Advertisement

Closed-form Approximate Solution for the Postbuckling Behavior of Orthotropic Shallow Shells Under Axial Compression

  • Matthias BeerhorstEmail author
  • Michael Seibel
  • Christian Mittelstedt
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 15)

Abstract

The current paper deals with a closed-form approximate solution for the postbuckling behavior of an unstiffened, singly-curved, orthotropic shell. As loading condition the case of uniform axial compression is treated. Concerning the boundary conditions all edges are supposed to be simply supported. Additionally, geometrical imperfections in form of an initial deflection of the shell can be accounted for. Choosing rather simple shape functions for the deflection a closed-form expression for the Airy stress function is obtained from the compatibility condition. As the equilibrium condition cannot be satisfied exactly the solution procedures of Galerkin as well as Ritz are employed to obtain an approximate solution. The resulting expressions from these procedures again allow for a closed-form solution of the load-deflection-relationship. After the force and the amplitude are known all other state variables such as stresses and displacements can be evaluated in a closed-form manner. Due to the rather simple formulation of the deflection shape the algorithm is limited to cases where the qualitative shape of the deflection does not change significantly. On the other hand the very high computational efficiency of the described solution procedure makes it ideally suited for use in the field of optimization and preliminary design, if the applied load does not exceed the linear buckling load too much.

keywords

Postbuckling Orthotropic shells Closed-form solution Analytical 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bürmann, P.: A semi-analytical model for the post-buckling analysis of stringer- and frame-stiffened cylindrical panels under combined loading. Ph.D. thesis, Technische Universität Braunschweig (2006)Google Scholar
  2. 2.
    Byklum, E.: Ultimate strength analysis of stiffened steel and aluminium panels using semi-analytical methods. Ph.D. thesis, NTNU Trondheim (2002)Google Scholar
  3. 3.
    Diaconu, C.G., Weaver, P.M.: Approximate solution and optimum design of compression-loaded, postbuckled laminated composite plates. AIAA Journal 43(4) (2005)Google Scholar
  4. 4.
    Diaconu, C.G., Weaver, P.M.: Postbuckling of long unsymmetrically laminated composite plates under axial compression. International Journal of Solids and Structures 43, 6978–6997 (2006)CrossRefGoogle Scholar
  5. 5.
    Donnell, L.H.: Stability of thin-walled tubes under torsion. NACA Report 479 (1933) Google Scholar
  6. 6.
    Flügge, W.: Stresses in shells. 2nd edn. Springer, Berlin (1973)Google Scholar
  7. 7.
    Jaunky, N., Knight, N.F.: An assessment of shell theories for buckling of circular cylindrical laminated composite panels loaded in axial compression. International Journal of Solids and Structures 36, 3799–3820 (1999)CrossRefGoogle Scholar
  8. 8.
    Koiter W.T.: A consistent first approximation in general theory of thin elastic shells. The theory of thin elastic shells. In: Proceedings IUTAM Symposium, pp. 12–33, Delft (1959)Google Scholar
  9. 9.
    Kollar, L., Springer, G.: Mechanics of composite structures. 2nd edn. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  10. 10.
    Love, A.: A treatise on the mathematical theory of elasticity. 4th edn. Dover Publication, New York (1927)Google Scholar
  11. 11.
    Marguerre, K.: Die mittragende breite der gedrückten platte. Luftfahrtsforschung 14 3, 121–128 (1937)Google Scholar
  12. 12.
    Mittelstedt, C., Schröder, K.-U.: Postbuckling of compressively loaded imperfect composite plates: closed-form approximate solutions. International Journal of Structural Stability and Dynamics 10, 761–778 (2010)CrossRefGoogle Scholar
  13. 13.
    Reddy, J.: Mechanics of laminated composite plates and shells, 2nd edn. CRC Press, Boca Raton (2004)Google Scholar
  14. 14.
    Sanders, J.: An improved first-approximation theory for thin shells. NASA-TR-R-24 (1959) Google Scholar
  15. 15.
    Shin, D.K., Griffin, O.H., Gürdal, Z.: Postbuckling response of laminated plates under uniaxial compression. International Journal of Non-Linear Mechanics 28, 95–115 (1993)CrossRefGoogle Scholar
  16. 16.
    Simitses, G.J., Shaw, D., Sheinman, I.: Stability of cylindrical shells, by various nonlinear shell theories. Zeitschrift für Angewandte Mathematik und Mechanik 65, 159–166 (1985)CrossRefGoogle Scholar
  17. 17.
    Wiedemann, J.: Leichtbau-Elemente und Konstruktion. 3rd edn. Klassiker der Technik. Springer, Berlin (2007)Google Scholar
  18. 18.
    Zou, G., Qiao, P.: Higher-Order finite strip method for postbuckling analysis of imperfect composite plates. Journal of Engineering Mechanics (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Matthias Beerhorst
    • 1
    Email author
  • Michael Seibel
    • 1
  • Christian Mittelstedt
    • 2
  1. 1.HAW HamburgHamburgGermany
  2. 2.ELAN GmbH, Team Method and ToolsHamburgGermany

Personalised recommendations