Deformation and Mechanical Behaviors of SCSF and CCCF Rectangular Thin Plates Loaded by Hydrostatic Pressure

Elastic rectangular thin plate problems are very important both in theoretical research and engineering applications. Based on this, the flexural deformation functions of the rectangular thin plates with two opposite edges simply supported, one edge clamped and one edge free (SCSF) and three edges clamped and one edge free (CCCF), loaded by hydrostatic pressure are determined by single trigonometric series. And the flexural deformation functions are solved via the principle of minimum potential energy. Next, the internal force and stress functions of rectangular thin plates with two boundary conditions are obtained based on the small deflection bending theory of thin plates. *e dimensionless deflection, dimensionless internal force, and dimensionless stress functions of the rectangular thin plates are established as well.*e analytic solution in this paper is validated by the finite element method. Finally, the influence of aspect ratio λ and Poisson’s ratio μ on the deformation and mechanical behaviors of the rectangle thin plates is analyzed in this paper. *is research can provide references for the plane water gate problem in seaports and channels.


Introduction
Bending of rectangular thin plates has been heavily researched and gained great achievements.Numerical and analytical methods are two research methods that are often used for the analysis of thin plate problems.It is well known that many effective numerical methods have been developed in recent years.Representative methods include the finite element method (FEM) [1,2], the finite difference method [3][4][5], the finite strip method [6], the meshless method [7], the spline element method [8], etc. ese numerical methods normally meet the engineering requirements with acceptable errors and are greatly applied in practice.Meanwhile, analytical solutions are regarded as the benchmarks for verification of various numerical methods and have been investigated by many researchers.e problem of a rectangular thin plate is first given by Dixon [9].e solution of double trigonometric series of a rectangular thin plate with four edges simply supported (SSSS) under arbitrary loading has been proposed by Navier (Navier's solution).e solution of single trigonometric series of a rectangular thin plate with two opposite edges simply supported and other two opposite edges free (SFSF) under transverse loading has been given by Levy (Levy's solution).Different methods are used to analyze the rectangular thin plate problem under different boundary conditions and loading, such as the Fourier series method, the Rayleigh-Ritz method, the superposition method, the semi-inverse method, the symplectic geometry method, the integral-transform method, etc. e bending problem of a square plate with two adjacent edges clamped and the others either simply supported or free (CCSS or CCFF) under uniform loading has been investigated by Huang and Conway [10].Many exact solutions for the bending problem of elastic rectangular thin plates have been obtained by Timoshenko and Woinwsky-Krieger [11].e bending problem of a rectangular thin plate with two opposite edges simply supported has been analyzed by Hutchinson [12].e buckling problem of clamped rectangular plates with different aspect ratios has been solved by El-Bayoumy using extended Kantorovich method [13].e problem of an isotropic rectangular thin plate with four edges clamped has been given by Imrak and Gerdemeli [14].e accurate solution for a rectangular thin plate with two adjacent edges clamped and the others free (CCFF) is proposed by Chang [15], and the solution is yielded by superposing six known solutions.e symplectic geometry method is employed by Lim et al. [16] to investigate the bending problem of a rectangular thin plate with two opposite edges simply supported and the others free (SFSF).Moreover, the symplectic geometry method is also employed by Zhong and Li [17] and Liu and Li [18] to solve the deflection function and bending moment of a rectangular thin plate with four edges clamped (CCCC) under arbitrary loading.e analytic bending solutions of free rectangular thin plates resting on elastic foundations are obtained by Li et al. [19] via a new accurate symplectic superposition method.Moreover, Li et al. [20] extend the approach to the free vibration problems of the same plates and obtain the analytic solutions which cannot be obtained by the conventional symplectic approach.Besides, the analytic bending solutions of rectangular thin plates with a corner pointsupported, its adjacent corner free, and their opposite edge clamped or simply supported are obtained by Li et al. [21] via the superposition method in the symplectic space.e method of symplectic geometry is more reasonable than traditional semi-inverse solution.Khan et al. [22] employed the variation method to obtain a higher approximate solution for a rectangular thin plate with four edges simply supported (SSSS) under uniform loading.Based on this, the bending problem of rectangular thin plates has also been investigated by some other researchers under different boundary conditions and loading [23][24][25][26].However, it is still difficult to obtain the exact solution through solving the differential equation for the bending problem of the rectangular thin plate with certain boundary conditions.us, many exact solutions for the bending of thin plates are obtained with simple boundary conditions and transverse loading, such as the bending of thin plates with four edges clamped or simply supported, two opposite edges clamped or simply supported, three edges clamped or simply supported under uniform loading, and transverse loading.Most approximate solutions have been obtained for the bending problem of rectangular thin plates with relatively complicated boundary conditions and transverse loading.
In this paper, the flexural deformation functions of two types of rectangular thin plates (two opposite edges simply supported, one edge clamped and one edge free (SCSF) and three edges clamped and one edge free (CCCF)) loaded by hydrostatic pressure are established with single trigonometric series.e flexural deformation functions are solved using the principle of minimum potential energy.Internal force and stress functions of the rectangular thin plates under the two boundary conditions are obtained using the small deflection bending theory of thin plates.
e dimensionless deflection, dimensionless internal force, and dimensionless stress functions of rectangular thin plates under the two boundary conditions are established in this paper.Moreover, the influence of aspect ratio and Poisson's ratio on the deformation and mechanical characteristics of rectangular thin plates under the two boundary conditions is analyzed in this paper.is research can provide references for the plane water gate problem of seaports and channels.e hydrostatic pressure q w � q 0 (1 − y/b) is loaded on the surface of the rectangular thin plate.e width is a along the x axis.e height is b along the y axis.e thickness is δ along the z axis.e dimensions and load condition of the rectangular thin plate are shown in Figure 1.

