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http://dx.doi.org/10.12989/sem.2018.68.4.385

The use of the strain approach to develop a new consistent triangular thin flat shell finite element with drilling rotation  

Guenfoud, Hamza (LGCH Laboratory, 8 Mai 1945 University of Guelma)
Himeur, Mohamed (LGCH Laboratory, 8 Mai 1945 University of Guelma)
Ziou, Hassina (Mohamed Khider University)
Guenfoud, Mohamed (LGCH Laboratory, 8 Mai 1945 University of Guelma)
Publication Information
Structural Engineering and Mechanics / v.68, no.4, 2018 , pp. 385-398 More about this Journal
Abstract
In the present paper, we offer a new flat shell finite element. It is the result of the combination of a membrane element and a bending element, both based on the strain-based formulation. It is known that $C^{\circ}$ plane membrane elements provide poor deflection and stress for problems where bending is dominant. In addition, they encounter continuity and compliance problems when they connect to C1 class plate elements. The reach of the present work is to surmount these problems when a membrane element is coupled with a thin plate element in order to construct a shell element. The membrane element used is a triangular element with four nodes, three nodes at the vertices of the triangle and the fourth one at its barycenter. Each node has three degrees of freedom, two translations and one rotation around the normal. The coefficients related to the degrees of freedom at the internal node are subsequently removed from the element stiffness matrix by using the static condensation technique. The interpolation functions of strain, displacements and stresses fields are developed from equilibrium conditions. The plate element used for the construction of the present shell element is a triangular four-node thin plate element based on Kirchhoff plate theory, the strain approach, the four fictitious node, the static condensation and the analytic integration. The shell element result of this combination is robust, competitive and efficient.
Keywords
finite element method; membrane; plate; shell; condensation; deformation approach; drilling rotation;
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1 Boutagouga, D. (2016), "A new enhanced assumed strain quadrilateral membrane element with drilling degree of freedom and modified shape functions'', Int. J. Numer. Meth. Eng.
2 Boutagouga, D., Gouasmia, A. and Djeghaba, K. (2010), "Geometrically nonlinear analysis of thin shell by a quadrilateral finite element with in-plane rotational degrees of freedom'', Eur. J. Comput. Mech./Rev. Eur. Mecan. Numer., 19(8), 707-724.   DOI
3 Bouzriba, A. and Bouzrira, C. (2015), "Sector element for analysis of thick cylinders exposed to internal pressure and change of temperature'', Građevin., 67(6), 547-555.
4 Burkardt, J. (2010), FEM Basis Functions for a Triangle, .
5 Carpenter, N., Stolarski, H. and Belytschko, T. (1985), "A flat triangular shell element with improved membrane interpolation", Int. J. Numer. Meth. Biomed. Eng., 1(4), 161-168.
6 Carpenter, N., Stolarski, H. and Belytschko, T. (1986), "Improvements in 3-node triangular shell elements'', Int. J. Numer. Meth. Eng., 23(9), 1643-1667.   DOI
7 Chetty, S. and Tottenham, H. (1964), "An investigation into the bending analysis of hyperbolic paraboloide shells'', Ind. Concrete J., 248-258.
8 Chinosi, C. (2005), "PSRI elements for the Reissner-Mindlin free plate", Comput. Struct., 83(31), 2559-2572.   DOI
9 Dhatt, G.S. (1970), "Instability of thin shells by the finite element method", Proc. IASS Symp., Vienna, Austria.
10 Jeon, H.M., Lee, P.S. and Bathe, K.J. (2014), "The MITC3 shell finite element enriched by interpolation covers", Comput. Struct., 134, 128-142.   DOI
11 Kim, D.N. and Bathe, K.J. (2009), "A triangular six-node shell element", Comput. Struct., 87(23), 1451-1460.   DOI
12 Ko, Y., Lee, Y., Lee, P.S. and Bathe, K.J. (2017), "Performance of the MITC3+ and the MITC4+ shell elements in widely-used benchmark problems", Comput. Struct., 193, 187-206.   DOI
13 Koiter, W. (1960), "A consistent first approximation in the general theory of thin elastic shells", Theor. Thin Elast. Shells, 12-33.
14 Koiter, W.T. and Simmonds, J.G. (1973), Foundations of Shell Theory, Theoretical and Applied Mechanics, Springer Berlin Heidelberg.
15 Kugler, S., Fotiu, P.A. and Murin, J. (2010), "A highly efficient membrane finite element with drilling degrees of freedom", Acta Mech., 213(3-4), 323-348.   DOI
16 Leicester, R.H. (1968), "Finite deformations of shallow shells (Shallow shell deformations based on nonlinear equations solved by Newton-Raphson iteration)", Am. Soc. Civil Eng. Eng. Mech. Div. J., 94, 1409-1423.
17 Lindberg, G.M., Olson, M.D. and Cowper, G.R. (1969), "New developments in the finite element analysis of shells", Quarterly Bulletin of the Division of Mechanical Engineering and The National Aeronautical Establishment, 4, 1-38.
