`Mathematical Problems in EngineeringVolume 2007 (2007), Article ID 62157, 17 pagesdoi:10.1155/2007/62157`
Research Article

# Luiz Carlos Facundo Sanches,1Euclides Mesquita,2Renato Pavanello,2 and Leandro Palermo Jr.3

1Department of Mathematics, Paulista State University, Al. Rio de Janeiro s/n, Ilha Solteira 15385-000, SP, Brazil
2Department of Computational Mechanics, State University of Campinas, Rua Mendeleiev s/n, Campinas 13083-970, SP, Brazil
3Department of Structures, State University of Campinas, Avenida Albert Einstein 951, Campinas 13083-970, SP, Brazil

Received 1 October 2006; Revised 7 February 2007; Accepted 26 February 2007

Copyright © 2007 Luiz Carlos Facundo Sanches et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A direct version of the boundary element method (BEM) is developed to model the stationary dynamic response of reinforced plate structures, such as reinforced panels in buildings, automobiles, and airplanes. The dynamic stationary fundamental solutions of thin plates and plane stress state are used to transform the governing partial differential equations into boundary integral equations (BIEs). Two sets of uncoupled BIEs are formulated, respectively, for the in-plane state (membrane) and for the out-of-plane state (bending). These uncoupled systems are joined to form a macro-element, in which membrane and bending effects are present. The association of these macro-elements is able to simulate thin-walled structures, including reinforced plate structures. In the present formulation, the BIE is discretized by continuous and/or discontinuous linear elements. Four displacement integral equations are written for every boundary node. Modal data, that is, natural frequencies and the corresponding mode shapes of reinforced plates, are obtained from information contained in the frequency response functions (FRFs). A specific example is presented to illustrate the versatility of the proposed methodology. Different configurations of the reinforcements are used to simulate simply supported and clamped boundary conditions for the plate structures. The procedure is validated by comparison with results determined by the finite element method (FEM).