Harmonic differential quadrature-finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation
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.
Section snippets
Differential quadrature (DQ) method
In the DQ method, a partial derivative of a function with respect to a space variable at a discrete point is approximated as a weighted linear sum of the function values at all discrete points in the region of that variable. For simplicity, we consider a one-dimensional function u(x) in the [−1, 1] domain, and N discrete points. Then the first derivatives at point i, at x=xi is given bywhere xj are the discrete points in the variable domain, u(xj) are the
Governing equations
We consider thin rectangular plates resting on Winkler–Pasternak elastic foundation of length a in x- direction, width b in the y-direction and thickness h in the z-direction. The geometry of a typical rectangular plate resting on Winkler–Pasternak elastic foundation is shown in Fig. 1. The foundation is modelled in terms of Winkler parameter kf and shear parameter Gf of the Pasternak model. More detailed information about the elastic and inelastic foundation models and the analysis of
Numerical applications
The title problem is analysed and some of HDQ–FD results are compared with results in the open literature [6], [7], [8] to show the applicability and efficiency of HDQ–FD coupled methodology. A uniform step load of infinite duration, sinusoidal loading of finite duration , and N-shaped pulse load of finite duration (Fig. 2) have been considered.
First of all, to check whether the purposed formulation and programming are correct, a clamped immovable plate without an elastic
Conclusions
The geometrically nonlinear dynamic analysis of rectangular plates on Winkler–Pasternak elastic foundation has been presented using the HDQ method. Typical results obtained by HDQ–FD coupled methodolgy are compared with the avaliable results for various foundation parameters. The following conclusions can be obtained from the study:
- 1.
It is appeared that the shear parameter G of the Pasternak foundation and stiffness parameter K of the Winkler foundation have been found to have a significant
Acknowledgements
The financial support of the Scientific Research Projects Unit of Akdeniz University is gratefully acknowledged.
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