Vector Process Out‐Crossing as Parallel System Sensitivity Measure
Publication: Journal of Engineering Mechanics
Volume 117, Issue 10
Abstract
The mean rate of vector processes out‐crossing safe domains is calculated using methods from time‐independent reliability theory. The method is founded on a result for scalar up‐crossing derived by Madsen. The out‐crossing is formulated as a zero down‐crossing of a continuously differentiable scalar process, and the mean crossing rate is obtained as a sensitivity measure of the probability for an associated parallel system domain. The vector process may be Gaussian, non‐Gaussian, stationary or nonstationary, and the failure function defining the boundary of the safe domain may be time‐dependent. A method for calculation of the expected number of crossings in a time interval through the introduction of an auxiliary uniformly distributed variable is presented. For stochastic failure surfaces the ensemble averaged rate is determined. A closed‐form expression for the mean crossing rate of a nonstationary Gaussian vector process crossing into a time‐dependent convex polyhedral set is derived. The method is demonstrated to give good results by examples.
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References
1.
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.
2.
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.
3.
Bjerager, P. (1988). “Probability integration by directional simulation.” J. Engrg. Mech., ASCE, 114(8), 1285–1302.
4.
Bjerager, P., and Krenk, S. (1987). “Sensitivity measures in structural reliability analysis.” Proc. of 1st IF1P Working Conference on Reliability and Optimization on Structural Systems, P. Thoft‐Christensen, ed., Springer‐Verlag, Berlin, Germany, 459–470.
5.
Bjerager, P., Løseth, R., Winterstein, S. R., Cornell, C. A. (1988). “Reliability methods for marine structures under multiple environmental load processes.” Proc. 5th Int. Conf. on Behavior of Offshore Structures, Norwegian Inst. of Tech., Trondheim, Norway, 1239–1253.
6.
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).
7.
Breitung, K. (1984). “Asymptotic crossing rates for stationary Gaussian vector processes.” Tech. Report, 1, Dept. of Math. and Statistics, Univ. of Lund, Lund, Sweden.
8.
Breitung, K. (1989). “The extreme value distributions of non‐stationary vector processes.” Structural Safety & Reliability. Proc. of ICOSSAR '89, ASCE, 1327–1332.
9.
Cramer, H., and Leadbetter, M. R. (1967). Stationary and related stochastic processes. John Wiley and Sons, Inc., New York, N. Y.
10.
Der Kiureghian, A., Lin, H.‐Z., and Hwang, S. J. (1987). “Second order reliability approximations.” J. Engrg. Mech., ASCE, 113(8), 1208–1225.
11.
Ditlevsen, O. (1971). “Extremes and first passage times with applications in civil engineering,” thesis presented to the Technical University of Denmark, at Lyngby, Denmark, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
12.
Ditlevsen, O. (1973). “Structural reliability and the invariance problem.” Research Report No. 22, Solid Mech. Div., Univ. of Waterloo, Waterloo, Canada.
13.
Ditlevsen, O. (1983). “Gaussian outcrossings from safe convex polyhedrons.” J. Engrg. Mech., 109(1), 127–148.
14.
Folland, G. B. (1984). Real analysis, John Wiley and Sons, New York, N.Y.
15.
Fujita, M., Schall, G., and Rackwitz, R. (1987). “Time‐variant component reliabilities by FORM/SORM and updating by importance sampling.” Proc. of ICASP5, N.C. Lind, ed., Inst. for Risk Res., Univ. of Waterloo, Ontario, Canada, 520–527.
16.
Grigoriu, M. (1984). “Crossing of non‐Gaussian translation processes.” J. Engrg. Mech. Div., ASCE, 110(4), 610–620.
17.
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, Israel, 153–170.
18.
Hohenbichler, M., Gollwitzer, S., Kruse, W., and Rackwitz, R. (1987). “New lighton first‐ and second‐order reliability methods.” Struct. Saf., 4(4), 267–284.
19.
