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Matrix description of differential relations of moment functions in structural reliability sensitivity analysis

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Abstract

In a structural system reliability analysis that lacks probabilistic information, calculating the numerical characteristics of the state functions, especially the first four moments of the state functions, is necessary. Based on that, the structural system reliability is analyzed with a fourth-order moment method. The reliability sensitivity is required to conduct the differential operation of the numerical characteristic functions. A reliability sensitivity analysis formula is then derived in combination with the relation of the differential operation. Based on the matrix theory and Kronecker algebra, this paper systematically derives a matrix expression of the first four moments of the state functions, and establishes the matrix relation between the first four moments of the state functions and those of the basic random variables. On this basis, a differential operation formula of the first four moments of the state functions is further derived against the first four moments of the basic random variables. The vector relation between the state functions and the multidimensional basic random variables is described by means of the matrix operation to extend the operation method. Finally, a concise and intuitive formula is obtained to explore the inherent essential relation between the numerical characteristics of the state functions and those of the basic random variables, leading to a universal equation for the two kinds of numerical characteristics.

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References

  1. Hasofer, A. M. and Lind, N. C. Exact and invariant second moment code format.. ASCE Journal of the Engineering Mechanics Division, 100, 111–121 (1974)

    Google Scholar 

  2. Parkinson, D. B. First-order reliability analysis employing translation systems. Engineering Structures, 1, 31–40 (1978)

    Article  Google Scholar 

  3. Hohenbichler, M. and Rackwitz, R. Non-normal dependent vectors in structural swafety. ASCE Journal of the Engineering Mechanics Division, 107, 1227–1249 (1981)

    Google Scholar 

  4. Hohenbichler, M. and Rackwitz, R. First-order concepts in system reliability. Structural Safety, 1, 177–188 (1983)

    Article  Google Scholar 

  5. Der Kiureghian, A. and Liu, P. L. Structural reliability under incomplete probability information.. ASCE Journal of Engineering Mechanics, 112, 85–104 (1986)

    Article  Google Scholar 

  6. Der Kiureghian, A. and de Stefano, M. Efficient algorithm for second-order reliability analysis.. ASCE Journal of Engineering Mechanics, 117, 2904–2923 (1991)

    Article  Google Scholar 

  7. Zhang, Y. M., Chen, S. H., Liu, Q. L., and Liu, T. Q. Stochastic perturbation finite elements.. Computers & Structures, 59, 425–429 (1996)

    Article  MATH  Google Scholar 

  8. Zhang, Y. M., Wen, B. C., and Chen, S. H. PFEM formalism in Kronecker notation.. Mathematics and Mechanics of Solids, 1, 445–461 (1996)

    Article  MATH  Google Scholar 

  9. Zhang, Y. M., Wen, B. C., and Liu, Q. L. First passage of uncertain single degree-of-freedom nonlinear oscillators.. Computer Methods in Applied Mechanics and Engineering, 165, 223–231 (1998)

    Article  MATH  Google Scholar 

  10. Zhang, Y. M., Liu, Q. L., and Wen, B. C. Quasi-failure analysis on resonant demolition of random structural systems. AIAA Journal, 40, 585–586 (2002)

    Article  Google Scholar 

  11. Zhang, Y. M. and Liu, Q. L. Reliability-based design of automobile components. Proceedings of the Institution of Mechanical Engineers, Part D, Journal of Automobile Engineering, 216, 455–471 (2002)

    Article  Google Scholar 

  12. Montgomery, D. C. Design and Analysis of Experimens, Wiley, New York (2005)

    Google Scholar 

  13. Hurtado, J. E. Structural Reliability: Statistical Learning Perspectives, Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  14. Faravelli, L. Response surface approach for reliability analysis.. ASCE Journal of Engineering Mechanics, 115, 2763–2781 (1989)

    Article  Google Scholar 

  15. Bucher, C. G. and Bourgund, U. A fast and efficient response surface approach for structural reliability problems.. Structural Safety, 7, 57–66 (1990)

    Article  Google Scholar 

  16. Rajashekhar, M. R. and Ellingwood, B. R. A new look at the response surface approach for reliability analysis.. Structural Safety, 12, 205–220 (1993)

    Article  Google Scholar 

  17. Kim, S. H. and Na, S. W. Response surface method using vector projected sampling points.. Structural Safety, 19, 3–19 (1997)

    Article  Google Scholar 

  18. Zheng, Y. and Das, P. K. Improved response surface method and its application to stiffened plate reliability analysis.. Engineering Structures, 22, 544–551 (2000)

    Article  Google Scholar 

  19. Das, P. K. and Zheng, Y. Cumulative formation of response surface and its use in reliability analysis.. Probabilistic Engineering Mechanics, 15, 309–315 (2000)

    Article  Google Scholar 

  20. Guan, X. L. and Melchers, R. E. Effect of response surface parameter variation on structural reliability estimates.. Structural Safety, 23, 429–444 (2001)

    Article  Google Scholar 

  21. Der Kiureghian, A. and Ke, J. B. The stochastic finite element method in structural reliability.. Probabilistic Engineering Mechanics, 3, 83–91 (1988)

