An accurate solution method for the static and dynamic deflections of orthotropic plates with general boundary conditions Henry Khov a , Wen L. Li b , Ronald F. Gibson c, * a Ecole Nationale Supérieur des Arts et Métiers (ENSAM), Rue Saint Dominique – BP 508, 51006 Châlons-en-Champagne, France b Mechanical Engineering Department, Wayne State University, 5050 Anthony Wayne Drive, Detroit, MI 48202, USA c Mechanical Engineering Department, University of Nevada-Reno, MS-312, Reno, NV 89557, USA article info Article history: Available online 18 April 2009 Keywords: Orthotropic plates Composite plates General boundary condition Dynamic model Vibration analysis abstract Extending the previous work on isotropic beams and plates by the second author [Li WL, et al. An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports. J Sound Vib 2009;321:254–69], this paper describes an accurate analytical method for calculating the sta- tic deflections and modal characteristics of orthotropic plates with general elastic boundary supports. The displacement function is expressed as a 2-D Fourier cosine series supplemented with several terms in the form of 1-D series. The series expansions for all the relevant derivatives can be directly obtained through term-by-term differentiations of the displacement series. Thus, a classical solution can be derived by let- ting the series exactly satisfy the governing differential equation at every field point and all the boundary conditions at every boundary point, respectively. Several numerical examples are presented to demon- strate the excellent accuracy and convergence of the current solutions. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Design of plate structures often requires understanding their static deflections and dynamic characteristics. As shown by Leissa [1], Whitney [2] and others, analytical solutions for plate deflec- tions and vibrations exist only for some simple boundary condi- tions with at least two simply supported opposite edges. It is widely believed that plates with other boundary conditions are only amenable to an approximate or numerical solution, such as, the Rayleigh–Ritz method [1–4], the differential quadrature meth- od [5], the modified spline functions method [6], or the superposi- tion method of Levy-type solutions [7–10]. As pointed out by Li [11], the approximate shape functions (e.g., beam shape functions) employed by many of these methods may result in tedious calcu- lations for general elastic boundary conditions, and there is a need for a method which can be universally applied to general boundary conditions. For such a purpose, Li [12] proposed a modified Fourier series method for the vibration analysis of elastically supported beams. This method was subsequently employed to analyze the vibrations of elastically supported isotropic plates [11,13,14]. The present paper extends the application of this method to the static and dynamic analyses of thin orthotropic plates with general boundary conditions. In the present method, the displacement is expressed in the form of Fourier series expansions. This series expansion can be directly used to obtain other meaningful quanti- ties like slopes, moment and shear forces at any point on the plate (including the boundary edges) through appropriate mathematical operations. It is evident from the following discussion that the present series solution is exact, that is, the governing differential equation and the boundary conditions are all satisfied exactly on a point-wise basis. 2. Model development 2.1. Static analysis The governing equation of a symmetrically laminated thin plate under a static loading can be written as [8]: D 11 @ 4 W @x 4 þ 4D 16 @ 4 W @x 3 @y þ 2ðD 12 þ 2D 66 Þ @ 4 W @x 2 @y 2 þ 4D 26 @ 4 W @x@y 2 þ D 22 @ 4 W @y 4 ¼ qðx; yÞ; ð1Þ where x and y are the orthogonal plane coordinates, W is the plate deflection, D ij are the standard bending rigidities in the classical lamination theory [15], h is the plate thickness, and q(x, y) is a dis- tributed transverse load. For an orthotropic plate, the stiffness constants are related to the lamina engineering constants and the plate thickness as [15]: 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.04.020 * Corresponding author. E-mail address: ronaldgibson@unr.edu (R.F. Gibson). Composite Structures 90 (2009) 474–481 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct