Joint First-Passage Probability and Reliability of Systems under Stochastic Excitation
Publication: Journal of Engineering Mechanics
Volume 132, Issue 1
Abstract
The first-passage probability, describing the probability that a scalar process exceeds a prescribed threshold during an interval of time, is of great engineering interest. This probability is essential for estimating the reliability of a structural component whose response is a stochastic process. When considering the reliability of an engineering system composed of several interdependent components, the probability that two or more response processes exceed their respective safe thresholds during the operation time of the system is an equally essential quantity. This paper proposes simple and accurate formulas for approximating this joint first-passage probability of a vector process. The order joint first-passage probability is obtained from a recursive formula involving lower order joint first-passage probabilities and the out-crossing probability of the vector process over a safe domain. Interdependence between the crossings is approximately accounted for by considering the clumping of these events. The accuracy of the proposed formulas is examined by comparing analytical estimates with those obtained from Monte Carlo simulations for stationary Gaussian processes. As an example application, the reliability of a system of interconnected equipment items subjected to a stochastic earthquake excitation is estimated by linear programming bounds employing marginal and joint component fragilities obtained by the proposed formulas.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This paper is based on research supported by the Lifelines Program of the Pacific Earthquake Engineering Research Center funded by the Pacific Gas & Electric Co. and the California Energy Commission. Partial support was also provided by the Earthquake Engineering Research Centers Program of the National Science Foundation under Award No. NSFEEC-9701568 and by the Taisei Chair in Civil Engineering. This support is gratefully acknowledged.
References
Belyaev, Y. K. (1968). “On the number of exits across the boundary of a region by a vector stochastic process.” Theor. Probab. Appl., 13, 320–324.
Clough, R., and Penzien, J. (1993). Dynamics of structures, McGraw-Hill, New York.
Cramer, H., and Leadbetter, M. R. (1967). Stationary and related stochastic processes, Wiley, New York.
Davenport, W. B., and Root, W. L. (1958). An introduction to theory of random signals and noise, McGraw-Hill, New York.
Der Kiureghian, A. (1980). “Structural response to stationary excitation.” J. Eng. Mech. Div., Am. Soc. Civ. Eng., 106(6), 1195–1213.
Ditlevsen, O. (1979). “Narrow reliability bounds for structural systems.” J. Struct. Mech., 7(4), 453–472.
Hohenbichler, M., and Rackwitz, R. (1983). “First-order concepts in system reliability.” Struct. Safety, 1(3), 177–188.
Igusa, T., Der Kiureghian, A., and Sackman, J. L. (1984). “Modal decomposition method for stationary response of non-classically damped systems.” Earthquake Eng. Struct. Dyn., 12(1), 121–136.
Liu, P.-L., and Der Kiureghian, A. (1986). “Multivariate distribution models with prescribed marginals and covariances.” Probab. Eng. Mech., 1(2), 105–112.
Lutes, L. D., and Sarkani, S. (2004). Random vibrations analysis of structural and mechanical systems, Elsevier, New York.
Middleton, D. (1960). An introduction to statistical communication theory, McGraw-Hill, New York.
Rice, S. O. (1944). “Mathematical analysis of random noise.” Bell Syst. Tech. J., 23, 282–332.
Rice, S. O. (1945). “Mathematical analysis of random noise.” Bell Syst. Tech. J., 24, 46–156.
Song, J. (2004). “Seismic response and reliability of electrical substation equipment and system.” PhD thesis, Univ. of California, Berkeley.
Song, J., and Der Kiureghian, A. (2003). “Bounds on system reliability by linear programming.” J. Eng. Mech., 129(6), 627–636.
Stone, C. J. (1996). A course in probability and statistics, Duxbury Press, Belmont, Calif.
VanMarcke, E. H. (1975). “On the distribution of the first-passage time for normal stationary random processes.” J. Appl. Mech., 42, 215–220.
Zhang, Y. C. (1993). “High-order reliability bounds for series systems and applications to structural systems.” Comput. Struct., 46(2), 381–386.
Information & Authors
Information
Published In
Copyright
© 2006 ASCE.
History
Received: Aug 18, 2004
Accepted: Apr 12, 2005
Published online: Jan 1, 2006
Published in print: Jan 2006
Notes
Note. Associate Editor: Gerhart I. Schueller
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.