New Accurate Flexural Analysis for Different Types of Plates in a Rectangular Sewage Tank by Utilizing a Unified Analytic Solution Procedure

Abstract

In the present paper, a modified Fourier series approach is developed for new precise flexural analysis of three different types of concrete plates in a rectangular sewage tank. The bending problems of the bottom plate, side-plate, and the fluid-guiding plate are not easily solved via using the traditional analytic approaches. Based on the Fourier series theory, the present approach provides a unified semi-inverse solving procedure for the above plates by means of choosing three different kinds of Fourier series as the trial functions. Although all the trial functions are quite similar to the classical Navier-form solution, new, precise analytic flexural solutions for plates without Navier-type edge conditions (all edges simply-supported) are achieved, which is mainly attributed to employing the Stoke’s transform technique. For each case, the plate-bending problems are finally altered to deal with linear algebra equations. Furthermore, owing to the orthogonality and completeness of the Fourier series, the obtained solutions perfectly satisfy both the edge conditions and the governing partial differential equation of plates, which paves an easily implemented and rational way for engineers and researchers to provide new, exact designs of plate structures. The main contribution of this study lies in the provision of a unified solution procedure for addressing complex plate-bending problems across diverse boundary conditions. By employing a range of Fourier series types, this approach offers a comprehensive solution framework that accommodates the complexities inherent in plate analysis. The correctness of the present analytic solutions is verified against precise finite element method (FEM) results and ones available in the literature. Finally, the influences of foundation, edge conditions, and aspect ratio on flexural behaviors of plates are discussed in detail.

1. Introduction

Plate structures are being utilized increasingly in various engineering fields due to their key role as a load bearing component [1,2,3]. In this study, a rectangular sewage tank is considered, which is widely used in municipal engineering. Such a type of structure is usually composed by a rectangular bottom plate, side plate. and the fluid-guiding plate. Flexural investigation on these plates is of much importance since it is closely related to the cracking behaviors of sewage tanks. Aiming to provide a more reasonable structural design and avoid cracks for sewage tanks, numerous efforts have been devoted for conducting precise flexural analysis of plates in rectangular tanks. According to the available literature, numerical/approximate methods and analytic methods are reported as powerful tools for the flexural analysis of plates. There exists a substantial literature study dedicated to exploring the dynamic behavior of related structures [4,5,6]. However, accurate flexural solutions, whether from analytic results or numerical/approximate ones, are far from complete.

The bending analysis of plates has witnessed significant advancements in recent years. Some representative numerical or approximate solutions are reviewed here for a brief knowledge of the advancements in the field. Notably, Liu et al. [7] simulated the bending problems of thin plates by developing an effective superposition of the homogenization functions method. The authors presented six numerical experiments to test the convergence, accuracy, and efficiency of the proposed method. Qu et al. [8] developed a modified localized method of fundamental solutions for the flexural problems of simply supported or clamped plates, and all the given approximate results agreed well with solutions given by using the meshless collocation method. Zhao et al. [9] developed a moving least-squares-based numerical manifold approach for handling the bending problems of plates with various edge restraints and resting on Winkler-type or Pasternak-type foundations. Based on a size-dependent plate theory, Li et al. [10] derived the nonlinear axisymmetric bending governing equation for the circular strain gradient plates via utilizing the principle of minimum potential energy. The authors provided new, precise numerical data for the bending problem of fully clamped plates and revealed how the strain gradient affects the flexural behaviors of plates. Via combing the FEM and the variational differential quadrature method, Ansari et al. [11] proposed an advanced numerical approach to solve the nonlinear flexural problem for arbitrary-shaped porous nanocomposite plates subjected to various edge conditions. Bhaskar et al. [12] adopted an effective FEM for a new bending investigation of simply supported sandwich plates on the basis of an inverse trigonometric shear deformation theory, and several new findings were presented for the stresses and displacements of plates. Yu [13] obtained new approximate bending solutions for simply supported/clamped variable-thickness plates lying on a nonlinear elastic foundation via developing a brand-new unified Wavelet homotopy method, and all the obtained results revealed that thickness discrepancies have limiting influence when the plate is in the large bending stage. Huang et al. [14] provided new approximate bending results for irregular Kirchhoff plates under mixed edge restraints through combing novel deep learning and the finite difference approximation, and the precision and the effectiveness of the approach were comprehensively confirmed by the given numerical examples. Ye et al. [15] developed a new numerical solution procedure for predicting the bending response of functionally graded material plates by utilizing an advanced scaled boundary finite element method. By good agreement with the available analytical and numerical solutions, the accuracy and efficiency of the solution was well illustrated. Thai et al. [16] presented a three-dimensional numerical approach to analyze the bending and free vibration behavior of multi-directional functionally graded plates under the effect of thermal environment. Some other relevant plates were also studied to investigate their mechanical behavior [17,18,19]. As such, the vibration problem of such a structure has been thoroughly examined in the literature [20,21,22,23]. However, numerical approaches also have some shortcomings regarding providing approximate solutions.

On the other hand, many efforts have been devoted to exploring analytic approaches for validating numerical/approximated approaches and fast parameter analysis values in practical engineering applications. Several representative and accurate analytic methods/solutions can be found for plate-bending problems in the literature. For example, within the framework of the classical semi-inverse solution procedure, Tash et al. [24] analyzed the bending response of simply supported isotropic thick plates by using displacement potential functions. Magnucka-Blandzi [25] derived close-form solutions for the flexural and stability problems of circular plates under simply supported edge conditions with symmetrically varying mechanical properties. Magnucki et al. [26] proposed a generalized analytical model for the bending analysis of rectangular sandwich plates with metal foam cores, and new, exact analytic data for the critical loads and maximum deflection of plates were presented for the better design of sandwich structures. Li et al. [27] successfully predicted the bending creep behaviors of Navier-type layered thick plates subjected to time-varying loads via using the Laplace transform technique and series expansions. Chen et al. [28] employed the Rayleigh–Ritz formulation to the thermomechanical buckling of variable angle tow composite plates with general in-plane boundary constraint. Based on a four-variable plate theory, Demirhan et al. [29] deduced the governing equations for flexural and vibration problems of porous, functionally graded material thick plates via using the state space approach. Thereafter, the authors provided new analytic results for porous plates under two opposite simply supported edges (Lévy-type boundaries) using the Lévy solution technique. Some other analytical methods have also been developed for plate-related structures such as bridges [30,31,32].

Through the above-mentioned investigations, it is easily seen that current analytic flexural solutions are primarily restricted to plates with simple boundaries. However, in most engineering applications, non-Lévy-type plates are the most commonly encountered, for example, the three different types of plates in a sewage tank. The shortage of exact analytical flexural solutions for such kinds of boundaries considerably narrows the application scope of concrete plates. For this reason, several representative analytic approaches for plates under more complex boundaries have been proposed. For instance, the generalized integral transform method [33,34,35] was developed for solving the static/dynamic problem of fully clamped plates, in which beam functions are selected as integral kernels to construct integral transformation pairs. He et al. [36] used the generalized integral transform method to analyze the bending analysis of rectangular orthotropic plates with rotationally restrained and free edges. Similarly, Zhang [37,38,39,40,41] provided a finite Fourier integral transform solution procedure for the mechanical problems of plates subjected to various non-Lévy-type boundaries, in which Fourier series was selected as the integral kernel for constructing integral pairs on the basis of the principle of integral transformation. Meng et al. [42] estimated the state of nonlinear generalized systems subject to algebraic constraints by generalized inverse technique. Li et al. [43,44,45,46,47,48] proposed a new analytic approach, which is the combination of a symplectic geometry approach and the superposition approach, to handle non-Lévy-type thin/thick plate problems. In proposing a novel analytic approach, it should be pointed out that although the above approaches can offer precise analytic data for flexural problems of non-Lévy-type plates, the vast majority of these methods are only suitable for some specific non-Lévy-type boundaries. Consequently, based on the authors’ knowledge, there are still very few available flexural solutions for the three types of plates in a rectangular tank. Therefore, the exploration of precise analytic solutions for such plates remains imperative for researchers. As a result, more analytic methods have been introduced to solve complex engineering problems [49,50,51].

Recently, a modified 2D Fourier series approach [52,53] was developed for accurate mechanical analysis of plates under classical/non-classical edge conditions through combining the Stoke’s transformation and the Navier solution technique. The main advantage of this approach is that it can overcome the boundary-continuous problems that usually cannot be avoided by other semi-inverse methods. In this solution procedure, a double-sine series is adopted for plate deflection. Actually, this approach could be extended to deal with flexural problems of plates subjected to other non-Lévy boundaries via selecting other types of Fourier series. Therefore, it is of great importance to extent this approach for processing mechanical problems of non-Lévy-type plates in rectangular sewage tanks.

In this study, we developed a modified 2D Fourier series approach for solving the flexural problems of the three types of plates in sewage tanks. In the present procedure, three different types of Fourier series were adopted for the deflection of the bottom plate, the side plate, and the fluid-guiding plate, respectively. These three types of Fourier series have the feature of automatically matching part of the boundaries of plates; for example, the double-sine form solution automatically satisfies the zero-deflection requirements at each edge of the bottom plate. Through the application of the Stoke’s transformation on the three types of Fourier series, we obtained expressions for the derivatives of plate’s deflection with several unknown constants. The unknown constants were obtained after allowing the gained solution to satisfy the remain part of the boundaries of the plates. Compared with other analytical methods, the present method is simpler and more reasonable. Finally, the efficiency and accuracy of this approach are demonstrated by comparing the results with available analytical solutions and FEM data provided by ABAQUS.

2. Solution Procedure for Concrete Thin Plates with Various Edge Restraints

Consider a rectangular sewage tank resting on an elastic foundation; it usually contains three types of plates, i.e., the bottom plate, the side plate, and the fluid-guiding plate, as illustrated in Figure 1. Such types of plates are usually made of concrete and treated as thin rectangular plates.

