A New Eight Nodes Brick Finite Element Based on the Strain Approach

Document Type : Research Paper


1 Department of Mechanical Engineering, Djelfa University, Batna, Algeria

2 NMISSI Laboratory, Biskra University, Biskra, Algeria


In this paper, a new three dimensional brick finite element based on the strain approach is presented with the purpose of identifying the most effective to analyze linear thick and thin plate bending problems. The developed element which has the three essential external degrees of freedom (U, V and W) at each of the eight corner nodes, is used with a modified elasticity matrix in order to satisfy the basic hypotheses of the theory of plates. The displacements field of the developed element is based on assumed functions for the various strains satisfying the compatibility and the equilibrium equations. New and efficient formulations of this element is discussed in detail, and the results of several examples related to thick and thin plate bending in linear analysis are used to demonstrate the effectiveness of the proposed element. The linear analyses using this developed element exhibit an excellent performance over a set of problems.


[1] Ait-Ali L., 1984, Développement d’ Eléments Finis de Coque Pour le Calcul des Ouvrages d’Art, Thèse de Doctorat, Ecole Nationale Des Ponts Chaussées Paris Tech.
[2] 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 Sciences 13(6): 507-517.
[3] Ayad R., Batoz J.L., Dhatt G., 1995, Un élément quadrilatéral de plaque basé sur une formulation mixte-hybride avec projection en cisaillement, Revue Européenne des Eléments Finis 4(4): 415-440.
[4] Bassayya K., Shrinivasa U., 2000, A 14-node brick element, PN5X1, for plates and shells, Computers & Structures 74(2): 176 -178.
[5] Bassayya k., Bahattacharya K., Shrinivasa U., 2000, Eight –Node brick, PN340, represents constant stress fields exactly, Computers & Structures 74(4): 441-460.
[6] Bathe K.J., Dvorkin E.N., 1985, A four‐node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation‏, International Journal for Numerical Methods in Engineering 21(2): 367-383.
[7] Belarbi M.T., Charif A., 1998, Nouvel élément secteur basé sur le modèle de déformation avec rotation Dans le plan, Revue Européenne des Eléments Finis 7(4): 439-458.
[8] Belarbi M.T., Charif A., 1999, Développement d'un nouvel élément hexaédrique simple basé sur le modèle en déformation pour l'étude des plaques minces et épaisses, Revue Européenne des Eléments Finis 8(2): 135-157.
[9] Belarbi M.T., Maalem T., 2005, On improved rectangular finite element for plane linear elasticity analysis, Revue Européenne des Elements Finis 14(8): 985-997.
[10] Belarbi M.T., Bourezane M., 2005, On improved Sabir triangular element with drilling rotation, Revue Européenne de Génie Civil 9(9-10): 1151-117.
[11] Belounar L., Guenfoud M., 2005, A new rectangular finite element based on the strain approach for plate bending, Thin-Walled Structures 43(1): 47-63.
[12] Belounar L., Guerraiche K., 2014, Anew strain based brick element for plate bending, Alexandria Engineering Journal 53(1): 95-105.
[13] Bull J.W., 1984, The Strain approach to the development of thin cylindrical shell finite element, Thin-Walled Structures 2(3): 195-205.
[14] Charhabi A., 1990, Calcul des Plaques Minces et Epaisses à L'aide des Eléments Finis Tridimensionnels, Annales de l'ITBTP.
[15] Chen Y.I., Wu G.Y., 2004, A mixed 8-node hexahedral element based on the Hu-Washizu principle and the field extrapolation technique, Structural Engineering and Mechanics 17(1): 113-140.
[16] Djoudi M.S., Bahai H., 2003, A shallow shell finite element for the linear and non-linear analysis of cylindrical shells, Engineering Structures 25(6): 769-778.
[17] De Rosa M.A., Franciosi C., 1990, Plate bending analysis by the cell method: numerical comparisons with finite element methods, Computers & Structures 37(5): 731-735.
[18] Djoudi M. S., Bahai H., 2004, Strain based finite element for the vibration of cylindrical panels with opening, Thin-Walled Structures 42(4): 575-588.
[19] Fredriksson M., Ottosen N.S., 2007, Accurate eight-node hexahedral element, International Journal for Numerical Methods and Engineering 72(6): 631-657.