Deflection and Internal Force Function of the SCSF Rectangular Thin Plate
e governing differential equation for the bending problem of the rectangular thin plate is as follows: where is the flexural rigidity.E, δ, υ are the elastic modulus, plate thickness, and Poisson's ratio, respectively.w(x, y) is the transverse deflection.q(x, y) is the distributed transverse load acting on the surface of the plate.(0 ). e edges of x � 0 and x � a are simply supported, y � 0 is clamped, and y � b is free.e boundary condition of the SCSF rectangular thin plate can be expressed as follows: ( It is difficult to obtain the deflection function if we solve the differential equation for the bending problem of the rectangular thin plate directly with the boundary conditions.us, the deflection of thin plates is solved via Rayleigh-Ritz method.

Flexural Function.
Based on the small deflection assumption for the thin plate-bending problems, the deflection w is the only unknown function, and other components can be expressed in terms of w. e expression of deflection w can be expressed as w �  ∞ m�1,3,5,... C m w m , where C m is the independent and undetermined coefficient and w m is the deflection function.
e deflection function of the SCSF rectangular thin plate loaded by hydrostatic pressure is as follows: (3) e deflection function satisfies the boundary conditions of equation (2), where C m is the undetermined constant.e expression for strain energy of the thin plate is as follows: 2 Advances in Civil Engineering where A is the area of the thin plate, ∇ 2 w � (z 2 w/zx 2 ) + (z 2 w/zy 2 ).
Solving the second derivative of the deflection function w versus x and y, respectively, and substituting them into equation ( 4), the expression for strain energy of thin plate can be written as follows: ( e first derivative of the strain energy V ε versus the coefficient C m is as follows: From equation (3), we get Based on the principle of minimum potential energy, the first derivative of the strain energy V ε versus the coefficient C m can be expressed as follows: Substituting equations ( 6) and (7) into equation ( 8), the coefficient equation can be written as ( e dimensionless deflection w ′ � Dw/q 0 b 4 can be formulated as where x ′ � x/a, y ′ � y/b, and λ � a/b.

