Abstract
In structural reliability analysis, the input variables are often nonnormal and correlated. A procedure for efficient normal transformation, i.e., transforming dependent nonnormal random variables into independent standard normal ones, is often required. In general, Rosenblatt transformation is available to realize the normal transformation when the joint probability density function (PDF) of basic random variables is available and Nataf transformation can be used when the marginal PDFs and correlation coefficients are known. However, the joint PDF and marginal PDFs of some random variables are often unknown in practice, and the probabilistic characteristics of these variables are easier to be expressed using the statistical moments and correlation matrix. It is in this regard that the objective of the present paper is to develop a methodology for normal transformation including correlated random variables with unknown joint PDF and marginal PDFs. Based on the third-moment transformation technique for transforming independent nonnormal random variables into independent standard normal ones, the third-moment transformation is further developed for transforming the correlated variables including unknown joint PDF and marginal PDFs into independent standard normal variables. A first-order reliability method for structural reliability analysis including correlated random variables with unknown joint PDF and marginal PDFs is developed based on the proposed transformation. Using the proposed method, the normal transformation and reliability analysis can also be achieved for correlated nonnormal random variables with knowledge of only the statistical moments and correlation matrix. The simplicity and efficiency of the proposed method is further demonstrated through several numerical examples.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The research reported in this paper is partially supported by the National Natural Science Foundation of China (Grant Nos. 51422814, U1134209, U1434204), the Project of Innovation-Driven Plan in Central South University (2015CXS014), and the Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT) (Grant No. IRT1296). The Fundamental Research Funds for the Central Universities of Central South University is also gratefully acknowledged.
References
Baecher, G. B., and Christian, J. T. (2003). Reliability and statistics in geotechnical engineering, Wiley, New York.
Bjerager, P. (1991). “Methods for structural reliability computation.” Reliability problems: General principles and applications in mechanics of solids and structures, F. Casciati, ed., Springer, New York, 89–136.
Breitung, K. (1984). “Asymptotic approximation for multi-normal integrals.” J. Eng. Mech., 357–366.
Chang, C. H., Tung, Y. K., and Yang, J. C. (1994). “Monte Carlo simulation for correlated variables with marginal distributions.” J. Hydraul. Eng, 313–331.
Chen, X. Y., and Tung, Y. K. (2003). “Investigation of polynomial normal transform.” Struct. Saf., 25(4), 423–445.
Der Kiureghian, A., and Liu, L. P. (1986). “Structural reliability under incomplete probability information.” J. Eng. Mech., 85–104.
Ditlevsen, O. (1979). “Generalized second moment reliability index.” Mech. Based. Des. Struc., 7(4), 435–451.
Ditlevsen, O., and Madsen, H. O. (1996). Structural reliability methods, Wiley, New York.
Fan, W. L., Wei, J. H., Ang, H-S., and Li, Z. L. (2016). “Adaptive estimation of statistical moments of the responses of random systems.” Probab. Eng. Mech., 43, 50–67.
Haldar, A., and Ayyub, B. M. (1984). “Risk models for correlated non-normal variables.” Proc., 5th Engineering Mechanics Division Specialty Conf., Vol. 2, ASCE, Reston, VA, 1237–1240.
Hasofer, A. M., and Lind, N. C. (1974). “Exact and invariant second-moment code format.” J. Eng. Mech. Div., 100(1), 111–121.
Headrick, T. C., and Sawilowsky, S. S. (1999). “Simulating correlated multivariate non-normal distributions: Extending the Fleishman power method.” Psychometrika, 64(1), 25–35.
Hohenbichler, M., and Rackwitz, R. (1981). “Non-normal dependent vectors in structural safety.” J. Eng. Mech. Div., 107(6), 1227–1238.
Hong, H. P., and Lind, N. C. (1996). “Approximate reliability analysis using normal polynomial and simulation results.” Struct. Saf., 18(4), 329–339.
Li, D. Q., Wu, S. B., Zhou, C. B., and Phoon, K. K. (2012). “Performance of translation approach for modeling correlated non-normal variables.” Struct. Saf., 39(4), 52–61.
Li, H. S., Lu, Z. Z., and Yuan, X. K. (2008). “Nataf transformation based point estimate method.” Sci. Bull., 53(17), 2586–2592.
Liu, J. M., Yu, B., and Yang, L. F. (2015). “Influence of orthogonal and Nataf transformations on precision of first order reliability method.” Chinese J. Appl. Mech., 32(1), 125–131 (in Chinese).
Liu, L. P., and Der Kiureghian, A. (1986). “Multivariate distribution models with prescribed marginals and covariances.” Probab. Eng. Mech., 1(2), 105–112.
Low, B. K. (2007). “Reliability analysis of rock slopes involving correlated nonnormals.” Int. J. Rock. Mech. Min. Sci., 44(6), 922–935.
Meijerink, J. A., and van der Vorst, H. A. (1977). “An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix.” Math. Comput., 31(137), 148–162.
Piric, K. (2015). “Reliability analysis method based on determination of the performance function’s PDF using the univariate dimension reduction method.” Struct. Saf., 57, 18–25.
Rackwitz, R., and Fiessler, B. (1978). “Structural reliability under combined load sequence.” Comput. Struct., 9(5), 489–494.
Shinozuka, M. (1983). “Basic analysis of structural safety.” J. Struct. Eng., 721–740.
Sorensen, J. D. (2004). Structural reliability theory and risk analysis, Aalborg Univ., Aalborg, Denmark.
Thoft-Christensen, P., and Sorenson, J. D. (1982). “Reliability of structural systems with correlated element.” Appl. Math. Modeling, 6(3), 171–178.
Xiao, Q. (2014). “Evaluating correlation coefficient for Nataf transformation.” Probab. Eng. Mech., 37(4), 1–6.
Zhao, Y. G., and Ono, T. (1998). “System reliability evaluation of ductile frame structure.” J. Struct. Eng., 678–685.
Zhao, Y. G., and Ono, T. (2000). “Third-moment standardization for structural reliability analysis.” J. Struct. Eng., 724–732.
Information & Authors
Information
Published In
Copyright
©2017 American Society of Civil Engineers.
History
Received: Jul 27, 2016
Accepted: Jan 13, 2017
Published online: Mar 24, 2017
Published in print: Aug 1, 2017
Discussion open until: Aug 24, 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.