Multivariate Extreme Value Distributions for Random Vibration Applications
Publication: Journal of Engineering Mechanics
Volume 131, Issue 7
Abstract
The problem of determining the joint probability distribution of extreme values associated with a vector of stationary Gaussian random processes is considered. A solution to this problem is developed by approximating the multivariate counting processes associated with the number of level crossings as a multivariate Poisson random process. This, in turn, leads to approximations to the multivariate probability distributions for the first passage times and extreme values over a given duration. It is shown that the multivariate extreme value distribution has Gumbel marginal and the first passage time has exponential marginal. The acceptability of the solutions developed is examined by performing simulation studies on bivariate Gaussian random processes. Illustrative examples include a discussion on the response analysis of a two span bridge subjected to spatially varying random earthquake support motions.
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Acknowledgment
The work reported in this paper forms a part of the research project entitled “Seismic probabilistic safety assessment of nuclear power plant structures,” funded by the Board of Research in Nuclear Sciences, Department of Atomic Energy, Government of India.
References
Abrahamson, N. A., Schneider, F. S., and Stepp, C. (1991). “Spatial coherency of shear waves from Lotung, Taiwan large-scale seismic test.” Struct. Safety, 10, 145–162.
Castillo, E. (1988). Extreme value theory in engineering, Academic, Boston.
Cramer, H. (1966). “On the intersections between the trajectories of a normal stationary stochastic process and a high level.” Ark. Mat., 6, 337–349.
Cramer, H., and Leadbetter, M. R. (1967). Stationary and related stochastic processes: Sample function properties and their applications, Wiley, New York.
Der Kiureghian, A. (1996). “A coherency model for spatially varying ground motions.” Earthquake Eng. Struct. Dyn., 25(1), 99–111.
Ditlevsen, O. (1979). “Narrow reliability bounds for structural systems.” J. Struct. Mech., 7(4), 453–472.
Ditlevsen, O. (1984). “First outcrossing probability bounds.” J. Eng. Mech., 110(2), 282–292.
Galambos, J. (1978). The asymptotic theory of extreme order statistics, Wiley, New York.
Hagen, O. (1992). “Conditional and joint failure surface crossing of stochastic processes.” J. Eng. Mech., 118(9), 1814–1839.
Hagen, O., and Tvedt, L. (1991). “Vector process out-crossing as parallel system sensitivity measure.” J. Eng. Mech., 117(10), 2201–2220.
Johnson, N. L., and Kotz, S. (1969). Discrete distributions, Wiley, New York.
Kotz, S., and Nadarajah, S. (2000). Extreme value distributions, Imperial College Press, London.
Leira, B. J. (1994). “Multivariate distributions of maxima and extremes for Gaussian vector processes.” Struct. Safety, 14, 247–265.
Leira, B. J. (2003). “Extremes of Gaussian and non-Gaussian vector processes: A geometric approach.” Struct. Safety, 25, 401–422.
Lin, Y. K. (1967). Probabilistic theory of structural dynamics, McGraw-Hill, New York.
Lutes, L. D., and Sarkani, S. (1997). Stochastic analysis of structural and mechanical vibrations, Prentice-Hall, Englewood Cliffs, N.J.
Madsen, H. O., Krenk, S., and Lind, N. C. (1986). Methods of structural safety, Prentice-Hall, Englewood Cliffs, N.J.
Melchers, R. E. (1999). Structural reliability analysis and prediction, Wiley, Chichester.
Naess, A. (1989). “A study of linear combination of load effects.” J. Sound Vib., 129(2), 83–98.
Nigam, N. C. (1983). Introduction to random vibration, MIT Press, Cambridge, Mass.
Roberts, J. B. (1986). “First passage probabilities for randomly excited systems: diffusion methods.” Probab. Eng. Mech., 1, 66–81.
Spencer, B. F., and Bergman, L. A. (1985). “The first passage problem in random vibration for a simple hysteretic oscillator.” Technical Rep. AAE 85-8, University of Illinois, Urbana, Ill.
Srinivasan, S. K., and Mehata, K. M. (1976). Stochastic processes, Tata-McGraw-Hill, New York.
Vanmarcke, E. H. (1972). “Properties of spectral moments with applications to random vibrations.” J. Eng. Mech. Div., 98(2), 425–46.
Vanmarcke, E. H. (1975). “On the distribution of the first passage time for normal stationary random processes.” J. Appl. Mech., 42, 215–220.
Veneziano, D., Grigoriu, M., and Cornell, C. A. (1977). “Vector-process models for system reliability.” J. Eng. Mech. Div., 103(3), 441–460.
Wen, Y. K., and Chen, H.-C. (1989). “System reliability under time varying loads: I.” J. Eng. Mech., 115(4), 808–823.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 131 • Issue 7 • July 2005
Pages: 712 - 720
Copyright
© 2005 ASCE.
History
Received: Mar 2, 2004
Accepted: Oct 20, 2004
Published online: Jul 1, 2005
Published in print: Jul 2005
Notes
Note. Associate Editor: Gerhart I. Schueller
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