Composite (FGM’s) Beam Finite Elements

  • Justín Murín
  • Vladimír Kutiš
  • Michal Masný
  • Rastislav Ďuriš
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 9)


The composite structures (e.g. laminate, sandwich structures, or FGM’s) are often used in engineering applications. Their FE analyses require creating very fine mesh of elements even for relatively small sized bodies, what increases computational time, particularly in nonlinear analyses. Macro-mechanical modelling of the composites is based on material properties homogenisation. The homogenisation of the material properties is made for three layers sandwich bar with constant material properties of middle layer and polynomial variation of effective elasticity modulus and volume fraction of fibre and matrix at the top/bottom layer. In derivation of the bar element matrices the effective longitudinal elasticity modulus have been considered. The uni-axially polynomial variation fibre elasticity modulus E f and the matrix elasticity modulus E m are given as polynomials. In the numerical examples/analyses we assume a three-layers two-node sandwich bar with double symmetric rectangular cross-section. As a typical example of geometric nonlinear behaviour the three-hinge mechanism was analysed. Two different approaches have been considered for calculation of the effective longitudinal elasticity modulus of the composite (FGM’s) bar with both polynomial variation of constituent’s volume fraction and polynomial longitudinal variation of the elasticity modulus. Stiffness matrix of the composite bar contains transfer constants, which accurately describe the polynomial uni-axially variation of effective Young’s modulus. The obtained results are compared with 3D analysis in the ANSYS simulation program. Findings show good accuracy and effectiveness of this new finite element. The results obtained with this element do not depend on the mesh density.


Stiffness Matrix Beam Element Sandwich Beam Effective Elasticity Modulus Transfer Constant 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Y. Bansal, M.J. Pindera, Efficient Reformulation of the Thermoelastic Higher-Order Theory for FGMs, NASA, 2002.Google Scholar
  2. 2.
    M. Koizumi, FGM activity in Japan, Composites, Part B(28B), 1997.Google Scholar
  3. 3.
    M. Koizumi, M, Niino, Overview of FGM Research in Japan, MRS Bulletin (20), 19–21, 1995.Google Scholar
  4. 4.
    J.C. Halpin, J.L. Kardos, The Halpin-Tsai equations. A review. Polymer Engineering and Science 16(5), 344–352, 1976.CrossRefGoogle Scholar
  5. 5.
    T. Mori, K. Tanaka, Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica 21, 571–574, 1973.CrossRefGoogle Scholar
  6. 6.
    J. Fish, W. Chen, Y. Tang, Generalized mathematical homogenisation of atomistic media at finite temperatures. International Journal of Multiscale Computational Engineering 3(4), 393–413, 2005.CrossRefGoogle Scholar
  7. 7.
    W.K. Liu, E.G. Karpov, H.S. Park, Nano Mechanics and Materials: Theory, Multiple Scale Analysis, and Applications. Wiley, 2005.Google Scholar
  8. 8.
    J. Murín, V. Kutiš, Beam element with continuous variation of the cross-sectional area, Computers and Structures 80, 329–338, 2002.CrossRefMathSciNetGoogle Scholar
  9. 9.
    V. Kutiš, J. Murín, Bar element with variation of cross-section for geometric non-linear analysis, Journal of Computational and Applied Mechanics 6, 83–94, 2005.MATHMathSciNetGoogle Scholar
  10. 10.
    J. Murín, V. Kutiš, M. Masný, An effective solution of electro-thermo-structural problem of uni-axially graded material, Structural Engineering and mechanics, 28, 695–713, 2007.Google Scholar
  11. 11.
    J. Murín, V. Kutiš, Extended mixture rules for the composite (FGM’s) beam finite elements, prepared for publication, 2007.Google Scholar
  12. 12.
    J. Aboudi, M.J. Pindera, S.M. Arnold. Higher-order theory for functionally graded materials. Composites Part B: Engineering 30(8): 777–832, December 1999.CrossRefGoogle Scholar
  13. 13.
    A. Chakraborty, S. Gopalakrishnan, J.N. Reddy. A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical Sciences, 45, 519–539, 2003.MATHCrossRefGoogle Scholar
  14. 14.
    J. Murín, Implicit non-incremental FEM equations for non-linear continuum. Strojnícky časopis 52(3), 2001.Google Scholar
  15. 15.
    H. Rubin, Analytische Lösung linearer Differentialgleichungen mit veränderlichen Koeffizienten und baustatische Anwendung. Bautechnik, 76, 1999.Google Scholar
  16. 16.
    V. Kutiš, Beam element with variation of cross-section satisfying local and global equilibrium conditions, Ph.D. thesis, Slovak University of Techno1ogy, Bratislava, 2001.Google Scholar
  17. 17.
    H. Altenbach, J. Altenbach, W. Kissing, Mechanics of Composite Structural Elements, Springer, 2004.Google Scholar
  18. 18.
    J. Murín, V. Kutiš, Geometrically non-linear truss element with varying stiffness, Engineering Mechanics 13(6), 435–452, 2006.Google Scholar
  19. 19.
    Ansys Theory Manual 2004.Google Scholar
  20. 20.
    J. Murín, V. Kutiš, Improved mixture rules for the composite (FGM’s) sandwich beam finite element. In: IX International Conference on Computational Plasticity COMPLAS IX, E. Onate and D.R.J. Owen (eds.), CIMNE, Barcelona, 2007.Google Scholar
  21. 21.
    B.M. Love, R.C. Batra, Determination of effective thermomechanical parameters of a mixture of two elastothermoviscoplastic constituents. International Journal of Plasticity 22, 1026–1061, 2006.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V 2008

Authors and Affiliations

  • Justín Murín
    • 1
  • Vladimír Kutiš
    • 1
  • Michal Masný
    • 1
  • Rastislav Ďuriš
    • 2
  1. 1.Department of Mechanics, Faculty of Electrical Engineering and Information TechnologySlovak University of TechnologyBratislavaSlovakia
  2. 2.Department of Applied Mechanics, Faculty of Material Sciences and TechnologySlovak University of TechnologyTrnavaSlovak Republic

Personalised recommendations