Research Papers

Efficient Assessment of Structural Reliability in Presence of Random and Fuzzy Uncertainties

[+] Author and Article Information
A. S. Balu

Assistant Professor
Department of Civil Engineering,
National Institute of Technology Karnataka,
Mangalore 575 025, India
e-mail: arunsbalu@gmail.com

B. N. Rao

Structural Engineering Division,
Department of Civil Engineering,
Indian Institute of Technology Madras,
Chennai 600 036, India
e-mail: bnrao@iitm.ac.in

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received October 7, 2013; final manuscript received January 27, 2014; published online March 21, 2014. Assoc. Editor: David Gorsich.

J. Mech. Des 136(5), 051008 (Mar 21, 2014) (11 pages) Paper No: MD-13-1452; doi: 10.1115/1.4026650 History: Received October 07, 2013; Revised January 27, 2014

This paper presents an efficient uncertainty analysis for estimating the possibility distribution of structural reliability in presence of mixed uncertain variables. The proposed method involves high dimensional model representation for the limit state function approximation, transformation technique to obtain the contribution of the fuzzy variables to the convolution integral and fast Fourier transform for solving the convolution integral. In this methodology, efforts are required in evaluating conditional responses at a selected input determined by sample points, as compared to full scale simulation methods, thus the computational efficiency is accomplished. The proposed method is applicable for structural reliability estimation involving any number of fuzzy and random variables with any kind of distribution.

