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Computational Mechanics

, Volume 15, Issue 1, pp 1–18 | Cite as

Nonlinear shell formulations for complete three-dimensional constitutive laws including composites and laminates

  • M. Braun
  • M. Bischoff
  • E. Ramm
Originals

Abstract

One objective of the present study is to use arbitrary complete 3-dimensional constitutive equations without reduction or manipulation in nonlinear plate and shell analysis. The obvious consequence, namely the extension of a conventional 5-parameter shell formulation with Reissner-Mindlin kinematics to a 6-parameter formulation including the full set of stress and strain state does not solve the problem because a significant error in bending dominated cases occurs. To avoid this error the transverse normal strain is allowed to vary linearly across the thickness. This so-called 7-parameter theory recently proposed in the group of the authors resorts to the Enhanced Assumed Strain concept and preserves the basic features of a displacement formulation.

The 7-parameter formulation is extended to the simulation of the response of laminated structures with arbitrarily large displacements and rotations. Following the main idea of the concept, it is sufficient to formulate a complete 3-dimensional material law which considers the layered setup of the shell.

Finally, a layer-wise so-called multidirector model is developed which is well suited to grasp local interlaminar effects. In this formulation the displacement interpolation across the thickness is extended to a C0-continuous field described by a layer-wise Reissner-Mindlin kinematics. The purpose of the multidirector formulation is twofold: Firstly the higher order kinematics satisfies the same requirements discussed for the 7-parameter theory and allows also to use complete 3-dimensional material laws. Secondly it is appropriate to simulate laminates with extreme differences of the thicknesses or dissimilar material properties of each layer with sufficient accuracy. These are situations where formulations with C1-continuous displacement fields across the thickness fail.

Keywords

Laminate Structure Shell Formulation Dissimilar Material Extreme Difference Displacement Formulation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Andelfinger, U. 1991: Untersuchungen zur Zuverlässigkeit hybrid gemischter finiter Elemente für Flächentragwerke. Ph.D. Dissertation. Bericht Nr. 13, Institut für Baustatik, Universität StuttgartGoogle Scholar
  2. Andelfinger, U.; Ramm, E. 1993: EAS-elements for 2D-,3D-, plate and shell structures and their equivalence to HR-elements. Int. J. Numer. Methods Engng. 36: 1311–1337Google Scholar
  3. Basar, Y.; Krätzig, W. B. 1985: Mechanik der Flächentragwerke. Vieweg & Sohn, BraunschweigGoogle Scholar
  4. Basar, Y.; Ding, Y.; Schultz, R. 1993: Refined shear-deformation models for composite laminates with finite rotations. Int. J. Solids Structures 30(19): 2611–2638Google Scholar
  5. Büchter, N.; Ramm, E. 1992a: Shell theory versus degeneration—a comparison in large rotation finite element analysis. Int. J. Numer. Methods Engng. 34: 39–59Google Scholar
  6. Büchter, N. 1992: Zusammenführung von Degenerationskonzept und Schalentheorie bei endlichen Rotationen. Ph.D. Dissertation, Bericht Nr. 14 des Instituts für Baustatik, Universität StuttgartGoogle Scholar
  7. Büchter, N.; Ramm, E. 1992b: 3D-extension of nonlinear shell equations based on the enhanced assumed strain concept. In: Hirsch, Ch. (ed.) Computational Methods in Applied Sciences. Elsevier Science Publishers B.V. 55-62Google Scholar
  8. Büchter, N.; Ramm, E.; Roehl, D. 1994: Three-dimensional extension of nonlinear shell formulation based on the enhanced assumed strain concept. (to appear in Int. J. Numerical Methods in Engineering)Google Scholar
  9. Di, S.; Ramm, E. 1993: Hybrid stress formulation for higher-order theory of laminated shell analysis. Computer Methods Appl. Mech. Engng. 109: 359–376Google Scholar
  10. DiSciuva, M. 1985: Development of an anisotropic, multilayered, shear-deformable rectangular plate element. Computers and Structures 21(4): 789–796Google Scholar
  11. Dorninger, K. 1991: A nonlinear layered shell finite lement with improved transverse shear behavior. Composites Engineering 1(4): 211–224Google Scholar
  12. Epstein, M.; Glockner, P. G. 1977: Nonlinear analysis of multilayered shells. Int. J. Solids Strucutres 13: 1081–1089Google Scholar
  13. Epstein, M.; Huttelmaier, H. P. 1983: A finite element formulation for multilayered and thick plates. Computers and Struct. 16(5): 645–650Google Scholar
  14. Erki, M. A.; Rizkalla, S. H. 1993: FRP reinforcement for concrete structures. Concrete International 15(6): 48–53Google Scholar
  15. Green, A. E.; Zerna, W. 1968: Theoretical elasticity. Clarendon Press, second editionGoogle Scholar
  16. Huttelmaier, H. P.; Epstein, M. 1985: A finite element formulation for multilayered and thick shells. Computers and Structures 21(6): 1181–1185Google Scholar
  17. Huttelmaier, H. P.; Epstein, M. 1990: A large displacement finite element for multilayered plates. Finite Elements in Analysis and Design 6: 189–196Google Scholar
  18. Khalifa, M. A.; Kuska, S. S. B.; Krieger, J. 1993: Bridges constructed using fiber reinforced plastics. Concrete International 15(6): 43–47Google Scholar
  19. Maier, M. 1990: Experimentelle Untersuchung und numerische Simulation des Crachverhaltens von Faserverbundwerkstoffen. Ph.D. Dissertation, Fachbereich Maschinenwesen, Universität KaiserslauternGoogle Scholar
  20. Noor, A. K.; Burton, W. S. 1989: Assessment of shear deformation theories for Multilayered composite plates. Appl. Mech. Reviews 42(1): 1–13Google Scholar
  21. Owen, D. R. J.; Li, Z. H. 1987: A refined analysis of laminated plates by finite element displacement methods—I. Fundamentals and static analysis. Computers and Structures 26(6): 907–914Google Scholar
  22. Pagano, N. J. 1970: Exact solutions for rectangular bidirectional composites and sandwich plates. J. Composite Materias 4: 20–34Google Scholar
  23. Pinsky, P. M.; Kim, K. O. 1986: A multi-director formulation for elastic-viscoelastic layered shells. Int. J. Numer. Methods Engng. 23: 2213–2244Google Scholar
  24. Reddy, J. N. 1984a: Energy and variational methods in applied mechanics (with an introduction to the finite element method). John Wiley, New YorkGoogle Scholar
  25. Reddy, J. N. 1984b: A simple higher-order theory for laminated composite plates. J. Appl. Mech. 51: 745–752Google Scholar
  26. Reddy, J. N.; Pandey, A. K. 1987: A first-ply failure analysis of composite laminates. Computers and Structures 25(4): 371–393Google Scholar
  27. Rohwer, K. 1992: Application of higher order theories to the bending analysis of layered composite plates. Int. J. Solids Structures 29(1): 105–119Google Scholar
  28. Simo, J. C.; Rifai, S. 1990: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numerical Methods in Engineering 29: 1595–1638Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • M. Braun
    • 1
  • M. Bischoff
    • 1
  • E. Ramm
    • 1
  1. 1.Institut für BaustatikUniversity of StuttgartStuttgartGermany

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