Simplified approach of free vibration analysis of plates supported in vicinity of the corners by BEM

Abstract

Free vibration analysis of Kirchhoff plate by the Boundary Element Method is presented in the paper. The boundary integral equation are derived according to the Bettie theorem. The collocation version of BEM with non-singular approach with one and double collocation points is used. The constant type of element is introduced. Boundary support at selected point is modelled as support in vicinity of point along single boundary element. Introduction Plates supported at points, in vicinity of selected points, for example by the columns, are often used in building structures. The bending problem of plates supported on boundary or rested on internal supports is the classic boundary issue in theory of plates and shells. The columns are often located on plate edges. The first well known work which included strict solutions of these problems is a publication of Nadai [1]. Iguchi [2] considered free vibrations of thin plates supported at the corners. Kączkowski [3] applied double Fourier range of sines to derive a general solution of boundary problem of orthotropic rectangular plates with optional boundary conditions including internal supports located at the plate corners. Timoshenko and Woinowsky-Krieger [4] analysed a static problem of a square plate rested on supports located at the corners by. The solutions as polynominals and range of cosines were assumed. Woźnica [5] solved analytically similar static and dynamic problem using Fourier transformation. A lot of papers present numerical approach, especially the Finite Difference Method (FDM), the Finite Element Method (FEM) and the Finite Strip Method (FSM), as a useful tool in plate bending analysis. Wide review of the literature is included and cited by Guminiak and Sygulski [6]. The Boundary Element Method (BEM) was proposed as independent numerical tool for applying for engineering applications [7, 8]. Bezine [9] and Stern [10] introduced BEM for analysis of plate bending according to Kirchhoff theory. Vander Weeen [11] proposed fundamental solutions for Reissner plate and used BEM to numerical analysis. Thick plates according to Reissner theory were analysed by Litewka and Sygulski [12] by the BEM. Authors applied fundamental solution derived by Ganowicz [13]. Myślecki [14, 15], Myślecki and Olenkiewicz [16] and Olenkiewicz [17] applied BEM tech-