International Journal of Reliability, Risk and Safety: Theory and Application

International Journal of Reliability, Risk and Safety: Theory and Application

Step Direction Computation of the First-Order Reliability Analysis Based on Two-Step Root-Finding Methods

Document Type : Original Research Article

Authors
mehrshad ghorbanzadeh
Peyman Homami
Mohsen Shahruzi
Department of Civil Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran
10.22034/IJRRS.2023.6.1.8
Abstract
The iHLRF algorithm is a popular iterative algorithm for determining the failure probability in structural reliability problems. It belongs to the family of first-order reliability methods (FORM) and is known for its fast convergence and remarkable simplicity. However, in cases where the limit state function oscillates significantly near the design point, which often occurs in high nonlinear limit state functions, the iHLRF algorithm may suffer from convergence issues. To address these convergence issues, this paper proposes three two-step direction determination techniques for first-order analysis. These techniques are based on two-step root-finding methods with a higher convergence rate than existing methods. The proposed techniques aim to improve the accuracy and robustness of the iHLRF algorithm, especially in cases where the limit state function shows highly nonlinear behavior. A numerical example with high nonlinear limit state functions in standard normal space is presented to demonstrate the proposed techniques' efficiency and capability. The performance of each proposed technique is compared with other existing methods, highlighting the advantages and limitations of each approach. Overall, this paper aims to contribute to developing more accurate and reliable methods for determining the reliability index in structural reliability problems, with the potential to be applied in various engineering fields.
Keywords
Reliability analysis method
Two-step root-finding
Step direction
Convergence rate
Nonlinear limit state function
Subjects
  1. Y. Rubinstein and D. P. Kroese, Simulation and the Monte Carlo Method: Third Edition. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2016.
  2. Engelund and R. Rackwitz, “A benchmark study on importance sampling techniques in structural reliability,” Struct. Saf., vol. 12, no. 4, pp. 255–276, Nov. 1993, doi: https://doi.org/10.1016/0167-4730(93)90056-7
  3. E. Melchers, “Importance sampling in structural systems,” Struct. Saf., vol. 6, no. 1, pp. 3–10, Jul. 1989, doi: https://doi.org/10.1016/0167-4730(89)90003-9
  4. G. Bucher, “Adaptive sampling - an iterative fast Monte Carlo procedure,” Struct. Saf., vol. 5, no. 2, pp. 119–126, Jun. 1988, doi: https://doi.org/10.1016/0167-4730(88)90020-3
  5. T. Wu, “Computational methods for efficient structural reliability and reliability sensitivity analysis,” AIAA J., vol. 32, no. 8, pp. 1717–1723, Aug. 1994, doi: https://doi.org/10.2514/3.12164
  6. Maihulla and I. Yusuf, “Reliability Modeling and Performance Evaluation of Sachet Water Production System Using Gumbel- Hougaard Family Copula,” Int. J. Reliab. Risk Saf. Theory Appl., vol. 5, no. 2, pp. 9–18, Dec. 2022, doi: https://doi.org/10.30699/IJRRS.5.2.2
  7. A. Farsi, “Optimal Multi-Level Redundancy Allocation Using a New Modified Bat Algorithm,” Int. J. Reliab. Risk Saf. Theory Appl., vol. 5, no. 2, pp. 41–48, Dec. 2022, doi: https://doi.org/10.30699/IJRRS.5.2.5
