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

Authors: Jun Gao; Faning Dang; Zongyuan Ma; Jie Ren
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

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. The dimensionless deflection, dimensionless internal force, and dimensionless stress functions of the rectangular thin plates are established as well. The analytic solution in this paper is validated by the finite element method. Finally, the influence of aspect ratioλ and Poisson’s ratio μ on the deformation and mechanical behaviors of the rectangle thin plates is analyzed in this paper. This research can provide references for the plane water gate problem in seaports and channels.


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## Body

## 1. 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 [3–5], the finite strip method [6], the meshless method [7], the spline element method [8], etc. These 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. The problem of a rectangular thin plate is first given by Dixon [9]. The 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). The 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. The 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]. The bending problem of a rectangular thin plate with two opposite edges simply supported has been analyzed by Hutchinson [12]. The buckling problem of clamped rectangular plates with different aspect ratios has been solved by El-Bayoumy using extended Kantorovich method [13]. The problem of an isotropic rectangular thin plate with four edges clamped has been given by Imrak and Gerdemeli [14]. The 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. The 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. The 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 point-supported, 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. The 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 [23–26]. 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. Thus, 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. The 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. The 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. This research can provide references for the plane water gate problem of seaports and channels.


## 2. Deflection and Internal Force Function of the SCSF Rectangular Thin Plate

### 2.1. Bending Equation and Boundary Condition of the SCSF Rectangular Thin Plate

The hydrostatic pressureqw=q01−y/b is loaded on the surface of the rectangular thin plate. The width is a along the x axis. The height is b along the y axis. The thickness is δ along the z axis. The dimensions and load condition of the rectangular thin plate are shown in Figure 1.Figure 1
Rectangular thin plate under hydrostatic pressure.
(a)
(b)The governing differential equation for the bending problem of the rectangular thin plate is as follows:(1)D∇4wx,y=qx,y,where D=Eδ3/121−υ2 is the flexural rigidity. E, δ, υ are the elastic modulus, plate thickness, and Poisson’s ratio, respectively. wx,y is the transverse deflection. qx,y is the distributed transverse load acting on the surface of the plate. (0≤x≤a, 0≤y≤b, −δ/2≤z≤δ/2).The edges ofx=0 and x=a are simply supported, y=0 is clamped, and y=b is free. The boundary condition of the SCSF rectangular thin plate can be expressed as follows:(2)wx=0=0,wx=a=0,wy=0=0,∂w∂yy=0=0.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. Thus, the deflection of thin plates is solved via Rayleigh–Ritz method.


### 2.2. Flexural Function

Based on the small deflection assumption for the thin plate-bending problems, the deflectionw is the only unknown function, and other components can be expressed in terms of w. The expression of deflection w can be expressed as w=∑m=1,3,5,…∞Cmwm, where Cm is the independent and undetermined coefficient and wm is the deflection function. The deflection function of the SCSF rectangular thin plate loaded by hydrostatic pressure is as follows:(3)wx,y=∑m=1,3,5,…∞Cmwm=∑m=1,3,5,…∞Cmsinmπxayb2.The deflection function satisfies the boundary conditions of equation (2), where Cm is the undetermined constant. The expression for strain energy of the thin plate is as follows:(4)Vε=D2∬A∇2w2−21−ν∂2w∂x2∂2w∂y2−∂2w∂x∂y2dxdy,where A is the area of the thin plate, ∇2w=∂2w/∂x2+∂2w/∂y2.Solving the second derivative of the deflection functionw versus x and y, respectively, and substituting them into equation (4), the expression for strain energy of thin plate can be written as follows:(5)Vε=∑m=1,3,5,…∞DCm222+43−2νmπba2+110mπba4ab3.The first derivative of the strain energyVε versus the coefficient Cm is as follows:(6)∂Vε∂Cm=DCm2+43−2νmπba2+110mπba4ab3.From equation (3), we get(7)∬Aqwmdxdy=∬Aq01−ybwmdxdy=∫0a∫0bq01−ybyb2sinmπxadxdy=q0ab6mπ.Based on the principle of minimum potential energy, the first derivative of the strain energyVε versus the coefficient Cm can be expressed as follows:(8)∂Vε∂Cm=∫0a∫0bqwmdxdy.Substituting equations (6) and (7) into equation (8), the coefficient equation can be written as(9)Cm=q0b46mπD2+4/3−2νmπb/a2+1/10mπb/a4.Substituting equation (9) into equation (3) allows the deflection function w to be written as(10)w=∑m=1,3,5,…∞q0b2sinmπx/ay26mπD2+4/3−2νmπb/a2+1/10mπb/a4.The dimensionless deflectionw′=Dw/q0b4 can be formulated as(11)w′=∑m=1,3,5,…∞sinmπx′⋅y′26mπ2+4/3−2νm2π2/λ2+m4π4/10λ4,where x′=x/a, y′=y/b, and λ=a/b.