Based on the small-deflection thin-plate theory, the general governing PDE to describe the flexural behavior of plates in a rectangular sewage tank is as follows:

$${\nabla }^4{W}_i\left(x_i,y_i\right)+\frac{K_i}{D_i}\cdot W_i\left(x_i,y_i\right)=\frac{q_i\left(x_i,y_i\right)}{D_i}$$

(1)

where $W_i(x_i,y_i)$ is the deflection for plates in the tank; the subscript i represents the type of the plate, for example, $W_1(x_1,y_1)$ is for the bottom plate, $W_2(x_2,y_2)$ is for the side plate, and $W_3\left(x_3,y_3\right)$ is for the fluid-guiding plate. Thus, $D_i=E_i{h}^3/12\left(1-{\mu }_i^2\right)$, ${\mu }_i$, $K_i$, and $q_i\left(x_i,y_i\right)$ are the rigidity, Poisson’s ratio, stiffness of the Winkler foundation, and external loads for different plates, respectively. According to the practical engineering, the thickness of the above different plates is uniformly to be h. $a_i$ and $b_i$ represent the length and width of different plates, the range of which is in different domains, $x_i{o}_i{y}_i$, respectively, as shown in Figure 2. It is noted that only the bottom rests on the foundation, which leads to $K_2$ and $K_3$ being equal to zero for the side plate and the fluid-guiding plate, respectively. As for the internal forces in the above plates, the formula can be given as described by Equation (2) using plate’s deflection and the above elastic parameters.

$$\begin{array}{l}{M}_{x_i}=-D_i\left[\frac{{\partial }^2{W}_i\left(x_i,y_i\right)}{\partial x_i^2}+{\mu }_i\frac{{\partial }^2{W}_i\left(x_i,y_i\right)}{\partial y_i^2}\right]\\ M_{y_i}=-D_i\left[\frac{{\partial }^2{W}_i\left(x_i,y_i\right)}{\partial y_i^2}+{\mu }_i\frac{{\partial }^2{W}_i\left(x_i,y_i\right)}{\partial x_i^2}\right]\\ M_{x_i{y}_i}=-D_i\left(1-{\mu }_i\right)\frac{{\partial }^2{W}_i\left(x_i,y_i\right)}{\partial x_i\partial y_i}\\ V_{x_i}=-D_i\left[\frac{{\partial }^3{W}_i\left(x_i,y_i\right)}{\partial x_i^3}+\left(2-{\mu }_i\right)\frac{{\partial }^3{W}_i\left(x_i,y_i\right)}{\partial x_i\partial y_i^2}\right]\\ V_{y_i}=-D_i\left[\frac{{\partial }^3{W}_i\left(x_i,y_i\right)}{\partial y_i^3}+\left(2-{\mu }_i\right)\frac{{\partial }^3{W}_i\left(x_i,y_i\right)}{\partial x_i^2\partial y_i}\right]\end{array}$$

(2)

2.1. General Bending Solution Procedure for the Bottom Plate of Rectangular Sewage Tanks

With the consideration that the tank is lying on an elastic foundation in practical engineering, the bottom plate, as presented in Figure 2, is treated as a fully clamped plate resting on a Winkler foundation. Consequently, the expressions for each edge can be described as follows:

$${\left.W_1\right|}_{x_1=0}={\left.W_1\right|}_{x_1=a_1}=0,{\left.W_1\right|}_{y_1=0}={\left.W_1\right|}_{y_1=b_1}=0$$

(3a)

$${\left.\frac{\partial W_1}{\partial x_1}\right|}_{x_1=0}={\left.\frac{\partial W_1}{\partial x_1}\right|}_{x_1=a_1}=0,{\left.\frac{\partial W_1}{\partial y_1}\right|}_{y_1=0}={\left.\frac{\partial W_1}{\partial y_1}\right|}_{y_1=b_1}=0$$

(3b)

Figure 3 shows the procedure and sequence of calculations for the bottom plate. A similar procedure is followed for the other plates. Based on the Fourier series theory [54], we can treat the double-sine series as the trial function for the deflection $W_1\left(x_1,y_1\right)$ of the bottom plate when the plate’s deflection at each edge is zero, which is given as follows:

$$W_1(x_1,y_1)={\displaystyle \sum _{m=1}{\displaystyle \sum _{n=1}{W}_{1mn}}}\sin \left({\varphi }_{1m}{x}_1\right)\sin \left({\phi }_{1n}{y}_1\right)$$

(4)

where ${\varphi }_{1m}=\frac{m\pi }{a_1}$, and ${\phi }_{1n}=\frac{n\pi }{b_1}$; $W_{1mn}=\frac{4}{a_1{b}_1}{\displaystyle {\int }_0^{a_1}{\displaystyle {\int }_0^{b_1}{W}_1(x_1,y_1)}}\sin \left({\varphi }_{1m}{x}_1\right)\sin \left({\phi }_{1n}{y}_1\right)\mathrm{d}{x}_1\mathrm{d}{y}_1$ is the Fourier coefficient of the bottom plate’s deflection.

From Equation (4), incorporating the Stoke’s transform technique to obtain term-by-term differentiation of Fourier series form solutions, we obtain the following:

$$\begin{array}{l}\frac{\partial W_1}{\partial x_1}={\displaystyle \sum _{m=0}^{+\infty }{\displaystyle \sum _{n=1}^{+\infty }\left\{\frac{{\epsilon }_m}{a_1}\left[{\left(-1\right)}^m{J}_{2n}-J_{1n}\right]+{\varphi }_{1m}{W}_{1mn}\right\}\cos \left({\varphi }_{1m}{x}_1\right)\sin \left({\phi }_{1n}{y}_1\right)}}\\ \frac{\partial W_1}{\partial y_1}={\displaystyle \sum _{m=0}^{+\infty }{\displaystyle \sum _{n=1}^{+\infty }\left\{\frac{{\epsilon }_n}{b_1}\left[{\left(-1\right)}^n{L}_{2m}-L_{1m}\right]+{\phi }_{1n}{W}_{1mn}\right\}\sin \left({\varphi }_{1m}{x}_1\right)\cos \left({\phi }_{1n}{y}_1\right)}}\end{array}$$

(5a)

$$\begin{array}{l}\frac{{\partial }^2{W}_1}{\partial x_1^2}={\displaystyle \sum _{m=1}{\displaystyle \sum _{n=1}\left\{-\frac{2{\varphi }_{1m}}{a_1}\left[{\left(-1\right)}^m{J}_{2n}-J_{1n}\right]-{\varphi }_{1m}^2{W}_{1mn}\right\}}}\sin \left({\varphi }_{1m}{x}_1\right)\sin \left({\phi }_{1n}{y}_1\right)\\ \frac{{\partial }^2{W}_1}{\partial y_1^2}={\displaystyle \sum _{m=1}{\displaystyle \sum _{n=1}\left\{-\frac{2{\phi }_{1n}}{b_1}\left[{\left(-1\right)}^n{L}_{2m}-L_{1m}\right]-{\phi }_{1n}^2{W}_{1mn}\right\}}}\sin \left({\varphi }_{1m}{x}_1\right)\sin \left({\phi }_{1n}{y}_1\right)\end{array}$$

(5b)

$$\begin{array}{l}\frac{{\partial }^4{W}_1}{\partial x_1^4}={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=1}^{+\infty }\left\{-\frac{2}{a_1}{\varphi }_{1m}\left\{\begin{array}{l}{\left(-1\right)}^m{J}_{4n}-J_{3n}\\ -{\varphi }_{1m}^2\left[{\left(-1\right)}^m{J}_{2n}-J_{1n}\right]\end{array}\right\}+{\varphi }_{1m}^4{W}_{1mn}\right\}}}\sin \left({\varphi }_{1m}{x}_1\right)\sin \left({\phi }_{1n}{y}_1\right)\\ \frac{{\partial }^4{W}_1}{\partial y_1^4}={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=1}^{+\infty }\left\{-\frac{2}{b_1}{\phi }_{1n}\left\{\begin{array}{l}{\left(-1\right)}^n{L}_{4m}-L_{3m}\\ -{\phi }_{1n}^2\left[{\left(-1\right)}^n{L}_{2m}-L_{1m}\right]\end{array}\right\}+{\phi }_{1n}^4{W}_{1mn}\right\}}}\sin \left({\varphi }_{1m}{x}_1\right)\sin \left({\phi }_{1n}{y}_1\right)\\ \frac{{\partial }^4{W}_1}{\partial x_1^2\partial y_1^2}={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=1}^{+\infty }\left\{\begin{array}{l}-\frac{2}{b_1}{\phi }_{1n}\left[{\left(-1\right)}^n{L}_{6m}-L_{5m}\right]\\ +\frac{2}{a_1}{\varphi }_{1m}{\phi }_{1n}^2\left[{\left(-1\right)}^m{J}_{2n}-J_{1n}\right]+{\varphi }_{1m}^2{\phi }_{1n}^2{W}_{1mn}\end{array}\right\}}}\sin \left({\varphi }_{1m}{x}_1\right)\sin \left({\phi }_{1n}{y}_1\right)\end{array}$$

(5c)

It is noted that we can derive expressions for any other partial differential derivatives of deflection $W_1\left(x_1,y_1\right)$. For simplification, we only demonstrate formulas for $\partial W_1/\partial x_1$, $\partial W_1/\partial y_1$ and the 4th-order differential terms in Equation (1). In Equations (5a)–(5c), the unknown constants are definite integrals, which are illustrated as follows:

$$\begin{array}{l}{J}_{1n}=\frac{2}{b_1}{\displaystyle {\int }_0^{b_1}{W}_1{|}_{x_1=0}}\sin \left({\phi }_{1n}{y}_1\right)\mathrm{d}{y}_1 J_{2n}=\frac{2}{b_1}{\displaystyle {\int }_0^{b_1}{W}_1{|}_{x_1=a_1}}\sin \left({\phi }_{1n}{y}_1\right)\mathrm{d}{y}_1\\ J_{3n}=\frac{2}{b_1}{\displaystyle {\int }_0^{b_1}{\left.\frac{{\partial }^2{W}_1}{\partial x_1^2}\right|}_{x_1=0}}\sin \left({\phi }_{1n}{y}_1\right)\mathrm{d}{y}_1 J_{4n}=\frac{2}{b_1}{\displaystyle {\int }_0^{b_1}{\left.\frac{{\partial }^2{W}_1}{\partial x_1^2}\right|}_{x_1=a_1}}\sin \left({\phi }_{1n}{y}_1\right)\mathrm{d}{y}_1\\ L_{1m}=\frac{2}{a_1}{\displaystyle {\int }_0^{a_1}{W}_1{|}_{y_1=0}\sin \left({\varphi }_{1m}{x}_1\right)}\mathrm{d}{x}_1 L_{2m}=\frac{2}{a_1}{\displaystyle {\int }_0^{a_1}{W}_1{|}_{y_1=b_1}\sin \left({\varphi }_{1m}{x}_1\right)}\mathrm{d}{x}_1\\ L_{3m}=\frac{2}{a_1}{\displaystyle {\int }_0^{a_1}{\left.\frac{{\partial }^2{W}_1}{\partial y_1^2}\right|}_{y_1=0}}\sin \left({\varphi }_{1m}{x}_1\right)\mathrm{d}{x}_1 L_{4m}=\frac{2}{a_1}{\displaystyle {\int }_0^{a_1}{\left.\frac{{\partial }^2{W}_1}{\partial y_1^2}\right|}_{y_1=b_1}}\sin \left({\varphi }_{1m}{x}_1\right)\mathrm{d}{x}_1\\ L_{5m}=\frac{2}{a_1}{\displaystyle {\int }_0^{a_1}{\left.\frac{{\partial }^2{W}_1}{\partial x_1^2}\right|}_{y_1=0}\sin \left({\varphi }_{1m}{x}_1\right)\mathrm{d}{x}_1} L_{6m}=\frac{2}{a_1}{\displaystyle {\int }_0^{a_1}{\left.\frac{{\partial }^2{W}_1}{\partial x_1^2}\right|}_{y_1=b_1}}\sin \left({\varphi }_{1m}{x}_1\right)\mathrm{d}{x}_1\end{array}$$

