Research Papers: Design Automation

Efficient Reliability Assessment With the Conditional Probability Method

[+] 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

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received November 8, 2017; final manuscript received April 18, 2018; published online June 1, 2018. Assoc. Editor: Samy Missoum.

J. Mech. Des 140(8), 081402 (Jun 01, 2018) (12 pages) Paper No: MD-17-1747; doi: 10.1115/1.4040170 History: Received November 08, 2017; Revised April 18, 2018

Reliability assessment is an important procedure in engineering design in which the probability of failure or equivalently the probability of survival is computed based on appropriate design criteria and model behavior. In this paper, a new approximate and efficient reliability assessment method is proposed: the conditional probability method (CPM). Focus is set on computational efficiency and the proposed method is applied to classical load-strength structural reliability problems. The core of the approach is in the computation of the probability of failure starting from the conditional probability of failure given the load. The number of function evaluations to compute the probability of failure is a priori known to be 3n + 2 in CPM, where n is the number of stochastic design variables excluding the strength. The necessary number of function evaluations for the reliability assessment, which may correspond to expensive computations, is therefore substantially lower in CPM than in the existing structural reliability methods such as the widely used first-order reliability method (FORM).

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Lemaire, M. , 2009, Structural Reliability, Wiley, Hoboken, NJ. [CrossRef]
Madsen, H. O. , Krenk, S. , and Lind, N. C. , 1986, Methods of Structural Safety, Prentice Hall, Englewood Cliffs, NJ.
Balu, A. S. , and Rao, B. N. , 2014, “ Efficient Assessment of Structural Reliability in Presence of Random and Fuzzy Uncertainties,” ASME J. Mech. Des., 136(5), p. 051008. [CrossRef]
Du, X. , and Hu, Z. , 2012, “ First Order Reliability Method With Truncated Random Variables,” ASME J. Mech. Des., 134(9), p. 091005. [CrossRef]
Zhao, Y. , and Ono, T. , 1999, “ New Approximations for SORM—Part 1,” J. Eng. Mech., 125(1), pp. 79–85. [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]
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]
Dersjo, T. , and Olsson, M. , 2011, “ Reliability Based Design Optimization Using a Single Constraint Approximation Point,” ASME J. Mech. Des., 133(3), p. 031006. [CrossRef]
Aoues, Y. , and Chateauneuf, A. , 2010, “ Benchmark Study of Numerical Methods for Reliability-Based Design Optimization,” Struct. Multidiscip. Optim., 41(2), pp. 277–294. [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,” A/S Veritas Research, Hovik, Norway, Report No. RDIV/20-004-83.
Cai, G. Q. , and Elishakoff, I. , 1994, “ Refined Second-Order Reliability Analysis,” Struct. Saf., 14(4), 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]
Hohenbichler, M. , and Rackwitz, R. , 1981, “ Non-Normal Dependent Vectors in Structural Safety,” J. Eng. Mech. Div., 107(6), pp. 1127–1138.
Mansour, R. , and Olsson, M. , 2014, “ A Closed-Form Second-Order Reliability Method Using Noncentral Chi-Squared Distributions,” ASME J. Mech. Des., 136(10), p. 101402. [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]
Mansour, R. , and Olsson, M. , 2016, “ Response Surface Single Loop Reliability-Based Design Optimization With Higher Order Reliability Assessment,” Struct. Multidiscip. Optim., 54(1), pp. 63–79. [CrossRef]
Hu, Z. , and Du, X. , 2018, “ Saddlepoint Approximation Reliability Method for Quadratic Functions in Normal Variables,” Struct. Saf., 71, pp. 24–32. [CrossRef]
Papaioannou, I. , Papadimitriou, C. , and Straub, D. , 2016, “ Sequential Importance Sampling for Structural Reliability Analysis,” Struct. Saf., 62, pp. 66–75. [CrossRef]
Sundar, V. S. , and Shields, M. D. , 2016, “ Surrogate-Enhanced Stochastic Search Algorithms to Identify Implicitly Defined Functions for Reliability Analysis,” Struct. Saf., 62, pp. 1–11. [CrossRef]
Au, S. K. , Ching, J. , and Beck, J. L. , 2007, “ Application of Subset Simulation Methods to Reliability Benchmark Problems,” Struct. Saf., 29(3), pp. 183–193. [CrossRef]
Meng, D. , Li, Y. L. , Huang, H.-Z. , Wang, Z. , and Liu, Y. , 2015, “ Reliability-Based Multidisciplinary Design Optimization Using Subset Simulation Analysis and Its Application in the Hydraulic Transmission Mechanism Design,” ASME J. Mech. Des., 137(5), p. 051402. [CrossRef]
Holtz, M. , 2011, Sparse Grid Quadrature in High Dimensions With Applications in Finance and Insurance, Springer-Verlag, Berlin. [CrossRef]
Eldred, M. S. , and Burkardt, J. , 2009, “ Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification,” AIAA Paper No. 2009-976.
Kameshwar, S. , and Chakraborty, A. , 2013, “ On Reliability Evaluation of Structures Using Hermite Polynomial Chaos,” International Symposium on Engineering Under Uncertainty: Safety Assessment and Management, pp. 1141–1152.
Rosenblatt, M. , 1952, “ Remarks on a Multivariate Transformation,” Ann. Math. Stat., 23(3), pp. 470–472. [CrossRef]
Hasofer, A. M. , and Lind, N. C. , 1974, “ Exact and Invariant Second Moment Code Format,” J. Eng. Mech. Div., 100(1), pp. 111–121.
Wang, L. , and Grandhi, R. V. , 1995, “ Structural Reliability Optimization Using an Efficient Safety Index Calculation Procedure,” Int. J. Numer. Methods Eng., 38(10), pp. 1721–1738. [CrossRef]
Wang, L. , and Grandhi, R. V. , 1995, “ Improved Two-Point Function Approximations for Design Optimization,” AIAA J., 33(9), pp. 1720–1727. [CrossRef]
Wu, Y.-T. , and Wirsching, P. H. , 1987, “ New Algorithm for Structural Reliability Estimation,” J. Eng. Mech., 9(9), pp. 1319–1336. [CrossRef]
Wirsching, P. H. , Torng, T. Y. , and Martin, W. , 1991, “ Advanced Fatigue Reliability Analysis,” Int. J. Fatigue, 13(5), pp. 389–394. [CrossRef]
Sacks, J. , Welch, W. J. , J, M. T. , and P, W. H. , 1989, “ Design and Analysis of Computer Experiments. statistical Science,” Stat. Sci., 4, pp. 409–435. [CrossRef]
Wu, Y.-T. , and Wirsching, P. H. , 1992, “ Kriging, Cokriging, Radial Basis Functions and the Role of Positive Definiteness,” Comput. Math. Appl., 24(12), pp. 139–148. [CrossRef]
Wild, S. , Regis, R. , and Shoemaker, C. , 2008, “ Orbit: Optimization by Radial Basis Function Interpolation in Trust-Regions,” SIAM J. Sci. Comput., 30(6), pp. 3197–3219. [CrossRef]
Amouzgar, K. , and Stromberg, N. , 2017, “ Radial Basis Functions as Surrogate Models With a Priori Bias in Comparison With a Posteriori Bias,” Struct. Multidiscip. Optim., 55(4), pp. 1453–1469. [CrossRef]
Choi, S.-K. , Grandhi, R. , and Canfield, R. , 2007, Reliability-Based Structural Design, Springer-Verlag, London.
Valdebenito, M. A. , and Schuëller, G. I. , 2010, “ A Survey on Approaches for Reliability-Based Optimization,” Struct. Multidiscip. Optim., 42(5), pp. 645–663. [CrossRef]
Du, X. , 2005, “ Probabilistic Engineering Design,” Missouri University of Science and Technology, Rolla, MO.
Hurtado, J. E. , Alvarez, D. A. , and Paredes, J. A. , 2017, “ Interval Reliability Analysis Under the Specification of Statistical Information on the Input Variables,” Struct. Saf., 65, pp. 35–48. [CrossRef]
Wormsen, A. , and Haerkegaard, G. , 2004, “ A Statistical Investigation of Fatigue Behaviour According to Weibull's Weakest Link Theory,” European Conference on Fracture (ECF15), Stockholm, Sweden, Aug. 11–13. http://www.gruppofrattura.it/ocs/index.php/esis/ECF15/paper/viewFile/8634/4706
Sandberg, D. , Mansour, R. , and Olsson, M. , 2017, “ Fatigue Probability Assessment Including Aleatory and Epistemic Uncertainty With Application to Gas Turbine Compressor Blades,” Int. J. Fatigue, 95, pp. 132–142. [CrossRef]
Faber, M. H. , 2005, “ On the Treatment of Uncertainties and Probabilities in Engineering Decision Analysis,” ASME J. Offshore Mech. Arct. Eng., 127(3), pp. 243–248. [CrossRef]
Kiureghian, A. D. , 1989, “ Measures of Structural Safety Under Imperfect States of Knowledge,” J. Struct. Eng., 115(5), pp. 1119–1140. [CrossRef]
Kostov, M. , 2000, “ Seismic Fragility Analyses (Case Study),” Regional Workshop on External Events PSA, Sofia, Bulgaria, Nov. 6–10. http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/34/014/34014809.pdf


