Application of the Stability and Weight Criteria

  • N. V. Banichuk


Stability constitutes one of the basic demands that must be satisfied in designing elastic structures. It is particularly important in designs of slender structures or structures made of high-strength materials.


Optimal Design Critical Load Weight Criterion Thickness Function Multiple Eigenvalue 
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References for Chapter 8

  1. 8.1.
    Abramov, IU.Sh., Variational Methods in the Theory of Operator Bundles. Spectral Optimization, Leningrad, Leningrad State University, 1983, 180 pp.Google Scholar
  2. 8.2.
    Adali, S., Optimal shape and nonhomogenuity of nonuniformly compressed column, Intern. J. Solids Struct.,1979, 15, pp. 935–949.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 8.3.
    Albul, A.V., Banichuk, N.V., and Barsuk, A.A., Optimization of stability for elastic rods with thermal loads, Izv. Akad. Nauk S.S.S.R., M.T.T., 1980, No. 3, pp. 127–133.Google Scholar
  4. 8.4.
    Alfumov, N.A., Foundations for Computation of Stability of Elastic Systems,Moscow, Mashinostroienie, 1978, 312 pp.Google Scholar
  5. 8.5.
    Alfumov, N.A., and Balabukh, L.I., Energy criteria for stability of elastic bodies which do not require determination of the initial stressed state, Prikl. Mat. Mekh.,1972, 32, No. 1, pp. 703–770.Google Scholar
  6. 8.6.
    Andreev, L.V., Mossakovskii, V.I., and Obodan, N.I., On optimal thickness of a cylindrical shell loaded by external pressure, Prikl. Mat. Mekh., 1972, 36, No. 4, pp. 717–726.Google Scholar
  7. 8.7.
    Arnol’d, V.I., Modes and quasi-modes, Functional Analysis and Its Applications, 1972, 6, No. 2, pp. 12–20.Google Scholar
  8. 8.8.
    Arnol’d, V.I., Mathematical Methods of Classical Mechanics (translated from Russian by K. Vogtman and A. Weinstein), 1978, Springer-Verlag, Berlin and Heidelberg, 464 pp.Google Scholar
  9. 8.9.
    Artunian, N.Kh., and Kolmanovskii, V.B., Theory of Creep for Inhomogeneous Bodies, Moscow, Nauka, 1983, 336 pp.Google Scholar
  10. 8.10.
    Ashley, H., and McIntosh, S.C., Application of aeroelastic constraints in Structural optimization. In: Proceedings 12th International Congress Applied Mechanics, Stanford University, 1969. Springer Verlag, Berlin, pp. 100–113.Google Scholar
  11. 8.11.
    Banichuk, N.V., Optimization of stability for elastically embedded rod, Izv. Akad. Nauk S.S.S.R., M.T.T., 1974, No. 4, pp. 150–154.Google Scholar
  12. 8.12.
    Banichuk, N.V., Minimization of the weight of a wing with bounded speed of divergence, Nauchn. Zap. TsAGI, 1978, 9, No. 5.Google Scholar
  13. 8.13.
    Banichuk, N.V., and Barsuk, A.A., On stability of torsioned elastic rods, Izv. Akad. Nauk S.S.S.R., M.T.T., 1982, No. 6, pp. 148–154.Google Scholar
  14. 8.14.
    Banichuk, N.V., and Barsuk, A.A., Optimization of stability of elastic beams against simultaneous compression and torsion. In: Applied Problems of Strength and Plasticity. Automation and algorithms for Solving Problems in Elastoplastic State (collection). Gorkii, Gorkii State University, 1982, pp. 122–126.Google Scholar
  15. 8.15.
    Banichuk, N.V., and Barsuk, A.A., Application of decomposition of the spectrum of eigenvalues to problems of optimal design. In: Problems of Stability and Critical Load Carrying Capacity of Structures,Leningrad, L.I.S.I., 1983, pp. 17–24.Google Scholar
  16. 8.16.
