GENETIC ALGORITHMS IN SIMULATING OPTIMAL STACKING SEQUENCE OF A COMPOSITE LAMINATE PLATE WITH CONSTANT THICKNESS

P.-F. Pai, S. Deng, C.-C. Lai, and P.-S. Wu

References

  1. [1] L.R. Tsai, Y.H. Chang, & F.L. Tsao, The design of optimal stacking sequence for laminated FRP plates with in plane loading, Composite Structures, 55(4), 1995, 565–580. doi:10.1016/0045-7949(94)00321-S
  2. [2] L.R. Tsai & C.H. Liu, A comparison between two optimization methods on the stacking sequence of fiber-reinforced composite laminate, Composite Structures, 55(3), 1995, 515–525. doi:10.1016/0045-7949(95)98877-S
  3. [3] C.J. Chen, An efficient method for the optimization of ply stacking sequence for constant thickness composite laminate plate, master’s thesis, Da-Yeh University, Taiwan, 2000.
  4. [4] J.H. Park, J.H. Hwang, C.S. Lee, & W. Hwang, Stacking sequence design of composite laminates for maximum strength using genetic algorithms, Composite Structures, 52, 2001, 217–231. doi:10.1016/S0263-8223(00)00170-7
  5. [5] V.B. Gantovnik, Z. Gurdal, & L.T. Watson, A genetic algorithm with memory for optimal design of laminated sandwich composite panels, Composite Structures, 58, 2002, 513–520. doi:10.1016/S0263-8223(02)00128-9
  6. [6] B. Liu, T.R. Haftka, M.A. Akgun, & A. Todoroki, Permutation genetic algorithm for stacking sequence design of composite laminates, Computer Methods in Applied Mechanics and Engineering, 186, 2000, 357–372. doi:10.1016/S0045-7825(99)90391-2
  7. [7] A. Muc & W. Gurba, Genetic algorithm and finite element analysis in optimization of composite structures, Composite Structures, 54, 2001, 275–281. doi:10.1016/S0263-8223(01)00098-8
  8. [8] G. Soremekun, Z. Gurdal, R.T. Haftka, & L.T. Watson, Composite laminate design optimization by genetic algorithms with generalized elitist selection, Composite Structures, 79, 2001, 131–143. doi:10.1016/S0045-7949(00)00125-5
  9. [9] G. Soremekun, Z. Gurdal, C. Kassapoglou, & D. Toni, Stacking sequence blending of multiple composite laminates using genetic algorithms, Composite Structures, 56, 2002, 53–62. doi:10.1016/S0263-8223(01)00185-4
  10. [10] M. Walker & R.E. Smith, A technique for the multi-objective optimization of laminated composite structures using genetic algorithms and finite element analysis, Composite Structures, 62, 2003, 123–128. doi:10.1016/S0263-8223(03)00098-9
  11. [11] C.-C. Lin & Y.-J. Lee, Stacking sequence optimization of laminated composite structures using genetic algorithm with local improvement, Composite Structures, 63, 2004, 339–345. doi:10.1016/S0263-8223(03)00182-X
  12. [12] R.B. Pipes & N.J. Pagano, Interlaminar stresses in composite laminates under uniform axial extension, Journal of Composite Materials, 4, 1970, 538–548.
  13. [13] N.J. Pagano & R.B. Pipes, The influence of stacking sequence on laminate strength, Journal of Composite Materials, 5, 1971, 50–57. doi:10.1177/002199837100500105
  14. [14] A.S.D. Wang & F.W. Crossman, Some new results on edge effect in symmetric composite materials, Journal of Composite Materials, 11, 1977, 92–106. doi:10.1177/002199837701100110
