Bending Analysis of Stepped Rectangular Plates Resting on an Elastic Half-Space Foundation

Abstract

In this paper, the bending behavior of rectangular plates with stepped thickness resting on an elastic half-space foundation is investigated through an analytic method. Combined with the bending theory of the rectangular thin and moderately thick plate, the stepped rectangular plate is divided into upper and lower plates, and the Fourier series is used to obtain the analytical solution of the deflection of the plate and the interaction force between the plate and foundation. The influence of the elastic modulus of the plate, plate theory, and the dimension of the plate on the deflection of the stepped rectangular plate is also discussed. The results show that the analytical solution is basically the same as the existing research results, and it is also verified by the analysis results of the models established by ABAQUS software. The deflection at the center of the stepped rectangular plate increases with the increase of the elastic modulus of the upper plate and the decrease of the side length of the upper plate, while the plate theory has little effect on the deflection of the plate. This method not only overcomes some of the disadvantages of numerical methods but also eliminates the assumptions of the Winkler foundation model and the two-parameter foundation model, thus obtaining a more reasonable and accurate bending performance of the stepped rectangular plate resting on the elastic half-space foundation.

1. Introduction

Plate-foundation systems have become a research hotspot in civil engineering. Much practical engineering, such as foundation plates in civil engineering [1], piezoelectric laminated plates in electronic engineering [2], and pavement plates supporting the traffic load [3], can be simplified into a mechanical model for analysis [4]. In order to provide necessary parameters for the model of these structures, two methods are usually used. One is to provide accurate models for various types of plates and the interaction between the plate and foundation. The other one is to develop a numerical analysis method and computer codes for the solution of practical engineering problems.

In the past few centuries, much work has been done to develop simplified models of foundations to promote applications in engineering design. The Winkler foundation model, in which the foundation is assumed as a single layer of vertical springs, was first proposed in the 1860s [5]. However, the accuracy of this model cannot be guaranteed in the analysis of nonlinear conditions of the foundation. Thus, the linear spring is replaced by a nonlinear type of model to overcome this shortage [6]; this model can be used in static analysis [7] and dynamic analysis [8], but it still cannot accurately analyze the frictional foundation due to the reason that the analysis model of the Winkler foundation eliminates the interfacial shear stress. Thus, researchers have proposed two-parameter and multi-parameter models to solve this problem. Compared to the Winkler model, the two-parameter Pasternak foundation model has shear and vertical stiffness with nonlinear properties, which is helpful to improve the efficiency of the model, without resulting in significant loss of the accuracy of calculation results [9,10]. In addition, the three-parameter model [11] and the frictional model [7] were also proposed by researchers. Although the refinement of the parameters could improve the applicability of the foundation model, the increase of the number of the parameters will enhance the complication of the calculation process.

The plate theory also plays an important role in the simulation of numerical models and the structure design of the plate-foundation system. The Kirchhoff plate theory [12] and the Mindlin plate theory [13] are the two types of classical plate theories widely used in the simulation of plate foundations. The in-plane deformation of the plate is assumed to be linearly changed in the classical plate theories, and these theories could only be used in the analysis of thin and moderately thick plates. Many studies have been carried out using the classical plate theories. Özdemir [14,15] investigated the static performance, free vibration properties, and transient response of thick plates resting on the elastic Winkler foundation with the finite element method. Farida et al. [16] presented the efficient analysis of plates resting on an elastic half-space foundation using the boundary element method. Ragba et al. [17] used convolution and indirect meshless techniques to obtain the vibration performance of irregular composite plates on the linear and parabolic Winkler foundations. Ferreira et al. [18] investigated the free vibration problem of the plate resting on the Winkler foundation using the wavelet collocation method. Investigations of the Kirchhoff plate on foundations are also reported in the literature. Yue et al. [19] analyzed the influence of soil heterogeneity on the bending of a circular thin plate using two modified Vlasov foundation models. Mohammadimehr et al. [20] obtained the buckling and free vibration response of functionally graded materials resting on a Pasternak foundation. To promote the development of the plate theory, higher-order shear deformation theory and quasi-3D theory have attracted the attention of researchers in recent years, and these theories take into account the nonlinear deformation of the plates. Singh et al. [21] used the stress-function Galerkin method to investigate the dynamic response of a sandwich functionally graded plate resting on a Pasternak elastic foundation. Huang et al. [22] studied the nonlinear dynamic performance of functionally graded material plates using an improved perturbation technique. Rachid et al. [23] and Vu et al. [24] used different quasi-3D theories to analyze the static response of the plate. Kumar et al. [25] investigated the vibration performance of stepped FGM plates using the dynamic stiffness method.

Based on the foundation models and the different plate theories, analytical models can be used to obtain the accurate solution for the plate-foundation systems. Compared with the literature using numerical methods, analytical methods are not commonly used due to the difficulties in the partial differential equations and various boundary conditions [26]. Yan [27] introduced the usage of the Fourier series in the analysis of the bending, stability, and vibration of the plate. The deflection of the plate was expressed as a double Fourier series, and the various boundary conditions were used to obtain the analytical solution of the plate resting on the Winkler foundation. Wang [28] used the method mentioned in [27] to investigate the interaction and deflection of the thin plate resting on the elastic half-space foundation. The foundation models eliminated the assumptions of the Winkler foundation to overcome the shortages of the Winkler model to obtain a more accurate and reasonable analytical solution. Li et al. [29] presented the analytic bending solution of a thin plate resting on the Winkler foundation. The governing differential equations of the plate were transferred into Hamilton canonical equations, and the analytic solution could be obtained with all edges slidingly supported. Bai et al. [30] studied the bending problem of the free orthotropic rectangular thin plate (RTP) on a two-parameter elastic foundation under a concentrated load by using the symplectic superposition method. Tenenbaum et al. [31] gave the analytical solutions for the buckling loads of thin rectangular plates with internal supports and different boundary conditions. The solution had a series form, and the coefficients were solved to match the edge conditions. With the development of economy and technology, multilayer structures under static loading or dynamic loading have been delt with in recent years. Multilayer structures, whose material properties are discontinuous at each interface of the plate, are different from traditional laminated composites. Functionally graded material (FGM) is one of the multilayer structures that is attracting tremendous research interest. The analytical solution for the bending performance of the FGM plate was first introduced in the 1990s [32]. Afterwards, FGM plates were widely used in engineering structures, and the buckling and dynamic response [33,34,35], post buckling behavior [36,37], and elastoplastic mechanical performances [38,39,40] of the FGM plates were introduced.

As far as the authors know, there are few studies about the kinematical equations suitable for a stepped rectangular plate. It also seems to be a fact that there are few research papers on the static bending analysis of stepped rectangular plates resting on the elastic half-space foundation. To bridge this gap, this paper used traditional plate theory that is common in the analysis of single plates resting on an elastic foundation. In order to promote the application of traditional plate theories, the stepped rectangular plate is considered to be composed of two plates with different dimensions and properties (upper and lower plates), and taking into account the thickness of the upper and lower plates, the analytical method is divided into three cases: (1) the upper and lower plates are both thin plates; (2) one plate is a thin plate, while the other one is a moderately thick plate; (3) the upper and lower plates are both moderately thick plates. A Fourier series is used to establish the basic equations and the coordination equation of the plate-foundation system, and the boundary conditions are also used to obtain the analytical solution. In addition, the influence of plate theory, the elastic modulus of the plate-foundation system and dimensions of the plate on the bending performance of the stepped rectangular plate are analyzed.

2. Governing Equations and Results

A stepped rectangular plate, whose upper and lower plate dimensions are a₁ × b₁ (length × width) and a₂ × b₂ (length × width), respectively, is referred to the Cartesian systems of coordinates x₁y₁ and x₂y₂ associated with the external surfaces of the upper and lower plates (Figure 1). The contact surface between the upper and lower plates is assumed to exclude their mutual slipping. A uniformly distributed load q(x₁,y₁) is applied on the external surface of upper plate.

2.1. Both Upper and Lower Parts of the Plate Are Thin Plates

2.1.1. Basic Equations and Boundary Conditions

Taking the upper plate as an example, the bending equation of the upper plate can be expressed as [27]:

$$D{\nabla }^2{\nabla }^{2}w(x,y)=q(x,y)-F(x,y)$$

where w(x,y) is the deflection of the plate, D is the bending stiffness, q(x,y) and F(x,y) are the loadings applied on upper and lower surface of the plate, respectively, and ${\nabla }^2=\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial x^2}$ is the Laplacian operator.

Therefore, in the analysis process of the stepped rectangular plate resting on the elastic half-space foundation (Figure 1), the force of the upper and lower plates is different. The upper and lower loadings applied on the upper plate are q(x₁,y₁) and F(x₁,y₁), respectively, while for the lower plate, these two values are F(x₁,y₁) and Q(x₂,y₂), respectively, in which F(x₁,y₁) and Q(x₂,y₂) are the interaction force between the upper and lower plates and the subgrade reaction, respectively. After that, the bending equation of the upper and lower plates can be written as below:

$$D_{x_1}\frac{{\partial }^4{w}_1}{\partial x_1{}^4}+2H_1\frac{{\partial }^4{w}_1}{\partial x_1{}^2\partial y_1{}^2}+D_{y_1}\frac{{\partial }^4{w}_1}{\partial y_1{}^4}+F(x_1,y_1)=q(x_1,y_1)$$

(1a)

$$D_{x_2}\frac{{\partial }^4{w}_2}{\partial x_2{}^4}+2H_2\frac{{\partial }^4{w}_2}{\partial x_2{}^2\partial y_2{}^2}+D_{y_2}\frac{{\partial }^4{w}_2}{\partial y_2{}^4}+Q(x_2,y_2)=F(x_1,y_1)$$

(1b)

where $D_{x_i}=E_{x_i}{h}_i^3/12(1-v_{x_i}^2)$ and $D_{y_i}=E_{y_i}{h}_i^3/12(1-v_{y_i}^2)$ are the bending stiffness of x_i-axis direction and y_i-axis direction, respectively. w_i is deflection of the plate, h_i is the thickness of plate, $v_{x_i}$ and $v_{y_i}$ are the Poisson ratio of x_i-axis direction and y_i-axis direction respectively, $H_i=D_{x_i}{v}_{y_i}+2D_{x_i{y}_i}$ is equivalent stiffness, and $D_{x_i{y}_i}$ is torsional stiffness. The upper and lower plates are numbered i = 1 and i = 2, respectively.

The internal force of the upper and lower plates could be written in terms of deflection functions:

$$M_{x_i}=-D_{x_i}(\frac{{\partial }^2{w}_i}{\partial x_i^2}+v_{y_i}\frac{{\partial }^2{w}_i}{\partial y_i^2})$$

$$M_{y_i}=-D_{y_i}(\frac{{\partial }^2{w}_i}{\partial y_i^2}+v_{x_i}\frac{{\partial }^2{w}_i}{\partial x_i^2})$$

$$M_{x_i{y}_i}=-2D_{x_i{y}_i}\frac{{\partial }^2{w}_i}{\partial x_i\partial y_i}$$

$$Q_{x_i}=-D_{x_i}\frac{{\partial }^3{w}_i}{\partial x_i^3}-(H_i+2D_{x_i{y}_i})\frac{{\partial }^3{w}_i}{\partial x_i\partial y_i^2}$$

$$Q_{y_i}=-D_{y_i}\frac{{\partial }^3{w}_i}{\partial y_i^3}-(H_i+2D_{x_i{y}_i})\frac{{\partial }^3{w}_i}{\partial y_i\partial x_i^2}$$

$$R_{x_i{y}_i}=-2D_{x_i{y}_i}(1-v)\frac{{\partial }^2{w}_i}{\partial x_i\partial y_i}$$

in which $M_{x_i}$ and $M_{y_i}$ are the bending moment of the x_i-axis direction and y_i-axis direction, respectively, $M_{x_i{y}_i}$ is the twisting moment, $Q_{x_i}$ and $Q_{y_i}$ are the shear force of the x_i-axis direction and y_i-axis direction, respectively, and R is the counterforce at the corner of the upper and lower plates.

