Robust Displacement and Mixed CUF-Based Four-Node and Eight-Node Quadrilateral Plate Elements

  • Thi Huyen Cham LeEmail author
  • Michele D’Ottavio
  • Philippe Vidal
  • Olivier Polit
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 81)


This paper presents two classes of new four-node and eight-node quadrilateral finite elements for composite plates. Variable kinematics plate models are formulated in the framework of Carrera’s Unified Formulation, which encompass Equivalent Single Layer as well as Layer-Wise models, with the variables that are defined by polynomials up to 4th order along the thickness direction z. The two classes refer to two variational formulations that are employed to derive the finite elements matrices, namely the Principle of Virtual Displacement (PVD) and Reissner’s Mixed Variational Theorem (RMVT). For the PVD based elements, the main novelty consists in the extension of two field compatible approximations for the transverse shear strain field, referred to as QC4 and CL8 interpolations, which eliminate the shear locking pathology by constraining only the ɀ–constant transverse shear strain terms, to all variable kinematics plate elements. Moreover, for the first time the QC4 and CL8 interpolations are introduced for the transverse shear stress field within RMVT based elements. Preliminary numerical studies are proposed on homogeneous isotropic plates that demonstrate the absence of spurious modes and of locking problems as well as the enhanced robustness with respect to distorted element shapes. The new QC4 and CL8 variable kinematics plate elements display excellent convergence rates and yield accurate responses for both, thick and thin plates.


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  1. Barut A, Madenci E, Tessler A (2013) C0-continuous triangular plate element for laminated composite and sandwich plates using the {2,2}-Refined Zigzag Theory. Compos Struct 106:835–853Google Scholar
  2. Bathe KJ, Dvorkin EN (1985) A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. Int J Numer Meth Eng 21:367–383Google Scholar
  3. Batoz JL, Dhatt G (1990) Modélisation des Structures par Eléments Finis, volume 2: Poutres et Plaques. HermèsGoogle Scholar
  4. Batoz JL, Lardeur P (1989) A discrete shear triangular nine d.o.f. element for the analysis of thick to very thin plates. Int J Numer Meth Eng 28:533–560Google Scholar
  5. Belytschko T, Liu WK, Moran B (2000) Nonlinear Finite Elements for Continua and Structures. John Wiley & Sons, LTD, 2 editionGoogle Scholar
  6. Bletzinger KU, Bischoff M, Ramm E (2000) A unified approach for shear locking free triangular and rectangular shell finite elements. Comput Struct 75:321–334Google Scholar
  7. Botshekanan Dehkordi M, Cinefra M, Khalili SMR, Carrera E (2013) Mixed LW/ESL models for the analysis of sandwich plates with composite faces. Compos Struct 98:330–339Google Scholar
  8. Carrera E (1998) Evaluation of layerwise mixed theories for laminated plates analysis. AIAA J 36:830–839Google Scholar
  9. Carrera E (2000) Single- vs multilayer plate modelings on the basis of Reissner’s mixed theorem. AIAA J 38:342–352Google Scholar
  10. Carrera E (2001) Developments, ideas and evaluations based upon Reissner’s Mixed Variational Theorem in the modeling of multilayered plates and shells. Appl Mech Rev 54:301–329Google Scholar
  11. Carrera E (2003a) Historical review of zig-zag theories for multilayered plates and shells. Appl Mech Rev 56:287–308Google Scholar
  12. Carrera E (2003b) Theories and finite elements for multilayered plates and shells: A unified compact formulation with numerical assessment and benchmarking. Arch Comput Meth Eng 10:215–296Google Scholar
  13. Carrera E (2004) On the use of Murakami’s zig-zag function in the modeling of layered plates and shells. Comput Struct 82:541–554Google Scholar
  14. Carrera E, Demasi L (2002a) Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 1: Derivation of finite element matrices. Int J Numer Meth Eng 55(2):191–231Google Scholar
  15. Carrera E, Demasi L (2002b) Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: Numerical implementations. Int J Numer Meth Eng 55(3):253–291Google Scholar
