Research Papers: Design Automation

A Closed-Form Second-Order Reliability Method Using Noncentral Chi-Squared Distributions

[+] Author and Article Information
Rami Mansour

Department of Solid Mechanics,
Royal Institute of Technology,
Stockholm SE-100 44, Sweden
e-mail: ramimans@kth.se

Mårten Olsson

Department of Solid Mechanics,
Royal Institute of Technology,
Stockholm SE-100 44, Sweden
e-mail: mart@kth.se

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received June 18, 2013; final manuscript received June 10, 2014; published online July 31, 2014. Assoc. Editor: David Gorsich.

J. Mech. Des 136(10), 101402 (Jul 31, 2014) (10 pages) Paper No: MD-13-1265; doi: 10.1115/1.4027982 History: Received June 18, 2013; Revised June 10, 2014

In the second-order reliability method (SORM), the probability of failure is computed for an arbitrary performance function in arbitrarily distributed random variables. This probability is approximated by the probability of failure computed using a general quadratic fit made at the most probable point (MPP). However, an easy-to-use, accurate, and efficient closed-form expression for the probability content of the general quadratic surface in normalized standard variables has not yet been presented. Instead, the most commonly used SORM approaches start with a relatively complicated rotational transformation. Thereafter, the last row and column of the rotationally transformed Hessian are neglected in the computation of the probability. This is equivalent to approximating the probability content of the general quadratic surface by the probability content of a hyperparabola in a rotationally transformed space. The error made by this approximation may introduce unknown inaccuracies. Furthermore, the most commonly used closed-form expressions have one or more of the following drawbacks: They neither do work well for small curvatures at the MPP and/or large number of random variables nor do they work well for negative or strongly uneven curvatures at the MPP. The expressions may even present singularities. The purpose of this work is to present a simple, efficient, and accurate closed-form expression for the probability of failure, which does not neglect any component of the Hessian and does not necessitate the rotational transformation performed in the most common SORM approaches. Furthermore, when applied to industrial examples where quadratic response surfaces of the real performance functions are used, the proposed formulas can be applied directly to compute the probability of failure without locating the MPP, as opposed to the other first-order reliability method (FORM) and the other SORM approaches. The method is based on an asymptotic expansion of the sum of noncentral chi-squared variables taken from the literature. The two most widely used SORM approaches, an empirical SORM formula as well as FORM, are compared to the proposed method with regards to accuracy and computational efficiency. All methods have also been compared when applied to a wide range of hyperparabolic limit-state functions as well as to general quadratic limit-state functions in the rotationally transformed space, in order to quantify the error made by the approximation of the Hessian indicated above. In general, the presented method was the most accurate for almost all studied curvatures and number of random variables.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Lemaire, M., 2009, Structural Reliability, Wiley, Hoboken, NJ.
Madsen, H. O., Krenk, S., and Lind, N. C., 1986, Methods of Structural Safety, Prentice-Hall, Englewood Cliffs, NJ.
Tvedt, L., 1988, “Distribution of Quadratic Forms in the Normal Space Application to Structural Reliability,” J. Eng. Mech., 116(6), pp. 1183–1197. [CrossRef]
Lee, I., Noh, Y., and Yoo, D., 2012, “A Novel Second Order Reliability Method (SORM) Using Noncentral or Generalized Chi-Squared Distributions,” ASME J. Mech. Des., 134(10), p. 100912. [CrossRef]
Valdebenito, M. A., and Schuller, G. I., 2010, “A Survey on Approaches for Reliability-Based Optimization,” Struct. Multidiscip. Optim., 42(5), pp. 645–663. [CrossRef]