### 2.3. Dimensionless Internal Force Function and Stress Function

Substituting the deflection functionw 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(12)Mx=∑m=1,3,5,…∞−CmD2νa2−m2π2y2a2b2sinmπxaMy=∑m=1,3,5,…∞−CmD2a2−νm2π2y2a2b2sinmπxaMxy=∑m=1,3,5,…∞−CmD21−νmπab2cosmπxay,FSx=∑m=1,3,5,…∞−CmDmπ2a2−m2π2y2a3b2cosmπxaFSy=∑m=1,3,5,…∞2CmDmπab2sinmπxay,σx=∑m=1,3,5,…∞−12CmD2νa2−m2π2y2a2b2δ3sinmπxazσy=∑m=1,3,5,…∞−12CmD2a2−νm2π2y2a2b2δ3sinmπxazτxy=∑m=1,3,5,…∞−24CmD1−νmπab2δ3cosmπxayz.Moreover, the dimensionless bending momentsMij′=a2Mij/q0b4, dimensionless shear forces FSi′=a3FSi/q0b4, and dimensionless stresses σij′=a2δ2σij/q0b4 can be established as follows:(13)Mx′=∑m=1,3,5,…∞−Cm′D2νλ2−m2π2y′2sinmπx′My′=∑m=1,3,5,…∞−Cm′D2λ2−νm2π2y′2sinmπx′Mxy′=∑m=1,3,5,…∞−2Cm′D1−νmπλcosmπx′⋅y′,FSx′=∑m=1,3,5,…∞−Cm′Dmπ2λ2−m2π2y′2cosmπx′FSy′=∑m=1,3,5,…∞2Cm′Dm2π2λsinmπx′⋅y′,σx′=∑m=1,3,5,…∞−12Cm′D2νλ2−m2π2y′2sinmπx′⋅z′σy′=∑m=1,3,5,…∞−12Cm′D2λ2−νm2π2y′2sinmπx′⋅z′τxy′=∑m=1,3,5,…∞−24Cm′D1−νmπλcosmπx′⋅y′z′,where Mij is the bending moment, FSi is the shear force, σij is the stress tensor, z′=z/δ, Cm′=Cm/q0b4, and i,j=x,y.


## 2.1. Bending Equation and Boundary Condition of the SCSF Rectangular Thin Plate

The hydrostatic pressureqw=q01−y/b is loaded on the surface of the rectangular thin plate. The width is a along the x axis. The height is b along the y axis. The thickness is δ along the z axis. The dimensions and load condition of the rectangular thin plate are shown in Figure 1.Figure 1
Rectangular thin plate under hydrostatic pressure.
(a)
(b)The governing differential equation for the bending problem of the rectangular thin plate is as follows:(1)D∇4wx,y=qx,y,where D=Eδ3/121−υ2 is the flexural rigidity. E, δ, υ are the elastic modulus, plate thickness, and Poisson’s ratio, respectively. wx,y is the transverse deflection. qx,y is the distributed transverse load acting on the surface of the plate. (0≤x≤a, 0≤y≤b, −δ/2≤z≤δ/2).The edges ofx=0 and x=a are simply supported, y=0 is clamped, and y=b is free. The boundary condition of the SCSF rectangular thin plate can be expressed as follows:(2)wx=0=0,wx=a=0,wy=0=0,∂w∂yy=0=0.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. Thus, the deflection of thin plates is solved via Rayleigh–Ritz method.


