A finite element method for stiffened plates
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 2, p. 291-315

The aim of this paper is to analyze a low order finite element method for a stiffened plate. The plate is modeled by Reissner-Mindlin equations and the stiffener by Timoshenko beams equations. The resulting problem is shown to be well posed. In the case of concentric stiffeners it decouples into two problems, one for the in-plane plate deformation and the other for the bending of the plate. The analysis and discretization of the first one is straightforward. The second one is shown to have a solution bounded above and below independently of the thickness of the plate. A discretization based on DL3 finite elements combined with ad-hoc elements for the stiffener is proposed. Optimal order error estimates are proved for displacements, rotations and shear stresses for the plate and the stiffener. Numerical tests are reported in order to assess the performance of the method. These numerical computations demonstrate that the error estimates are independent of the thickness, providing a numerical evidence that the method is locking-free.

DOI : https://doi.org/10.1051/m2an/2011011
Classification:  65N30,  74K10,  74K20
Keywords: stiffened plates, Reissner-Mindlin model, Timoshenko beam, finite elements, error estimates, locking
@article{M2AN_2012__46_2_291_0,
author = {Dur\'an, Ricardo and Rodr\'\i guez, Rodolfo and Sanhueza, Frank},
title = {A finite element method for stiffened plates},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {46},
number = {2},
year = {2012},
pages = {291-315},
doi = {10.1051/m2an/2011011},
zbl = {1272.74399},
mrnumber = {2855644},
language = {en},
url = {http://www.numdam.org/item/M2AN_2012__46_2_291_0}
}

Durán, Ricardo; Rodríguez, Rodolfo; Sanhueza, Frank. A finite element method for stiffened plates. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 2, pp. 291-315. doi : 10.1051/m2an/2011011. http://www.numdam.org/item/M2AN_2012__46_2_291_0/

[1] D.N. Arnold, Discretization by finite element of a model parameter dependent problem. Numer. Math. 37 (1981) 405-421. | MR 627113 | Zbl 0446.73066

[2] D.N. Arnold and R.S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate. SIAM J. Numer. Anal. 26 (1989) 1276-1290. | MR 1025088 | Zbl 0696.73040

[3] K. Arunakirinathar and B.D. Reddy, Mixed finite element methods for elastic rods of arbitrary geometry. Numer. Math. 64 (1993) 13-43. | MR 1191321 | Zbl 0794.73070

[4] K.-J. Bathe, F. Brezzi and S.W. Cho, The MITC7 and MITC9 plate bending elements, Comput. Struct. 32 (1984) 797-814. | Zbl 0705.73241

[5] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991). | MR 1115205 | Zbl 0788.73002

[6] F. D'Hennezel, Domain decomposition method and elastic multi-structures: the stiffened plate problem. Numer. Math. 66 (1993) 181-197. | MR 1245010 | Zbl 0794.73077

[7] R.G. Durán and E. Liberman, On the mixed finite element methods for the Reissner-Mindlin plate model. Math. Comput. 58 (1992) 561-573. | MR 1106965 | Zbl 0763.73054

[8] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Springer-Verlag, New York (2004). | MR 2050138 | Zbl 1059.65103

[9] R. Falk, Finite element methods for linear elasticity, in Mixed Finite Elements, Compatibility Conditions, and Applications. Springer-Verlag, Berlin, Heidelberg (2006) 159-194. | Zbl pre05310105

[10] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin, Heidelberg (1986). | MR 851383 | Zbl 0585.65077

[11] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985). | MR 775683 | Zbl 0695.35060

[12] T.P. Holopainen, Finite element free vibration analysis of eccentrically stiffened plates. Comput. Struct. 56 (1995) 993-1007. | Zbl 0900.73770

[13] V. Janowsky, and P. Procházka, The nonconforming finite element method in the problem of clamped plate with ribs. Appl. Math. 21 (1976) 273-289. | MR 413548 | Zbl 0357.65087

[14] A. Mukherjee and M. Mukhopadhyay, Finite element free vibration of eccentrically stiffened plates. Comput. Struct. 30 (1988) 1303-1317. | Zbl 0664.73037

[15] J. O'Leary and I. Harari, Finite element analysis of stiffened plates. Comput. Struct. 21 (1985) 973-985. | Zbl 0587.73115

[16] 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. | MR 483555 | Zbl 0362.65089

[17] L. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483-493. | MR 1011446 | Zbl 0696.65007