Buckled plate vibrations and large amplitude vibrations using high-order triangular elements

Abstract

A high-order triangular membrane finite element is combined with a fully conforming triangular plate bending element to solve the geometrically nonlinear problems of plates where the membrane and flexural behaviors are coupled and the effect of the inplane boundary conditions is as significant as the flexural boundary conditions. Each of the three orthogonal displacement components is represented by a two-dimensional polynomial of the same quintic order with no bias against one another giving the element a total of 54 degrees of freedom. The nonlinear stiffness matrices are formulated and a Newton-Raphson iteration procedure is used. Examples include the analyses of plane stresses of a parabolically loaded square plate, large deflections of a square plate under lateral pressure, postbuckling of a square plate, linear free vibration of a buckled rectangular plate, large amplitude free vibration of a square plate with and without inplane stresses. Various flexural and inplane boundary conditions are considered. Results are compared with those obtained by alternative finite element methods, analytical approximate methods, and an experiment. Physical interpretations of the results and explanations of the discrepancies among various solutions are provided. The results indicate that the present development is capable of accurately solving a wide variety of geometrically nonlinear plate problems.