Outcrossing of Stationary Gaussian Process from 2D Elliptical Region
Publication: Journal of Engineering Mechanics
Volume 119, Issue 5
Abstract
An exact expression for the mean outcrossing rate of a Gaussian vector process from two‐dimensional elliptical regions is derived. The expression is exact under the assumption that the cross correlation between process component 1 and velocity component 2 is zero, and it generally yields a good approximation for nonzero cross correlations. A correction for a nonzero cross correlation is shown to be vanishing to first order in the cross‐correlation coefficient. Closed‐form expressions are derived for the ensemble‐averaged rate for various types of uncertain failure surfaces, and a upper bound for the expected number of outcrossings of an exponentially deteriorating elliptical surface is determined. The analytical formulas are demonstrated to give good results by examples.
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References
1.
Abramowitz, M., and Stegun, I. A. (1965). Handbook of mathematical functions. Dover Publications, Inc., New York, N.Y.
2.
Belyaev, Y. K. (1968). “On the number of exits across the boundary of a region by a vector stochastic process.” Theory of Probability Applications, 13, 320–324.
3.
Belyaev, Y. K., and Nosko, V. P. (1969). “Characteristics of excursions above a high level for a Gaussian process and its envelope.” Theory of Probability Applications, 14, 296–309.
4.
Bolotin, V. V. (1971). Primenenie Metodov Teorii Verovatnostei i Teorii Nadejnosti v Raschetah Soorujenni. Izdatelstvo Literaturi po Stroitelstvi, Moscow, U.S.S.R. (in Russian).
5.
Breitung, K. (1984). “Asymptotic crossing rates for stationary Gaussian vector processes.” Tech. Rep. 1, Dept. of Math. Statistics, University of Lund, Sweden.
6.
Cramer, H., and Leadbetter, M. R. (1967). Stationary and related stochastic processes. John Wiley & Sons, Inc., New York, N.Y.
7.
Ditlevsen, O. (1973). “Structural reliability and the invariance problem.” Res. Rep. No. 22, Solid Mechanics Div., University of Waterloo, Waterloo, Canada.
8.
Ditlevsen, O. (1983). “Gaussian outcrossings from safe convex polyhedrons.” J. Engrg. Mech., ASCE, 109(1), 127–148.
9.
Gradshteyn, I. S., and Ryzhik, I. M. (1980). Tables of integrals, series, and products. Academic Press, New York, N.Y.
10.
Grigoriu, M. (1984). “Crossing of non‐Gaussian translation processes.” J. Engrg. Mech., ASCE, 110(4), 610–620.
11.
Hagen, Ø. (1992a). “Structural reliability methods under time dependency,” PhD thesis, University of Oslo, Norway.
12.
Hagen, Ø. (1992b). “Conditional and joint crossing of stochastic processes.” J. Engrg. Mech., ASCE, 118(9), 1814–1839.
13.
Hagen, Ø., and Tvedt, L.(1991). “Vector process outcrossing as a parallel system sensitivity measure.” J. Engrg. Mech., ASCE, 117(10), 2201–2220.
14.
Hasofer, A. M. (1974). “The upcrossing rate of a class of stochastic processes.” Studies in probability and statistics, E. J. Williams, ed., Jerusalem Academic Press, Jerusalem, 153–170.
15.
Lindgren, G. (1984). “Extremal ranks and transformation of variables or extremes of functions of multivariate Gaussian processes.” Stochastic Processes and Their Applications, 17, 285–312.
16.
Madsen, H. O. (1985). “Extreme‐value statistics for nonlinear stress combination.” J. Engrg. Mech., ASCE, 111(9), 1121–1129.
17.
Naeæss, A. (1984). “Technical note: On the long‐term statistics of extremes.” Appl. Ocean Res., 6(4), 227–228.
18.
Rice, S. O. (1944). “Mathematical analysis of random noise.” Bell System Tech. J., 23, 282–332.
19.
Veneziano, G., Grigoriu, M., and Cornell, C. A. (1977). “Vector‐process models for system reliability.” J. Engrg. Mech. Div., ASCE, 103(3), 441–460.
20.
Wen, Y. K., (1987). “Approximate methods for nonlinear time‐variant reliability analysis.” J. Engrg. Mech., ASCE, 113(12), 1826–1839.
21.
Wen, Y. K., and Chen, H.‐C. (1987). “On fast integration for time variant structural reliability.” Prob. Engrg. Mech., 2(3), 156–162.
Information & Authors
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Published In
Journal of Engineering Mechanics
Volume 119 • Issue 5 • May 1993
Pages: 973 - 996
Copyright
Copyright © 1993 American Society of Civil Engineers.
History
Received: Jun 1, 1992
Published online: May 1, 1993
Published in print: May 1993
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