Harmonic differential quadrature-finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation

Abstract

The geometrically nonlinear static and dynamic analysis of thin rectangular plates resting on elastic foundation has been studied. Winkler–Pasternak foundation model is considered. Dynamic analogues Von Karman equations are used. The governing nonlinear partial differential equations of the plate are discretized in space and time domains using the harmonic differential quadrature (HDQ) and finite differences (FD) methods, respectively. The analysis provides for both clamped and simply supported plates with immovable inplane boundary conditions at the edges. Various types of dynamic loading, namely a step function, a sinusoidal pulse and an N-wave, are investigated and the results are presented graphically. The accuracy of the proposed HDQ–FD coupled methodology is demonstrated by the numerical examples.

Introduction

The practical importance of vibration analysis of plates with or without on elastic foundation has been increased in structural, aerospace, civil and mechanical engineering applications. Nonlinear static and dynamic analyses of plates of various shapes have been carried out by various researchers [1], [2], [3], [4]. More detailed information can be found in a recent review paper by Sathyamorth [5]. Nath et al. [6] presented the finite differences methods for spatial discretization and Houbolt's time marching discretization to study the dynamic analysis of rectangular plates resting on elastic foundation. Dumir [7] and Dumir and Bhaskar [8] have investigated nonlinear static and dynamic analysis of rectangular plates on elastic foundation employing the orthogonal point collocation method. A few studies concerning to static and dynamic analysis of rectangular plates resting on elastic foundation have been carried out, namely by Cheung and Zienkiewicz [9] and Liew et al. [10] and Liu [11].

In structural mechanics, differential quadrature (DQ) methods are becoming popular as many important researchers demonstrated their successful applications of the method to the static, vibration and buckling analysis of various type beams, plates and shells. These applications include the work of Bert and his co-workers [12], [13], [14], [15], Liew et al. [16], [17], [18], Shu et al. [19], [20], Striz et al. [21], [22], Civalek et al. [23], [24], [25], [26], [27], and Wu and Liu [28]. Details on the development of the DQ methods and on its applications to the structural and fluid mechanics problems may be found in a well-known paper by Bert and Malik [14].

The practical importance of dynamic analysis of plates and shells on elastic foundation has been increased in structural, aerospace, biomechanics, civil and mechanical engineering applications. There are many situations such as seismic tests, nuclear explosions, earthquakes, etc. in which these structures are subjected to transient loads and large amplitudes of motion may occur. The objective of this study is to present an approximate numerical solution of the Von Karman–Donnel type governing equations for the geometrically nonlinear analysis of rectangular plates resting on Winkler–Pasternek elastic foundations under the various types of dynamic loading. To the author's knowledge, it is the first time the DQ method has been successfully applied to thin, isotropic rectangular plates resting on an elastic foundation problem for the geometrically nonlinear dynamic analysis.