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Mathematical Modelling of Instability Phenomena

  • M. Pignataro
  • G. C. Ruta
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 470)

Abstract

Liapunov theory is first presented and discrete mechanical systems are in particular analysed. Then buckling and postbuckling analysis of continuous mechanical system using the general theory formulated by Koiter are discussed in some detail following Budiansky presentation. Finally, the influence of multiple interactive buckling modes on postbuckling behaviour is analysed in some detail for frames, thin-walled members and panels.

Keywords

Equilibrium Path Local Buckling Initial Imperfection Collapse Load Instability Phenomenon 
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Bibliography

  1. S.S. Antman. Bifurcation problems for nonlinear elastic structures. In P.H. Rabinowitz, editor, Applications of Bifurcation Theory. Academic Press, New York, 1977.Google Scholar
  2. R. Benito. Static and Dynamic Buckling of Plate Assemblies. PhD thesis, Washington University in St Louis, 1983.Google Scholar
  3. R. Benito and S. Sridharan. Mode interaction in thin-walled structural members. J. Struct. Mech., 12:517–542, 1984–1985.Google Scholar
  4. R. Benito and S. Sridharan. Interactive buckling: a novel approach and some new results. In Third Intern. Colloq. on Stability of Metal Structures, pages 91–111, Toronto 1983.Google Scholar
  5. E. S. Bernard, R. Q. Bridge, and G.J. Hancock. Design method for profiled steel decks with intermediate stiffeners. J. Constr. Steel Res., 38:61–68, 1996.CrossRefGoogle Scholar
  6. E. S. Bernard, R.Q. Bridge, and G. J. Hancock. Test on profiled steel decks with v-stiffeners. J. Struct. Engng, ASCE, 119:2277–2293, 1993.CrossRefGoogle Scholar
  7. E.S. Bernard, R.Q. Bridge, and G.J. Hancock. Test on profiled steel decks with flat hat stiffeners. J. Struct. Engng, ASCE, 121:1175–1182, 1995.CrossRefGoogle Scholar
  8. M.A. Bradford and G.J. Hancock. Elastic interaction of local and lateral buckling in beams. Thin-Walled Structures, 2:1–25, 1984.CrossRefGoogle Scholar
  9. B. Budiansky. Theory of buckling and post-buckling behaviour of elastic structures. In Advances in Applied Mechanics Advances in Applied Mechanics. Chia-Shun Yih editor, Vol. 14 1–65 Academic Press, New York 1974.Google Scholar
  10. E. Byskov. Elastic buckling problem with infinitely many local modes. Technical Report 327, Technical University of Denmark, Lyngby, 1986.Google Scholar
  11. E. Byskov and J.W. Hutchinson. Mode interaction in axially stiffened cylindrical shells. AIAA J., 15:941–948, 1977.CrossRefGoogle Scholar
  12. R. Casciaro, A. Di Carlo, and Pignataro M. A finite element technique for bifurcation analysis. Technical Report 11-192, Istituto di Scienza della Costruzioni, University of Rome, 1976.Google Scholar
  13. R. Casciaro, G. Garcea, G. Attanasio, and F. Giordano. Perturbation approach to elastic post-buckling analysis. Comput. & Struct., 66/5:585–595, 1998.MATHCrossRefGoogle Scholar
  14. A. Di Carlo. A non-standard format for continuum mechanics. In Batra R. C. and Beatty M. F., editors, Contemporary Research in the Mechanics and Mathematics of Materials. CIMNE, Barcelona, 1996.Google Scholar
  15. A. Di Carlo, M. Pignataro, and N. Rizzi. On the proper treatment of axial and shear undeformability constraints in post-buckling analysis of beams. Int. J. Non-Linear Mechanics, 16(2):221–229, 1981.MATHCrossRefGoogle Scholar
  16. A. Di Lanzo and G. Garcea. Koiter’s analysis of thin-walled structures by a finite element approch. Int. J. Numer. Methods Engng., 39:3007–3031, 1996.MATHCrossRefGoogle Scholar
  17. M. Epstein. Thin-walled beams as directed curves. Acta Mechanica, 33:229–242, 1979.MATHCrossRefMathSciNetGoogle Scholar