The boundary restrictions are given as:

At x = 0, x = a

$$\frac{{\partial }^2{w}_i}{\partial x_i^2}+v_{y_i}\frac{{\partial }^2{w}_i}{\partial y_i^2}=0 (M_{x_i}=0)$$

(2a)

$$D_{x_i}\frac{{\partial }^3{w}_i}{\partial x_i^3}+(H_i+2D_{x_i{y}_i})\frac{{\partial }^3{w}_i}{\partial x_i\partial y_i^2}=0 (Q_{x_i}=0)$$

(2b)

At y = 0, y = a

$$\frac{{\partial }^2{w}_i}{\partial y_i^2}+v_{x_i}\frac{{\partial }^2{w}_i}{\partial x_i^2}=0 (M_{y_i}=0)$$

(2c)

$$D_{y_i}\frac{{\partial }^3{w}_i}{\partial y_i^3}+(H_i+2D_{x_i{y}_i})\frac{{\partial }^3{w}_i}{\partial y_i\partial x_i^2}=0 (Q_{x_i}=0)$$

(2d)

At the corner of the plate

$$\frac{{\partial }^2{w}_i}{\partial x_i\partial y_i}=0 (R_{x_i{y}_i}=0)$$

(2e)

2.1.2. Coordination Equation and Analytical Solution

The deflections of the upper and lower plates can be expressed as a double cosine series with supplementary terms [27] (Appendix A):

$$\begin{array}{l}{w}_i={\displaystyle \sum _{m_i=0}^{\infty }{\displaystyle \sum _{n_i=0}^{\infty }{w}_{m_i{n}_i}\cos \frac{m_i\pi x_i}{a_i}\cos \frac{n_i\pi y_i}{b_i}}+}{\displaystyle \sum _{m_i=0}^{\infty }\left\{\left[{\mu }_{2m_i{y}_i}\frac{m_i{}^2{\pi }^2{b}_i{}^2}{a_i{}^2}\cdot \frac{4b_i{y}_i{}^3-4b_i{}^2{y}_i{}^2-y_i{}^4}{24b_i{}^4}\right.\right.}\\ \left.+\frac{2b_i{y}_i-y_i{}^2}{2b_i{}^2}\right]C_{m_i}+\left[{\mu }_{2m_i{y}_i}\frac{m_i{}^2{\pi }^2{b}_i{}^2}{a_i{}^2}\cdot \frac{y_i{}^4-2b_i{}^2{y}_i{}^2}{24b_i{}^4}\right.\left.\left.+\frac{y_i{}^2}{2b_i{}^2}\right]D_{m_i}\right\}\cos \frac{m_i\pi x_i}{a_i}\\ +{\displaystyle \sum _{n_i=0}^{\infty }\left\{\left[{\mu }_{2m_i{x}_i}\frac{n_i{}^2{\pi }^2{a}_i{}^2}{b_i{}^2}\right.\right.}\left.\cdot \frac{4a_i{x}_i{}^3-4a_i{}^2{x}_i{}^2-x_i{}^4}{24a_i{}^4}+\frac{2a_i{x}_i-x_i{}^2}{2a_i{}^2}\right]G_{n_i}+\left[{\mu }_{2n_i{x}_i}\frac{n_i{}^2{\pi }^2{a}_i{}^2}{b_i{}^2}\right.\\ \left.\left.\cdot \frac{x_i{}^4-2a_i{}^2{x}_i{}^2}{24a_i{}^4}+\frac{x_i{}^2}{2a_i{}^2}\right]H_{n_i}\right\}\cos \frac{n_i\pi y_i}{b_i}\\ (m_i=0,1,2,\cdot \cdot \cdot ; n_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

(3)

where ${\mu }_{2n_i{x}_i}=\frac{H_i+2D_{x_i{y}_i}}{D_{x_i}}$, ${\mu }_{2m_i{y}_i}=\frac{H_i+2D_{x_i{y}_i}}{D_{y_i}}$, $w_{m_i{n}_i}$, $C_{m_i}$, $D_{m_i}$, $G_{n_i}$, $H_{n_i}$ are undetermined parameters.

Equation (4) has four steps’ derivation for a rectangular plate with four free edges, which could satisfy the boundary conditions, such as the shear force at the boundary (Equation (2b,d)) and the corner condition (Equation (2e)). If the plate is made of isotropic material, the Equation (4) can degenerate into an expression of an isotropic rectangular plate.

Based on Equation (1a,b), it could be found that F(x₁,y₁) is related to the control differential equations of the upper and lower plates, so F(x₁,y₁) can be expanded into a double cosine series represented by x₁, y₁ and x₂, y₂.

$$F(x_1,y_1)={\displaystyle \sum _{m_1=0}^{\infty }{\displaystyle \sum _{n_1=0}^{\infty }{\lambda }_{m_1{n}_1}{F}_{m_1{n}_1}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1^{}\pi y_1}{b_1}}}$$

(4)

$$\begin{gathered} F(x_2,y_2)={\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}{F}_{m_2{n}_2}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}}} \\ (m_i=0,1,2,\cdot \cdot \cdot ; n_i=0,1,2,\cdot \cdot \cdot ) \end{gathered}$$

(5)

where

$${\lambda }_{m_i{n}_i}=\left\{\begin{array}{l}1/4, m_i=n_i=0\\ 1/2, m_i=0, n_i>0 or m_i>0, n_i=0\\ 1/4, m_i>0, n_i>0 \end{array}\right.$$

$$F_{m_1{n}_1}=\frac{4}{a_1{b}_1}{\displaystyle {\int }_0^{b_1}{\displaystyle {\int }_0^{a_1}{F}_1(x_1,y_1)\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}}}d{x}_{1}d{y}_1$$

$$\begin{array}{l}{F}_{m_2{n}_2}=\frac{4}{a_2{b}_2}{\displaystyle {\int }_0^{b_2}{\displaystyle {\int }_0^{a_2}F(x_1,y_1)\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}}}d{x}_{2}d{y}_2\\ =\frac{4}{a_2{b}_2}{\displaystyle {\int }_0^{b_2}{\displaystyle {\int }_0^{a_2}{\displaystyle \sum _{m_1=0}^{\infty }{\displaystyle \sum _{n_1=0}^{\infty }{\lambda }_{m_1{n}_1}{F}_{m_1{n}_1}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}}}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}}}d{x}_{2}d{y}_2\\ =\frac{4}{a_2{b}_2}{\displaystyle \sum _{m_1=0}^{\infty }{\displaystyle \sum _{n_1=0}^{\infty }{\displaystyle {\int }_0^{b_2}{\displaystyle {\int }_0^{a_2}{\lambda }_{m_1{n}_1}{F}_{m_1{n}_1}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}}d}}}{x}_{2}d{y}_2\\ =\frac{4}{a_2{b}_2}{\displaystyle \sum _{m_1=0}^{\infty }{\displaystyle \sum _{n_1=0}^{\infty }{\lambda }_{m_1{n}_1}{F}_{m_1{n}_1}{\displaystyle {\int }_0^{b_2}{\displaystyle {\int }_0^{a_2}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}}d}}}{x}_{2}d{y}_2\\ (m_i=0,1,2,\cdot \cdot \cdot ; n_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

x₀, y₀ represent the relationship between the x₁y₁ and x₂y₂ coordinate systems, and substituting x₂ = x₁ + x₀, y₂ = y₁ + y₀ into the integral part of expression of $F_{m_2{n}_2}$:

$$\begin{array}{l}{\displaystyle {\int }_0^{b_2}{\displaystyle {\int }_0^{a_2}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}d{x}_{2}d{y}_2}}\\ ={\displaystyle {\int }_{x_0}^{x_0+a_1}{\displaystyle {\int }_{y_0}^{y_0+b_1}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}d{x}_{2}d{y}_2}}\\ ={\displaystyle {\int }_{x_0}^{x_0+a_1}{\displaystyle {\int }_{y_0}^{y_0+b_1}\cos \frac{m_1\pi (x_2-x_0)}{a_1}\cos \frac{n_1\pi (y_2-y_0)}{b_1}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}d{x}_{2}d{y}_2}}\\ =\left\{\frac{1}{2}\cos \frac{m_1\pi x_0}{a_1}\frac{a_2{a}_1}{\pi (m_1{a}_2+m_2{a}_1)}\right.\left[\sin \frac{\pi (m_1{a}_2+m_2{a}_1)(x_0+a_1)}{a_2{a}_1}\right.-\left.\sin \frac{\pi (m_1{a}_2+m_1{a}_1)x_0}{a_2{a}_1}\right]\\ +\frac{1}{2}\cos \frac{m_1\pi x_0}{a_1}\frac{a_2{a}_1}{\pi (m_1{a}_2-m_2{a}_1)}\left[\sin \frac{\pi (m_1{a}_2-m_2{a}_1)(x_0+a_1)}{a_2{a}_1}\right.-\sin \left.\frac{\pi (m_1{a}_2-m_2{a}_1)x_0}{a_2{a}_1}\right]\\ +\frac{1}{2}\sin \frac{m_1\pi x_0}{a_1}\frac{a_2{a}_1}{\pi (m_1{a}_2+m_2{a}_1)}\left[\cos \frac{\pi (m_1{a}_2+m_2{a}_1)x_0}{a_2{a}_1}\right.-\left.\cos \frac{\pi (m_1{a}_2+m_2{a}_1)(x_0+a_1)}{a_2{a}_1}\right]\\ +\frac{1}{2}\sin \frac{m_1\pi x_0}{a_1}\frac{a_2{a}_1}{\pi (m_1{a}_2-m_2{a}_1)}\left[\cos \frac{\pi (m_1{a}_2-m_2{a}_1)x_0}{a_2{a}_1}\right.\left.-\left.\cos \frac{\pi (m_1{a}_2-m_2{a}_1)(x_0+a_1)}{a_2{a}_1}\right]\right\}\\ \cdot \left\{\frac{1}{2}\cos \frac{n_1\pi y_0}{b_1}\frac{b_2{b}_1}{\pi (n_1{b}_2+n_2{b}_1)}\right.\left[\sin \frac{\pi (n_1{b}_2+n_2{b}_1)(y_0+b_1)}{b_2{b}_1}\right.-\left.\sin \frac{\pi (n_1{b}_2+n_1{b}_1)y_0}{b_2{b}_1}\right]\\ +\frac{1}{2}\cos \frac{n_1\pi y_0}{b_1}\frac{b_2{b}_1}{\pi (n_1{b}_2-n_2{b}_1)}\left[\sin \frac{\pi (n_1{b}_2-n_2{b}_1)(y_0+b_1)}{b_2{b}_1}\right.\left.-\sin \frac{\pi (n_1{b}_2-n_2{b}_1)y_0}{b_2{b}_1}\right]\\ +\frac{1}{2}\sin \frac{n_1\pi y_0}{b_1}\frac{b_2{b}_1}{\pi (n_1{b}_2+n_2{b}_1)}\left[\cos \frac{\pi (n_1{b}_2+n_2{b}_1)y_0}{b_2{b}_1}\right.-\left.\cos \frac{\pi (n_1{b}_2+n_2{b}_1)(y_0+b_1)}{b_2{b}_1}\right]\\ +\frac{1}{2}\sin \frac{n_1\pi y_0}{b_1}\frac{b_2{b}_1}{\pi (n_1{b}_2-n_2{b}_1)}\left[\cos \frac{\pi (n_1{b}_2-n_2{b}_1)y_0}{b_2{b}_1}\right.\left.-\left.\cos \frac{\pi (n_1{b}_2-n_2{b}_1)(y_0+b_1)}{b_2{b}_1}\right]\right\}\\ =B\\ (m_i=0,1,2,\cdot \cdot \cdot ; n_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