  16. Carrera E, Cinefra M, Nali P (2010) MITC technique extended to variable kinematic multilayered plate elements. Compos Struct 92:1888–1895Google Scholar
  17. Carrera E, Pagani A, Petrolo M (2013) Use of Lagrange multipliers to combine 1D variable kinematic finite elements. Comput Struct 129:194–206Google Scholar
  18. Carrera E, Cinefra M, Petrolo M, Zappino E (2014) Finite Element Analysis of Structures through Unified Formulation. John Wiley & Sons, LtdGoogle Scholar
  19. Carrera E, Cinefra M, Lamberti A, Petrolo M (2015) Results on best theories for metallic and laminated shells including layer-wise models. Compos Struct 126:285–298Google Scholar
  20. Carrera E, Pagani A, Valvano S (2017) Multilayered plate elements accounting for refined theories and node-dependent kinematics. Compos B 114:189–210Google Scholar
  21. Chinosi C, Cinefra M, Della Croce L, Carrera E (2013) Reissner’s mixed variational theorem toward MITC finite elements for multilayered plates. Compos Struct 99:443–452Google Scholar
  22. Cinefra M, Chinosi C, Della Croce L (2013) MITC9 shell elements based on refined theories for the analysis of isotropic cylindrical structures. Mech Adv Mater Struct 20:91–100Google Scholar
  23. Cinefra M, Chinosi C, Della Croce L, Carrera E (2014) Refined shell finite elements based on RMVT and MITC finite elements for the analysis of laminated structures. Compos Struct 113:492–497Google Scholar
  24. Demasi L (2008) ∞3 hierarchy plate theories for thick and thin composite plates: The generalized unified formulation. Compos Struct 84:256–270Google Scholar
  25. Demasi L (2010) Invariant finite element model for composite structures: The generalized unified formulation. AIAA J 48:1602–1619Google Scholar
  26. Demasi L (2012) Partially Zig-Zag advanced higher order shear deformation theories based on the Generalized Unified Formulation. Compos Struct 94:363–375Google Scholar
  27. Demasi L (2013) Partially LayerWise advanced Zig Zag and HSDT models based on the Generalized Unified Formulation. Eng Struct 53:63–91Google Scholar
  28. D’Ottavio M (2016) A Sublaminate Generalized Unified Formulation for the analysis of composite structures and its application to sandwich plates bending. Compos Struct 142:187–199Google Scholar
  29. D’Ottavio M, Ballhause D, Wallmersperger T, Kröplin B (2006) Considerations on higher-order finite elements for multilayered plates based on a unified formulation. Comput Struct 84:1222–1235Google Scholar
  30. D’Ottavio M, Dozio L, Vescovini R, Polit O (2016) Bending analysis of composite laminated and sandwich structures using sublaminate variable-kinematic Ritz models. Compos Struct 155:45–62Google Scholar
  31. Feng W, Hoa SV (1998) Partial hybrid finite elements for composite laminates. Finite Elem Anal Des 30:365–382Google Scholar
  32. Ferreira AJM (2005) Analysis of composite plates using a layerwise shear deformation theory and multiquadrics discretization. Mech Adv Mater Struct 12:99–112Google Scholar
  33. Hoa SV, Feng W (1998) Hybrid Finite Element Method for Stress Analysis of Laminated Composites. Springer Science+Business Media, LLC, New YorkGoogle Scholar
  34. Hu H, Belouettar S, Potier-Ferry M, Daya EM (2009) Multi-scale modelling of sandwich structures using the Arlequin method - Part I: Linear modelling. Finite Elem Anal Des 45:37–51Google Scholar
  35. Hughes TJR (1987) The Finite Element Method. Prentice-HallGoogle Scholar
  36. Hughes TJR, Tezduyar T (1981) Finite elements based upon Mindlin plate theory with particular reference to the four node bilinear isoparametric element. J Appl Mech 46:587–596Google Scholar
  37. Kulikov GM, Plotnikova SV (2016) A hybrid-mixed four-node quadrilateral plate element based on sampling surfaces method for 3D stress analysis. Int J Numer Meth Eng 108:26–54Google Scholar
  38. Le THC, D’Ottavio M, Vidal P, Polit O (2017) A new robust quadrilateral four-node variable kinematics plate element for composite structures. Finite Elem Anal Des 113:10–24Google Scholar
  39. Li MS (1989) Higher order laminated composite plate analysis by hybrid finite element method. PhD thesis, Massachusetts Institute of TechnologyGoogle Scholar
