Abstract
The problem of estimating the time variant reliability of randomly parametered dynamical systems subjected to random process excitations is considered. Two methods, based on Monte Carlo simulations, are proposed to tackle this problem. In both the methods, the target probability of failure is determined based on a two-step approach. In the first step, the failure probability conditional on the random variable vector modelling the system parameter uncertainties is considered. The unconditional probability of failure is determined in the second step, by computing the expectation of the conditional probability with respect to the random system parameters. In the first of the proposed methods, the conditional probability of failure is determined analytically, based on an approximation to the average rate of level crossing of the dynamic response across a specified safe threshold. An augmented space of random variables is subsequently introduced, and the unconditional probability of failure is estimated by using variance-reduced Monte Carlo simulations based on the Markov chain splitting methods. A further improvement is developed in the second method, in which, the conditional failure probability is estimated by using Girsanov’s transformation-based importance sampling, instead of the analytical approximation. Numerical studies on white noise-driven single degree of freedom linear/nonlinear oscillators and a benchmark multi-degree of freedom linear system under non-stationary filtered white noise excitation are presented. The probability of failure estimates obtained using the proposed methods shows reasonable agreement with the estimates from existing Monte Carlo simulation strategies.
Similar content being viewed by others
References
Schueller, G.I.: Developments in stochastic structural mechanics. Arch. Appl. Mech. 75, 755–773 (2006)
Goller, B., Pradlwarter, H.J., Schueller, G.I.: Reliability assessment in structural dynamics. J. Sound Vib. 332, 2488–2499 (2013)
Soize, C.: Stochastic modeling of uncertainties in computational structural dynamics—recent theoretical advances. J. Sound Vib. 332, 2379–2395 (2013)
Rice, S.O.: Mathematical analysis of random noise. Bell Syst. Tech. J. 23(3), 282–332 (1944)
Yang, J.N., Shinozuka, M.: On the first excursion probability in stationary narrow-band random vibration. ASME J. Appl. Mech. 38, 1017–22 (1971)
Vanmarcke, E.H.: On the distribution of the first-passage time for normal stationary random processes. ASME J. Appl. Mech. 42, 215–20 (1975)
Roberts, J.B.: First passage probability for nonlinear oscillators. ASCE J. Eng. Mech. 102, 851–66 (1976)
Spencer, B.F., Bergman, L.A.: On the numerical solution of the Fokker–Planck equation for nonlinear stochastic systems. Nonlinear Dyn. 4, 357–72 (1993)
Wu, Y.J., Luo, M., Zhu, W.Q.: First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations. Arch. Appl. Mech. 78, 501–515 (2008)
Chen, L.C., Zhu, W.Q.: First passage failure of quasi non-integrable generalized Hamiltonian systems. Arch. Appl. Mech. 80, 883–893 (2010)
Wen, Y.K., Chen, H.C.: On fast integration for time variant structural reliability. Probab. Eng. Mech. 2(3), 156–162 (1987)
Papadimitriou, C., Beck, J.L., Katafygiotis, L.S.: Asymptotic expansions for reliability and moments of uncertain systems. ASCE J. Eng. Mech. 123(12), 1219–29 (1997)
Au, S.K., Papadimitriou, C., Beck, J.L.: Reliability of uncertain dynamical systems with multiple design points. Struct. Saf. 21, 113–133 (1999)
Gupta, S., Manohar, C.S.: Reliability analysis of randomly vibrating structures with parameter uncertainties. J. Sound Vib. 297, 1000–24 (2006)
Chen, J.B., Li, J.: The extreme value distribution and dynamic reliability analysis of non-linear structures with uncertain parameters. Struct. Saf. 29, 77–93 (2007)
Zhou, Q., Fan, W., Li, Z., Ohsaki, M.: Time variant system reliability assessment by probability density evolution method. ASCE J. Eng. Mech. 143(11), 04017131:1-10 (2017)
Grigoriu, M.: Stochastic Calculus: Applications in Science and Engineering. Birkhauser, Boston (2002)
