Advertisement

Classical Tools in Structural Optimization

  • Raphael T. Haftka
  • Zafer Gürdal
Chapter
Part of the Solid Mechanics And Its Applications book series (SMIA, volume 11)

Abstract

Classical optimization tools used for finding the maxima and minima of functions and functionals have direct applications in the field of structural optimization. The words ‘classical tools’ are implied here to encompass the classical techniques of ordinary differential calculus and the calculus of variations. Exact solutions to a few relatively simple unconstrained or equality constrained problems have been obtained in the literature using these two techniques. It must be pointed out, however, that such problems are often the result of simplifying assumptions which at times lack realism, and result in unreasonable configurations. Still, the consideration of such problems is not a purely academic exercise, but is very helpful in the process of solving more realistic problems.

Keywords

Design Variable Lagrange Multiplier Differential Calculus Truss Structure Variational Calculus 
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]
    Hancock, H., Theory of Maxima and Minima. Ginn and Company, New York, 1917.Google Scholar
  2. [2]
    Gelfand, I.M. and Fomin, S.V., Calculus of Variations. Prentice Hall, Inc., Englewood Cliffs, NJ, 1963.Google Scholar
  3. [3]
    Pars, L.A., An Introduction to the Calculus of Variations. Heinmann, London, 1962.zbMATHGoogle Scholar
  4. [4]
    Hildebrand, F. B., Methods of Applied Mathematics. Prentice-Hall, New Jersey, 1965.zbMATHGoogle Scholar
  5. [5]
    Reddy, J.N., Energy and Variational Methods in Applied Mechanics. John Wiley and Sons, New York, 1984.zbMATHGoogle Scholar
  6. [6]
    Barnett, R.L., “Minimum Weight Design of Beams for Deflection,” J. EM Division, ASCE, Vol. EMI, 1961, pp. 75–95.Google Scholar
  7. [7]
    Makky, S.M. and Ghalib, M.A., “Design for Minimum Deflection,” Eng. Opt., 4, pp. 9–13, 1979.CrossRefGoogle Scholar
  8. [8]
    Taylor, J.E., and Bendsøe, M.P.,, “An Interpretation for Min-Max Structural Design Problems Including a Method for Relaxing Constraints,” International Journal of Solids and Structures, 30, 4, pp. 301–314, 1984.CrossRefGoogle Scholar
  9. [9]
    Prager, W. and Taylor, J.E., “Problems of Optimal Structural Design,” J. Appl. Mech. 35, pp. 102–106, 1968.zbMATHCrossRefGoogle Scholar
  10. [10]
    Prager, W., “Optimization of Structural Design,” J. Optimization Theory and Applications, 6, pp. 1–21, 1979.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Washizu, K., Variational Methods in Elasticity and Plasticity. 2nd ed. Pergamon Press, 1975.Google Scholar
  12. [12]
    Keller, J.B., “The Shape of the Strongest Column,” Arch. Rat. Mech. Anal. 5, pp. 275–285, 1960.CrossRefGoogle Scholar
  13. [13]
    Tadjbaksh, I. and Keller, J.B., “Strongest Columns and Isoperimetric Inequalities for Eigenvalues,” J. Appl. Mech. 29, pp. 159–164, 1962.CrossRefGoogle Scholar
  14. [14]
    Keller, J.B. and Niordson, F.I., “The Tallest Column,” J. Math. Mech., 29, pp. 433–446, 1966.MathSciNetGoogle Scholar
  15. [15]
    Huang, N.C. and Sheu, C.Y., “Optimal Design of an Elastic Column of Thin-Walled Cross Section,” J. Appl. Mech., 35, pp. 285–288, 1968.CrossRefGoogle Scholar
  16. [16]
    Taylor, J.E., “The Strongest Column — An Energy Approach,” J. Appl. Mech., 34, pp. 486–487, 1967.CrossRefGoogle Scholar
