The finite element method has been extended to determine the response of large amplitude forced vibrations of thin plates. A harmonic force matrix of a rectangular element under uniform harmonic excitation is developed for nonlinear forced vibration analysis. Inplane deformation and inertia are both considered in the formulation. Results obtained are compared with simple elliptic response, perturbation and other approximation solutions.

A finite element method for nonlinear forced vibrations of rectangular plates

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.

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

A reinvestigation of post-buckling behaviour of elastic circular plates using a simple finite element formulation

Formulations and computational procedures are presented for studying the geometrically nonlinear behavior, including buckling, postbuckling, and nonlinear vibrations of perfect and imperfect, isotropic and laminated thin plates. The finite-element method is used. The element used is a 48 degree-of-freedom thin flat plate rectangular element capable of modeling arbitrary imperfections. The incremental and total stiffness matrices for large displacement behavior are derived based on the total Lagrangian approach in conjunction with the Hamilton's principle. The geometric imperfections are treated by considering additional terms in the straindisplacement relations. Numerical results are presented for a variety of examples including: 1) postbuckling analysis of an isotropic, imperfect flat plate; 2) postbuckling analysis of a thin, cross-ply laminated imperfect plate; 3) free vibrations of an angle-ply laminated plate; 4) free-vibrations of an imperfect isotropic plate under the action of axial loads; 5) nonlinear free-vibrations of isotropic perfect flat plates; 6) nonlinear vibrations of axially loaded, isotropic perfect flat plates; and 7) nonlinear vibrations of an imperfect laminated plate. The results obtained in this study are compared with existing solutions and a good agreement is seen.

Mei Chuh

Papers

Large amplitude free flexural vibrations of thin plates of arbitrary shape