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Mathematical Problems in Engineering
Volume 2015, Article ID 456740, 12 pages
http://dx.doi.org/10.1155/2015/456740
Review Article

Developments of Mindlin-Reissner Plate Elements

Song Cen1,2,3 and Yan Shang1,2

1Department of Engineering Mechanics, School of Aerospace Engineering, Tsinghua University, Beijing 100084, China
2High Performance Computing Center, School of Aerospace Engineering, Tsinghua University, Beijing 100084, China
3Key Laboratory of Applied Mechanics, School of Aerospace Engineering, Tsinghua University, Beijing 100084, China

Received 8 September 2014; Accepted 13 January 2015

Academic Editor: Xin-Lin Gao

Copyright © 2015 Song Cen and Yan Shang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. J. Melosh, “A stiffness matrix for the analysis of thin plates in bending,” Journal of the Aerospace Sciences, vol. 28, no. 1, pp. 34–42, 1961. View at Google Scholar
  2. R. W. Clough and J. L. Tocher, “Finite element stiffness matrices for analysis of plates in bending,” in Proceedings of Conference on Matrix Methods in Structural Analysis, pp. 515–545, 1965.
  3. O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method for Solid and Structural Mechanics, Elsevier, Oxford, UK, 6th edition, 2005.
  4. A. J. Fricker, “A simple method for including shear deformations in thin plate elements,” International Journal for Numerical Methods in Engineering, vol. 23, no. 7, pp. 1355–1366, 1986. View at Publisher · View at Google Scholar · View at Scopus
  5. M. M. Hrabok and T. M. Hrudey, “A review and catalogue of plate bending finite elements,” Computers & Structures, vol. 19, no. 3, pp. 479–495, 1984. View at Publisher · View at Google Scholar · View at Scopus
  6. R. D. Mindlin, “Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates,” Journal of Applied Mechanics—Transactions of the ASME, vol. 18, no. 1, pp. 31–38, 1951. View at Google Scholar
  7. E. Reissner, “The effect of transverse shear deformation on the bending of elastic plates,” Journal of Applied Mechanics-Transactions of the ASME, vol. 12, no. 2, pp. A69–A77, 1945. View at Google Scholar · View at MathSciNet
  8. Y.-Q. Long, S. Cen, and Z.-F. Long, Advanced Finite Element Method in Structural Engineering, Tsinghua University Press, Beijing, China, Springer, Berlin, Germany, 2009.
  9. D. N. Arnold, “Discretization by finite elements of a model parameter dependent problem,” Numerische Mathematik, vol. 37, no. 3, pp. 405–421, 1981. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. O. C. Zienkiewicz, R. L. Taylor, and J. M. Too, “Reduced integration technique in general analysis of plates and shells,” International Journal for Numerical Methods in Engineering, vol. 3, no. 2, pp. 275–290, 1971. View at Google Scholar · View at Scopus
  11. E. D. L. Pugh, E. Hinton, and O. C. Zienkiewicz, “A study of quadrilateral plate bending elements with ‘reduced’ integration,” International Journal for Numerical Methods in Engineering, vol. 12, no. 7, pp. 1059–1079, 1978. View at Publisher · View at Google Scholar · View at Scopus
  12. T. J. R. Hughes, M. Cohen, and M. Haroun, “Reduced and selective integration techniques in the finite element analysis of plates,” Nuclear Engineering and Design, vol. 46, no. 1, pp. 203–222, 1978. View at Publisher · View at Google Scholar · View at Scopus
  13. D. S. Malkus and T. J. R. Hughes, “Mixed finite element methods—reduced and selective integration techniques: a unification of concepts,” Computer Methods in Applied Mechanics and Engineering, vol. 15, no. 1, pp. 63–81, 1978. View at Publisher · View at Google Scholar · View at Scopus
  14. C. A. Xenophontos, “Finite element computations for the Reissner-Mindlin plate model,” Communications in Numerical Methods in Engineering, vol. 14, no. 12, pp. 1119–1131, 1998. View at Publisher · View at Google Scholar · View at Scopus
  15. D. N. Arnold and R. S. Falk, “A uniformly accurate finite element method for the Reissner-Mindlin plate,” SIAM Journal on Numerical Analysis, vol. 26, no. 6, pp. 1276–1290, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  16. H. R. H. Kabir, “Shear locking free isoparametric three-node triangular finite element for moderately-thick and thin plates,” International Journal for Numerical Methods in Engineering, vol. 35, no. 3, pp. 503–519, 1992. View at Publisher · View at Google Scholar · View at Scopus
  17. S. Ahmad, B. M. Irons, and O. C. Zienkiewicz, “Analysis of thick and thin shell structures by curved finite elements,” International Journal for Numerical Methods in Engineering, vol. 2, no. 3, pp. 419–451, 1970. View at Publisher · View at Google Scholar · View at Scopus
  18. O. C. Zienkiewicz and E. Hinton, “Reduced integration, function smoothing and non-conformity in finite element analysis (with special reference to thick plates),” Journal of the Franklin Institute, vol. 302, no. 5-6, pp. 443–461, 1976. View at Publisher · View at Google Scholar · View at Scopus
  19. T. J. R. Hughes, R. L. Taylor, and W. Kanoknukulchai, “Simple and efficient finite element for plate bending,” International Journal for Numerical Methods in Engineering, vol. 11, no. 10, pp. 1529–1543, 1977. View at Publisher · View at Google Scholar · View at Scopus
  20. R. L. Spilker and N. I. Munir, “The hybrid-stress model for thin plates,” International Journal for Numerical Methods in Engineering, vol. 15, no. 8, pp. 1239–1260, 1980. View at Publisher · View at Google Scholar · View at MathSciNet
  21. R. L. Spilker and N. I. Munir, “A hybrid-stress quadratic serendipity displacement mindlin plate bending element,” Computers and Structures, vol. 12, no. 1, pp. 11–21, 1980. View at Publisher · View at Google Scholar · View at Scopus
  22. G. Prathap and G. R. Bhashyam, “Reduced integration and the shear-flexible beam element,” International Journal for Numerical Methods in Engineering, vol. 18, no. 2, pp. 195–210, 1982. View at Publisher · View at Google Scholar · View at Scopus
