fuzzy input set i.. into a new fundamental set Z with the aid of the mapping z =f(x l , ... ,xn) (Fig. 13). 1 Fuzziness - Definitions and Arithmetic 29 its membership function IlB(Z) is gained (Eq. 32)). The membership values IlB(Z) are computed with the aid of the max-min operator (Eq. 33)). On this basis the extension principle can be defined: Given are: - n+ 1 fundamental sets XI' ... , Xi' ... , X n, Z - n fuzzy sets Ai on the fundamental sets Xi with the membership functions ll(xJ = IlAi(XJ; i = 1, ...
Interactive dependencies exist between them. When the mapping model is stepwise applied, these interactive dependencies must be considered. Nonobservance of this interaction leads to the defect that, starting from the intermediate results, nonpennissible parameter combinations are generated and processed forward. Thereby additional, artificial uncertainty is introduced into the mapping in each step. With a rising number of mapping steps the uncertainty of the intermediate results increases exponentially, the solution "diverges" - the results are deficient [19,20].
29). Between the crisp the fuzzy process x(r) is linearly interpolated. Owing to the points in time special fuzzy functional values at the the obtained fuzzy process possesses linear, homogeneous increments. 81) is fulfilled. +Or ('t) I 4 Fig. 29. , which constitutes the fundamental set I, if these are continuous. As in structural-engineering applications generally continuous fuzzy functions appear, point and time discretization are thus necessary in addition to a-discretization. Point and time discretization means that the deterministic argument !