# A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam

Carlo Lovadina; David Mora; Rodolfo Rodríguez

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

- Volume: 45, Issue: 4, page 603-626
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topLovadina, Carlo, Mora, David, and Rodríguez, Rodolfo. "A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam." ESAIM: Mathematical Modelling and Numerical Analysis 45.4 (2011): 603-626. <http://eudml.org/doc/276344>.

@article{Lovadina2011,

abstract = {
The aim of this paper is to develop a finite element method which allows computing
the buckling coefficients and modes of a non-homogeneous Timoshenko beam.
Studying the spectral properties of a non-compact operator,
we show that the relevant buckling coefficients correspond to isolated
eigenvalues of finite multiplicity.
Optimal order error estimates are proved for the eigenfunctions
as well as a double order of convergence for
the eigenvalues using classical abstract spectral approximation theory for non-compact operators.
These estimates are valid independently of the thickness of the beam, which
leads to the conclusion that the method is locking-free.
Numerical tests are reported in order to assess the performance of the method.
},

author = {Lovadina, Carlo, Mora, David, Rodríguez, Rodolfo},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Finite element approximation; eigenvalue problems; Timoshenko beams; finite element approximation},

language = {eng},

month = {1},

number = {4},

pages = {603-626},

publisher = {EDP Sciences},

title = {A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam},

url = {http://eudml.org/doc/276344},

volume = {45},

year = {2011},

}

TY - JOUR

AU - Lovadina, Carlo

AU - Mora, David

AU - Rodríguez, Rodolfo

TI - A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2011/1//

PB - EDP Sciences

VL - 45

IS - 4

SP - 603

EP - 626

AB -
The aim of this paper is to develop a finite element method which allows computing
the buckling coefficients and modes of a non-homogeneous Timoshenko beam.
Studying the spectral properties of a non-compact operator,
we show that the relevant buckling coefficients correspond to isolated
eigenvalues of finite multiplicity.
Optimal order error estimates are proved for the eigenfunctions
as well as a double order of convergence for
the eigenvalues using classical abstract spectral approximation theory for non-compact operators.
These estimates are valid independently of the thickness of the beam, which
leads to the conclusion that the method is locking-free.
Numerical tests are reported in order to assess the performance of the method.

LA - eng

KW - Finite element approximation; eigenvalue problems; Timoshenko beams; finite element approximation

UR - http://eudml.org/doc/276344

ER -

## References

top- D.N. Arnold, Discretization by finite elements of a model parameter dependent problem. Numer. Math.37 (1981) 405–421. Zbl0446.73066
- I. Babuška and J. Osborn, Eigenvalue Problems, in Handbook of Numerical AnalysisII, P.G. Ciarlet and J.L. Lions Eds., North-Holland, Amsterdam (1991) 641–787. Zbl0875.65087
- F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). Zbl0788.73002
- M. Dauge and M. Suri, Numerical approximation of the spectra of non-compact operators arising in buckling problems. J. Numer. Math.10 (2002) 193–219. Zbl1099.74545
- J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 1: The problem of convergence. RAIRO Anal. Numér.12 (1978) 97–112. Zbl0393.65024
- J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 2: Error estimates for the Galerkin method. RAIRO Anal. Numér.12 (1978) 113–119. Zbl0393.65025
- R.S. Falk, Finite Elements for the Reissner-Mindlin Plate, in Mixed Finite Elements, Compatibility Conditions, and Applications, D. Boffi and L. Gastaldi Eds., Springer-Verlag, Berlin (2008) 195–230. Zbl1167.74041
- E. Hernández, E. Otárola, R. Rodríguez and F. Sanhueza, Approximation of the vibration modes of a Timoshenko curved rod of arbitrary geometry. IMA J. Numer. Anal.29 (2009) 180–207. Zbl1155.74042
- T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1966). Zbl0148.12601
- C. Lovadina, D. Mora and R. Rodríguez, Approximation of the buckling problem for Reissner-Mindlin plates. SIAM J. Numer. Anal.48 (2010) 603–632. Zbl05866183
- J.N. Reddy, An Introduction to the Finite Element Method. McGraw-Hill, New York (1993).
- B. Szabó and G. Királyfalvi, Linear models of buckling and stress-stiffening. Comput. Methods Appl. Mech. Eng.171 (1999) 43–59. Zbl0944.74028

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.