  23. R. C. Averill and J. N. Reddy, “Behaviour of plate elements based on the first-order shear deformation theory,” Engineering Computations, vol. 7, no. 1, pp. 57–74, 1990. View at Publisher · View at Google Scholar · View at Scopus
  24. S. F. Pawsey and R. W. Clough, “Improved numerical integration of thick shell finite elements,” International Journal for Numerical Methods in Engineering, vol. 3, no. 4, pp. 575–586, 1971. View at Publisher · View at Google Scholar · View at Scopus
  25. S. Brasile, “An isostatic assumed stress triangular element for the Reissner-Mindlin plate-bending problem,” International Journal for Numerical Methods in Engineering, vol. 74, no. 6, pp. 971–995, 2008. View at Publisher · View at Google Scholar · View at Scopus
  26. T. J. R. Hughes and M. Cohen, “The ‘heterosis’ finite element for plate bending,” Computers & Structures, vol. 9, no. 5, pp. 445–450, 1978. View at Publisher · View at Google Scholar · View at Scopus
  27. A. F. Saleeb and T. Y. Chang, “An efficient quadrilateral element for plate bending analysis,” International Journal for Numerical Methods in Engineering, vol. 24, no. 6, pp. 1123–1155, 1987. View at Publisher · View at Google Scholar · View at Scopus
  28. G. Prathap and B. R. Somashekar, “Field- and edge-consistency synthesis of a 4-noded quadrilateral plate bending element,” International Journal for Numerical Methods in Engineering, vol. 26, no. 8, pp. 1693–1708, 1988. View at Publisher · View at Google Scholar · View at Scopus
  29. J. L. Batoz, K. J. Bathe, and L. W. Ho, “A study of 3-node triangular plate bending elements,” International Journal for Numerical Methods in Engineering, vol. 15, no. 12, pp. 1771–1812, 1980. View at Publisher · View at Google Scholar · View at Scopus
  30. G. Prathap and S. Viswanath, “An optimally integrated 4-node quadrilateral plate bending element,” International Journal for Numerical Methods in Engineering, vol. 19, no. 6, pp. 831–840, 1983. View at Publisher · View at Google Scholar · View at Scopus
  31. T. Belytschko, H. Stolarski, and N. Carpenter, “A C0 triangular plate element with one-point quadrature,” International Journal for Numerical Methods in Engineering, vol. 20, no. 5, pp. 787–802, 1984. View at Publisher · View at Google Scholar · View at Scopus
  32. R. L. Spilker, “Invariant 8-node hybrid-stress elements for thin and moderately thick plates,” International Journal for Numerical Methods in Engineering, vol. 18, no. 8, pp. 1153–1178, 1982. View at Publisher · View at Google Scholar
  33. T. Belytschko, C. S. Tsay, and W. K. Liu, “A stabilization matrix for the bilinear Mindlin plate element,” Computer Methods in Applied Mechanics and Engineering, vol. 29, no. 3, pp. 313–327, 1981. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. T. Belytschko and C. S. Tsay, “A stabilization procedure for the quadrilateral plate element with one point quadrature,” International Journal for Numerical Methods in Engineering, vol. 19, no. 3, pp. 405–419, 1983. View at Publisher · View at Google Scholar · View at Scopus
  35. T. Belytschko, J. S. J. Ong, and W. K. Liu, “A consistent control of spurious singular modes in the 9-node Lagrange element for the laplace and mindlin plate equations,” Computer Methods in Applied Mechanics and Engineering, vol. 44, no. 3, pp. 269–295, 1984. View at Publisher · View at Google Scholar · View at Scopus
  36. W. K. Liu, J. S. Ong, and R. A. Uras, “Finite element stabilization matrices—a unification approach,” Computer Methods in Applied Mechanics and Engineering, vol. 53, no. 1, pp. 13–46, 1985. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. D. P. Flanagan and T. Belytschko, “A uniform strain hexahedron and quadrilateral with orthogonal hourglass control,” International Journal for Numerical Methods in Engineering, vol. 17, no. 5, pp. 679–706, 1981. View at Publisher · View at Google Scholar · View at Scopus
  38. T. J. R. Hughes and T. E. Tezduyar, “Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element,” Transactions ASME—Journal of Applied Mechanics, vol. 48, no. 3, pp. 587–596, 1981. View at Publisher · View at Google Scholar · View at Scopus
  39. R. H. MacNeal, “Derivation of element stiffness matrices by assumed strain distributions,” Nuclear Engineering and Design, vol. 70, no. 1, pp. 3–12, 1982. View at Publisher · View at Google Scholar · View at Scopus
  40. F. Brezzi, K.-J. Bathe, and M. Fortin, “Mixed-interpolated elements for Reissner-Mindlin plates,” International Journal for Numerical Methods in Engineering, vol. 28, no. 8, pp. 1787–1801, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice Hall, Englewood Cliffs, NJ, USA, 1987. View at MathSciNet
  42. A. Tessler and T. J. R. Hughes, “An improved treatment of transverse shear in the mindlin-type four-node quadrilateral element,” Computer Methods in Applied Mechanics and Engineering, vol. 39, no. 3, pp. 311–335, 1983. View at Publisher · View at Google Scholar · View at Scopus
  43. A. Tessler and T. J. R. Hughes, “A three-node mindlin plate element with improved transverse shear,” Computer Methods in Applied Mechanics and Engineering, vol. 50, no. 1, pp. 71–101, 1985. View at Publisher · View at Google Scholar · View at Scopus
  44. K. Y. Sze, D. Zhu, and D.-P. Chen, “Quadratic triangular C0 plate bending element,” International Journal for Numerical Methods in Engineering, vol. 40, no. 5, pp. 937–951, 1997. View at Publisher · View at Google Scholar · View at Scopus
  45. K. Y. Sze and D. Zhu, “A quadratic assumed natural strain triangular element for plate bending analysis,” Communications in Numerical Methods in Engineering, vol. 14, no. 11, pp. 1013–1025, 1998. View at Publisher · View at Google Scholar · View at Scopus
  46. K. J. Bathe and F. Brezzi, “On the convergence of a four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation,” in Proceedings of the 5th Conference on Mathematics of Finite Elements and Applications, New York, NY, USA, 1985.