Hence, the expression for $F_{m_2{n}_2}$ is rewritten as

$$F_{m_2{n}_2}=\frac{4}{a_2{b}_2}{\displaystyle \sum _{m_1=0}^{\infty }{\displaystyle \sum _{n_1=0}^{\infty }{\lambda }_{m_1{n}_1}{F}_{m_1{n}_1}B}}$$

q(x₁,y₁) is expanded into a double cosine series represented by x₁ and y₁.

$$\begin{gathered} q(x_1,y_1)={\displaystyle \sum _{m_1=0}^{\infty }{\displaystyle \sum _{n_1=0}^{\infty }{\lambda }_{m_1{n}_1}}{q}_{m_1{n}_1}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}} \\ (m_i=0,1,2,\cdot \cdot \cdot ; n_i=0,1,2,\cdot \cdot \cdot ) \end{gathered}$$

(6)

where

$$q_{m_1{n}_1}=\frac{4}{a_1{b}_1}{\displaystyle {\int }_0^{b_1}{\displaystyle {\int }_0^{a_1}q(x_1,y_1)}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}d{x}_{1}d{y}_1}$$

The subgrade reaction can be expressed in terms of the double cosine series as

$$Q(x_2,y_2)={\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}{Q}_{m_2{n}_2}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}}}$$

where

$$Q_{m_2{n}_2}=\frac{4}{a_2{b}_2}{\displaystyle {\int }_0^{b_2}{\displaystyle {\int }_0^{a_2}Q(x_2,y_2)\cos \frac{m_2\pi x_2}{a_2}}}\cos \frac{n_2\pi y_2}{b_2}d{x}_{2}d{y}_2$$

Substituting Equations (3)–(6) into Equation (1a,b), and then expanding the polynomial of the supplementary terms in the formulas to the cosine series. Comparing the coefficients of the corresponding items on both sides of the Equation (1a,b), the expressions can be obtained as:

$$\begin{array}{l}\left[D_{x_1}\right.{\alpha }_{m_1}{}^4+2H_1{\alpha }^4\left.{{}_{m_1}}^2{\beta }_{n_1}{}^2+D_{y_1}{\beta }_{n_1}{}^4\right]w_{m_1{n}_1}+\\ \left\{2D_{x_1}{\alpha }_{m_1}{}^4\left[\frac{H_1+2D_{x_1{y}_1}}{D_{y_1}}\right.\cdot \frac{m_1{}^2{\pi }^2{b}_1{}^2}{a_1{}^2}\left(\frac{h_{n_1}}{{\beta }_{n_1}{}^4{b}_1{}^4}-\frac{\overline{h_{n_1}}}{90}\right)\right.-\frac{h_{n_1}}{{\beta }_{n_1}{}^2{b}_1{}^2}+\left.\frac{\overline{h_{n_1}}}{6}\right]+\\ 2H_1{\alpha }_{m_1}{}^4\cdot \frac{2(H_1+2D_{x_1{y}_1})}{D_{y_1}{\beta }_{n_1}{}^2{b}_1{}^2}{h}_{n_1}+\left.\frac{{\alpha }_{m_1}{}^2}{b_1{}^2}\left[H_1-2D_{x_1{y}_1}\right]\overline{h_{n_1}}\right\}{C}_{m_1}+\\ \left\{2D_{x_1}{\alpha }_{m_1}{}^4\left[\frac{H_1+2D_{x_1{y}_1}}{D_{y_1}}\right.\cdot \frac{m_1{}^2{\pi }^2{b}_1{}^2}{a_1{}^2}\left(\frac{h_{n_1}}{{\beta }_{n_1}{}^4{b}_1{}^4}+\frac{7\overline{h_{n_1}}}{720}\right)\right.-\frac{h_{n_1}}{{\beta }_{n_1}{}^2{b}_1{}^2}-\left.\frac{\overline{h_{n_1}}}{12}\right]+\\ 2H_1{\alpha }_{m_1}{}^4\cdot \frac{2(H_1+2D_{x_1{y}_1})}{D_{y_1}{\beta }_{n_1}{}^2{b}_1{}^2}{h}_{n_1}+\left.\frac{{\alpha }_{m_1}{}^2}{b_1{}^2}\left[H_1-2D_{x_1{y}_1}\right]\overline{h_{n_1}}\right\}{(-1)}^{n_1+1}{D}_{m_1}+\\ \left\{2D_{y_1}{\beta }_{n_1}{}^4\left[\frac{H_1+2D_{x_1{y}_1}}{D_{x_1}}\right.\cdot \frac{n_1{}^2{\pi }^2{a}_1{}^2}{b_1{}^2}\left(\frac{h_{m_1}}{{\alpha }_{m_1}{}^4{a}_1{}^4}-\frac{\overline{h_{m_1}}}{90}\right)\right.-\frac{h_{m_1}}{{\alpha }_{m_1}{}^2{a}_1{}^2}+\left.\frac{\overline{h_{m_1}}}{6}\right]+\\ 2H_1{\beta }_{n_1}{}^4\cdot \frac{2(H_1+2D_{x_1{y}_1})}{D_{x_1}{\alpha }_{m_1}{}^2{a}_1{}^2}{h}_{m_1}+\left.\frac{{\beta }_{n_1}{}^2}{a_1{}^2}\left[H_1-2D_{x_1{y}_1}\right]\overline{h_{m_1}}\right\}{G}_{n_1}+\\ \left\{2D_{y_1}{\beta }_{n_1}{}^4\left[\frac{H_1+2D_{x_1{y}_1}}{D_{x_1}}\right.\cdot \frac{n_1{}^2{\pi }^2{a}_1{}^2}{b_1{}^2}\left(\frac{h_{m_1}}{{\alpha }_{m_1}{}^4{a}_1{}^4}+\frac{7\overline{h_{m_1}}}{720}\right)\right.-\frac{h_{m_1}}{{\alpha }_{m_1}{}^2{a}_1{}^2}-\left.\frac{\overline{h_{m_1}}}{12}\right]+\\ 2H_1{\beta }_{n_1}{}^4\cdot \frac{2(H_1+2D_{x_1{y}_1})}{D_{x_1}{\alpha }_{m_1}{}^2{a}_1{}^2}{h}_{m_1}+\left.\frac{{\beta }_{n_1}{}^2}{a_1{}^2}\left[H_1-2D_{x_1{y}_1}\right]\overline{h_{m_1}}\right\}{(-1)}^{m_1+1}{H}_{n_1}\\ ={\lambda }_{m_1{n}_1}(q_{m_1{n}_1}-F_{m_1{n}_1})\\ (m_1=0,1,2,\cdot \cdot \cdot ; n_1=0,1,2,\cdot \cdot \cdot )\end{array}$$

(7)

$$\begin{array}{l}\left[D_{x_2}\right.{\alpha }_{m_2}{}^4+2H_2{\alpha }^4\left.{{}_{m_2}}^2{\beta }_{n_2}{}^2+D_{y_2}{\beta }_{n_2}{}^4\right]w_{m_2{n}_2}+\\ \left\{2D_{x_2}{\alpha }_{m_2}{}^4\left[\frac{H_2+2D_{x_2{y}_2}}{D_{y_2}}\right.\cdot \frac{m_2{}^2{\pi }^2{b}_2{}^2}{a_2{}^2}\left(\frac{h_{n_2}}{{\beta }_{n_2}{}^4{b}_2{}^4}-\frac{\overline{h_{n_2}}}{90}\right)\right.-\frac{h_{n_2}}{{\beta }_{n_2}{}^2{b}_2{}^2}+\left.\frac{\overline{h_{n_2}}}{6}\right]+\\ 2H_2{\alpha }_{m_2}{}^4\cdot \frac{2(H_2+2D_{x_2{y}_2})}{D_{y_2}{\beta }_{n_2}{}^2{b}_2{}^2}{h}_{n_2}+\left.\frac{{\alpha }_{m_2}{}^2}{b_2{}^2}\left[H_2-2D_{x_2{y}_2}\right]\overline{h_{n_2}}\right\}{C}_{m_2}+\\ \left\{2D_{x_2}{\alpha }_{m_2}{}^4\left[\frac{H_2+2D_{x_2{y}_2}}{D_{y_2}}\right.\cdot \frac{m_2{}^2{\pi }^2{b}_2{}^2}{a_2{}^2}\left(\frac{h_{n_2}}{{\beta }_{n_2}{}^4{b}_2{}^4}+\frac{7\overline{h_{n_2}}}{720}\right)\right.-\frac{h_{n_2}}{{\beta }_{n_2}{}^2{b}_2{}^2}-\left.\frac{\overline{h_{n_2}}}{12}\right]+\\ 2H_2{\alpha }_{m_2}{}^4\cdot \frac{2(H_2+2D_{x_2{y}_2})}{D_{y_2}{\beta }_{n_2}{}^2{b}_2{}^2}{h}_{n_2}+\left.\frac{{\alpha }_{m_2}{}^2}{b_2{}^2}\left[H_2-2D_{x_2{y}_2}\right]\overline{h_{n_2}}\right\}{(-1)}^{n_2+1}{D}_{m_2}+\\ \left\{2D_{y_2}{\beta }_{n_2}{}^4\left[\frac{H_2+2D_{x_2{y}_2}}{D_{x_2}}\right.\cdot \frac{n_2{}^2{\pi }^2{a}_2{}^2}{b_2{}^2}\left(\frac{h_{m_2}}{{\alpha }_{m_2}{}^4{a}_2{}^4}-\frac{\overline{h_{m_2}}}{90}\right)\right.-\frac{h_{m_2}}{{\alpha }_{m_2}{}^2{a}_2{}^2}+\left.\frac{\overline{h_{m_2}}}{6}\right]+\\ 2H_2{\beta }_{n_2}{}^4\cdot \frac{2(H_2+2D_{x_2{y}_2})}{D_{x_2}{\alpha }_{m_2}{}^2{a}_2{}^2}{h}_{m_2}+\left.\frac{{\beta }_{n_2}{}^2}{a_2{}^2}\left[H_2-2D_{x_2{y}_2}\right]\overline{h_{m_2}}\right\}{G}_{n_2}+\\ \left\{2D_{y_2}{\beta }_{n_2}{}^4\left[\frac{H_2+2D_{x_2{y}_2}}{D_{x_2}}\right.\cdot \frac{n_2{}^2{\pi }^2{a}_2{}^2}{b_2{}^2}\left(\frac{h_{m_2}}{{\alpha }_{m_2}{}^4{a}_2{}^4}+\frac{7\overline{h_{m_2}}}{720}\right)\right.-\frac{h_{m_2}}{{\alpha }_{m_2}{}^2{a}_2{}^2}-\left.\frac{\overline{h_{m_2}}}{12}\right]+\\ 2H_2{\beta }_{n_2}{}^4\cdot \frac{2(H_2+2D_{x_2{y}_2})}{D_{x_2}{\alpha }_{m_2}{}^2{a}_2{}^2}{h}_{m_2}+\left.\frac{{\beta }_{n_2}{}^2}{a_2{}^2}\left[H_2-2D_{x_2{y}_2}\right]\overline{h_{m_2}}\right\}{(-1)}^{m_2+1}{H}_{n_2}\\ ={\lambda }_{m_2{n}_2}(F_{m_2{n}_2}-Q_{m_2{n}_2})\\ (m_2=0,1,2,\cdot \cdot \cdot ; n_2=0,1,2,\cdot \cdot \cdot )\end{array}$$

(8)

where ${\alpha }_{m_i}=\frac{m_i\pi }{a_i}$, ${\beta }_{m_i}=\frac{n_i\pi }{b_i}$, $h_{m_i}=h_{n_i}=\left\{\begin{array}{l}0,i=0 \\ 1,i\ne 0\end{array}\right.$, and $\overline{h_{m_i}}=\overline{h_{n_i}}=\left\{\begin{array}{l}1,i=0 \\ 0,i\ne 0\end{array}\right.$.