18 Papanicolopulos, S.A., Zervos, A. and Vardoulakis, I. (2009), "A three-dimensional $C^{1}$ finite element for gradient elasticity", Int. J. Numer. Meth. Eng., 77(10), 1396-1415.   DOI
19 Providas, E. and Kattis, M.A. (2000), "An assessment of two fundamental flat triangular shell elements with drilling rotations", Comput. Struct., 77(2), 129-139.   DOI
20 Donnell, L.H. (1933), Stability of Thin Walled Tubes under Torsion, NACA Report No. 479.
21 El-Khaldi, F. (1987), "Contribution au traitement des phenomenes de blocage de membrane et cisaillement dans la modelisation des arcs et des coques minces en theorie de marguerre", Ph.D. Dissertation, Villeurbanne, INSA.
22 Forsberg, K. and Flugge, W. (1966), "Point load on a shallow elliptic paraboloid'', J. Appl. Mech., 33(3), 575-585.   DOI
23 Fezans, G. (1981), "Analyse lineaire et non lineaire geometrique des coques par elements finis isoparametriques tridimensionnels degeneres", Ph.D. Dissertation.
24 Flugge, W. (1960), Stresses in Shells, Springer Verlag, Berlin/Heidelberg/New York.
25 Forsberg, K. and Hartung, R. (1970), "An evaluation of finite difference and finite element techniques for analysis of general shells", Proceedings of the Symposium on High Speed Computation of Elastic Structures, IUTAM, Liege, Belgium.
26 Frey, F. (2000), Analyse des Structures et Milieux Continus : Mecanique des Structures, PPUR Presses Polytechniques.
27 Gallagher, R.H. (1975), "Shell elements", Proceedings of the 1st World Congress on Finite Element Methods in Structural Mechanics, Bournemouth, U.K.
28 Geoffroy, P. (1983), "Developpement et evaluation d'un element fini pour l'analyse non lineaire statique et dynamique de coques minces'', Ph.D. Dissertation.
29 Rezaiee-Pajand, M. and Yaghoobi, M. (2014), "An efficient formulation for linear and geometric non-linear membrane elements", Lat. Am. J. Sol. Struct., 11(6), 1012-1035.   DOI
30 Rezaiee-Pajand, M. and Karkon, M. (2014), "Hybrid stress and analytical functions for analysis of thin plates bending", Lat. Am. J. Sol. Struct., 11(4), 556-579.   DOI
31 Sabir, A.B. (1985), "A rectangular and triangular plane elasticity element with drilling degrees of freedom", Proceedings of the 2nd International Conference on Variational Methods in Engineering.
32 Sanders, J.L. (1959), An Improved First Approximation Theory for Thin Shells (NASA TR-R24), US Government Printing Office, Washington, U.S.A.
33 Sanders, Jr, J.L. (1959), An Improved First-Approximation Theory for Thin Shells.
34 Scordelis, A.C. and Lo, K.S. (1964), "Computer analysis of cylindrical shells", J. Proc., 61(5), 539-562.
35 Serpik, I.N. (2010), "Development of a new finite element for plate and shell analysis by application of generalized approach to patch test", Fin. Elem. Analy. Des., 46(11), 1017-1030.   DOI
36 Guenfoud, M. (1993), "Presentation de l'element DSTM pour le calcul lineaire des coques d'epaisseur quelconque'', Ann. ITBTP, (515), 25-52.
37 Gileva, L., Shaydurov, V. and Dobronets, B. (2013), "The triangular Hermite finite element complementing the Bogner-Fox-Schmit rectangle'', Appl. Math., 4(12), 50.   DOI
38 Guenfoud, M. (2000), "A new three nodes shell element with transverse shear'', Eng. J. Qatar Univ., 13, 193-221.
39 Guenfoud, M. (1990), "Deux elements Triangulaires Nouveaux pour L'analyse Lineaire et Non Lineaire Geometrique des Coques'', Ph.D. Dissertation, Villeurbanne, INSA, France.
40 Guenfoud, M. (1996), "A new three nodes shell element with transverse shear'', J. Int. Assoc. Shell Spat. Struct., 37(3), 193-220.
41 Thomas, G.R. and Gallagher, R.H. (1975), A Triangular Thin Shell Finite Element: Linear Analysis.
42 Shin, C.M. and Lee, B.C. (2014), "Development of a strainsmoothed three-node triangular flat shell element with drilling degrees of freedom", Fin. Elem. Analy. Des., 86, 71-80.   DOI
43 Tahiani, C. and Lachance, L. (1975), "Linear and non-linear analysis of thin shallow shells by mixed finite elements", Comput. Struct., 5(2-3), 167-177.   DOI
44 Teodorescu, P. (1982), "Grands elements finis GEF pour l'elasticite plane'', These no 462 de doctorat presentee au departement de genie civil, Ecole polytechnique federale de Lausanne Suisse.
45 Hamadi, D. (1989), "Numerical and experimental investigation of an elliptical parboloid thin shell structures", Ph.D. Dissertation, City University London, U.K.
