Fourth-Moment Standardization for Structural Reliability Assessment
Publication: Journal of Structural Engineering
Volume 133, Issue 7
Abstract
In structural reliability analysis, the uncertainties related to resistance and load are generally expressed as random variables that have known cumulative distribution functions. However, in practical applications, the cumulative distribution functions of some random variables may be unknown, and the probabilistic characteristics of these variables may be expressed using only statistical moments. In the present paper, in order to conduct structural reliability analysis without the exclusion of random variables having unknown distributions, the third-order polynomial normal transformation technique using the first four central moments is investigated, and an explicit fourth-moment standardization function is proposed. Using the proposed method, the normal transformation for independent random variables with unknown cumulative distribution functions can be realized without using the Rosenblatt transformation or Nataf transformation. Through the numerical examples presented, the proposed method is found to be sufficiently accurate in its inclusion of the independent random variables which have unknown cumulative distribution functions, in structural reliability analyses with minimal additional computational effort.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The study was partially supported by Grant-in-Aid from the Ministry of ESCST, Japan (Grant No. 17560501). The support is gratefully acknowledged.
References
Ang, A. H.-S., and Tang, W. H. (1984). Probability concepts in engineering planning and design, Vol. II: Decision, risk, and reliability, Wiley, New York.
Cai, G. Q., and Elishakoff, I. (1994). “Refined second-order reliability analysis.” Struct. Safety, 14(4), 267–276.
Chen, X., and Tung, Y. K. (2003). “Investigation of polynomial normal transform.” Struct. Safety, 25(4), 423–455.
Der Kiureghian, A., and De Stefano, M. (1991). “Efficient algorithm for second-order reliability analysis.” J. Eng. Mech., 117(12), 2904–2923.
Der Kiureghian, A., Lin, H. Z., and Hwang, S. J. (1987). “Second-order reliability approximations.” J. Eng. Mech., 113(8), 1208–1225.
Der Kiureghian, A., and Liu, P. L. (1986). “Structural reliability under incomplete probability information.” J. Eng. Mech., 112(1), 85–104.
Fisher, R. A., and Cornish, E. A. (1960). “The percentile points of distributions having known cumulants.” Technometrics, 2, 209.
Fleishman, A. L. (1978). “A method for simulating non-normal distributions.” Psychometrika, 43(4), 521–532.
Fu, G. (1994). “Variance reduction by truncated multimodal importance sampling.” Struct. Safety, 13(4), 267–283.
Grigoriu, M. (1983). “Approximate analysis of complex reliability analysis.” Struct. Safety, 1(4), 277–288.
Hasofer, A. M., and Lind, N. C. (1974). “Exact and invariant second moment code format.” J. Engrg. Mech. Div., 100(1), 111–121.
Hohenbichler, M., and Rackwitz, R. (1981). “Non-normal dependent vectors in structural safety.” J. Engrg. Mech. Div., 107(6), 1227–1238.
Hong, H. P. (1996). “Point-estimate moment-based reliability analysis.” Civ. Eng. Syst., 13, 281–294.
Hong, H. P., and Lind, N. C. (1996). “Approximation reliability analysis using normal polynomial and simulation results.” Struct. Safety, 18(4), 329–339.
Johnson, N. L., and Kotz, S. (1970). Continuous univariate distributions—1, Wiley, New York.
Liu, P. L., and Der Kiureghian, A. (1986). “Multivariate distribution models with prescribed marginals and covariances.” Probab. Eng. Mech., 1(2), 105–116.
Liu, Y. W., and Moses, F. (1994). “A sequential response surface method and its application in the reliability analysis of aircraft structural systems.” Struct. Safety, 16(1), 39–46.
Melchers, R. E. (1990). “Radial importance sampling for structural reliability.” J. Eng. Mech., 116(1), 189–203.
Nie, J., and Ellingwood, B. R. (2000). “Directional methods for structural reliability analysis.” Struct. Safety, 22(3), 233–249.
Ono, T., Idota, H., and Kawahara, H. (1986). “A statistical study on resistances of steel columns and beams using higher order moments.” J. Struct. Constr. Eng., 370, 19–37 (in Japanese).
Pearson, E. S., Johnson, N. L., and Burr, I. W. (1979). “Comparison of the percentage points of distributions with the same first four moments, chosen from eight different systems of frequency curves.” Commun. Stat.-Simul. Comput., B8(3), 191–229.
Rackwitz, R. (1976). “Practical probabilistic approach to design—First-order reliability concepts for design codes.” Bull. d’Information, No. 112, Comite European du Beton, Munich, Germany.
Rajashekhar, M. R., and Ellingwood, B. R. (1993). “A new look at the response surface approach for reliability analysis.” Struct. Safety, 12(3), 205–220.
Ramberg, J., and Schmeiser, B. (1974). “An approximate method for generating asymmetric random variables.” Commun. ACM, 17(2), 78–82.
Shinozuka, M. (1983). “Basic analysis of structural safety.” J. Struct. Eng., 109(3), 721–740.
Stuart, A., and Ord, J. K. (1987). Kendall’s advanced theory of statistics, Charles Griffin and Company Ltd., London, Vol. 1, 210–275.
Tichy, M. (1994). “First-order third-moment reliability method.” Struct. Safety, 16(3), 189–200.
Tung, Y. K. (1999). “Polynomial normal transform in uncertainty analysis.” ICASP 8—Application of probability and statistics, Balkema, The Netherlands, 167–174.
Winterstein, S. R. (1988). “Nonlinear vibration models for extremes and fatigue.” J. Eng. Mech., 114(10), 1769–1787.
Zhao, Y. G., and Ang, A. H.-S. (2003). “System reliability assessment by method of moments.” J. Struct. Eng., 129(10), 1341–1349.
Zhao, Y. G., and Ono, T. (1999). “New approximations for SORM: Part II.” J. Eng. Mech., 125(1), 86–93.
Zhao, Y. G., and Ono, T. (2000). “Third-moment standardization for structural reliability analysis.” J. Struct. Eng., 126(6), 724–732.
Zhao, Y. G., Ono, T., and Kato, M. (2002). “Second-order third-moment reliability method.” J. Struct. Eng., 128(8), 1087–1090.
Zong, Z., and Lam, K. Y. (1998). “Estimation of complicated distribution using B-spline functions.” Struct. Safety, 20(4), 341–355.
Information & Authors
Information
Published In
Journal of Structural Engineering
Volume 133 • Issue 7 • July 2007
Pages: 916 - 924
Copyright
© 2007 ASCE.
History
Received: Aug 25, 2006
Accepted: Dec 14, 2006
Published online: Jul 1, 2007
Published in print: Jul 2007
Notes
Note. Associate Editor: Shahram Sarkani
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.