  40. MacNeal RH (1982) Derivation of element stiffness matrices by assumed strain distributions. Nucl Eng Des 70:3–12Google Scholar
  41. Mijuca D (2004) On hexahedral finite element HC8/27 in elasticity. Comput Mech 33:466–480Google Scholar
  42. Murakami H (1986) Laminated composite plate theory with improved in-plane response. J Appl Mech 53:661–666Google Scholar
  43. N N (2016) Abaqus Theory Manual. Dassault SystèmesGoogle Scholar
  44. Park KC, Pramono E, Stanley GM, Cabiness HA (1989) The ANS shell elements: earlier developments and recent improvements. vol 3 of CED, pp 217–239Google Scholar
  45. Pian THH, Li MS (1990) Stress analysis of laminated composites by hybrid finite elements. Springer-VerlagGoogle Scholar
  46. Pian THH, Sumihara K (1995) State-of-the-art development of hybrid/mixed finite element method. Finite Elem Anal Des 21:5–20Google Scholar
  47. Polit O, Touratier M, Lory P (1994) A new eight-node quadrilateral shear-bending plate finite element. Int J Numer Meth Eng 37:387–411Google Scholar
  48. Polit O, Vidal P, D’Ottavio M (2012) Robust C 0 high-order plate finite element for thin to very thick structures: mechanical and thermo-mechanical analysis. Int J Numer Meth Eng 90:429–451Google Scholar
  49. Rao MK, Desai YM (2004) Analytical solutions for vibrations of laminated and sandwich plates using mixed theory. Compos Struct 63:361–373Google Scholar
  50. Reddy JN (1984) A simple higher-order theory for laminated composite plates. J Appl Mech 51:745–752Google Scholar
  51. Reddy JN (1987) A generalization of two-dimensional theories of laminated composite plates. Comm Appl Numer Meth 3:173–180Google Scholar
  52. Reddy JN (1993) An evaluation of equivalent-single-layer and layerwise theories of composite laminates. Compos Struct 25:21–35Google Scholar
  53. Reddy JN (2004) Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, 2nd editionGoogle Scholar
  54. Reissner E (1984) On a certain mixed variational theorem and a proposed application. Int J Numer Meth Eng 20:1366–1368Google Scholar
  55. Robbins Jr DH, Reddy JN (1993) Modelling of thick composites using a layerwise laminate theory. Int J Numer Meth Eng 36:655–677Google Scholar
  56. Robbins Jr DH, Reddy JN (1996) Variable kinematic modeling of laminated composite plates. Int J Numer Meth Eng 39:2283–2317Google Scholar
  57. Sayyad AS, Ghugal YM (2015) On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results. Compos Struct 129:177–201Google Scholar
  58. Somashekar BR, Prathap G, Babu CR (1987) A field-consistent, four-noded, laminated, anisotropic plate/shell element. Comput Struct 25:345–353Google Scholar
  59. Sun CT, Whitney JM (1973) Theories for the dynamic response of laminated plates. AIAA J 11:178–183Google Scholar
  60. Tessler A (2015) Refined zigzag theory for homogeneous, laminated composite, and sandwich beams derived from Reissner’s mixed variational principle. Meccanica 50:2621–2648Google Scholar
  61. Timoshenko SP, Woinowsky-Krieger S (1959) Theory of Plates and Shells. McGraw-Hill, 2 editionGoogle Scholar
  62. Toledano A, Murakami H (1987) A composite plate theory for arbitrary laminate configurations. J Appl Mech 54:181–189Google Scholar
  63. Touratier M (1991) An efficient standard plate theory. Int J Eng Sci 29:901–916Google Scholar
  64. Vidal P, Polit O (2011) A sine finite element using a zig-zag function for the analysis of laminated composite beams. Compos B 42:1671–1682Google Scholar
  65. Wenzel C, Vidal P, D’Ottavio M, Polit O (2014) Coupling of heterogeneous kinematics and finite element approximations applied to composite beam structures. Compos Struct 116:177–192Google Scholar
  66. Zienkiewicz OC, Taylor RL (2000) The Finite Element Method, volume 2: Solid Mechanics. Butterworth-Heinemann, 5 editionGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Thi Huyen Cham Le
    • 1
    Email author
  • Michele D’Ottavio
    • 1
  • Philippe Vidal
    • 1
  • Olivier Polit
    • 1
  1. 1.UPL, Univ Paris Nanterre, LEMEVille d’AvrayFrance

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