    Banichuk, N.V., and Gura, N.M., On a dynamic problem of optimal design. In: Mechanics of Deformable Solid Bodies, Novosibirsk, 1979, 41, pp. 20–24.Google Scholar
  17. 8.17.
    Banichuk, N.V., and Kobelev V.V., Optimization of structures made from chaotically reinforced composites, Mekh. komp. materialov,1981, No. 4, pp. 668–676.Google Scholar
  18. 8.18.
    Bochenek, B., and Gajewski, A., Certain problems of unimodal and bimodal optimal design of structures. In: Proceedings Euromech. In:Optimization Methods in Structural Design, H. Eschenauer and N. Olhoff (eds.), Mannheim, Bibl. Institute, 1983, pp. 204–209.Google Scholar
  19. 8.19.
    Bolotin, V.V., Dynamic Stability of Elastic Systems, Moscow, Gostekhizdat, 1956.Google Scholar
  20. 8.20.
    Bolotin, V.V., Non-conservative Problems in the Theory of Elastic Stability,Moscow, Fizmatgiz, 1961, 339 pp.Google Scholar
  21. 8.21.
    Bolotin, V.V., Variational Principles in Theory of Elastic Stability. In: Problems in Mechanics of Solid Deformable Bodies, Moscow, Subostroienie, 1973, pp. 83–88.Google Scholar
  22. 8.22.
    Budianski, B., Frauenthal, J.C., and Hutchinson, J.W., On optimal arches, J. Applied Mech., Trans. A.S.M.E.,1969, 36, No. 4, pp. 239–240.Google Scholar
  23. 8.23.
    Bushnell, D., Buckling of shells; Pitfall for Designers, A.I.A.A. J., 1981, 19, No. 9, pp. 1183–1226. (Russian translation in Raket. Tekh. Kosmon.). Google Scholar
  24. 8.24.
    Chentsov, P.G., Minimal weight supports, Trudy TsAGI, 1936, 265, pp. 1–48.Google Scholar
  25. 8.25.
    Choi, K.K., and Haug, E.J., Optimization of structures with repeated eigenvalues. In: Optimization of Distributed Parameter Structures, E.J. Haug and J. Cea (eds.), Alphen aan den Rijn, Sijthoff and Noordhoff, 1981, pp. 219–277.Google Scholar
  26. 8.26.
    Clausen, T., Uber die Formarchitektonischen Saulen, Bull. Phys. Math. Acad. St. Petersburg, 1851, 9, pp. 279–294.Google Scholar
  27. 8.27.
    Claudon J.-L., and Sunakawa M., Optimizing distributed structures for maximum flutter load, AIAA J., 1981, 19, No. 7, pp. 957–959.zbMATHCrossRefGoogle Scholar
  28. 8.28.
    Danielov, E.R., Girder with uniform strength against longitudinal-transverse bending, Tr. Khabar. Politechn. In. 1967, 8, pp. 26–34.Google Scholar
  29. 8.29.
    Fedoseev, V.I., Selected Problems and Exercises on Strength of Materials,Moscow, Nauka, 1973, 400 pp.Google Scholar
  30. 8.30.
    Frauenthal, J.C., Constrained optimal design of circular plates against buckling, J. Structural Mech.,1972, 1, pp. 159–186.CrossRefGoogle Scholar
  31. 8.31.
    Gajewski, A., and Lyczkowski, M., Optimal design of elastic columns subject to general conservative behavior of loading, ZAMP, 1970, 21, No. 5, pp. 806–818.zbMATHCrossRefGoogle Scholar
  32. 8.32.
    Geniev, G.A., On the equigradient principle and its applications to problems of optimizing stability of systems of rods, Stroit. Mekh. i Raschet Sooruzhenii, 1979, No. 6, pp. 8–13.Google Scholar
  33. 8.33.