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


Ditlevsen, O., and Madsen, H. O., 1996, Structural Reliability Methods, Wiley, Chichester, UK.
Madsen, H. O., Krenk, S., and Lind, N. C., 1986, Methods of Structural Safety, Prentice-Hall, Englewood Cliffs, NJ.
Rackwitz, R., 2001, “Reliability Analysis—A Review and Some Perspectives,” Struct. Saf., 23(4), pp. 365–395. [CrossRef]
Breitung, K., 1984, “Asymptotic Approximations for Multinormal Integrals,” ASCE J. Eng. Mech., 110(3), pp. 357–366. [CrossRef]
Der Kiureghian, A., and De Stefano, M., 1991, “Efficient Algorithm for Second-Order Reliability Analysis,” ASCE J. Eng. Mech., 117(12), pp. 2904–2923. [CrossRef]
Du, X., and Hu, Z., 2012, “First Order Reliability Method With Truncated Random Variables,” ASME J. Mech. Des., 134(9), p. 091005. [CrossRef]
Chiralaksanakul, A., and Mahadevan, S., 2005, “First-Order Approximation Methods in Reliability-Based Design Optimization,” ASME J. Mech. Des., 127(5), pp. 851–857. [CrossRef]
Zhang, J., and Du, X., 2010, “A Second-Order Reliability Method With First-Order Efficiency,” ASME J. Mech. Des., 132(10), p. 101006. [CrossRef]
Liu, P. L., and Der Kiureghian, A., 1991, “Finite Element Reliability of Geometrically Nonlinear Uncertain Structures,” ASCE J. Eng. Mech., 117(8), pp. 1806–1825. [CrossRef]
Impollonia, N., and Sofi, A., 2003, “A Response Surface Approach for the Static Analysis of Stochastic Structures With Geometrical Nonlinearities,” Comput. Meth. Appl. Mech. Eng., 192(37–38), pp. 4109–4129. [CrossRef]
Au, S. K., and Beck, J. L., 2001, “Estimation of Small Failure Probabilities in High Dimensions by Subset Simulation,” Probab. Eng. Mech., 16(4), pp. 263–277. [CrossRef]
Melchers, R. E., 1989, “Importance Sampling in Structural Systems,” Struct. Saf., 6(1), pp. 3–10. [CrossRef]
Rubinstein, R. Y., 1981, Simulation and the Monte Carlo Method, Wiley, New York.
Faravelli, L., 1989, “Response-Surface Approach for Reliability Analysis,” ASCE J. Eng. Mech., 115(12), pp. 2763–2781. [CrossRef]
Gavin, H. P., and Yau, S. C., 2008, “High-Order Limit State Functions in the Response Surface Method for Structural Reliability Analysis,” Struct. Saf., 30(2), pp. 162–179. [CrossRef]
Wu, Y. T., and Torng, T. Y., 1990, “A Fast Convolution Procedure for Probabilistic Engineering Analysis,” Proceedings of First International Symposium Uncertainty Modeling and Analysis, College Park, MD, pp. 670–675.
Sakamoto, J., Mori, Y., and Sekioka, T., 1997, “Probability Analysis Method Using Fast Fourier Transform and Its Application,” Struct. Saf., 19(1), pp. 21–36. [CrossRef]
Penmetsa, R. V., and Grandhi, R., 2003, “Adaptation of Fast Fourier Transformations to Estimate Structural Failure Probability,” Finite Elem. Anal. Des., 39(5–6), pp. 473–485. [CrossRef]
Grandhi, R., and Wang, L., 1998, “Reliability-Based Structural Optimization Using Improved Two-Point Adaptive Nonlinear Approximations,” Finite Elem. Anal. Des., 29(1), pp. 35–48. [CrossRef]
Rao, B. N., and Chowdhury, R., 2008, “Probabilistic Analysis Using High Dimensional Model Representation and Fast Fourier Transform,” Int. J. Comput. Methods Eng. Sci. Mech., 9(6), pp. 342–357. [CrossRef]
Huang, B., and Du, X., 2006, “Uncertainty Analysis by Dimension Reduction Integration and Saddlepoint Approximations,” ASME J. Mech. Des., 128(1), pp. 26–33. [CrossRef]
Dong, W. M., and Wong, F. S., 1987, “Fuzzy Weighted Averages and Implementation of the Extension Principle,” Fuzzy Sets Syst., 21(2), pp. 183–199. [CrossRef]
Muhanna, R. L., and Mullen, R. L., 2001, “Uncertainty in Mechanics Problems-Interval-Based Approach,” ASCE J. Eng. Mech., 127(6), pp. 557–566. [CrossRef]
Penmetsa, R. V., and Grandhi, R., 2003, “Uncertainty Propagation Using Possibility Theory and Function Approximations,” Mech. Based Des. Struct. Mach., 31(2), pp. 257–279. [CrossRef]
Qiu, Z., Wang, X., and Chen, J., 2006, “Exact Bounds for the Static Response Set of Structures With Uncertain-But-Bounded Parameters,” Int. J. Solids Struct., 43(21), pp. 6574–6593. [CrossRef]
Balu, A. S., and Rao, B. N., 2012, “High Dimensional Model Representation Based Formulations for Fuzzy Finite Element Analysis of Structures,” Finite Elem. Anal. Des., 50, pp. 217–230. [CrossRef]
Mourelatos, Z. P., and Zhou, J., 2006, “A Design Optimization Method Using Evidence Theory,” ASME J. Mech. Des., 128(4), pp. 901–908. [CrossRef]
Zhou, J., and Mourelatos, Z. P., 2008, “A Sequential Algorithm for Possibility-Based Design Optimization,” ASME J. Mech. Des., 130(1), p. 011001. [CrossRef]