## 2.2. Flexural Function

Based on the small deflection assumption for the thin plate-bending problems, the deflectionw is the only unknown function, and other components can be expressed in terms of w. The expression of deflection w can be expressed as w=∑m=1,3,5,…∞Cmwm, where Cm is the independent and undetermined coefficient and wm is the deflection function. The deflection function of the SCSF rectangular thin plate loaded by hydrostatic pressure is as follows:(3)wx,y=∑m=1,3,5,…∞Cmwm=∑m=1,3,5,…∞Cmsinmπxayb2.The deflection function satisfies the boundary conditions of equation (2), where Cm is the undetermined constant. The expression for strain energy of the thin plate is as follows:(4)Vε=D2∬A∇2w2−21−ν∂2w∂x2∂2w∂y2−∂2w∂x∂y2dxdy,where A is the area of the thin plate, ∇2w=∂2w/∂x2+∂2w/∂y2.Solving the second derivative of the deflection functionw versus x and y, respectively, and substituting them into equation (4), the expression for strain energy of thin plate can be written as follows:(5)Vε=∑m=1,3,5,…∞DCm222+43−2νmπba2+110mπba4ab3.The first derivative of the strain energyVε versus the coefficient Cm is as follows:(6)∂Vε∂Cm=DCm2+43−2νmπba2+110mπba4ab3.From equation (3), we get(7)∬Aqwmdxdy=∬Aq01−ybwmdxdy=∫0a∫0bq01−ybyb2sinmπxadxdy=q0ab6mπ.Based on the principle of minimum potential energy, the first derivative of the strain energyVε versus the coefficient Cm can be expressed as follows:(8)∂Vε∂Cm=∫0a∫0bqwmdxdy.Substituting equations (6) and (7) into equation (8), the coefficient equation can be written as(9)Cm=q0b46mπD2+4/3−2νmπb/a2+1/10mπb/a4.Substituting equation (9) into equation (3) allows the deflection function w to be written as(10)w=∑m=1,3,5,…∞q0b2sinmπx/ay26mπD2+4/3−2νmπb/a2+1/10mπb/a4.The dimensionless deflectionw′=Dw/q0b4 can be formulated as(11)w′=∑m=1,3,5,…∞sinmπx′⋅y′26mπ2+4/3−2νm2π2/λ2+m4π4/10λ4,where x′=x/a, y′=y/b, and λ=a/b.


## 2.3. Dimensionless Internal Force Function and Stress Function

Substituting the deflection functionw 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(12)Mx=∑m=1,3,5,…∞−CmD2νa2−m2π2y2a2b2sinmπxaMy=∑m=1,3,5,…∞−CmD2a2−νm2π2y2a2b2sinmπxaMxy=∑m=1,3,5,…∞−CmD21−νmπab2cosmπxay,FSx=∑m=1,3,5,…∞−CmDmπ2a2−m2π2y2a3b2cosmπxaFSy=∑m=1,3,5,…∞2CmDmπab2sinmπxay,σx=∑m=1,3,5,…∞−12CmD2νa2−m2π2y2a2b2δ3sinmπxazσy=∑m=1,3,5,…∞−12CmD2a2−νm2π2y2a2b2δ3sinmπxazτxy=∑m=1,3,5,…∞−24CmD1−νmπab2δ3cosmπxayz.Moreover, the dimensionless bending momentsMij′=a2Mij/q0b4, dimensionless shear forces FSi′=a3FSi/q0b4, and dimensionless stresses σij′=a2δ2σij/q0b4 can be established as follows:(13)Mx′=∑m=1,3,5,…∞−Cm′D2νλ2−m2π2y′2sinmπx′My′=∑m=1,3,5,…∞−Cm′D2λ2−νm2π2y′2sinmπx′Mxy′=∑m=1,3,5,…∞−2Cm′D1−νmπλcosmπx′⋅y′,FSx′=∑m=1,3,5,…∞−Cm′Dmπ2λ2−m2π2y′2cosmπx′FSy′=∑m=1,3,5,…∞2Cm′Dm2π2λsinmπx′⋅y′,σx′=∑m=1,3,5,…∞−12Cm′D2νλ2−m2π2y′2sinmπx′⋅z′σy′=∑m=1,3,5,…∞−12Cm′D2λ2−νm2π2y′2sinmπx′⋅z′τxy′=∑m=1,3,5,…∞−24Cm′D1−νmπλcosmπx′⋅y′z′,where Mij is the bending moment, FSi is the shear force, σij is the stress tensor, z′=z/δ, Cm′=Cm/q0b4, and i,j=x,y.


## 3. 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. The flexural deformation function coefficient is solved using the Rayleigh–Ritz method and the principle of minimum potential energy. The internal force and stress functions are obtained via the small deflection bending theory of thin plate.


### 3.1. Bending Equation and Boundary Condition of the CCCF Rectangular Thin Plate

The rectangular thin plate surface is loaded by the hydrostatic pressureqw=q01−y/b (along the direction of y). The width, height, and thickness are shown in Figure 1. The governing differential equation for the bending problem of rectangular thin plate is given in equation (1). The edges of x=0, x=a, and y=0 are clamped and y=b is free. The boundary condition of the CCCF rectangular thin plate can be expressed as follows:(14)wx=0,a=0,∂w∂xx=0,a=0,wy=0=0,∂w∂yy=0=0.