  15. [15] C. Kassapoglou & P.A. Lagace, An efficient method for the calculation of interlaminar stresses in composite materials, Journal of Applied Mechanics, 53, 1986, 744–760.
  16. [16] E.F. Rybicki, Approximation three-dimensional solutions for symmetric laminates under in-plane loading, Journal of Composite Materials, 5, 1971, 354–361. doi:10.1177/002199837100500305
  17. [17] J. Holland, Adaptation in natural and artificial systems (AnnArbor, MI: University of Michigan Press, 1975).
  18. [18] D.E. Goldberg & Jr. R. Lingle, Alleles, loci and the Traveling Salesman Problem, Proc. Int. Conf. on Genetic Algorithms, ed. 66 J.J. Grefenstette (Hillsdale, NJ: Lawrence Erlbaum Associates, 1985).
  19. [19] S.A. Brah & J.L. Hunsucker, Branch and bound algorithm for the flow shop with multiple processors, European Journal Operational Research, 51, 1991, 88–99. Appendix A ¯S(k ) =               ¯S (k) 11 ¯S (k) 12 ¯S (k) 13 0 0 ¯S (k) 16 ¯S (k) 12 ¯S (k) 22 ¯S (k) 23 0 0 ¯S (k) 26 ¯S (k) 13 ¯S (k) 23 ¯S (k) 33 0 0 ¯S (k) 36 0 0 0 ¯S (k) 44 ¯S (k) 45 0 0 0 0 ¯S (k) 45 ¯S (k) 55 0 ¯S (k) 16 ¯S (k) 26 ¯S (k) 36 0 0 ¯S (k) 66               (A1) ¯S(k) : the compliance tensor of the k-th ply (A2) ¯S (k) 11 = S (k) 11 m4 + (2S (k) 12 + S (k) 66 )m2 n2 + S (k) 22 n4 (A3) ¯S (k) 12 = (S (k) 11 + S (k) 22 − S (k) 66 )m2 n2 + S (k) 12 (m4 + n4 ) (A4) ¯S (k) 13 = S (k) 13 m2 + S (k) 23 n2 (A5) ¯S (k) 22 = S (k) 11 n4 + (2S (k) 12 + S (k) 66 )m2 n2 + S (k) 22 m4 (A6) ¯S (k) 23 = S (k) 13 n2 + S (k) 23 m2 (A7) ¯S (k) 33 = S (k) 33 (A8) ¯S (k) 16 = (2S (k) 11 − 2S (k) 12 − S (k) 66 )nm3 − (2S (k) 22 − 2S (k) 12 − S (k) 66 )mn3 (A9) ¯S (k) 26 = (2S (k) 11 − 2S12 − S66)n3 m − (2S22 − 2S12 − S66)m3 n (A10) ¯S (k) 36 = 2(S (k) 13 − S (k) 23 )mn (A11) ¯S (k) 44 = S (k) 44 m2 + S (k) 55 n2 (A12) ¯S (k) 45 = (S (k) 55 − S (k) 44 )mn (A13) ¯S (k) 55 = S (k) 44 n2 + S (k) 55 m2 (A14) ¯S (k) 66 = 4(S (k) 11 − 2S (k) 12 + S (k) 22 )m2 n2 + S (k) 66 (m2 − n2 )2 (A15) where: S (k) ij : the interlaminar strain of the k-th ply i, j: directions m: cos θ n: sin θ Appendix B f1 = 1 2 nk =1 [˜σ (k) 22 ]2 ¯ S (k) 22 − [ ¯S (k) 12 ]2 ¯S (k) 11 ( B1) f2 = 1 120 nk =1 {3[˜σ (k) 22 ]2 + 15˜σ (k) 22 B (k) 4 + 20˜σ (k) 22 B (k) 5 + 20[B (k) 4 ]2 + 60B (k) 4 B (k) 5 + 60[B (k) 5 ]2 } × ¯ S (k) 22 − [ ¯S (k) 13 ]2 ¯S (k) 11 ( B2) f3 = 1 6 nk =1 {[˜σ (k) 22 ]2 + 3˜σ (k) 22 B (k) 4 + 3[B (k) 4 ]2 } ¯S (k) 44 (B3) f4 = 1 6 nk =1 {[˜σ (k) 12 ]2 + 3˜σ (k) 12 B (k) 2 + 3[B (k) 2 ]2 } ¯S (k) 55 (B4) f5 = 3 2 nk =1 [˜σ (k) 12 ]2 ¯ S (k) 66 − [ ¯S (k) 16 ] ¯S (k) 11 ( B5) f6 = 1 2 nk =1 ˜σ (k) 22 [˜σ (k) 11 ¯S (k) 11 + ˜σ (k) 22 ¯S (k) 12 + ˜σ (k) 12 ¯S (k) 16 ] (B6) f7 = 1 2 nk =1 ˜σ (k) 12 [˜σ (k) 11 ¯S (k) 11 + ˜σ (k) 22 ¯S (k) 12 + ˜σ (k) 12 ¯S (k) 16 ] (B7) f8 = 1 6 nk =1 {[˜σ (k) 22 ]2 + 3˜σ (k) 22 B (k) 4 + 6˜σ (k) 22 B (k) 4 } × ¯ S (k) 22 − ¯S (k) 12 ¯S (k) 13 ¯S (k) 11 ( B8) f9 = ˜σ (k) 22 ˜σ (k) 12 ¯ S (k) 26 − ¯S (k) 12 ¯S (k) 16 ¯S (k) 11 ( B9) f10 = 1 6 nk =1 ˜σ (k) 12 [˜σ (k) 22 + 3B (k) 4 + 6B (k) 5 ] ¯ S (k) 36 − ¯S (k) 13 ¯S (k) 16 ¯S (k) 11 ( B10) f11 = 1 6 nk =1 [2˜σ (k) 22 ˜σ (k) 12 + 3˜σ (k) 22 B (k) 2 + 3˜σ (k) 12 B (k) 4 + 6B (k) 2 B (k) 4 ] ¯S (k) 45 (B11) 67 doi:10.1016/0377-2217(91)90148-O

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