Dimensionless Internal Force Function and Stress
Function.Substituting the deflection function w of the SCSF rectangular thin plate loaded by the hydrostatic pressure into the internal force equations and stress equations of classical elastic thin plate, the internal force equation and stress equation can be rewritten as Figure 1: Rectangular thin plate under hydrostatic pressure.

Advances in Civil Engineering
Moreover, the dimensionless bending moments M ij ′ � a 2 M ij /q 0 b 4 , dimensionless shear forces F Si ′ � a 3 F Si /q 0 b 4 , and dimensionless stresses σ ij ′ � a 2 δ 2 σ ij /q 0 b 4 can be established as follows: where M ij is the bending moment, F Si is the shear force, σ ij is the stress tensor, z ′ � z/δ, C m ′ � C m /q 0 b 4 , and i, j � x, y.

Deflection and Internal Force Function of the CCCF Rectangular Thin Plate
Similarly, the flexural deformation function of the CCCF rectangular thin plate loaded by hydrostatic pressure can also be established by single trigonometric series.e flexural deformation function coefficient is solved using the Rayleigh-Ritz method and the principle of minimum potential energy.e internal force and stress functions are obtained via the small deflection bending theory of thin plate.

Bending Equation and Boundary Condition of the CCCF Rectangular in Plate.
e rectangular thin plate surface is loaded by the hydrostatic pressure q w � q 0 (1 − y/b) (along the direction of y ).
e width, height, and thickness are shown in Figure 1.
e governing differential equation for the bending problem of rectangular thin plate is given in equation ( 1).
e edges of x � 0, x � a, and y � 0 are clamped and y � b is free.e boundary condition of the CCCF rectangular thin plate can be expressed as follows:

Flexural Function.
Based on the small deflection assumption for thin plate-bending problems, the deflection w is the only unknown function, and other components can be expressed in terms of w. e expression of deflection w is w �  ∞ m�1 C m w m , where C m is the undetermined coefficients and w m is the deflection function.e deflection function of the CCCF rectangular thin plate loaded by the hydrostatic pressure is as follows: us, the deflection function satisfies the boundary condition of equation (14).Solving the second derivative of the deflection function w versus x and y and substituting them into equation ( 4), the expression for strain energy of the thin plate can be rewritten as follows: e first derivative of the strain energy V ε versus C m can be deduced as follows: From equation ( 15), we get Based on the principle of minimum potential energy equation (8), substituting equations ( 17) and ( 18) into equation (8) gives the expression of the coefficient C m as follows: 4

Advances in Civil Engineering
Substituting equation (19) into equation ( 15), the deflection function w can be written as follows: e dimensionless deflection w ′ � Dw/q 0 b 4 is as follows: . (21)

Dimensionless Internal Force Function and Stress
Function.Substituting the deflection function w of the CCCF rectangular thin plate loaded by hydrostatic pressure into the internal force equation and stress equation of elastic thin plate, the internal force equation and stress equation can be rewritten as follows: en, the dimensionless bending moments M ij ′ � a 2 M ij /q 0 b 4 , dimensionless shear forces F Si ′ � a 3 F Si /q 0 b 4 , and dimensionless stresses σ ij ′ � a 2 δ 2 σ ij /q 0 b 4 can be established as follows:

Results and Discussion
e influence of aspect ratio λ (0.5, 1.0, 1.5, and 2.0) and Poisson's ratio μ (0.25, 0.30, and 0.35) on the deformation and mechanical properties of the rectangle thin plates with two boundary conditions is analyzed in this paper.e physical and mechanical parameters of the two types of rectangular thin plates are shown in Table 1.It is known that larger values of m give calculation accuracy.e distribution regularities of dimensionless deflection, dimensionless internal force, and dimensionless stress of the two rectangular thin plates loaded by the hydrostatic pressure are shown in Figures 2-7.

