A unified analytic solution approach to static bending and free vibration problems of rectangular thin plates

A unified analytic solution approach to both static bending and free vibration problems of rectangular thin plates is demonstrated in this paper, with focus on the application to corner-supported plates. The solution procedure is based on a novel symplectic superposition method, which transforms the problems into the Hamiltonian system and yields accurate enough results via step-by-step rigorous derivation. The main advantage of the developed approach is its wide applicability since no trial solutions are needed in the analysis, which is completely different from the other methods. Numerical examples for both static bending and free vibration plates are presented to validate the developed analytic solutions and to offer new numerical results. The approach is expected to serve as a benchmark analytic approach due to its effectiveness and accuracy.

Static bending and free vibration problems of thin plates are two types of fundamental issues in mechanical and civil engineering as well as in applied mathematics, with extensive applications such as floor slabs for buildings, bridge decks, and flat panels for aircrafts. In view of their importance, the problems have received considerable attention. Since the governing equations as well as boundary conditions for thin plates have been established long ago, the main focus has been on the solutions, which has brought in a variety of solution methods for various plates. Most of these methods are approximate/numerical ones such as the finite difference method 1,2 , the finite strip method 3,4 , the finite element method (FEM) 5,6 , the boundary element method 7,8 , the differential quadrature method 9,10 , the discrete singular convolution method [11][12][13][14] , the meshless method [15][16][17] , the collocation method [18][19][20] , the Illyushin approximation method 21,22 , the Rayleigh-Ritz method and Galerkin method 23 .
In comparison with the prosperity of approximate/numerical methods, analytic methods are scarce for both static bending and free vibration problems of rectangular thin plates. The reason is that the governing partial differential equation for the problems is very difficult to solve analytically except the cases of plates with two opposite edges simply supported, which have the classical Lévy-type semi-inverse solutions. For the plates without two opposite edges simply supported, there exist several representative analytic methods such as the semi-inverse superposition method 24,25 , series method 23 , integral transform method 26 , and symplectic elasticity method [27][28][29][30][31][32] .
It should be noted that many of previous analytic methods are only suitable for one type of static bending and free vibration problems. In this paper, a unified analytic solution approach to static bending and free vibration problems of rectangular thin plates is developed. The approach is implemented in the symplectic space within the framework of the Hamiltonian system. Superposition of two fundamental problems, which are solved analytically, is applied. Therefore, it is referred to as the symplectic +  Herein Ω denotes the plate domain; x and y are the Cartesian coordinates; w is the plate's transverse deflection for static bending problems and is mode shape function for free vibration; M y is the bending moment; ν is the Poisson's ratio; D is the flexural rigidity; θ and T will be interpreted after equation (4). χ equals Kw 2 /2 − qw for static bending problems and K * w 2 /2 for free vibration, where K is the Winkler-type foundation modulus; q is the distributed transverse load; K * = K − ρhω 2 , in which ρ is the plate mass density, h is the plate thickness, and ω is the circular frequency. The variations with respect to the independent w, θ, T, and M y , respectively, lead to a matrix equation for static bending problems and where I 2 is the 2 × 2 unit matrix. One could find from equations (3) and (4) that θ = ∂w/∂y, and T = − V y , where V y is the equivalent shear force 24 . Equations (3) and (4) are the Hamiltonian system-based governing matrix equations for static bending problems and free vibration of a thin plate, respectively. It is interesting to note that the two governing equations are similar in form; only equation (4) is homogeneous while equation (3) is inhomogeneous. Accordingly, as will be shown in the following, the solution approaches to these two problems are also similar, only different in solving the final simultaneous algebraic equations because one group is homogeneous while the other one is inhomogeneous. We will start with the solution of the inhomogeneous equation (3) and then reduce to the homogeneous case based on the unified analytic approach.

Symplectic analytic solutions for fundamental problems
Fundamental problem 1. To solve a rectangular thin foundation plate as shown in Fig. 1a, the foundation plate with two opposite edges simply supported and with given deflections distributed along the other two simply supported edges is regarded as the fundamental problem (Fig. 1b). Our goal is to construct the fundamental solutions for superposition. Without loss of generality, the static bending problem of such a rectangular thin plate subjected to a concentrated load P is considered. In the Cartesian coordinate system (x, y), x ∈     Y is determined by by substituting equation (7) into equation (3) and using HX = XM and f = XG, where M = diag(···, μ n1 , μ n2 , μ n3 , μ n4 , ···), and G = [···, g n1 , g n2 , g n3 , g n4 , ···] T is the expansion coefficients of f. For the concentrated load P, is the Dirac delta function. Thus we have, from equation (10),  where H(y − y 0 ) is the Heaviside theta function. The constants c n1 -c n4 are determined by substituting equations (6) and (11) into equations (8) and (9) then equation (7), and using the boundary conditions at y = 0 and y = b: In this way we obtain the analytic solution of the first fundamental problem:   Fundamental problem 2. When P = 0, the solution of the first fundamental problem reduces to that of the second fundamental problem, i.e., an unloaded rectangular thin foundation plate with the same boundary conditions as in the first fundamental problem (Fig. 1c). By interchanging x and y as well as a and b, and replacing E n and F n with G n and H n , respectively, we have the solution of the plate simply supported along the edges y = 0 and y = b, with the bending moment M x vanishing along the edges x = 0 and x = a but the deflections represented by Setting P = 0 and using K * instead of K in equations (13) and (14), the corresponding mode shape solutions for free vibration problems are readily obtained, i.e.,    where the quantities with an asterisk are those with K * instead of K, i.e., =

Symplectic superposition for analytic solutions of static bending and free vibration problems of corner-supported plates
The analytic solutions of the two fundamental problems have been obtained in section 3. The original problem's solution is given by where the constants E m , F m , G n , and H n (m, n = 1, 2, 3, ···) are to be determined by imposing the original boundary conditions along each edge. Here the subscripts "m" and "n" are used to differentiate between the constants of the two fundamental problems.