### 3.2. Flexural Function

Based on the small deflection assumption for thin plate-bending problems, the deflectionw is the only unknown function, and other components can be expressed in terms of w. The expression of deflection w is w=∑m=1∞Cmwm, where Cm is the undetermined coefficients and wm is the deflection function. The deflection function of the CCCF rectangular thin plate loaded by the hydrostatic pressure is as follows:(15)wx,y=∑m=1∞Cmwm=∑m=1∞Cmsin2mπxayb2.Thus, 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:(16)Vε=∑m=1∞DCm2232+43−2νmπba2+25mπba4ab3.The first derivative of the strain energyVε versus Cm can be deduced as follows:(17)∂Vε∂Cm=DCm32+43−2νmπba2+25mπba4ab3.From equation (15), we get(18)∫0a∫0bqwmdxdy=∬Aq01−ybwmdxdy=∫0a∫0bq01−ybyb2sin2mπxadxdy=q0ab24.Based on the principle of minimum potential energy equation (8), substituting equations (17) and (18) into equation (8) gives the expression of the coefficient Cm as follows:(19)Cm=q0b424D3/2+4/3−2νmπb/a2+2/5mπb/a4.Substituting equation (19) into equation (15), the deflection function w can be written as follows:(20)w=∑m=1∞q0b2sin2mπx/ay224D3/2+4/3−2νmπb/a2+2/5mπb/a4.The dimensionless deflectionw′=Dw/q0b4 is as follows:(21)w′=∑m=1∞sin2mπx′⋅y′2243/2+4/3−2νm2π2/λ2+2m4π4/5λ4.


### 3.3. Dimensionless Internal Force Function and Stress Function

Substituting the deflection functionw 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:(22)Mx=∑m=1∞−2CmDνb2sin2mπxa+m2π2a2b2cos2mπxay2My=∑m=1∞−2CmD1b2sin2mπxa+νm2π2a2b2cos2mπxay2Mxy=Myx=∑m=1∞−2CmD1−νmπab2sin2mπxay,FSx=∑m=1∞−2CmDmπa2−2m2π2y2a3b2sin2mπxaFSy=∑m=1∞−4CmDm2π2a2b2cos2mπxay,σx=∑m=1∞−24CmDδ3νb2sin2mπxa+m2π2a2b2cos2mπxay2zσy=∑m=1∞−24CmDδ31b2sin2mπxa+νm2π2a2b2cos2mπxay2zτxy=τyx=∑m=1∞−241−νmπCmDab2δ3sin2mπxayz.Then, the dimensionless bending momentsMij′=a2Mij/q0b4, dimensionless shear forces FSi′=a3FSi/q0b4, and dimensionless stresses σij′=a2δ2σij/q0b4 can be established as follows:(23)Mx′=∑m=1∞−2Cm′Dνλ2sin2mπx′+m2π2y′2cos2mπx′My′=∑m=1∞−2Cm′Dλ2sin2mπx′+νm2π2y′2cos2mπx′Mxy′=Myx′=∑m=1∞−2Cm′D1−νmπλsin2mπx′⋅y′,FSx′=∑m=1∞−2Cm′Dmπλ2−2m2π2y′2sin2mπx′FSy′=∑m=1∞−4Cm′Dλm2π2cos2mπx′⋅y′,σx′=∑m=1∞−24Cm′Dνλ2sin2mπx′+m2π2y′2cos2mπx′z′σy′=∑m=1∞−24Cm′Dλ2sin2mπx′+νm2π2y′2cos2mπx′z′τxy′=τyx′=∑m=1∞−24Cm′D1−νmπλsin2mπx′⋅y′z′.


## 3.1. Bending Equation and Boundary Condition of the CCCF Rectangular Thin Plate

The rectangular thin plate surface is loaded by the hydrostatic pressureqw=q01−y/b (along the direction of y). The width, height, and thickness are shown in Figure 1. The governing differential equation for the bending problem of rectangular thin plate is given in equation (1). The edges of x=0, x=a, and y=0 are clamped and y=b is free. The boundary condition of the CCCF rectangular thin plate can be expressed as follows:(14)wx=0,a=0,∂w∂xx=0,a=0,wy=0=0,∂w∂yy=0=0.


