After a very short description of Pontryagin’s maximum principle and sensitivity analysis as applied to eigenvalue problems, a unified approach to column optimization has been presented. Particular attention has been paid to multimodal solutions obtained for compressed columns in an elastic medium and elastically clamped columns for buckling in two planes. Next, a general statement of the optimization of arches has been formulated. The necessity of multimodal optimization was pointed out, especially if in-plane and out-of-plane buckling was taken into account. The sensitivity analysis has been applied to a new optimization problem of annular plates compressed by uniformly distributed non-conservative forces. Both the precritical membrane state and the small transverse vibration has been taken into account. Finally, the parametrical optimization of a visco-elastic column compressed by a follower force with respect to its dynamic stability, as well as the optimization of a plane bar system in conditions of internal resonance has been considered.


Design Variable Annular Plate Stability Constraint Follower Force Optimal Structural Design 
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  1. 1.
    Gajewski, A., Zyczkowski, M.: Optimal structural design under stability constraints, Kluwer Academic Publishers, Dordrecht-Boston-London 1988.CrossRefMATHGoogle Scholar
  2. 2.
    Pontryagin, L.S., Boltyanskii, V. G., Gamkrelidze, R.V., Mishchenko, E.F.: The mathematical theory of optimal processes (in Russian), Fizmatgiz, Moskva 1961. English translation: Wiley, New York 1962.Google Scholar
  3. 3.
    Pedersen, P.: A unified approach to optimal design, in: Optimization methods in structural design, Euromech-Colloquium 164, Siegen 1982 (Eds. H. Eschenauer, N. Olhoff), Bibliogr. Inst., Mannheim-Wien-Zurich 1983, 182–187.Google Scholar
  4. 4.
    Adelman, H. M., Haftka, R. T.: Sensitivity analysis of discrete structural systems, AIAA Journal, 24 (1986), 5, 823–832.CrossRefGoogle Scholar
  5. 5.
    Haug, E.J., Arora, J.S.: Applied optimal design, Viley, New York 1979.Google Scholar
  6. 6.
    Haug, E. J., Choi, K. K., Komkov, V.: Design sensitivity analysis of structural systems, Academic Press, New York 1986.MATHGoogle Scholar
  7. 7.
    Wittrick, W. H.: Rates of change of eigenvalues with reference to buckling and vibration problems, J. Roy. Aero. Soc., 66 (1962), 590–591.Google Scholar
  8. 8.
    Plaut, R.H., Huseyin, K.: Derivatives of eigenvalues and eigenvectors in non-selfadjoint systems, AIAA Journal, 11 (1973), 2, 250–251.CrossRefMathSciNetGoogle Scholar
  9. 9.
    Farshad, M.: Variation of eigenvalues and eigen-functions in continuum mechanics, AIAA Journal, 12 (1974), 4, 560–561.CrossRefMATHGoogle Scholar
  10. 10.
    Haug, E.J., Rousse1et, B.: Design sensitivity analysis of eigenvalue variations, in: Optimization of distributed parameter structures (Eds. E.J. Haug and J. Cea), Vol. I–II, Sijthoff and Noordhoff, Alphen aan den Rijn 1981.Google Scholar
  11. 11.
    Seyranian, A.P., Sharanyuk, A.V.: Sensitivity and optimization of critical parameters in dynamic stability problems (in Russian), Mekh.Tv.Tela, 18 (1983), 5,174–183.Google Scholar
  12. 12.
    Claudon, J.-L., Sunakawa, M.: Sensitivity analysis for continuous mechanical systems governed by double eigenvalue problems, in: Optimization of distributed parameter structures (Eds. E.J. Haug and J. Cea), Vol.I–II, Sijthoff and Noordhoff, Alphen aan den Rijn 1981.Google Scholar
  13. 13.
    Pedersen, P., Seyranian, A.P.: Sensitivity analysis for problems of dynamic stability, Int.J.Solids Structures 19 (1983), 4, 315–335.CrossRefMATHGoogle Scholar
  14. 14.
    Szefer, G.: Analiza wrazliwoéci i optymalizacja ukladów dynamicznych z roziozonymi parametrami, Zesz.Nauk. AGH, Krakow, Mechanika, 1 (1982), 4, 5–36.Google Scholar
  15. 15.
    Demyanov, V.F., Malozemov, V.N.: Introduction to minimax (in Russian), Nauka, Moskva 1972. English translation: Wiley 1974.Google Scholar
  16. 16.
    Blachut, J., Gajewski, A.: A unified approach to optimal design of columns, Solids Mech.Archives, 5 (1980), 4, 363–413.MATHGoogle Scholar
  17. 17.
    Haug, E.J.: Optimization of distributed parameter structures with repeated eigenvalues, in: New Approaches to Nonlinear Problems in Dynamics, (ed. P.J. Holmes), SIAM Publications, 1980.Google Scholar
  18. 18.