Influence of Aspect Ratio λ on the Deformation and Mechanical Properties of the Rectangular in
Plates.e comparison between the analytic solution presented in this paper and the finite element method (FEM) via the software package ABAQUS for the SCSF rectangular thin plate with aspect ratio λ � 2.0 is shown in Figures 2 and 3. e finite element types used in ABAQUS are C3D8R and 22400 uniform elements.Figures 2 and 3 show that there are some errors between the analytical solutions presented in this paper and the numerical solutions obtained by the FEM.e errors are mainly caused by the values of m and the selection of the solution method.e larger the values of m, the closer the analytical solutions to the exact solutions and the smaller the error.Moreover, the flexural deformation function of the SCSF rectangular thin plate loaded by the hydrostatic pressure is obtained via the Rayleigh-Ritz method in this paper.It is known that the thin plate-bending problems are solved by the Rayleigh-Ritz method; only the flexural deformation function is required to meet the displacement boundary conditions, but Advances in Civil Engineering not the internal force boundary conditions (if it can partially or completely meet the internal force boundary conditions, the accuracy of solutions can be improved).Equation ( 2) is the displacement boundary conditions.us, the accuracy of the solutions can be improved via the larger values of m and the selection of the trial functions that meet the displacement boundary conditions and internal force boundary conditions.
As shown in Figures 2 and 3, it is obvious that the results of FEM computation well agree with that of the analytical computation, which demonstrates the correctness of the present method for the SCSF rectangular thin plate.e distribution regularities of the dimensionless de ection, dimensionless internal force, and dimensionless stress of the SCSF rectangular thin plate with di erent values of aspect ratio are given in Figure 4.
As shown in Figure 4, w ′ , M x ′ , M y ′ , F Sy ′ , σ x ′ , and σ y ′ of the SCSF rectangular thin plate are symmetrically distributed, which is divided by x ′ 0.5, and M xy ′ , F Sx ′ , and τ xy ′ are antisymmetrically distributed.e maximum of w ′ , M x ′ , F Sy ′ , and σ x ′ appears at the point of (0.5,1).F Sx ′ is shown in fourquadrant chart, and the extremum of F Sx ′ is appeared at the four corners of the SCSF rectangular thin plate.Besides, the horizontal axis moves upward with the increase of aspect ratio λ. e extremum of M y ′ , σ y ′ appears at the points of (0.5, 0) and (0.5, 1), and the positive M y ′ , σ y ′ gradually disappear with the increase of aspect ratio λ. e extremum of M xy ′ , τ xy ′ appears at the points of (0, 1) and (1,1).
e maximums of dimensionless de ection, dimensionless internal force, and dimensionless stress of the SCSF rectangular thin plate loaded by the hydrostatic pressure with di erent values of aspect ratio are shown in Table 2.
e comparison between the analytic solution presented in this paper and the well-accepted nite element method (FEM) via software package ABAQUS for the CCCF rectangular thin plate with aspect ratio λ 2.0 is shown in Figures 5 and 6. e nite element types used in ABAQUS are C3D8R and 22400 uniform elements.Similar to Figures 2  and 3, Figures 5 and 6 show that there are some errors between the analytical solutions and the numerical solutions.
e errors are mainly caused by the values of m and the selection of the solution method.e larger the values of m, the closer the analytical solutions to the exact solutions and the smaller the error.Moreover, the exural deformation function of the CCCF rectangular thin plate loaded by the hydrostatic pressure is obtained via the Rayleigh-Ritz method.It is known that the thin plate-bending problems are solved by the Rayleigh-Ritz method, only the exural deformation function is required to meet the displacement boundary conditions, but not the internal force boundary conditions (if it can partially or completely meet the internal force boundary conditions, the accuracy of solutions can be improved).Equation ( 14) is the displacement boundary conditions.us, the accuracy of the solutions can be improved via the larger values of m and the selection of the trial functions that meet the displacement boundary conditions and internal force boundary conditions.
As shown in Figures 5 and 6, it is obvious that the FEM computation is almost equivalent to the analytical computation, which the correctness of the present method for the CCCF rectangular thin plate is veri ed.us, the present method of this research is viable.
e distribution regularities of the dimensionless de ection, dimensionless internal force and dimensionless stress of the CCCF rectangular thin plate with di erent values of aspect ratio are given in Figure 7.
Similarly, as shown in Figure 7, w′, M x ′ , M y ′ , F Sy ′ , σ x ′ , and σ y ′ of the CCCF rectangular thin plate are symmetrically distributed (divided by x ′ 0.5), and M xy ′ , F Sx ′ , and τ xy ′ are also antisymmetrically distributed (divided by x ′ 0.5).e maximum of w ′ is appeared at the points of (0.5 1).e extremum of M x ′ , M y ′ , σ x ′ and σ y ′ are appeared at the four corners of the SCSF rectangular thin plate, which is the points of (0.5, 0) and (0.5, 1).e extremum of F Sx ′ is appeared at the points of (0.25, 0), (0.75, 0), (0.25, 1), and (0.75, 1).e extremum of F Sy ′ is appeared at the points of (0,       Advances in Civil Engineering 1), (0.5, 1), and (1, 1).e extremum of M xy ′ , τ xy ′ is appeared at the points of (0.25, 1) and (0.75, 1).It should be noted that the distribution regularity of the dimensionless internal force and dimensionless stress are di erent, except the dimensionless de ection.And the dimensionless de ection of CCCF rectangular thin plate is small due to the stronger boundary constraint condition.e maximums of dimensionless de ection, dimensionless internal force, and dimensionless stress of the CCCF rectangular thin plate loaded by the hydrostatic pressure with di erent values of aspect ratio are shown in Table 3.