## 3.2. Flexural Function

Based on the small deflection assumption for thin plate-bending problems, the deflectionw is the only unknown function, and other components can be expressed in terms of w. The expression of deflection w is w=∑m=1∞Cmwm, where Cm is the undetermined coefficients and wm is the deflection function. The deflection function of the CCCF rectangular thin plate loaded by the hydrostatic pressure is as follows:(15)wx,y=∑m=1∞Cmwm=∑m=1∞Cmsin2mπxayb2.Thus, 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:(16)Vε=∑m=1∞DCm2232+43−2νmπba2+25mπba4ab3.The first derivative of the strain energyVε versus Cm can be deduced as follows:(17)∂Vε∂Cm=DCm32+43−2νmπba2+25mπba4ab3.From equation (15), we get(18)∫0a∫0bqwmdxdy=∬Aq01−ybwmdxdy=∫0a∫0bq01−ybyb2sin2mπxadxdy=q0ab24.Based on the principle of minimum potential energy equation (8), substituting equations (17) and (18) into equation (8) gives the expression of the coefficient Cm as follows:(19)Cm=q0b424D3/2+4/3−2νmπb/a2+2/5mπb/a4.Substituting equation (19) into equation (15), the deflection function w can be written as follows:(20)w=∑m=1∞q0b2sin2mπx/ay224D3/2+4/3−2νmπb/a2+2/5mπb/a4.The dimensionless deflectionw′=Dw/q0b4 is as follows:(21)w′=∑m=1∞sin2mπx′⋅y′2243/2+4/3−2νm2π2/λ2+2m4π4/5λ4.


## 3.3. Dimensionless Internal Force Function and Stress Function

Substituting the deflection functionw 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:(22)Mx=∑m=1∞−2CmDνb2sin2mπxa+m2π2a2b2cos2mπxay2My=∑m=1∞−2CmD1b2sin2mπxa+νm2π2a2b2cos2mπxay2Mxy=Myx=∑m=1∞−2CmD1−νmπab2sin2mπxay,FSx=∑m=1∞−2CmDmπa2−2m2π2y2a3b2sin2mπxaFSy=∑m=1∞−4CmDm2π2a2b2cos2mπxay,σx=∑m=1∞−24CmDδ3νb2sin2mπxa+m2π2a2b2cos2mπxay2zσy=∑m=1∞−24CmDδ31b2sin2mπxa+νm2π2a2b2cos2mπxay2zτxy=τyx=∑m=1∞−241−νmπCmDab2δ3sin2mπxayz.Then, the dimensionless bending momentsMij′=a2Mij/q0b4, dimensionless shear forces FSi′=a3FSi/q0b4, and dimensionless stresses σij′=a2δ2σij/q0b4 can be established as follows:(23)Mx′=∑m=1∞−2Cm′Dνλ2sin2mπx′+m2π2y′2cos2mπx′My′=∑m=1∞−2Cm′Dλ2sin2mπx′+νm2π2y′2cos2mπx′Mxy′=Myx′=∑m=1∞−2Cm′D1−νmπλsin2mπx′⋅y′,FSx′=∑m=1∞−2Cm′Dmπλ2−2m2π2y′2sin2mπx′FSy′=∑m=1∞−4Cm′Dλm2π2cos2mπx′⋅y′,σx′=∑m=1∞−24Cm′Dνλ2sin2mπx′+m2π2y′2cos2mπx′z′σy′=∑m=1∞−24Cm′Dλ2sin2mπx′+νm2π2y′2cos2mπx′z′τxy′=τyx′=∑m=1∞−24Cm′D1−νmπλsin2mπx′⋅y′z′.


## 4. Results and Discussion

The influence of aspect ratioλ (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. The 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. The 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.Table 1
Physical and mechanical parameters of the SCSF and CCCF rectangular thin plates.
Height (m)
Thickness (m)
Elastic modulusE (GPa)
Loadingq0 (kN)
Series itemm
2.0
0.1
25.0
20.0
1.0Figure 2
Relationship between dimensionless deflection andx′ at y′=1.0 (SCSF).Figure 3
Relationship between dimensionless deflection andy′ at x′=0.5 (SCSF).Figure 4
Dimensionless deflection, internal force, and stress contour maps of the SCSF at different aspect ratio.Figure 5
Relationship between dimensionless deflection andx′ at y′=1.0 (CCCF).Figure 6
Relationship between dimensionless deflection andy′ at x′=0.5 (CCCF).Figure 7
Dimensionless deflection, internal force, and stress contour maps of the CCCF at different aspect ratio.