(6)

Substituting Equation (3a) into the above definite integrals, we can deduce that the following constants are zero:

$$J_{1n}=J_{2n}=L_{1m}=L_{2m}=L_{5m}=L_{6m}=0$$

(7)

Furthermore, the bending moment at each edge can be directly acquired by using constants $J_{3n},J_{4n},L_{3m}$, and $L_{4m}$, which is shown in Equation (8). As for bending moments at other places along the plates, this can be determined by using the formulas in Equations (2) and (5b).

$$\begin{array}{l}{\left.M_{y_1}\right|}_{y_1=0}=-D_1{\left.\frac{{\partial }^2{W}_1}{\partial y_1^2}\right|}_{y_1=0}=-D_1{\displaystyle \sum _{m=1}^{+\infty }{L}_{3m}\sin \left({\varphi }_{1m}{x}_1\right)}\\ {\left.M_{y_1}\right|}_{y_1=b_1}=-D_1{\left.\frac{{\partial }^2{W}_1}{\partial y_1^2}\right|}_{y_1=b_1}=-D_1{\displaystyle \sum _{m=1}^{+\infty }{L}_{4m}\sin \left({\varphi }_{1m}{x}_1\right)}\\ {\left.M_{x_1}\right|}_{x_1=0}=-D_1{\left.\frac{{\partial }^2{W}_1}{\partial x_1^2}\right|}_{x_1=0}=-D_1{\displaystyle \sum _{n=1}^{+\infty }{J}_{3n}\sin \left({\phi }_{1n}{y}_1\right)}\\ {\left.M_{x_1}\right|}_{x_1=a_1}=-D_1{\left.\frac{{\partial }^2{W}_1}{\partial x_1^2}\right|}_{x_1=a_1}=-D_1{\displaystyle \sum _{n=1}^{+\infty }{J}_{4n}\sin \left({\phi }_{1n}{y}_1\right)}\end{array}$$

(8)

Substituting Equations (4) and (5c) into the general governing bending PDE, we obtain the following:

$${\displaystyle \sum _{m=1}{\displaystyle \sum _{n=1}\left\{\begin{array}{l}\left[{\left({\varphi }_{1m}^2+{\phi }_{1n}^2\right)}^2+\frac{K}{D_1}\right]W_{1mn}\\ -\frac{2}{b_1}{\phi }_{1n}\left[{\left(-1\right)}^n{L}_{4m}-L_{3m}\right]\\ -\frac{2}{a_1}{\varphi }_{1m}\left[{\left(-1\right)}^m{J}_{4n}-J_{3n}\right]-\frac{q_{1mm}}{D_1}\end{array}\right\}}}\sin \left({\varphi }_{1m}{x}_1\right)\sin \left({\phi }_{1n}{y}_1\right)=0$$

(9)

where $q_{1mn}=\frac{4}{a_1{b}_1}{\displaystyle {\int }_0^{a_1}{\displaystyle {\int }_0^{b_1}{q}_1(x_1,y_1)}}\sin \left({\varphi }_{1m}{x}_1\right)\sin \left({\phi }_{1n}{y}_1\right)\mathrm{d}{x}_1\mathrm{d}{y}_1$. With the requirement of non-zero bending solutions for Equation (1), we can derive the formula of the coefficient $W_{1mn}$, which can be expressed by $J_{3n},J_{4n},L_{3m}$, and $L_{4m}$ as follows:

$$W_{1mn}=f_{1mn}\left\{\begin{array}{l}\frac{q_{1mm}}{D_1}+\frac{2}{b_1}{\phi }_{1n}\left[{\left(-1\right)}^n{L}_{4m}-L_{3m}\right]\\ +\frac{2}{a_1}{\varphi }_{1m}\left[{\left(-1\right)}^m{J}_{4n}-J_{3n}\right]\end{array}\right\}$$

(10)

where $f_{1mn}=\frac{1}{{\left({\varphi }_{1m}^2+{\phi }_{1n}^2\right)}^2-K_1/D_1}$. Substituting the obtained $W_{1mn}$ into the Equation (4), we finally obtain the explicit solution for the flexural behaviors of the bottom plate resting on an elastic foundation.

$$W_1\left(x_1,y_1\right)={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=1}^{+\infty }{f}_{1mn}\left\{\begin{array}{l}\frac{q_{1mm}}{D_1}+\frac{2}{b_1}{\phi }_{1n}\left[{\left(-1\right)}^n{L}_{4m}-L_{3m}\right]\\ +\frac{2}{a_1}{\varphi }_{1m}\left[{\left(-1\right)}^m{J}_{4n}-J_{3n}\right]\end{array}\right\}}}\sin \left({\varphi }_{1m}{x}_1\right)\sin \left({\phi }_{1n}{y}_1\right)$$

(11)

Equation (9) can match the edge condition in Equation (3a), as indicated above. Substituting the deflection $W_1\left(x_1,y_1\right)$ of Equation (11) into the Equation (5a) and allowing them to be equal to zero makes the obtained $W_1\left(x_1,y_1\right)$ satisfy the remaining edge conditions in Equation (3b). The above procedure produces the following linear algebraic equations, as represented by Equations (12a)–(12d), which can determine solutions for the constants $J_{3n},J_{4n},L_{3m}$, and $L_{4m}$.

$${\displaystyle \sum _{m=1}^{+\infty }{\varphi }_{1m}{f}_{1mn}\left\{\begin{array}{l}\frac{2}{b_1}{\phi }_{1n}\left[{\left(-1\right)}^n{L}_{4m}-L_{3m}\right]\\ +\frac{2}{a_1}{\varphi }_{1m}\left[{\left(-1\right)}^m{J}_{4n}-J_{3n}\right]\end{array}\right\}}=-{\displaystyle \sum _{m=1}^{+\infty }\frac{{\varphi }_{1m}{f}_{1mn}{q}_{1mm}}{D_1}}$$

(12a)

$${\displaystyle \sum _{m=1}^{+\infty }{(-1)}^m{\varphi }_{1m}{f}_{1mn}\left\{\begin{array}{l}\frac{2}{b_1}{\phi }_{1n}\left[{\left(-1\right)}^n{L}_{4m}-L_{3m}\right]\\ +\frac{2}{a_1}{\varphi }_{1m}\left[{\left(-1\right)}^m{J}_{4n}-J_{3n}\right]\end{array}\right\}}=-{\displaystyle \sum _{m=1}^{+\infty }\frac{{(-1)}^m{\varphi }_{1m}{f}_{1mn}{q}_{1mm}}{D_1}}$$

(12b)

for n equal to 1, 2, 3, ….

$${\displaystyle \sum _{n=1}^{+\infty }{\phi }_{1n}{f}_{1mn}\left\{\begin{array}{l}\frac{2}{b_1}{\phi }_{1n}\left[{\left(-1\right)}^n{L}_{4m}-L_{3m}\right]\\ +\frac{2}{a_1}{\varphi }_{1m}\left[{\left(-1\right)}^m{J}_{4n}-J_{3n}\right]\end{array}\right\}}=-{\displaystyle \sum _{n=1}^{+\infty }\frac{{\phi }_{1n}{f}_{1mn}{q}_{1mm}}{D_1}}$$

(12c)

$${\displaystyle \sum _{n=1}^{+\infty }{(-1)}^n{\phi }_{1n}{f}_{1mn}\left\{\begin{array}{l}\frac{2}{b_1}{\phi }_{1n}\left[{\left(-1\right)}^n{L}_{4m}-L_{3m}\right]\\ +\frac{2}{a_1}{\varphi }_{1m}\left[{\left(-1\right)}^m{J}_{4n}-J_{3n}\right]\end{array}\right\}}=-{\displaystyle \sum _{n=1}^{+\infty }\frac{{(-1)}^n{\phi }_{1n}{f}_{1mn}{q}_{1mm}}{D_1}}$$

(12d)

for m equal to 1, 2, 3, ….

2.2. General Bending Solution Procedure for the Side Plate of Rectangular Sewage Tanks

In a rectangular sewage tank, the side plate, as presented in Figure 4, can be regarded as having three edges clamped and one edge free. Thus, the expressions for each edge can be described as follows:

$${\left.W_2\right|}_{x_2=0}={\left.W_2\right|}_{x_2=a_2}={\left.W_2\right|}_{y_2=0}=0, {\left.V_{y_2}\right|}_{y_2=b_2}=0$$

(13a)

$${\left.\frac{\partial W_2}{\partial x_2}\right|}_{x_2=0}={\left.\frac{\partial W_2}{\partial x_2}\right|}_{x_2=a_2}={\left.\frac{\partial W_2}{\partial y_2}\right|}_{y_2=0}=0, {\left.M_{y_2}\right|}_{y_2=b_2}=0$$

(13b)

Along the $x_2$ direction, the deflection for both opposite edges is zero, which leads to use of the sine series form solution. In the $y_2$ direction, the value for ${\left.W_2\right|}_{y_2=0}$ is zero but not for ${\left.W_2\right|}_{y_2=b_2}$, which leads to the use of the half-sinusoidal series form solution. Thus, we choose the double-sine half-sinusoidal series as the trial function for the deflection of the side plate, which is given below:

$$W_2(x_2,y_2)={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }{W}_{2mn}\sin \left({\varphi }_{2m}{x}_2\right)\sin \frac{{\phi }_{2n}{y}_2}{2}}}$$

(14)

where ${\varphi }_{2m}=\frac{m\pi }{a_2}$, and ${\phi }_{2n}=\frac{n\pi }{b_2}$. Performing the Stoke’s transform technique on Equation (14) results in the following expressions:

$$\begin{array}{l}\frac{\partial W_2}{\partial x_2}={\displaystyle \sum _{m=0}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}\frac{{\epsilon }_m}{a_2}\left[{\left(-1\right)}^m{Q}_{2n}-Q_{1n}\right]\\ +{\varphi }_{2m}{W}_{2mn}\end{array}\right\}\cos \left({\varphi }_{2m}{x}_2\right)\sin \frac{{\phi }_{2n}{y}_2}{2}}}\\ \frac{\partial W_2}{\partial y_2}={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1,3,}^{\infty }\left(-\frac{2}{b_2}{R}_{1m}+\frac{{\phi }_{2n}}{2}{W}_{2mn}\right)\sin \left({\varphi }_{2m}{x}_2\right)\cos \frac{{\phi }_{2n}{y}_2}{2}}}\end{array}$$

