## Abstract

Analytic bending solutions of free rectangular thin plates resting on elastic foundations, based on the Winkler model, are obtained by a new symplectic superposition method. The proposed method offers a rational elegant approach to solve the problem analytically, which was believed to be difficult to attain. By way of a rigorous but simple derivation, the governing differential equations for rectangular thin plates on elastic foundations are transferred into Hamilton canonical equations. The symplectic geometry method is then introduced to obtain analytic solutions of the plates with all edges slidingly supported, followed by the application of superposition, which yields the resultant solutions of the plates with all edges free on elastic foundations. The proposed method is capable of solving plates on elastic foundations with any other combinations of boundary conditions. Comprehensive numerical results validate the solutions by comparison with those obtained by the finite element method.

## 1. Introduction

In a vast range of engineering applications, rectangular thin plates resting on elastic foundations are of considerable importance because they represent a class of commonly used structural elements that normally serve as the key load-bearing components in, for example, rigid pavements, bridge decks, mat and raft foundations. Their bending under external loading thus becomes the mechanical behaviour of crucial significance, and has received sufficient attention for many years. The research continues because some critical issues, seeking analytic solutions, for example, are still worthy of investigation. The analytic model not only provides insights into accurate solutions, but is also very useful for design guidelines.

Analytic bending solutions of rectangular plates on elastic foundations having all edges free are hard to solve due to the imposition of the free boundary conditions on the governing equations, which increased the mathematically impenetrable complexity of the solution procedure. Therefore, some numerical methods have to be used to handle the problem, such as the finite difference method [1], the finite strip method [2], the finite element method (FEM) [3–5], the boundary element method [6–12], the Galerkin method [9–11] and the boundary integral equation method [12–14]. In recent years, there have been some new numerical methods for bending of plates on elastic foundations, such as the iterative method [15,16], the differential quadrature method [17], the discrete singular convolution method [18], the method of fundamental solutions [19], Illyushin's method [20] and the Fourier differential quadrature method [21]. Thus, numerical methods could be used to solve most plate bending problems. However, only the analytic method can give the benchmark solution, which is the basis for verification of various numerical methods.

In the 1990s, a novel symplectic methodology for elasticity was proposed by Zhong *et al*. [22–24]. The methodology was rapidly developed in applied mechanics for its capability of going beyond the limitation of the semi-inverse method and extending the scope of analytic solutions. It has been applied in a number of research fields, including symplectic numerical methods [25], symplectic elasticity [24], fracture mechanics [26], perturbation [27], viscoelasticity [28], fluid mechanics [29], control [30], thermal effects [31], functionally graded effects [32], piezoelectricity [33], electromagnetism and waveguide [34], magneto-electro-elasticity [35], etc. For further details, the reader is referred to the review article by Lim & Xu [36], which systematically introduces the theory and applications of the symplectic approach with reference to a lot of works concerned.

It should be noted that the symplectic method has been successfully used in analysing some plate problems. Among these applications, Yao *et al.* [37] conducted systematic research on bending analysis for rectangular thin plates. Some analytic results were presented in their work such as fully simply supported plates, fully clamped plates and semi-infinite cantilever plates. The investigation of the related subjects has been performed by many researchers using the symplectic approach. Zhong & Yao [38] obtained analytic solutions of the Saint-Venant problem for layered plates under plane stress. Zou [39] derived the exact solutions of Reissner plates with two opposite edges simply supported. Yao *et al.* [40] directed the Hamiltonian system to orthotropic plate bending problems. Yao & Yang [41] presented Hamiltonian system-based Saint-Venant solutions for the problem of multi-layered composite plane anisotropic plates. Zhong & Zhang [42] attempted the semi-analytic solutions for rectangular thin plates on foundations, which satisfy the boundary conditions only at two opposite edges. Hu *et al.* [43] investigated the elastic waves and vibrations when the two lateral sides of a strip plate are free of traction. Lim *et al.* [44] presented analytic solutions of rectangular thin plates with two opposite sides simply supported. Lim *et al.* [45] gave analytic solutions for bending of a rectangular thin plate supported only at its four corners. Lim *et al.* [46] obtained exact frequency equations for Lévy-type thin plates. Zhong & Li [47] performed exact bending analysis for fully clamped rectangular thin plates subjected to arbitrary loads. Zhong *et al.* [48] extended the symplectic method to analytic bending solutions of moderately thick rectangular plates. Lim [49] carried out a study on analytic solutions to some basic problems in free vibration of rectangular thin plates with clamped or free boundary conditions. Li & Zhong [50] obtained exact bending solutions of orthotropic rectangular thin plates with two opposite sides clamped.

