Nonlinear Vibration of Thin Elastic Plates, Part 1: Generalized Incremental Hamilton’s Principle and Element Formulation

Abstract

The finite element method has been widely used for analyzing nonlinear problems, but it is surprising that so far only a few papers have been devoted to nonlinear periodic structural vibrations. In Part 1 of this paper, a generalized incremental Hamilton’s principle for nonlinear periodic vibrations of thin elastic plates is presented. This principle is particularly suitable for the formulation of finite elements and finite strips in geometrically nonlinear plate problems due to the fact that the nonlinear parts of inplane stress resultants are functions subject to variation and that the Kirchhoff assumption is included as part of its Euler equations. Following a general formulation method given in this paper, a simple triangular incremental modified Discrete Kirchhoff Theory (DKT) plate element with 15 stretching and bending nodal displacements is derived. The accuracy of this element is demonstrated via some typical examples of nonlinear bending and frequency response of free vibrations. Comparisons with previous results are also made. In Part 2 of this paper, this incremental element is applied to the computation of complicated frequency responses of plates with existence of internal resonance and very interesting seminumerical results are obtained.