Grahic Jump Location
Fig. 1

Presentation of the main steps in the proposed CPM and the FORM

Grahic Jump Location
Fig. 2

An example of a (a) highly nonlinear and (c) linear conditional probability of failure Pf|u(u) with constant gradient direction; and corresponding integrand Pf|u(u)fu(u) for the (b) nonlinear and (d) linear case

Grahic Jump Location
Fig. 3

Illustration of the CPM: (a) linearize the load, Lu, at u = 0, (b) find umax,lin according to Eq. (29), (c) interpolate a polyharmonic surrogate model Lupolyh(u) of the form given in Eq. (31), and (d) compute probability of failure using Eq. (35)

Grahic Jump Location
Fig. 4

Illustration of the steps used to compute the total probability of failure using CPM for the problem defined in Eq. (36)

Grahic Jump Location
Fig. 5

True load and conditional probability of failure for the problem defined in Eq. (36)

Grahic Jump Location
Fig. 6

(a) CDF computed using MV and AMV and (b) performance function evaluated at the nominal value of the strength

Grahic Jump Location
Fig. 8

(a) Polyharmonic spline approximation of the load, (b) conditional probability of failure computed using the polyharmonic approximation of the load, (c) true load, and (d) true conditional probability for the problem defined in Eq. (44)

Grahic Jump Location
Fig. 9

A shaft subject to torsion and flexure: (a) structure and (b) cross section

Grahic Jump Location
Fig. 10

(a) Polyharmonic spline approximation of the load, (b) conditional probability of failure computed using the polyharmonic approximation of the load, (c) true load, and (d) true conditional probability for the problem defined in Eq. (45)

Grahic Jump Location
Fig. 12

(a) Generalized reliability index computed using different mean values of the strength for bar 5 and bar 7, (b) relative error in probability of failure, and (c) number of function evaluations

Grahic Jump Location
Fig. 13

A box plot of the absolute relative error of the generalized reliability index for the problems studied in this paper




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