  47. K.-J. Bathe and E. N. Dvorkin, “A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation,” International Journal for Numerical Methods in Engineering, vol. 21, no. 2, pp. 367–383, 1985. View at Publisher · View at Google Scholar · View at Scopus
  48. E. N. Dvorkin and K.-J. Bathe, “A continuum mechanics based four-node shell element for general non-linear analysis,” Engineering computations, vol. 1, no. 1, pp. 77–88, 1984. View at Publisher · View at Google Scholar · View at Scopus
  49. K.-J. Bathe and E. N. Dvorkin, “A formulation of general shell elements—the use of mixed interpolation of tensorial components,” International Journal for Numerical Methods in Engineering, vol. 22, no. 3, pp. 697–722, 1986. View at Publisher · View at Google Scholar · View at Scopus
  50. F. Brezzi, M. Fortin, and R. Stenberg, “Error analysis of mixed-interpolated elements for Reissner-Mindlin plates,” Mathematical Models & Methods in Applied Sciences, vol. 1, no. 2, pp. 125–151, 1991. View at Publisher · View at Google Scholar
  51. C. T. Wu and H. P. Wang, “An enhanced cell-based smoothed finite element method for the analysis of Reissner-Mindlin plate bending problems involving distorted mesh,” International Journal for Numerical Methods in Engineering, vol. 95, no. 4, pp. 288–312, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  52. W. K. Liu, E. S. Law, D. Lam, and T. Belytschko, “Resultant-stress degenerated-shell element,” Computer Methods in Applied Mechanics and Engineering, vol. 55, no. 3, pp. 259–300, 1986. View at Publisher · View at Google Scholar · View at Scopus
  53. Y. Lee, P.-S. Lee, and K.-J. Bathe, “The MITC3+ shell element and its performance,” Computers and Structures, vol. 138, pp. 12–23, 2014. View at Publisher · View at Google Scholar · View at Scopus
  54. H.-M. Jeon, P.-S. Lee, and K.-J. Bathe, “The MITC3 shell finite element enriched by interpolation covers,” Computers and Structures, vol. 134, pp. 128–142, 2014. View at Publisher · View at Google Scholar · View at Scopus
  55. K. J. Bathe, F. Brezzi, and L. D. Marini, “The MITC9 shell element in plate bending: mathematical analysis of a simplified case,” Computational Mechanics, vol. 47, no. 6, pp. 617–626, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  56. P.-S. Lee and K.-J. Bathe, “The quadratic MITC plate and MITC shell elements in plate bending,” Advances in Engineering Software, vol. 41, no. 5, pp. 712–728, 2010. View at Publisher · View at Google Scholar · View at Scopus
  57. H. Kebari, “A one point integrated assumed strain 4-node Mindlin plate element,” Engineering Computations, vol. 7, no. 4, pp. 284–290, 1990. View at Publisher · View at Google Scholar · View at Scopus
  58. K.-U. Bletzinger, M. Bischoff, and E. Ramm, “A unified approach for shear-locking-free triangular and rectangular shell finite elements,” Computers & Structures, vol. 75, no. 3, pp. 321–334, 2000. View at Publisher · View at Google Scholar · View at Scopus
  59. G. Falsone and D. Settineri, “A Kirchhoff-like solution for the Mindlin plate model: a new finite element approach,” Mechanics Research Communications, vol. 40, pp. 1–10, 2012. View at Publisher · View at Google Scholar · View at Scopus
  60. N. Nguyen-Thanh, T. Rabczuk, H. Nguyen-Xuan, and S. Bordas, “An alternative alpha finite element method with discrete shear gap technique for analysis of isotropic Mindlin-Reissner plates,” Finite Elements in Analysis and Design, vol. 47, no. 5, pp. 519–535, 2011. View at Publisher · View at Google Scholar · View at Scopus
  61. H. Nguyen-Xuan, T. Rabczuk, N. Nguyen-Thanh, T. Nguyen-Thoi, and S. Bordas, “A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates,” Computational Mechanics, vol. 46, no. 5, pp. 679–701, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  62. X. Cui, G. R. Liu, G. Y. Li, G. Zhang, and G. Zheng, “Analysis of plates and shells using an edge-based smoothed finite element method,” Computational Mechanics, vol. 45, no. 2-3, pp. 141–156, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  63. C. V. Le, “A stabilized discrete shear gap finite element for adaptive limit analysis of Mindlin-Reissner plates,” International Journal for Numerical Methods in Engineering, vol. 96, no. 4, pp. 231–246, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  64. J. L. Batoz and M. B. Bentahar, “Evaluation of a new quadrilateral thin plate bending element,” International Journal for Numerical Methods in Engineering, vol. 18, no. 11, pp. 1655–1677, 1982. View at Google Scholar · View at Scopus
  65. J. L. Batoz, “An explicit formulation for an efficient triangular plate-bending element,” International Journal for Numerical Methods in Engineering, vol. 18, no. 7, pp. 1077–1089, 1982. View at Google Scholar
  66. O. C. Zienkiewicz, R. L. Taylor, P. Papadopoulos, and E. Oñate, “Plate bending elements with discrete constraints: new triangular elements,” Computers & Structures, vol. 35, no. 4, pp. 505–522, 1990. View at Publisher · View at Google Scholar · View at Scopus
  67. J. Aalto, “From Kirchhoff to Mindlin plate elements,” Communications in Applied Numerical Methods, vol. 4, no. 2, pp. 231–241, 1988. View at Google Scholar · View at Scopus
  68. J.-L. Batoz and P. Lardeur, “A discrete shear triangular nine D.O.F. element for the analysis of thick to very thin plates,” International Journal for Numerical Methods in Engineering, vol. 28, no. 3, pp. 533–560, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  69. J.-L. Batoz and I. Katili, “On a simple triangular reissner/mindlin plate element based on incompatible modes and discrete constraints,” International Journal for Numerical Methods in Engineering, vol. 35, no. 8, pp. 1603–1632, 1992. View at Publisher · View at Google Scholar · View at Scopus
  70. Y. C. Cai, L. G. Tian, and S. N. Atluri, “A simple locking-free discrete shear triangular plate element,” Computer Modeling in Engineering & Sciences, vol. 77, no. 3-4, pp. 221–238, 2011. View at Google Scholar · View at Scopus
  71. P. Lardeur, Développement et Évaluation de Deux Nouveaux Éléments Finis de Plaques et Coques Composites Avec Influence Du Cisaillement Transversal, UTC, Compiègne, France, 1990.
  72. J. L. Batoz and G. Dhatt, Modélisation des Structures Par Éléments Finis: Solides Élastiques, Université Laval, Paris, France, 1990.