In uence of Poisson's Ratio υ on the Deformation and Mechanical Properties of the Rectangular in Plates.
e maximums of dimensionless de ection, internal force, and     and 5.
As shown in Tables 4 and 5, the dimensionless deflections, internal forces, and stresses of SCSF and CCCF rectangular thin plates loaded by hydrostatic pressure are increased linearly with the values of Poisson's ratio υ increased.

Conclusions
e deflection, internal force, and stress functions of the SCSF and CCCF rectangular thin plates loaded by the hydrostatic pressure are obtained via the Rayleigh-Ritz method.e method presented in this paper is correct and viable validated by FEM.Moreover, the dimensionless deflection, dimensionless internal force, and dimensionless stress functions of two types of the rectangular thin plates loaded by the hydrostatic pressure are established, which makes this research more general.
e dimensionless deflection of the CCCF rectangular thin plate is smaller than the dimensionless deflection of the SCSF rectangular thin plate.e values of dimensionless deflection, dimensionless internal force, and dimensionless stress of the SCSF and CCCF rectangular thin plates are increased with the increasing values of aspect ratio λ and Poisson's ratio μ. e results obtained by this paper can provide the references for the similar thin plates and plane gates in hydraulic engineering.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Additional Points
Highlights. (i) e deflection, internal force, and stress functions of the SCSF and CCCF rectangular thin plates loaded by hydrostatic pressure are established and solved.(ii) e dimensionless deflection, dimensionless internal force, and dimensionless stress functions of the rectangular thin plates with two boundary conditions are obtained.(iii) e influence of aspect ratio λ and Poisson's ratio μ on the deformations and mechanical behaviors of the rectangle thin plate is analyzed.

Figure 7 :
Figure 7: Dimensionless de ection, internal force, and stress contour maps of the CCCF at di erent aspect ratio.

Table 1 :
Physical and mechanical parameters of the SCSF and CCCF rectangular thin plates.

Table 2 :
Dimensionless de ection, internal force, and stress of the SCSF rectangular thin plate at μ 0.30.
stress of the SCSF rectangular thin plate and the CCCF rectangular thin plate loaded by the hydrostatic pressure with different values of Poisson's ratio are given in Tables4