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

Abstract

top
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.

How to cite

top

Lovadina, 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
  1. D.N. Arnold, Discretization by finite elements of a model parameter dependent problem. Numer. Math.37 (1981) 405–421.  Zbl0446.73066
  2. 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
  3. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).  Zbl0788.73002
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1966).  Zbl0148.12601
  10. 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
  11. J.N. Reddy, An Introduction to the Finite Element Method. McGraw-Hill, New York (1993).  
  12. B. Szabó and G. Királyfalvi, Linear models of buckling and stress-stiffening. Comput. Methods Appl. Mech. Eng.171 (1999) 43–59.  Zbl0944.74028

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.