    Grinev, V.B., and Filippov, A.P., On optimal shape of beams in problems of stability, Stroit. Mekh. i Raschet Sooruzhenii, 1977, No. 2, pp. 21–27.Google Scholar
  34. 8.34.
    Grinev, V.B., and Filippov, A.P., Optimization of circular plates in problems of stability, Stroit. Mekh. i Raschet Sooruzhenii, 1977, No. 2, pp. 16–20.Google Scholar
  35. 8.35.
    Haug, E.J., Optimization of distributed parameter structures with repeated eigenvalues. In: New Approaches to Nonlinear Problems in Dynamics, P.J. Holmes (eds), Philadelphia, S.I.A.M., 1980, pp. 511–520.Google Scholar
  36. 8.36.
    Haug, E.J., and Rousselet, B., Design sensitivity analysis in structural mechanics II: Eigenvalues variations, J. Struct. Mech., 1980, 8.Google Scholar
  37. 8.37.
    Hu, K.K., and Kirmser, P.G., Numerical solution of a nonlinear differentialintegral equation for the optimal shape of the tallest column, Intern. J. Eng. Science, 1980, 18, pp. 333–339.zbMATHCrossRefGoogle Scholar
  38. 8.38.
    Iasinskii, F.S., Collected Works on Stability of Compressed Columns,Moscow and Leningrad, Gostekhizdat, 1952, 427 pp.Google Scholar
  39. 8.39.
    Karihaloo, B.L., and Parberry, R.D., Optimal design of beam-columns subjected to concentrated moments, Eng. Optim., 1980, 5, pp. 59–66.CrossRefGoogle Scholar
  40. 8.40.
    Keller, J.B., The shape of the strongest column, Arch. Rational Mechanics and Analysis, 1960, 5, No. 4, pp. 275–285.CrossRefGoogle Scholar
  41. 8.41.
    Keller, J.B., and Niordson, F.I., The tallest column. J. Math. and Mech.,1966, 16, No. 5, pp. 433–446.MathSciNetzbMATHGoogle Scholar
  42. 8.42.
    Komkov, V., Application of invariant principles to the optimal design of a column, ZAMM, 1981, 61, pp. 75–80.MathSciNetzbMATHCrossRefGoogle Scholar
  43. 8.43.
    Komkov, V., and Haug, E.J., On the optimum shape of columns. In: Optimization of Distributed Parameter Structures, E.J. Haug and J. Cea (ed.), Alphen aan den Rijn, Sijthoff and Noordhoff, 1981, pp. 219–277.Google Scholar
  44. 8.44.
    Kruzelecki, J., Optimization of shells under combined loading via the concept of uniform stability. In: Optimization of Distributed Parameter Structures, E.J. Haug and J. Cea (eds.), Alphen aan den Rijn, Sijthoff and Noordhoff, 1981, pp. 929–950.Google Scholar
  45. 8.45.
    Lagrange, J.L., Sur la figure des colonnes, Miscellanea Taurinensia, 1770–1773, 5.Google Scholar
  46. 8.46.
    Kuznetsov, Iu.M., Compression and combined tension and bending of beams with minimal weight, Dopovidi Akad. Nauk Ukr.S.S.R., 1959, No. 14, pp. 372–378.Google Scholar
  47. 8.47.
    Larichev, A.D., Finding a minimum volume for a beam on an elastic foundation, for a given magnitude of a critical load. In: Applied and Theoretical Research into Building Structures, Moscow, Kucherenko TsNIINSK, 1981, pp. 19–25.Google Scholar
  48. 8.48.
    Leipholz, H.H.E., On conservative elastic systems of the first and second kind, Ing. Arch., 1974, 43, No. 5, pp. 255–271.MathSciNetzbMATHCrossRefGoogle Scholar
  49. 8.49.