In this paper, a novel symplectic superposition method [51] is proposed to yield analytic solutions of rectangular thin plates with all four free edges resting on Winkler foundations. The solution consists of several steps, i.e. derivation of the Hamiltonian canonical equations, separation of variables, symplectic eigenfunction expansion and superposition. The most significant advantages of this method over the techniques previously reported are (i) it yields the benchmark analytic solutions that cannot be acquired by any numerical methods; (ii) no pre-selected or trial functions are used in the course of analysis, thus it provides a completely rational theoretical model, which prevails over the conventional semi-inverse methods by enabling one to explore more solutions; and (iii) it is computationally efficient with fast convergence and sufficient accuracy, which is validated by FEM via the numerical results in §5.

## 2. Derivation of the Hamiltonian canonical equations from the governing equations of a rectangular thin plate resting on an elastic foundation

Figure 1 illustrates the coordinate system of a free rectangular thin plate resting on an elastic Winkler foundation, with the dimensions *a* in the *x*-direction and *b* in the *y*-direction.

The basic equations of the plate are
2.1
2.2
and
2.3whereas the internal forces are
2.4a
2.4band
2.4cand
2.5a,b
2.6a,bwhere *K* denotes the foundation modulus, *W* is the transverse displacement of the plate midplane, *D* is the flexural rigidity, *q* is the distributed transverse load, *M*_{x} and *M*_{y} are the bending moments, *M*_{xy} is the torsional moment, *Q*_{x} and *Q*_{y} are the shear forces, *V* _{x} and *V* _{y} are the equivalent shear forces, respectively.

Equations (2.3) and (2.6*a*,*b*) yield
2.7By introducing
2.8and equation (2.4*b*), we have
2.9From equation (2.4c),
2.10Equations (2.4*b*,*c*), (2.5*a*), (2.6*a*) and (2.5) lead to
2.11From equations (2.2), (2.6*b*) and (2.8), we obtain
2.12Introducing *V* _{y}=−*T*, equations (2.6), (2.7), (2.9) and (2.10) are integrated in the matrix form
2.13where
and . **Z**=[*W*,*θ*,*T*,*M*_{y}]^{T} is the state vector of the plate. **f**=[0,0,*q*,0]^{T} is the vector with respect to the external load *q*. Observing **H**^{T}=**JHJ**, where
is the symplectic matrix in which **I**_{2} is 2×2 unit matrix, **H** is a Hamiltonian operator matrix [37]. Via the above derivation, we obtain the Hamiltonian canonical equations for a rectangular thin plate on a Winkler foundation in the form of equation (2.13).

## 3. Symplectic analytic bending solution of a plate with two opposite edges slidingly supported resting on an elastic foundation

In this section, the analytic bending solution of the plate slidingly supported at *x*=0 and *x*=*a* is obtained via the symplectic approach, which prepares for the superposition toward the resultant solution.

First, consider the homogeneous equation of equation (2.11):
3.1In the symplectic geometry, the method of separation of variables in **Z** needs to be applied, which gives
3.2where **X**(*x*)=[*W*(*x*),*θ*(*x*),*T*(*x*),*M*_{y}(*x*)]^{T}. Substituting equation (3.2) into equation (3.1), we obtain
3.3where *μ* is the eigenvalue and **X**(*x*) is the corresponding eigenvector.

As an eigenvalue problem, equation (3.3*b*) has its characteristic equation
3.4with the roots denoted by *λ*. Expanding the determinant of equation (3.4), we get
3.5The roots of equation (3.5) are *λ*=±*ξi* and *λ*=±*ηi*, where , and . Accordingly, the general solutions are obtained:
3.6Substituting equation (3.6) into equation (3.3*b*) yields the relation of the constants:
3.7The boundary conditions of a plate slidingly supported at *x*=0 and *x*=*a* are
3.8Substitution of equations (3.6) and (3.7) into equation (3.8) then equating the determinant of the coefficient matrix to zero, we arrive at the transcendental equation of the non-zero eigenvalues:
3.9which gives two groups of roots, i.e.
3.10and
3.11in which *α*_{n}=*nπ*/*a*.