    Plaut, N.: Optimal design with respect to structural eigenvalues, Proc.15th Int.Congr.Theor. and Appl.Mech. (Toronto 1980), Prepr. Amsterdam 1980, 133–149.Google Scholar
  19. 19.
    Weishaar, T.A., Plaut, R.H.: Structural optimization under nonconservative loading, in: Optimization of distributed parameter structures (Eds. E.J. Haug and J. Cea), Vol.I–II, Sijthoff and Noordhoff, Alphen aan den Rijn 1981.Google Scholar
  20. 20.
    Olhoff, N., Rasmussen, S.H.: On single and bimodal optimum buckling loads of clamped columns, Int.J.Solids Structures, 13 (1977), 7, 605–614.CrossRefMATHGoogle Scholar
  21. 21.
    Tadjbakhsh, I., Keller, J.B.: Strongest columns and isope-rimetric inequalities for eigenvalues, J.Appl.Mech, 29 (1962), 1, 159–164.CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Gajewski, A., Życzkowski, M.: Optimal design of elastic columns subject to the general conservative behaviour of loading, ZAMP, 21 (1970), 5, 806–818.CrossRefMATHGoogle Scholar
  23. 23.
    Farshad, M., Tadjbakhsh, I.: Optimum shape of columns with general conservative end loading, JOTA, 11 (1973), 4, 413–420.CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Gajewski, A.: A note on unimodal and bimodal optimal design of vibrating compressed columns, Int.J.Mech.Sei. 23 (1981), 1, 11–16.CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Bochenek, B., Gajewski, A.: Jednomodalna i dwumodalna optymalizacja éciskanych prętów drgających, Mech.Teor.Stos., 22 (1984), 1/2, 185–195.Google Scholar
  26. 26.
    Gajewski, A.: Bimodal optimization of a column in an elastic medium with respect to buckling or vibration, Int.J.Mech.Sci., 21 (1985), 1/2, 45–53.CrossRefGoogle Scholar
  27. 27.
    Kiusalaas, J.: Optimal design of structures with buckling constraints, Int.J.Solids Structures, 9 (1973), 7, 863–878.CrossRefMathSciNetGoogle Scholar
  28. 28.
    Repin, S.I.: Shape optimization of a bar on elastic foundation for multiple solutions (in Russian), Prikl. Mat., Tula 1979, 44–30.Google Scholar
  29. 29.
    Larichev, A.D.: Problem of optimization of a clamped beam on elastic foundation (in Russian), Issled. po Stroit.Konstr., Moskva, 1982.Google Scholar
  30. 30.
    Plaut, R.H., Johnson, L.W., Olhoff, N.: Bimodal optimization of compressed columns on elastic foundations, J.Appl. Mech., 53 (1986), 3, 130–134.CrossRefMATHGoogle Scholar
  31. 31.
    Shin, Y.S., Plaut, R.H., Haftka, R.T.: Simultaneous analysis and design for eigenvalue maximization, Proc. of the AIAA/ASME/ASGE/AHS Structures, Structural Dynamics and Materials Conference, Monterey, California, Vol.1, April 1987, 334–342.Google Scholar
  32. 32.
    Shin, Y.S., Haftka, R.T., Vatson, L.T., Plaut, R.H.: Tracing structural optima as a function of available resources by a homotopy method, Int. J.Comp.Meth.Appl.Mech. and Ing., (in print).Google Scholar
  33. 33.
    Bochenek, B.: Multimodal optimal design of a compressed column with respect to buckling in two planes, Int.J. Solids Structures, 23 (1987), 3, 599–603.CrossRefMATHGoogle Scholar
  34. 34.
    Bochenek, B., Nowak, M.: Optymalne kształtowanie slupów z uwagi na wyboczenie w dwóch piaszczyznach (submitted to print).Google Scholar
  35. 35.
    Prager, S., Prager, W.: A note on optimal design of columns, Int.J.Mech.Sci., 21 (1979), 4, 249–231.CrossRefGoogle Scholar
  36. 36.
    O1hoff, N.: Optimization of columns against buckling, in: Optimization of distributed parameter structures (Eds. E.J. Haug and J. Cea), Vol.I–II, Sijthoff and Noordhoff, Alphen aan den Rijn 1981.Google Scholar
  37. 37.
    Lam, H.L., Haug, E. J., Choi, K. K.: Optimal design of structures with constraints on eigenvalues, Materials Division, The University of Iowa, Techn. Report No 79, Jan. 1981, 1–71.Google Scholar
  38. 38.
    Choi, K.K., Haug, E.J.: Repeated eigenvalues in mechanical optimization problems, (Meeting on Probl.Elastic Stab. and Vibr., Pittsburgh 1981), Providence 1981, 61–86.Google Scholar
  39. 39.