  8. Deng, “Direct Quantile Function Estimation Using Information Principles and Its Applications in Reliability Analysis,” Int. J. Reliab. Risk Saf. Theory Appl., vol. 5, no. 2, pp. 63–77, Feb. 2023, doi: https://doi.org/10.30699/IJRRS.5.2.7
  9. ali Karimi, E. Zarei, and R. Hokmabadi, “Analyzing Reliability of CGS Station by Continuous Time Markov Chains (CTMC),” J. Reliab. Risk Saf. Theory Appl., vol. 4, no. 2, pp. 91–96, Dec. 2021, doi: https://doi.org/10.30699/IJRRS.4.2.10
  10. Nadjafi and A. Najafi ARK, “Improving Accuracy in Importance Sampling: An Integrated Approach with Fuzzy-Strata Sampling,” Int. J. Reliab. Risk Saf. Theory Appl., vol. 4, no. 1, pp. 61–67, Jan. 2021, doi: https://doi.org/10.30699/IJRRS.4.1.8
  11. M. Hasofer and N. C. Lind, “Exact and Invariant Second-Moment Code Format.,” ASCE J Eng Mech Div, vol. 100, no. EM1, pp. 111–121, 1974. doi: https://doi.org/10.1061/JMCEA3.0001848
  12. Xiang, B. Pan, and L. Luo, “A most probable point method for probability distribution construction,” Struct. Multidiscip. Optim., vol. 62, no. 5, pp. 2537–2554, Nov. 2020, doi: https://doi.org/10.1007/s00158-020-02623-2
  13. Rackwitz and B. Flessler, “Structural reliability under combined random load sequences,” Comput. Struct., vol. 9, no. 5, pp. 489–494, Nov. 1978, doi: https://doi.org/10.1016/0045-7949(78)90046-9
  14. T. Zhou, Q. Xiao, J. M. Zhou, and H. G. Li, “Improvements of Rackwitz-Fiessler Method for Correlated Structural Reliability Analysis,” Int. J. Comput. Methods, vol. 17, no. 6, p. 1950077, Aug. 2020, doi: https://doi.org/10.1142/S0219876219500774
  15. Yang, D. Zhang, and X. Han, “New efficient and robust method for structural reliability analysis and its application in reliability-based design optimization,” Comput. Methods Appl. Mech. Eng., vol. 366, p. 113018, Jul. 2020, doi:. https://doi.org/10.1016/j.cma.2020.113018
  16. Zhang and A. Kiureghian, “Two Improved Algorithms for Reliability Analysis,” in Reliability and Optimization of Structural Systems, Boston, MA: Springer US, 1995, pp. 297–304. doi: https://doi.org/10.1007/978-0-387-34866-7_32
  17. Gholami, M. A. Kouchakzadeh, and M. A. Farsi, “A Continuum Damage Mechanics Approach for Reliability Analysis of Composite Laminates Containing a Central Circular Hole,” Int. J. Reliab. Risk Saf. Theory Appl., vol. 4, no. 1, pp. 23–30, 2021, doi: https://doi.org/10.30699/IJRRS.4.1.4
  18. Rahman and D. Wei, “A univariate approximation at most probable point for higher-order reliability analysis,” Int. J. Solids Struct., vol. 43, no. 9, pp. 2820–2839, May 2006, doi: https://doi.org/10.1016/j.ijsolstr.2005.05.053
  19. -H. Lu, C.-H. Cai, and Y.-G. Zhao, “Structural Reliability Analysis Including Correlated Random Variables Based on Third-Moment Transformation,” J. Struct. Eng., vol. 143, no. 8, p. 04017067, Aug. 2017, doi: https://doi.org/10.1061/(ASCE)ST.1943-541X.0001801
  20. -G. Zhao and T. Ono, “Third-Moment Standardization for Structural Reliability Analysis,” J. Struct. Eng., vol. 126, no. 6, pp. 724–732, Jun. 2000, doi: https://doi.org/10.1061/(ASCE)0733-9445(2000)126:6(724)
  21. -H. Lu, C.-H. Cai, Y.-G. Zhao, Y. Leng, and Y. Dong, “Normalization of correlated random variables in structural reliability analysis using fourth-moment transformation,” Struct. Saf., vol. 82, p. 101888, Jan. 2020, https://doi.org/10.1016/j.strusafe.2019.101888