    Article  Google Scholar 

  22. Ghanem, R. G. and Spanos, P. D. Spectral stochastic finite-element formulation for reliability analysis.. ASCE Journal of Engineering Mechanics, 117, 2351–2372 (1991)

    Article  Google Scholar 

  23. Haldar, A. and Mahadevan, S. Reliability Assessment Using Stochastic Finite Element Analysis, John Wiley & Sons, Inc., New York (2000)

    Google Scholar 

  24. Hurtado, J. E. and Alvarez, D. A. Classification approach for reliability analysis with stochastic finite-element modeling.. ASCE Journal of Structural Engineering, 129, 1141–1149 (2003)

    Article  Google Scholar 

  25. Vanmarcke, E., Shinozuka, M., Nakagiri, S., Schuëller, G. I., and Grigoriu, M. Random fields and stochastic finite elements.. Structural Safety, 3, 143–166 (1986)

    Article  Google Scholar 

  26. Au, S. K. Reliability-based design sensitivity by efficient simulation. Computers & Structures, 83, 1048–1061 (2005)

    Article  Google Scholar 

  27. Haukaas, T. and der Kiureghian, A. Parameter sensitivity and importance measures in nonlinear finite element reliability analysis. ASCE Journal of Engineering Mechanics, 131, 1013–1026 (2005)

    Article  Google Scholar 

  28. Sudret, B. Global sensitivity analysis using polynomial chaos expansion.. Reliability Engineering and System Safety, 93, 964–979 (2007)

    Article  Google Scholar 

  29. Ibrahim, R. A. Structural dynamics with parameter uncertainties.. Applied Mechanics Reviews, 40, 309–328 (1987)

    Article  Google Scholar 

  30. Benaroya, H. and Rehak, M. Finite element methods in probabilistic structural analysis: a selective review.. Applied Mechanics Reviews, 41, 201–213 (1988)

    Article  MATH  Google Scholar 

  31. Hohenbichler, M. and Rackwitz, R. Sensitivity and importance measures in structural reliability.. Civil Engneering Systems, 3, 203–209 (1986)

    Article  Google Scholar 

  32. Bjerager, P. and Krenk, S. Parametric sensitivity in first-order reliability analysis.. ASCE Journal of Engineering Mechanics, 115, 1577–1582 (1989)

    Article  Google Scholar 

  33. Sues, R. H. and Cesare, M. A. System reliability and sensitivity factors via the MPPSS method.. Probabilistic Engineering Mechanics, 20, 148–157 (2005)

    Article  Google Scholar 

  34. Wu, Y. T. Computational methods for efficient structural reliability and reliability sensitivity analysis.. AIAA Journal, 32, 1717–1723 (1994)

    Article  MATH  Google Scholar 

  35. Alibrandi, U. and Koh, C. G. First-order reliability method for structural reliability analysis in the presence of random and interval variables. ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering, 1, 041006 (2015)

    Article  Google Scholar 

  36. Du, X. P. and Hu, Z. First-order reliability method with truncated random variables. Journal of Mechanical Design, 134, 091005 (2012)

    Article  Google Scholar 

  37. Yao, W., Chen, X. Q., Huang, Y. Y., and Tooren, M. An enhanced unified uncertainty analysis approach based on first-order reliability method with single-level optimization.. Reliability Engineering and System Safety, 116, 28–37 (2013)

    Article  Google Scholar 

  38. Kang, W. H., Lee, Y. J., Song, J., and Gencturk, B. Further development of matrix-based system reliability method and applications to structural systems.. Structure and Infrastructure Engineering, 8, 441–457 (2012)

    Article  Google Scholar 

  39. Guo, S. X. and Lu, Z. Z. A non-probabilistic robust reliability method for analysis and design optimization of structures with uncertain-but-bounded parameters.. Applied Mathematical Modelling, 39, 1985–2002 (2015)

    Article  MathSciNet  Google Scholar 

  40. Zhao, Y. G. and Ono, T. Moment methods for structural reliability.. Structural Safety, 23, 47–75 (2001)

    Article  Google Scholar 

  41. Zhang, T. X. The Reliability Design and Reliability Sensitivity Analysis of Hydraulic Component (in Chinese), Ph.D. dissertation, Jilin Unversity, Jilin (2014)

    Google Scholar 

  42. Wu, Y. T. and Mohanty, S. Variable screening and ranking using sampling-based sensitivity measures.. Reliability Engineering and System Safety, 91, 634–647 (2005)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Illinois, 60607-7022, USA

    Tianxiao Zhang

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  1. Tianxiao Zhang
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Correspondence to Tianxiao Zhang.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 51135003 and U1234208) and the Major State Basic Research Development Program of China (973 Program) (No. 2014CB046303)

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Zhang, T. Matrix description of differential relations of moment functions in structural reliability sensitivity analysis. Appl. Math. Mech.-Engl. Ed. 38, 57–72 (2017). https://doi.org/10.1007/s10483-017-2160-6

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  • DOI: https://doi.org/10.1007/s10483-017-2160-6

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

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