46 Abderrahmani, S., Maalam, T. and Hamadi, D. (2016), "On improved thin plate bending rectangular finite element based on the strain approach'', Int. J. Eng. Res. Afr., 27, 76-86.   DOI
47 Barik, M. and Mukhopadhyay, M. (2002), "A new stiffened plate element for the analysis of arbitrary plates'', Thin-Wall. Struct., 40(7), 625-639.   DOI
48 Batoz, J.L. (1977), "Analyse non lineaire de coques minces elastiques de formes arbitraires par elements triangulaires courbes'', These de Doctorat, Faculte des sciences et de genie civil, Quebec, 372.
49 Batoz, J.L. and Dhatt, G. (1990), Modelisation Des Structures Par Elements Finis: Solides Elastiques, Presses Universite Laval, 1.
50 Belarbi M.T. (2000), "Developpement de nouveaux elements a modele en deformation: Application lineaire et non lineaire'', These de Doctorat, Universite de Constantine, Algerie.
51 Hamadi, D. (2006), "Analysis of structures by non-conforming finite elements" analyse des structures par elements finis non conformes", Ph.D. Dissertation, Universite Mohamed Khider Biskra.
52 Hamadi, D., Ayoub, A. and Abdelhafid, O. (2016), "A new flat shell finite element for the linear analysis of thin shell structures'', Eur. J. Comput. Mech., 1-24.
53 Belytschko, T., Stolarski, H., Liu, W.K., Carpenter, N. and Ong, J.S. (1985), "Stress projection for membrane and shear locking in shell finite elements'', Comput. Meth. Appl. Mech. Eng., 51(1), 221-258.   DOI
54 Timoshenko, S. and Woinowsky-Krieger, S. (1959), Theory of Plates and Shells, McGraw-Hill.
55 Zienkiewicz, O.C., Taylor, R.L., Zienkiewicz, O.C. and Taylor, R.L. (1977), The Finite Element Method, McGraw-Hill, London, U.K.
56 Zweiling, K. (1952), Grundlagen Einer Theorie der Biharmonischen Polynome, Verlag Technik.
57 Belarbi, M.T. and Charif, A. (1999), "Developpement d'un nouvel element hexaedrique simple base sur le modele en deformation pour l'etude des plaques minces et epaisses'', Revue Europeenne des elements Finis, 8(2), 135-157.   DOI
58 Belytschko, T., Ong, J.S.J., Liu, W.K. and Kennedy, J.M. (1984), "Hourglass control in linear and nonlinear problems", Comput. Meth. Appl. Mech. Eng., 43(3), 251-276.   DOI
59 Bentaher, M. (1981), "Analyse elastoplastique des plaques et coques minces par elements finis'', These de 3eme cycle, Universite de Technologie de Compiegne, 130.
60 Bhothikhun, P. and Dechaumphai, P. (2014), "Adaptive DKT finite element for plate bending analysis of built-up structures'', Int. J. Mech. Mechatron. Eng., 14(1), 12-20.
61 Bonnes, G. (1969), "Analyse des Voiles Mz par elements finis courbes'', Ph.D. Dissertation, Universite Laval, Canada.
62 Boutagouga D. (2008), "Analyse non-lineaire geometrique et materielle des coques par un element quadrilatere avec ddl rotationnel dit (Drilling rotation)'', Memoire de Magister, Universite Badji mokhtar Annaba.
63 Boutagouga, D. and Djeghaba, K. (2016), "Nonlinear dynamic corotational formulation for membrane elements with in-plane drilling rotational degree of freedom", Eng. Comput., 33(3), 667-697.   DOI
64 Himeur, M. and Guenfoud, M. (2011), "Bending triangular finite element with a fictitious fourth node based on the strain approach", Eur. J. Comput. Mech./Rev. Eur. Mecan. Numer., 20(7-8), 455-485.   DOI
65 Hamadi, D., Ayoub, A. and Maalem, T. (2016), "A new strainbased finite element for plane elasticity problems'', Eng. Comput., 33(2), 562-579.   DOI
66 Himeur, M. (2008), "Developpement d'elements membranaires nouveaux d'elasticite plane bases sur la formulation en deformation'', Ph.D. Dissertation, These de magistere, Universite de Guelma (Algerie), Departement de Genie Civil.
67 Himeur, M. and Guenfoud, M. (2008), Element Fini Triangulaire Nouveau a Noeud Central Perturbe en Formulation Deformation Avec Drilling Rotation, CIFMA 3, 21-23, Alep.
68 Himeur, M., Benmarce, A. and Guenfoud, M. (2014), "A new finite element based on the strain approach with transverse shear effect", Struct. Eng. Mech., 49(6), 793-810.   DOI
69 Himeur, M., Zergua, A. and Guenfoud, M. (2015), "A finite element based on the strain approach using Airy's function'', Arab. J. Sci. Eng., 40(3), 719-733.   DOI
70 Huang, M., Zhao, Z. and Shen, C. (2010), "An effective planar triangular element with drilling rotation", Fin. Elem. Analy. Des., 46(11), 1031-1036.   DOI