  22. L. Liu and A. Der Kiureghian, “Optimization algorithms for structural reliability,” Struct. Saf., vol. 9, no. 3, pp. 161–177, Feb. 1991, doi: https://doi.org/10.1016/0167-4730(91)90041-7
  23. [J. M. Hegseth, E. E. Bachynski, and B. J. Leira, “Effect of environmental modelling and inspection strategy on the optimal design of floating wind turbines,” Eng. Syst. Saf., vol. 214, p. 107706, Oct. 2021, doi: https://doi.org/10.1016/j.ress.2021.107706
  24. Chang, C. Zhou, P. Wei, Y. Zhang, and Z. Yue, “A new non-probabilistic time-dependent reliability model for mechanisms with interval uncertainties,” Reliab. Eng. Syst. Saf., vol. 215, p. 107771, Nov. 2021, doi: https://doi.org/10.1016/j.ress.2021.107771
  25. Razaaly and P. M. Congedo, “Extension of AK-MCS for the efficient computation of very small failure probabilities,” Reliab. Eng. Syst. Saf., vol. 203, p. 107084, Nov. 2020, doi: https://doi.org/10.1016/j.ress.2020.107084
  26. Li, X. Liu, and X. Zhang, “A Few Iterative Methods by Using [1,n]-Order Padé Approximation of Function and the Improvements,” Mathematics, vol. 7, no. 1, p. 55, Jan. 2019, doi: https://doi.org/10.3390/math7010055
  27. Ali, W. Aslam, K. Ali, M. A. Anwar, and A. Nadeem, “New Family of Iterative Methods for Solving Nonlinear Models,” Discret. Dyn. Nat. Soc., vol. 2018, pp. 1–12, 2018, doi: https://doi.org/10.1155/2018/9619680
  28. Wang, Y. Qin, W. Qian, S. Zhang, and X. Fan, “A family of Newton type iterative methods for solving nonlinear equations,” Algorithms, vol. 8, no. 3, pp. 786–798, Sep. 2015, doi: https://doi.org/10.3390/a8030786
  29. M. Young, J. F. Traub, D. M. Young, J. F. Traub, D. M. Young, and J. F. Traub, “Iterative Methods for the Solution of Equations.,” Am. Math. Mon., vol. 74, no. 3, p. 346, Mar. 1967, doi: https://doi.org/10.2307/2316079
  30. Chun, “Some fourth-order iterative methods for solving nonlinear equations,” Appl. Math. Comput., vol. 195, no. 2, pp. 454–459, Feb. 2008, doi: https://doi.org/10.1016/j.amc.2007.04.105
  31. Potra and V. Pták, Nondiscrete induction and iterative processes. Number‏. Boston: Pitman Advanced Pub. Program, 1984.
  32. R. Sharma, “A composite third order Newton-Steffensen method for solving nonlinear equations,” Appl. Math. Comput., vol. 169, no. 1, pp. 242–246, Oct. 2005, doi: https://doi.org/10.1016/j.amc.2004.10.040
  33. K. Argyros and S. Hilout, “On the semilocal convergence of damped Newton’s method,” Appl. Math. Comput., vol. 219, no. 5, pp. 2808–2824, Nov. 2012, doi: https://doi.org/10.1016/j.amc.2012.09.011
  34. Xiaojian, “Modified Chebyshev-Halley methods free from second derivative,” Appl. Math. Comput., vol. 203, no. 2, pp. 824–827, Sep. 2008, doi: https://doi.org/10.1016/j.amc.2008.05.092
  35. Der Kiureghian, “First- and second-order reliability methods,” in Engineering Design Reliability Handbook, CRC Press, 2004, pp. 14-1-14–24.
  36. Ditlevsen and H. O. Madsen, Structural Reliability Methods, vol. 178, no. July. Wiley New York, 2005.
  37. P. Hong and N. C. Lind, “Approximate reliability analysis using normal polynomial and simulation results,” Struct. Saf., vol. 18, no. 4, pp. 329–339, Jan. 1996, doi: https://doi.org/10.1016/S0167-4730(96)00018-5
International Journal of Reliability, Risk and Safety: Theory and Application
Volume 6, Issue 1
July 2023
Pages 71-76

  • Receive Date 13 June 2023
  • Revise Date 07 August 2023
  • Accept Date 08 August 2023