Introduction

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 method1,2, the finite strip method3,4, the finite element method (FEM)5,6, the boundary element method7,8, the differential quadrature method9,10, the discrete singular convolution method11,12,13,14, the meshless method15,16,17, the collocation method18,19,20, the Illyushin approximation method21,22, the Rayleigh-Ritz method and Galerkin method23.

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 method24,25, series method23, integral transform method26 and symplectic elasticity method27,28,29,30,31,32.

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 superposition approach33. It was first proposed to solve the static bending problems34,35 and was successfully extended to free vibration problems recently36. We thus find a way to analytically solve both static bending and free vibration problems in a unified procedure. When the static bending solutions are obtain by the current approach, the free vibration solutions can be readily obtained without extra methodological effort.

To provide new benchmark solutions, we focus on the rectangular thin plates with four corners point-supported, which could rest on an elastic foundation. The investigations on such problems are less common than those on the plates with combinations of free, clamped and simply supported boundary conditions. Several related references are reviewed here, which provide the solutions by numerical results for validation of our approach. Rajaiah & Rao37 used the collocation method with equidistant points along the plate edge to present a series solution to the problem of laterally loaded square plates simply supported at discrete points around its periphery. Lim et al.29 developed the analytic solutions for bending of a uniformly loaded rectangular thin plate supported only at its four corners, where the symplectic elasticity method was employed and the free boundaries with corner supports were dealt with using the variational principle. Abrate38 presented a general approach based on the Rayleigh-Ritz method and the Lagrange multiplier technique to study the free vibrations of point-supported rectangular composite plates. Cheung & Zhou39 proposed a new set of admissible functions which were composed of static beam functions to give numerical solutions for the free vibrations of rectangular composite plates with point-supports. It was then further improved to obtain optimal convergence40. In an important technical report by Leissa23, many conventional solution methodologies for free vibration of plates were introduced and comprehensive numerical results for the frequencies and mode shapes were presented, including those of corner-supported plates.

In this paper, accurate analytic results for both the static bending and free vibration solutions, validated by the FEM and other solution methods (if any), are tabulated or plotted to serve as the benchmarks for validation and error analysis of various new methods developed in future.

Hamiltonian system-based governing equations for static bending and free vibration problems of a thin plate

The Hamiltonian variational principle for static bending and free vibration problems of a thin plate on an elastic foundation is in the form36,41

where the mixed energy functional ΠH is

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; My 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 Kw2/2 − qw for static bending problems and K*w2/2 for free vibration, where K is the Winkler-type foundation modulus; q is the distributed transverse load; K* = K − ρhω2, 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 My, respectively, lead to a matrix equation

for static bending problems and

for free vibration, where Z = [w, θ, T, My]T, f = [0, 0, q, 0]T, , , , , , . H and H* are both the Hamiltonian operator matrices, which satisfy HT = JHJ and H*T = JH*J, respectively. is the symplectic matrix where I2 is the 2 × 2 unit matrix. One could find from equations (3) and (4) that θ = ∂w/∂y and T = −Vy, where Vy is the equivalent shear force24. 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 [0, a] and y [0, b]. (x0, y0) is the coordinate of load position.

Figure 1
figure 1

Symplectic superposition for static bending problem of a rectangular thin foundation plate with four corners point-supported.

The plate is simply supported along the edges x = 0 and x = a. The bending moment My vanishes along the edges y = 0 and y = b but the deflections represented by and are distributed along the two edges, respectively.

An eigenvalue problem HX(x) = μX(x) in combination with dY(y)/dy = μY(y) determines a variable separation solution, Z = X(x) Y(y), for ∂Z/∂y = HZ. Herein X(x) = [w(x), θ(x), T(x), My(x)]T depends only on x and Y(y) only on y. The eigenvalues μ and corresponding eigenvectors X(x) satisfying the boundary conditions are

and

for n = 1, 2, 3, ···, where αn = /a and , j is the imaginary unit.

The solution of equation (3) is

where

Y is determined by

by substituting equation (7) into equation (3) and using HX = XM and f = XG, where M = diag(···, μn1, μn2, μn3, μn4, ···) and G = [···, gn1, gn2, gn3, gn4, ···]T is the expansion coefficients of f.

For the concentrated load P, and , where δ(y − y0) is the Dirac delta function. Thus we have, from equation (10),

where H(y − y0) is the Heaviside theta function. The constants cn1cn4 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:

where , , , , ϕ = b/a, , , , and .

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 En and Fn with Gn and Hn, respectively, we have the solution of the plate simply supported along the edges y = 0 and y = b, with the bending moment Mx vanishing along the edges x = 0 and x = a but the deflections represented by and distributed along the two edges, respectively. This solution is

where , , , , and , in which βn = /b.

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.,

and

where the quantities with an asterisk are those with K* instead of K, i.e., , , , , and , in which .

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 Em, Fm, Gn and Hn (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 Vy must be zero along the free edges y = 0 and y = b and Vx be zero along the free edges x = 0 and x = a, we obtain a set of 2M + 2N simultaneous algebraic equations to determine Em, Fm, Gn and Hn after truncating the infinite series at m = M terms and n = N terms, respectively. These equations are

and

for i = 1, 2, 3, ···, M, and

and

for i = 1, 2, 3, ···, N, where , , , , , , , , and , in which αi = /a and βi = /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 Em, Fm, Gn and Hn 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 Fm = 0 and eliminating equation (19) to solve for Em, Gn and Hn, 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 Fm = 0 and Hn = 0 and eliminating equations (19) and (21) to solve for Em and Gn, 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 literature37, 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 ABAQUS42 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 singularity24.

Table 1 Bending solutions of a square thin plate with four corners point-supported, having a concentrated load at the plate center.

The second example is a square thin foundation plate with four corners point-supported under a central concentrated load, with the nondimensional foundation modulus Ka4/D = 102. 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 Ka4/D = 102, 5 × 102, 103, 5 × 103 and 104. Again, excellent agreement with FEM is observed.

Table 2 Bending solutions of a square thin foundation plate with four corners point-supported, having a concentrated load at the plate center (Ka4/D = 102).
Figure 2
figure 2

Distribution of (a) nondimensional deflections and (b) nondimensional bending moments along the diagonal of a square thin foundation plate with four corners point-supported, with Ka4/D = 102, 5 × 102, 103, 5 × 103 and 104, respectively.

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 literature23,38,39,40 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.

Table 3 Frequency parameters, , of some rectangular thin foundation plates with four corners point-supported.
Figure 3
figure 3

First ten mode shapes of a square thin plate with four corners point-supported.

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.

Figure 4
figure 4

Convergence of the normalized bending and free vibration solutions of a square thin plate with four corners point-supported.

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 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.

Additional Information

How to cite this article: Li, R. et al. A unified analytic solution approach to static bending and free vibration problems of rectangular thin plates. Sci. Rep. 5, 17054; doi: 10.1038/srep17054 (2015).