  18. F. Gantmacher. Lectures in Analytical Mechanics. MIR, Moscow, 1970.Google Scholar
  19. G. Garcea. Mixed formulation in koiter analysis of thin-walled beams. Comput. Methods Appl. Mech. Engng., 190:3369–3399, 2001.MATHCrossRefGoogle Scholar
  20. P. Germain. La méthode des puissances virtuelles en mécanique des milieux continus, lère partie: La théorie du second gradient. Journal de Mécanique, 12:235–274, 1973a.MATHMathSciNetGoogle Scholar
  21. P. Germain. The method of virtual power in continuum mechanics, part II, application to continuous media with microstructure. SIAM Journal of Applied Mathematics, 23:556–575, 1973b.CrossRefGoogle Scholar
  22. V. Gioncu. General theory of elastic stability. Thin-Walled Structures, 19:81–127, 1994.CrossRefGoogle Scholar
  23. T.R. Graves-Smith. The ultimate strength of locally buckled columns of arbitrary length. In K.C. Rochey and H.V. Hill, editors, Thin-Walled Steel Constructions. Crosy Lockwood, London, 1967.Google Scholar
  24. T.R. Graves-Smith and S. Sridharan. A finite strip method for the buckling of plate structures under arbitrary loading. Int. J. Mech. Sci., 20:685–693, 1978.CrossRefGoogle Scholar
  25. A. Grimaldi and M. Pignataro. Postbuckling behaviour of thin-walled open cross-section compression members. J. Struct. Mech., 7:143–159, 1979.Google Scholar
  26. G.J. Hancock. Local, distorsional and lateral buckling of i-beams. In Proc. ASCE, 104, ST11, pages 1787–1798, 1978.Google Scholar
  27. G.J. Hancock. Nonlinear analysis of thin sections in compression. In Proc. ASCE, 107 ST3, pages 455–471, 1981.Google Scholar
  28. American Iron and Steel Institute. Specifications for the Design of Cold-Formed Steel Structural Members, Washington: AISI, 1980.Google Scholar
  29. W. T. Koiter. Post-buckling analysis of a simple two-bar frame. In The Folke Odkvist volume. Almqvist and Wiksell, Stockholm, John Wiley and Sons, 1967.Google Scholar
  30. W.T. Koiter. On the Stability of Elastic Equilibrium (in Dutch). PhD thesis, H.J. Paris, Amsterdam, 1945, English translation as NASA TT F-10,833, 1967 and AFFDL report TR 70-25, 1970.Google Scholar
  31. W.T. Koiter and G. D. C. Kuiken. The interaction between local buckling and overall buckling on the behaviour of built-up columns. WTHD Report 23, Delft, 1971.Google Scholar
  32. W.T. Koiter and M. Pignataro. A general theory of the interaction between local and overall buckling of stiffened panels. WTHD report 83, Delft, 1976. Technical report.Google Scholar
  33. W.T. Koiter and M. Pignataro. An alternative approach to the interaction between local and overall buckling in stiffened panels. In B. Budiansky, editor, Buckling of structures. Springer-Verlag, New York, 133–148, 1976.Google Scholar
  34. W.T. Koiter and M. Skaloud. Interventión sur le comportement postcritique des plaques utilisées en constructions metalliques. Mem. Soc. R. Sci. Liège, [5] 8(5):64–68,103,104, 1963.Google Scholar
  35. M. Kotelko and Z. Kolakowski. Coupled instabilty related collapse behaviour of channel-section beam-column. In Proc. CIMS 2000, Lisbon, (2000), 205–212.Google Scholar
  36. M. Królak, Z. Kolakowski, and M. Kotelko. Influence of load-nonuniformity and eccentricity on the stability and load carrying of orthotropic tubular columns of regular hexagonal cross-sections. Thin-Walled Structures, 39:483–498, 2001.CrossRefGoogle Scholar
  37. J. La Salle and S. Lefschetz. Stability by Liapunov’s Direct Method with Applications. Academic Press, New York, 1961.MATHGoogle Scholar
  38. L. Leiva-Aravena. Trapezoidally corrugated panels. buckling behaviour under axial compression and shear. Chalmers Univ. of Technology at Gothenburg, Sweden, Publ. 87,1, 1987.Google Scholar
  39. A.M. Liapunov. Problème genéral de la stabilité du mouvement (in Russian)., Karkov, 1892; French traslation in Ann. Fac. Sci. Univ. Tolouse,9,1907; English traslation: Stability of motion, Academic Press, New York, 1966.Google Scholar
