Structural Engineering and Mechanics

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Problem-dependent cubic linked interpolation for Mindlin plate four-node quadrilateral finite elements

  • Received : 2016.04.06
  • Accepted : 2016.06.15
  • Published : 2016.09.25
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Abstract

We employ the so-called problem-dependent linked interpolation concept to develop two cubic 4-node quadrilateral plate finite elements with 12 external degrees of freedom that pass the constant bending patch test for arbitrary node positions of which the second element has five additional internal degrees of freedom to get polynomial completeness of the cubic form. The new elements are compared to the existing linked-interpolation quadratic and nine-node cubic elements presented by the author earlier and to the other elements from literature that use the cubic linked interpolation by testing them on several benchmark examples.

Keywords

Acknowledgement

Grant : Configuration-dependent Approximation in Non-linear Finite-element Analysis of Structures

Supported by : Croatian Science Foundation

References

  1. Auricchio, F. and Taylor, R.L. (1994), "A shear deformable plate element with an exact thin limit", Comput. Meth. Appl. Mech. Eng., 118(3), 393-412. https://doi.org/10.1016/0045-7825(94)90009-4
  2. Bathe, K.J. and Dvorkin, E.N. (1986), "A formulation of general shell elements-the use of mixed interpolation of tensorial components", Int. J. Numer. Meth. Eng., 22(3), 697-722. https://doi.org/10.1002/nme.1620220312
  3. Cen, S., Shang, Y., Li, C.F. and Li, H.G. (2014), "Hybrid displacement function element method: a simple hybrid-Trefftz stress element method for analysis of Mindlin-Reissner plate", Int. J. Numer. Meth. Eng., 98(3), 203-234. https://doi.org/10.1002/nme.4632
  4. Chen, W.J., Wang, J.Z. and Zhao, J. (2009), "Functions for patch test in finite element analysis of the Mindlin plate and the thin cylindrical shell", Sci. China Ser. G: Phys. Mech. Astron., 52(5), 762-767. https://doi.org/10.1007/s11433-009-0097-y
  5. Choo, Y.S., Choi, N. and Lee, B.C. (2010), "A new hybrid-Trefftz triangular and quadrilateral plate elements", Appl. Math. Model., 34(1), 14-23. https://doi.org/10.1016/j.apm.2009.03.022
  6. Crisfield, M.A. (1984), "A quadratic Mindlin element using shear constraints", Comput. Struct., 18(5), 833-852. https://doi.org/10.1016/0045-7949(84)90030-0
  7. Dukic-Papa, E. and Jelenic, G. (2014), "Exact solution of 3D Timoshenko beam problem: problemdependent formulation", Arch. Appl. Mech., 84(3), 375-384. https://doi.org/10.1007/s00419-013-0805-y
  8. Hughes, T.J.R. and Tezduyar, T.E. (1981), "Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element", J. Appl. Mech., 48(3), 587-596. https://doi.org/10.1115/1.3157679
  9. Ibrahimbegovic, A. (1993), "Quadrilateral finite elements for analysis of thick and thin plates", Comput. Meth. Appl. Mech. Eng., 110(3), 195-209. https://doi.org/10.1016/0045-7825(93)90160-Y
  10. Jelenic, G. and Papa, E. (2011), "Exact solution of 3D Timoshenko beam problem using linked interpolation of arbitrary order", Arch. Appl. Mech., 81(2), 171-183. https://doi.org/10.1007/s00419-009-0403-1
  11. Lee, P.S. and Bathe, K.J. (2010), "The quadratic MITC plate and MITC shell elements in plate bending", Adv. Eng. Softw., 41(5), 712-728. https://doi.org/10.1016/j.advengsoft.2009.12.011
