A finite element method for stiffened plates
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 2, pp. 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 : 10.1051/m2an/2011011
Classification : 65N30, 74K10, 74K20
Mots-clés : 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 },
     pages = {291--315},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {2},
     year = {2012},
     doi = {10.1051/m2an/2011011},
     mrnumber = {2855644},
     zbl = {1272.74399},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2011011/}
}
TY  - JOUR
AU  - Durán, Ricardo
AU  - Rodríguez, Rodolfo
AU  - Sanhueza, Frank
TI  - A finite element method for stiffened plates
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2012
SP  - 291
EP  - 315
VL  - 46
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2011011/
DO  - 10.1051/m2an/2011011
LA  - en
ID  - M2AN_2012__46_2_291_0
ER  - 
%0 Journal Article
%A Durán, Ricardo
%A Rodríguez, Rodolfo
%A Sanhueza, Frank
%T A finite element method for stiffened plates
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2012
%P 291-315
%V 46
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2011011/
%R 10.1051/m2an/2011011
%G en
%F 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 , Tome 46 (2012) no. 2, pp. 291-315. doi : 10.1051/m2an/2011011. http://www.numdam.org/articles/10.1051/m2an/2011011/

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

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

Cité par Sources :