(15a)

$$\begin{array}{l}\frac{{\partial }^2{W}_2}{\partial x_2^2}={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}-\frac{2}{a_2}{\varphi }_{2m}\left[{\left(-1\right)}^m{Q}_{2n}-Q_{1n}\right]\\ -{\varphi }_{2m}^2{W}_{2mn}\end{array}\right\}\sin \left({\varphi }_{2m}{x}_2\right)\sin \frac{{\phi }_{2n}{y}_2}{2}}}\\ \frac{{\partial }^2{W}_2}{\partial y_2^2}={\displaystyle \sum _{m=1,3}^{+\infty }{\displaystyle \sum _{n=1,3,}^{+\infty }\left\{\begin{array}{l}\frac{2}{b_2}\left[{\left(-1\right)}^{\frac{n-1}{2}}{R}_{2m}+\frac{{\phi }_{2n}}{2}{R}_{1m}\right]\\ -{\left(\frac{{\phi }_{2n}}{2}\right)}^2{W}_{2mn}\end{array}\right\}\sin \left({\varphi }_{2m}{x}_2\right)\sin \frac{{\phi }_{2n}{y}_2}{2}}}\end{array}$$

(15b)

$$\begin{array}{l}\frac{{\partial }^4{W}_2}{\partial x_2^4}={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}-\frac{2}{a_2}{\varphi }_{2m}\left\{\begin{array}{l}{\left(-1\right)}^m{Q}_{4n}-Q_{3n}\\ -{\varphi }_m^2\left[{\left(-1\right)}^m{Q}_{2n}-Q_{1n}\right]\end{array}\right\}\\ +{\varphi }_{2m}^4{W}_{2mn}\end{array}\right\}\sin \left({\varphi }_{2m}{x}_2\right)\sin \frac{{\phi }_{2n}{y}_2}{2}}}\\ \end{array}$$

(15c)

$$\begin{array}{l}\frac{{\partial }^4{W}_2}{\partial y_2^4}={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}\frac{2}{b_2}\left[\begin{array}{l}{\left(-1\right)}^{\frac{n-1}{2}}{R}_{4m}+\frac{{\phi }_{2n}}{2}{R}_{3m}\\ -{\left(-1\right)}^{\frac{n-1}{2}}{\left(\frac{{\phi }_{2n}}{2}\right)}^2{R}_{2m}-{\left(\frac{{\phi }_{2n}}{2}\right)}^3{R}_{1m}\end{array}\right]\\ +{\left(\frac{{\phi }_{2n}}{2}\right)}^4{W}_{2mn}\end{array}\right\}\sin \left({\varphi }_{2m}{x}_2\right)\sin \frac{{\phi }_{2n}{y}_2}{2}}}\\ \end{array}$$

(15d)

$$\frac{{\partial }^4{W}_2}{\partial x_2^2\partial y_2^2}={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}-\left\{\begin{array}{l}\frac{2}{b_2}{\varphi }_{2m}^2\left[{\left(-1\right)}^{\frac{n-1}{2}}{R}_{2m}+\frac{{\phi }_{2n}}{2}{R}_{1m}\right]\\ +\frac{2}{a_2}{\varphi }_{2m}\left[{\left(-1\right)}^m{Q}_{6n}-Q_{5n}\right]\end{array}\right\}\\ +{\varphi }_{2m}^2{\left(\frac{{\phi }_{2n}}{2}\right)}^2{W}_{2mn}\end{array}\right\}\sin \left({\varphi }_{2m}{x}_2\right)\sin \frac{{\phi }_{2n}{y}_2}{2}}}$$

(15e)

In the same way, we can use the following unknown constants to define the definite integrals in Equations (15a)–(15e):

$$\begin{array}{l}{Q}_{1n}=\frac{2}{b_2}{\displaystyle {\int }_0^{b_2}{\left.W_2\right|}_{x_2=0}\sin \frac{{\phi }_{2n}{y}_2}{2}\mathrm{d}{y}_2} Q_{2n}=\frac{2}{b_2}{\displaystyle {\int }_0^{b_2}{\left.W_2\right|}_{x_2=a_2}\sin \frac{{\phi }_{2n}{y}_2}{2}\mathrm{d}{y}_2}\\ Q_{3n}=\frac{2}{b_2}{\displaystyle {\int }_0^{b_2}{\left.\frac{{\partial }^2{W}_2}{\partial x_2^2}\right|}_{x_2=0}\sin \frac{{\phi }_{2n}{y}_2}{2}\mathrm{d}{y}_2} Q_{4n}=\frac{2}{b_2}{\displaystyle {\int }_0^{b_2}{\left.\frac{{\partial }^2{W}_2}{\partial x_2^2}\right|}_{x_2=a_2}}\sin \frac{{\phi }_{2n}{y}_2}{2}\mathrm{d}{y}_2\\ Q_{5n}=\frac{2}{b_2}{\displaystyle {\int }_0^{b_2}{\left.\frac{{\partial }^2{W}_2}{\partial y_2^2}\right|}_{x_2=0}\sin \frac{{\phi }_n{y}_2}{2}\mathrm{d}{y}_2} Q_{6n}=\frac{2}{b_2}{\displaystyle {\int }_0^{b_2}{\left.\frac{{\partial }^2{W}_2}{\partial y_2^2}\right|}_{x_2=a_2}\sin \frac{{\phi }_{2n}{y}_2}{2}\mathrm{d}{y}_2}\\ R_{1m}=\frac{2}{a_2}{\displaystyle {\int }_0^{a_2}{\left.W_2\right|}_{y_2=0}}\sin \left({\varphi }_{2m}{x}_2\right)\mathrm{d}{x}_2 R_{2m}=\frac{2}{a_2}{\displaystyle {\int }_0^{a_2}{\left.\frac{\partial W_2}{\partial y_2}\right|}_{y_2=b_2}}\sin \left({\varphi }_{2m}{x}_2\right)\mathrm{d}{x}_2\\ R_{3m}=\frac{2}{a_2}{\displaystyle {\int }_0^{a_2}{\left.\frac{{\partial }^2{W}_2}{\partial y_2^2}\right|}_{y_2=0}}\sin \left({\varphi }_{2m}{x}_2\right)\mathrm{d}{x}_2 R_{4m}=\frac{2}{a_2}{\displaystyle {\int }_0^{a_2}{\left.\frac{{\partial }^3{W}_2}{\partial y_2^3}\right|}_{y_2=b_2}}\sin \left({\varphi }_{2m}{x}_2\right)\mathrm{d}{x}_2\end{array}$$

(16)

Substituting Equation (13a) into Equation (16), we can deduce the following relationships:

$$Q_{1n}=Q_{2n}=Q_{5n}=Q_{6n}=R_{1m}=0, R_{4m}=\left(2-{\mu }_2\right){\varphi }_{2m}^2{R}_{2m}$$

(17)

Similarly, the bending moments along the plate’s clamped edges can be directly obtained using constants $R_{3m}$, $Q_{3n}$, and $Q_{4n}$.

$$\begin{array}{l}{\left.M_{y_2}\right|}_{y_2=0}=-D_2{\displaystyle \sum _{m=1}^{+\infty }{R}_{3m}\sin \left({\varphi }_{2m}{x}_2\right)}\\ {\left.M_{x_2}\right|}_{x_2=0}=-D_2{\displaystyle \sum _{n=1}^{+\infty }{Q}_{3n}\sin \frac{{\phi }_{2n}{y}_2}{2}}\\ {\left.M_{x_2}\right|}_{x_2=a_2}=-D_2{\displaystyle \sum _{n=1}^{+\infty }{Q}_{4n}\sin \frac{{\phi }_{2n}{y}_2}{2}}\end{array}$$

(18)

Substituting Equations (15a)–(15e) into the general governing bending PDE, we have the following:

$${\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}\frac{{\left(4{\varphi }_{2m}^2+{\phi }_{2n}^2\right)}^2}{16}{W}_{2mn}-\frac{q_{2mn}}{D_2}\\ -\frac{2}{a_2}{\varphi }_{2m}\left[{\left(-1\right)}^m{Q}_{4n}-Q_{3n}\right]\\ -\frac{2}{b_2}{\left(-1\right)}^{\frac{n-1}{2}}\frac{4{\mu }_2{\varphi }_{2m}^2+{\phi }_{2n}^2}{4}{R}_{2m}+\frac{2}{b_2}\frac{{\phi }_{2n}}{2}{R}_{3m}\end{array}\right\}\sin \left({\varphi }_{2m}{x}_2\right)\sin \frac{{\phi }_{2n}{y}_2}{2}}}=0$$

(19)

where $q_{2mn}=\frac{4}{a_2{b}_2}{\displaystyle {\int }_0^{a_2}{\displaystyle {\int }_0^{b_2}{q}_2\left(x_2,y_2\right)\sin \left({\varphi }_{2m}{x}_2\right)\sin \frac{{\phi }_{2n}{y}_2}{2}\mathrm{d}{x}_2\mathrm{d}{y}_2}}$. Through the same procedure, we can obtain expression for the $W_{2mn}$ with terms $R_{2m}$ $R_{3m}$, $Q_{3n}$, and $Q_{4n}$.

$$W_{2mn}=f_{2mn}\left\{\frac{q_{2mn}}{D_2}+\frac{2}{b_2}{\zeta }_{mn}{R}_{2m}-\frac{2}{b_2}\frac{{\phi }_{2n}}{2}{R}_{3m}+\frac{2}{a_2}{\varphi }_{2m}\left[{\left(-1\right)}^m{Q}_{4n}-Q_{3n}\right]\right\}$$

(20)

where $f_{2mn}=\frac{16}{{\left(4{\varphi }_{2m}^2+{\phi }_{2n}^2\right)}^2},{\zeta }_{mn}={\left(-1\right)}^{\frac{n-1}{2}}\frac{4{\mu }_2{\varphi }_{2m}^2+{\phi }_{2n}^2}{4}$. Substituting the obtained $W_{2mn}$ into the Equation (14), we finally obtain the explicit solution for the flexural behaviors of the side plate in a tank.

$$W_2(x_2,y_2)={\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }{f}_{2mn}\left[\begin{array}{l}\frac{q_{2mn}}{D_2}+\frac{2}{b_2}{\zeta }_{mn}{R}_{2m}-\frac{2}{b_2}\frac{{\phi }_{2n}}{2}{R}_{3m}\\ +\frac{2}{a_2}{\varphi }_{2m}\left[{\left(-1\right)}^m{Q}_{4n}-Q_{3n}\right]\end{array}\right]\sin \left({\varphi }_{2m}{x}_2\right)\sin \frac{{\phi }_{2n}{y}_2}{2}}}$$