### 4.1. Influence of Aspect Ratioλ on the Deformation and Mechanical Properties of the Rectangular Thin Plates

The 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 ratioλ = 2.0 is shown in Figures 2 and 3. The 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. The errors are mainly caused by the values of m and the selection of the solution method. The 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 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. Thus, 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 Figures2 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. The distribution regularities of the dimensionless deflection, dimensionless internal force, and dimensionless stress of the SCSF rectangular thin plate with different values of aspect ratio are given in Figure 4.As shown in Figure4, w′, Mx′, My′, FSy′, σx′, and σy′ of the SCSF rectangular thin plate are symmetrically distributed, which is divided by x′=0.5, and Mxy′, FSx′, and τxy′ are antisymmetrically distributed. The maximum of w′, Mx′, FSy′, and σx′ appears at the point of (0.5,1). FSx′ is shown in four-quadrant chart, and the extremum of FSx′ 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. The extremum of My′, σy′ appears at the points of (0.5, 0) and (0.5, 1), and the positive My′, σy′ gradually disappear with the increase of aspect ratio λ. The extremum of Mxy′, τxy′ appears at the points of (0, 1) and (1, 1). The maximums of dimensionless deflection, dimensionless internal force, and dimensionless stress of the SCSF rectangular thin plate loaded by the hydrostatic pressure with different values of aspect ratio are shown in Table 2.Table 2
Dimensionless deflection, internal force, and stress of the SCSF rectangular thin plate atμ = 0.30.
λ = a/b
w
′ (0.5, 1)/10−3
M
x
′ (0.5, 1)
M
y
′ (0.5, 1)
M
x
y
′ (1, 1)
F
S
x
′ (0, 1)
F
S
y
′ (0.5, 1)
σ
x
′ (0.5, 1)
σ
y
′ (0.5, 1)
τ
x
y
′ (1, 1)
0.5
0.2839
−0.0028
0.0007
0.0006
0.0084
0.0028
−0.0171
0.0042
0.0037
1.0
2.7953
−0.0293
0.0027
0.0123
0.0691
0.0552
−0.1756
0.0161
0.0738
1.5
7.4293
−0.0834
−0.0114
0.0490
0.1253
0.2200
−0.5001
−0.0685
0.2941
2.0
12.0074
−0.1473
−0.0605
0.1056
0.0705
0.4740
−0.8840
−0.3630
0.6337The comparison between the analytic solution presented in this paper and the well-accepted finite element method (FEM) via software package ABAQUS for the CCCF rectangular thin plate with aspect ratioλ = 2.0 is shown in Figures 5 and 6. The finite 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. The errors are mainly caused by the values of m and the selection of the solution method. The 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 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 flexural 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. Thus, 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 Figures5 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 verified. Thus, the present method of this research is viable. The distribution regularities of the dimensionless deflection, dimensionless internal force and dimensionless stress of the CCCF rectangular thin plate with different values of aspect ratio are given in Figure 7.Similarly, as shown in Figure7, w′, Mx′, My′, FSy′, σx′, and σy′ of the CCCF rectangular thin plate are symmetrically distributed (divided by x′ = 0.5), and Mxy′, FSx′, and τxy′ are also antisymmetrically distributed (divided by x′ = 0.5). The maximum of w′ is appeared at the points of (0.5 1). The extremum of Mx′, My′, σx′ and σy′ are appeared at the four corners of the SCSF rectangular thin plate, which is the points of (0.5, 0) and (0.5, 1). The extremum of FSx′ is appeared at the points of (0.25, 0), (0.75, 0), (0.25, 1), and (0.75, 1). The extremum of FSy′ is appeared at the points of (0, 1), (0.5, 1), and (1, 1). The extremum of Mxy′, τxy′ 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 different, except the dimensionless deflection. And the dimensionless deflection of CCCF rectangular thin plate is small due to the stronger boundary constraint condition. The maximums of dimensionless deflection, dimensionless internal force, and dimensionless stress of the CCCF rectangular thin plate loaded by the hydrostatic pressure with different values of aspect ratio are shown in Table 3.Table 3
Dimensionless deflection, internal force, and stress of the CCCF rectangular thin plate atμ = 0.30.
λ = a/b
w
′ (0.5, 1)/10−3
M
x
′ (0.5, 1)
M
y
′ (0.5, 1)
M
x
y
′ (0.25, 1)
F
S
x
′ (0.75, 1)
F
S
y
′ (0.5, 1)
σ
x
′ (0.5, 1)
σ
y
′ (0.5, 1)
τ
x
y
′ (0.75, 1)
0.5
0.0637
0.0013
0.0003
−0.0001
−0.0078
0.0013
0.0075
0.0021
0.0008
1.0
0.8735
0.0167
0.0034
−0.0038
−0.1028
0.0345
0.1003
0.0206
0.0231
1.5
3.3566
0.0617
0.0048
−0.0221
−0.3689
0.1988
0.3704
0.0286
0.1329
2.0
7.2531
0.1258
−0.0151
−0.0638
−0.7173
0.5727
0.7546
−0.0904
0.3828