    Haug, E. J., Choi, K. K.: Systematic occurence of repeated eigenvalues in structural optimization, JOTA, 38 (1982), 2, 251–274.CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Teschner, W.: Minimum weight design for structural eigenvalue problems by optimal control theory, in: Optimization methods in structural design, Euromech-Golloquium 164, Siegen 1982 (Eds. H. Eschenauer, N. Olhoff), Bibliogr. Inst., Mannheim-Wien-Zurich 1983, 424–429.Google Scholar
  41. 41.
    Banichuk, N.V., Barsuk, A.A.: On a certain method of optimization of elastic stability in the case of multiple critical loadings (in Russian), Prikl.Probl.Prochn.Plast., 24 (1983), 85–89.Google Scholar
  42. 42.
    Seyranian, A.P.: On a certain solution of a problem of Lagrange (in Russian), Dokl.AN SSSR, 271 (1983), 3, 337–340.Google Scholar
  43. 43.
    Masur, E.F.: Optimal structural design under multiple eigenvalue constraints, Int.J.Solids Structures, 20 (1984), 3, 211–231.CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Biachut, J., Gajewski, A.: On unimodal and bimodal optimal design of funicular arches, Int.J.Solids Structures, 17 (1981), 7, 653–667.CrossRefGoogle Scholar
  45. 45.
    Błachut, J., Gajewski, A.: Unimodal and bimodal optimal design of extensible arches with respect to buckling and; vibration, Optimal Control Appl. Meth., 2 (1981), 4, 383–402.CrossRefMATHGoogle Scholar
  46. 46.
    Błachut, J.: Unimodalna optymalizacja drgających i nararażonych na utratę statecznoéci iuków o osi wydluéalnej, Rozpr.Inz., 30 (1982), 1, 37–55.MATHGoogle Scholar
  47. 47.
    Bochenek, B., Gajewski, A.: Optimal design of funicular arches with respect to in-plane and out-of-plane buckling, J. Struct. Mech., 14 (1986), 3, 257–274.CrossRefGoogle Scholar
  48. 48.
    Bochenek, B.,Gajewski, A.: Multimodal optimization of arches under stability constraints with two independent design functions, Int.J.Solids Structures, (in print).Google Scholar
  49. 49.
    Bochenek, B.: On multimodal parametrical optimization of arches against plane and spatial buckling, Eng. Optim., 14 (1988), 27–37.CrossRefGoogle Scholar
  50. 50.
    Plaut, R.H., Olhoff, N.: Optimal forms of shallow arches with respect to vibration and stability, J.Struct.Mech., 11 (1983), 1, 81–100.CrossRefGoogle Scholar
  51. 51.
    Olhoff, N., Plaut, R.H.: Bimodal optimization of vibrating shallow arches, Int.J.Solids Structures, 19 (1983), 6, 553–570.CrossRefMATHGoogle Scholar
  52. 52.
    Rakowski, G., Solecki, R.: Pręty zakrzywione: obliczenia statyczne, Arkady, Warszawa 1963.Google Scholar
  53. 53.
    Schmidt, R.: Postbuckling behaviour of uniformly compressed circular arches with clamped ends, ZAMP, 30 (1979), 353–356.CrossRefGoogle Scholar
  54. 54.
    Irie, T., Yamada, G., Takahashi, I.: In plane vibration of Timoshenko arcs with variable cross-section, Ing.-Archiv, 48 (1979), 5, 337–346.CrossRefMATHGoogle Scholar
  55. 55.
    Blachut, J.: Analiza stateczności pryzmatycznych łuków o osi odkształcalnej, Mech.Teor.Stos., 20 (1982), 1/2, 141–157.MATHGoogle Scholar
  56. 56.
    Suzuki, K., Kosawada, T., Takahashi, S.: Out-of-plane vibrations of curved bars with varying cross-section, Bull. JSME, 26 (1983), 212, 268–275.CrossRefGoogle Scholar
  57. 57.
    Nikolai, E.L.: On the stability of a circular ring and of a circular arch under uniformly distributed normal loading (in Russian), Izv.Petrogradskogo Polit.Inst., 27 (1918).Google Scholar
  58. 58.
    Ponomarev, S. D., Biderman, V.L., Likharev, K.K., Makushin, V.M., Malinin, N.N., Feodosyev, V.I.: Fundamentals of contemporary methods of strength calculations in mashine design (in Russian), Mashgiz, Moskva 1952/1954, C1957/1959).Google Scholar
  59. 59.
    Ojalvo, M., Demuts, E., Tokarz, F.: Out-of-plane buckling of curved members, Proc.ASCE, J.Struct.Div., 96 (1969), ST10, 2305–2316.Google Scholar
  60. 60.