  73. I. Katili, “New discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields. Part II. An extended DKQ element for thick-plate bending analysis,” International Journal for Numerical Methods in Engineering, vol. 36, no. 11, pp. 1885–1908, 1993. View at Publisher · View at Google Scholar · View at Scopus
  74. I. Katili, “New discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields. Part I: an extended DKT element for thick-plate bending analysis,” International Journal for Numerical Methods in Engineering, vol. 36, no. 11, pp. 1859–1883, 1993. View at Publisher · View at Google Scholar · View at Scopus
  75. O. C. Zeinkiewicz, Z. N. Xu, L. F. Zeng, A. Samuelsson, and N.-E. Wiberg, “Linked interpolation for Ressiner-Mindlin plate element: part I—a simple quadrilateral,” International Journal for Numerical Methods in Engineering, vol. 36, no. 18, pp. 3043–3056, 1993. View at Publisher · View at Google Scholar · View at Scopus
  76. R. L. Taylor and F. Auricchio, “Linked interpolation for Reissner-Mindlin plate elements: part II—a simple triangle,” International Journal for Numerical Methods in Engineering, vol. 36, no. 18, pp. 3057–3066, 1993. View at Publisher · View at Google Scholar · View at Scopus
  77. F. Auricchio and R. L. Taylor, “A shear deformable plate element with an exact thin limit,” Computer Methods in Applied Mechanics and Engineering, vol. 118, no. 3-4, pp. 393–412, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  78. X. Zhongnian, “A thick-thin triangular plate element,” International Journal for Numerical Methods in Engineering, vol. 33, no. 5, pp. 963–973, 1992. View at Publisher · View at Google Scholar · View at Scopus
  79. R. G. Durán and E. Liberman, “On the convergence of a triangular mixed finite element method for Reissner-Mindlin plates,” Mathematical Models and Methods in Applied Sciences, vol. 6, no. 3, pp. 339–352, 1996. View at Publisher · View at Google Scholar · View at Scopus
  80. D. Ribarić and G. Jelenić, “Higher-order linked interpolation in quadrilateral thick plate finite elements,” Finite Elements in Analysis and Design, vol. 51, pp. 67–80, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  81. D. Ribarić and G. Jelenić, “Distortion-immune nine-node displacement-based quadrilateral thick plate finite elements that satisfy constant-bending patch test,” International Journal for Numerical Methods in Engineering, vol. 98, no. 7, pp. 492–517, 2014. View at Publisher · View at Google Scholar · View at Scopus
  82. F. Auricchio and R. L. Taylor, “A triangular thick plate finite element with an exact thin limit,” Finite Elements in Analysis and Design, vol. 19, no. 1-2, pp. 57–68, 1995. View at Publisher · View at Google Scholar · View at Scopus
  83. P. Papadopoulos and R. L. Taylor, “A triangular element based on Reissner-Mindlin plate theory,” International Journal for Numerical Methods in Engineering, vol. 30, no. 5, pp. 1029–1049, 1990. View at Publisher · View at Google Scholar · View at Scopus
  84. Z. Xu, O. C. Zienkiewicz, and L. F. Zeng, “Linked interpolation for Reissner-Mindlin plate elements: Part III. An alternative quadrilateral,” International Journal for Numerical Methods in Engineering, vol. 37, no. 9, pp. 1437–1443, 1994. View at Publisher · View at Google Scholar · View at Scopus
  85. F. Auricchio and C. Lovadina, “Analysis of kinematic linked interpolation methods for Reissner-Mindlin plate problems,” Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 18-19, pp. 2465–2482, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  86. C. Lovadina, “Analysis of a mixed finite element method for the Reissner-Mindlin plate problems,” Computer Methods in Applied Mechanics and Engineering, vol. 163, no. 1–4, pp. 71–85, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  87. M. Lyly, “On the connection between some linear triangular Reissner-Mindlin plate bending elements,” Numerische Mathematik, vol. 85, no. 1, pp. 77–107, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  88. S. W. Lee and T. H. H. Pian, “Improvement of plate and shell finite elements by mixed formulations,” AIAA Journal, vol. 16, no. 1, pp. 29–34, 1978. View at Publisher · View at Google Scholar · View at Scopus
  89. S. W. Lee and J. C. Zhang, “A 6-node finite element for plate bending,” International Journal for Numerical Methods in Engineering, vol. 21, no. 1, pp. 131–143, 1985. View at Publisher · View at Google Scholar · View at Scopus
  90. S. W. Lee and S. C. Wong, “Mixed formulation finite elements for Mindlin theory plate bending,” International Journal for Numerical Methods in Engineering, vol. 18, no. 9, pp. 1297–1311, 1982. View at Google Scholar · View at Scopus
  91. T. H. H. Pian and K. Sumihara, “Hybrid SemiLoof elements for plates and shells based upon a modified Hu-Washizu principle,” Computers & Structures, vol. 19, no. 1-2, pp. 165–173, 1984. View at Publisher · View at Google Scholar · View at Scopus
  92. S. L. Weissman and R. L. Taylor, “Resultant fields for mixed plate bending elements,” Computer Methods in Applied Mechanics and Engineering, vol. 79, no. 3, pp. 321–355, 1990. View at Publisher · View at Google Scholar · View at Scopus
  93. R. D. Cook, “Two hybrid elements for analysis of thick, thin and sandwich plates,” International Journal for Numerical Methods in Engineering, vol. 5, no. 2, pp. 277–288, 1972. View at Publisher · View at Google Scholar · View at Scopus
  94. J. Robinson and G. W. Haggenmacher, “LORA-an accurate four node stress plate bending element,” International Journal for Numerical Methods in Engineering, vol. 14, no. 2, pp. 296–306, 1979. View at Publisher · View at Google Scholar · View at Scopus
  95. A. F. Saleeb, T. Y. Chang, and S. Yingyeunyong, “A mixed formulation of C0-linear triangular plate/shell element-the role of edge shear constraints,” International Journal for Numerical Methods in Engineering, vol. 26, no. 5, pp. 1101–1128, 1988. View at Publisher · View at Google Scholar · View at Scopus
  96. M. Gellert, “A new method for derivation of locking-free plate bending finite elements via mixed hybrid formulation,” International Journal for Numerical Methods in Engineering, vol. 26, no. 5, pp. 1185–1200, 1988. View at Publisher · View at Google Scholar · View at Scopus
  97. O. C. Zienkiewicz and D. Lefebvre, “A robust triangular plate bending element of the Reissner–Mindlin type,” International Journal for Numerical Methods in Engineering, vol. 26, no. 5, pp. 1169–1184, 1988. View at Publisher · View at Google Scholar · View at Scopus
  98. P. M. Pinsky and R. V. Jasti, “A mixed finite element formulation for Reissner-Mindlin plates based on the use of bubble functions,” International Journal for Numerical Methods in Engineering, vol. 28, no. 7, pp. 1677–1702, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  99. R. Ayad, G. Dhatt, and J. L. Batoz, “A new hybrid-mixed variational approach for Reissner-Mindlin plates. The MISP model,” International Journal for Numerical Methods in Engineering, vol. 42, no. 7, pp. 1149–1179, 1998. View at Google Scholar · View at MathSciNet · View at Scopus
  100. S. de Miranda and F. Ubertini, “A simple hybrid stress element for shear deformable plates,” International Journal for Numerical Methods in Engineering, vol. 65, no. 6, pp. 808–833, 2006. View at Publisher · View at Google Scholar · View at Scopus
  101. H.-Y. Duan and G.-P. Liang, “Analysis of some stabilized low-order mixed finite element methods for Reissner-Mindlin plates,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 3–5, pp. 157–179, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  102. G. Y. Shi and P. Tong, “Assumed stress C0 quadrilateral triangular plate elements by interrelated edge displacements,” International Journal for Numerical Methods in Engineering, vol. 39, no. 6, pp. 1041–1051, 1996. View at Google Scholar · View at Scopus