Du, X., and Chen, W., 2004, “Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design,” ASME J. Mech. Des., 126(2), pp. 225–233. [CrossRef]
Youn, B. D., Choi, K. K., Yang, R., and Gu, L., 2004, “Reliability-Based Design Optimization for Crashworthiness of Vehicle Side Impact,” Struct. Multidiscip. Optim., 26(3), pp. 272–283. [CrossRef]
Youn, B. D., Xi, Z., and Yang, P., 2008, “Eigenvector Dimension Reduction Method for Sensitivity-Free Probability Analysis,” Struct. Multidiscip. Optim., 37(1), pp. 13–28. [CrossRef]
Breitung, K., 1984, “Asymptotic Approximations for Multi-Normal Integrals,” J. Eng. Mech. Div., 110(3), pp. 357–366. [CrossRef]
Tvedt, L., 1983, “Two Second-Order Approximations to the Failure Probability-Section on Structural Reliability,” Technical Report No. RDIV/20-004-83, A/S Veritas Research, Hovik.
Ditlevsen, O., and Madsen, H. O., 2005, Structural Reliability Methods, Internet edition 2.2.5, Denmark.
Zhao, Y., and Ono, T., 1999, “New Approximations for SORM: Part 1,” J. Eng. Mech., 125(1), pp. 79–85. [CrossRef]
Choi, S.-K., Grandhi, R., and Canfield, R., 2007, Reliability-Based Structural Design, Springer-Verlag, London.
Jin, R., Chen, W., and Simpson, T. W., 2000, “Comparative Studies of Metamodeling Techniques Under Multiple Modeling Criteria,” Struct. Multidiscip. Optim., 23(1), pp. 1–13. [CrossRef]
Konishi, S., Naoto, N., and Gupta, A. K., 1988, “Asymptotic Expansion for the Distribution of Quadratic Forms,” Ann. Inst. Stat. Math., 40(2), pp. 279–296. [CrossRef]
Hasofer, A. M., and Lind, N. C., 1974, “Exact and Invariant Second Moment Code Format,” ASCE J. Eng. Mech. Div., 100(1), pp. 111–121.
Cai, G. Q., and Elishakoff, I., 1994, “Refined Second-Order Reliability Analysis,” Struct. Saf., 14, pp. 267–276. [CrossRef]
Koyluoglu, H. U., and Nielsen, S. R. K., 1994, “New Approximations for SORM Integrals,” Struct. Saf., 13(4), pp. 235–246. [CrossRef]
Du, X., and Chen, W., 2000, “Towards a Better Understanding of Modeling Feasibility Robustness in Engineering Design,” ASME J. Mech. Des., 122(4), pp. 385–394. [CrossRef]
Phoon, K., and Huang, S., 2007, “Uncertainty Quantification Using Multi-Dimensional Hermite Polynomials,” Probabilistic Applications in Geotechnical Engineering, American Society of Civil Engineers, pp. 1–10. [CrossRef]
Eldred, M. S., and Burkardt, J., 2009, “Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification,” AIAA Paper No. 2009-0976, pp. 1–20.
Kameshwar, S., and Chakraborty, A., 2013, “On Reliability Evaluation of Structures Using Hermite Polynomial Chaos,” Proceedings of the International Symposium on Engineering Under Uncertainty: Safety Assessment and Management, Springer India, pp. 1141–1152. [CrossRef]
Montgomery, D. C., 2001, Design and Analysis of Experiments, John Wiley and Sons, Inc., New York.
ansys, 2011. 14.0 ed. s.l., SAS IP, Inc.
Holtz, M., 2011, Sparse Grid Quadrature in High Dimensions With Applications in Finance and Insurance, Springer-Verlag, Heidelberg.
Rahman, S., and Wei, D., 2006, “A Univariate Approximation at Most Probable Point for Higher-Order Reliability Analysis,” Int. J. Solids Struct., pp. 2820–2839. [CrossRef]
Kiureghian, A., and Dakessian, T., 1998, “Multiple Design Points in First and Second-Order Reliability,” Struct. Saf., 20(1), pp. 37–49. [CrossRef]


Grahic Jump Location
Fig. 1

Flowchart of the methods proposed by Breitung/Tvedt, Zhao, and the present method

Grahic Jump Location
Fig. 2

Reinforced concrete cross section subjected to pure bending

Grahic Jump Location
Fig. 3

Generalized reliability index comparison for a parabolic limit-state according to (a)–(c) Eq. (44) and (d) Eq. (45)

Grahic Jump Location
Fig. 4

Generalized reliability index comparison for a quadratic limit-state according to (a)–(c) Eq. (46) and (d) Eq. (47)

Grahic Jump Location
Fig. 5

Generalized reliability index comparison for a quadratic limit-state according to (a)–(c) Eq. (48) and (d) Eq. (49)




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In