  17. [17]
    Salinas, D., On Variational Formulations for Optimal Structural Design. Ph.D. Dissertation, University of California, Los Angeles, 1968.Google Scholar
  18. [18]
    Simitses, G.J., Kamat, M.P. and Smith, C.V., Jr., “The Strongest Column by the Finite Element Displacement Method,” AIAA Paper No: 72-141, 1972.Google Scholar
  19. [19]
    Hornbuckle, J.C., On the Automated Optimal Design of Constrained Structures. Ph.D. Dissertation, University of Florida, 1974.Google Scholar
  20. [20]
    Turner, H.K. and Plaut, R.H., “Optimal Design for Stability under Multiple Loads,” J. EM Div. ASCE 12, pp. 1365–1382, 1980.Google Scholar
  21. [21]
    Olhoff, N.J. and Rasmussen, H., “On Single and Bimodal Optimal Buckling Modes of Clamped Columns,” Int. J. Solids and Structures, 13, pp. 605–614, 1977.zbMATHCrossRefGoogle Scholar
  22. [22]
    Masur, E.F., “Optimal Structural Design under Multiple Eigenvalue Constraints,” Int. J. Solids Structures, 20, pp. 211–231, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Masur, E.F., “Some Additional Comments on Optimal Structural Design under Multiple Eigenvalue Constraints,” Int. J. Solids Structures, 21, pp. 117–120, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Olhoff, N.J., “Structural Optimization by Variational Methods,” in Computer Aided Structural Design: Structural and Mechanical Systems (C.A. Mota Soares, Editor), Springer Verlag, pp. 87–164, 1987.Google Scholar
  25. [25]
    Plaut, R.H., Johnson, L.W. and Olhoff, N., “Bimodal Optimization of Compressed Columns on Elastic Foundations,” J. Appl. Mech., 53, pp. 130–134, 1986.zbMATHCrossRefGoogle Scholar
  26. [26]
    Frauenthal, J.C., “Constrained Optimal Design of Circular Plates against Buckling,” J. Struct. Mech., 1, pp. 159–186, 1972.CrossRefGoogle Scholar
  27. [27]
    Kamat, M.P., Optimization of Structural Elements for Stability and Vibration. Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA, 1972.Google Scholar
  28. [28]
    Armand, J.L. and Lodier, B., “Optimal Design of Bending Elements,” Int. J. Num. Meth. Eng., 13, pp. 373–384, 1978.zbMATHCrossRefGoogle Scholar
  29. [29]
    Simitses, G.J., “Optimal Versus the Stiffened Circular Plate,” AIAA J., 11, pp. 1409–1412, 1973.CrossRefGoogle Scholar
  30. [30]
    Haftka, R.T. and Prasad, B., “Optimum Structural Design with Plate Bending Elements — A Survey,” AIAA J., 19, pp. 517–522, 1981.zbMATHCrossRefGoogle Scholar
  31. [31]
    Olhoff, N., “On Singularities, Local Optima and Formation of Stiffeners in Optimal Design of Plates,” In: Optimization in Structural Design, A. Sawczuk and Z. Mroz (eds.). Springer-Verlag, 1975, pp. 82–103.Google Scholar
  32. [32]
    Gajewski, A., and Zyczkowski, M., Optimal Structural Design under Stability Constraints, Springer Science+Business Media Dordrecht, 1988.Google Scholar
  33. [33]
    Niordson, F.I., “On the Optimal Design of a Vibrating Beam,” Quart. Appl. Math., 23, pp. 47–53, 1965.MathSciNetGoogle Scholar
  34. [34]
    Turner, M.J., “Design of Minimum-Mass Structures with Specified Natural Frequencies,” AIAA J., 5, pp. 406–412, 1967.zbMATHCrossRefGoogle Scholar
  35. [35]
    Taylor, J.E., “Minimum-Mass Bar for Axial vibration at Specified Natural Frequency,” AIAA J., 5, pp. 1911–1913, 1967.CrossRefGoogle Scholar
  36. [36]
