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Shape optimization of axisymmetric shells using a higher-order shear deformation theory

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In this paper structural and sensitivity analysis for the optimization of laminated axisymmetric shells subjected to static constraints and arbitrary loading is presented. The shell thickness, radial coordinate of a nodal point, lamina thickness and the angle of orientation of the fibers are the design variables. The objective of the design optimization is the minimization of the volume of the shell or the strain energy. The model is based on a three-node axisymmetric finite element with 24 degrees of freedom. A higher-order theory is developed for the nonlinear distribution of the meridional displacement component through the thickness of the shell. The sensitivities of the discrete model developed are evaluated analytically using a symbolic manipulator. The efficiency and accuracy of the proposed model is discussed with reference to the applications.

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  1. Barbosa, J.I.; Mota Soares, C.M.; Mota Soares, C.A. 1991: Sensitivity analysis and shape optimal design of axisymmetric shell structures.Comp. Syst. Engng. 2, 525–533

  2. Bernadou, M.; Palma, F.J.; Rousselet, B. 1991: Shape optimization of elastic thin shell under various criteria.Struct. Optim. 3 7–21

  3. Char, B.W.; Geddes, K.O.; Leong, B.L.; Monagan, M.B.; Watt, S.W 1991:Maple V library reference manual. Berlin, Heidelberg, New York: Springer

  4. Char, B.W.; Geddes, K.O.; Leong, B.L.; Monagan, M.B.; Watt, S.W 1992:First leaves: a tutorial introduction to Maple V. Berlin, Heidelberg, New York: Springer

  5. Chenais, D. 1987: Shape optimization in shell theory: design sensitivity of the continuous problem.Eng. Opt. 11, 289–303

  6. Chenais, D. 1994: Discrete gradient and discretized continuum gradient methods for shape optimization of shells.Mech. Struct. & Mach. 22, 73–115

  7. Csonka, B. 1993: Sensitivity analysis of a higher-order model for axisymmetric shells using a symbolic manipulator.Report ID-MEC, IST, PROJ. STRD/TPR/592/92, Lisbon

  8. Haftka, R.; Gurdal, Z. 1993:Elements of structural optimization. Dordrecht: Kluwer

  9. Haug, E.J.; Choi, K.K.; Komkov, V. 1986:Design sensitivity analysis of structural systems. Orlando: Academic Press

  10. Kant, T.; Pandya, B.N. 1988: A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates.Comp. Struct. 9, 215–246

  11. Kraus, H. 1976:Thin elastic shells. New York: John Wiley & Sons

  12. Lund, E.; Olhoff, N. 1993: Reliable and efficient finite element based design sensitivity analysis of eigenvalues. In: Herskovits, J. (ed.)Structural optimization 93, pp. 197–204. Coppe: Rio de Janeiro

  13. Marcelin, J.L.; Trompette, P. 1988: Optimal shape design of thin axisymmetric shells.Eng. Opt. 13, 108–117

  14. Mehrez, S.; Rousselet, B. 1989: Analysis and optimization of a shell of revolution. In: Brebbia, C.A.; Hernandez, S. (eds.)Computer aided optimum design of structures: applications, pp. 123–133. Berlin, Heidelberg, New York: Springer

  15. Mindlin, R.D. 1952: Influence of rotary and shear deformation on flexural motions of isotropic plates.ASME, J. Appl. Mech. 18, 31–38

  16. Moriano, S. 1988:Optimisation de forme de coques. Thesis, University of Nice (in French)

  17. Mota Soares, C.A.; Leal, R.P. 1992: Mixed elements in shape optimal design of structures based on global criteria. In: Rozvany, G.I.N. (ed.)Shape and layout optimization of structural systems and optimality criteria methods, pp. 279–300. Vienna: Springer

  18. Mota Soares, C.M.; Franco Correia, V.M.; Herskovits, J. 1993: A discrete model for the optimal design of thin composite plate-shell type structures. In: Herkovits, J. (ed.)Structural Optimization 93, pp. 171–180. Coppe: Rio de Janeiro

  19. Mota Soares, C.M.; Mota Soares, C.A.; Barbosa, J.I. 1994: Sensitivity analysis and optimal design of thin shells of revolution.AIAA J. 32, 1034–1042

  20. Noor, A.K.; Burton, W.S. 1989: Assessment of shear deformation theories for multilayered composite plates.Appl. Mech. Rev. 42, 1–13

  21. Olhoff, N.; Rasmussen, J.; Lund, E. 1993: A method of “exact” numerical differentiation for error elimination in finite-elementbased semi-analytical shape sensitivity analysis.Mech. Struct. Mach. 21, 1–66

  22. Plaut, R.H.; Johnson, L.W.; Parbery, R. 1984: Optimal form of shallow shells with circular boundary.Trans. ASME 51, 526–538

  23. Reddy, J.N. 1990: A review of refined theories of laminated composite plates.Shock and Vibrations Digest 22, 3–17

  24. Reissner, E. 1945: The effect of transverse shear deformation on the bending of elastic plates.ASME, J. Applied Mech. 12, A69-A77

  25. Sheinman, I.; Weissman, S. 1978: Coupling between symmetric and antisymmetric modes in shell of revolution.J. Comp. Mat. 21, 988–1007

  26. Tsai, S.W.; Hahn, H.T. 1980:Introduction to composite materials. Technomic Publishing Co., Inc.

  27. Vanderplaats, G.N. 1987:ADS. A Fortran program for automated design synthesis, version 2.01. St. Barbara: Engineering Design Optimization Inc.

  28. Vanderplaats, G.N. 1984:Numerical optimization techniques for engineering design. New York: McGraw-Hill

  29. Yamaguchi, F. 1988:Curves and surfaces in computer aided geometric design. Berlin, Heidelberg, New York: Springer

  30. Zienkiewicz, O.C. 1977:The finite element method in engineering science (3rd ed.). London: McGraw-Hill

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Csonka, B., Kozák, I., Mota Soares, C.M. et al. Shape optimization of axisymmetric shells using a higher-order shear deformation theory. Structural Optimization 9, 117–127 (1995). https://doi.org/10.1007/BF01758828

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  • Design Variable
  • Shear Deformation
  • Symbolic Manipulator
  • Static Constraint
  • Nodal Point