  40. R. Luo and B. Edlund. Buckling analysis of trapazeoidally corrugated panels using spline finite strip method. Thin-Walled Structures, 18:209–224, 1994.CrossRefGoogle Scholar
  41. A. Luongo and M. Pignataro. On the use of finite strip method in the analysis of nontraditional problems of thin-walled members. In Proc. VII Italian Corgress of Theoretical and Applied Mechanics AIMETA (in Italian), pages 193–204, 1984.Google Scholar
  42. A. Luongo and M. Pignataro. Buckling and post-buckling analysis of stiffened channels under uniform compression. Costruzioni Metalliche, 4:242–249, 1986.Google Scholar
  43. A. Luongo and M. Pignataro. Multiple interaction and localization phenomenon in post-buckling of compressed thin-walled members. AIAA J., 26:1395–1400, 1988.MATHGoogle Scholar
  44. R. Maquoi and C. Massonnet. Interaction between local plate buckling and overall buckling in thin-walled compression members. theories and experiments. In B. Budiansky, editor, Buckling of structures, pages 350–382. Springer-Verlag New York, 1976.Google Scholar
  45. C.M. Menken, W.J. Groot, and Stallenberg G.A.J. Interactive buckling of beams in bending. Thin-Walled Structures, 12:415–434, 1991.CrossRefGoogle Scholar
  46. J.J. Meyer and A. Van der Neut. The interaction of local buckling and column failure of imperfect thin-walled compression members. Technical Report 160, WTHD, report 160, Delft, 1970.Google Scholar
  47. H. Mollmann. Theory of thin-walled beams with finite displacements. In W. Pietraszkiewicz, editor, EUROMECH Colloquium 197. Springer-Verlag, New York, 195–209, 1986.Google Scholar
  48. K.E. Moxham. Buckling tests on individual welded steel plates in compressions. Technical Report CUED/C-Structures/TR3, Cambridge University Engineering Department, 1971.Google Scholar
  49. Commission of the European Community. Eurocode 8, Design of Steel Structures Part 1.3 Brussels, 1992.Google Scholar
  50. J.P. Papangelis and G. J. Hancock. Computer analysis of thin-walled structural members. Computer Struct., 56:157–176, 1995.MATHCrossRefGoogle Scholar
  51. M. Pignataro, A. Di Carlo, and R. Casciaro. On the non-linear beam models from the point of view of computational post-buckling analysis. Int. J. Solids Struct., 18(4): 327–347, 1982.MATHCrossRefGoogle Scholar
  52. M. Pignataro and A. Luongo. Asymmetric interactive buckling of thin-walled columns with initial imperfections. Thin-Walled Structures, 5:365–386, 1987.CrossRefGoogle Scholar
  53. M. Pignataro and A. Luongo. Simultaneous buckling modes and imperfection sensitivity of channels in compression. In J. Szabó, editor, EUROMECH Colloquium 200, 233–252 1985.Google Scholar
  54. M. Pignataro, A. Luongo, and N. Rizzi. On the effect of the local-overall interaction on the postbuckling of uniformly compressed channels. Thin-Walled Structures, 3: 293–321, 1985.CrossRefGoogle Scholar
  55. M. Pignataro, M. Pasca, and P. Franchin. Post-buckling analysis of corrugated panels in the presence of multiple interacting modes. Thin-Walled Structures, 36:47–66, 2000.CrossRefGoogle Scholar
  56. M. Pignataro and N. Rizzi. On the interaction between local and overall buckling of an asymmetric portal frame. Meccanica, 18:92–96, 1983.MATHCrossRefGoogle Scholar
  57. M. Pignataro and G. C. Ruta. Coupled instabilities in thin-walled beams: a qualitative approach. European Journal of Mechanics A/Solids, 22:139–149, 2002.CrossRefMathSciNetGoogle Scholar
  58. R.J. Plank. Developments in the finite strip method for the buckling analysis of compression members. In Trosième Colloque International “Stabilité des Structures Metalliques”, Paris. 1983.Google Scholar
  59. R.J. Plank and W.H. Wittrick. Buckling under combined loading of thin-flat-walled structures by a complex finite strip method. Int. J. Num. Meth. Engng., 8:323–339, 1974.MATHCrossRefGoogle Scholar
