Structural and Multidisciplinary Optimization

, Volume 59, Issue 5, pp 1483–1501 | Cite as

Coordinative optimization method of composite laminated structures based on system reliability

  • Yujia Ma
  • Xiaojun WangEmail author
  • Qinghe Shi
  • Qiang Ren
Research Paper


This paper develops the coordinative optimization method based on system reliability for laminated structures. The proposed method improves the rough RBO based on first layer failure (FLF) criterion for composite laminates, and the coupling optimization method of thickness and sequence in traditional RBO strategy based on last layer failure criterion (LLF) is improved. In this paper, the finite element analysis is used to obtain the response for the failure based on two-dimension Hashin failure criterion (the limit function). Obviously, the stiffness of composite materials will decline due to destruction of elements. Therefore, stiffness degradation is considered to describe the process of damage evolution. Subsequently, combining with the branch-bound method (B&B), we can complete the search of main failure sequences and calculate the system reliability with the help of the second-order upper bound theory. In order to guarantee the efficiency and accuracy of optimization, the adaptive GA algorithm is introduced in the whole optimization procedure. After the proposed optimization policy is given in detail, two laminated structures are presented and the results are compared with the traditional optimal method based on safety factor, which demonstrates the validity and reasonability of the developed methodology.


System reliability optimization Last layer failure criterion Laminated structures Hashin failure criterion Finite element analysis 



The authors would like to thank the National Key Research and Development Program (No. 2016YFB0200700), the Defense Industrial Technology Development Program (No. JCKY2016601B001, No. JCKY2017601B001) P. R. China (11432002, 11572024, 11872089) and for the financial supports. Besides, the authors wish to express their many thanks to the reviewers for their useful and constructive comments.


