An initial stability of plates in various conservative load conditions by the boundary element method

Abstract

Abstract. An initial stability of Kirchhoff plates by the Boundary Element Method (BEM) is presented in the paper. A plate is subjected by external in-plane normal and tangential conservative loadings acting in two perpendicular directions. The Betti's theorem is used to derive the boundary-domain integral equations. The direct version of the Boundary Element Method is presented with combination to simplified boundary conditions. The singular and non-singular approach of the boundary integrals derivation is used.

The Stability of a Steel Welded Girder with Bending and Shear Forces Included

“…The initial stability problem of the plate subjected only to in-plane forces is solved as the simple benchmark test in reference to the FEM. The solution of differential equation (1) can be expressed as the integral representation of two boundary-domain integral equations formulated according to the simplified approach (Guminiak and Sygulski, 2003;Guminiak, 2014):…”

“…The expression ̃( )denotes shear force for clamped and for simply-supported edges: ̃( ) = ( ) is the shear force (distributed reaction of the support) on the boundary far from the plate corner or ̃( ) = ( ) the distributed reaction along the small fragment of the boundary close to the corner. Because the relation between ( ) and the deflection is known: ( ) = ( )⁄ , the angle of rotation ( ) can be evaluated using a finite difference scheme of the deflection with two or more adjacent nodal values (Guminiak and Sygulski, 2003;Guminiak, 2014). …”

“…Integration of suitable fundamental functions is done in a local coordinate system ni, si connected with ith boundary element and next, these integrals must be transformed to nk, sk coordinate system, connected with kth element (Guminiak and Sygulski, 2003;Guminiak, 2014). For a non-singular approach, the localization of a collocation point is defined by the parameter ̃ which determines the distance from a plate edge or by non-dimensional parameter =̃⁄ where d is the element length (Guminiak and Sygulski, 2003;Guminiak, 2014).…”

“…The analysis of plates with external in-plane loading acting directly along free edges (edges without any constraints) requires a broader analysis. Elimination of boundary variables and from matrix equation (8) allows to obtain the standard eigenvalue problem (Guminiak and Sygulski, 2003;Guminiak, 2014):…”

“…where elements of the eigenvector κ are obtained after solution of the standard eigenvalue problem (9). The first equation (11)1 is obtained from the first equation of (8) …”