Static bending problems.
To satisfy the conditions that the equivalent shear force V y must be zero along the free edges y = 0 and y = b and V x be zero along the free edges x = 0 and x = a, we obtain a set of 2M + 2N simultaneous algebraic equations to determine E m , F m , G n , and H n after truncating the infinite series at m = M terms and n = N terms, respectively. These equations are , in which α i = iπ/a and β i = iπ/b. For simplification, we take M = N in calculation.

Free vibration problems.
Based on the solutions we have obtained for static bending problems, it is easy to solve free vibration problems by setting P = 0 and using K * instead of K throughout the solution procedure. The updated equations of equations (18)-(21) become homogeneous, and the frequency parameters are within the coefficient matrix. This is different from static bending problems where the inhomogeneous equations are directly solved with a unique solution. The determinant of the coefficient matrix is set to be zero to yield the frequency equation. Substituting one of the frequency solutions back into the homogeneous equations, a nonzero solution comprising a set of E m , F m , G n , and H n is obtained. Substituting them into equation (17) gives the corresponding mode shape function.
It should be noted that proper manipulation of the above simultaneous algebraic equations will lead to analytic solutions of more static bending and free vibration problems of point-supported plates with simply supported edges. For example, setting F m = 0 and eliminating equation (19) to solve for E m , G n , and H n , we obtain the analytic solution of the plate with two adjacent corners point-supported and their opposite edge simply supported by using equation (17). Setting F m = 0 and H n = 0, and eliminating equations (19) and (21) to solve for E m and G n , we obtain the analytic solution of the plate with a corner point-supported and its two opposite edges simply supported.

Numerical examples and Discussion
A square thin plate with four corners point-supported under a central concentrated load is solved as the first representative static bending problem. Poisson's ratio ν = 0.3 is taken throughout the study. Nondimensional deflections and bending moments at different locations are tabulated in Table 1. We first compare the analytic results with those available in the literature 37 , where the collocation method was applied to obtain the deflection distribution along the central line of the plate. Very good agreement is observed. The small errors are probably due to the approximation of the collocation method itself. Noting that there are only six data available in ref. 37 for the current problem, we perform the refined finite element analysis by the FEM software package ABAQUS 42 to further validate our solutions. The 4-node shell element S4R and 200 × 200 uniform mesh (i.e., with the element size of 1/200 of the plate width) are adopted. Excellent agreement is observed between all the current solutions and those by FEM. It should be noted in Table 1 that the bending moment at the concentrated load position does not converge due to singularity 24 .
The second example is a square thin foundation plate with four corners point-supported under a central concentrated load, with the nondimensional foundation modulus Ka 4 /D = 10 2 . The analytic results are tabulated in Table 2 by comparison with those by FEM only since we did not find any such solutions in the literature. Excellent agreement is also observed for all the results. It is convenient to use the above analytic solutions to investigate the effect of K on the plate solutions. As shown in Fig. 2, nondimensional deflections (Fig. 2a) and bending moments (Fig. 2b) along the diagonal of a square thin foundation plate are plotted for Ka 4 /D = 10 2 , 5 × 10 2 , 10 3 , 5 × 10 3 , and 10 4 . Again, excellent agreement with FEM is observed.
To illustrate the applicability of the method to free vibration, we calculate the first ten frequency parameters of corner-supported rectangular foundation plates with the aspect ratios ranging from 1 to 5, as shown in Table 3. The validity and accuracy of the current method are confirmed in view of the excellent agreement with the literature 23,[38][39][40] and, especially, with FEM. The first ten mode shapes of a square thin plate are shown in Fig. 3, which have also been validated by FEM.
An important issue concerned in solving the above problems is the convergence of the solutions. To examine it, we investigate a square corner-supported plate. Figure 4 illustrates the normalized central bending deflection and fundamental frequency versus the series terms adopted in calculation. It is seen that both the bending and free vibration solutions converge rapidly since only dozens of terms are enough to furnish satisfactory convergence. Actually rapid convergence is found for most solutions. The maximum number of series terms is taken to be 100 to achieve the convergence of all current numerical results to the last digit of five significant figures.

Conclusions
A unified analytic approach is developed in this paper to solve static bending and free vibration problems of rectangular thin plates. The approach combines the Hamiltonian system-based symplectic method and the superposition, and has the exceptional advantage that no trial solutions are needed in the analysis. Therefore, it provides a rational way to yield the analytic solutions. The procedures for the two kinds of problems are similar except in solving the equations in terms of undetermined constants. For static bending problems, the equations are inhomogeneous and a unique solution could be directly obtained, while for free vibration problems, the equations are homogeneous and the condition of having nonzero solutions is imposed to give the frequencies before solving for nonzero solutions, for which the determinant  , of some rectangular thin foundation plates with four corners point-supported. a From the finite difference method. b From the Rayleigh-Ritz method. c From the series method. * The results are divided by (b/a) 2 to yield the same form of nondimensional solutions as that of the present ones. of the coefficient matrix is set to be zero to yield the frequency equation. The resultant key quantities for static bending problems are the transverse deflection and its derivatives while those for free vibration problems are the frequencies and associated mode shapes. It is seen that the proposed approach is very effective and accurate for rectangular thin plate problems. It is expected to serve as a benchmark analytic approach to similar problems.