Considering the boundary conditions of the bending moment, we could obtain:

(1)

When $x_i=0$, $M_{x_i}=0$ (Equation (2a)):

$$\begin{array}{l}{\displaystyle \sum _{m_i=0}^{\infty }({\alpha }_{m_i}^2+v_{y_i}{\beta }_{n_i}^2)w_{m_i{n}_i}+2}{\displaystyle \sum _{m_i=0}^{\infty }\left\{\frac{H_i+2D_{x_i{y}_i}}{D_{y_i}}{b}_i{}^2{\alpha }_{m_i}^4\left(\frac{h_{n_i}}{{\beta }_{n_i}^4{b}_i{}^4}-\frac{{\overline{h}}_{n_i}}{90}\right)\right.}\\ \left.+{\alpha }_{m_i}^2\left[-\frac{1}{{\beta }_{n_i}^2{b}_i{}^2}{h}_{n_i}+\frac{{\overline{h}}_{n_i}}{6}\right]+{\alpha }_{m_i}^2{v}_{y_i}\cdot \frac{H_i+2D_{x_i{y}_i}}{D_{y_i}}\cdot \frac{1}{{\beta }_{n_i}^2{b}_i{}^2}{h}_{n_i}+\frac{v_{y_i}{\overline{h}}_{n_i}}{2b_i{}^2}\right\}{C}_{m_i}\\ +2{(-1)}^{n_i+1}{\displaystyle \sum _{m_i=0}^{\infty }\left\{\frac{H_i+2D_{x_i{y}_i}}{D_{y_i}}{b}_i{}^2{\alpha }_{m_i}^4\left(\frac{h_{n_i}}{{\beta }_{n_i}^4{b}_i{}^4}+\frac{7{\overline{h}}_{n_i}}{720}\right)\right.}\\ \left.-{\alpha }_{m_i}^2\left[\frac{1}{{\beta }_{n_i}^2{b}_i{}^2}{h}_{n_i}+\frac{{\overline{h}}_{n_i}}{12}\right]+{\alpha }_{m_i}^2{v}_{y_i}\cdot \frac{H_i+2D_{x_i{y}_i}}{D_{y_i}}\cdot \frac{1}{{\beta }_{n_i}^2{b}_i{}^2}{h}_{n_i}+\frac{v_{y_i}{\overline{h}}_{n_i}}{2b_i{}^2}\right\}{D}_{m_i}\\ +\left(\frac{H_i+2D_{x_i{y}_i}}{3D_{x_i}}{\beta }_{n_i}^2+\frac{1}{a_i{}^2}\right)G_{n_i}+\left(\frac{H_i+2D_{x_i{y}_i}}{6D_{x_i}}{\beta }_{n_i}^2-\frac{1}{a_i{}^2}\right)H_{n_i}=0\\ (n_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

(9)

(2)

When $x_i=a_i$, $M_{x_i}=0$ (Equation (2a)):

$$\begin{array}{l}{\displaystyle \sum _{m_i=0}^{\infty }{(-1)}^{m_i}({\alpha }_{m_i}^2+v_{y_i}{\beta }_{n_i}^2)w_{m_i{n}_i}+2}{\displaystyle \sum _{m_i=0}^{\infty }\left\{{(-1)}^{m_i}\frac{H_i+2D_{x_i{y}_i}}{D_{y_i}}{b}_i{}^2{\alpha }_{m_i}^4\left(\frac{h_{n_i}}{{\beta }_{n_i}^4{b}_i{}^4}-\frac{{\overline{h}}_{n_i}}{90}\right)\right.}\\ \left.+{\alpha }_{m_i}^2\left[-\frac{1}{{\beta }_{n_i}^2{b}_i{}^2}{h}_{n_i}+\frac{{\overline{h}}_{n_i}}{6}\right]+{\alpha }_{m_i}^2{v}_{y_i}\cdot \frac{H_i+2D_{x_i{y}_i}}{D_{y_i}}\cdot \frac{1}{{\beta }_{n_i}^2{b}_i{}^2}{h}_{n_i}+\frac{v_{y_i}{\overline{h}}_{n_i}}{2b_i{}^2}\right\}{C}_{m_i}\\ +2{(-1)}^{n_i+1}{\displaystyle \sum _{m_i=0}^{\infty }\left\{{(-1)}^{m_i}\frac{H_i+2D_{x_i{y}_i}}{D_{y_i}}{b}_i{}^2{\alpha }_{m_i}^4\left(\frac{h_{n_i}}{{\beta }_{n_i}^4{b}_i{}^4}+\frac{7{\overline{h}}_{n_i}}{720}\right)\right.}\\ \left.-{\alpha }_{m_i}^2\left[\frac{1}{{\beta }_{n_i}^2{b}_i{}^2}{h}_{n_i}+\frac{{\overline{h}}_{n_i}}{12}\right]+{\alpha }_{m_i}^2{v}_{y_i}\cdot \frac{H_i+2D_{x_i{y}_i}}{D_{y_i}}\cdot \frac{1}{{\beta }_{n_i}^2{b}_i{}^2}{h}_{n_i}+\frac{v_{y_i}{\overline{h}}_{n_i}}{2b_i{}^2}\right\}{D}_{m_i}\\ +\left(-\frac{v_{y_i}(H_i+2D_{x_i{y}_i})}{24D_{x_i}}{a}_i{}^2{\beta }_{n_i}^4+\left(\frac{v_{y_i}}{2}-\frac{H_i+2D_{x_i{y}_i}}{6D_{x_i}}\right){\beta }_{n_i}^2+\frac{1}{a_i{}^2}\right)G_{n_i}+\\ \left(-\frac{v_{y_i}(H_i+2D_{x_i{y}_i})}{24D_{x_i}}{a}_i{}^2{\beta }_{n_i}^4+\left(\frac{v_{y_i}}{2}-\frac{H_i+2D_{x_i{y}_i}}{3D_{x_i}}\right){\beta }_{n_i}^2-\frac{1}{a_i{}^2}\right)H_{n_i}=0\\ (n_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

(10)

(3)

When $y_i=0$, $M_{y_i}=0$ (Equation (2c)):

$$\begin{array}{l}{\displaystyle \sum _{n_i=0}^{\infty }({\beta }_{n_i}^2+v_{x_i}{\alpha }_{m_i}^2)w_{m_i{n}_i}+}\left(\frac{H_i+2D_{x_i{y}_i}}{3D_{y_i}}{\alpha }_{m_i}^2+\frac{1}{b_i{}^2}\right)C_{m_i}+\left(\frac{H_i+2D_{x_i{y}_i}}{6D_{y_i}}{\alpha }_{m_i}^2-\frac{1}{b_i{}^2}\right)D_{m_i}\\ +2{\displaystyle \sum _{n_i=0}^{\infty }\left\{\frac{H_i+2D_{x_i{y}_i}}{D_{x_i}}{a}_i{}^2{\beta }_{n_i}^4\left(\frac{h_{m_i}}{{\alpha }_{m_i}^4{a}_i{}^4}-\frac{{\overline{h}}_{m_i}}{90}\right)\right.}+{\beta }_{n_i}^2\left[-\frac{1}{{\alpha }_{m_i}^2{a}_i{}^2}{h}_{m_i}+\frac{{\overline{h}}_{m_i}}{6}\right]\\ \left.+{\beta }_{n_i}^2{v}_{x_i}\cdot \frac{H_i+2D_{x_i{y}_i}}{D_{x_i}}\cdot \frac{1}{{\alpha }_{m_i}^2{a}_i{}^2}{h}_{m_i}+\frac{v_{x_i}{\overline{h}}_{m_i}}{2a_i{}^2}\right\}{G}_{n_i}\\ +2{(-1)}^{m_i+1}{\displaystyle \sum _{n_i=0}^{\infty }\left\{\frac{H_i+2D_{x_i{y}_i}}{D_{x_i}}{a}_i{}^2{\beta }_{n_i}^4\left(\frac{h_{m_i}}{{\alpha }_{m_i}^4{a}_i{}^4}+\frac{7{\overline{h}}_{m_i}}{720}\right)\right.}\\ \left.-{\beta }_{n_i}^2\left[\frac{1}{{\alpha }_{m_i}^2{a}_i{}^2}{h}_{m_i}+\frac{{\overline{h}}_{m_i}}{12}\right]+{\beta }_{n_i}^2{v}_{x_i}\cdot \frac{H_i+2D_{x_i{y}_i}}{D_{x_i}}\cdot \frac{1}{{\alpha }_{m_i}^2{a}_i{}^2}{h}_{m_i}+\frac{v_{x_i}{\overline{h}}_{m_i}}{2a_i{}^2}\right\}{H}_{n_i}=0\\ (m_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

(11)

(4)

When $y_i=b_i$, $M_{y_i}=0$ (Equation (2c)):

$$\begin{array}{l}+2{\displaystyle \sum _{n_i=0}^{\infty }\left\{{(-1)}^{n_i}\frac{H+2D_{x_i{y}_i}}{D_{x_i}}{a}_i{}^2{\beta }_{n_i}^4\left(\frac{h_{m_i}}{{\alpha }_{m_i}^4{a}_i{}^4}-\frac{{\overline{h}}_{m_i}}{90}\right)\right.}+{\beta }_{n_i}^2\left[-\frac{1}{{\alpha }_{m_i}^2{a}_i{}^2}{h}_{m_i}+\frac{{\overline{h}}_{m_i}}{6}\right]\\ \left.+{\beta }_{n_i}^2{v}_{x_i}\cdot \frac{H_i+2D_{x_i{y}_i}}{D_{x_i}}\cdot \frac{1}{{\alpha }_{m_i}^2{a}_i{}^2}{h}_{m_i}+\frac{v_{x_i}{\overline{h}}_{m_i}}{2a_i{}^2}\right\}{G}_{n_i}\\ +2{(-1)}^{m_i+1}{\displaystyle \sum _{n_i=0}^{\infty }\left\{{(-1)}^{n_i}\frac{H_i+2D_{x_i{y}_i}}{D_{x_i}}{a}_i{}^2{\beta }_{n_i}^4\left(\frac{h_{m_i}}{{\alpha }_{m_i}^4{a}_i{}^4}+\frac{7{\overline{h}}_{m_i}}{720}\right)\right.}\\ \left.-{\beta }_{n_i}^2\left[\frac{1}{{\alpha }_{m_i}^2{a}_i{}^2}{h}_{m_i}+\frac{{\overline{h}}_{m_i}}{12}\right]+{\beta }_{n_i}^2{v}_{x_i}\cdot \frac{H_i+2D_{x_i{y}_i}}{D_{x_i}}\cdot \frac{1}{{\alpha }_{m_i}^2{a}_i{}^2}{h}_{m_i}+\frac{v_{x_i}{\overline{h}}_{m_i}}{2a_i{}^2}\right\}{H}_{n_i}=0\\ (m_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

