Closed-Form Solution to First Passage Probability for Nonstationary Lognormal Processes
Publication: Journal of Engineering Mechanics
Volume 142, Issue 12
Abstract
In time-dependent reliability analysis, the calculation of the mean outcrossing or upcrossing rate of a stochastic process from a safe domain or barrier level based on the Rice formula continues to present serious challenge to researchers in the field. Furthermore, the derivation of closed-form analytical solutions to the first passage probability for nonstationary processes has not been very successful except for Gaussian process. The intention of this paper is to drive a closed-form solution for the calculation of mean upcrossing rate of a scalar nonstationary lognormal process from a barrier level. The applicability of this new solution is illustrated in a time-dependent reliability analysis of corrosion-induced concrete cracking. It is found that the results of first passage probability calculated from the derived closed-form solution are in very good agreement with those from Monte Carlo simulation and safety index methods. A merit of the derived solution is that it eliminates unrealistic negative values for inherently positive values of physical properties as a normal distribution would otherwise assume. The paper concludes that the derived closed-form solution for lognormal processes can predict the first passage probability with accuracy. Accurate prediction of first passage probability is of significance in preventing failures of engineering structures during their lifetime.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
Financial support from Australian Research Council under DP140101547 and LP150100413, Natural Science Foundation of Hubei Province of China under 2015CFB510, and Fundamental Research Funds by Wuhan University of Technology of China under 2015IVA012 is gratefully acknowledged.
References
ACI (American Concrete Institute). (2014). “Building code requirements for structural concrete.” ACI 318-14, Farmington Hills, MI.
Andrade, C., Molina, F. J., and Alonso, C. (1993). “Cover cracking as a function of rebar corrosion. I: Experiment test.” Mater. Struct., 26(8), 453–464.
Andrieu-Renaud, C., Sudret, B., and Lemaire, M. (2004). “The PHI2 method: A way to compute time-variant reliability.” Reliab. Eng. Syst. Saf., 84(1), 75–86.
Beck, A. T., and Melchers, R. E. (2005). “Barrier failure dominance in time variant reliability analysis.” Probab. Eng. Mech., 20(1), 79–85.
Breitung, K. (1988). “Asymptotic approximations for the outcrossing rates of stationary vector processes.” Stochastic Process. Appl., 29(2), 195–207.
BSI (British Standards Institute). (1997). “Structural use of concrete—Code practice for design and construction: Part 1.” London.
Calif, R., and Schmitt, F. G. (2012). “Modeling of atmospheric wind speed sequence using a lognormal continuous stochastic equation.” J. Wind Eng. Ind. Aerodyn., 109, 1–8.
Desmond, A. F., and Guy, B. T. (1991). “Crossing theory for non-Gaussian stochastic processes with an application to hydrology.” Water Resour. Res., 27(10), 2791–2797.
Ditlevsen, O. (1983). “Gaussian outcrossings from safe convex polyhedrons.” J. Eng. Mech., 127–148.
Engelund, S., Rackwitz, R., and Lange, C. (1995). “Approximations of first-passage times for differentiable processes based on higher-order threshold crossings.” Probab. Eng. Mech., 10(1), 53–60.
Ferrante, F. J., Arwade, S. R., and Graham-Brady, L. L. (2005). “A translation model for non-stationary, non-Gaussian random processes.” Probab. Eng. Mech., 20(3), 215–228.
Grigoriu, M. (1984). “Crossing of non-Gaussian translation processes.” J. Eng. Mech., 610–620.
Hagen, O., and Tvedt, L. (1991). “Vector process outcrossing as parallel system sensitivity measure.” J. Eng. Mech., 2201–2220.
Hu, Z., and Du, X. (2013). “Time-dependent reliability analysis with joint upcrossing rates.” Struct. Multidiscip. Optim., 48(5), 893–907.
Li, C. Q., and Mahmoodian, M. (2013). “Risk based service life prediction of underground cast iron pipes.” Reliab. Eng. Syst. Saf., 119, 102–108.
Li, C. Q., and Melchers, R. E. (1993). “Outcrossing from polyhedrons for nonstationary Gaussian processes.” J. Eng. Mech., 2354–2361.
Li, C. Q., and Melchers, R. E. (2005). “Time-dependent reliability analysis of corrosion-induced concrete cracking.” ACI Struct. J., 102(4), 543–549.
Madhavan, P. T. M., and Veena, G. (2006). “Fatigue reliability analysis of fixed offshore structures: A first passage problem approach.” J. Zhejiang Univ. Sci. A, 7(11), 1839–1845.
Madsen, H. O., and Tvedt, L. (1990). “Methods for time dependent reliability and sensitivity analysis.” J. Eng. Mech., 2118–2135.
Melchers, R. E. (1992). “Load space formulation for time-dependent structural reliability.” J. Eng. Mech., 853–870.
Melchers, R. E. (1999). Structural reliability analysis and prediction, 2nd Ed., Wiley, Chichester, U.K.
Papoulis, A. (1965). Probability, random variables, and stochastic processes, McGraw-Hill, New York.
Rackwitz, R. (2001). “Reliability analysis—A review and some perspectives.” Struct. Saf., 23(4), 365–395.
Rice, S. O. (1944). “Mathematical analysis of random noise.” Bell Syst. Tech. J., 23(3), 282–332.
Shinozuka, M. (1964). “Probability of failure under random loading.” J. Eng. Mech., 90(EM5), 147–171.
Sudret, B. (2008). “Analytical derivation of the outcrossing rate in time-variant reliability problems.” Struct. Infrastruct. Eng., 4(5), 353–362.
Thoft-Christensen, P. (2001). “Corrosion crack based assessment of the life-cycle reliability of concrete structures.” Structural Safety and Reliability: Proc., Eighth Int. Conf., ICOSSAR ’01, R. B. Corotis, et al., eds., CRC Press, Boca Raton, FL, 1–7.
Vanmarke, E. H. (1975). “On the distribution of the first passage time for normal stationary random processes.” J. Appl. Mech., 42(1), 215–220.
Wu, W. F., and Ni, C. C. (2004). “Probabilistic models of fatigue crack propagation and their experimental verification.” Probab. Eng. Mech., 19(3), 247–257.
Xing, J., Zhong, Q. P., and Hong, Y. J. (1997). “A simple lognormal random process approach of the fatigue crack growth considering the distribution of initial crack size and loading condition.” Int. J. Pressure Vessels Piping, 74(1), 7–12.
Yang, J. N., and Manning, S. D. (1990). “Stochastic crack growth analysis methodologies for metallic structures.” Eng. Fract. Mech., 37(5), 1105–1124.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 142 • Issue 12 • December 2016
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Jul 24, 2015
Accepted: Jun 29, 2016
Published online: Sep 28, 2016
Published in print: Dec 1, 2016
Discussion open until: Feb 28, 2017
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.