  103. K. J. Bathe, Finite Element Procedures, Prentice Hall, Upper Saddle River, NJ, USA, 1996.
  104. M. A. Aminpour, “Direct formulation of a hybrid 4-node shell element with drilling degrees of freedom,” International Journal for Numerical Methods in Engineering, vol. 35, no. 5, pp. 997–1013, 1992. View at Publisher · View at Google Scholar · View at Scopus
  105. J. Jirousek, A. Wroblewski, and B. Szybinski, “New 12 DOF quadrilateral element for analysis of thick and thin plates,” International Journal for Numerical Methods in Engineering, vol. 38, no. 15, pp. 2619–2638, 1995. View at Publisher · View at Google Scholar · View at Scopus
  106. F. S. Jin and Q. H. Qin, “A variational principle and hybrid Trefftz finite element for the analysis of Reissner plates,” Computers & Structures, vol. 56, no. 4, pp. 697–701, 1995. View at Publisher · View at Google Scholar · View at Scopus
  107. Y. F. Dong and J. A. T. Defreitas, “A quadrilateral hybrid stress element for mindlin plates based on incompatible displacements,” International Journal for Numerical Methods in Engineering, vol. 37, no. 2, pp. 279–296, 1994. View at Google Scholar
  108. T. H. H. Pian and D. P. Chen, “Alternative ways for formulation of hybrid stress elements,” International Journal for Numerical Methods in Engineering, vol. 18, no. 11, pp. 1679–1684, 1982. View at Google Scholar · View at Scopus
  109. G. Shi and G. Z. Voyiadjis, “Efficient and accurate four-node quadrilateral C0 plate bending element based on assumed strain fields,” International Journal for Numerical Methods in Engineering, vol. 32, no. 5, pp. 1041–1055, 1991. View at Publisher · View at Google Scholar · View at Scopus
  110. G. Shi and G. Z. Voyiadjis, “Simple and efficient shear flexible two-node arch/beam and four-node cylindrical shell/plate finite elements,” International Journal for Numerical Methods in Engineering, vol. 31, no. 4, pp. 759–776, 1991. View at Publisher · View at Google Scholar · View at Scopus
  111. F. Gruttmann and W. Wagner, “A stabilized one-point integrated quadrilateral Reissner-Mindlin plate element,” International Journal for Numerical Methods in Engineering, vol. 61, no. 13, pp. 2273–2295, 2004. View at Publisher · View at Google Scholar · View at Scopus
  112. A. K. Noor and C. M. Andersen, “Mixed models and reduced selective integration displacement models for non-linear shell analysis,” International Journal for Numerical Methods In Engineering, vol. 18, no. 10, pp. 1429–1454, 1982. View at Google Scholar · View at Scopus
  113. R. Ayad and A. Rigolot, “An improved four-node hybrid-mixed element based upon Mindlin's plate theory,” International Journal for Numerical Methods in Engineering, vol. 55, no. 6, pp. 705–731, 2002. View at Publisher · View at Google Scholar · View at Scopus
  114. R. Ayad, A. Rigolot, and N. Talbi, “An improved three-node hybrid-mixed element for Mindlin/Reissner plates,” International Journal for Numerical Methods in Engineering, vol. 51, no. 8, pp. 919–942, 2001. View at Publisher · View at Google Scholar · View at Scopus
  115. E. M. B. R. Pereira and J. A. T. Freitas, “Numerical implementation of a hybrid-mixed finite element model for Reissner-Mindlin plates,” Computers & Structures, vol. 74, no. 3, pp. 323–334, 2000. View at Publisher · View at Google Scholar · View at Scopus
  116. E. M. B. R. Pereira and J. A. T. Freitas, “A hybrid-mixed finite element model based on Legendre polynomials for Reissner-Mindlin plates,” Computer Methods in Applied Mechanics and Engineering, vol. 136, no. 1-2, pp. 111–126, 1996. View at Publisher · View at Google Scholar · View at Scopus
  117. J. C. Simo and M. S. Rifai, “A class of mixed assumed strain methods and the method of incompatible modes,” International Journal for Numerical Methods in Engineering, vol. 29, no. 8, pp. 1595–1638, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  118. Y. S. Choo, N. Choi, and B. C. Lee, “A new hybrid-Trefftz triangular and quadrilateral plate elements,” Applied Mathematical Modelling, vol. 34, no. 1, pp. 14–23, 2010. View at Publisher · View at Google Scholar · View at Scopus
  119. K.-Y. Yuan, Y.-S. Huang, and T. H. H. Pian, “New strategy for assumed stresses for 4-node hybrid stress membrane element,” International Journal for Numerical Methods in Engineering, vol. 36, no. 10, pp. 1747–1763, 1993. View at Publisher · View at Google Scholar · View at Scopus
  120. D. N. Arnold, F. Brezzi, and L. D. Marini, “A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate,” Journal of Scientific Computing, vol. 22-23, no. 1, pp. 25–45, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  121. F. Brezzi and L. D. Marini, “A nonconforming element for the Reissner-Mindlin plate,” Computers and Structures, vol. 81, no. 8-11, pp. 515–522, 2003. View at Publisher · View at Google Scholar · View at Scopus
  122. C. Chinosi, C. Lovadina, and L. D. Marini, “Nonconforming locking-free finite elements for Reissner-Mindlin plates,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 25–28, pp. 3448–3460, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  123. C. Lovadina, “A low-order nonconforming finite element for Reissner-Mindlin plates,” SIAM Journal on Numerical Analysis, vol. 42, no. 6, pp. 2688–2705, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  124. D. N. Arnold, F. Brezzi, R. S. Falk, and L. D. Marini, “Locking-free Reissner-Mindlin elements without reduced integration,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 37–40, pp. 3660–3671, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  125. P. R. Boesing, A. L. Madureira, and I. Mozolevski, “A new interior penalty discontinuous Galerkin method for the Reissner-Mindlin model,” Mathematical Models and Methods in Applied Sciences, vol. 20, no. 8, pp. 1343–1361, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  126. P. Hansbo and M. G. Larson, “Locking free quadrilateral continuous/discontinuous finite element methods for the Reissner-Mindlin plate,” Computer Methods in Applied Mechanics and Engineering, vol. 269, pp. 381–393, 2014. View at Publisher · View at Google Scholar · View at Scopus