(12)

The deflection of the upper and lower plates could be expressed by formula [27]:

$$w_i(x_i,y_i)={\displaystyle \sum _{m_i=0}^{\infty }{\displaystyle \sum _{n_i=0}^{\infty }{\lambda }_{m_i{n}_i}{w}_i{}_{m_i{n}_i}\cos \frac{m_i\pi x_i}{a_i}\cos \frac{n_i\pi y_i}{b_i}}}$$

where

$$w_i{}_{m_i{n}_i}=\frac{4}{a_i{b}_i}{\displaystyle {\int }_0^{b_i}{\displaystyle {\int }_0^{a_i}{w}_i(x_i,y_i)}\cos \frac{m_i\pi x_i}{a_i}\cos \frac{n_i\pi y_i}{b_i}d{x}_{i}d{y}_i}$$

$$\begin{array}{l}{w}_i{}_{m_i{n}_i}=w_{m_i{n}_i}+2\left[{\mu }_{2m_i{y}_i}{b}_i{}^2{\alpha }_{m_i}^2\left(\frac{h_{n_i}}{{\beta }_{n_i}^4{b}_i{}^4}-\frac{{\overline{h}}_{n_i}}{90}\right)\right.-\left.\frac{h_{n_i}}{{\beta }_{n_i}^2{b}_i{}^2}+\frac{{\overline{h}}_{n_i}}{6}\right]C_{m_i}\\ +2\left[{\mu }_{2m_i{y}_i}{b}_i{}^2{\alpha }_{m_i}^2\left(\frac{h_{n_i}}{{\beta }_{n_i}^4{b}_i{}^4}+\frac{7{\overline{h}}_{n_i}}{720}\right)\right.-\left.\frac{h_{n_i}}{{\beta }_{n_i}^2{b}_i{}^2}-\frac{{\overline{h}}_{n_i}}{12}\right]{(-1)}^{n_i+1}{D}_{m_i}\\ +2\left[{\mu }_{2m_i{\mathrm{x}}_i}{a}_i{}^2{\beta }_{n_i}^2\left(\frac{h_{m_i}}{{\alpha }_{m_i}^4{a}_i{}^4}-\frac{{\overline{h}}_{m_i}}{90}\right)\right.-\left.\frac{h_{m_i}}{{\alpha }_{m_i}^2{a}_i{}^2}+\frac{{\overline{h}}_{m_i}}{6}\right]G_{n_i}\\ +2\left[{\mu }_{2m_i{x}_i}{a}_i{}^2{\beta }_{n_i}^2\left(\frac{h_{m_i}}{{\alpha }_{m_i}^4{a}_i{}^4}+\frac{7{\overline{h}}_{m_i}}{720}\right)\right.-\left.\frac{h_{m_i}}{{\alpha }_{m_i}^2{a}_i{}^2}-\frac{{\overline{h}}_{m_i}}{12}\right]{(-1)}^{m_i+1}{H}_{n_i}\\ (m_i=0,1,2,\cdot \cdot \cdot ; n_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

Expanding the deflections of the lower plate into the double cosine series at the contacting position with the upper plate:

$$w_2(x_1,y_1)={\displaystyle \sum _{m_1=0}^{\infty }{\displaystyle \sum _{n_1=0}^{\infty }{\lambda }_{m_1{n}_1}{W}_2{}_{m_1{n}_1}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}}}$$

$$\begin{array}{l}{w}_2{}_{m_1{n}_1}=\frac{4}{a_1{b}_1}{\displaystyle {\int }_0^{b_1}{\displaystyle {\int }_0^{a_1}{W}_2(x_2,y_2)}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}d{x}_{1}d{y}_1}\\ =\frac{4}{a_1{b}_1}{\displaystyle {\int }_0^{b_1}{\displaystyle {\int }_0^{a_1}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}{W}_2{}_{m_2{n}_2}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}}}}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}d{x}_{1}d{y}_1}\\ =\frac{4}{a_1{b}_1}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}{W}_2{}_{m_2{n}_2}}}{\displaystyle {\int }_0^{b_1}{\displaystyle {\int }_0^{a_1}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}d{x}_{1}d{y}_1}\\ (m_i=0,1,2,\cdot \cdot \cdot ; n_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

Substituting x₂ = x₁ + x₀ and y₂ = y₁ + y₀ into the expression of $w_{2m_1{n}_1}$, we obtain:

$$\begin{array}{l}{\displaystyle {\int }_0^{b_1}{\displaystyle {\int }_0^{a_1}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}d{x}_{1}d{y}_1}\\ ={\displaystyle {\int }_0^{b_1}{\displaystyle {\int }_0^{a_1}\cos \frac{m_2\pi (x_1+x_0)}{a_2}\cos \frac{n_2\pi (y_1+y_0)}{b_2}}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}d{x}_{1}d{y}_1}\\ =\left\{\frac{1}{2}\right.\cos \frac{m_2\pi x_0}{a_2}\frac{a_2{a}_1}{\pi (m_2{a}_1+m_1{a}_2)}\sin \frac{\pi a_1(m_2{a}_1+m_1{a}_2)}{a_2{a}_1}\\ +\frac{1}{2}\cos \frac{m_2\pi x_0}{a_2}\frac{a_2{a}_1}{\pi (m_2{a}_1-m_1{a}_2)}\sin \frac{\pi c(m_2{a}_1-m_1{a}_2)}{a_2{a}_1}\\ -\frac{1}{2}\sin \frac{m_2\pi x_0}{a_2}\frac{a_2{a}_1}{\pi (m_2{a}_1+m_1{a}_2)}\left[1\left.-\cos \frac{\pi c(m_2{a}_1+m_1{a}_2)}{a_2{a}_1}\right]\right.\\ -\frac{1}{2}\sin \frac{m_2\pi x_0}{a_2}\frac{a_2{a}_1}{\pi (m_2{a}_1-m_1{a}_2)}\left.\left[1\left.-\cos \frac{\pi a_1(m_2{a}_1-m_1{a}_2)}{a_2{a}_1}\right]\right.\right\}\\ \cdot \left\{\frac{1}{2}\right.\cos \frac{n_2\pi y_0}{b_2}\frac{b_2{b}_1}{\pi (n_2{b}_1+n_1{b}_2)}\sin \frac{\pi b_1(n_2{b}_1+n_1{b}_2)}{b_2{b}_1}\\ +\frac{1}{2}\cos \frac{n_2\pi y_0}{b_2}\frac{b_2{b}_1}{\pi (n_2{b}_1-n_1{b}_2)}\sin \frac{\pi b_1(n_2{b}_1-n_1{b}_2)}{b_2{b}_1}\\ -\frac{1}{2}\sin \frac{n_2\pi y_0}{b_2}\frac{b_2{b}_1}{\pi (n_2{b}_1+n_1{b}_2)}\left[1\left.-\cos \frac{\pi d(n_2{b}_1+n_1{b}_2)}{b_2{b}_1}\right]\right.\\ -\frac{1}{2}\sin \frac{n_2\pi y_0}{b_2}\frac{b_2{b}_2}{\pi (n_2{b}_1-n_1{b}_2)}\left.\left[1\left.-\cos \frac{\pi b_1(n_2{b}_1-n_1{b}_2)}{b_2{b}_1}\right]\right.\right\}=D\\ (m_i=0,1,2,\cdot \cdot \cdot ; n_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

The expression of $w_{2m_1{n}_1}$ can be rewritten as:

$$W_2{}_{m_1{n}_1}=\frac{4}{a_1{b}_1}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}{W}_2{}_{m_2{n}_2}D}}$$

Taking into account that the deflections of the upper and lower plates at the contact position are the same, the deformation equation of compatibility is expressed as:

$$\begin{array}{l}{w}_{m_1{n}_1}+2\left[{\mu }_{2m_1{y}_1}{b}_1{}^2{\alpha }_{m_1}^2\left(\frac{h_{n_1}}{{\beta }_{n_1}^4{b}_1{}^4}-\frac{{\overline{h}}_{n_1}}{90}\right)\right.-\left.\frac{h_{n_1}}{{\beta }_{n_1}^2{b}_1{}^2}+\frac{{\overline{h}}_{n_1}}{6}\right]C_{m_1}\\ +2\left[{\mu }_{2m_1{y}_1}{b}_1{}^2{\alpha }_{m_1}^2\left(\frac{h_{n_1}}{{\beta }_{n_1}^4{b}_1{}^4}+\frac{7{\overline{h}}_{n_1}}{720}\right)\right.-\left.\frac{h_{n_1}}{{\beta }_{n_1}^2{b}_1{}^2}-\frac{{\overline{h}}_{n_1}}{12}\right]{(-1)}^{n_1+1}{D}_{m_1}\\ +2\left[{\mu }_{2m_1{\mathrm{x}}_1}{a}_1{}^2{\beta }_{n_1}^2\left(\frac{h_{m_1}}{{\alpha }_{m_1}^4{a}_1{}^4}-\frac{{\overline{h}}_{m_1}}{90}\right)\right.-\left.\frac{h_{m_1}}{{\alpha }_{m_1}^2{a}_1{}^2}+\frac{{\overline{h}}_{m_1}}{6}\right]G_{n_1}\\ +2\left[{\mu }_{2m_1{\mathrm{x}}_1}{a}_1{}^2{\beta }_{n_1}^2\left(\frac{h_{m_1}}{{\alpha }_{m_1}^4{a}_1{}^4}+\frac{7{\overline{h}}_{m_1}}{720}\right)\right.-\left.\frac{h_{m_1}}{{\alpha }_{m_1}^2{a}_1{}^2}-\frac{{\overline{h}}_{m_1}}{12}\right]{(-1)}^{m_1+1}{H}_{n_1}\\ =\frac{4}{a_1{b}_1}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}{w}_{m_2{n}_2}D}}+\frac{4}{a_1{b}_1}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}D\cdot 2\left[{\mu }_{2m_2{y}_2}{b}_2{}^2{\alpha }_{m_2}^2\left(\frac{h_{n_2}}{{\beta }_{n_2}^4{b}_2{}^4}-\frac{{\overline{h}}_{n_2}}{90}\right)\right.-\left.\frac{h_{n_2}}{{\beta }_{n_2}^2{b}_2{}^2}+\frac{{\overline{h}}_{n_2}}{6}\right]C_{m_2}}}\\ +\frac{4}{a_1{b}_1}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}D\cdot 2\left[{\mu }_{2m_2{y}_2}{b}_2{}^2{\alpha }_{m_2}^2\left(\frac{h_{n_2}}{{\beta }_{n_2}^4{b}_2{}^4}+\frac{7{\overline{h}}_{n_2}}{720}\right)\right.-\left.\frac{h_{n_2}}{{\beta }_{n_2}^2{b}_2{}^2}-\frac{{\overline{h}}_{n_2}}{12}\right]{(-1)}^{n_2+1}{D}_{m_2}}}\\ +\frac{4}{a_1{b}_1}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}D\cdot 2\left[{\mu }_{2m_2{x}_2}{a}_2{}^2{\beta }_{n_2}^2\left(\frac{h_{m_2}}{{\alpha }_{m_2}^4{a}_2{}^4}-\frac{{\overline{h}}_{m_2}}{90}\right)\right.-\left.\frac{h_{m_2}}{{\alpha }_{m_2}^2{a}_2{}^2}+\frac{{\overline{h}}_{m_2}}{6}\right]G_{n_2}}}\\ +\frac{4}{a_1{b}_1}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}D\cdot 2\left[{\mu }_{2m_2{x}_2}{a}_2{}^2{\beta }_{n_2}^2\left(\frac{h_{m_2}}{{\alpha }_{m_2}^4{a}_2{}^4}+\frac{7{\overline{h}}_{m_2}}{720}\right)\right.-\left.\frac{h_{m_2}}{{\alpha }_{m_2}^2{a}_2{}^2}-\frac{{\overline{h}}_{m_2}}{12}\right]{(-1)}^{m_2+1}}}{H}_{n_2}\\ (m_i=0,1,2,\cdot \cdot \cdot ; n_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

