Structural Shape Optimization of Vibrating Axisymmetric and Prismatic Shells

  • Ernest Hinton
  • Johann Sienz
  • Mustafa Özakça


Chapter 9 is concerned with structural shape and thickness optimization of vibrating axisymmetric and prismatic plates and shells. The basic algorithm for SSO is described first. Details are subsequently presented concerning the problem definition. Attention is then focused on the sensitivity calculations and problems connected with their accuracy. In the penultimate section some details regarding mathematical programming are given. Finally, various practical aspects of SSO are discussed, such as specification of end conditions, scaling and bounds on design variables. This chapter is also concerned with the SSO of vibrating axisymmetric and prismatic shells. Natural frequencies and mode shapes are determined using curved, variable-thickness, MR FSs and FEs introduced and bench-marked in Chapters 7 and 8. The whole shape optimization process is carried out by integrating FE and FS analysis, cubic spline shape and thickness definitions, sensitivity analysis and mathematical programming. In most cases, the objective is either the maximization of the fundamental frequency or the minimization of the volume by changing the shape or thickness variation of the cross-section of the structure with constraints on the volume or natural frequencies. The SAM and FDM are used to determine the sensitivities of the objective function and constraints to changes in the design variables. Several examples are considered to illustrate and highlight various features of the optimization, including plates, conical shells, box-girder bridges with rectangular or curved planform, axisymmetric branched shells, bells and cylindrical shells.


