Abstract
This study is devoted to two objectives to illustrate that the probability and convexity concepts are not antagonistic and to introduce a new non-probabilistic convex model for structural reliability analysis. It is shown that the new measure of safety is easier to evaluate than the corresponding measure utilizing the interval analysis. Moreover, interrelation between the classical probabilistic method and convex modeling method is demonstrated. The purpose of this study is not to replace the probabilistic approach by the convex modeling method, but to illustrate that the probability and convexity concepts are compatible. Some numerical examples are presented to illustrate the feasibility of the proposed methodology.
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References
Wang G.Y.: On the development of uncertain structural mechanics (in Chinese). Adv. Appl. Mech. 32(2), 205–211 (2002)
Oberkampf W.L., Helton J.C., Joslyn C.A., Wojtkiewicz S.F., Ferson S.: Challenge problems: uncertainty in system response given uncertain parameters. Reliab. Eng. Syst. Saf. 85, 11–19 (2004)
Elishakoff I.: Three versions of the finite element method based on concepts of either stochasticity, fuzziness or anti-optimization. Appl. Mech. Rev. 51(3), 209–218 (1998)
Ben-Haim Y., Elishakoff I.: Convex Models of Uncertainty in Applied Mechanics. Elsevier, Amsterdam (1990)
Qiu Z.P., Wang X.J.: Set-Theoretical Convex Methods for Problems in Structural Mechanics with Uncertainties (in Chinese). Science Press, Beijing (2008)
Elishakoff I., Ohsaki M.: Optimization and Anti-Optimization of Structures Under Uncertainty. Imperial College Press, London (2010)
Ben-Haim Y.: Robust reliability of structures. Adv. Appl. Mech. 33, 1–41 (1997)
Guo S.X., Lu Z.Z., Feng S.Y.: A non-probabilistic model of structural reliability based on interval analysis (in Chinese). Chin. J. Comput. Mech. 18(1), 56–60 (2001)
Elishakoff I.: Discussion on the paper: a non-probabilistic concept of reliability. Struct. Saf. 17(3), 195–199 (1995)
Guo X.S., Lu Z.Z.: Procedure for analyzing the fuzzy reliability of mechanical structures when parameters of probabilistic models are fuzzy (in Chinese). J. Mech. Strength 25(5), 527–529 (2003)
Qiu P.Z., Mueller C.P., Frommer A.: The new non-probabilistic criterion of failure for dynamical systems based on convex models. Math. Comput. Model. 40(1–2), 201–215 (2004)
Qiu P.Z., Chen Q.S., Wang J.X.: Criterion of the non-probabilistic robust reliability for structures (in Chinese). Chin. J. Comput. Mech. 21(1), 1–6 (2004)
Wang X.J.: Robust reliability of structural vibration. J. Beijing Univ. Aeronaut. Astronaut. 29(11), 1006–1010 (2003) (in Chinese)
Wang X.J., Qiu P.Z., Elishakoff I.: Non-probabilistic set-model for structural safety measure. Acta Mech. 198, 51–64 (2008)
Elishakoff, I.: Are probabilistic and antioptimization methods interrelated? In: Elishakoff, I. (ed.) Whys and Hows in Uncertainty Modeling: Probability Fuzziness and Anti-Optimization, Springer, Wien (1999)
Ziegler F.: Private communication (2010)
Entropy (information theory) From Wikipedia, the free encyclopedia. http://en.wikipedia.org/wiki/Entropy_%28information_theory%29
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Wang, X., Wang, L., Elishakoff, I. et al. Probability and convexity concepts are not antagonistic. Acta Mech 219, 45–64 (2011). https://doi.org/10.1007/s00707-010-0440-4
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DOI: https://doi.org/10.1007/s00707-010-0440-4