Au, S.K., Beck, J.L.: First excursion probabilities for linear systems by very efficient importance sampling. Probab. Eng. Mech. 16, 193–207 (2001)
Au, S.K., Lam, H.F., Ng, C.T.: Reliability analysis of single-degree-of-freedom elastoplastic systems I: critical excitations. ASCE J. Eng. Mech. 133(10), 1072–1080 (2007)
Au, S.K., Beck, J.L.: Estimation of small failure probabilities in high dimensions by subset simulation. Probab. Eng. Mech. 16, 263–277 (2001)
Au, S.K., Beck, J.L.: Subset simulation and its applications to seismic risk based on dynamic analysis. ASCE J. Eng. Mech. 129(8), 901–917 (2003)
Katafygiotis, L.S., Cheung, S.H., Yuen, K.: Spherical subset simulation (S\(^{\rm 3})\) for solving non-linear dynamical reliability problems. Int. J. Reliab. Saf. 4(2–3), 122–138 (2010)
Koutsourelakis, P.S., Pradlwarter, H.J., Schueller, G.I.: Reliability of structures in high dimensions, part I: algorithms and applications. Probab. Eng. Mech. 19, 409–417 (2004)
Kanjilal, O., Manohar, C.S.: Markov chain splitting methods in structural reliability integral estimation. Probab. Eng. Mech. 40, 42–51 (2015)
Macke, M., Bucher, C.: Importance sampling for randomly excited dynamical systems. J. Sound Vib. 268, 269–290 (2003)
Nayek, R., Manohar, C.S.: Girsanov transformation-based reliability modelling and testing of actively controlled structures. ASCE J. Eng. Mech. 141(6), 04014168:1-8 (2015)
Kanjilal, O., Manohar, C.S.: Girsanov’s transformation-based variance reduced Monte Carlo simulation schemes for reliability estimation in non-linear stochastic dynamics. J. Comput. Phys. 341, 278–294 (2017)
Olsen, A.I., Naess, A.: An importance sampling procedure for estimating failure probabilities of non-linear dynamic systems subjected to random noise. Int. J. Nonlinear Mech. 42, 848–863 (2007)
Kanjilal, O., Manohar, C.S.: State dependent Girsanov’s controls in time variant reliability estimation in randomly excited dynamical systems. Struct. Saf. 72, 30–40 (2018)
Au, S.K.: Stochastic control approach to reliability of elasto-plastic structures. Struct. Eng. Mech. 32(1), 21–36 (2009)
Schueller, G.I., Pradlwarter, H.J.: Benchmark study on reliability estimation in higher dimensions of structural systems—an overview. Struct. Saf. 29, 167–182 (2007)
Jensen, H.A., Valdebenito, M.A.: Reliability analysis of linear dynamical systems using approximate representations of performance functions. Struct. Saf. 29, 222–237 (2007)
Sundar, V.S., Manohar, C.S.: Estimation of time variant reliability of randomly parametered non-linear vibrating systems. Struct. Saf. 47, 59–66 (2014)
Valdebenito, M.A., Jensen, H.A., Labarca, A.A.: Estimation of first excursion probabilities for uncertain stochastic linear systems subjected to Gaussian load. Comput. Struct. 138, 36–48 (2014)
Pradlwarter, H.J., Schueller, G.I.: Uncertain linear structural systems in dynamics: efficient stochastic reliability assessment. Comput. Struct. 88, 74–86 (2010)
Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics: Advanced Theory and Applications. McGraw-Hill, Singapore (1995)
Melchers, R.E.: Structural Reliability Analysis and Prediction. Wiley, Chichester (1999)
Firouzi, A., Yang, W., Li, C.Q.: Efficient solution for calculation of up-crossing rate of non-stationary Gaussian processes. ASCE J. Eng. Mech. 144(4), 04018015:1-9 (2018)
Botev, Z.I., Kroese, D.P.: Efficient Monte Carlo simulation via the generalized splitting method. Stat. Comput. 22, 1–16 (2012)
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was carried out when the first author was a research student at the Indian Institute of Science.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Kanjilal, O., Manohar, C.S. Time variant reliability estimation of randomly excited uncertain dynamical systems by combined Markov chain splitting and Girsanov’s transformation. Arch Appl Mech 90, 2363–2377 (2020). https://doi.org/10.1007/s00419-020-01726-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-020-01726-y