    Tadjbakhsh, I., Farshad, M.: On conservatively loaded funicular arches and their optimal design, in: Optimization in structural design (Eds. A. Sawczuk and Z. Mróz), IUTAM Symposium, Warsaw 1973, Springer, Berlin-New York 1973.Google Scholar
  61. 61.
    Olhoff, N.: Bimodality in optimizing the shape of a vibrating shallow arch, in: Optimization methods in structural design, Euromech-Colloquium 164, Siegen 1982 (Eds. H. Eschenauer and N. Olhoff), Bibliogr. Inst., Mannheim-Wien-Zurich 1983, 182–187.Google Scholar
  62. 62.
    Biachut, J.: Parametrical optimal design of funicular arches against buckling and vibration, Int.J.Mech.Sci., 26 (1984), 5, 305–310.CrossRefGoogle Scholar
  63. 63.
    Pierson, B.L.: Panel flutter optimization by gradient projection, Int. J. Num. Meth. Engng., 9 (1975), 271–296.CrossRefMATHGoogle Scholar
  64. 64.
    Seyranian, A.P.; Optimization of structures subjected to aeroelastic instability phenomena, Arch.Mech.Stos., 34 (1982), 2, 133–146.MATHGoogle Scholar
  65. 65.
    Frauenthal, J.C.: Constrained optimal design of circular plates against buckling, J.Struct.Mech., 1 (1972), 2, 115–127.CrossRefGoogle Scholar
  66. 66.
    Grinev, V.B., Filippov, A.P.: Optimal design of circular plates against buckling (in Russian), Stroit.Mekh.Rasch. Sooruzh., (1972), 2, 16–20.Google Scholar
  67. 67.
    Rzegocihska-Peiech, K., Waszczyszyn, Z.: Numerical optimum design of elastic annular plates with respect to buckling, Computers and Structures, 18 (1984), 2, 369–378.CrossRefGoogle Scholar
  68. 68.
    Irie, T., Yamada, G., Kaneko, Y.: Vibration and stability of a non-uniform annular plate subjected to a follower force, J.Sound and Vibration, 73 (1980), 2, 261–269.CrossRefMATHGoogle Scholar
  69. 69.
    Gajewski, A., Cupiał P.: Optimal structural design of an annular plate compressed by non-conservative forces (submitted to print).Google Scholar
  70. 70.
    Bolotin, B.B.: Dynamic stability of elastic systems (in Russian), Gostekhizdat, Moskva 1956. English translation: Holden-Day, San Francisco 1964.Google Scholar
  71. 71.
    Volmir, A.S.: Stability of deformable systems (in Russian), Nauka, Moskva 1967.Google Scholar
  72. 72.
    Zyczkowski, M., Gajewski, A.: Optimal structural design in non-conservative problems of elastic stability, in: Instability of continuous systems (Ed. H.H.E. Leipholz), IUTAM Symposium, Herrenalb 1969, Springer, Berlin-Heidelberg-New York 1971.Google Scholar
  73. 73.
    Claudon, J.-L.: Characteristic curves and optimum design of two structures subjected to circulatory loads, Journal de Mécanique 14 (1975), 3, 531–543.MATHGoogle Scholar
  74. 74.
    Hanaoka, M., Washizu, K.: Optimum design of Beck’s column, Comp, and Struct., 11 (1980), 6, 473–480.CrossRefMATHMathSciNetGoogle Scholar
  75. 75.
    Foryś, A., Gajewski, A.: Parametryczna optymalizacja pręta lepkosprężystego ze względu na stateczność dynamiczną, Rozpr.Inż., 35 (1987), 2, 297–308.MATHGoogle Scholar
  76. 76.
    Foryś, Anna: Vibrations and dynamical stability of some system of rods in nonlinear approach, Nonlin.Vibr.Problems, 22 (1984), 213–231.Google Scholar
  77. 77.
    Foryś, Anna: Periodic and non-periodic combination resonance in a non-linear system of rods, J.Sound and Vibration, 105 (1986), 3, 461–472.CrossRefGoogle Scholar
  78. 78.
    Foryś, Anna, Nizioł, J.: Internal resonance in a plane system of rods, J.Sound and Vibration, 95 (1984), 3, 361–374.CrossRefMATHGoogle Scholar
  79. 79.
    Foryś, Anna, Gajewski, A.: Analiza i optymalizacja układu prętowego o zmiennych przekrojach w warunkach rezonansu wewnętrznego, Rozpr.Inż., 32 (1984), 4, 575–598.MATHGoogle Scholar
  80. 80.
    Foryś, Anna, Foryś, A.: Optymalizacja parametryczna układu prętów z uwzględnieniem nieliniowości. Rozpr. Inż., 34 (1986), 4, 399–518.MATHGoogle Scholar

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© Springer-Verlag Wien 1989

Authors and Affiliations

  • A. Gajewski
    • 1
  1. 1.Technical University of CracowCracowPoland

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