Advertisement

A Variational Principle, the Finite Element Method, and Optimal Structural Design for Given Deflection

  • G. A. Hegemier
  • H. T. Tang
Part of the International Union of Theoretical and Applied Mechanics book series (IUTAM)

Abstract

General Remarks. The optimal design of continuous or pieeewise continuous structural systems can be approached along two avenues. On one hand the system can be discretized at the outset and a sequence of problems involving refined domains may be studied in an effort to extract an approximate optimal solution. On the other hand, the continuous nature of the problem may be recognized and optimality criteria for the continua sought directly.

Keywords

Variational Principle Optimality Criterion Inequality Constraint Sandwich Plate Kirchhoff Plate 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Sheu, C. Y., Prager, W.: Recent Developments in Optimal Structural Design. Appl. Mech. Rev. 21, 1968.Google Scholar
  2. 2.
    Niordson, F. I., Pedersen, P.: A Review of Optimal Structural Design, DCAMM Report 31, Technical University of Denmark, September 1972.Google Scholar
  3. 3.
    Fiaceo, A. V., McCormick, G. P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley 1968.Google Scholar
  4. 4.
    Pontryagin, L. S., Eoltyanski, V. G., Gamkrelidze, R. V., Meshchenko, E. F.: The Mathematical Theory of Optimal Processes, (trans. by K. N. Trinogoff), Interscience 1962.MATHGoogle Scholar
  5. 5.
    Bellman, R.: Dynamic Programming, Princeton Univ. Press 1957.MATHGoogle Scholar
  6. 6.
    Smith, D. R.: Variational Methods in Optimization, Prentice-Hall 1974.MATHGoogle Scholar
  7. 7.
    de Silva, B. M. E.: Application of Optimal Control Theory to Some Structural Optimization Problems, ASME Research 71-Vibr-66, Sept. 1971 (Presented at the Vibration Conference and the International Design Automation Conference, Toronto, Sept. 8–10, 1971).Google Scholar
  8. 8.
    Haug, E. J., Kirmser, P. G.: Minimum-Weight Design with Inequality Constraints on Stress and Deflection. J. Appl. Mech. 34 (1967).Google Scholar
  9. 9.
    Huang, N. C., Tang, H. T.: Minimum Weight Design of Elastic Sandwich Beams with Deflection Constraints. J. of Optimization Theory and Applications 4, No. 4 (1969).Google Scholar
  10. 10.
    Frauenthal, J. C.: Constrained Optimal Design of Circular Plates Against Buckling. J. Struc. Mech. 1 (1972).Google Scholar
  11. 11.
    Valentine, F. A.: The Problem of Lagrange with Differential Inequalities as Added Side Conditions, Contribution to the Calculus of Variations, 1933 – 1937, University of Chicago Press 1937.Google Scholar
  12. 12.
    Bolza, O.: Lectures on the Calculus of Variations, Dover 1961.MATHGoogle Scholar
  13. 13.
    Bliss, G. A.: Lectures on the Calculus of Variations, University of Chicago Press 1946.MATHGoogle Scholar
  14. 14.
    Cesari, L.: Optimization with Partial Differential Equations in Dieudonné-Rashevsky Form and Conjugate Problems, Report No. 14, US-AFOSR Research Project 942-65, University of Michigan, 1968.Google Scholar
  15. 15.
    Prager, W., Taylor, J. E.: Problems of Optimal Structural Design. J. Appl. Mech. 85 (1968).Google Scholar
  16. 16.
    Prager, W.: Optimality Criteria in Structural Design. Proc. Nat’l Acad. Sci. 61 (1968).Google Scholar
  17. 17.
    Masur, E. F.: Optimum Stiffness and Strength of Elastic Structures. Proc. ASCE 96, EM5 (1970).Google Scholar
  18. 18.
    Shield, R. T., Prager, W.: Optimal Structural Design for Given Deflection. ZAMP 21 (1970).Google Scholar
  19. 19.
    Shield, R. T.: Optimal Design of Structures through Variational Principles. Lecture Notes in Physics (21): Optimization and Stability Problems in Continuum Mechanics (Ed. P. R. C. Wang), Berlin, Heidelberg, New York: Springer-Verlag 1973.Google Scholar
  20. 20.
    Barnet, R. L.: Minimum-Weight Design of Beams for Deflection. Trans. ASCE 128, Part I (1963).Google Scholar
  21. 21.
    Tang, H. T.: Optimal Design of Elastic Structures, Ph. D. Dissertation, University of California, San Diego, 1971.Google Scholar
  22. 22.
    Huang, N. C.: On a Principle of Stationary Mutual Complementary Energy and its Application to Optimal Structural Design. ZAMP 22 (1971).Google Scholar
  23. 23.
    Prager, W.: Variational Principles of Linear Elastostatics for Discontinuous Displacements, Strains and Stress, Recent Progress in Applied Mechanics, The Folke-Odqvist Volume (Ed. B. Broberg, J. Hult and F. Niordson), Stockholm: Olmqvist & Wiksell 1967.Google Scholar
  24. 24.
    Nemat-Nasser, S.: General Variational Methods for Waves in Elastic Composites. J. Elasticity 2, No. 2 (1972).Google Scholar
  25. 25.
    Oden, J. T.: Finite Elements of Nonlinear Continua, New York: McGraw-Hill 1972.MATHGoogle Scholar
  26. 26.
    Zienkiewicz, O. C., Cheung, Y. K.: The Finite Element Method in Structural and Continuum Mechanics, New York: McGraw-Hill 1967.MATHGoogle Scholar
  27. 27.
    Pian, T. H. H., Tong, P.: Finite Element Methods in Continuum Mechanics. Advances in Applied Mechanics (Ed. Chia-Shun Yih), Vol. 12, Academic Press 1972.Google Scholar
  28. 28.
    Dixon, L. C. W.: Pontryagin’s Maximum Principle Applied to the Profile of a Beam. Aeronautical Journal of the Royal Aeronautical Society 71 (July 1967).Google Scholar
  29. 29.
    Dixon, L. C. W.: Further Comments on Pontryagin’s Maximum Principle Applied to the Profile of a Beam. Aeronautical Journal of the Royal Aeronautical Society 72 (June 1968).Google Scholar
  30. 30.
    Pistefano, N.: Dynamic Programming and the Optimum Design of Rotating Disks. J. Optimization Theory and Applications 10, No. 2 (1972).Google Scholar
  31. 31.
    Armand, J.-L.: Minimum-Mass Design of a Plate-Like Structure for Specified Fundamental Frequency. AIAA J. 9, No. 9 (1971).Google Scholar
  32. 32.
    Charrett, D. E., Rozvany, G. I. N.: Extensions of the Prager-Shield Theory of Optimal Plastic Design. Int. J. Nonlinear Mechanics 7 (1972).Google Scholar

Copyright information

© Springer-Verlag, Berlin/Heidelberg 1975

Authors and Affiliations

  • G. A. Hegemier
    • 1
  • H. T. Tang
    • 2
  1. 1.University of California, San DiegoLa JollaUSA
  2. 2.General Electric Co.San JoséUSA

Personalised recommendations