  60. H. Poincaré. Sur l’equilibre d’une masse fluide animée d’un mouvement de rotation. Acta Math., 7:259, 1885.CrossRefMathSciNetGoogle Scholar
  61. L. Pontriaguine. Equations Differentielles Ordinaires. MIR, Moscow, 1969.MATHGoogle Scholar
  62. M. Potier-Ferry. Wavelength selection and pattern localization in buckling problems. In J.E. Wesfreid and S. Zalesky, editors, Cellular Structures in Instability Problems. Lecture Notes in Phisics, Springer-Verlag Berlin, 1984.Google Scholar
  63. E. Reissner. On a simple variational analysis of small finite deformations of prismatical beams. ZAMP, 34:642–648, 1983.MATHCrossRefGoogle Scholar
  64. N. Rizzi and M. Pignataro. The effect of multiple buckling modes on the postbuckling behaviour of plane elastic frames. part I: Symmetric frames, part II: Asymmetric frames. J. Struct. Mech., 10:437–458 and 459–474, 1982.Google Scholar
  65. N. Rizzi and A. Tatone. Nonstandard models for thin-walled beams with a view to applications. J. Appl. Mech., 63:399–403, 1996.MATHGoogle Scholar
  66. J. Roorda. Stability of structures with small imperfections. J. Eng. Mech. Div. ASCE, 91, No. EMl,:Proc. paper 4230,87, 1965.Google Scholar
  67. J. C. Simo and L. Vu-Quoc. A geometrically exact rod model incorporating shear and torsion-warping deformation. Int. J. Solids Struct., 27:371–393, 1991.MATHCrossRefMathSciNetGoogle Scholar
  68. S. Sridharan. Doubly symmetric interactive buckling of plate structures. Int. J. Solids Struct, 19:625–641, 1983.MATHCrossRefGoogle Scholar
  69. S. Sridharan and A. Ali. Interactive buckling in thin-walled beam-columns. ASCE, 111,EM12:1470–1486, 1985.Google Scholar
  70. S. Sridharan and R. Benito. Columns: static and dynamic interactive buckling. ASCE, 110EMl:49–65, 1984.Google Scholar
  71. A. Tatone and N. Rizzi. A one-dimensional model for thin-walled beams. In H. Troger W. Schneider and F. Ziegler, editors, Trends in Applications of Mathematics to Mechanics. Longman, Avon: 312–320, 1991.Google Scholar
  72. J. M. T. Thompson, J. D. Tulk, and A. C. Walker. An experimental study of imperfection-sensitivity in the interactive buckling of stiffened plates. In B. Budiansky, editor, Buckling of structures. Springer-Verlag, New York, 149–159, 1976.Google Scholar
  73. S. P. Timoshenko and J. M. Gere. Theory of Elastic Stability. McGraw-Hill New York, 1961.Google Scholar
  74. C. Truesdell and W. Noll. The non-linear field theories of mechanics. In Handbuch der Physik III/3. Springer-Verlag, New York, 1965.Google Scholar
  75. V. Tvergaard. Imperfection sensitivity of a wide integrally stiffened panel under compression. Int. J. Solids Struct, 9:177–192, 1973.CrossRefMATHGoogle Scholar
  76. V. Tvergaard and A. Needleman. On the localization of buckling patterns. J. Appl. Mech., 47:613–619, 1980.CrossRefGoogle Scholar
  77. A. Van Der Neut. Mode interaction with stiffened panels. In B. Budiansky, editor, Buckling of Structures. Springer-Verlag, New York, 117–132, 1976.Google Scholar
  78. A. Van der Neut. The interaction of local buckling and column failure of thin-walled compression members. In Proc. 12th International Congress of Applied Mechanics, pages 389–399, Springer Verlag, Berlin 1969.Google Scholar
  79. B. L. Van der Waerden. Modern Algebra, Vol. II. Ungar, 1950.Google Scholar
  80. V.I. Vlasov. Thin-walled elastic beams, 2nd ed. the Israel Program for Scientific Translations, Jerusalem 1961.Google Scholar
  81. H. Yoshida and K. Maegawa. Local and member buckling of h-columns. J. Struct. Mech., 6:1–27, 1978.Google Scholar

Copyright information

© CISM, Udine 2005

Authors and Affiliations

  • M. Pignataro
    • 1
  • G. C. Ruta
    • 1
  1. 1.University of Roma “La Sapienza”RomaItaly

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