  127. P. Hansbo, D. Heintz, and M. G. Larson, “A finite element method with discontinuous rotations for the Mindlin-Reissner plate model,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 5–8, pp. 638–648, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  128. S. Cen, Y.-Q. Long, Z.-H. Yao, and S.-P. Chiew, “Application of the quadrilateral area co-ordinate method: a new element for Mindlin-Reissner plate,” International Journal for Numerical Methods in Engineering, vol. 66, no. 1, pp. 1–45, 2006. View at Publisher · View at Google Scholar · View at Scopus
  129. Y. Q. Long, J. X. Li, Z. F. Long, and S. Cen, “Area co-ordinates used in quadrilateral elements,” Communications in Numerical Methods in Engineering, vol. 15, no. 8, pp. 533–545, 1999. View at Google Scholar · View at MathSciNet · View at Scopus
  130. Z. F. Long, J. X. Li, S. Cen, and Y. Q. Long, “Some basic formulae for area co-ordinates in quadrilateral elements,” Communications in Numerical Methods in Engineering, vol. 15, no. 12, pp. 841–852, 1999. View at Google Scholar · View at MathSciNet · View at Scopus
  131. A. Ibrahimbegović, “Plate quadrilateral finite element with incompatible modes,” Communications in Applied Numerical Methods, vol. 8, no. 8, pp. 497–504, 1992. View at Publisher · View at Google Scholar
  132. A. Ibrahimbegović, “Quadrilateral finite elements for analysis of thick and thin plates,” Computer Methods in Applied Mechanics and Engineering, vol. 110, no. 3-4, pp. 195–209, 1993. View at Publisher · View at Google Scholar · View at Scopus
  133. A. K. Soh, Z. F. Long, and S. Cen, “A new nine dof triangular element for analysis of thick and thin plates,” Computational Mechanics, vol. 24, no. 5, pp. 408–417, 1999. View at Publisher · View at Google Scholar · View at Scopus
  134. A.-K. Soh, S. Cen, Y.-Q. Long, and Z.-F. Long, “A new twelve DOF quadrilateral element for analysis of thick and thin plates,” European Journal of Mechanics. A. Solids, vol. 20, no. 2, pp. 299–326, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  135. C. S. Wang, P. Hu, and Y. Xia, “A 4-node quasi-conforming Reissner-Mindlin shell element by using Timoshenko's beam function,” Finite Elements in Analysis and Design, vol. 61, pp. 12–22, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  136. H. X. Zhang and J. S. Kuang, “Eight-node Reissner-Mindlin plate element based on boundary interpolation using Timoshenko beam function,” International Journal for Numerical Methods in Engineering, vol. 69, no. 7, pp. 1345–1373, 2007. View at Publisher · View at Google Scholar · View at Scopus
  137. G. R. Liu, K. Y. Dai, and T. T. Nguyen, “A smoothed finite element method for mechanics problems,” Computational Mechanics, vol. 39, no. 6, pp. 859–877, 2007. View at Publisher · View at Google Scholar · View at Scopus
  138. G. R. Liu, “A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods,” International Journal of Computational Methods, vol. 5, no. 2, pp. 199–236, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  139. J. S. Chen, C. T. Wu, S. Yoon, and Y. You, “A stabilized conforming nodal integration for Galerkin mesh-free methods,” International Journal for Numerical Methods in Engineering, vol. 50, no. 2, pp. 435–466, 2001. View at Publisher · View at Google Scholar
  140. T. Nguyen-Thoi, P. Phung-Van, H. Luong-Van, H. Nguyen-Van, and H. Nguyen-Xuan, “A cell-based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of plates,” Computational Mechanics, vol. 51, no. 1, pp. 65–81, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  141. H. Nguyen-Xuan, T. Rabczuk, S. Bordas, and J. F. Debongnie, “A smoothed finite element method for plate analysis,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 13–16, pp. 1184–1203, 2008. View at Publisher · View at Google Scholar · View at Scopus
  142. C. M. Shin and B. C. Lee, “Development of a strain-smoothed three-node triangular flat shell element with drilling degrees of freedom,” Finite Elements in Analysis and Design, vol. 86, pp. 71–80, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  143. H. Nguyen-Xuan, G. R. Liu, C. Thai-Hoang, and T. Nguyen-Thoi, “An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 9–12, pp. 471–489, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  144. G. R. Liu and G. Y. Zhang, “Upper bound solution to elasticity problems: a unique property of the linearly conforming point interpolation method (LC-PIM),” International Journal for Numerical Methods in Engineering, vol. 74, no. 7, pp. 1128–1161, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  145. G. R. Liu, T. Nguyen-Thoi, and K. Y. Lam, “A novel alpha finite element method (αFEM) for exact solution to mechanics problems using triangular and tetrahedral elements,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 45–48, pp. 3883–3897, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  146. G. R. Liu, H. Nguyen-Xuan, T. Nguyen-Thoi, and X. Xu, “A novel Galerkin-like weakform and a superconvergent alpha finite element method (SαFEM) for mechanics problems using triangular meshes,” Journal of Computational Physics, vol. 228, no. 11, pp. 4055–4087, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  147. G. R. Liu, T. Nguyen-Thoi, and K. Y. Lam, “A novel FEM by scaling the gradient of strains with factor α (αFEM),” Computational Mechanics, vol. 43, no. 3, pp. 369–391, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  148. L. Yu-Qiu and X. Ke-Gui, “Generalized conforming element for bending and buckling analysis of plates,” Finite Elements in Analysis and Design, vol. 5, no. 1, pp. 15–30, 1989. View at Publisher · View at Google Scholar · View at Scopus
  149. Y. L. Chen, S. Cen, Z. H. Yao, Y. Q. Long, and Z. F. Long, “Development of triangular flat-shell element using a new thin-thick plate bending element based on semiLoof constrains,” Structural Engineering and Mechanics, vol. 15, no. 1, pp. 83–114, 2003. View at Publisher · View at Google Scholar · View at Scopus
