Research Papers

Imprecise Probabilities in Fatigue Reliability Assessment of Hydraulic Turbines

[+] Author and Article Information
Mounia Berdai

École de Technologie Supérieure, 1100, Rue Notre-Dame Ouest, Montréal, Québec H3C 1K3, Canada e-mail: mounia.berdai.1@ens.etsmtl.ca

Antoine Tahan

École de Technologie Supérieure,
1100, rue Notre-Dame Ouest, Montréal, Québec H3C 1K3, Canada e-mail: antoine.tahan@etsmtl.ca

Martin Gagnon

1800 Boulevard Lionel-Boulet, Varennes, Québec J3X 1S1, Canada e-mail: gagnon.martin@ireq.ca

1Corresponding author.

Manuscript received May 16, 2016; final manuscript received September 4, 2016; published online November 21, 2016. Assoc. Editor: Konstantin Zuev.

ASME J. Risk Uncertainty Part B 3(1), 011006 (Nov 21, 2016) (8 pages) Paper No: RISK-16-1083; doi: 10.1115/1.4034690 History: Received May 16, 2016; Accepted September 04, 2016

Risk analyses are often performed for economic reasons and safety purposes. In some cases, these studies are biased by epistemic uncertainties due to the lack of information and knowledge, which justifies the need for expert opinion. In such cases, experts can follow different approaches for the elicitation of epistemic data, using probabilistic or imprecise theories. But how do these theories affect the reliability calculation? What are the influences of using a mixture of theories in a multivariable system with a nonexplicit limit model? To answer these questions, we propose an approach for the comparison of these theories, which was performed based on a reliability model using the first-order reliability method (FORM) approach and having the Kitagawa–Takahashi diagram as limit state. We also propose an approach, appropriate to this model, to extend the reliability calculation to variables derived from imprecise probabilities. For the chosen reliability model, obtained results show that there is a certain homogeneity among the considered theories. The study also concludes that priority should be given to expert opinions formulated according to unbounded distributions, in order to achieve better reliability calculation accuracy.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Gagnon, M., Tahan, A., Bocher, P., and Thibault, D., 2013, “A Probabilistic Model for the Onset of High Cycle Fatigue (HCF) Crack Propagation: Application to Hydroelectric Turbine Runner,” Int. J. Fatigue, 47, pp. 300–307. 10.1016/j.ijfatigue.2012.09.011
Thibault, D., Gagnon, M., and Godin, S., 2015, “The Effect of Materials Properties on the Reliability of Hydraulic Turbine Runners,” Int. J. Fluid Mach. Syst., 8(4), pp. 254–263. 10.5293/IJFMS.2015.8.4.254
Aven, T., Baraldi, P., Flage, R., and Zio, E., 2014, Uncertainty in Risk Assessment: The Representation and Treatment of Uncertainties by Probabilistic and Non-Probabilistic Methods, Wiley, Hoboken, NJ.
Ayyub, B. M., 2001, Elicitation of Expert Opinions for Uncertainty and Risks, CRC Press, Boca Raton, FL.
O’Hagan, A., 2012, “Probabilistic Uncertainty Specification: Overview, Elaboration Techniques and Their Application to a Mechanistic Model of Carbon Flux,” Environ. Modell. Software, 36, pp. 35–48. 10.1016/j.envsoft.2011.03.003
Möller, B., Graf, W., Beer, M., and Sickert, J., 2001, “Fuzzy Probabilistic Method and Its Application for the Safety Assessment of Structures,” Proceedings of the European Conference on Computational Mechanics (EECM-2001), Kraków, Poland.
Limbourg, P., and De Rocquigny, E., 2010, “Uncertainty Analysis Using Evidence Theory: Confronting Level-1 and Level-2 Approaches With Data Availability and Computational Constraints,” Reliab. Eng. Syst. Saf., 95(5), pp. 550–564. 10.1016/j.ress.2010.01.005
Zadeh, L. A., 1965. “Fuzzy Sets,” Inform. Control, 8(3), pp. 338–353. 0892-3876 10.1016/S0019-9958(65)90241-X
Zadeh, L. A., 1978, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets Syst., 1(1), pp. 3–28. 0165-0114 10.1016/0165-0114(78)90029-5
Dubois, D., and Prade, H., 1998, “Possibility Theory: Qualitative and Quantitative Aspects,” Quantified Representation of Uncertainty and Imprecision, Springer, Berlin, pp. 169–226.