### 4.2. Influence of Poisson’s Ratioυ on the Deformation and Mechanical Properties of the Rectangular Thin Plates

The maximums of dimensionless deflection, internal force, and 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 and 5.Table 4
Dimensionless deflection, internal force, and stress of the SCSF rectangular thin plate atλ = 2.0.
υ
w
′ (0.5, 1)/10−3
M
x
′ (0.5, 1)
M
y
′ (0.5, 1)
M
x
y
′ (1, 1)
F
S
x
′ (0, 1)
F
S
y
′ (0.5, 1)
σ
x
′ (0.5, 1)
σ
y
′ (0.5, 1)
τ
x
y
′ (1, 1)
0.25
11.3723
−0.1349
−0.0629
0.1072
0.0668
0.4490
−0.8099
−0.3775
0.6431
0.30
12.0074
−0.1473
−0.0605
0.1056
0.0705
0.4740
−0.8840
−0.3630
0.6337
0.35
12.7177
−0.1611
−0.0578
0.1039
0.0747
0.5021
−0.9668
−0.3469
0.6233Table 5
Dimensionless deflection, internal force, and stress of the CCCF rectangular thin plate atλ = 2.0.
υ
w
′ (0.5, 1)/10−3
M
x
′ (0.5, 1)
M
y
′ (0.5, 1)
M
x
y
′ (0.25, 1)
F
S
x
′ (0.75, 1)
F
S
y
′ (0.5, 1)
σ
x
′ (0.5, 1)
σ
y
′ (0.5, 1)
τ
x
y
′ (0.75, 1)
0.25
6.9544
0.1234
−0.0213
−0.0655
−0.6877
0.5491
0.7402
−0.1279
0.3933
0.30
7.2531
0.1258
−0.0151
−0.0638
−0.7173
0.57527
0.7546
−0.0904
0.3828
0.35
7.5862
0.1284
−0.0083
−0.0619
−0.7495
0.5984
0.7703
−0.0496
0.3714As shown in Tables4 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.


