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A catenoidal patch test for the inextensional bending of thin shell finite elements. (English) Zbl 0742.73020
Summary: The history of the problem of inextensional bending of thin shells is briefly introduced. A comparison between Novoshilov’s and Sander’s shell theories is performed and it is shown that, for a catenoid of revolution, both theories will yield identical results. The problem of inextensional bending is then defined and discussed, with the catenoid of revolution as the focus. The theory of inextensional bending of a catenoid of revolution is then established and a simple patch test for constant inextensional bending is created. The critical size of a quadratic thin shell Sanders-type finite element is then predicted, above which parasitic membrane action will prevail. A very surprising result was obtained, that is, it appears to be practically impossible for quadratic thin shell finite elements to achieve a membrane stress free state along the meridional direction of the catenoid under inextensional bending. Finally, the patch test in question is examined, in both an ‘ undistorted’ and a ’distorted’ mode, with quadratic Mindlin isoparametric shell elements and the stress results obtained are presented and discussed. The superiority in the performance of the Gauss point stresses under reduced integration as compared to that of the nodal stresses is highlighted and integrated with results of previous publications.

74S05 Finite element methods applied to problems in solid mechanics
74K15 Membranes
Full Text: DOI
[1] Morley, L.S.D., Polynomial stress states in first approximation theory of circular cylindrical shells, Quart. J. mech. appl. math., XXV, 1, (February 1972)
[2] Morley, L.S.D., Analysis of developable shells with special reference to the finite element method and circular cylinders, Philos. trans. roy. soc. London ser. A, 281, 113-170, (1976) · Zbl 0319.73040
[3] Morley, L.S.D.; Morris, A.J., Conflict between finite elements and shell theory, ()
[4] Morley, L.S.D., Approximate/displacements of finite membrane actions in a shell triangular element, Aeronaut. quart., XXXIV, 282-302, (November 1983)
[5] Morley, L.S.D., Finite element criteria for some shells, Internat. J. numer. methods engrg., 20, 1711-1728, (1984) · Zbl 0544.73102
[6] Morley, L.S.D., Fortran computer program for inextensional bending of a doubly curved shell triangular element, Internat. J. numer. methods engrg., 19, 647-664, (1983) · Zbl 0508.73067
[7] Morley, L.S.D., Bending of bilinear quadrilateral shell elements, Internat. J. numer. methods engrg., 20, 1373-1376, (1984) · Zbl 0537.73065
[8] Morley, L.S.D., A curvilinear triangular finite element for plate bending by a hybrid method of assumed displacement supplemented with assumed stress, Comput. & structures, 18, 2, 311-318, (1984) · Zbl 0523.73058
[9] Morley, L.S.D., Inextensional bending in shells of explicit cubic representation, Internat. J. solids and structures, 20, 7, 631-635, (1984) · Zbl 0542.73088
[10] Morley, L.S.D., Inextensional bending of a shell triangular element in quadratic parametric representation, Internat. J. solids and structures, 18, 11, 919-935, (1982) · Zbl 0488.73079
[11] Novoshilov, V.V., The theory of thin elastic shells, (1959), Noordhoff Groningen, The Netherlands
[12] Maxwell, J.C., On the transformation of surfaces by bending, (), 9, 81-114, (1954), Reprinted, in:
[13] Gauss, K.F., General investigation of curved surfaces, (1828), English Translation 1902 by J.C. Morehead and A.M. Hiltebeitel, Princeton; Reprinted 1965 with introduction by R. Courant. Hewlett (Raven Press, New York) · Zbl 0137.00204
[14] Love, A.E.H., A treatise on the mathematical theory of elasticity, (1944), Dover New York · Zbl 0063.03651
[15] Novoshilov, V.V., Certain remarks regarding the theory of shells, Prikl. mat. mekh. SSSR, V, 3, (1941)
[16] Brebbia, C.A.; Connor, J.J., Fundamentals of finite element techniques, (1973), Butterworths London · Zbl 0431.76001
[17] B. Budianski and J.L. Sanders. On the ‘best’ first-order linear shell theory, in: Progress in Applied Mechanics (The Prager Anniversary Volume) (Macmillan, New York) 129-140.
[18] Sanders, J.L., An improved first-approximation theory for thin shells, ()
[19] Koiter, W.T., The theory of thin elastic shells (Proceedings of the symbosium on) I.U.T.A.M., Delft, (), 24-28
[20] Ashwell, D.G.; Gallagher, R.H., Finite elements for thin shells and curved members, (1976), Wiley New York · Zbl 0397.73062
[21] Cook, R.D., Concepts and applications of finite element analysis, (1974), Wiley New York, Second Edition, 1981
[22] Calladine, C.R., Theory of shell structures, (1983), Cambridge Univ. Press Cambridge · Zbl 0507.73079
[23] Kamoulakos, A., Understanding and improving the reduced integration of Mindlin shell elements, Internat. J. numer. methods engrg., 26, 2009-2029, (1988) · Zbl 0662.73047
[24] Kamoulakos, A.; Hitchings, D., Designing benchmark tests for shell finite elements, ()
[25] Meriam, J.L., Statics - second edition - SI version, (1975), Wiley New York
[26] C.R. Calladine, The static-geometric analogy in the equations of thin shell structures, Math. Prec, Cambridge Philos. Soc. 82, 335-351. · Zbl 0359.73073
[27] Gol’Denveizer, A.L., Theory of elastic thin shells, (1961), Pergamon Oxford
[28] Flugge, W., Stresses in shells, (1960), Springer Berlin · Zbl 0092.41504
[29] Green, A.E.; Zerna, W., Theoretical elasticity, (1954), Clarendon Press Oxford · Zbl 0056.18205
[30] A. Kamoulakos, G.A.O. Davies and D. Hitchings, Benchmark Tests for Various Finite Element Assemblies, N.A.F.E.M.S. Publication C1 and Progress Report of August 1985.
[31] A. Kamoulakos, D. Hitchings and G.A.O. Davies, Benchmark Tests for Various Finite Element Assemblies - Thin Shells, N.A.F.E.M.S. Publication TSBM and Progress Report of April 1986.
[32] A.I. Lur‘e. On the static-geometric analogue of shell theory, in: J.R. Radok, ed., Problems of Continuum Mechanics (The Muskhelishvili Anniversary Volume) (SIAM, Philadelphia, PA) 267-274.
[33] Morris, A.J., Shell finite element evaluation tests, N.A.F.E.M.S., report C4, (August 1985)
[34] Muskhelishvili, N.I., Some basic problems in the mathematical theory of elasticity, (1954), translated by J.R.M. Radok (Noordhoff, Groningen, The Netherlands, 1963) · Zbl 0057.16805
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