    Zarghamee, M.S., “Optimum Frequency of Structures,” AIAA J., 6, pp. 749–750, 1968.CrossRefGoogle Scholar
  37. [37]
    Brach, R.M., “On Optimal Design of Vibrating Structures,” J. Optimization Theory and Applications, 11, pp. 662–667, 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    Miele, A., Mangiavacchi, A., Mohanty, B.P. and Wu, A.K., “Numerical Determination of Minimum Mass Structures with Specified Natural Frequencies,” Int. J. Num. Meth. Engng., 13, pp. 265–282, 1978.zbMATHCrossRefGoogle Scholar
  39. [39]
    Kamat, M.P. and Simitses, G.J., “Optimum Beam Frequencies by the Finite Element Displacement Method,” Int. J. Solids and Structures, 9, pp. 415–429, 1973.zbMATHCrossRefGoogle Scholar
  40. [40]
    Kamat, M.P., “Effect of Shear Deformations and Rotary Inertia on Optimum Beam Frequencies,” Int. J. Num. Meth. Engng., 9, pp. 51–62, 1975.zbMATHCrossRefGoogle Scholar
  41. [41]
    Pierson, B.L., “A Survey of Optimal Structural Design under Dynamic Constraints,” Int. J. Num. Meth. Engng., 4, pp. 491–499, 1972.CrossRefGoogle Scholar
  42. [42]
    Kiusalaas, J., “An Algorithm for Optimal Structural Design with Frequency Constraints,” Int. J. Num. Meth. Engng., 13, pp. 283–295, 1978.zbMATHCrossRefGoogle Scholar
  43. [43]
    Icerman, L.J., “Optimal Structural Design for given Dynamic Deflection,” Int. J. Solids and Structures, 5, pp. 473–490, 1969.CrossRefGoogle Scholar
  44. [44]
    Plaut, R.H., “Optimal Structural Design for given Deflection under Periodic Loading,” Quart. Appl. Math., 29, pp. 315–318, 1971.MathSciNetzbMATHGoogle Scholar
  45. [45]
    Shield, R.T. and Prager, W., “Optimal Structural Design for given Deflection,” Z. Angew. Math. Phys., 21, pp. 513–523, 1970.zbMATHCrossRefGoogle Scholar
  46. [46]
    Mróz, Z., “Optimal Design of Elastic Structures subjected to Dynamic, Harmonically Varying Loads,” Z. Angew. Math. Mech., 50, pp. 303–309, 1970.zbMATHCrossRefGoogle Scholar
  47. [47]
    Olhoff, N., “Optimal Design of Vibrating Circular Plates,” Int. J. Solids and Structures, 6, pp. 139–156, 1970.zbMATHCrossRefGoogle Scholar
  48. [48]
    Olhoff, N., “Optimal Design of Vibrating Rectangular Plates,” Int. J. Solids and Structures, 10, p. 93–109, 1974.zbMATHCrossRefGoogle Scholar
  49. [49]
    Kamat, M.P., “Optimal Thin Rectangular Plates for Vibration,” Recent Advances in Engineering Science, Vol. 3. Proceedings of the 10th Annual Meeting of the Society of Engineering Science, pp. 101–108, 1973.Google Scholar
  50. [50]
    Armand, J.L., Lurie, K.A. and Cherkaev, A.V., “Existence of Solutions of the Plate Optimization Problem,” Proceedings of the International Symposium Structural Design, Tucson, AZ, pp. 3.1-3.2, 1981.Google Scholar
  51. [51]
    Balasubramanyam, K. and Spillers, W.R., “Examples of the Use of Fourier Series in Structural Optimization,” Quart. of Appl. Math., 3, pp. 559–566, 1986.MathSciNetGoogle Scholar
  52. [52]
    Parbery, R.D., “On Minimum-Area Convex Shapes of given Torsional and Flexural Rigidity,” Eng. Opt., 13, pp. 189–196, 1988.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1992

Authors and Affiliations

  • Raphael T. Haftka
    • 1
  • Zafer Gürdal
    • 2
  1. 1.Department of Aerospace and Ocean EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  2. 2.Department of Engineering Science and MechanicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA

Personalised recommendations