  1. AIAA, System reliability-based optimization of composite structures, Probabilistic Mechanics & Structural Reliability (1996), 1998Google Scholar
  2. António CAC, Marques AT, Gonçalves JF (1996) Reliability based design with a degradation model of laminated composite structures. Struct Optim 12:16–28CrossRefGoogle Scholar
  3. Cederbaum G, Elishakoff I, Librescu L (1990) Reliability of laminated plates via the first-order second-moment method. Compos Struct 15:161–167CrossRefGoogle Scholar
  4. Chen NZ, Guedes Soares C (2007) Progressive failure analysis for prediction of post-buckling compressive strength of laminated composite plates and stiffened panels. J Reinf Plast Compos 26:1021–1042CrossRefGoogle Scholar
  5. Chen S, Lin Z, An H, Huang H, Kong C (2013) Stacking sequence optimization with genetic algorithm using a two-level approximation. Struct Multidiscip Optim 48:795–805MathSciNetCrossRefGoogle Scholar
  6. Dolinski K, Dolinski K (1982) First-order second-moment approximation in reliability of structural systems: critical review and alternative approach. Struct Saf 1:211–231CrossRefGoogle Scholar
  7. Farshi B, Herasati S (2006) Optimum weight design of fiber composite plates in flexure based on a two level strategy. Compos Struct 73:495–504CrossRefGoogle Scholar
  8. Gavin HP, Yau SC (2008) High-order limit state functions in the response surface method for structural reliability analysis. Struct Saf 30:162–179CrossRefGoogle Scholar
  9. Hou JW, Gumbert CR, Newman PA (2004) A Most probable point-based method for reliability analysis, Sensitivity Analysis and Design OptimizationGoogle Scholar
  10. Jeong HK, Shenoi RA (2000) Probabilistic strength analysis of rectangular FRP plates using Monte Carlo simulation. Comput Struct 76:219–235CrossRefGoogle Scholar
  11. Jianqiao C, Tang Y, Qunli X (2013) Reliability design optimization of composite structures based on PSO together with FEA. Chin J Aeronaut 26:343–349CrossRefGoogle Scholar
  12. Kang Z, Luo Y (2009) Non-probabilistic reliability-based topology optimization of geometrically nonlinear structures using convex models. Comput Methods Appl Mech Eng 198:3228–3238MathSciNetCrossRefzbMATHGoogle Scholar
  13. Lee OS, Park YC, Dong HK (2007) Reliability estimation of solder joints under thermal fatigue with varying parameters by using FORM and MCS. J Mech Sci Technol 2166:683–688Google Scholar
  14. Li HS (2013) Reliability-based design optimization via high order response surface method. J Mech Sci Technol 27:1021–1029CrossRefGoogle Scholar
  15. Lim J, Lee B (2016) A semi-single-loop method using approximation of most probable point for reliability-based design optimization. Struct Multidiscip Optim 53:745–757MathSciNetCrossRefGoogle Scholar
  16. Lin SC, Kam TY (2000) Probabilistic failure analysis of transversely loaded laminated composite plates using first-order second moment method. J Eng Mech 126:812–820CrossRefGoogle Scholar
  17. Liu B, Haftka RT, Akgün MA (2000) Two-level composite wing structural optimization using response surfaces, Structural & Multidisciplinary Optimization, 20:87–96Google Scholar
  18. Liu X, Mahadevan S, Liu X, Mahadevan S (2013) System reliability of composite laminates, Structures, Structural Dynamics, and Materials ConferenceGoogle Scholar
  19. López C, Bacarreza O, Baldomir A, Hernández S, Aliabadi MHF (2016) Reliability-based design optimization of composite stiffened panels in post-buckling regime. Struct Multidiscip Optim:1–21Google Scholar
  20. Luo Y, Li A, Kang Z (2011) Reliability-based design optimization of adhesive bonded steel–concrete composite beams with probabilistic and non-probabilistic uncertainties. Eng Struct 33:2110–2119CrossRefGoogle Scholar
  21. Ma J, Ren Z, Zhao G, Zhang Y, Koh CS (2017) A new reliability analysis method combining adaptive kriging with weight index Monte Carlo simulation, IEEE transactions on magnetics, PP 1–4Google Scholar
  22. Mahadevan S, Liu X, Xiao Q, Probabilistic Progressive A (1997) Failure model of composite laminates. J Reinf Plast Compos 16:1020–1038CrossRefGoogle Scholar
  23. Melchers RE, Ahammed M, Middleton C (2003) FORM for discontinuous and truncated probability density functions. Struct Saf 25:305–313CrossRefGoogle Scholar
  24. Mitra J, Xu X (2010) Composite system reliability analysis using particle swarm optimization, IEEE International Conference on Probabilistic Methods Applied To Power Systems, pp. 548–552Google Scholar
  25. Nafday AM, Corotis R.B (1987) Failure mode enumeration for system reliability assessment by optimization algorithms, Springer Berlin HeidelbergGoogle Scholar
  26. Onkar AK, Upadhyay CS, Yadav D (2007) Probabilistic failure of laminated composite plates using the stochastic finite element method. Compos Struct 77:79–91CrossRefGoogle Scholar
  27. Park JS, Kim CG, Hong CS (2015) Bimodal bound of system reliability for random composite structures. AIAA J 34:1494–1500CrossRefzbMATHGoogle Scholar
  28. Qiu Z (2003) Comparison of static response of structures using convex models and interval analysis method. Int J Numer Methods Eng 56:1735–1753CrossRefzbMATHGoogle Scholar
  29. Qiu Z, Elishakoff I (1998) Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis. Comput Methods Appl Mech Eng 152:361–372CrossRefzbMATHGoogle Scholar
  30. Qiu Z, Yang D, Elishakoff I (2008) Probabilistic interval reliability of structural systems. International Journal of Solids & Structures 45:2850–2860CrossRefzbMATHGoogle Scholar
  31. Rahman S, Wei D (2006) A univariate approximation at most probable point for higher-order reliability analysis. Int J Solids Struct 43:2820–2839CrossRefzbMATHGoogle Scholar
  32. Rais-Rohani M, Singh MN (2004) Comparison of global and local response surface techniques in reliability-based optimization of composite structures. Struct Multidiscip Optim 26:333–345CrossRefGoogle Scholar
  33. Salas P, Venkataraman S (2007) Controlling failure using structural fuses for predictable progressive failure of composite laminates. Struct Multidiscip Optim 34:473–489CrossRefGoogle Scholar
  34. Thoft-Christensen P, Murotsu Y (1989) Application of structural system reliability theory. J Am Stat Assoc 84Google Scholar
  35. Todoroki A, Sekishiro M (2007) Two-level optimization of dimensions and stacking sequences for hat-stiffened composite panel. J Comput Sci Technol 1:22–33CrossRefGoogle Scholar
  36. Venugopal SM (2003) Stochastic mechanics and reliability of composite laminates based on experimental investigation and stochastic FEM. Mech Ind EngGoogle Scholar
  37. Wang X, Guo B (2004) Optimization of laminated composite plates based on the last-ply failure criterion. J Mech StrengthGoogle Scholar
  38. Wang X, Li Z (2009) Reliability-based robust of symmetric laminated plates subject to last-ply failure, International Conference on Industrial Mechatronics and Automation, pp. 464–467Google Scholar
  39. Wang L, Liang J, Wu D (2018) A non-probabilistic reliability-based topology optimization (NRBTO) method of continuum structures with convex uncertainties. Struct Multidiscip Optim:1–20Google Scholar
  40. Zhang X, Chang X (2008) A Probabilistic Progressive Failure model for reliability of composite laminates, 2008 national academic symposium for ph. d. candidates and tri-university workshop on aero-structural mechanics & aerospace engineeringGoogle Scholar
  41. Zhao Y, Su GS, Yan LB (2014) KNN classification based MCS method of structural reliability analysis. Appl Mech Mater 638-640:136–139CrossRefGoogle Scholar
  42. Zhu ZQ, Chen JJ, Song ZF, Lin LG (2010) Non-probabilistic reliability index of bar structures with interval parameters based on modified affine arithmetic, Engineering Mechanics, 27:49–54Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Solid MechanicsBeihang UniversityBeijingChina

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