Nikolaidis, E., Chen, S., Cudney, H., Haftka, R. T., and Rosca, R., 2004, “Comparison of Probability and Possibility for Design Against Catastrophic Failure Under Uncertainty,” ASME J. Mech. Des., 126(3), pp. 386–394. [CrossRef]
Zaman, K., McDonald, M., and Mahadevan, S., 2011, “Probabilistic Framework for Uncertainty Propagation With Both Probabilistic and Interval Variables,” ASME J. Mech. Des., 133(2), p. 021010. [CrossRef]
Jiang, C., Han, X., Li, W. X., Liu, J., and Zhang, Z., 2012, “A Hybrid Reliability Approach Based on Probability and Interval for Uncertain Structures,” ASME J. Mech. Des., 134(3), p. 031001. [CrossRef]
Langley, R. S., 2000, “Unified Approach to Probabilistic and Possibilistic Analysis of Uncertain Systems,” ASCE J. Eng. Mech., 126(11), pp. 1163–1172. [CrossRef]
Möller, B., Graf, W., and Beer, M., 2003, “Safety Assessment of Structures in View of Fuzzy Randomness,” Comput. Struct., 81(15), pp. 1567–1582. [CrossRef]
Qiu, Z., Yang, D., and Elishakoff, I., 2008, “Probabilistic Interval Reliability of Structural Systems,” Int. J. Solids Struct., 45(10), pp. 2850–2860. [CrossRef]
Penmetsa, R. V., and Grandhi, R., 2002, “Efficient Estimation of Structural Reliability for Problems With Uncertain Intervals,” Comput. Struct., 80(12), pp. 1103–1112. [CrossRef]
Adduri, P. R., and Penmetsa, R. C., 2007, “Bounds on Structural System Reliability in the Presence of Interval Variables,” Comput. Struct., 85(5–6), pp. 320–329. [CrossRef]
Adduri, P. R., and Penmetsa, R. C., 2008, “Confidence Bounds on Component Reliability in the Presence of Mixed Uncertain Variables,” Int. J. Mech. Sci., 50(3), pp. 481–489. [CrossRef]
Elishakoff, I., and Ferracuti, B., 2006, “Four Alternative Definitions of the Fuzzy Safety Factor,” ASCE J. Aerosp. Eng., 19(4), pp. 281–287. [CrossRef]
Elishakoff, I., and Ferracuti, B., 2006, “Fuzzy Sets Based Interpretation of the Safety Factor,” Fuzzy Sets and Systems, 157, pp. 2495–2512. [CrossRef]
Balu, A. S., and Rao, B. N., 2012, “Multicut-High Dimensional Model Representation for Structural Reliability Bounds Estimation Under Mixed Uncertainties,” Comput. Aided Civil Infrastruct. Eng., 27, pp. 419–438. [CrossRef]
Alis, O. F., and Rabitz, H., 2001, “Efficient Implementation of High Dimensional Model Representations,” J. Math. Chem., 29(2), pp. 127–142. [CrossRef]
Li, G., Wang, S. W., Rabitz, H., Wang, S., and Jaffé, P., 2002, “Global Uncertainty Assessments by High Dimensional Model Representations (HDMR),” Chem. Eng. Sci., 57(21), pp. 4445–4460. [CrossRef]
Rabitz, H., Alis, O. F., Shorter, J., and Shim, K., 1999, “Efficient Input-Output Model Representations,” Comput. Phys. Commun., 117(1–2), pp. 11–20. [CrossRef]
Sobol, I. M., 2003, “Theorems and Examples on High Dimensional Model Representations,” Reliab. Eng. Syst. Saf., 79(2), pp. 187–193. [CrossRef]
Wang, S. W., LevyII, H., Li, G., and Rabitz, H., 1999, “Fully Equivalent Operational Models for Atmospheric Chemical Kinetics Within Global Chemistry-Transport Models,” J. Geophy. Res., 104(D23), pp. 30417–30426. [CrossRef]
Chowdhury, R., Rao, B. N., and Prasad, A. M., 2009, "High Dimensional Model Representation for Structural Reliability Analysis," Commun. Numer. Meth. Eng., 25, pp. 301–337. [CrossRef]
Lin, Y. K., 1967, Probabilistic Theory of Structural Dynamics, Krieger, Florida.
Arora, J. S., 2004, Introduction to Optimum Design, Elsevier Academic Press, San Diego.
Lim, O. K., and Arora, J. S., 1986, “An Active Set RQP Algorithm for Engineering Design Optimization,” Comput. Meth. Appl. Mech. Eng., 57(1), pp. 51–65. [CrossRef]


Grahic Jump Location
Fig. 1

Response bounds using transformation techniques

Grahic Jump Location
Fig. 2

Flowchart of coupled HDMR–FFT based reliability analysis

Grahic Jump Location
Fig. 3

Sampling scheme for first-order HDMR: (a) for a function having one variable (x); (b) for a function having two variables (x1 and x2)

Grahic Jump Location
Fig. 4

Membership function of reliability for hypothetical limit state function

Grahic Jump Location
Fig. 6

Membership function of reliability for cantilever beam

Grahic Jump Location
Fig. 8

Membership function of reliability for ten-bar truss

Grahic Jump Location
Fig. 9

Edge-cracked plate: (a) Geometry, loads, and domain size (b) FEM discretization

Grahic Jump Location
Fig. 10

Membership function of reliability for edge-cracked plate




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