Applied Mathematics and Mechanics

, Volume 38, Issue 1, pp 57–72

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

• Tianxiao Zhang
Article

## 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.

## Keywords

structural system reliability analysis state function numerical characteristic matrix description Kronecker product functional differential reliability sensitivity

O213.2

45A55

## References

1. [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. [2]
Parkinson, D. B. First-order reliability analysis employing translation systems. Engineering Structures, 1, 31–40 (1978)
3. [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. [4]
Hohenbichler, M. and Rackwitz, R. First-order concepts in system reliability. Structural Safety, 1, 177–188 (1983)
5. [5]
Der Kiureghian, A. and Liu, P. L. Structural reliability under incomplete probability information.. ASCE Journal of Engineering Mechanics, 112, 85–104 (1986)
6. [6]
Der Kiureghian, A. and de Stefano, M. Efficient algorithm for second-order reliability analysis.. ASCE Journal of Engineering Mechanics, 117, 2904–2923 (1991)
7. [7]
Zhang, Y. M., Chen, S. H., Liu, Q. L., and Liu, T. Q. Stochastic perturbation finite elements.. Computers & Structures, 59, 425–429 (1996)
8. [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)
9. [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)
10. [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)
11. [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)
12. [12]
Montgomery, D. C. Design and Analysis of Experimens, Wiley, New York (2005)Google Scholar
13. [13]
Hurtado, J. E. Structural Reliability: Statistical Learning Perspectives, Springer, Berlin (2004)
14. [14]
Faravelli, L. Response surface approach for reliability analysis.. ASCE Journal of Engineering Mechanics, 115, 2763–2781 (1989)
15. [15]
Bucher, C. G. and Bourgund, U. A fast and efficient response surface approach for structural reliability problems.. Structural Safety, 7, 57–66 (1990)
16. [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)
17. [17]
Kim, S. H. and Na, S. W. Response surface method using vector projected sampling points.. Structural Safety, 19, 3–19 (1997)
18. [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)
19. [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)
20. [20]
Guan, X. L. and Melchers, R. E. Effect of response surface parameter variation on structural reliability estimates.. Structural Safety, 23, 429–444 (2001)
21. [21]
Der Kiureghian, A. and Ke, J. B. The stochastic finite element method in structural reliability.. Probabilistic Engineering Mechanics, 3, 83–91 (1988)
22. [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)
23. [23]
Haldar, A. and Mahadevan, S. Reliability Assessment Using Stochastic Finite Element Analysis, John Wiley & Sons, Inc., New York (2000)Google Scholar
24. [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)
25. [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)
26. [26]
Au, S. K. Reliability-based design sensitivity by efficient simulation. Computers & Structures, 83, 1048–1061 (2005)
27. [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)
28. [28]
Sudret, B. Global sensitivity analysis using polynomial chaos expansion.. Reliability Engineering and System Safety, 93, 964–979 (2007)
29. [29]
Ibrahim, R. A. Structural dynamics with parameter uncertainties.. Applied Mechanics Reviews, 40, 309–328 (1987)
30. [30]
Benaroya, H. and Rehak, M. Finite element methods in probabilistic structural analysis: a selective review.. Applied Mechanics Reviews, 41, 201–213 (1988)
31. [31]
Hohenbichler, M. and Rackwitz, R. Sensitivity and importance measures in structural reliability.. Civil Engneering Systems, 3, 203–209 (1986)
32. [32]
Bjerager, P. and Krenk, S. Parametric sensitivity in first-order reliability analysis.. ASCE Journal of Engineering Mechanics, 115, 1577–1582 (1989)
33. [33]
Sues, R. H. and Cesare, M. A. System reliability and sensitivity factors via the MPPSS method.. Probabilistic Engineering Mechanics, 20, 148–157 (2005)
34. [34]
Wu, Y. T. Computational methods for efficient structural reliability and reliability sensitivity analysis.. AIAA Journal, 32, 1717–1723 (1994)
35. [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)
36. [36]
Du, X. P. and Hu, Z. First-order reliability method with truncated random variables. Journal of Mechanical Design, 134, 091005 (2012)
37. [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)
38. [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)
39. [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)
40. [40]
Zhao, Y. G. and Ono, T. Moment methods for structural reliability.. Structural Safety, 23, 47–75 (2001)
41. [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. [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)