(21)

Equation (21) can match the edge condition in Equation (3a), as indicated above. Substituting Equation (21) into the Equations (15a) and (15b) and allowing them to match the edge conditions in Equation (13b) produces another four sets of linear algebraic equations, as represented by Equations (22a)–(22d).

$${\displaystyle \sum _{m=1}^{+\infty }{\varphi }_{2m}{f}_{2mn}\left\{\begin{array}{l}\frac{2}{b_2}{\zeta }_{mn}{R}_{2m}-\frac{2}{b_2}\frac{{\phi }_{2n}}{2}{R}_{3m}\\ +\frac{2}{a_2}{\varphi }_{2m}\left[{\left(-1\right)}^m{Q}_{4n}-Q_{3n}\right]\end{array}\right\}}=-{\displaystyle \sum _{m=1}^{+\infty }\frac{{\varphi }_{2m}{f}_{2mn}{q}_{2mm}}{D_2}}$$

(22a)

$${\displaystyle \sum _{m=1}^{+\infty }{(-1)}^m{\varphi }_{2m}{f}_{2mn}\left\{\begin{array}{l}\frac{2}{b_2}{\zeta }_{mn}{R}_{2m}-\frac{2}{b_2}\frac{{\phi }_{2n}}{2}{R}_{3m}\\ +\frac{2}{a_2}{\varphi }_{2m}\left[{\left(-1\right)}^m{Q}_{4n}-Q_{3n}\right]\end{array}\right\}}=-{\displaystyle \sum _{m=1}^{+\infty }\frac{{(-1)}^m{\varphi }_{2m}{f}_{2mn}{q}_{2mm}}{D_2}}$$

(22b)

for n equal to 1, 3, 5, ….

$${\displaystyle \sum _{n=1,3}^{+\infty }{\phi }_{2n}{f}_{2mn}\left\{\begin{array}{l}\frac{2}{b_2}{\zeta }_{mn}{R}_{2m}-\frac{2}{b_2}\frac{{\phi }_{2n}}{2}{R}_{3m}\\ +\frac{2}{a_2}{\varphi }_{2m}\left[{\left(-1\right)}^m{Q}_{4n}-Q_{3n}\right]\end{array}\right\}}=-{\displaystyle \sum _{n=1,3}^{+\infty }\frac{{\phi }_{2n}{f}_{2mn}{q}_{2mm}}{D_2}}$$

(22c)

$${\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}\frac{2}{b_2}\left[{\left(-1\right)}^{n-1}-{\left({\zeta }_{mn}\right)}^2{f}_{2mn}\right]R_{2m}\\ +{\zeta }_{mn}{f}_{2mn}\frac{2}{b_2}\frac{{\phi }_{2n}}{2}{R}_{3m}\\ -{\zeta }_{mn}{f}_{2mn}\frac{2}{a_2}{\varphi }_{2m}\left[{\left(-1\right)}^m{Q}_{4n}-Q_{3n}\right]\end{array}\right\}}={\displaystyle \sum _{n=1,3}^{+\infty }\frac{{\zeta }_{mn}{f}_{2mn}{q}_{2mm}}{D_2}}$$

(22d)

for m equal to 1, 2, 3, ….

2.3. General Bending Solution Procedure for the Fluid-Guiding Plate of Rectangular Sewage Tanks

In a rectangular sewage tank, the fluid-guiding plate, as presented in Figure 5, can be regarded as two having adjacent edges free and the other edges clamped. Thus, the expressions for each edge can be described as follows:

$${\left.W_3\right|}_{x_3=0}={\left.W_3\right|}_{y_3=0}=0, {\left.V_{x_3}\right|}_{x_3=a_3}={\left.V_{y_3}\right|}_{y_3=b_3}=0, {\left.M_{x_3{y}_3}\right|}_{x_3=a_3,y_3=b_3}=0$$

(23a)

$${\left.\frac{\partial W_3}{\partial x_3}\right|}_{x_3=0}={\left.\frac{\partial W_3}{\partial y_3}\right|}_{y_3=0}=0, {\left.M_{x_3}\right|}_{x_3=a_3}=0, {\left.M_{y_3}\right|}_{y_3=b_3}=0$$

(23b)

Along both the $x_3$ and $y_3$ directions, the value of plate deflection for one end is zero but not for the other end. Thus, we choose the double half-sinusoidal series as the trial function for the deflection of the fluid-guiding plate, which is given below:

$$W_3(x_3,y_3)={\displaystyle \sum _{m=1,3}{\displaystyle \sum _{n=1,3}{W}_{3mn}}}\sin \frac{{\varphi }_{3m}{x}_3}{2}\sin \frac{{\phi }_{3n}{y}_3}{2}$$

(24)

where ${\varphi }_{3m}=\frac{m\pi }{a_3}$, and ${\phi }_{3n}=\frac{n\pi }{b_3}$. Similarly, after applying the Stokes’ transform technique in Equation (24), we can obtain the following:

$$\begin{array}{l}\frac{\partial W_3}{\partial x_3}={\displaystyle \sum _{m=1,3}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left(-\frac{2}{a_3}{S}_{1n}+\frac{{\varphi }_{3m}}{2}{W}_{3mn}\right)\cos \frac{{\varphi }_{3m}{x}_3}{2}\sin \frac{{\phi }_{3n}{y}_3}{2}}}\\ \frac{\partial W_3}{\partial y_3}={\displaystyle \sum _{m=1,3}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left(-\frac{2}{b_3}{T}_{1m}+\frac{{\phi }_{3n}}{2}{W}_{3mn}\right)\sin \frac{{\varphi }_{3m}{x}_3}{2}\cos \frac{{\phi }_{3n}{y}_3}{2}}}\end{array}$$

(25a)

$$\begin{array}{l}\frac{{\partial }^2{W}_3}{\partial x_3^2}={\displaystyle \sum _{m=1,3}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}\frac{2}{a_3}\left[{\left(-1\right)}^{\frac{m-1}{2}}{S}_{2n}+\frac{{\varphi }_{3m}}{2}{S}_{1n}\right]\\ -{\left(\frac{{\varphi }_{3m}}{2}\right)}^2{W}_{3mn}\end{array}\right\}\sin \frac{{\varphi }_{3m}{x}_3}{2}\sin \frac{{\phi }_{3n}{y}_3}{2}}}\\ \frac{{\partial }^2{W}_3}{\partial y_3^2}={\displaystyle \sum _{m=1,3}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}\frac{2}{b_3}\left[{\left(-1\right)}^{\frac{n-1}{2}}{T}_{2m}+\frac{{\phi }_{3n}}{2}{T}_{1m}\right]\\ -{\left(\frac{{\phi }_{3n}}{2}\right)}^2{W}_{3mn}\end{array}\right\}\sin \frac{{\varphi }_{3m}{x}_3}{2}\sin \frac{{\phi }_{3n}{y}_3}{2}}}\end{array}$$

(25b)

$$\frac{{\partial }^4{W}_3}{\partial x_3^4}={\displaystyle \sum _{m=1,3}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}\frac{2}{a_3}\left[\begin{array}{l}{\left(-1\right)}^{\frac{m-1}{2}}{S}_{4n}+\frac{{\varphi }_{3m}}{2}{S}_{3n}\\ -{\left(-1\right)}^{\frac{m-1}{2}}{\left(\frac{{\varphi }_{3m}}{2}\right)}^2{S}_{2n}-{\left(\frac{{\varphi }_{3m}}{2}\right)}^3{S}_{1n}\end{array}\right]\\ +{\left(\frac{{\varphi }_{3m}}{2}\right)}^4{W}_{3mn}\end{array}\right\}\sin \frac{{\varphi }_{3m}{x}_3}{2}\sin \frac{{\phi }_{3n}{y}_3}{2}}}$$

(25c)

$$\frac{{\partial }^4{W}_3}{\partial y_3^4}={\displaystyle \sum _{m=1,3}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}\frac{2}{b_3}\left[\begin{array}{l}{\left(-1\right)}^{\frac{n-1}{2}}{T}_{4m}+\frac{{\phi }_{3n}}{2}{T}_{3m}\\ -{\left(-1\right)}^{\frac{n-1}{2}}{\left(\frac{{\phi }_{3n}}{2}\right)}^2{T}_{2m}-{\left(\frac{{\phi }_{3n}}{2}\right)}^3{T}_{1m}\end{array}\right]\\ +{\left(\frac{{\phi }_{3n}}{2}\right)}^4{W}_{3mn}\end{array}\right\}\sin \frac{{\varphi }_{3m}{x}_3}{2}\sin \frac{{\phi }_{3n}{y}_3}{2}}}$$

(25d)

$$\frac{{\partial }^4{W}_3}{\partial x_3^2\partial y_3^2}={\displaystyle \sum _{m=1,3}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}\frac{4}{a_3{b}_3}{\left(-1\right)}^{\frac{m-1}{2}}\left[{\left.{\left(-1\right)}^{\frac{n-1}{2}}\frac{{\partial }^2{W}_3}{\partial x_3\partial y_3}\right|}_{\begin{array}{l}{x}_3=a_3\\ y_3=b_3\end{array}}+\frac{{\phi }_{3n}}{2}{\left.\frac{\partial W_3}{\partial x_3}\right|}_{\begin{array}{l}{x}_3=a_3\\ y_3=0\end{array}}\right]\\ -\frac{2}{a_3}\left[{\left(-1\right)}^{\frac{m-1}{2}}{\left(\frac{{\phi }_{3n}}{2}\right)}^2{S}_{2n}-\frac{2}{b_3}\frac{{\varphi }_{3m}}{2}{S}_{5n}\right]\\ -\frac{2}{b_3}{\left(\frac{{\varphi }_{3m}}{2}\right)}^2\left[{\left(-1\right)}^{\frac{n-1}{2}}{T}_{2m}+\frac{{\phi }_{3n}}{2}{T}_{1m}\right]\\ +{\left(\frac{{\varphi }_{3m}}{2}\right)}^2{\left(\frac{{\phi }_{3n}}{2}\right)}^2{W}_{3mn}\end{array}\right\}\sin \frac{{\varphi }_{3m}{x}_3}{2}\sin \frac{{\phi }_{3n}{y}_3}{2}}}$$

(25e)