    Litvinov, V.G., and Medvedev, N.G., Stability problem for shells of revolution and the Ritz-Galerkin techniques, Prikl. Mekhanika,1977, 13, No. 7, pp. 8–16.MathSciNetGoogle Scholar
  50. 8.50.
    Masur, E.F., Optimal design of symmetric structures against postbuckling collapse, Intern. J. Solids Struct., 1978, 14, pp. 319–326.zbMATHCrossRefGoogle Scholar
  51. 8.51.
    Masur, E.F., and Mrôz, Z., On nonstationary optimality conditions in structural design, Intern. J. Solids Struct., 1979, 15, pp. 503–512.zbMATHCrossRefGoogle Scholar
  52. 8.52.
    Masur, E.F., Optimal structural design under multiple eigenvalue constraint, Intern. J. Solids Struct.,1984, 20, No. 3, pp. 211–231.MathSciNetzbMATHCrossRefGoogle Scholar
  53. 8.53.
    McIntosh, S.C., and Ashley, H., On optimization of discrete structures with aeroelastic constraints, Comput. and Structures, 1978, 8, pp. 411–419.zbMATHCrossRefGoogle Scholar
  54. 8.54.
    Medvedev, N.G., Some spectral properties of optimal stability problems for shells with variable thickness, Dokl. Akad. Nauk U.S.S.R., Ser. A, No. 9, pp. 59–63.Google Scholar
  55. 8.55.
    Medvedev, N.G., and Totskii, N.P., Multiplicity of eigenvalues in the spectrum in optimal stability problems for cylindrical shells with variable thickness, Prikl. Mekhanika,1984, 20, No. 6, pp. 113–116.Google Scholar
  56. 8.56.
    Mikisheva, V.I., Optimal weaving of shells made of glassy plastics, subjected to loss of stability due to external pressure or to axial compression, Polymer Mechanics,1968, No. 5, pp. 864–875.Google Scholar
  57. 8.57.
    Naimark, M.A., Linear Differential Operators, Moscow, Nauka, 1969, 526 pp.Google Scholar
  58. 8.58.
    Nemirovskii, Iu.V., and Samsonov, V.I., On rational reinforcement of cylindrical shells compressed by an axial force, Izv. Akad. Nauk S.S.S.R., M.T.T., 1974, No. 1, pp. 103–112.Google Scholar
  59. 8.59.
    Nemirovskii, Iu.V., and Samsonov, V.I., Cylindrical reinforced shells most stable against external pressure, Polymers Mechanics,1974, No. 1, pp. 75–83.Google Scholar
  60. 8.60.
    Nicolai, E.L., Lagrange’s problem on the best shapes of columns, Izv. Peterburg. Polytekh. In., 1907, 5.Google Scholar
  61. 8.61.
    Nicolai, E.L., Research Works in Mechanics, Moscow, Gostekhizdat, 1955, 584 pp.Google Scholar
  62. 8.62.
    Olhoff, N., and Rasmussen, S.H., On single and bimodal optimum buckling loads of clamped columns, Intern. J. Solids Struct., 1977, 13, No. 7, pp. 605–614.zbMATHCrossRefGoogle Scholar
  63. 8.63.
    Olhoff, N., and Taylor, J.E., Designing continuous columns for minimum total cost of material and interior supports, J. Struct. Mech., 1978, 6, pp. 367–382.CrossRefGoogle Scholar
  64. 8.64.
    Papkovich, P.F., Studies on Shipbuilding Structural Mechanics,vol. 4, Leningrad, (Sudostroienie), 1963, 552 pp.Google Scholar
  65. 8.65.
    Pedersen, P., and Seyranian, A.P., Sensitivity analysis for problems of dynamic stability, Intern. J. Solids Struct., 1983, 19, No. 4, pp. 315–335.zbMATHCrossRefGoogle Scholar
  66. 8.66.
    Pierson, B.L., Pannel flutter optimization by gradient projection, Intern. J. Numer. Meth Eng., 1975, 9, pp. 271–296.zbMATHCrossRefGoogle Scholar
  67. 8.67.