(13)

The double Fourier transformation of the ground reaction force is expressed as:

$$Q{(\xi ,\eta )}_{}=-\frac{1}{2\pi }{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}}{Q}_{m_2{n}_2}}\frac{\left[e^{i\xi a}{(-1)}^{m_2}-1\right]\left[{(-1)}^{n_2}{e}^{i\eta b}-1\right]}{\xi \eta \left[1-{(\frac{m_2\pi }{a\xi })}^2\right]\left[1-{(\frac{n_2\pi }{b\eta })}^2\right]}$$

(14)

Expanding $w\left|{}_{z_2=0}\right.$ into the double cosine series form:

$$w\left|{}_{z_2=0}\right.={\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}}{w}_{z_2}{}_{m_2{n}_2}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}}$$

(15)

in which:

$$w_{z_2{m}_2{n}_2}=\frac{4}{a_2{b}_2}{\displaystyle {\int }_0^{a_2}{\displaystyle {\int }_0^{b_2}w\left|{}_{z_2=0}\right.\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}d{x}_{2}d{y}_2}}$$

(16)

$$w_{z_2{m}_2{n}_2}=\frac{1}{2{\pi }^2{a}_2{b}_2}\frac{(\lambda +2\mu )}{(\lambda +\mu )\mu }{\displaystyle \sum _{p=0}^{\infty }{\displaystyle \sum _{q=0}^{\infty }{Q}_{pq}{\lambda }_{pq}{\eta }_{pq{m}_2{n}_2}}}$$

(17)

where

$${\eta }_{pq{m}_2{n}_2}={\displaystyle {\int }_{-\infty }^{\infty }}{\displaystyle {\int }_{-\infty }^{\infty }\frac{1}{q_1}\frac{[{(-1)}^p{e}^{i\xi a}-1][{(-1)}^q{e}^{i\eta b}-1][{(-1)}^{m_2}{e}^{-i\xi a}-1][{(-1)}^{n_2}{e}^{-i\eta b}-1]}{{\xi }^2{\eta }^2{\left[1-{(\frac{m_2\pi }{a\xi })}^2\right]}^{}\left[1-{(\frac{n_2\pi }{b\eta })}^2\right]{\left[1-{(\frac{p\pi }{a\xi })}^2\right]}^{}{\left[1-{(\frac{q\pi }{b\eta })}^2\right]}^{}}}d\xi d\eta$$

Equation (4) could be rewritten in a double cosine series form, and considering that the plate and surface of the elastic foundation have the same vertical displacement, thus the coefficients of the corresponding items are also the same. The deformation equation of compatibility is given as:

$$\begin{array}{l}\frac{1}{2{\pi }^2{a}_2{b}_2}\frac{(\lambda +2\mu )}{(\lambda +\mu )\mu }{\displaystyle \sum _{p=0}^{\infty }{\displaystyle \sum _{q=0}^{\infty }{Q}_{pq}{\lambda }_{pq}{\eta }_{pq{m}_2{n}_2}}}{\lambda }_{m_2{n}_2}\\ =w_{m_2{n}_2}+2\left[{\mu }_{2m_2{y}_2}{b}_2{}^2{\alpha }_{m_2}^2\left(\frac{h_{n_2}}{{\beta }_{n_2}^4{b}^4}-\frac{{\overline{h}}_{n_2}}{90}\right)\right.-\left.\frac{h_{n_2}}{{\beta }_{n_2}^2{b}_2{}^2}+\frac{{\overline{h}}_{n_2}}{6}\right]C_{m_2}\\ +2\left[{\mu }_{2m_2{y}_2}{b}_2{}^2{\alpha }_{m_2}^2\left(\frac{h_{n_2}}{{\beta }_{n_2}^4{b}_2{}^4}+\frac{7{\overline{h}}_{n_2}}{720}\right)\right.-\left.\frac{h_{n_2}}{{\beta }_{n_2}^2{b}_2{}^2}-\frac{{\overline{h}}_{n_2}}{12}\right]{(-1)}^{n_2+1}{D}_{m_2}\\ +2\left[{\mu }_{2m_2{x}_2}{a}_2{}^2{\beta }_{n_2}^2\left(\frac{h_{m_2}}{{\alpha }_{m_2}^4{a}_2{}^4}-\frac{{\overline{h}}_{m_2}}{90}\right)\right.-\left.\frac{h_{m_2}}{{\alpha }_{m_2}^2{a}_2{}^2}+\frac{{\overline{h}}_{m_2}}{6}\right]G_{n_2}\\ +2\left[{\mu }_{2m_2{x}_2}{a}_2{}^2{\beta }_{n_2}^2\left(\frac{h_{m_2}}{{\alpha }_{m_2}^4{a}_2{}^4}+\frac{7{\overline{h}}_{m_2}}{720}\right)\right.-\left.\frac{h_{m_2}}{{\alpha }_{m_2}^2{a}_2{}^2}-\frac{{\overline{h}}_{m_2}}{12}\right]{(-1)}^{m_2+1}{H}_{n_2}\\ (m_i=0,1,2,\cdot \cdot \cdot ; n_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

(18)

Therefore, the undetermined coefficients $w_{m_1{n}_1}$, $F_{m_1{n}_1}$, $C_{m_1}$, $D_{m_1}$, $G_{n_1}$, $H_{n_1}$, $w_{m_2{n}_2}$, $Q_{m_2{n}_2}$, $C_{m_2}$, $D_{m_2}$, $G_{n_2}$, and $H_{n_2}$ could be obtained by twelve equations (Equations (7)–(18)), in which N₁, N₂, M₁, and M₂ are the maximum values of n₁, n₂, m₁, and m₂, respectively. Thus, the number of the governing equation is 2(M₁ + 1)(N₁ + 1) + 2(M₁ + 1) + 2(N₁ + 1) + 2(M₂ + 1)(N₂ + 1) + 2(M₂ + 1) + 2(N₂ + 1), which can be used to solve the same number of undetermined coefficients. In order to ensure the correctness of the calculation results, the maximum values of M₁, M₂, N₁, and N₂ are taken to 20. Substituting the solved coefficients into Equation (1a,b), the deflection of the plate can be obtained, and further, the internal force and the foundation reaction can also be calculated.

2.1.3. Example

Case 1: We consider a stepped rectangular plate resting on the surface of an elastic half-space foundation. The dimensions of the upper and lower plates are 4.0 m × 0.2 m (side length × thickness) and 4 m × 0 m (side length × thickness), respectively, and the uniform load $q(x_1,y_1)$ on the plate is 0.98 MPa. In this case, the stepped rectangular plate degenerates into a rectangular plate. The performance parameters of the plate and the foundation are given in

Table 1. Performance parameters.

	Poisson Ratio	Elastic Modulus (MPa)
Plate	0.167	34,300
Foundation	0.4	343

.

The subgrade reaction, bending moment, and deflection of the plate could be obtained by the solution, as shown in Figure 2.

Table 2. Comparison of calculation results of thin plate.

	In This Paper	[28]
Maximum deflection (m)	0.0107	0.0107
Maximum bending moment (kN·m)	35.558	35.558

shows that the calculated results are consistent with the results in [28]. This comparison proves the effectiveness of the theory proposed in this paper.

Case 2: We consider a stepped rectangular plate resting on the surface of an elastic half-space foundation. The dimensions of the upper and lower plates are 4.0 m × 0.2 m (side length × thickness) and 4 m × 0.2 m (side length × thickness), respectively, and the concentrated loading on the plate is F = 100 kN. The performance parameters of the stepped plate and the foundation are given in

Table 3. Performance parameters of the stepped rectangular plate.

Component Name	Poisson Ratio	Elastic Modulus (MPa)
Upper plate	0.167	26,000
Lower plate	0.167	26,000
Foundation	0.4	343

.

Figure 3 shows the comparison of the analytical solution and the simulation results, in which the model is created by ABAQUS software. For the deflection at the center of the lower plate, the result of the finite element model is smaller than that of the analytical solution when it is close to the edge of the plate, while when it is close to the center of the plate, the situation is just the opposite (Figure 3a). For the bending moment of the lower plate at x = 0.5 m and y = 0.5 m, the curve increases slowly at the edge of the plate, and as it approaches the center of the plate, the curve increases first and then decreases (Figure 3b,c). It can be concluded that the results of finite element model well support the present results using the thin plate theory.

2.2. Both Upper and Lower Parts of the Plate Are Moderately Thick Plates

2.2.1. Basic Equations and Boundary Conditions

The governing differential equations of the moderately thick plate are determined by formulas [27]:

$$D_{11}\frac{{\partial }^2{\mathrm{\Phi }}_{x_1}}{\partial x_1{}^2}+D_{66}\frac{{\partial }^2{\mathrm{\Phi }}_{x_1}}{\partial y_1{}^2}+(D_{12}+D_{66})\frac{{\partial }^2{\mathrm{\Phi }}_{y_1}}{\partial x_1\partial y_1}+C_{11}\frac{\partial w_1}{\partial x_1}-C_{11}{\mathrm{\Phi }}_{x_1}=0$$

$$D_{22}\frac{{\partial }^2{\mathrm{\Phi }}_{y_1}}{\partial y_1{}^2}+D_{66}\frac{{\partial }^2{\mathrm{\Phi }}_{y_1}}{\partial x_1{}^2}+(D_{12}+D_{66})\frac{{\partial }^2{\mathrm{\Phi }}_{x_1}}{\partial x_1\partial y_1}+C_{22}\frac{\partial w_1}{\partial y_1}-C_{22}{\mathrm{\Phi }}_{y_1}=0$$

$$C_{11}\frac{{\partial }^2{w}_1}{\partial x_1{}^2}+C_{22}\frac{{\partial }^2{w}_1}{\partial y_1{}^2}-C_{11}\frac{\partial {\mathrm{\Phi }}_{x_1}}{\partial x_1}-C_{22}\frac{\partial {\mathrm{\Phi }}_{y_1}}{\partial y_1}+q-F=0$$

$$D_{11}\frac{{\partial }^2{\mathrm{\Phi }}_{x_2}}{\partial x_2{}^2}+D_{66}\frac{{\partial }^2{\mathrm{\Phi }}_{x_2}}{\partial y_2{}^2}+(D_{12}+D_{66})\frac{{\partial }^2{\mathrm{\Phi }}_{y_2}}{\partial x_2\partial y_2}+C_{11}\frac{\partial w_2}{\partial x_2}-C_{11}{\mathrm{\Phi }}_{x_2}=0$$

$$D_{22}\frac{{\partial }^2{\mathrm{\Phi }}_{y_2}}{\partial y_2{}^2}+D_{66}\frac{{\partial }^2{\mathrm{\Phi }}_{y_2}}{\partial x_2{}^2}+(D_{12}+D_{66})\frac{{\partial }^2{\mathrm{\Phi }}_{x_2}}{\partial x_2\partial y_2}+C_{22}\frac{\partial w_2}{\partial y_2}-C_{22}{\mathrm{\Phi }}_{y_2}=0$$

$$C_{11}\frac{{\partial }^2{w}_2}{\partial x_2{}^2}+C_{22}\frac{{\partial }^2{w}_2}{\partial y_2{}^2}-C_{11}\frac{\partial {\mathrm{\Phi }}_{x_2}}{\partial x_2}-C_{22}\frac{\partial {\mathrm{\Phi }}_{y_2}}{\partial y_2}+F-Q=0$$

where ${\mathrm{\Phi }}_{x_i}$, ${\mathrm{\Phi }}_{y_i}$, $w_i$, F, $Q$ are unknown coefficients.