Using the following constants to define the definite integrals of Equations (25a)–(25e), we obtain the following:

$$\begin{array}{l}{S}_{1n}=\frac{2}{b_3}{\displaystyle {\int }_0^{b_3}{\left.W_3\right|}_{x_3=0}}\sin \frac{{\phi }_{3n}{y}_3}{2}\mathrm{d}{y}_3, S_{2n}=\frac{2}{b_3}{\displaystyle {\int }_0^{b_3}{\left.\frac{\partial W_3}{\partial x_3}\right|}_{x_3=a_3}}\sin \frac{{\phi }_{3n}{y}_3}{2}\mathrm{d}{y}_3\\ S_{3n}=\frac{2}{b_3}{\displaystyle {\int }_0^{b_3}{\left.\frac{{\partial }^2{W}_3}{\partial x_3^2}\right|}_{x_3=0}}\sin \frac{{\phi }_{3n}{y}_3}{2}\mathrm{d}{y}_3, S_{4n}=\frac{2}{b_3}{\displaystyle {\int }_0^{b_3}{\left.\frac{{\partial }^3{W}_3}{\partial x_3^3}\right|}_{x_3=a_3}}\sin \frac{{\phi }_{3n}{y}_3}{2}\mathrm{d}{y}_3\\ S_{5n}=\frac{2}{b_3}{\displaystyle {\int }_0^{b_3}{\left.\frac{{\partial }^2{W}_3}{\partial y_3^2}\right|}_{x_3=0}}\sin \frac{{\phi }_{3n}{y}_3}{2}\mathrm{d}{y}_3, T_{1m}=\frac{2}{a_3}{\displaystyle {\int }_0^{a_3}{\left.W_3\right|}_{y_3=0}}\sin \frac{{\varphi }_{3m}{x}_3}{2}\mathrm{d}{x}_3\\ T_{2m}=\frac{2}{a_3}{\displaystyle {\int }_0^{a_3}{\left.\frac{\partial W_3}{\partial y_3}\right|}_{y_3=b_3}}\sin \frac{{\varphi }_{3m}{x}_3}{2}\mathrm{d}{x}_3, T_{3m}=\frac{2}{a_3}{\displaystyle {\int }_0^{a_3}{\left.\frac{{\partial }^2{W}_3}{\partial y_3^2}\right|}_{y_3=0}}\sin \frac{{\varphi }_{3m}{x}_3}{2}\mathrm{d}{x}_3\\ T_{4m}=\frac{2}{a_3}{\displaystyle {\int }_0^{a_3}{\left.\frac{{\partial }^3{W}_3}{\partial y_3^3}\right|}_{y_3=b_3}}\sin \frac{{\varphi }_{3m}{x}_3}{2}\mathrm{d}{x}_3, T_{5m}=\frac{2}{a_3}{\displaystyle {\int }_0^{a_3}{\left.\frac{{\partial }^2{W}_3}{\partial x_3^2}\right|}_{y_3=0}}\sin \frac{{\varphi }_{3m}{x}_3}{2}\mathrm{d}{x}_3\end{array}$$

(26)

Substituting Equation (23a) into above constants, it is easy to derive the following:

$$S_{1n}=S_{5n}=T_{1m}=T_{5m}=0,S_{4n}=\left(2-{\mu }_3\right){\left(\frac{{\phi }_{3n}}{2}\right)}^2{S}_{2n},T_{4m}=\left(2-{\mu }_3\right){\left(\frac{{\varphi }_{3m}}{2}\right)}^2{T}_{2m}$$

(27)

The bending moments along the clamped edges can be determined by the following:

$$\begin{array}{l}{\left.M_{y_2}\right|}_{y_2=0}=-D_2{\displaystyle \sum _{m=1}^{+\infty }{T}_{3m}\sin \frac{{\varphi }_{3m}{x}_3}{2}}\\ {\left.M_{x_2}\right|}_{x_2=0}=-D_2{\displaystyle \sum _{n=1}^{+\infty }{S}_{3n}\sin \frac{{\phi }_{3n}{y}_3}{2}}\end{array}$$

(28)

Substituting Equations (25c)–(25e) into the general governing bending PDE, we obtain the following:

$${\displaystyle \sum _{m=1,3}^{+\infty }{\displaystyle \sum _{n=1,3}^{+\infty }\left\{\begin{array}{l}\frac{16}{{\left({\varphi }_{3m}^2+{\phi }_{3n}^2\right)}^2}{W}_{3mn}-\frac{q_{3mn}}{D_3}\\ -\frac{2}{b_3}{\left(-1\right)}^{\frac{n-1}{2}}\frac{{\mu }_3{\varphi }_{3m}^2+{\phi }_{3n}^2}{4}{T}_{2m}+\frac{2}{b_3}\frac{{\phi }_{3n}}{2}{T}_{3m}\\ -\frac{2}{a_3}{\left(-1\right)}^{\frac{m-1}{2}}\frac{{\varphi }_{3m}^2+{\mu }_3{\phi }_{3n}^2}{4}{S}_{2n}+\frac{2}{a_3}\frac{{\varphi }_{3m}}{2}{S}_{3n}\end{array}\right\}\sin \frac{{\varphi }_{3m}{x}_3}{2}\sin \frac{{\phi }_{3n}{y}_3}{2}}}=0$$

(29)

where $q_{3mn}=\frac{4}{a_3{b}_3}{\displaystyle {\int }_0^{a_3}{\displaystyle {\int }_0^{b_3}{q}_3(x_3,y_3)}}\sin \frac{{\varphi }_{3m}{x}_3}{2}\sin \frac{{\phi }_{3n}{y}_3}{2}\mathrm{d}{x}_3\mathrm{d}{y}_3$. In a same way, one can derive the formula for $W_{3mn}$ with terms $T_{2m}$, $T_{3m}$, $S_{2n}$, and $S_{3n}$.

$$W_{3mn}=f_{3mn}\left[\frac{q_{3mn}}{D_3}+\frac{2}{b_3}{\xi }_{2mn}{T}_{2m}-\frac{2}{b_3}\frac{{\phi }_{3n}}{2}{T}_{3m}+\frac{2}{a_3}{\xi }_{1mn}{S}_{2n}-\frac{2}{a_3}\frac{{\varphi }_{3m}}{2}{S}_{3n}\right]$$

(30)

where $f_{3mn}=\frac{16}{{\left({\varphi }_{3m}^2+{\phi }_{3n}^2\right)}^2},{\xi }_{1mn}={\left(-1\right)}^{\frac{m-1}{2}}\frac{{\varphi }_{3m}^2+{\mu }_3{\phi }_{3n}^2}{4},{\xi }_{2mn}={\left(-1\right)}^{\frac{n-1}{2}}\frac{{\mu }_3{\varphi }_{3m}^2+{\phi }_{3n}^2}{4}$.

Substitution of Equation (30) into Equation (24) yields the explicit solution for the flexural behaviors of the fluid-guiding plate in a tank.

$$W_3\left(x_3,y_3\right)={\displaystyle \sum _{m=1,3}^{\infty }{\displaystyle \sum _{n=1,3}^{\infty }{f}_{3mn}\left[\begin{array}{l}\frac{q_{3mn}}{D_3}+\frac{2}{b_3}{\xi }_{2mn}{T}_{2m}-\frac{2}{b_3}\frac{{\phi }_{3n}}{2}{T}_{3m}\\ +\frac{2}{a_3}{\xi }_{1mn}{S}_{2n}-\frac{2}{a_3}\frac{{\varphi }_{3m}}{2}{S}_{3n}\end{array}\right]\sin \frac{{\varphi }_{3m}{x}_3}{2}\sin \frac{{\phi }_{3n}{y}_3}{2}}}$$

(31)

Equation (31) can match the edge condition in Equation (23a), as indicated above. Substituting Equation (31) into the Equations (25a) and (25b) and allowing them to match the edge conditions in Equation (23b) produces the following linear algebraic equations, as represented by Equations (32a)–(32d).

$$\begin{array}{l}{\displaystyle \sum _{m=1,3}^{+\infty }{\varphi }_{3m}{f}_{3mn}\frac{2}{b_3}{\xi }_{2mn}{T}_{2m}-{\displaystyle \sum _{m=1,3}^{+\infty }{\varphi }_{3m}{f}_{3mn}\frac{2}{b_3}\frac{{\phi }_{3n}}{2}{T}_{3m}}}\\ +{\displaystyle \sum _{m=1,3}^{+\infty }{\varphi }_{3m}{f}_{3mn}\frac{2}{a_3}{\xi }_{1mn}{S}_{2n}}\\ -{\displaystyle \sum _{m=1,3}^{+\infty }{\varphi }_{3m}{f}_{3mn}\frac{2}{a_3}\frac{{\varphi }_{3m}}{2}{S}_{3n}}=-{\displaystyle \sum _{m=1,3}^{+\infty }\frac{{\varphi }_{3m}{f}_{3mn}{q}_{3mn}}{D_3}}\end{array}$$

(32a)

$$\begin{array}{l}{\displaystyle \sum _{m=1,3}^{\infty }\frac{2}{b_3}\left[{\left(-1\right)}^{\frac{m+n-2}{2}}{\mu }_3-{\xi }_{3mn}{\xi }_{2mn}\right]}{T}_{2m}+{\displaystyle \sum _{m=1,3}^{\infty }{\xi }_{3mn}\frac{2}{b_3}\frac{{\phi }_{3n}}{2}}{T}_{3m}\\ +{\displaystyle \sum _{m=1,3}^{\infty }\frac{2}{a_3}\left[{\left(-1\right)}^{m-1}-{\xi }_{3mn}{\xi }_{1mn}\right]}{S}_{2n}+{\displaystyle \sum _{m=1,3}^{\infty }{\xi }_{3mn}\frac{2}{a_3}\frac{{\varphi }_{3m}}{2}}{S}_{2n}\\ ={\displaystyle \sum _{m=1,3}^{\infty }}\frac{{\xi }_{3mn}{q}_{3mn}}{D_3}\end{array}$$

(32b)

for n equal to 1, 3, 5, ….

$$\begin{array}{l}{\displaystyle \sum _{m=1,3}^{\infty }{\phi }_{3n}{f}_{3mn}\frac{2}{b_3}{\xi }_{2mn}{T}_{2m}-{\displaystyle \sum _{m=1,3}^{\infty }{\phi }_{3n}{f}_{3mn}\frac{2}{b_3}\frac{{\phi }_{3n}}{2}{T}_{3m}}}\\ +{\displaystyle \sum _{m=1,3}^{\infty }{\phi }_{3n}{f}_{3mn}\frac{2}{a_3}{\xi }_{1mn}{S}_{2n}}\\ -{\displaystyle \sum _{m=1,3}^{\infty }{\phi }_{3n}{f}_{3mn}\frac{2}{a_3}\frac{{\varphi }_{3m}}{2}{S}_{3n}}=-{\displaystyle \sum _{m=1,3}^{\infty }\frac{{\phi }_{3n}{f}_{3mn}{q}_{3mn}}{D_3}}\end{array}$$

(32c)

$$\begin{array}{l}{\displaystyle \sum _{n=1,3}^{\infty }\frac{2}{b_3}\left[{\left(-1\right)}^{n-1}-{\xi }_{4mn}{\xi }_{2mn}\right]}{T}_{2m}+{\displaystyle \sum _{n=1,3}^{\infty }{\xi }_{4mn}\frac{2}{b_3}\frac{{\phi }_{3n}}{2}}{T}_{3m}\\ +{\displaystyle \sum _{n=1,3}^{\infty }\frac{2}{a_3}\left[{\left(-1\right)}^{\frac{m+n-2}{2}}{\mu }_3-{\xi }_{4mn}{\xi }_{1mn}\right]}{S}_{2n}\\ +{\displaystyle \sum _{n=1,3}^{\infty }{\xi }_{4mn}\frac{2}{a_3}\frac{{\varphi }_{3m}}{2}}{S}_{3n}={\displaystyle \sum _{n=1,3}^{\infty }}\frac{{\xi }_{4mn}{q}_{3mn}}{D_3}\end{array}$$

(32d)

for m equal to 1, 3, 5, ….

