Use of the Probability Transformation Method in Some Random Mechanic Problems
Publication: ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering
Volume 7, Issue 1
Abstract
This work addresses the use of the probability transformation method (PTM) and of some of its extensions for solving mechanical and structural systems for which the response is modeled as random fields or variables that cannot be well-approximated as Gaussian. In particular, the static and dynamic stochastic analyses of linear structural systems excited by non-Gaussian excitations were considered. Moreover, the stochastic structures, the geometric and/or material properties of which are random, were analyzed by coupling the PTM with another approach introduced by one of the authors, the approximated principal deformation mode (APDM) method. Some of the results were reported in other works, and some results are shown here for the first time. This work gathered all the typologies of stochastic structural analyses in which the PTM can be advantageously applied, both in terms of accuracy and in terms of efficiency.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
References
Ariaratnam, S., G. I. Schuëller, I. Elishakoff, and Y. Lin. 1988. Stochastic structural dynamics London: Elsevier.
Chen, C., W. Wu, B. Zhang, and H. Sun. 2015. “Correlated probabilistic load flow using a point estimate method with Nataf transformation.” Int. J. Electr. Power Energy Syst. 65 (Feb): 325–333. https://doi.org/10.1016/j.ijepes.2014.10.035.
Contreras, H. 1980. “The stochastic finite-element method.” Comput. Struct. 12 (3): 341–348. https://doi.org/10.1016/0045-7949(80)90031-0.
D’Agostini, G. 1995. “A multidimensional unfolding method based on Bayes’ theorem.” Nucl. Instrum. Methods Phys. Res., Sect. A 362 (2–3): 487–498.
Di Paola, M., and G. Falsone. 1994. “Non-linear oscillators under parametric and external poisson pulses.” Nonlinear Dyn. 5 (3): 337–352. https://doi.org/10.1007/BF00045341.
Doostan, A., R. G. Ghanem, and J. Red-Horse. 2007. “Stochastic model reduction for chaos representations.” Comput. Methods Appl. Mech. Eng. 196 (37–40): 3951–3966. https://doi.org/10.1016/j.cma.2006.10.047.
Elishakoff, I., Y. Ren, and M. Shinozuka. 1995. “Improved finite element method for stochastic problems.” Chaos, Solitons Fractals 5 (5): 833–846. https://doi.org/10.1016/0960-0779(94)00157-L.
Falsone, G. 2005. “An extension of the Kazakov relationship for non-Gaussian random variables and its use in the non-linear stochastic dynamics.” Probab. Eng. Mech. 20 (1): 45–56. https://doi.org/10.1016/j.probengmech.2004.06.001.
Falsone, G., and N. Impollonia. 2002. “A new approach for the stochastic analysis of finite element modelled structures with uncertain parameters.” Comput. Methods Appl. Mech. Eng. 191 (44): 5067–5085. https://doi.org/10.1016/S0045-7825(02)00437-1.
Falsone, G., and R. Laudani. 2019a. “Exact response probability density functions of some uncertain structural systems.” Arch. Mech. 71 (4–5): 315–336.
Falsone, G., and R. Laudani. 2019b. “Matching the principal deformation mode method with the probability transformation method for the analysis of uncertain systems.” Int. J. Numer. Methods Eng. 118 (7): 395–410. https://doi.org/10.1002/nme.6018.
Falsone, G., and D. Settineri. 2013a. “Explicit solutions for the response probability density function of linear systems subjected to random static loads.” Probab. Eng. Mech. 33 (Jul): 86–94. https://doi.org/10.1016/j.probengmech.2013.03.001.
Falsone, G., and D. Settineri. 2013b. “Explicit solutions for the response probability density function of nonlinear transformations of static random inputs.” Probab. Eng. Mech. 33 (Jul): 79–85. https://doi.org/10.1016/j.probengmech.2013.03.003.
Field, R., Jr., and M. Grigoriu. 2004. “On the accuracy of the polynomial chaos approximation.” Probab. Eng. Mech. 19 (1–2): 65–80. https://doi.org/10.1016/j.probengmech.2003.11.017.
Ghanem, R. G., and P. D. Spanos. 1991. “Stochastic finite element method: Response statistics.” In Stochastic finite elements: A spectral approach, 101–119. New York: Springer.
Hurtado, J. E., and A. H. Barbat. 1998. “Monte Carlo techniques in computational stochastic mechanics.” Arch. Comput. Methods Eng. 5 (1): 3. https://doi.org/10.1007/BF02736747.
Kahn, H. 1955. Use of different Monte Carlo sampling techniques. Santa Monica, CA: Rand Corporation.
