Plate stability by boundary element method

Elzein, Abbas (1990) Plate stability by boundary element method. University of Southampton, Doctoral Thesis.

Abstract

The Boundary Element Method is applied to problems involving plates loaded in their own plane. Initially, the membrane stress distribution is obtained through a boundary integral formulation of the plane stress problem in terms of the Airy stress function. Subsequently, a boundary integral formulation of the eigenvalue problem of plate buckling is developed which contains no curvatures and requires the modelling of the domain deflections only. A numerical implementation of the formulation discretizes both domain and boundary with one degree of freedom per domain node and two per boundary node. Various boundary and domain element types are tested. The transverse moments at corners, appearing in jump terms, are evaluated using polynomial approximations of the slope normal to the boundary. The deflection and normal slope of free boundaries are evaluated from the domain deflection model. The dual-reciprocity technique is applied to the domain integral appearing in the formulation for the critical buckling loads. The deflections are modelled using either trigonometric deflection shapes or a distribution of nodal unknowns related by a Fourier series approximation. A Fourier series scheme is then applied to the curvatures expression in the domain integrals and the Rayleigh-Green reciprocity equation is used to transform these integrals to the boundary. Incremental boundary integral equations of the large deflection problem of imperfect plates are derived. The curvatures are eliminated from the equation governing the bending deflection. The curvatures appearing in the equation for the membrane stresses are expressed in terms of the deflections using various nonlinear domain deflection models. Four systems of equations are obtained that are solved iteratively to account for the second-order terms. Computer programs written in FORTRAN-77 and run on a VAX/VMS microcomputer are developed. Convergence tests and parametric optimisation studies are performed. Results obtained from these programs are compared to analytical, numerical and experimental results. The boundary element method yielded very accurate results in almost all examples solved throughout the project.

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