Abstract
A new approach to evaluate the extreme value distribution (EVD) of the response and reliability of general multi-DOF nonlinear stochastic structures is proposed. The approach is based on the recently developed probability density evolution method, which enables the instantaneous probability density functions of the stochastic responses to be captured. In the proposed method, a virtual stochastic process is first constructed to satisfy the condition that the extreme value of the response equals the value of the constructed process at a certain instant of time. The probability density evolution method is then applied to evaluate the instantaneous probability density function of the response, yielding the EVD. The reliability is therefore available through a simple integration over the safe domain. A numerical algorithm is developed using the Number Theoretical Method to select the discretized representative points. Further, a hyper-ball is imposed to sieve the points from the preceding point set in the hypercube. In the numerical examples, the EVD of random variables is evaluated and compared with the analytical solution. A frame structure is analyzed to capture the EVD of the response and the dynamic reliability. The investigations indicate that the proposed approach provides reasonable accuracy and efficiency.
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References
Ang A H-S and Tang WH (1984), Probabilistic Concepts in Engineering Planning and Design, John Wiley & Sons.
Balakrishnan N and Cohen AC (1991), Order Statistics and Inference, Academic Press, New York.
Chen JB and Li J (2004), “Extreme Value Distribution and Dynamic Reliability of Stochastic Structures,” Lecture in the 21st International Congress of Theoretical and Applied Mechanics, August 15–21, 2004, Warsaw, Poland, in: Gutkowski W, Kowalewski TA (Ed.) ICTAM04 Abstracts and CD-ROM Proceedings, ISBN 83-89697-01-1, p.366.
Chen JB and Li J (2005), “Dynamic Response and Reliability Analysis of Nonlinear Stochastic Structures,” Probabilistic Engineering Mechanics, 20 (1): 33–44.
Cheong HF (1995), “Estimation of the Minimum Pressure Coefficient due to Gusts,” Structural Safety, 17: 1–16.
Cramer SH (1966), “On the Intersection Between the Trajectories of a Normal Stationary Stochastic Process and a High Level,” Arkiv for Matematik, 6: 337–349.
Crandall SH (1970), “First-crossing Probability of the Linear Oscillator,” Journal of Sound and Vibration, 12: 285–299.
Ditlevsen O (2004), “Extremes of Random Fields Over Arbitrary Domains with Application to Concrete Rupture Stresses,” Probabilistic Engineering Mechanics, 19: 373–384.
Ditlevsen O and Lindgren G (1988), “Empty Envelope Excursions in Stationary Gaussian Processes,” Journal of Sound and Vibration, 122 (3): 571–587.
Dostupov BG and Pugachev VS (1957), “The Equation for the Integral of a System of Ordinary Differential Equations Containing Random Parameters,” Automatikai Telemekhanika, 18: 620–630.
Fisher RA and Tippett LHC (1928), “Limiting Forms of the Frequency Distribution of the Largest and Smallest Member of a Sample,” Proc. Cambridge Phil. Soc., 24: 180–190.
Gumbel EJ (1958), Statistics of Extremes, Columbia University Press.
Hua LK and Wang Y (1960), “Remarks Concerning Numerical Integration,” Sci. Rec. New Ser., 4 (1): 8–11.
Hua LK and Wang Y (1981), Applications of Number Theory to Numerical Analysis, Springer-Verlag, Berlin.
Li J and Chen JB (2004), “Probability Density Evolution Method for Dynamic Response Analysis of Structures with Uncertain Parameters,” Computational Mechanics, 34: 400–409.
Li J and Chen JB (2005a), “Dynamic Response and Reliability Analysis of Structures with Uncertain Parameters,” International Journal of Numerical Methods in Engineering, 62: 289–315.
Li J and Chen JB (2005b), “The Number Theoretical Method in Response Analysis of Nonlinear Stochastic Structures,” Chinese Journal of Theoretical and Applied Mechanics, 37(4):460–466.
Lin YK and Cai GQ (1995), Probability Structural Dynamics: Advanced Theory and Application, McGraw Hill College Div.
Loeve M (1977), Probability Theory, Springer-Verlag, Berlin.
Michaelov G, Lutes LD and Sarkani S (2001), “Extreme Value of Response to Nonstationary Excitation,” Journal of Engineering Mechanics, 127 (4): 352–363.
Newland DE (1975), An Introduction to Random Vibration and Spectral Analysis, Longmans.
Rice SO (1944), “Mathematical Analysis of Random Noise,” Bell Sys. Tech. J., 23: 282–332.
Rychlik I, Johannesson P and Leadbetter MR (1997), “Modeling and Statistical Analysis of Ocean-wave Data Using Transformed Gaussian Processes,” Marine Structures, 10: 13–47.
Schroeder MR (1984), “Number Theory in Science and Communication: with Applications in Cryptography,” Physics, Biology, Digital Information, and Computing, Springer-Verlag, Berlin.
Senthilnathan A and Lutes LD (1991), “Nonstationary Maximum Response Statistics for Linear Structures,” Journal of Engineering Mechanics, 117 (2): 294–311.
Soong TT (1973), Random Differential Equations in Science and Engineering, Academic Press, New York.
Spencer BF Jr. and Elishakoff I (1988), “Reliability of Uncertain Linear and Nonlinear Systems,” Journal of Engineering Mechanics, 114 (1): 135–149.
Vanmarcke EH (1975), “On the Distribution of the First-passage Time for Normal Stationary Processes,” Journal of Applied Mechanics, 42: 215–220.
Vanmarcke EH (1983), Random Field: Analysis and Synthesis, MIT Press.
Zaremba SK (1972), Applications of Number Theory to Numerical Analysis, Academic Press, New York.
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Supported by: National Natural Science Foundation of China for Innovative Research Groups Under Grant No. 50321803; National Natural Science Foundation of China for Young Scholars Under Grant No. 10402030
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Chen, J., Li, J. Extreme value distribution and reliability of nonlinear stochastic structures. Earthq. Engin. Engin. Vib. 4, 275–286 (2005). https://doi.org/10.1007/s11803-005-0010-2
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DOI: https://doi.org/10.1007/s11803-005-0010-2