Kamiński, M. 2007. “Generalized perturbation-based stochastic finite element method in elastostatics.” Comput. Struct. 85 (10): 586–594. https://doi.org/10.1016/j.compstruc.2006.08.077.
Kamiński, M. 2013. The stochastic perturbation method for computational mechanics. New York: Wiley.
Kleiber, M., and T. Hien. 1992. The stochastic finite element method. New York: Wiley.
Lavielle, M., and E. Lebarbier. 2001. “An application of MCMC methods for the multiple change-points problem.” Signal Process. 81 (1): 39–53. https://doi.org/10.1016/S0165-1684(00)00189-4.
Lebrun, R., and A. Dutfoy. 2009. “An innovating analysis of the Nataf transformation from the copula viewpoint.” Probab. Eng. Mech. 24 (3): 312–320. https://doi.org/10.1016/j.probengmech.2008.08.001.
Li, J., and J. Chen. 2009. Stochastic dynamics of structures. New York: Wiley.
Liu, W. K., A. Mani, and T. Belytschko. 1987. “Finite element methods in probabilistic mechanics.” Probab. Eng. Mech. 2 (4): 201–213. https://doi.org/10.1016/0266-8920(87)90010-5.
Lutes, L. D., and S. Sarkani. 2004. Random vibrations: Analysis of structural and mechanical systems. London: Butterworth–Heinemann.
Matthies, H. G., C. E. Brenner, C. G. Bucher, and C. G. Soares. 1997. “Uncertainties in probabilistic numerical analysis of structures and solids-stochastic finite elements.” Struct. Saf. 19 (3): 283–336. https://doi.org/10.1016/S0167-4730(97)00013-1.
Metropolis, N., and S. Ulam. 1949. “The Monte Carlo method.” J. Am. Stat. Assoc. 44 (247): 335–341. https://doi.org/10.1080/01621459.1949.10483310.
Papadrakakis, M., and V. Papadopoulos. 1996. “Robust and efficient methods for stochastic finite element analysis using Monte Carlo simulation.” Comput. Methods Appl. Mech. Eng. 134 (3–4): 325–340. https://doi.org/10.1016/0045-7825(95)00978-7.
Sachdeva, S. K., P. B. Nair, and A. J. Keane. 2006. “Comparative study of projection schemes for stochastic finite element analysis.” Comput. Methods Appl. Mech. Eng. 195 (19–22): 2371–2392. https://doi.org/10.1016/j.cma.2005.05.010.
Schuëller, G. I. 2001. “Computational stochastic mechanics–recent advances.” Comput. Struct. 79 (22–25): 2225–2234. https://doi.org/10.1016/S0045-7949(01)00078-5.
Schuëller, G. I., C. G. Bucher, U. Bourgund, and W. Ouypornprasert. 1989. “On efficient computational schemes to calculate structural failure probabilities.” Probab. Eng. Mech. 4 (1): 10–18. https://doi.org/10.1016/0266-8920(89)90003-9.
Schuëller, G. I., and H. Pradlwarter. 2009. “Uncertain linear systems in dynamics: Retrospective and recent developments by stochastic approaches.” Eng. Struct. 31 (11): 2507–2517. https://doi.org/10.1016/j.engstruct.2009.07.005.
Soong, T. 1973. Random differential equations in science and engineering. New York: Academic Press.
Sudret, B., and A. Der Kiureghian. 2000. Stochastic finite element methods and reliability: A state-of-the-art report. Berkeley, CA: Dept. of Civil and Environmental Engineering, Univ. of California.
Wu, W., and Y. Lin. 1984. “Cumulant-neglect closure for non-linear oscillators under random parametric and external excitations.” Int. J. Non Linear Mech. 19 (4): 349–362. https://doi.org/10.1016/0020-7462(84)90063-5.
Yamazaki, F., M. Shinozuka, and G. Dasgupta. 1988. “Neumann expansion for stochastic finite element analysis.” J. Eng. Mech. 114 (8): 1335–1354. https://doi.org/10.1061/(ASCE)0733-9399(1988)114:8(1335).
Zhu, S.-P., Q. Liu, W. Peng, and X.-C. Zhang. 2018. “Computational-experimental approaches for fatigue reliability assessment of turbine bladed disks.” Int. J. Mech. Sci. 142 (Jul): 502–517. https://doi.org/10.1016/j.ijmecsci.2018.04.050.
Information & Authors
Information
Published In
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering
Volume 7 • Issue 1 • March 2021
Copyright
© 2020 American Society of Civil Engineers.
History
Received: Jul 7, 2020
Accepted: Sep 29, 2020
Published online: Dec 1, 2020
Published in print: Mar 1, 2021
Discussion open until: May 1, 2021
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.