Boundary restrictions are given as:

$$M_{x_i}=M_{x_i{y}_i}=Q_{x_i}=0(x_i=0,x_i=\mathrm{a})$$

$$M_{y_i}=M_{x_i{y}_i}=Q_{y_i}=0(y_i=0,y_i=\mathrm{b})$$

2.2.2. Coordination Equation and Analytical Solution

Expanding ${\mathrm{\Phi }}_{x_i}$, ${\mathrm{\Phi }}_{y_i}$, $w_i$, F, $Q$, q into a Fourier series

$${\mathrm{\Phi }}_{x_i}={\displaystyle \sum _{m_i}{\displaystyle \sum _{n_i}{\phi }_{m_i{n}_i}\sin {\alpha }_{m_i}{x}_i\cos {\beta }_{n_i}{y}_i}}$$

$${\mathrm{\Phi }}_{y_i}={\displaystyle \sum _{m_i}{\displaystyle \sum _{n_i}{\psi }_{m_i{n}_i}\cos {\alpha }_{m_i}{x}_i\sin {\beta }_{n_i}{y}_i}}$$

$$w_i={\displaystyle \sum _{m_i}{\displaystyle \sum _{n_i}{w}_{m_i{n}_i}\cos {\alpha }_{m_i}{x}_i\cos {\beta }_{n_i}{y}_i}}$$

$$F={\displaystyle \sum _{m_2}{\displaystyle \sum _{n_2}{F}_{m_2{n}_2}\cos {\alpha }_{m_2}{x}_2\cos {\beta }_{n_2}{y}_2}}$$

$$Q={\displaystyle \sum _{m_i}{\displaystyle \sum _{n_i}{Q}_{m_i{n}_i}\cos {\alpha }_{m_i}{x}_i\cos {\beta }_{n_i}{y}_i}}$$

$$q={\displaystyle \sum _{m_1}{\displaystyle \sum _{n_1}{q}_{m_1{n}_1}\cos {\alpha }_{m_1}{x}_1\cos {\beta }_{n_1}{y}_1}}$$

in which ${\mathrm{\Phi }}_{m_i{n}_i}$, ${\psi }_{m_i{n}_i}$, $w_{m_i{n}_i}$, $F_{m_2{n}_2}$, $q_{m_1{n}_1}$ and $Q_{m_i{n}_i}$ are unknown coefficients.

Taking into account the continuous differentiability of the formula on the boundary of the plate, we can write:

$$\frac{\partial w_i(0,y_i)}{\partial x_i}=-\frac{a_i}{4}{\displaystyle \sum _{n_i}{a}_{n_i}\cos {\beta }_{n_i}{y}_i;}\frac{\partial w_i(x_i,0)}{\partial y_i}=-\frac{b_i}{4}{\displaystyle \sum _{n_i}{b}_{m_i}\cos {\alpha }_{m_i}{x}_i}$$

$${\mathrm{\Phi }}_{x_i}(0,y_i)=-\frac{a_i}{4}{\displaystyle \sum _{n_i}{c}_{n_i}\cos {\beta }_{n_i}{y}_i;}{\mathrm{\Phi }}_{y_i}(x_i,0)=-\frac{b_i}{4}{\displaystyle \sum _{n_i}{d}_{m_i}\cos {\alpha }_{m_i}{x}_i}$$

$$\frac{\partial w_i(a_i,y_i)}{\partial x_i}=-\frac{a_i}{4}{\displaystyle \sum _{n_i}{e}_{n_i}\cos {\beta }_{n_i}{y}_i;}\frac{\partial w_i(x_i,b_i)}{\partial y_i}=-\frac{b_i}{4}{\displaystyle \sum _{n_i}{f}_{m_i}\cos {\alpha }_{m_i}{x}_i}$$

$${\mathrm{\Phi }}_{x_i}(a_i,y_i)=-\frac{a_i}{4}{\displaystyle \sum _{n_i}{g}_{n_i}\cos {\beta }_{n_i}{y}_i;}{\mathrm{\Phi }}_{y_i}(x_i,b_i)=-\frac{b_i}{4}{\displaystyle \sum _{n_i}{h}_{m_i}\cos {\alpha }_{m_i}{x}_i}$$

where $a_{n_i}$, $b_{m_i}$, $c_{n_i}$, $d_{m_i}$, $e_{n_i}$, $f_{m_i}$, $g_{n_i}$, and $h_{m_i}$ are unknown coefficients. According to boundary conditions, we obtain the expressions:

$$a_{n_i}=c_{n_i}, e_{n_i}=g_{n_i}, b_{m_i}=d_{m_i}, f_{m_i}=h_{m_i}$$

(1)

Derived from $M_{x_i}=0$ on the edge x_i = 0, we obtain the expression:

$${\displaystyle \sum _{m_i}\left[\frac{{\epsilon }_{m_i}}{2}{D}_{11}\left(e_{n_i}\cos m_i\pi -a_{n_i}\right)-{\alpha }_{m_i}{\phi }_{m_i{n}_i}{D}_{11}+\frac{{\epsilon }_{m_i}}{2}\left(f_{m_i}\cos m_i\pi -b_{m_i}\right)D_{12}-{\beta }_{n_i}{\psi }_{m_i{n}_i}{D}_{12}\right]=0}$$

(2)

Derived from $M_{x_i}=0$ on the edge x_i = a_i, we obtain the expression:

$${\displaystyle \sum _{m_i}\left[\frac{{\epsilon }_{m_i}}{2}{D}_{11}\left(e_{n_i}\cos m_i\pi -a_{n_i}\right)-{\alpha }_{m_i}{\phi }_{m_i{n}_i}{D}_{11}+\frac{{\epsilon }_{m_i}}{2}\left(f_{m_i}\cos m_i\pi -b_{m_i}\right)D_{12}-{\beta }_{n_i}{\psi }_{m_i{n}_i}{D}_{12}\right]\cos {\alpha }_{m_i}{a}_i=0}$$

(3)

Derived from $M_{y_i}=0$ on the edge y_i = 0, we obtain the expression:

$${\displaystyle \sum _{n_i}\left[\frac{{\epsilon }_{m_i}}{2}{D}_{12}\left(e_{n_i}\cos m_i\pi -a_{n_i}\right)-{\alpha }_{m_i}{\phi }_{m_i{n}_i}{D}_{12}+\frac{{\epsilon }_{m_i}}{2}\left(f_{m_i}\cos m_i\pi -b_{m_i}\right)D_{22}-{\beta }_{n_i}{\psi }_{m_i{n}_i}{D}_{22}\right]=0}$$

(4)

Derived from $M_{y_i}=0$ on the edge y_i = b_i, we obtain the expression:

$${\displaystyle \sum _{n_i}\left[\frac{{\epsilon }_{m_i}}{2}{D}_{12}\left(e_{n_i}\cos m_i\pi -a_{n_i}\right)-{\alpha }_{m_i}{\phi }_{m_i{n}_i}{D}_{12}+\frac{{\epsilon }_{m_i}}{2}\left(f_{m_i}\cos m_i\pi -b_{m_i}\right)D_{22}-{\beta }_{n_i}{\psi }_{m_i{n}_i}{D}_{22}\right]\cos {\beta }_{n_i}{b}_i=0}$$

Substituting the unfolded Fourier form of ${\mathrm{\Phi }}_{x_i}$, ${\mathrm{\Phi }}_{y_i}$, and w into the governing differential equations, the expressions can be written as:

$$\begin{array}{l}\frac{{\alpha }_{m_1}{\epsilon }_{m_1}}{2}{D}_{11}{a}_{n_1}+\frac{{\alpha }_{m_1}{\epsilon }_{n_1}}{2}{D}_{12}{b}_{m_1}-\frac{{\alpha }_{m_1}{\epsilon }_{m_1}}{2}{D}_{11}\cos m_1\pi e_{n_1}-\frac{{\alpha }_{m_1}{\epsilon }_{n_1}}{2}{D}_{12}\cos n_1\pi f_{m_1}\\ +\left(D_{11}{\alpha }_{m_1}^2+D_{66}{\beta }_{n_1}^2+C_{11}\right){\phi }_{m_1{n}_1}+\left(D_{12}+D_{66}\right){\alpha }_{m_1}{\beta }_{n_1}{\psi }_{m_1{n}_1}+{\alpha }_{m_1}{C}_{11}{w}_{m_1{n}_1}=0\end{array}$$

$$\begin{array}{l}\frac{{\beta }_{n_1}{\epsilon }_{m_1}}{2}{D}_{12}{a}_{n_1}+\frac{{\beta }_{n_1}{\epsilon }_{n_1}}{2}{D}_{22}{b}_{m_1}-\left(\frac{{\beta }_{n_1}{\epsilon }_{m_1}}{2}{D}_{12}\cos m_1\pi \right)e_{n_1}-\left(\frac{{\beta }_{n_1}{\epsilon }_{n_1}}{2}{D}_{22}\cos n_1\pi \right)f_{m_1}\\ +{\alpha }_{m_1}{\beta }_{n_1}\left(D_{12}+D_{66}\right){\phi }_{m_1{n}_1}+\left({\beta }_{n_1}^2{D}_{22}+{\alpha }_{m_1}^2{D}_{66}+C_{22}\right){\psi }_{m_1{n}_1}+{\beta }_{n_1}{C}_{22}{w}_{m_1{n}_1}=0\end{array}$$

$$\begin{array}{l}{\alpha }_{m_1}{C}_{11}{\phi }_{m_1{n}_1}+{\beta }_{n_1}{C}_{22}{\psi }_{m_1{n}_1}+\left({\alpha }_{m_1}^2{C}_{11}+{\beta }_{n_1}^2{C}_{22}\right)w_{m_1{n}_1}+F_{m_1{n}_1}=q_{m_1{n}_1}\end{array}$$

$$\begin{array}{l}\frac{{\alpha }_{m_2}{\epsilon }_{m_2}}{2}{D}_{11}{a}_{n_2}+\frac{{\alpha }_{m_2}{\epsilon }_{n_2}}{2}{D}_{12}{b}_{m_2}-\frac{{\alpha }_{m_2}{\epsilon }_{m_2}}{2}{D}_{11}\cos m_2\pi e_{n_2}-\frac{{\alpha }_{m_2}{\epsilon }_{n_2}}{2}{D}_{12}\cos n_2\pi f_{m_2}\\ +\left(D_{11}{\alpha }_{m_2}^2+D_{66}{\beta }_{n_2}^2+C_{11}\right){\phi }_{m_2{n}_2}+\left(D_{12}+D_{66}\right){\alpha }_{m_2}{\beta }_{n_2}{\psi }_{m_2{n}_2}+{\alpha }_{m_2}{C}_{11}{w}_{m_2{n}_2}=0\end{array}$$

$$\begin{array}{l}\frac{{\beta }_{n_2}{\epsilon }_{m_2}}{2}{D}_{12}{a}_{n_2}+\frac{{\beta }_{n_2}{\epsilon }_{n_2}}{2}{D}_{22}{b}_{m_2}-\left(\frac{{\beta }_{n_2}{\epsilon }_{m_2}}{2}{D}_{12}\cos m_2\pi \right)e_{n_2}-\left(\frac{{\beta }_{n_2}{\epsilon }_{n_2}}{2}{D}_{22}\cos n_2\pi \right)f_{m_2}\\ +{\alpha }_{m_2}{\beta }_{n_2}\left(D_{12}+D_{66}\right){\phi }_{m_2{n}_2}+\left({\beta }_{n_2}^2{D}_{22}+{\alpha }_{m_2}^2{D}_{66}+C_{22}\right){\psi }_{m_2{n}_2}+{\beta }_{n_2}{C}_{22}{w}_{m_2{n}_2}=0\end{array}$$

$$\begin{array}{l}{\alpha }_{m_2}{C}_{11}{\phi }_{m_2{n}_2}+{\beta }_{n_2}{C}_{22}{\psi }_{m_2{n}_2}+\left({\alpha }_{m_2}^2{C}_{11}+{\beta }_{n_2}^2{C}_{22}\right)w_{m_2{n}_2}+Q_{m_2{n}_2}=F_{m_2{n}_2}\end{array}$$