Design Variable Fundamental Frequency Circular Plate Conical Shell Stiffened Panel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Hartmann D, Neumann M. Structural Optimization of a Box-girder Bridge by Means of the Finite Strip Method. Computer Aided Optimum Design of Structures. In: Brebbia CA, Hernandez S, editors. Computational Mechanics Publications. Heidelberg: Springer Verlag; 1989.Google Scholar
  2. [2]
    Hinton E, Rao NVR. Analysis and Shape Optimization of variable Thickness prismatic folded Plates and curved Shells, Part 2: Shape Optimization. Thin Walled Struct 1993;17:81–111.CrossRefGoogle Scholar
  3. [3]
    Ramm E, Bletzinger KU, Kimmich S. Strategies in Shape Optimization of free Form Shells. In: Wriggers P, Wagner W, editors. Festschrift Erwin Stein: Nonlinear Computational Mechanics — a State of the Art. Heidelberg: Springer; 1991.Google Scholar
  4. [4]
    Niordson FI. On the Optimum Design of vibrating Beams. Q Appl Mech 1965;23:47–53.MathSciNetGoogle Scholar
  5. [5]
    Karihaloo BL, Niordson FI. Optimum Design of vibrating Cantilevers. J Optim Theory Appl 1973;11:638–54.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Olhoff N, Plaut RH. Bimodal Optimization of vibrating shallow Arches. Int J Solids Struct 1991;19(6):553–70.CrossRefGoogle Scholar
  7. [7]
    Blachut J, Gajewski A. Unimodal and bimodal Optimum Designs of extensible Arches with respect to Buckling and Vibrations. Optim Contr Appl Methods 1983;2:112–34.Google Scholar
  8. [8]
    Olhoff N. Optimal Design of vibrating circular Plates. Int J Solids Struct 1970;6:139–56.MATHCrossRefGoogle Scholar
  9. [9]
    Thambiratnam DP, Thevendran V. Optimum vibrating Shapes of Beams and circular Plates. J Sound Vib 1988;121:13–23.MATHCrossRefGoogle Scholar
  10. [10]
    Plaut RH, Johnson LW, Parbery R. Optimal Forms of shallow Shells with circular Boundary, Part I: Maximum fundamental Frequency. J Appl Mech 1984;51:526–30.MATHCrossRefGoogle Scholar
  11. [11]
    Hinton E, Özakça M, Rao NVR. Structural Shape Optimization of vibrating Shells and folded Plates using two noded Finite Strips. Eng Computations 1993;10:139–57.CrossRefGoogle Scholar
  12. [12]
    Hinton E, Özakça M, Rao NVR. Free Vibration Analysis and Shape optimization of variable Thickness prismatic folded Plates and curved Shells, Part 2: Shape Optimization. J Sound Vib 1995;181:567–81.CrossRefGoogle Scholar
  13. [13]
    Haftka RT, Adelman HM. Recent Development in structural Sensitivity Analysis. Struct Optim 1989;1:137–151.CrossRefGoogle Scholar
  14. [14]
    Petiau C. Structural Optimization of Aircraft. Thin Walled Struct 1991;11:43–6.CrossRefGoogle Scholar
  15. [15]
    Botkin ME. Shape Optimization using fully automatic 3-D Mesh Generator. In: Proceedings of Conference on Structures, Dynamics and Materials; 1991; Baltimore, MD.Google Scholar
  16. [16]
    Bletzinger KU, Kimmich S, Ramm E. Interactive Shape Optimization of Shells. In: Proceedings International Conference on Numerical Methods In Engineering — Theory and Applications; 1990; Elsevier Applied Science, London; 1990. p. 464–473.Google Scholar
  17. [17]
    Bletzinger KU, Kimmich S, Ramm E. Efficient Modelling in Shape optimal Design. Comput Syst Eng 1991;2:483–496.CrossRefGoogle Scholar
  18. [18]
    Bletzinger KU, Reitinger R. Shape Optimization of Shells. In: Proceedings of International Symposium on Natural Structures-Principles, Strategies and Models in Architecture and Nature, 1991 Oct 1-4, Stuttgart, Germany. Stuttgart: Universität Stuttgart; 1991.Google Scholar
  19. [19]
    Barbosa JI, Mota Soares CM, Mota Soares CA. Sensitivity Analysis and Shape optimal Design of axisymmetric Shell Structures. Comput Syst Eng 1991;2:525–34.CrossRefGoogle Scholar
  20. [20]
    Hinton E, Özakça M, Sienz J. Optimum Shapes of vibrating axisymmetric Shell Structures. J Sound Vib 1993;167:511–28.MATHCrossRefGoogle Scholar
  21. [21]
    Plaut RH, Johnson LW, Parbery R. Optimal Form of shallow Shell with circular Boundary, Part 2: Maximum Buckling Load. J Appl Mech 1984;51:531–6.MATHCrossRefGoogle Scholar
  22. [22]
    Roozen-Kroon PJM. Structural optimization of bells [PhD thesis]. Eindhoven: Technische Universiteit Eindhoven; 1992.Google Scholar
  23. [23]
    Kimmich S, Reitinger R, Ramm E. Integration of different numerical Techniques in Shape Optimization. Struct Optim 1992;4:149–55.CrossRefGoogle Scholar
  24. [24]
    Roozen-Kroon PJM, Schoofs AJG, Van Campen DH. Fast numerical Shape Optimization of Bells using Design of Experiment and Regression Techniques. In: Brebbia CA, Hernandez S, editors. Computer aided Optimum Design of Structures, Computational Mechanics Publications. Heidelberg: Springer Verlag; 1989.Google Scholar
  25. [25]
    Murra IA, Gladwell GML. On a Search for major-third and other non-standard Bells. J Sound Vib 1991;149:330–40.CrossRefGoogle Scholar
  26. [26]
    Hinton E, Öz akça M. Structural Optimization of Bells. ACME 93. Proceedings of Computational Mechanics in the UK; Swansea; 1993. p. 149–53.Google Scholar
  27. [27]
    Perrin R, Charnley T, Depont J. Normal Modes of the modern English Church Bell. J Sound Vib 1983;90:29–49.CrossRefGoogle Scholar
  28. [28]
    Hinton E, Rao NVR. Structural Shape Optimization of Shells and folded Plates using two noded Finite Strips. Comput Struct 1993;46:1055–71.MATHCrossRefGoogle Scholar
  29. [29]
    Özakça M, Hinton E, Rao NVR. Free Vibration Analysis and Shape Optimization of prismatic folded Plates and Shells with curved Planform, Part 2: Shape Optimization. Int J Numer Methods Eng 1994;37:1713–39.CrossRefGoogle Scholar
  30. [30]
    Dawe DJ. Finite Strip Models for Vibration of Mindlin Plates. J Sound Vib 1978;59:441–52.CrossRefGoogle Scholar
  31. [31]
    Hinton E, Öz akça M, Jantan MH. A computational Tool for structural Shape Optimization of vibrating Arches. Struct Eng Rev 1992;4:163–74.Google Scholar
  32. [32]
    Morris IR, Dawe DJ. Free Vibration of curved-plate Assemblies with Diaphragm Ends. J Sound Vib 1980;73:l–17.CrossRefGoogle Scholar
  33. [33]
    Cheung YK, Cheung MS. Free Vibration of curved and straight Beam-slab or Box-girder Bridges. J IABSE 1972;32(II):41–52.Google Scholar

Copyright information

© Springer-Verlag London 2003

Authors and Affiliations

  • Ernest Hinton
    • 1
  • Johann Sienz
    • 2
  • Mustafa Özakça
    • 3
  1. 1.Department of Civil EngineeringUniversity of Wales SwanseaSwanseaUK
  2. 2.Department of Mechanical EngineeringUniversity of Wales SwanseaSwanseaUK
  3. 3.Department of Civil Engineering, Faculty of EngineeringUniversity of GaziantepGaziantepTurkey

Personalised recommendations