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

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


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.


Postbuckling Orthotropic shells Closed-form solution Analytical 


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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

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