    Popelar, C.H., Optimal design of beams against buckling; a potential energy approach, J. Struct. Mech., 1976, 4, pp. 181–196.CrossRefGoogle Scholar
  68. 8.68.
    Popelar, C.H., Optimal design of structures against buckling; a complementary energy approach, J. Struct. Mech., 1977, 5, pp. 45–66.CrossRefGoogle Scholar
  69. 8.69.
    Prager, S., and Prager, W., A note on optimal designs of columns, Intern. J. Mech. Science, 1979, 21, pp. 249–251.CrossRefGoogle Scholar
  70. 8.70.
    Prasad, S.N., and Herman, G., The usefulness of adjoint system in solving nonconservative problems of elastic continua, Intern. J. Solids Struct., 1969, 5.Google Scholar
  71. 8.71.
    Roorda, J., and Reiss A.J., Nonlinear interactive buckling; sensitivity and optimality, J. Struct. Mech.,1977, 5, pp. 207–232.CrossRefGoogle Scholar
  72. 8.72.
    Rzhanitsyn, A.P., Stability of Equilibrium for Elastic Bodies, Moscow, Gostekhizdat, 1955, 476 pp.Google Scholar
  73. 8.73.
    Sandararaian, C., Optimization of a nonconservative elastic system with stability constraint, J. Optimiz. Theory Appl., 1975, 16, pp. 355–378.CrossRefGoogle Scholar
  74. 8.74.
    Schmit, L.A., Jr., and Ramanathan, R., Multilevel approach to minimum weight design including buckling constraints, AIAA J., 1978, 16, No. 2, pp. 97–104.CrossRefGoogle Scholar
  75. 8.75.
    Seyranian, A.P., Optimization of stability for a plate in a supersonic gas stream, Izv. Akad. Nauk S.S.S.R.,M.T.T., 1974, No. 1, pp. 103–112.Google Scholar
  76. 8.76.
    Seyranian, A.P., On a problem of Lagrange, Izv. Akad. Nauk S.S.S.R, M.T.T., 1974, No. 1, pp. 103–112.Google Scholar
  77. 8.77.
    Seyranian, A.P., and Sharaniuk, A.V., Seneitivity analysis for optimization of critical parameters in problems of dynamic stability, lzv. Akad. Nauk S.S.S.R., M.T.T., 1983, No. 5, pp. 173–182.Google Scholar
  78. 8.78.
    Seyranian, A.P., Sensitivity analysis and optimization of aeroelastic stability characteristics, Intern. J. Solids Struct., 1982, 18, No. 9, pp. 791–808.zbMATHCrossRefGoogle Scholar
  79. 8.79.
    Shanley, F.R., Optimum design for eccentrically loaded columns, J. Structural Division, Proc. Amer. Soc. Civ. Eng., 1967, 93, pp. 201–226.Google Scholar
  80. 8.80.
    Smirnov, A.F., Stability and Vibration of Reinforcements, Moscow, Transzheldorizdat, 1958, 572 pp.Google Scholar
  81. 8.81.
    Smirnov, A.F., Beams and arches of least weight in longitudinal bending, Tr. Mosk. In. Zheldor. Transp., 1950, 74, pp. 3–40.Google Scholar
  82. 8.82.
    Sofonov, Iu.D., Computation of the least weight of beams, with consideration of stability for a planar shape of deformed shape, Zap, Kazan’. Aviat. In., Kazan’, 1974, 168, pp. 3–43.Google Scholar
  83. 8.83.
    Solodovnikov, V.N., On techniques of optimization of shells for stability of the stressed state. In: Dynamics of Continuous Media,Novosibirsk, Institute of Hydrodynamics, 1976, 27, pp. 135–143.Google Scholar
  84. 8.84.