Where ${\xi }_{3mn}={\xi }_{1mn}{f}_{3mn}, {\xi }_{4mn}={\xi }_{2mn}{f}_{3mn}$. In the above-mentioned derivations, we utilize three different types of Fourier series form solutions for the bending problems of the bottom plate, the side plate, and the fluid-guiding plate, respectively. In each case, the complex plate-bending problem is altered to solve the linear algebra equations represented by Equations (12a)–(12d), Equations (22a)–(22d), and Equations (32a)–(32d), respectively. This unified solution procedure is quite easily mastered by engineers and researchers. It is noted that we can obtain non-zero solutions for each case since the number of unknown constants is equal to the number of equations. After obtaining results for unknown constants such as $J_{3n},J_{4n},L_{3m}$, and $L_{4m}$ in case 2.1, we can determine the deflection at any position of plates. In cases 2.1–2.3, the bending along the clamped edges can be determined by using Equation (8), Equation (18), and Equation (28), respectively. And the bending moments at the other positions of the plates can be determined by using expressions of the second-order differential derivatives of the plate’s deflection. Although the obtained solution are exact when m and n approach $+\infty$, accurate results meeting the engineering accuracy requirements are achieved when finite terms are chosen for both m and n.

3. Numerical Examples and Discussion

The above derivations in Section 2 are present for illustrating the unified solving procedures of the present approach on handling the flexural problems of three non-Levy-type plates; such problems are usually considered to be hard issues for the accurate flexural analysis of plates. To certify the accuracy of the analytical solutions derived, we carried out the following examinations on plates subjected to different types of loads/supports conditions in a sewage tank. For each type of plate, the aspect ratio ranges from 0.5 to 3. It is noted that the present solutions are suitable for elastic plates of any kind of materials, and the non-dimensional deflections and internal forces provided can be considered as benchmark data. It is worth noting that reliable FEM results for comparison are obtained by utilizing ABAQUS 6.13, where 4-node thin-shell element S4R is employed. The material and geometric properties within the FEM model remain consistent across CCCC, CCCF, and CCFF plates when compared to the analytical model. The boundary conditions of the numerical model are also same with the above cases; for example, for clamped edges, both the deflection and slope are restricted to zero. It is well known that the accuracy of numerical results given by the FEM depends on the finite element size. To provide precise FEM results and taking CCCC (without resting on a Winkler foundation), CCCF, and CCFF square plates as examples, the present paper conducts the convergence study shown in Figure 6 to examine the effect of mesh size on the accuracy of the bending solutions at a plate’s center, in which the mesh sizes of the CCCC, CCCF, and CCFF square plates are set to be a/10, a/20, a/50, a/100, a/150, a/200, a/300, and a/400, respectively. The FEM results for both the deflection and bending moments are convergent when the mesh size is set to be a/400. Thus, the uniform mesh size of a/400 is taken throughout the present work. The contour displacement results are also presented for the CCCC (without resting on a Winkler foundation), CCCF, and CCFF square plates shown in Figure 7 to illustrate the accuracy of the FEM.

(1) The first example is the fully clamped bottom plate without/with foundation; all the obtained solutions are demonstrated in

Table 1. Non-dimensional bending solutions for the bottom plate without/with foundation.

$\frac{\mathit{K}{\mathit{a}}_1^4}{{\mathit{D}}_1}$	${\mathit{b}}_1/{\mathit{a}}_1$	Method	$\begin{array}{l} {\mathit{D}}_1{\mathit{W}}_1/\left(\mathit{q}{\mathit{a}}_1^4\right)\\ \left({\mathit{x}}_1=\frac{{\mathit{a}}_1}{2},{\mathit{y}}_1=\frac{{\mathit{b}}_1}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{x}1}/\left(\mathit{q}{\mathit{a}}_1^2\right)\\ \left({\mathit{x}}_1=\frac{{\mathit{a}}_1}{2},{\mathit{y}}_1=\frac{{\mathit{b}}_1}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{y}1}/\left(\mathit{q}{\mathit{a}}_1^2\right)\\ \left({\mathit{x}}_1=\frac{{\mathit{a}}_1}{2},{\mathit{y}}_1=\frac{{\mathit{b}}_1}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{x}1}/\left(\mathit{q}{\mathit{a}}_1^2\right)\\ \left({\mathit{x}}_1=0,{\mathit{y}}_1=\frac{{\mathit{b}}_1}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{y}1}/\left(\mathit{q}{\mathit{a}}_1^2\right)\\ \left({\mathit{x}}_1=\frac{{\mathit{a}}_1}{2},{\mathit{y}}_1=0\right)\end{array}$
0	0.5	Present	0.000158310	0.00398458	0.0103285	−0.0142466	−0.0207165
		FEM	0.000158308	0.00395156	0.0102880	−0.0139588	−0.0203956
	1.0	Present	0.00126532	0.0230113	0.0230113	−0.0513336	−0.0513336
		FEM	0.00126534	0.0229047	0.0229047	−0.0507844	−0.0507844
		Ref. [33]	0.001265	0.0231	—	−0.0513	—
		Ref. [55]	0.00126	—	—	−0.0513	−0.0513
	2.0	Present	0.00253296	0.0413141	0.0159383	−0.0828658	−0.0569863
		FEM	0.00253294	0.0411541	0.0158075	−0.0822225	−0.0564088
		Ref. [33]	0.00254	0.0412	—	−0.0829	—
		Ref. [55]	0.00254	—	—	−0.0829	−0.0570
	3.0	Present	0.00261723	0.0420618	0.0128233	−0.0837763	−0.0568852
		FEM	0.00261720	0.0419002	0.0126924	−0.0831492	−0.0563085
1	0.5	Present	0.000158292	0.00398402	0.0103273	−0.0142456	−0.0207144
		FEM	0.000158290	0.0039510	0.0102867	−0.0139579	−0.0203935
	1.0	Present	0.00126430	0.0229902	0.0229902	−0.0512992	−0.0512992
		FEM	0.00126432	0.0228839	0.0228839	−0.0507506	−0.0507506
	2.0	Present	0.00252843	0.0412339	0.0159026	−0.0827329	−0.0569252
		FEM	0.00252842	0.0410743	0.0157721	−0.0820904	−0.0563481
	3.0	Present	0.00261199	0.0419716	0.0127943	−0.0836272	−0.0568243
		FEM	0.00261195	0.0418103	0.0126636	−0.0830010	−0.0562481
100	0.5	Present	0.000156555	0.00392934	0.0102043	−0.0141517	−0.0205105
		FEM	0.000156553	0.00389658	0.0101642	−0.0138654	−0.0201922
	1.0	Present	0.00117046	0.0210548	0.0210548	−0.0481280	−0.0481280
		FEM	0.00117048	0.0209551	0.0209551	−0.0476074	−0.0476074
	2.0	Present	0.00214471	0.0344562	0.0129151	−0.0714785	−0.0516440
		FEM	0.00214471	0.0343172	0.0127984	−0.0709082	−0.0511090
	3.0	Present	0.00217720	0.0345027	0.0104182	−0.0712687	−0.0515751
		FEM	0.00217718	0.0343640	0.0103018	−0.0707157	−0.0510407
1000	0.5	Present	0.000142270	0.00348298	0.00919453	−0.0133699	−0.0188344
		FEM	0.000142269	0.00345234	0.00915757	−0.0130961	−0.0185376
	1.0	Present	0.000690091	0.0112338	0.0112338	−0.0317595	−0.0317595
		FEM	0.000690107	0.0111682	0.0111682	−0.0313842	−0.0313842
	2.0	Present	0.000866129	0.0121797	0.00379482	−0.0339969	−0.0312459
		FEM	0.000866134	0.0121108	0.00372886	−0.0336631	−0.0308779
	3.0	Present	0.000849931	0.0119061	0.00356552	−0.0334182	−0.0312519
		FEM	0.000849935	0.0118381	0.00349988	−0.0330884	−0.0308840

. We first provide analytical flexural solutions for the bottom plate subjected to uniform loading and without foundation, and all the obtained non-dimensional results are compared with analytical results from the literatures [33,55] and numerical data offered by FEM, in which excellent agreement is observed. We then investigated the effect of foundation stiffness on flexural behaviors of the bottom plate. Attributing to the shortage of analytic results, all the obtained results are compared with FEM data. As expected, results for the deflection and bending moments are reduced with the increasing of the foundation stiffness. It was still found that both the deflection and internal bending moments increase with the increase in the aspect ratio, which indicates the plate’s geometry size affects the flexural performance of the plate;

(2) According to the practical engineering, the second example is on the side plate subjected to hydrostatic pressure. In the coordinate $x_2{o}_2{y}_2$, the hydrostatic pressure can be described as a linear distributed load with the expression $q_2\left(x_2,y_2\right)=q\left(1-b_2/y_2\right)$. In this case, similar to the bottom plate with foundation, the present solutions are only matched with FEM data, as shown in

Table 2. Non-dimensional bending solutions for the side plate.