The eigenvectors are
3.12
3.13
3.14
and
3.15for *μ*_{1}−*μ*_{4} and
3.16
3.17
3.18
and
3.19for *μ*_{n1}−*μ*_{n4}.

The eigenvectors (3.12)–(3.15) and (3.16)–(3.19), respectively, satisfy the symplectic conjugacy and orthogonality, i.e. , , and but all the other combinations of the eigenvectors are orthogonal, for example, .

Based on the above derivation (3.19), the solution of the inhomogeneous equation (2.11) is represented in the form
3.20where
3.21and
3.22Substituting equation (3.20) into equation (2.13) yields
3.23We find from equation (3.3*b*) that
3.24where **M**=*diag*(…,**P**_{0},…,**Q**_{n},…), **P**_{0}=*diag*(*μ*_{1},*μ*_{2},*μ*_{3},*μ*_{4}) and **Q**_{n}=*diag*(*μ*_{n1},*μ*_{n2},*μ*_{n3},*μ*_{n4}). The vector with respect to the external load can be expanded by the symplectic eigenvectors, i.e.
3.25where **G**=[*g*_{1},*g*_{2},*g*_{3},*g*_{4},…,*g*_{n1},*g*_{n2},*g*_{n3},*g*_{n4},…]^{T} is the column matrix of the expansion coefficients.

Substituting equations (3.24) and (3.25) into equation (3.23), we have
3.26i.e.
3.27Multiplying both sides of equation (3.25) by **X**^{T}**J***dx* and integrating from 0 to *a*, we obtain
3.28By expanding equation (3.28), the components of **G** are obtained.

For the plate with a concentrated load *P* acting at a point with variable position coordinates (*x*_{0}, *y*_{0}) on the plate surface, we obtain
3.29where *δ*(*y*−*y*_{0}) is the Dirac delta function. After substituting equation (3.29) into equation (3.27), we get
3.30where *H*(*y*−*y*_{0}) is the Heaviside theta function. Imposing the remaining boundary conditions at *y*=0 and *y*=*b*, the constants in equation (3.30) are obtained.

## 4. Analytic bending solutions of free rectangular thin plates resting on elastic foundations

As depicted in figure 2, the problem of a free plate, subjected to a concentrated load *P* at (*x*_{0}, *y*_{0}), resting on an elastic foundation with the modulus *K* is solved by superposing three sub-problems, which are shown as follows.

(1) Fully slidingly supported plate, subjected to the concentrated load *P* at (*x*_{0}, *y*_{0}), on the elastic foundation (figure 2(1)).

The boundary conditions at *y*=0 and *y*=*b* are
4.1Substituting equations (3.12)–(3.19), (3.21), (3.22) and (3.30) into equation (3.20) then using equation (4.1), the undetermined constants are obtained and hereby we get the analytic expression of the deflection, referred to as *W*_{1}(*x*,*y*):
4.2

(2) Plate with the slopes and , where *α*_{n}=*nπ*/*a*, distributed along the edges with zero equivalent shear forces, *y*=0 and *y*=*b*, respectively (figure 2(2)).

The boundary conditions at *y*=0 and *y*=*b* are
4.3Proceeding as in the first sub-problem, we obtain the analytic expression of the deflection, referred to as *W*_{2}(*x*, *y*):
4.4

(3) Plate with the slopes and , where *β*_{n}=*nπ*/*b*, distributed along the edges with zero equivalent shear forces, *x*=0 and *x*=*a*, respectively (figure 2(3)).

The boundary conditions at *x*=0 and *x*=*a* are
4.5Proceeding as in the first sub-problem but interchanging the variables in the derived expressions (i.e. replacing *x*, *y*, *a* and *b* by their inverse, respectively), we obtain the analytic expression of the deflection, referred to as *W*_{3}(*x*,*y*):
4.6The other physical quantities of these three sub-problems such as the bending moments can be readily derived via the deflections as given above. Below, the original problem is solved.