Considering the same displacement of the upper and lower plates at the contacting position, thus, the deflection equations of the upper and lower plates could be expanded into a double cosine series as:

$$\begin{array}{l}{w}_i(x_i,y_i)={\displaystyle \sum _{m_i=0}^{\infty }{\displaystyle \sum _{n_i=0}^{\infty }{\lambda }_{m_i{n}_i}{w}_i{}_{m_i{n}_i}\cos \frac{m_i\pi x_i}{a_i}\cos \frac{n_i\pi y_i}{b_i}}}\\ w_i{}_{m_i{n}_i}=\frac{4}{a_i{b}_i}{\displaystyle {\int }_0^{b_i}{\displaystyle {\int }_0^{a_i}{w}_i(x_i,y_i)}\cos \frac{m_i\pi x_i}{a_i}\cos \frac{n_i\pi y_i}{b_i}d{x}_{i}d{y}_i}\\ w_2{}_{m_1{n}_1}=\frac{4}{a_1{b}_1}{\displaystyle {\int }_0^{b_1}{\displaystyle {\int }_0^{a_1}{W}_2(x_2,y_2)}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}d{x}_{1}d{y}_1}\\ =\frac{4}{a_1{b}_1}{\displaystyle {\int }_0^{b_1}{\displaystyle {\int }_0^{a_1}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}{w}_2{}_{m_2{n}_2}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}}}}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}d{x}_{1}d{y}_1}\\ =\frac{4}{a_1{b}_1}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}{w}_2{}_{m_2{n}_2}}}{\displaystyle {\int }_0^{b_1}{\displaystyle {\int }_0^{a_1}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}}\cos \frac{m_2\pi x_2}{a_2}\cos \frac{n_2\pi y_2}{b_2}d{x}_{1}d{y}_1}\\ =\frac{4}{a_1{b}_1}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}{w}_2{}_{m_2{n}_2}}}{\displaystyle {\int }_0^{b_1}{\displaystyle {\int }_0^{a_1}\cos \frac{m_1\pi x_1}{a_1}\cos \frac{n_1\pi y_1}{b_1}}\cos \frac{m_2\pi (x_1+x_0)}{a_2}\cos \frac{n_2\pi (y_1+y_0)}{b_2}d{x}_{1}d{y}_1}\\ =\frac{4}{a_2{b}_2}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}{w}_{m_2{n}_2}}}D\\ (m_i=0,1,2,\cdot \cdot \cdot ; n_i=0,1,2,\cdot \cdot \cdot )\end{array}$$

The deformation coordination equation between the upper and the lower plates is expressed as:

$$w_1{}_{m_1{n}_1}=\frac{4}{a_1{b}_1}{\displaystyle \sum _{m_2=0}^{\infty }{\displaystyle \sum _{n_2=0}^{\infty }{\lambda }_{m_2{n}_2}{w}_2{}_{m_2{n}_2}}}D$$

The deformation coordination equation between the lower plate and the foundation is given as:

$$\frac{1}{2{\pi }^2{a}_2{b}_2}\frac{(\lambda +2\mu )}{(\lambda +\mu )\mu }{\displaystyle \sum _{p=0}^{\infty }{\displaystyle \sum _{q=0}^{\infty }{Q}_{pq}{\lambda }_{pq}{\eta }_{pq{m}_2{n}_2}}}{\lambda }_{m_2{n}_2}=w_{m_2{n}_2}$$

Using the same method in Section 2.1, the undetermined coefficients $a_{n_i}$, $b_{m_1}$, $e_{n_1}$, $f_{m_1}$, ${\phi }_{m_i{n}_i}$, ${\psi }_{m_i{n}_i}$, $w_{m_i{n}_i}$, $F_{m_i{n}_i}$, and $Q_{m_2{n}_2}$ could be simultaneously solved. Substituting the solved coefficients into related formulas, the subgrade reaction, deflection, and internal force of the plate could be obtained.

2.2.3. Example

Case 3: Recalculating case 1 in Section 2.1.3. According to the theory mentioned in this section, the results are given in Figure 4. Through the comparison of Figure 4 and Figure 2, it can be seen that the subgrade reaction, deflection, and internal force of the plate are basically the same.

Meanwhile, the analytical solution in this paper is consistent with the results in [28], as shown in

Table 4. Comparison of calculation results of thick plate.

	In This Paper	Thin Plate Theory	[28]
Maximum deflection (m)	0.0107	0.0107	0.0107
Maximum bending moment (kN·m)	35.551	35.558	35.558

. This comparison proves the effectiveness of the theory proposed in this paper.

Case 4: The recalculated results of case 2 in Section 2.1.3 also indicate that the influence of plate theory on the internal force and deflection of the plate is relatively small. Due to the limited length of this article, the specific curve is no longer listed.

When the stepped rectangular plate is simultaneously analyzed using both moderately thick plate theory and thin plate theory, the same method (as shown in Section 2.1 and Section 2.2) could be used to obtain the internal force and deflection of the stepped rectangular plate.

3. Discussion

3.1. Effect of Elastic Modulus on the Deflection of Plate

The dimensions, Poisson ratios, and elastic modulus of the rectangular stepped plate and foundation are given in

Table 5. Dimensions and properties of plates with different elastic modulus.

Component Name	Side Length (m)	Thickness (m)	Poisson Ratio	Elastic Modulus (MPa)
Upper plate	2.0	0.1	0.167	Variable
Lower plate	4.0	0.3	0.167	34,300
Foundation	-	-	0.4	343

. The vertical uniform load value is 0.98 MPa. The deflection curve of the center line of the plate is shown in Figure 5, in which the number of the curves 1, 2, 3, and 4 indicate that the elastic modulus of the upper plate is 34,300 MPa, 343,000 MPa, 686,000 MPa, and 1,029,000 MPa, respectively.

Through the analysis of Figure 5, it can be seen that the variation trend of the deflection curve of the lower plate is the same as that of Figure 3a. The deflection value of the lower plate is largest when the elastic modulus of the upper and lower plates is the same. The deflection along the center line of the plate decreases with the increase of the elastic modulus of the upper plate, while the deflection at the edge of the plate is just the opposite.

3.2. Influence of Plate Theory on Calculation Results of Plate Deflection

The dimensions, Poisson ratios, and elastic modulus of the rectangular stepped plate and foundation are given in

Table 6. Dimensions and properties of plates with different thickness.

Component Name	Side Length (m)	Thickness (m)	Poisson Ratio	Elastic Modulus (MPa)
Upper plate	2.0	0.2	0.167	34,300
Lower plate	2.0	Variable	0.167	34,300
Foundation	-	-	0.4	343

. The vertical uniform load value is 0.98 MPa, and the thickness of the lower plate is 0.2 m, 0.3 m, 0.4 m, 0.5 m, 0.6 m, 0.7 m, 0.8 m, 0.9 m, and 1.0 m, respectively. The deflection of the center of the lower plate is given in

Table 7. Deflection at the center of the lower plate.

Thickness of Lower Plate (m)	w1 (m)	w2 (m)
0.2	0.0053	0.0052
0.3	0.0042	0.0041
0.4	0.0035	0.0034
0.5	0.0030	0.0030
0.6	0.0027	0.0027
0.7	0.0025	0.0025
0.8	0.0024	0.0024
0.9	0.0023	0.0023
1.0	0.0022	0.0023

, in which $w_1$ and $w_2$ are the deflection values calculated by thin plate theory and moderately thick plate theory, respectively.

Through the calculation results in Table 7, it can be obtained that when the thickness of the upper plate is constant, the maximum deflection of the lower plate decreases with the increase in the thickness. Meanwhile, the comparison of w₁ and w₂ shows that the plate theory has little influence on the analytical solution of the maximum deflection of the plate.

3.3. Influence of Side Length of Upper and Lower Plates on Calculation Results of Plate Deflection

The dimensions of the stepped rectangular plates are given in

Table 8. Dimensions and properties of plates with different side length.

Component Name	Side Length (m)	Thickness (m)	Poisson Ratio	Elastic Modulus (MPa)
Upper plate	Variable	0.2	0.167	34,300
Lower plate	4.0	0.3	0.167	34,300
Foundation	-	-	0.4	343

. The vertical uniform load value is 0.98 MPa. Calculation results can be seen in Figure 6, in which 1, 2, 3, and 4 indicate the side lengths of the upper plate are 3.0 m, 2.0 m, 1.0 m, and 0.5 m, respectively.

It can be seen from Figure 6 that the deflection of the center of the plate increases as the size of the upper plate increases. It can also be found that when the coordinates of x are between 0–1 or 3–4, the deflection growth rate is relatively fast, while the growth rate of deflection is slow when 2 < x < 4, indicating that the existence of the upper plate increases the stiffness of the lower plate.

4. Conclusions

This paper presents a new solving method to obtain the bending moment and deflection of the stepped rectangular plate using traditional thin plate theory and moderately thick plate theory. The stepped rectangular plate is divided into an upper plate and lower plate, and the analytical solution could be obtained through the differential equation and boundary conditions of the plate-foundation system. Several conclusions can be drawn as follows.

(1)

The analytical solution is basically the same as those in the existing literature and the simulation results using ABAQUS software, indicating that the traditional plate theory could be used to analyze the bending property of stepped rectangular plate. The analysis results also demonstrate that the analytical solution of the stepped rectangular plate is not related to the plate theory adopted.

(2)

The increase in the elastic modulus of the upper plate can effectively reduce the deflection at the center of the plate and slightly increase the deflection at the edge of the plate, showing that the increase in the elastic modulus of the upper plate could effectively improve the stiffness of the stepped rectangular plate.

(3)

The thickness and maximum deflection of the plate are negatively related when the thin plate theory or moderately thick plate theory is used for analysis, and the analytical solution obtained by using these two types of plate theory is basically the same.

(4)

The greater the difference in the side length of the upper and lower plates, the greater the deflection of the stepped rectangular plate. The increase in the deflection at the edge of the plate is not as significant as that at the center of the plate.

Due to the limitations of time, level, and vision, the research on the topics of this paper is not yet complete. In the future, more in-depth research can be conducted regarding the following aspects:

(1)

The interaction between the plate and the elastic half-space foundation is discussed in this paper, but the characteristics of the stepped rectangular plate resting on a two-parameter foundation should be analyzed in the future.

(2)

The bending performance of the plate resting on an elastic half-space foundation under static load is analyzed in this paper, while the dynamic characteristics of the plate should be studied in future research.

(3)

The characteristics of the stepped rectangular plate embedded in the foundation should be studied in follow-up research.