${\mathit{b}}_2/{\mathit{a}}_2$	Method	$\begin{array}{l} {\mathit{D}}_2{\mathit{W}}_2/\left(\mathit{q}{\mathit{a}}_2^4\right)\\ \left({\mathit{x}}_2=\frac{{\mathit{a}}_2}{2},{\mathit{y}}_2=\frac{{\mathit{b}}_2}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{x}2}/\left(\mathit{q}{\mathit{a}}_2^2\right)\\ \left({\mathit{x}}_2=\frac{{\mathit{a}}_2}{2},{\mathit{y}}_2=\frac{{\mathit{b}}_2}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{y}2}/\left(\mathit{q}{\mathit{a}}_2^2\right)\\ \left({\mathit{x}}_2=\frac{{\mathit{a}}_2}{2},{\mathit{y}}_2=\frac{{\mathit{b}}_2}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{x}2}/\left(\mathit{q}{\mathit{a}}_2^2\right)\\ \left({\mathit{x}}_2=0,{\mathit{y}}_2=\frac{{\mathit{b}}_2}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{y}2}/\left(\mathit{q}{\mathit{a}}_2^2\right)\\ \left({\mathit{x}}_2=\frac{{\mathit{a}}_2}{2},{\mathit{y}}_2=0\right)\end{array}$
0.5	Present	0.000271967	0.00444556	0.00395108	−0.0125170	−0.0216466
	FEM	0.000272048	0.00445657	0.00394951	−0.0123956	−0.0213606
1.0	Present	0.000797320	0.0138366	0.0104131	−0.0297117	−0.0349494
	FEM	0.000797530	0.0138065	0.0103610	−0.0294154	−0.0345369
2.0	Present	0.00127272	0.0206824	0.00771801	−0.0415135	−0.0458357
	FEM	0.00127302	0.0206158	0.00766163	−0.0411447	−0.0453265
3.0	Present	0.00130783	0.0209865	0.00637767	−0.0418724	−0.0495410
	FEM	0.00130812	0.0209230	0.00632879	−0.0415213	−0.0489908

. It is seen that the present solutions show satisfactory accuracy as well as excellent agreement with FEM data;

(3) The third example is the fluid-guiding plate subjected to hydrostatic pressure, for which all the obtained results are tabulated in

Table 3. Non-dimensional bending solutions for the fluid-guiding plate.

${\mathit{b}}_3/{\mathit{a}}_3$	Method	$\begin{array}{l} {\mathit{D}}_3{\mathit{W}}_3/\left(\mathit{q}{\mathit{a}}_3^4\right)\\ \left({\mathit{x}}_3=\frac{{\mathit{a}}_3}{2},{\mathit{y}}_3=\frac{{\mathit{b}}_3}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{x}3}/\left(\mathit{q}{\mathit{a}}_3^2\right)\\ \left({\mathit{x}}_3=\frac{{\mathit{a}}_3}{2},{\mathit{y}}_3=\frac{{\mathit{b}}_3}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{y}3}/\left(\mathit{q}{\mathit{a}}_3^2\right)\\ \left({\mathit{x}}_3=\frac{{\mathit{a}}_3}{2},{\mathit{y}}_3=\frac{{\mathit{b}}_3}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{x}3}/\left(\mathit{q}{\mathit{a}}_3^2\right)\\ \left({\mathit{x}}_3=0,{\mathit{y}}_3=\frac{{\mathit{b}}_3}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{y}3}/\left(\mathit{q}{\mathit{a}}_3^2\right)\\ \left({\mathit{x}}_3=\frac{{\mathit{a}}_3}{2},{\mathit{y}}_3=0\right)\end{array}$
0.5	Present	0.000484410	0.00182225	0.000295615	−0.0140546	−0.0297067
	FEM	0.000484525	0.00182781	0.000306302	−0.0139717	−0.0294607
1.0	Present	0.00288467	0.00417148	0.00900662	−0.0497505	−0.0601081
	FEM	0.00288533	0.00415715	0.00899613	−0.0495322	−0.0598352
2.0	Present	0.0103312	0.00809036	0.0150303	−0.139248	−0.0965970
	FEM	0.0103330	0.00809363	0.0150143	−0.138719	−0.0963735
3.0	Present	0.0160482	0.0273581	0.00478074	−0.198426	−0.114160
	FEM	0.0160504	0.0273056	0.00475685	−0.197736	−0.113988

. Through the comparison in Table 3, it is seen again that all the proposed results agree well with FEM data. By comparing the results listed in Table 2 and Table 3, when the side plate and the fluid-guiding plate have the same aspect ratio, the non-dimensional deflection of the side plate is smaller than that of the fluid-guiding plate, which implies the boundary constraint also heavily influences the flexural performance of plates.

Since the obtained solutions are in the form of a Fourier series, the accuracy of the obtained results is closely related to the number of series terms adopted. In the present study, we chose the same term t for both m and n in each case, i.e., m = 1, 2, 3, …, t and n = 1, 2, 3, …, t for the CCCC plate; m = 1, 2, 3, …, t and n = 1, 3, 5, …, (2t − 1)/2 for the CCCF plate; and m = 1, 3, 5, …, (2t − 1)/2 and n = 1, 3, 5, …, (2t − 1)/2 for the CCFF plate. The convergence study was carried out for the non-dimensional deflection and bending moments at typical positions, as illustrated in

Table 4. Convergence study for the non-dimensional bending solutions of CCCC (without resting on a Winkler foundation), CCCF, and CCFF square plates.

BCs	t	$\begin{array}{l} {\mathit{D}}_{\mathit{i}}{\mathit{W}}_{\mathit{i}}/\left(\mathit{q}{\mathit{a}}_{\mathit{i}}^4\right)\\ \left({\mathit{x}}_{\mathit{i}}=\frac{{\mathit{a}}_{\mathit{i}}}{2},{\mathit{y}}_{\mathit{i}}=\frac{{\mathit{b}}_{\mathit{i}}}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{x}\mathit{i}}/\left(\mathit{q}{\mathit{a}}_{\mathit{i}}^2\right)\\ \left({\mathit{x}}_{\mathit{i}}=\frac{{\mathit{a}}_{\mathit{i}}}{2},{\mathit{y}}_{\mathit{i}}=\frac{{\mathit{b}}_{\mathit{i}}}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{y}\mathit{i}}/\left(\mathit{q}{\mathit{a}}_{\mathit{i}}^2\right)\\ \left({\mathit{x}}_{\mathit{i}}=\frac{{\mathit{a}}_{\mathit{i}}}{2},{\mathit{y}}_{\mathit{i}}=\frac{{\mathit{b}}_{\mathit{i}}}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{x}\mathit{i}}/\left(\mathit{q}{\mathit{a}}_{\mathit{i}}^2\right)\\ \left({\mathit{x}}_{\mathit{i}}=0,{\mathit{y}}_{\mathit{i}}=\frac{{\mathit{b}}_{\mathit{i}}}{2}\right)\end{array}$	$\begin{array}{l} {\mathit{M}}_{\mathit{y}\mathit{i}}/\left(\mathit{q}{\mathit{a}}_{\mathit{i}}^2\right)\\ \left({\mathit{x}}_{\mathit{i}}=\frac{{\mathit{a}}_{\mathit{i}}}{2},{\mathit{y}}_{\mathit{i}}=0\right)\end{array}$
CCCC	50	0.00126391	0.0220441	0.0220441	−0.0513659	−0.0513659
	100	0.00126465	0.0233237	0.0233237	−0.0513493	−0.0513493
	300	0.00126509	0.0231145	0.0231145	−0.0513416	−0.0513416
	500	0.00126527	0.0230447	0.0230447	−0.0513390	−0.0513390
	700	0.00126532	0.0230188	0.0230188	−0.0513349	−0.0513349
	1000	0.00126532	0.0230113	0.0230113	−0.0513336	−0.0513336
CCCF	50	0.000794842	0.0133083	0.0099375	−0.0297504	−0.0350134
	100	0.000796193	0.0140291	0.0105889	−0.0297300	−0.0349792
	300	0.000797074	0.0138794	0.0104522	−0.0297158	−0.0349560
	500	0.000797251	0.0138494	0.0104248	−0.0297129	−0.0349514
	700	0.000797320	0.0138366	0.0104131	−0.0297117	−0.0349494
	1000	0.000797377	0.0138270	0.0104043	−0.0297108	−0.0349479
CCFF	50	0.00287932	0.00349033	0.00824854	−0.0497615	−0.0601492
	100	0.00288217	0.00442029	0.00928010	−0.0497582	−0.0601297
	300	0.00288410	0.00422683	0.00906739	−0.0497520	−0.0601126
	500	0.00288450	0.00418809	0.00902485	−0.0497509	−0.0601094
	700	0.00288467	0.00417148	0.00900662	−0.0497505	−0.0601081
	1000	0.00288479	0.00415901	0.00899294	−0.0497501	−0.0601071

, respectively, where CCCC (without resting on a Winkler foundation), CCCF, and CCFF square plates were selected as extreme cases. Through Table 4, it is found that the obtained deflections converge faster than the bending moments, which is because the bending moments are obtained by the second derivatives of $W_i\left(x_i,y_i\right)$. Obviously, all the given solutions are precise up to six significant figures despite a slower convergence rate. But all the present solutions can be easily calculated by using Mathematica11.3 with less calculation time. Take the CCCC plate (without resting on a Winkler foundation) with b/a = 3 as an example, with the requirement of the same precision, and on a workstation with Intel Processor Intel(R) Core(TM) i9-10980XE CPU 3.00 GHz: The computational times for the present method utilizing Wolfram Mathematica software version 11.3 compared to the FEM results using ABAQUS software version 6.13 are 10.11 s and 150.32 s, respectively. It is also observed that all the present results are accurate enough when t equals 700; therefore, 700 series terms were adopted throughout the study.

The FEM deformation diagram and 3D deformation plot for the CCCC without foundation, CCCF, and CCFF square plate are also presented in Figure 8, Figure 9 and Figure 10.

The above numerical/graphical results and comparison works enable us to demonstrate the precision and efficiency of the proposed approach and to offer new reference data for other newly developed approaches.

4. Conclusions

The present research provides new progress in the 2D modified Fourier series approach on the accurate flexural analysis of three different types of plates in a rectangular sewage tank. On basis of the Fourier series theory, we provide a unified, straightforward solution scheme, and the flexure problem of the bottom plate, side plate, and fluid-guiding plate in a sewage tank can thus be handled directly. Due to the theoretical and rigorous solution procedure, new, benchmark non-dimensional deflection and internal forces are obtained for the three types of plates, and all the results are confirmed to be precise enough since all of them match well with few FEM data and solutions from the literature. The effects of foundation stiffness, aspect ratio, and boundary conditions on the flexural behaviors of plates were also studied using the present analytic results. The present parameter study revealed the following observations: (1) Increase in the stiffness of a Winkler foundation leads to the decrease in both non-dimensional deflection and bending moments of the bottom plate; (2) the high aspect ratio corresponds to an increase in non-dimensional deflection across all cases; and (3) bending moment distributions are influenced by the boundary conditions. These findings provide valuable insights for guiding structural design and analysis. In view of the advantages of the present approach, it is promising for adoption in dealing with the mechanical problems of thick plates subjected to various non-Levy-type boundaries. Furthermore, the method can be easy for both engineers and scientists to implement, and the present results are believed to provide a benchmark reference for the validation of other numerical and analytical methods.