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Using fuzzy numbers in fatigue reliability estimation

Bednarek T., Prokopowicz P., Sosnowski W.

Computer Methods in Materials Science

2009

Vol. 9, No. 1

43-48

Introduction Increasing safety requirements of structures induces necessity of taking into consideration the random nature of variables in the fatigue design. Fatigue analysis of real structures in connection with probabilistic analysis needs a big computational effort. In this paper a new approach and methodology to fatigue analysis of structures using fuzzy numbers are presented. The method has been developed. A new efficient tool of fatigue analysis was created. The calculation of fatigue life using fuzzy numbers set is illustrated with analysis of hydraulic cylinders. Local fatigue approach is used at critical locations on hydraulic cylinders. The purpose of the probabilistic design is to estimate the probability of failure for critical locations on hydraulic cylinder. Fuzzy numbers idea In common real-life problems often both, parameters and data used in mathematical modeling are vague. In mathematical modeling vagueness can be described by a fuzzy sets and numbers (main idea was started in 1965 by L. Zadeh). Fuzzy set theory has its well-known achievements in many branches of science and technology. Fuzzy concept was introduced in order to model such vague terms, as observed values of some physical or economical terms, pressure values or stock market rates, that can be inaccurate, can be noisy or can be difficult to measure with an appropriate precision because of technical reasons. Fuzzy numbers are a special kind of fuzzy sets, with restrictions, which makes them proper in analysis of imprecise data. As the fatigue is a parameter, which is really hard to determine precisely, so using here the idea of fuzzy numbers looks like the direction worth to investigate. However, in classical fuzzy numbers the problem is which usefulness of results after many calculations. Thus a new model of fuzzy numbers ? ordered fuzzy numbers ? were used in this paper in fatigue analysis of structures. New model has a good and flexible concept of calculations, which makes them easier, and have no basics problems from the classical fuzzy numbers. Thanks to the flexibility of the new concept the four algebraic operations: addition, subtraction, multiplication and division of such fuzzy numbers have been constructed in a way that renders their algebra. Operations rising to a power and finding the logarithm are also defined using numerical methods. Fatigue analysis of structures using fuzzy numbers idea The fatigue life calculations were performed using standard FEM program. The idea of the equivalent amplitude stress described in [fatigue] is used, in order to calculate the number of cycles to failure of the analyzed structure loaded by arbitrary non-symmetric load with any stress ratio. Previously presented equivalent amplitude stress idea and analytical form of S-N fatigue curve were modified. Fuzzy numbers idea was applied to define material parameters and value of external load. In consequence a fatigue life of structure is also fuzzy number. Analysis of fuzzy fatigue life could be estimation of reliability.

Ciągle zwiększające się wymogi bezpieczeństwa wywołują konieczność wzięcia pod uwagę stochastyczną charakterystykę zmiennych w analizie zmęczeniowej konstrukcji. Analizę zmęczeniową rzeczywistych konstrukcji w połączeniu z analizą niezawodności w klasycznym probabilistycznym ujęciu cechuje niezwykle duży koszt obliczeniowy. W artykule przedstawiono nową metodę i oceny niezawodności konstrukcji za pomocą zbioru ukierunkowanych liczb rozmytych. Obliczenia niezawodności konstrukcji i narażonych na zmęczenie materiału przedstawiono na przykładzie analizy cylindra hydraulicznego. Zidentyfikowano miejsca narażone na pękanie zmęczeniowe. Obliczono trwałość zmęczeniową cylindra hydraulicznego za pomocą zbioru liczb rozmytych. Celem analizy zmęczeniowej z wykorzystaniem liczb rozmytych jest oszacowanie prawdopodobieństwa zniszczenia zmęczeniowego dla danej trwałości projektowej lub wyznaczenie okresów serwisowych przy zadanym poziomie niezawodności konstrukcji.

Probabilistic assessment of the fatigue reliability

Nechval K. N., Nechval N. A., Berzins G., Purgailis M.

Eksploatacja i Niezawodność

2007

nr 3

3-6

The prediction of stochastic crack growth accumulation is important for the reliability analysis of structures as well as the scheduling of inspection and repair/replacement maintenance. Because the initial crack size, the stress, the material properties and other factors that may affect the fatigue crack growth are statistically distributed, the fi rst-order second-moment technique is often adopted to calculate the fatigue reliability of industrial structures. In this paper, a second-order thirdmoment technique is presented and a three-parameter Weibull distribution is adopted to refl ect the infl uences of skewness of the probability density function. The second-order third-moment technique that has more characteristics of those random variables that are concerned in reliability analysis is obviously more accurate than the traditional fi rst-order second-moment technique.