  150. L. Cai, T. Rong, and D. Chen, “Generalized mixed variational methods for Reissner plate and its applications,” Computational Mechanics, vol. 30, no. 1, pp. 29–37, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  151. T.-Y. Rong, “Generalized mixed variational principles and new FEM models in solid mechanics,” International Journal of Solids and Structures, vol. 24, no. 11, pp. 1131–1140, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  152. P. G. Bergan and L. Hanssen, “A new approach for deriving ‘good’ finite elements,” The Mathematics of Finite Elements and Applications, vol. 2, pp. 483–498, 1975. View at Google Scholar
  153. P. G. Bergan and X. Wang, “Quadrilateral plate bending elements with shear deformations,” Computers and Structures, vol. 19, no. 1-2, pp. 25–34, 1984. View at Publisher · View at Google Scholar · View at Scopus
  154. P. G. Bergan and M. K. Nygard, “Finite elements with increased freedom in choosing shape functions,” International Journal for Numerical Methods in Engineering, vol. 20, no. 4, pp. 643–663, 1984. View at Publisher · View at Google Scholar · View at Scopus
  155. I. W. Liu, T. Kerh, and C. C. Lin, “A conforming quadrilateral plate bending element with shear deformations,” Computers & Structures, vol. 56, no. 1, pp. 93–100, 1995. View at Publisher · View at Google Scholar · View at Scopus
  156. C. A. Felippa and P. G. Bergan, “A triangular bending element based on an energy-orthogonal free formulation,” Computer Methods in Applied Mechanics and Engineering, vol. 61, no. 2, pp. 129–160, 1987. View at Publisher · View at Google Scholar · View at Scopus
  157. P. G. Bergan, “Finite elements based on energy orthogonal functions,” International Journal for Numerical Methods in Engineering, vol. 15, no. 10, pp. 1541–1555, 1980. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  158. E. Hinton and H. C. Huang, “A family of quadrilateral Mindlin plate elements with substitute shear strain fields,” Computers and Structures, vol. 23, no. 3, pp. 409–431, 1986. View at Publisher · View at Google Scholar · View at Scopus
  159. O. L. Roufaeil, “A new four-node quadrilateral plate bending element,” Computers & Structures, vol. 54, no. 5, pp. 871–879, 1995. View at Publisher · View at Google Scholar · View at Scopus
  160. S. Holzer, E. Rank, and H. Werner, “An implementation of the hp-version of the finite element method for Reissner-Mindlin plate problems,” International Journal for Numerical Methods in Engineering, vol. 30, no. 3, pp. 459–471, 1990. View at Publisher · View at Google Scholar · View at Scopus
  161. Y. K. Cheung and W. J. Chen, “Refined nine-parameter triangular thin plate bending element by using refined direct stiffness method,” International Journal for Numerical Methods in Engineering, vol. 38, no. 2, pp. 283–298, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  162. W. J. Chen and Y. K. Cheung, “Refined quadrilateral element based on Mindlin/Reissner plate theory,” International Journal for Numerical Methods in Engineering, vol. 47, no. 1–3, pp. 605–627, 2000. View at Google Scholar · View at MathSciNet · View at Scopus
  163. C. Wanji and Y. K. Cheung, “Refined 9-Dof triangular Mindlin plate elements,” International Journal for Numerical Methods in Engineering, vol. 51, no. 11, pp. 1259–1282, 2001. View at Publisher · View at Google Scholar · View at Scopus
  164. G. Castellazzi and P. Krysl, “Displacement-based finite elements with nodal integration for Reissner-Mindlin plates,” International Journal for Numerical Methods in Engineering, vol. 80, no. 2, pp. 135–162, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  165. C.-K. Choi and S.-H. Kim, “Coupled use of reduced integration and non-conforming modes in quadratic Mindlin plate element,” International Journal for Numerical Methods in Engineering, vol. 28, no. 8, pp. 1909–1928, 1989. View at Publisher · View at Google Scholar · View at Scopus
  166. E. Hinton and N. Bićanić, “A comparison of lagrangian and serendipity mindlin plate elements for free vibration analysis,” Computers and Structures, vol. 10, no. 3, pp. 483–493, 1979. View at Publisher · View at Google Scholar · View at Scopus
  167. G. Castellazzi and P. Krysl, “A nine-node displacement-based finite element for Reissner-Mindlin plates based on an improved formulation of the NIPE approach,” Finite Elements in Analysis and Design, vol. 58, pp. 31–43, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  168. X. Y. Zhuang, R. Q. Huang, H. H. Zhu, H. Askes, and K. Mathisen, “A new and simple locking-free triangular thick plate element using independent shear degrees of freedom,” Finite Elements in Analysis and Design, vol. 75, pp. 1–7, 2013. View at Publisher · View at Google Scholar · View at Scopus
  169. C. Carstensen, X. P. Xie, G. Z. Yu, and T. X. Zhou, “A priori and a posteriori analysis for a locking-free low order quadrilateral hybrid finite element for Reissner-Mindlin plates,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 9–12, pp. 1161–1175, 2011. View at Publisher · View at Google Scholar · View at Scopus
  170. E. A. W. Maunder and J. P. M. de Almeida, “A triangular hybrid equilibrium plate element of general degree,” International Journal for Numerical Methods in Engineering, vol. 63, no. 3, pp. 315–350, 2005. View at Publisher · View at Google Scholar · View at Scopus
  171. B. Hu, Z. Wang, and Y. Xu, “Combined hybrid method applied in the Reissner-Mindlin plate model,” Finite Elements in Analysis and Design, vol. 46, no. 5, pp. 428–437, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  172. Z. Wang and B. Hu, “Research of combined hybrid method applied in the Reissner-Mindlin plate model,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 49–66, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  173. P. Ming and Z.-C. Shi, “Analysis of some low order quadrilateral Reissner-Mindlin plate elements,” Mathematics of Computation, vol. 75, no. 255, pp. 1043–1065, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  174. H.-Y. Duan and G.-P. Liang, “Mixed and nonconforming finite element approximations of Reissner-Mindlin plates,” Computer Methods in Applied Mechanics and Engineering, vol. 192, no. 49-50, pp. 5265–5281, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  175. J. P. Pontaza and J. N. Reddy, “Mixed plate bending elements based on least-squares formulation,” International Journal for Numerical Methods in Engineering, vol. 60, no. 5, pp. 891–922, 2004. View at Publisher · View at Google Scholar · View at Scopus