[20] Gallagher R.H., 1976, Introduction aux Eléments Finis, Edition Pluralise.
[21] Himeur M., Guenfoud M., 2011, Bending triangular finite element with a fictitious fourth node based on the strain approach, European Journal of Computational Mechanics 20(7-8): 455-485.
[22] Himeur M., Benmarce A., Guenfoud M., 2014, A new finite element based on the strain approach with transverse shear effect, Structural Engineering and Mechanics 49(6): 793-810.
[23] Himeur M., Zergua A., Guenfoud M., 2015, A Finite Element Based on the Strain Approach Using Airy’s Function, Arabian Journal for Science and Engineering 40(3): 719-733.
[24] Hamadi D., Ayoub A., Toufik M., 2016, A new strain-based finite element for plane elasticity problems, Engineering Computations 33(2): 562-579.
[25] Jirousek J., Wroblewski A., Qin Q., He X., 1995, A family of quadrilateral hybrid –Trefftz p-elements for thick plate analysis, Computer Methods in Applied Mechanics Engineering 127(1-4): 315-344.
[26] Lemosse D., 2000, Eléments Finis Isoparamétriques Tridimensionnels Pour L’étude des Structures Minces, Thèse de Doctorat, Ecole Doctorale SPMI/INSA-Rouen.
[27] Li H.G., Cen S., Cen Z.Z., 2008, Hexahedral volume coordinate method (HVCM) and improvements on 3D Wilson hexahedral element, Computer Methods in Applied Mechanics and Engineering 197(51-52): 4531-4548.
[28] Lo S.H., Ling C., 2000, Improvement on the 10-node tetrahedral element for three-dimensional problems, Computer Methods in Applied Mechanics and Engineering 189(3): 961-974.
[29] MacNeal R.H., Harder R. L., 1985, A Proposed Standard Set of Problems to Test Finite Element Accuracy, Finite Element in Analysis and Design 1(1): 3-20.
[30] Mindlin R.D., 1951, Influence of rotary inertia and shear on flexural motions of isotropic elastic plates, Journal of Applied Mechanics 18: 31-38.
[31] Ooi E.T., Rajendran S., Yeo J.H., 2004, A 20-node hexahedron element with enhanced distortion tolerance, International Journal for Numerical Methods in Engineering 60(15): 2501-2530.
[32] Reissner E., 1945, The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics ASME 12: 69-77.
[33] Sabir A.B., Lock A.C.,1972, Curved Cylindrical Shell Finite element, International Journal of Mechanical Sciences 14(2): 125-135.
[34] Sabir A.B., 1983, A new class of finite elements for plane elasticity problems, CAFEM7, 7th International Conference on Structural Mechanics in Reactor Technology, Chicago.
[35] Sabir A.B., Sfendji A., 1995, Triangular and rectangular plane elasticity finite elements, Thin-Walled Structures 21(3): 225-232.
[36] Sabir A.B., Moussa A.I., 1997, Analysis of fluted conical shell roofs using the finite element method, Computers & Structures 64(1-4): 239-251.
[37] Smith I.M., Griffith D.V., 1988, Programming the Finite Element Method, John Wiley & Sons, UK.
[38] Smith I.M., Griffith D.V., 2004, Programming the Finite Element Method, John Wiley & Sons, UK.
[39] Sze K.Y., Chan W.K., 2001, A six-node pentagonal assumed natural strain solid-shell element, Finite Elements in Analysis and Design 37(8): 639-655.
[40] Timoshenko S., Woinowsky-Krieger S., 1959, Theory of Plates and Shells, London, McGraw-Hill.
[41] Timoshenko S., Goodier J. N., 1951, Theory of Elasticity, McGraw-Hill.
[42] Trinh V.D., 2009, Formulation, Développement et Validation d’Eléments Finis de Type Coques Volumiques Sous Intégrés Stabilisés Utilisables Pour des Problèmes a Cinématique et Comportement Non Linéaires, Thèse de Doctorat, Ecole Doctorale, ENSAM-Paris.
[43] Trinh V.D., Abed-Meraim F., A. Combescure, 2011, Assumed strain solid–shell formulation “SHB6” for the six-node prismatic, Journal of Mechanical Science and Technology 25(9): 2345-2364.
[44] Venkatesh D.N., Shrinivasa U., 1996, Plate bending with hexahedral with PN elements, Computers & Structures 60(4): 635-641.
[45] Yuan F., Miller R.E., 1988, A rectangular finite element for moderately thick flat plates, Computers & Structures 30(6): 1375-1387.
[46] Zienkiewicz O.C., Taylor R.L., 1977, The Finite Element Method, McGraw-Hill.
[47] Zienkiewicz O.C., Taylor R.L., 1989, The Finite Element Method, McGraw–Hill.
[48] Zienkiewicz O.C., Taylor R.L., 2000, The Finite Element Method, Butterworth-Heinemann.