## Preprint 2010-23

## Ricardo Durán, Rodolfo Rodríguez, Frank Sanhueza:

### A finite element method for stiffened plates

### Abstract:

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.

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

**Ricardo DURáN, Rodolfo RODRíGUEZ, Frank SANHUEZA: ***A finite element method for stiffened plates*. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 46, 2, pp. 291-315 (2012).