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A finite element method for stiffened plates

Published online by Cambridge University Press:  12 October 2011

Ricardo Durán ,
Rodolfo Rodríguez  and
Frank Sanhueza
Ricardo Durán
Affiliation:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina. [email protected]
Rodolfo Rodríguez
Affiliation:
CI2MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Chile. [email protected]
Frank Sanhueza
Affiliation:
Escuela de Obras Civiles, Universidad Andres Bello, Autopista Concepción, Talcahuano 7100, Concepción, Chile. [email protected]
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Abstract

The aim of this paper is to analyze a low order finite element methodfor a stiffened plate. The plate is modeled by Reissner-Mindlinequations and the stiffener by Timoshenko beams equations. Theresulting problem is shown to be well posed. In the case of concentricstiffeners it decouples into two problems, one for the in-plane plate deformation and the other for the bending of the plate. The analysisand discretization of the first one is straightforward. The second oneis shown to have a solution bounded above and below independently of thethickness of the plate. A discretization based on DL3 finite elementscombined with ad-hoc elements for the stiffener is proposed.Optimal order error estimates are proved for displacements, rotationsand shear stresses for the plate and the stiffener. Numerical tests arereported in order to assess the performance of the method. Thesenumerical computations demonstrate that the error estimates areindependent of the thickness, providing a numerical evidence that themethod is locking-free.


Keywords

Stiffened platesReissner-Mindlin modelTimoshenko beamfinite elementserror estimateslocking
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 2 , March 2012 , pp. 291 - 315
Copyright
© EDP Sciences, SMAI, 2011

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References

Arnold, D.N., Discretization by finite element of a model parameter dependent problem. Numer. Math. 37 (1981) 405–421. CrossRef
Arnold, D.N. and Falk, R.S., A uniformly accurate finite element method for the Reissner-Mindlin plate. SIAM J. Numer. Anal. 26 (1989) 1276–1290. CrossRef
Arunakirinathar, K. and Reddy, B.D., Mixed finite element methods for elastic rods of arbitrary geometry. Numer. Math. 64 (1993) 13–43. CrossRef
K.-J. Bathe, F. Brezzi and S.W. Cho, The MITC7 and MITC9 plate bending elements, Comput. Struct. 32 (1984) 797–814. CrossRef
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991).
d'Hennezel, F., Domain decomposition method and elastic multi-structures: the stiffened plate problem. Numer. Math. 66 (1993) 181–197. CrossRef
Durán, R.G. and Liberman, E., On the mixed finite element methods for the Reissner-Mindlin plate model. Math. Comput. 58 (1992) 561–573. CrossRef
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Springer-Verlag, New York (2004).
R. Falk, Finite element methods for linear elasticity, in Mixed Finite Elements, Compatibility Conditions, and Applications. Springer-Verlag, Berlin, Heidelberg (2006) 159–194.
V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin, Heidelberg (1986).
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985).
Holopainen, T.P., Finite element free vibration analysis of eccentrically stiffened plates. Comput. Struct. 56 (1995) 993–1007. CrossRef
Janowsky, V., and Procházka, P., The nonconforming finite element method in the problem of clamped plate with ribs. Appl. Math. 21 (1976) 273–289.
Mukherjee, A. and Mukhopadhyay, M., Finite element free vibration of eccentrically stiffened plates. Comput. Struct. 30 (1988) 1303–1317. CrossRef
O'Leary, J. and Harari, I., Finite element analysis of stiffened plates. Comput. Struct. 21 (1985) 973–985. CrossRef
P.A. Raviart and J.M. Thomas, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics, Springer, Berlin, Heidelberg (1977) 292–315.
Scott, L. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. CrossRef