Dubois, D., Prade, H., and Smets, P., 2008, “A Definition of Subjective Possibility,” Int. J. Approx. Reason., 48(2), pp. 352–364. 10.1016/j.ijar.2007.01.005
Dubois, D., Foulloy, L., Mauris, G., and Prade, H., 2004, “Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities,” Reliable Comput., 10(4), pp. 273–297. 10.1023/B:REOM.0000032115.22510.b5
Destercke, S., Dubois, D., and Chojnacki, E., 2006, “Aggregation of Expert Opinions and Uncertainty Theories,” Rencontres Francophones sur la Logique Floue et ses Applications, Cepadues, pp. 295–302.
Klir, G. J., and Smith, R. M., 2001, “On Measuring Uncertainty and Uncertainty-Based Information: Recent Developments,” Ann. Math. Artif. Intell., 32(1–4), pp. 5–33. 10.1023/A:1016784627561
Baraldi, P., Popescu, I. C., and Zio, E., 2010, “Methods of Uncertainty Analysis in Prognostics,” Int. J. Perform. Eng., 6(4), pp. 303–330.
Yager, R. R., and Liu, L., 2008, Classic Works of the Dempster-Shafer Theory of Belief Functions, Springer Science & Business Media, Berlin, p. 219.
Baudrit, C., and Dubois, D., 2005, “Comparing Methods for Joint Objective and Subjective Uncertainty Propagation With an Example in a Risk Assessment,” ISIPTA, Vol. 5.
Beer, M., Ferson, S., and Kreinovich, V., 2013, “Imprecise Probabilities in Engineering Analyses,” Mech. Syst. Sig. Process., 37(1–2), pp. 4–29. 10.1016/j.ymssp.2013.01.024
Flage, R., Baraldi, P., Zio, E., and Aven, T., 2013, “Probability and Possibility-Based Representations of Uncertainty in Fault Tree Analysis,” Risk Anal., 33(1), pp. 121–133. 0272-4332 10.1111/risk.2013.33.issue-1 [PubMed]
Masson, M.-H., 2005, “Apports de la Théorie des Possibilités et des Fonctions de Croyance à l’analyse de Données Imprécises,” Mémoire de Direction de Recherche, p. 126.
Ferson, S., Joslyn, C. A., Helton, J. C., Oberkampf, W. L., and Sentz, K., 2004, “Summary from the Epistemic Uncertainty Workshop: Consensus Amid Diversity,” Reliab. Eng. Syst. Saf., 85(1), pp. 355–369. 10.1016/j.ress.2004.03.023
Beck, A. T., 2016, “Strategies for Finding the Design Point under Bounded Random Variables,” Struct. Saf., 58, pp. 79–93. 10.1016/j.strusafe.2015.08.006
Castillo, E., O’Connor, A. J., Nogal, M., and Calviño, A., 2014, “On the Physical and Probabilistic Consistency of Some Engineering Random Models,” Struct. Saf., 51, pp. 1–12. 10.1016/j.strusafe.2014.05.003
Oussalah, M., 2000, “On the Probability/Possibility Transformations: A Comparative Analysis,” Int. J. General Syst., 29(5), pp. 671–718. 10.1080/03081070008960969
Cobb, B. R., and Shenoy, P. P., 2006, “On the Plausibility Transformation Method for Translating Belief Function Models to Probability Models,” Int. J. Approx. Reason., 41(3), p 314–330. 10.1016/j.ijar.2005.06.008
Lasserre, V., 1999, Modélisation floue des incertitudes de mesures de capteurs, Doctoral dissertation, Chambéry.
Hahn, G. J., and Shapiro, S. S., 1968, Statistical Models in Engineering, Wiley, New York.


Grahic Jump Location
Fig. 1

(a) Reliability model based on Kitagawa–Takahashi diagram and (b) reliability index in the iso-probabilistic space

Grahic Jump Location
Fig. 2

Example of the reliability index distribution, resulting from loading stress following a possibility distribution and defect size formulated according to a triangular distribution

Grahic Jump Location
Fig. 3

Theories used in the case matrix

Grahic Jump Location
Fig. 4

Points from the probabilistic domain used in the matrix case (Table 1). In this figure, defect size and loading stress are expressed with Gumbel distribution

Grahic Jump Location
Fig. 5

Example of determining the mode of distribution#2 (D2) from distribution#1 (D1)

Grahic Jump Location
Fig. 6

(a) Example of a belief function expressed as an evidence distribution and (b) example of a continuous belief function for the example in (a)

Grahic Jump Location
Fig. 7

Reliability indices calculated according to different theories combinations (Table 1) for defect size and loading stress for G0, G01, G02, G03, G04, and G05 (Fig. 8)

Grahic Jump Location
Fig. 8

Distributions used in the case matrix




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
Articles from Part A: Civil Engineering
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