To satisfy the boundary conditions at the free edge *y*=0, the sum of the bending moments *M*_{y} of the three sub-problems must vanish at *y*=0, which yields
4.7and
4.8where *α*_{i}=*iπ*/*a*, *i*=1,2,3,….

To satisfy the boundary conditions at the free edge *y*=*b*, the sum of the bending moments *M*_{y} of the three sub-problems must vanish at *y*=*b*, which yields
4.9and
4.10To satisfy the boundary conditions at the free edge *x*=0, the sum of the bending moments *M*_{x} of the three sub-problems must vanish at *x*=0, which yields
4.11and
4.12where *β*_{i}=*iπ*/*b*, *i*=1,2,3,….

To satisfy the boundary conditions at the free edge *x*=*a*, the sum of the bending moments *M*_{x} of the three sub-problems must vanish at *x*=*a*, which yields
4.13and
4.14

The infinite systems of simultaneous equations (4.7)–(4.14) are solved to obtain *E*_{0},*F*_{0},*G*_{0},*H*_{0},*E*_{m},*F*_{m},*G*_{n} and *H*_{n} (*m*,*n*=1,2,3,…), after substituting which into equations (4.2), (4.4) and (4.6) the resultant solution to the original problem is obtained by
4.15The results are theoretically exact when an infinite number of constants are obtained in solving equations (4.7)–(4.14), while, in practice, we obtain desired accuracy by taking only a limited number of them. For the sake of convenience, the same number of four set of constants *N* are taken in the calculation, i.e. setting *m*,*n*=1,2,3,…,*N*. The solution procedures for any other load conditions are similar to that for concentrated loading.

## 5. Numerical calculation and discussion

To verify the accuracy of the developed method, two cases of concentrated loading of intensity *P* for a square thin plate with *ν*=0.3 resting on an elastic foundation with *Ka*/*D*=10^{2} are examined.

(1) The load is applied at (

*a*/2,*a*/2), the centre of the plate.(2) The load is applied at (0,0), the corner of the plate.

The results are tabulated in tables 1–4 for comparison with those obtained by the FEM software package ANSYS, where the Shell 63 element is used for a plate with the length and width 1 m, thickness 0.001 m, Young's modulus *E*=10.92 *MPa* and the foundation modulus is set to be *K*=100 *N* *m*^{−3}. There are 102 400 square elements with the same size for the plate, i.e. there are 320 elements for each edge, which is set to be free in ANSYS.

Convergence study shows that sufficient accuracy is obtained when only taking *N*=10 for the deflections while taking *N*=30 for the bending moments. It should be pointed out that the moment at (*a*/2,*a*/2), i.e. the concentrated loading position, in table 2, does not converge just as it does in a simply supported plate [52]. It is clear that the solutions by the symplectic superposition method agree quite well with those obtained by FEM, especially for the deflections, which confirms the validity and accuracy of the present method. The error by FEM generates slight differences of some bending moments from the analytic solutions.

Using the present solutions, it is convenient to investigate the effects of the normalized foundation modulus *Ka*^{4}/*D* on the deflections and bending moments of a square plate with the above loading conditions. Figure 3*a*,*b* shows the normalized maximum deflections, , while figure 3*c* and *d* the normalized bending moments at the edge centre, 100*M*_{x}|_{(a/2,0)}/*P*, versus *Ka*^{4}/*D* for the loading cases (1) and (2), respectively.

## 6. Conclusions

The symplectic superposition method is proposed, in this paper, to obtain analytic bending solutions of free rectangular thin plates resting on Winkler elastic foundations that were difficult to solve before. The solution approach reveals several advantages with respect to bending problems of rectangular plates on elastic foundations. First, the symplectic superposition method provides a totally rational way to obtain analytic solutions, which starts from the basic elasticity equations of the problem and proceeds without any pre-selected solutions. The second advantage is that the method gives us a systematic solution procedure, which can be applied to plates with all possible combinations of clamped, free and simply supported boundary conditions. In addition, the method is expected to be extended to vibration and buckling problems, which will be developed in the follow-up work.

## Acknowledgements

This work was supported by China Postdoctoral Science Foundation (2012M520619) and Fundamental Research Funds for the Central Universities of China (DUT12RC(3)44).

- Received November 14, 2012.
- Accepted February 1, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.