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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

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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

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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

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  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.  
  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.  
  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.  
  9. T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1966).  
  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.  
  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.  

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