    Solodovnikov, V.N., Optimization of elastic shells of revolution, Prikl. Math. Mekh. 1978, 42, No. 3, pp. 511–520.MathSciNetGoogle Scholar
  85. 8.85.
    Stadler, W., Stability of the natural shapes of sinusoidally loaded uniform shallow arches, Quart. J. Mech. Appl. Math., 1983, 33, pt. 3, pp.MathSciNetGoogle Scholar
  86. 8.86.
    Swisterski, W., Wroblewski, W., and Zyczkowski, M., Geometrically nonlinear excentrically compressed columns of uniform creep strength vs optimal columns, Intern. J. Nonlinear Mechanics,1983, 18, No. 4, pp. 287–296.zbMATHCrossRefGoogle Scholar
  87. 8.87.
    Tadjbaksh, I., and Keller, J.B., Strongest columns and isoperimetric inequalities for eigenvalues, J. Appl. Mech., 1962, 29, No. 1, pp. 159–164.Google Scholar
  88. 8.88.
    Taylor, J.E., The strongest column; an energy approach. J. Appl. Mechanics, Trans. A.S.M.E., 1967, 34, No. 2, pp. 486–487.Google Scholar
  89. 8.89.
    Taylor, J.E., Optimal prestress against buckling; an energy approach, Intern. J. Solids Struct., 1971, 7, pp. 213–223.CrossRefGoogle Scholar
  90. 8.90.
    Taylor, J.E., and Liu, C.Y., Optimal design of columns, A.I.A.A. J. 1968, 6, pp. 1496–1502.Google Scholar
  91. 8.91.
    Thomson, J.M.T., Optimization as a generator of structural instability, Intern. J. Mech. Science, 1972, 14, No. 9, pp. 627–630.CrossRefGoogle Scholar
  92. 8.92.
    Thomson, J.M.T., and Hunt, G.W., Dangers of structural optimization Eng. Optim., 1974, 1, pp. 99–100.CrossRefGoogle Scholar
  93. 8.93.
    Thomson, J.M.T., and Supple, W.D., Erosion of optimal design by compound branching phenomena, J. Mech. Phys. Solids, 1972, 21, No. 3, pp. 135–144.CrossRefGoogle Scholar
  94. 8.94.
    Timoshenko, S.P., Stability of beams plates and shells. Selected Works,Moscow, Nauka, 1971, 808 pp.Google Scholar
  95. 8.95.
    Vol’mir, A.S., Stability of Deformed Systems, Moscow, Nauka, 1967, 984 pp.Google Scholar
  96. 8.96.
    Weisshaar, T.A., and Plaut, R.H., Structural optimization under nonconservative loading. In: Optimization of Distributed Parameter Structures, E.J. Haug and J. Cea (eds), Alphen aan den Rijn, Sijthoff and Noordhoff, 1981, pp. 843–864.Google Scholar
  97. 8.97.
    Ziegler, G., On stability of elastic systems. In: Problems of Mechanics, Moscow, Foreign Literature, 1959, pp. 116–160.Google Scholar
  98. 8.98.
    Ziegler, G., Foundations of the Theory of Structural Stability, Moscow, Mir, 1971, 192 pp.Google Scholar
  99. 8.99.
    Zyczkowski, M., Optimal design of shells with respect to their elastic stability. In: Stability of Elastic Structures, H.H.E. Leipholz (ed.), Wien, Springer-Verlag, 1978, pp. 268–291.Google Scholar
  100. 8.100.
    Zyczkowski, M., and Kruzelewski, J., Optimal design of shells with respect to their stability. In: Optimization in Structural Design, IUTAM, A. Sawczuk and Z. Mroz (eds.), Springer-Verlag, Berlin and New York, 1975, pp. 229–247.Google Scholar

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© Springer-Verlag New York, Inc. 1990

Authors and Affiliations

  • N. V. Banichuk
    • 1
  1. 1.Institute for Problems in MechanicsUSSR Academy of SciencesMoscowSoviet Union

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