×

The problem of membrane locking in finite element analysis of cylindrical shells. (English) Zbl 0768.73079

Membrane (or shear-membrane) locking may arise in a thin shell problem if the deformation state of the shell is such that bending deformations carry the dominating part of the total deformation energy. In this paper finite element schemes are studied where the displacement fields is approximated using \(C^ 0\) finite elements of degree \(\geq 1\) on a rectangular or quadrilateral grid. Two alternative strategies are considered: one where the energy is simply minimized as it is given, and a special implementation of selective reduced integration strategy, where the strain components that cause locking are underintegrated in a special way. The basic idea for the second strategy is due to K.-J. Bathe and E. N. Dvorkin [Int. J. Numer. Methods Eng. 22, 697-722 (1986; Zbl 0585.73123)]. The main results of the paper are error estimates for the two families of finite element schemes in relative energy norm.
Reviewer: V.Arnautu (Iaşi)

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74K15 Membranes

Citations:

Zbl 0585.73123
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Arnold, D.N., Falk, R.S. (1989): A Uniformly Accurate Finite Element Method for the Reissner-Mindlin Plate. SIAM J. Numer. Anal.26, 1276-1290 · Zbl 0696.73040
[2] Ashwell, D.G., Gallagher, R.H. (eds.) (1976): Finite Elements for Thin Shells & Curved Members. Wiley, London New York Sydney Toronto · Zbl 0397.73062
[3] Babu?ka, I., Griebel, M., Pitkäranta, J. (1989): The Problem of Selecting the Shape Functions for ap-Type Finite Element. Int. J. Numer. Methods Eng.28, 1891-1908 · Zbl 0705.73246
[4] Babu?ka, I., Suri, M. (1987): The Optimal Convergence Rate of thep-Version of the Finite Element Method. SIAM J. Numer. Anal.24, 750-776 · Zbl 0637.65103
[5] Babu?ka, I., Suri, M. (1987): Theh-p Version of the Finite Element Method with Quasiuniform Meshes. Math. Modelling Numer. Anal.21, 199-238 · Zbl 0623.65113
[6] Babu?ka, I., Suri, M. (1990): On Locking and Robustness in the Finite Element Method. Preprint, Institute for Physical Science and Technology, University of Maryland · Zbl 0731.73078
[7] Bathe, K.J., Brezzi, F. (1987): A Simplified Analysis of Two Plate Bending Elements ? The MITC4 and MITC9 Elements. In: G.N. Pande, J. Middleton, eds. NUMETA 87, Numerical Techniques for Engineering Analysis and Design, Vol. 1, D46/1. Martinus Nijhoff, Amsterdam
[8] Bathe, K.J., Brezzi, F., Fortin, M. (1989): Mixed-Interpolated Elements for Reissner-Mindlin Plates. Int. J. Numer. Methods Eng.28, 1787-1801 · Zbl 0705.73238
[9] Bathe, K.J., Dvorkin, E. (1985): A Four-Node Plate Bending Element Based on Mindlin-Reissner Plate Theory and Mixed Interpolation. Int. J. Numer. Methods Eng.21, 367-383 · Zbl 0551.73072
[10] Bathe, K.J., Dvorkin, E. (1986): A Formulation of General Shell Elements ? The Use of Mixed Interpolation of Tensorial Components. Int. J. Numer. Methods Eng.22, 697-722 · Zbl 0585.73123
[11] Bernadou, M., Trouvé, P. (1989/1990): Approximation of general shell problems by flat elements. Parts 1-3. Comput. Mech.5, 175-208;6, 359-378;7, 1-11 · Zbl 0698.73050
[12] Brezzi, F., Fortin, M., Stenberg, R. (1990): Error Analysis of Mixed-Interpolated Elements for Reissner-Mindlin Plates. Preprint, Laboratory of Strength of Materials, Helsinki University of Technology · Zbl 0751.73053
[13] Ciarlet, P.G. (1978): The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam · Zbl 0383.65058
[14] Hughes, T.J.R. (1987): The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs, N.J. · Zbl 0634.73056
[15] Leino, Y., Pitkäranta, J. (1992): On the Membrane Locking ofh-p Finite Elements in a Cylindrical Shell Problem. To appear · Zbl 0799.73071
[16] Novozhilov, V.V. (1964): Thin Shell Theory. Translated by P.G. Lowe, J.R.M. Radok, ed. Noordhoff, Groningen · Zbl 0135.43602
[17] Piila, J., Pitkäranta, J. (1990/1991): Energy Estimates Relating Different Linear Elastic Models of a Cylindrical Shell. Part I: The Membrane-Dominated Case, Part 2: The Bending-Dominated Case, Part 3: The Soft Membrane Case. Reports A286, A293 and A299, Institute of Mathematics, Helsinki University of Technology · Zbl 0768.73045
[18] Pitkäranta, J. (1988): Analysis of Some Low-Order Finite Element Schemes for Mindlin-Reissner and Kirchhoff Plates. Numer. Math.53, 237-254 · Zbl 0654.73043
[19] Stolarski, H., Belytschko, T. (1982): Membrane Locking and Reduced Indegration for Curved Elements. J. Appl. Mech.49, 172-176 · Zbl 0482.73060
[20] Stolarski, H., Belytschko, T. (1983): Shear and Membrane Locking in CurvedC{\(\deg\)} Elements. Comput. Methods Appl. Mech. Eng.41, 279-296 · Zbl 0517.73068
[21] Vinson, J.R. (1974): Structural Mechanics: The Behavior of Plates and Shells. Wiley, New York London Sydney Toronto
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.