  12. MacNeal, R.H. (1982), "Derivation of element stiffness matrices by assumed strain distributions", Nucl. Eng. Des., 70(1), 3-12. https://doi.org/10.1016/0029-5493(82)90262-X
  13. Papadopoulos, P. and Taylor, R.L. (1990), "A triangular element based on Reissner-Mindlin plate theory", Int. J. Numer. Meth. Eng., 30(5), 1029-1049. https://doi.org/10.1002/nme.1620300506
  14. Rezaiee-Pajand, M. and Karkon, M. (2012), "Two efficient hybrid-Trefftz elements for plate bending analysis", Latin Am. J. Solid. Struct., 9(1), 43-67. https://doi.org/10.1590/S1679-78252012000100003
  15. Ribaric, D. (2012), "Higher-order linked interpolation in moderately thick plate and facet shell finite elements", PhD Thesis, Faculty of Civil Engineering, University of Rijeka.
  16. Ribaric, D. and Jelenic, G. (2012), "Higher-order linked interpolation in quadrilateral thick plate finite elements", Finite Elem. Anal. Des., 51, 67-80. https://doi.org/10.1016/j.finel.2011.10.003
  17. Ribaric, D. and Jelenic, G. (2014), "Distortion-immune nine-node displacement-based quadrilateral thick plate finite elements that satisfy constant-bending patch test", Int. J. Numer. Meth. Eng., 98(7), 492-517. https://doi.org/10.1002/nme.4639
  18. Soh, A.K., Cen, S., Long, Y.Q. and Long, Z.F. (2001), "A new twelve DOF quadrilateral element for analysis of thick and thin plates", Eur. J. Mech. A/Solid., 20(2), 299-326. https://doi.org/10.1016/S0997-7538(00)01129-3
  19. Taylor, R.L. and Auricchio, F. (1993), "Linked interpolation for Reissner-Mindlin plate elements: Part II-a simple triangle", Int. J. Numer. Meth. Eng., 36(18), 3057-3066. https://doi.org/10.1002/nme.1620361803
  20. Tessler, A. and Dong, S.B. (1981), "On a hierarchy of conforming Timoshenko beam elements", Comput. Struct., 14(3), 335-344. https://doi.org/10.1016/0045-7949(81)90017-1
  21. Tessler, A. and Hughes, T.J.R. (1983), "An improved treatment of transverse shear in the Mindlin-type fournode quadrilateral element", Comput. Meth. Appl. Mech. Eng., 39(3), 311-335. https://doi.org/10.1016/0045-7825(83)90096-8
  22. Wang, C.M., Reddy, J.N. and Lee, K.H. (2000), Shear deformable beams and plates: Relationships with classical solutions, Elsevier.
  23. WanJi, C. and Cheung, Y.K. (2000), "Refined quadrilateral element based on Mindlin/Reissner plate theory" Int. J. Numer. Meth. Eng., 47(1-3), 605-627. https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<605::AID-NME785>3.0.CO;2-E
  24. Xu, Z., Zienkiewicz, O.C. and Zeng L.F. (1994), "Linked interpolation for Reissner-Mindlin plate elements: Part III-An alternative quadrilateral", Int. J. Numer. Meth. Eng., 37(9), 1437-1443. https://doi.org/10.1002/nme.1620370902
  25. Zhang, H. and Kuang, J.S. (2007), "Eight-node Reissner-Mindlin plate element based on boundary interpolation using Timoshenko beam function", Int. J. Numer. Meth. Eng., 69(7), 1345-1373. https://doi.org/10.1002/nme.1809
  26. Zienkiewicz, O.C. and Taylor, R.L. (2000), The finite element method: Solid mechanics, Fifth Edition, Butterworth-Heinemann.
  27. Zienkiewicz, O.C., Xu, Z., Zeng, L.F., Samuelsson, A. and Wiberg, N.E. (1993), "Linked interpolation for Reissner-Mindlin plate elements: Part I-A simple quadrilateral", Int. J. Numer. Meth. Eng., 36(18), 3043-3056. https://doi.org/10.1002/nme.1620361802