## Preprint 2009-01

## Carlo Lovadina, David Mora, Rodolfo Rodríguez:

### Approximation of the buckling problem for Reissner-Mindlin plates

### Abstract:

This paper deals with the approximation of the buckling coefficients and modes of a clamped plate modeled by the Reissner-Mindlin equations. These coefficients are related with the eigenvalues of a non-compact operator. We give a spectral characterization of this operator and show that the relevant buckling coefficients correspond to isolated nondefective eigenvalues. Then we consider the numerical solution of the buckling problem. For the finite element approximation of Reissner-Mindlin equations, it is well known that some kind of reduced integration or mixed interpolation has to be used to avoid locking. In particular we consider Duran-Liberman elements, which have been already proved to be locking-free for load and vibration problems. We adapt the classical approximation theory for non-compact operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. These estimates are valid with constants independent of the plate thickness. We report some numerical experiments confirming the theoretical results. Finally, we refine the analysis in the case of a uniformly compressed plate.

This preprint gave rise to the following definitive publication(s):

**Carlo LOVADINA, David MORA, Rodolfo RODRíGUEZ: ***Approximation of the buckling problem for Reissner-Mindlin plates*. SIAM Journal on Numerical Analysis, vol. 48, 2, pp. 603-632, (2010)