A Plate Bending Kirchhoff Element Based on Assumed Strain Functions

Subject Areas : Mechanical Engineering

F  Boussem ¹ (Biskra University, BP145 Biskra 07000, Algeria)
L  Belounar ² (NMISSI Laboratory, Biskra University, BP145 Biskra 07000, Algeria)

Received: 2020-08-05 Accepted : 2020-10-05 Published : 2020-12-01

Keywords: Plate bending, Kirchhoff plate theory, Strain approach, Free vibration, Higher-order strain field,

Abstract :

To investigate static and free vibration for thin plate bending structures, a four-node quadrilateral finite element is proposed in this research paper. This element has been formulated by using both the assumptions of thin plates theory (Kirchhoff plate theory) and strain approach. The suggested element which possesses only three degrees of freedom (one transverse displacement and two normal rotations) at each of four corner nodes is based on assumed higher-order functions for the various components of strain field that satisfies the compatibility equation. The displacement functions of the developed element are obtained by integrating the assumed strains functions and satisfy the exact representation of the rigid body modes. Several numerical tests in both static and free vibration analysis are presented to assess the performance of the new element. The obtained results show high solution accuracy, especially for coarse meshes, of the developed element compared with analytical and other numerical solutions available in the literature.

References:

[1] Timoshenko S., Woinowsky-Krieger S., 1959, Theory of Plates and Shells, London, McGraw-Hill.
[2] Hrabok M. M., Hrudey T. M., 1984, A review and catalogue of plate bending finite elements, Computure Structure 19(3): 479-495.
[3] Melosh R. J., 1962, A stiffness matrix for the analysis of thin plates in bending, Journal of the Aerospace Science 29(1): 102-103.
[4] Batoz J-L., Bathe K-Jür., Ho L-W., 1980, A study of three-node triangular plate bending elements, International Journal for Numerical Methods in Engineering 15(12): 1771-1812.
[5] Batoz J.L., Tahar M. Ben., 1982, Evaluation of a new quadrilateral thin plate bending element, International Journal for Numerical Methods in Engineering 18(11): 1655-1677.
[6] Jirousek J., Leon N., 1977, A powerful finite element for plate bending, Computer Methods in Applied Mechanics Engineering 12(1): 77-96.
[7] Rezaiee-Pajand M., Yaghoobi M., Karkon M., 2012, Hybrid trefftz formulation for thin plate analysis, International Journalof Computuational Methods 9(4).
[8] Belarbi M. T., Charif A., 1998, Nouvel elément secteur basé sur le modèle de déformation avec rotation dans le plan, Revue Européenne des Elements Finis 7(4): 439-458.
[9] Rezaiee-Pajand M., Gharaei-Moghaddam N., Ramezani, M., 2019, A new higher-order strain-based plane element, Scientia Iranica A 26(4): 2258-2275.
[10] Ashwell D. G., Sabir A. B., Roberts T. M., 1971, Further studies in the application of curved finite elements to circular arches, International Journal of Mechanical Science 13(6): 507-517.
[11] Ashwell D.G., Sabir A. B., 1972, A new cylindrical shell finite element based on simple independant strain, International Journal of Mechanical Science 14: 171-183.
[12] Belounar L., Guenfoud M., 2005, A new rectangular finite element based on the strain approach for plate bending, Thin-Walled Structure 43(1): 47-63.
[13] Belarbi M. T., Charif A., 1999, Développement d’un nouvel elément hexaédrique simple basé sur le modèle en déformation pour l’étude des plaques minces et epaisses, Revue Européenne des Elements Finis 8(2): 135-157.
[14] Guerraiche K. H., Belounar L., Bouzidi L., 2018, A new eight nodes brick finite element based on the strain approach, Journal of Solid Mechanics 10(1): 186-199.
[15] Messai A., Belounar L., Merzouki T., 2018, Static and free vibration of plates with a strain based brick element, European Journal of Computational Mechanics 99(1): 1-21.
[16] Belounar L., Guerraiche K., 2014, A new strain based brick element for plate bending, Alexandria Engineering Journal 53(1): 95-105.
[17] Khiouani H.E., Belounar L., Houhou M. N., 2020, A new three-dimensional sector element for circular curved structures analysis, Journal of Solid Mechanics 12(1): 165-174.
[18] Belounar A., Benmebarek S., Belounar L., 2018, Strain based triangular finite element for plate bending analysis, Mechanics of Advanced Materials and Structures 1: 1-13.
[19] Belounar A., Benmebarek S., Houhou M. N., Belounar L., 2019, Static, free vibration, and buckling analysis of plates using strain-based reissner–mindlin elements, International Journal of Advanced Structural Engineering 11(2): 211-230.
[20] Belounar A., Benmebarek S., Houhou M. N., Belounar L., 2020, Free vibration with mindlin plate finite element based on the strain approach, Journal of the Institutionof Engineers Series C 101(2): 331-346.
[21] Clough R., Tocher J., 1965, Finite element stiffness matices for analysis of plate bending, First Conference Matrix Methods in Strucrural Mechanics.
[22] Dvorkin E.N., Bathe K.J., 1984, A continuum mechanics based four-node shell element for general nonlinear analysis, Engineering Computuer 1(1): 77-88.
[23] Leissa A.W., 1973, The free vibration of rectangular plates, Journal of Sound and Vibration 31(3): 257-293.
[24] Lee S.J., Han S.E., 2001, Free-vibration analysis of plates and shells with a nine-node assumed natural degenerated shell element, Journal of Sound and Vibration 241(4): 605-633.
[25] Janabi B., Hinton E., Vuksanovic D., 1989, Free vibrations of mindlin plates using the finite element method: Part 1. Square Plates with Various Edge Conditions, Engineering Computer 6: 90-96.
[26] Den Hartog J.P, 1985, Mechanical Vibrations, Dover Publications, New York.
[27] Zouari W., Assarar M., Meftah K., Ayad R., 2015, Free vibration analysis of homogeneous piezoelectric structures using specific hexahedral elements with rotational DOFs, Acta Mechanica 226(6): 1737-1756.
[28] Nair P.S., Durvasula S., 1973, Vibration of skew plates, Journal of Sound and Vibration 26(1):1-19.