Structural Engineering and Mechanics

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An assumed-stress finite element for static and free vibration analysis of Reissner-Mindlin plates

  • Darilmaz, Kutlu (Department of Civil Engineering, Istanbul Technical University)
  • Received : 2004.06.07
  • Accepted : 2004.09.16
  • Published : 2005.01.30
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Abstract

An assumed stress quadrilateral thin/moderately thick plate element HQP4 based on the Mindlin/Reissner plate theory is proposed. The formulation is based on Hellinger-Reissner variational principle. Static and free vibration analyses of plates are carried out. Numerical examples are presented to show that the validity and efficiency of the present element for static and free vibration analysis of plates. Satisfactory accuracy for thin and moderately thick plates is obtained and it is free from shear locking for thin plate analysis.

Keywords

References

  1. Akoz, A.Y and Uzcan, N. (1992), 'The new functional for Reissner plates and its application', Comput. Struet., 44(5), 1139-1144 https://doi.org/10.1016/0045-7949(92)90334-V
  2. Akoz, A.Y and Erath, N. (2000), 'A sectorial element based on Reissner plate theory', Struet. Eng. Meeh., 9(6), 519-540
  3. ANSYS (1997), Swanson Analysis Systems, Swanson J. ANSYS 5.4. USA
  4. Ayad, R, Rigolot, A and TaIbi, N. (2001), 'An improved three-node hybrid-mixed element for MindlinlReissner plates', Int. J. Hum. Meth. Eng., 51(8), 919-942 https://doi.org/10.1002/nme.188
  5. Ayad, R. and Rigolot, A (2002), 'An improved four-node hybrid-mixed element based upon Mindlin's plate theory', Int. J. Hum. Meth. Eng., 55, 705-731 https://doi.org/10.1002/nme.528
  6. Bathe, K.J. and Dvorkin, E.H. (1985), 'A four-node plate bending element based on Mindlin/Reissner plate theory and mixed interpolation', Int. J. Hum. Meth. Eng., 21, 367-383 https://doi.org/10.1002/nme.1620210213
  7. Bathe, K.J., Brezzi, F. and Cho, S.W (1989), 'The MITC7 and MITC9 plate bending elements', Comput. Struet., 32(3-4), 797-814 https://doi.org/10.1016/0045-7949(89)90365-9
  8. Batoz, J.L. and Tahar, M.B. (1982), 'Evaluation of a new quadrilateral thin plate bending element', Int. J. Num. Meth. Eng., 18,1655-1677 https://doi.org/10.1002/nme.1620181106
  9. Bergan, P.G. and Wang, X. (1984), 'Quadrilateral plate bending elements with shear deformations', Comput. Struet., 19(1-2),25-34 https://doi.org/10.1016/0045-7949(84)90197-4
  10. Bhashyam, G.R and Gallagher, R.H. (1983), 'A triangular shear flexible finite element for moderately thick laminated composite plates', Comp. Methods Appl. Meeh. Eng., 40, 309-326 https://doi.org/10.1016/0045-7825(83)90104-4
  11. Brezzi, F. and Marini, L.D. (2003), 'A nonconforming element for the Reissner-Mindlin plate', Comput. Struet., 81(8-11), 515-522 https://doi.org/10.1016/S0045-7949(02)00418-2
  12. Choi, C.K. and Park, Y.M. (1999), 'Quadratic NMS Mindlin-Plate-Bending Element', Int. J. Hum. Meth. Eng., 46, 1273-1289 https://doi.org/10.1002/(SICI)1097-0207(19991120)46:8<1273::AID-NME754>3.0.CO;2-N
  13. Cheung, Y.K. and Wanji, C. (1989), 'Hybrid quadrilateral element based on Mindlin-Reissner plate theory', Comput. Struet., 32(2), 327-339 https://doi.org/10.1016/0045-7949(89)90044-8
  14. Dong, Y.F., Wu, C.C. and Defreitas, J.A.T. (1993), 'The hybrid stress model for Mindlin-Reissner plates based on a stress optimization condition', Comput. Struet., 46(5), 877-897 https://doi.org/10.1016/0045-7949(93)90150-C
  15. Erath, N. and Akoz, A.Y. (2002), 'Free vibration analysis of Reissner plates by mixed finite element', Struet. Eng. Meeh., 13(3), 277-298
  16. Feng, W, Hoa, S.V and Huang, Q. (1997), 'Classification of stress modes in assumed stress elds of hybrid finite elements', Int. J. Hum. Meth. Eng., 40, 4313-4339 https://doi.org/10.1002/(SICI)1097-0207(19971215)40:23<4313::AID-NME259>3.0.CO;2-N
  17. Hughes, T.J.R, Cohen, M. and Haroun, M. (1978), 'Reduced and selective integration techniques in finite element analysis of plates', Nue. Eng. Des., 46, 203-222 https://doi.org/10.1016/0029-5493(78)90184-X
  18. 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., 46, 587-596
  19. Ibrahimbegovic, A (1993), 'Quadrilateral finite elements for analysis of thick and thin plates', Comput. Meth. Appl. Mech. Eng., 110, 195-209 https://doi.org/10.1016/0045-7825(93)90160-Y
