Advertisement

Plate Bending Macroelements

  • Christopher G. Provatidis
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 256)

Abstract

This chapter deals with plate bending analysis applying several CAD-based interpolations. First the performance of boundary-only Coons interpolation is studied; it will be shown that its simplest form coincides with the well-known BFS element of mid-1960s. Then Gordon interpolation is used (i.e., internal nodes are inserted) in order to improve the accuracy of the numerical solution; it will be shown that the Hermite tensor-product element is a special case. The applicability of Bernstein–Bézier interpolation, as a substitute of Lagrange and Hermite polynomials, is discussed in detail. Also, the use of B-splines is examined and it is clearly shown that the barriers are broken when a control points-based tensor product is applied to curvilinear domains. Numerical examples include rectangular and circular thin plates which are solved using a single macroelement.

Keywords

Plate bending Blending functions C1-continuity Coons formula BFS element Transfinite Tensor-product Hermites Eigenvalues Bernstein–Bézier Beam bending B-splines Rational macroelement Test cases 

References

  1. 1.
    Angelidis D (2005) Development of Coons macroelements for static thin plate-bending analysis, Diploma Work. National Technical University of Athens, School of Mechanical Engineering, July 2005Google Scholar
  2. 2.
    Blaauwendraad J (2010) Plates and FEM: surprises and pitfalls, solid mechanics and its applications (Book 171). Springer, BerlinCrossRefGoogle Scholar
  3. 3.
    Blevins RD (1979) Formulas for natural frequency and mode shape. Van Nostrand, New YorkGoogle Scholar
  4. 4.
    Bogner FK, Fox RL, Schmit LA Jr (1965) The generation of inter-element compatible stiffness and mass matrices by use of interpolation formulas. In: Proceedings of 1st conference on matrix method in structural mechanics, Ohio, pp 397–443 (Oct 1965)Google Scholar
  5. 5.
    Buffa A, Sangalli G (eds) (2016) IsoGeometric analysis: a new paradigm in the numerical approximation of PDEs. Springer International Publishing, SwitzerlandzbMATHGoogle Scholar
  6. 6.
    Carey GF, Oden JT (1983) Finite elements: a second course, vol II. Prentice-Hall, Englewood Cliffs, New Jersey, p 68zbMATHGoogle Scholar
  7. 7.
    Coons SA (1964) Surfaces for computer aided design of space form, Project MAC, MIT (1964), revised for MAC-TR-41 (1967). Springfield, VA 22161, USA: Available as AD 663 504 from the National Technical Information Service (CFSTI), Sills Building, 5285 Port Royal Road. Available online: http://publications.csail.mit.edu/lcs/pubs/pdf/MIT-LCS-TR-041.pdf
  8. 8.
    Cottrell J, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, New YorkCrossRefGoogle Scholar
  9. 9.
    Farin G (1990) Curves and surfaces for computer aided geometric design, 2nd edn. Academic Press, New YorkzbMATHGoogle Scholar
  10. 10.
    Gorman DJ (1978) Free vibration analysis of the completely free rectangular plate by the method of superposition. J Sound Vib 57(3):437–447CrossRefGoogle Scholar
  11. 11.
    Höllig K (2003) Finite element methods with B-Splines. SIAM, PhiladelphiaCrossRefGoogle Scholar
  12. 12.
    Hrabok MM, Hrudey TM (1984) A review and catalogue of plate bending finite elements. Comput Struct 19(3):479–495CrossRefGoogle Scholar
  13. 13.
    Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195MathSciNetCrossRefGoogle Scholar
  14. 14.
    Leissa AW (1973) The free vibration of rectangular plates. J Sound Vib 31(2):257–293CrossRefGoogle Scholar
  15. 15.
    Melosh RJ (1965) A stiffness matrix for the analysis of thin plates in bending. J Aeronaut Sci 28(1):34–42MathSciNetzbMATHGoogle Scholar
  16. 16.
    Oñate E (2013) Structural analysis with the finite element method. linear statics: volume 2: beams, plates and shells. Lecture Notes on Numerical methods in engineering and sciences, vol 2. Springer, BerlinGoogle Scholar
  17. 17.
    Provatidis CG (2018) Engineering analysis with CAD-based macroelements. Arch Appl Mech 88(1–2):121–140CrossRefGoogle Scholar
  18. 18.
    Provatidis CG (2017) B-splines collocation for plate bending eigenanalysis. J Mech Mater Struct 12(4):353–371MathSciNetCrossRefGoogle Scholar
  19. 19.
    Provatidis CG (2012) Two-dimensional elastostatic analysis using Coons-Gordon interpolation. Meccanica 47(4):951–967MathSciNetCrossRefGoogle Scholar
  20. 20.
    Provatidis CG (2008) Plate bending analysis using transfinite interpolation. In: Talaslidis D, Manolis G (eds) Proceedings 6th GRACM international congress on computational mechanics. Thessaloniki, GreeceGoogle Scholar
  21. 21.
    Provatidis CG (2003) Coons-patch macroelements for boundary-value problems governed by fourth-order partial differential equation. In: Lipitakis EA (ed) Proceedings of the ‘sixth hellenic-european conference on computer mathematics and its applications’ (HERCMA 2003), Sept 25–27, 2003, Athens, Greece, pp 634–643, Lea Publications. ISBN: 960-87275-1-0Google Scholar
  22. 22.
    Provatidis CG, Angelidis D (2014) Performance of Coons’ macroelements in plate bending analysis. Int J Comput Methods Eng Sci Mech 15:110–125MathSciNetCrossRefGoogle Scholar
  23. 23.
    Provatidis CG, Isidorou SK (2012) Solution of one-dimensional hyperbolic problems using cubic B-Splines collocation. Int J Comput Sci Appl (IJCSA) 1(1):12–18Google Scholar
  24. 24.
    Ugural AC (1981) Stresses in plates and shells. McGraw-Hill, New YorkGoogle Scholar
  25. 25.
    Weaver W Jr, Timoshenko SP, Young DH (1990) Vibration problems in engineering, 5th edn. Wiley-Interscience, New YorkGoogle Scholar
  26. 26.
    Zhou ZH, Wong KW, Xu XS, Leung AYT (2011) Natural vibration of circular and annular thin plates by Hamiltonian approach. J Sound Vib 330(5):1005–1017CrossRefGoogle Scholar
  27. 27.
    Zienkiewicz OC, Cheung YK (1964) The finite element method for analysis of elastic isotropic and orthotropic slabs. Proc Inst Civ Eng 28:471–488Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringNational Technical University of AthensAthensGreece

Personalised recommendations