Skip to main content
SpringerLink
Log in

Strength modeling using Weibull distributions

  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

The two-parameter Weibull distribution is the most popular model for material strength. However, it may not be a good model for all materials over a wide range of sizes. In this note, a comprehensive review of the known variations of the two-parameter Weibull distribution is provided to help providing better modeling. Over 20 variations are reviewed. The appropriateness of the variations is discussed as models for brittle versus ductile strength. A comparison study of a selection of the variations is also provided. It is hoped that this review will also serve as an important reference and encourage developments of further variations of the two-parameter Weibull distribution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. C. Cohen, The reflected Weibull distribution. Technometrics, 15 (1973) 867–873.

    Article  MATH  MathSciNet  Google Scholar 

  2. N. Balakrishnan and S. Kocherlakota, On the double Weibull distribution: order statistics and estimation. Sankhyā, 47 (1985) 161–176.

    MATH  MathSciNet  Google Scholar 

  3. J. S. White, The moments of log-Weibull order statistics. Technometrics, 11 (1969) 373–386.

    Article  MATH  MathSciNet  Google Scholar 

  4. A. Drapella, Complementary Weibull distribution: unknown or just forgotton. Quality and Reliability Engineering International, 9 (1993) 383–385.

    Article  Google Scholar 

  5. G. S. Mudholkar and G. D. Kollia, Generalized Weibull family: a structural analysis. Communications in Statistics—Theory and Methods, 23 (1994) 1149–1171.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. P. McEwen and B. R. Parresol, Moment expressions and summary statistics for the complete and truncated Weibull distribution. Communications in Statistics—Theory and Methods, 20 (1991) 1361–1372.

    Article  Google Scholar 

  7. A. W. Marshall and I. Olkin, A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84 (1997) 641–652.

    Article  MATH  MathSciNet  Google Scholar 

  8. V. G. Voda, New models in durability tool-testing: pseudo-Weibull distribution. Kybernetika, 25 (1989) 209–215.

    MATH  MathSciNet  Google Scholar 

  9. E. W. Stacy, A generalization of gamma distribution. Annals of Mathematical Statistics, 33 (1962) 1187–1192.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. E. Ghitany, On a recent generalization of a gamma distribution. Communications in Statistics—Theory and Methods, 27 (1998) 223–233.

    Article  MATH  MathSciNet  Google Scholar 

  11. G. S. Mudholkar, D. K. Srivastava and M. Freimer, The exponentiated Weibull family. Technometrics, 37 (1995) 436–445.

    Article  MATH  Google Scholar 

  12. G. S. Mudholkar and A. D. Huston, The exponentiated Weibull family: some properties and a flood data application. Communications in Statistics—Theory and Methods, 25 (1996) 3059–3083.

    Article  MATH  MathSciNet  Google Scholar 

  13. M. M. Nassar and F. H. Eissa, On the exponentiated Weibull distribution. Communications in Statistics—Theory and Methods, 32 (2003) 1317–1336.

    Article  MATH  MathSciNet  Google Scholar 

  14. S. Nadarajah and A. K. Gupta, On the moments of the exponentiatedWeibull distribution. Communications in Statistics—Theory and Methods, 34 (2005) 253–256.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Xie, Y. Tang and T. N. Goh, A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering & System Safety, 76 (2002) 279–285.

    Article  Google Scholar 

  16. Z. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statistics and Probability Letters, 49 (2000) 155–161.

    Article  MATH  MathSciNet  Google Scholar 

  17. Y. Tang, M. Xie and T. N. Goh, Statistical analysis of a Weibull extension model. Communications In Statistics—Theory and Methods, 32 (2003) 913–928.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. W. Wu, H. L. Lu, C. H. Chen, and C. H. Wu, Statistical inference about the shape parameter of the new two-parameter bathtub-shaped lifetime distribution. Quality and Reliability Engineering International, 20 (2004) 607–616.

    Article  MATH  Google Scholar 

  19. S. Nadarajah, On the moments of the modified—Weibull distribution. Reliability Engineering and System Safety, 90 (2005) 114–117.

    Article  Google Scholar 

  20. C. D. Lai, M. Xie and D. N. P. Murthy, Modified Weibull model. IEEE Transactions on Reliability, 52 (2003) 33–37.

    Article  Google Scholar 

  21. G. S. Mudholkar and D. K. Srivastava, Exponentiated Weibull family analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 42 (1993) 299–302.

    Article  MATH  Google Scholar 

  22. K. K. Phani, A new modifed Weibull distribution. Communications of the American Ceramic Society, 70 (1987) 182–184.

    Google Scholar 

  23. J. A. Kies, The strength of glass. Naval Research Lab Report No 5093, Washington DC, (1958).

  24. J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, New Jersey, (1983).

    MATH  Google Scholar 

  25. R. B. Schnabel, J. E. Koontz and B. E. Weiss, A modular system of algorithms for unconstrained minimization. ACM Transactions on Mathematical Software, 11 (1985) 419–440.

    Article  MATH  MathSciNet  Google Scholar 

  26. R. Ihaka and R. Gentleman, R: A language for data analysis and graphics. Journal of Computational and Graphical Statistics, 5 (1996) 299–314.

    Article  Google Scholar 

  27. G. Blom, Statistical Estimates and Transformed Beta-variables. John Wiley and Sons, New York, (1958).

    MATH  Google Scholar 

  28. J. Chambers, W. Cleveland, B. Kleiner and P. Tukey, Graphical Methods for Data Analysis. Chapman and Hall, (1983).

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Manchester, Manchester, M60 1QD, UK

    Saralees Nadarajah

  2. Department of Engineering Management and Systems Engineering, The George Washington University, Washington DC, 20052, USA

    Samuel Kotz

Authors
  1. Saralees Nadarajah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Samuel Kotz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saralees Nadarajah.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nadarajah, S., Kotz, S. Strength modeling using Weibull distributions. J Mech Sci Technol 22, 1247–1254 (2008). https://doi.org/10.1007/s12206-008-0426-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-008-0426-5

Keywords

Search

Navigation