  20. Lee, S.W and Pian, T.H.H. (1978), 'Improvement of plate and shell finite-elements by mixed formulations', AIAA J., 16(1),29-34 https://doi.org/10.2514/3.60853
  21. Leissa, A.W (1973), 'The free vibration of rectangular plates', J. Sound Vib., 31, 257-293 https://doi.org/10.1016/S0022-460X(73)80371-2
  22. Malkus, D.S. and Hughes, T.J.R (1978), 'Mixed finite element methods-reduced and selective integration techniques: a unification of concepts', Comput. Meth. Appl. Mech. Eng., 15,63-81 https://doi.org/10.1016/0045-7825(78)90005-1
  23. Mindlin, R.D. (1951), 'Influence of rotatory inertia and shear on flexural motion of isotropic, elastic plates', J. Appl. Mech., ASME, 18, 31-38
  24. Morley, L.S.D. (1963), Skew Plates and Stnlctures, Series of Monographs in Aeronautics and Astronautics, MacMillan, New York
  25. Omurtag, M.H., Ozutok, A and Akoz, A.Y (1997), 'Free vibration analysis of Kirchhoff plates resting on elastic foundation by mixed fmite element formulation based on Gateaux differential', Int. J. Num. Meth. Eng., 40(2), 295-317 https://doi.org/10.1002/(SICI)1097-0207(19970130)40:2<295::AID-NME66>3.0.CO;2-2
  26. Pian, T.H.H. and Chen, D.P (1983), 'On the suppression of zero energy deformation modes'. Int. J. Num. Meth. Eng., 19, 1741-1752 https://doi.org/10.1002/nme.1620191202
  27. Pryor, C.W., Jr., Barker, R.M. and Frederick, D. (1970), 'Finite element bending analysis of Reissner plate', J. Eng. Mech. Div. ASCE, 96, 967-983
  28. Punch, E.F. and Aduri, S.N. (1984), 'Development and testing of stable, isoparametric curvilinear 2 and 3-D hybrid stress elements', Comput. Meth. Appl. Mech. Eng., 47, 331-356 https://doi.org/10.1016/0045-7825(84)90083-5
  29. Rao, G.V, Venkataramana, J. and Raju, I.S. (1974), 'A high precision triangular plate bending element for the analysis of thick plates', Nuc. Eng. Des., 30, 408-412 https://doi.org/10.1016/0029-5493(74)90225-8
  30. Reissner, E. (1945), 'The effect of transverse shear deformation on the bending of plates', J. Appl. Mech., ASME, 12, A69-A77
  31. Sydenstricker, R.M. and Landau, L. (2000), 'A study of some triangular discrete Reissner-Mindlin plate and shell elements', Comput. Struct., 78(1-3), 21-33 https://doi.org/10.1016/S0045-7949(00)00103-6
  32. Timoshenko, S.P. and Woinowsky-Krieger, S. (1959), Theory of Plates and Shells (2nd edn). McGraw-Hill: New York
  33. Wang, Y.L., Wang, X.W. and Zhou, Y (2004), 'Static and free vibration analyses of rectangular plates by the new version of the differential quadrature element method', Int. J. Num. Meth. Eng., 59(9), 1207-1226 https://doi.org/10.1002/nme.913
  34. Wanji, C and Cheung, Y.K (2000), 'Refined quadrilateral element based on Mindlin/Reissner plate theory', Int. J. Num. Meth. Eng., 47(1-3), 605-627 https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<1::AID-NME810>3.0.CO;2-Q
  35. Wanji, C and Cheung, Y.K (2001), 'Refined 9-dof triangular Mindlin plate elements', Int. J. Num. Meth. Eng., 51, 1259-1281 https://doi.org/10.1002/nme.196
  36. Woo, K.S., Hong, C.H., Basu, P.K and Seo, C.G. (2003), 'Free vibration of skew Mindlin plates by p-version of F.E.M.', J. Sound Vib., 268, 637-656 https://doi.org/10.1016/S0022-460X(02)01536-5
  37. Xu, Z., Zienkiewicz, O.C and Zeng, L.F. (1994), 'Linked interpolation for Reissner-Mindlin plate elements: Part III- An alternative quadrilateral', Int. J. Num. Meth. Eng., 37,1437-1443 https://doi.org/10.1002/nme.1620370902
  38. Yuan, F. and R. Miller E. (1989), 'A cubic triangular finite element for flat plates with shear', Int. J. Num. Meth. Eng., 28, 109-126 https://doi.org/10.1002/nme.1620280109
  39. Zengjie, G. and Wanji, C. (2003), 'Refined triangular discrete Mindlin flat shell elements', Comput. Mech., 33(1),52-60 https://doi.org/10.1007/s00466-003-0499-z
  40. Zienkiewicz, O.C (1971), The Finite Element Method in Engineering Science, McGraw-Hill
  41. Zienkiewicz, O.C, Taylor, R.L. and Too, J.M. (1971), 'Reduced integration technique in geneal analysis of plates and shells', Int. J. Num. Meth. Eng., 3, 275-290 https://doi.org/10.1002/nme.1620030211
  42. 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. Num. Meth. Eng., 36, 3043-3056 https://doi.org/10.1002/nme.1620361802

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