Hohenbichler, M., and Rackwitz, R. (1981). “Nonnormal dependent vectors in structural safety.” J. Engrg. Mech. Div., ASCE, 107(6), 1227–1247.
20.
Hohenbichler, M., and Rackwitz, R. (1986a). “Asymptotic crossing rate of Gaussian vector processes into intersections of failure domains.” Probabilistic Engineering Mechanics, I(3), 177–179.
21.
Hohenbichler, M., and Rackwitz, R. (1986b). “Sensitivity and importance measures in structural reliability.” Civ. Engrg. Syst., 3(4), 203–209.
22.
Johnson, N. L., and Kotz, S. (1970). Distributions in statistics: Continuous univariate distributions 2, John Wiley and Sons, New York, N.Y.
23.
Lindgren, G. (1984). “Extremal ranks and transformation of variables or extremes of functions of multivariate Gaussian processes.” Stoch. Proc. and Their Applications, 17, 285–312.
24.
Madsen, H. O., Krenk, S., and Lind, N. C. (1986). Methods of structural safety, Prentice‐Hall Inc., Englewood Cliffs, N.J.
25.
Madsen, H. O., and Løseth, R. (1986). “Extreme value distribution and fatigue damage for combined wave and curent loading.” A. S. Veritas Research Report No. 86‐2006, Det norske Veritas, Høvik, Norway.
26.
Madsen, H. O., and Tvedt, L. (1988). “Efficient methods in time‐dependent reliability.” Proc. 5th ASCE Specialty Conference on Probability Methods in Civil Engineering, ASCE, 432–435.
27.
Madsen, H. O., and Tvedt, L. (1990). “Methods for time‐dependent reliability and sensitivity analysis.” J. Engrg. Mech., 116(10), 2118–2135.
28.
Olesen, R. (1990). “PROBAN Version 2 Users Manual.” A. S. Veritas Research Report No. 89‐2024, Det norske Veritas, Høvik, Norway.
29.
Plackett, R. L. (1954). “A reduction formula for normal multivariate integrals.” Biometrika, 41, 351–360.
30.
Plantec, J.‐Y., and Rackwitz, R. (1989). “Structural reliability under nonstationary Gaussian vector process loads.” Proc. Eighth Int. Conf. on Offshore Mechanics and Arctic Engineering, The Hague 1989.
31.
Rice, S. O. (1944). “Mathematical analysis of random noise.” Bell System Tech. J., 23(282), 24(46).
32.
Rosenblatt, M. (1954). “Remarks on a multivariate transformation.” Ann. Math. Stat., 23, 470–472.
33.
Shinozuka, M. (1964). “Probability of failure under random loading.” J. of Engrg. Mech., ASCE, 90(5), 147–171.
34.
Vanmarcke, E. H. (1975). “On the distribution of the first‐passage time for normal stationary random processes.” J. Appl. Mech. Trans. ASME, 42, Mar., 215–220.
35.
Veneziano, D., Grigoriu, M., and Cornell, C. A. (1977). “Vector‐process models for system reliability.” J. Engrg. Mech. Div., ASCE, 103(3), 441–460.
36.
Wen, Y. K., and Chen, H.‐C. (1987). “On fast integration for time variant structural reliability.” Prob. Engrg. Mech., 2(3), 156–162.
37.
Wen, Y. K., and Chen, H.‐C. (1989a). “System reliability under time varying loads: I.” J. Engrg. Mech., ASCE, 115(4), 808–823.
38.
Wen, Y. K., and Chen, H.‐C. (1989b). “System reliability under time varying loads: II.” J. Engrg. Mech., ASCE, 115(4), 823–839.
39.
Winterstein, S. (1985). “Non‐normal responses and fatigue damage.” J. Engrg. Mech., ASCE, 111(10), 1291–1295.
40.
Winterstein, S. (1988). “Nonlinear vibration models for extremes and fatigue.” J. Engrg. Mech., ASCE, 114, 1772–1790.
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Copyright © 1991 ASCE.
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Published online: Oct 1, 1991
Published in print: Oct 1991
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