  176. T. X. Zhou, “The partial projection method in the finite element discretization of the Reissner-Mindlin plate model,” Journal of Computational Mathematics, vol. 13, no. 2, pp. 172–191, 1995. View at Google Scholar · View at MathSciNet
  177. D. N. Arnold and F. Brezzi, “Some new elements for the Reissner-Mindlin plate model,” in Boundary Value Problems for Partial Differential Equations and Applications, pp. 287–292, 1993. View at Google Scholar · View at MathSciNet
  178. C. Chinosi and C. Lovadina, “Numerical analysis of some mixed finite element methods for Reissner-Mindlin plates,” Computational Mechanics, vol. 16, no. 1, pp. 36–44, 1995. View at Publisher · View at Google Scholar · View at Scopus
  179. O. Polit, M. Touratier, and P. Lory, “A new 8-node quadrilateral shear-bending plate finite element,” International Journal for Numerical Methods in Engineering, vol. 37, no. 3, pp. 387–411, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  180. H. R. Dhananjaya, J. Nagabhushanam, P. C. Pandey, and M. Z. Jumaat, “New twelve node serendipity quadrilateral plate bending element based on Mindlin-Reissner theory using integrated force method,” Structural Engineering and Mechanics, vol. 36, no. 5, pp. 625–642, 2010. View at Publisher · View at Google Scholar · View at Scopus
  181. H. R. Dhananjaya, P. C. Pandey, and J. Nagabhushanam, “New eight node serendipity quadrilateral plate bending element for thin and moderately thick plates using integrated force method,” Structural Engineering and Mechanics, vol. 33, no. 4, pp. 485–502, 2009. View at Publisher · View at Google Scholar · View at Scopus
  182. J. Hu and Z.-C. Shi, “Error analysis of quadrilateral Wilson element for Reissner-Mindlin plate,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 6–8, pp. 464–475, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  183. M. Lyly, R. Stenberg, and T. Vihinen, “A stable bilinear element for the Reissner-Mindlin plate model,” Computer Methods in Applied Mechanics and Engineering, vol. 110, no. 3-4, pp. 343–357, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  184. F. Kikuchi and K. Ishii, “Improved 4-node quadrilateral plate bending element of the Reissner-Mindlin type,” Computational Mechanics, vol. 23, no. 3, pp. 240–249, 1999. View at Publisher · View at Google Scholar · View at Scopus
  185. D. Sohn and S. Im, “Variable-node plate and shell elements with assumed natural strain and smoothed integration methods for nonmatching meshes,” Computational Mechanics, vol. 51, no. 6, pp. 927–948, 2013. View at Publisher · View at Google Scholar · View at Scopus
  186. C. K. Choi and Y. M. Park, “Conforming and nonconforming transition plate bending elements for an adaptive h-refinement,” Thin-Walled Structures, vol. 28, no. 1, pp. 1–20, 1997. View at Publisher · View at Google Scholar · View at Scopus
  187. C.-K. Choi and Y.-M. Park, “Transition plate-bending elements for compatible mesh gradation,” Journal of Engineering Mechanics, vol. 118, no. 3, pp. 462–480, 1992. View at Publisher · View at Google Scholar · View at Scopus
  188. H. Sofuoglu and H. Gedikli, “A refined 5-node plate bending element based on Reissner-Mindlin theory,” Communications in Numerical Methods in Engineering, vol. 23, no. 5, pp. 385–403, 2007. View at Publisher · View at Google Scholar · View at Scopus
  189. S. Li, “On the micromechanics theory of Reissner-Mindlin plates,” Acta Mechanica, vol. 142, no. 1, pp. 47–99, 2000. View at Publisher · View at Google Scholar · View at Scopus
  190. A. Eijo, E. Oñate, and S. Oller, “A four-noded quadrilateral element for composite laminated plates/shells using the refined zigzag theory,” International Journal for Numerical Methods in Engineering, vol. 95, no. 8, pp. 631–660, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  191. H. M. Ma, X.-L. Gao, and J. N. Reddy, “A non-classical Mindlin plate model based on a modified couple stress theory,” Acta Mechanica, vol. 220, no. 1–4, pp. 217–235, 2011. View at Publisher · View at Google Scholar · View at Scopus
  192. H. M. Ma, X.-L. Gao, and J. N. Reddy, “A microstructure-dependent Timoshenko beam model based on a modified couple stress theory,” Journal of the Mechanics and Physics of Solids, vol. 56, no. 12, pp. 3379–3391, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  193. S. K. Park and X.-L. Gao, “Bernoulli-Euler beam model based on a modified couple stress theory,” Journal of Micromechanics and Microengineering, vol. 16, no. 11, article 2355, 2006. View at Publisher · View at Google Scholar · View at Scopus
  194. S. Cen, Y. Shang, C.-F. Li, and H.-G. Li, “Hybrid displacement function element method: a simple hybrid-Trefftz stress element method for analysis of Mindlin-Reissner plate,” International Journal for Numerical Methods in Engineering, vol. 98, no. 3, pp. 203–234, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  195. H. C. Hu, Variational Principles of Theory of Elasticity with Applications, Science Press, Beijing, China; Gordan and Breach Science, New York, NY, USA, 1984.
  196. Y. Shang, S. Cen, C.-F. Li, and J.-B. Huang, “An effective hybrid displacement function element method for solving the edge effect of Mindlin-Reissner plate,” International Journal for Numerical Methods in Engineering, 2015. View at Publisher · View at Google Scholar
  197. Y. Shang, S. Cen, C.-F. Li, and X.-R. Fu, “Two generalized conforming quadrilateral Mindlin-Reissner plate elements based on the displacement function,” Finite Elements in Analysis and Design, vol. 99, pp. 24–28, 2015. View at Publisher · View at Google Scholar