High-order finite elements reduced models for modal analysis of stiffened panels

  • Paolo Lisandrin
  • Michel van Tooren


A method is presented to build reduced (equivalent) models of stiffened panels made of thin-walled composite materials. The technique is developed to be used in the modal analysis of panels and wing boxes, allowing finite element modelling and analysis using a single-type, three-dimensional orthotropic p-element. The use of a single element guarantees speed and flexibility in the (re)modelling of the structure and reduces the modelling and analysis errors connected to finite element analysis in preliminary-design/multidisciplinary-optimization environments. The method is tested on two types of representative wing boxes. Different approaches for the equivalencing are tested and compared to each other. The results show that the equivalent models give results within few percent from those obtained running a full model, saving as much as one order of magnitude in the number of degrees of freedom employed.


P-elements Modal analysis Equivalent/reduced models Stiffened panels 


  1. Babuska, I., Suri, M.: Locking effects in the finite element approximation of elasticity problems. Numerische Matematik 62, 439–463 (1992a)MATHCrossRefMathSciNetGoogle Scholar
  2. Babuska, I., Suri, M.: On locking and robustness in the finite element method. SIAM J. Numer. Anal. 29, 1261–1293 (1992b)MATHCrossRefMathSciNetGoogle Scholar
  3. Babuska, I., Szabo, B., Katz, I.: The p-version of the finite element method. SIAM J. Numer. Anal. 18(3), 515–545 (1981)MATHCrossRefMathSciNetGoogle Scholar
  4. Bruhn, E.: Analysis and Design of Flight Vehicle Structures. Jacobs Publishing, Inc., USA (1973)Google Scholar
  5. Collier, C.: Stiffness, thermal expansion, and thermal bending formulation of stiffened, fiber-reinforced composite panels. In: AIAA (ed.) AIAA/ASME/ASCE/AHS/ACS 34th Structures, Dynamics, & Materials Conference, La Jolla, CA. Paper No. AIAA-93-1569 (1993)Google Scholar
  6. Collier, C.: Thermoelastic formulation of stiffened, unsymmetric composite panels for finite element analysis of high speed aircraft. In: AIAA/ASME/ASCE/AHS/ACS 35th Structures, Dynamics, & Materials Conference, Hilton Head, SC. Paper AIAA 94-1579 (1994)Google Scholar
  7. Düster, A.: High order finite elements for three dimensional, thin-walled nonlinear continua. PhD dissertation, Technische Universität München – Lehrstuhl für Bauinformatik (2001)Google Scholar
  8. Gere, J., Timoshenko, S.: Mechanics of Materials. Van Nostrand Reinhold Co. Ltd (1987)Google Scholar
  9. Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice Hall, Englewood Cliffs, NJ (1987. Reprinted by Dover, NY, 2000)Google Scholar
  10. Jones, R.: Mechanics of Composite Materials, Chapter 2–4. Taylor & Francis, Philadelphia, PA (1999)Google Scholar
  11. Lisandrin, P., van Tooren, M.: High-order finite elements reduced models for use in a flutter design tool. J. Aircraft 42(3), 748–754 (2005)Google Scholar
  12. Morino, L., Mastroddi, F., Bernardini, G., Piccirilli, M.: A hermite interpolation finite element for structural dynamics. In: AIDAA (ed.) XVI Congresso Nazionale AIDAA, Palermo, Italy. Associazione Italiana di Aeronautica e Astronautica (2001)Google Scholar
  13. Stroud, J.: Elastic constants for bending and twisting of corrugation-stiffened panels. Technical report, NASA TR R-166, Langley, VA (1963)Google Scholar
  14. Stroud, J.: Pasco: structural panel analysis and sizing code, capability and analytical foundations. Technical report, NASA TM 80181, Langley, VA (1981)Google Scholar
  15. Stroud, J., Agranoff, N.: Minimum mass design of filamentary composite panels under combined loads: design procedure based on simplified buckling equations. Technical report, NASA TN D 8257, Langley, VA (1976)Google Scholar
  16. Szabo, B., Sahrmann, G.: Hierarchic plate and shell models based on p-extension. Int. J. Numer. Methods Eng. 26, 1855–1881 (1988)MATHCrossRefGoogle Scholar
  17. Timoshenko, S.P., Woinowsky-Krieger, S.: Theory of Plates and Shells, 2nd edn. McGraw-Hill Book Company, Inc. (1959)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Faculty of Aerospace EngineeringDelft University of TechnologyDelftThe Netherlands

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