## 4.1. Influence of Aspect Ratioλ on the Deformation and Mechanical Properties of the Rectangular Thin Plates

The 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 ratioλ = 2.0 is shown in Figures 2 and 3. The 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. The errors are mainly caused by the values of m and the selection of the solution method. The 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 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. Thus, 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 Figures2 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. The distribution regularities of the dimensionless deflection, dimensionless internal force, and dimensionless stress of the SCSF rectangular thin plate with different values of aspect ratio are given in Figure 4.As shown in Figure4, w′, Mx′, My′, FSy′, σx′, and σy′ of the SCSF rectangular thin plate are symmetrically distributed, which is divided by x′=0.5, and Mxy′, FSx′, and τxy′ are antisymmetrically distributed. The maximum of w′, Mx′, FSy′, and σx′ appears at the point of (0.5,1). FSx′ is shown in four-quadrant chart, and the extremum of FSx′ 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. The extremum of My′, σy′ appears at the points of (0.5, 0) and (0.5, 1), and the positive My′, σy′ gradually disappear with the increase of aspect ratio λ. The extremum of Mxy′, τxy′ appears at the points of (0, 1) and (1, 1). The maximums of dimensionless deflection, dimensionless internal force, and dimensionless stress of the SCSF rectangular thin plate loaded by the hydrostatic pressure with different values of aspect ratio are shown in Table 2.Table 2
Dimensionless deflection, internal force, and stress of the SCSF rectangular thin plate atμ = 0.30.
λ = a/b
w
′ (0.5, 1)/10−3
M
x
′ (0.5, 1)
M
y
′ (0.5, 1)
M
x
y
′ (1, 1)
F
S
x
′ (0, 1)
F
S
y
′ (0.5, 1)
σ
x
′ (0.5, 1)
σ
y
′ (0.5, 1)
τ
x
y
′ (1, 1)
0.5
0.2839
−0.0028
0.0007
0.0006
0.0084
0.0028
−0.0171
0.0042
0.0037
1.0
2.7953
−0.0293
0.0027
0.0123
0.0691
0.0552
−0.1756
0.0161
0.0738
1.5
7.4293
−0.0834
−0.0114
0.0490
0.1253
0.2200
−0.5001
−0.0685
0.2941
2.0
12.0074
−0.1473
−0.0605
0.1056
0.0705
0.4740
−0.8840
−0.3630
0.6337The comparison between the analytic solution presented in this paper and the well-accepted finite element method (FEM) via software package ABAQUS for the CCCF rectangular thin plate with aspect ratioλ = 2.0 is shown in Figures 5 and 6. The finite 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. The errors are mainly caused by the values of m and the selection of the solution method. The 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 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 flexural 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. Thus, 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 Figures5 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 verified. Thus, the present method of this research is viable. The distribution regularities of the dimensionless deflection, dimensionless internal force and dimensionless stress of the CCCF rectangular thin plate with different values of aspect ratio are given in Figure 7.Similarly, as shown in Figure7, w′, Mx′, My′, FSy′, σx′, and σy′ of the CCCF rectangular thin plate are symmetrically distributed (divided by x′ = 0.5), and Mxy′, FSx′, and τxy′ are also antisymmetrically distributed (divided by x′ = 0.5). The maximum of w′ is appeared at the points of (0.5 1). The extremum of Mx′, My′, σx′ and σy′ are appeared at the four corners of the SCSF rectangular thin plate, which is the points of (0.5, 0) and (0.5, 1). The extremum of FSx′ is appeared at the points of (0.25, 0), (0.75, 0), (0.25, 1), and (0.75, 1). The extremum of FSy′ is appeared at the points of (0, 1), (0.5, 1), and (1, 1). The extremum of Mxy′, τxy′ 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 different, except the dimensionless deflection. And the dimensionless deflection of CCCF rectangular thin plate is small due to the stronger boundary constraint condition. The maximums of dimensionless deflection, dimensionless internal force, and dimensionless stress of the CCCF rectangular thin plate loaded by the hydrostatic pressure with different values of aspect ratio are shown in Table 3.Table 3
Dimensionless deflection, internal force, and stress of the CCCF rectangular thin plate atμ = 0.30.
λ = a/b
w
′ (0.5, 1)/10−3
M
x
′ (0.5, 1)
M
y
′ (0.5, 1)
M
x
y
′ (0.25, 1)
F
S
x
′ (0.75, 1)
F
S
y
′ (0.5, 1)
σ
x
′ (0.5, 1)
σ
y
′ (0.5, 1)
τ
x
y
′ (0.75, 1)
0.5
0.0637
0.0013
0.0003
−0.0001
−0.0078
0.0013
0.0075
0.0021
0.0008
1.0
0.8735
0.0167
0.0034
−0.0038
−0.1028
0.0345
0.1003
0.0206
0.0231
1.5
3.3566
0.0617
0.0048
−0.0221
−0.3689
0.1988
0.3704
0.0286
0.1329
2.0
7.2531
0.1258
−0.0151
−0.0638
−0.7173
0.5727
0.7546
−0.0904
0.3828


## 4.2. Influence of Poisson’s Ratioυ on the Deformation and Mechanical Properties of the Rectangular Thin Plates

The maximums of dimensionless deflection, internal force, and 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 and 5.Table 4
Dimensionless deflection, internal force, and stress of the SCSF rectangular thin plate atλ = 2.0.
υ
w
′ (0.5, 1)/10−3
M
x
′ (0.5, 1)
M
y
′ (0.5, 1)
M
x
y
′ (1, 1)
F
S
x
′ (0, 1)
F
S
y
′ (0.5, 1)
σ
x
′ (0.5, 1)
σ
y
′ (0.5, 1)
τ
x
y
′ (1, 1)
0.25
11.3723
−0.1349
−0.0629
0.1072
0.0668
0.4490
−0.8099
−0.3775
0.6431
0.30
12.0074
−0.1473
−0.0605
0.1056
0.0705
0.4740
−0.8840
−0.3630
0.6337
0.35
12.7177
−0.1611
−0.0578
0.1039
0.0747
0.5021
−0.9668
−0.3469
0.6233Table 5
Dimensionless deflection, internal force, and stress of the CCCF rectangular thin plate atλ = 2.0.
υ
w
′ (0.5, 1)/10−3
M
x
′ (0.5, 1)
M
y
′ (0.5, 1)
M
x
y
′ (0.25, 1)
F
S
x
′ (0.75, 1)
F
S
y
′ (0.5, 1)
σ
x
′ (0.5, 1)
σ
y
′ (0.5, 1)
τ
x
y
′ (0.75, 1)
0.25
6.9544
0.1234
−0.0213
−0.0655
−0.6877
0.5491
0.7402
−0.1279
0.3933
0.30
7.2531
0.1258
−0.0151
−0.0638
−0.7173
0.57527
0.7546
−0.0904
0.3828
0.35
7.5862
0.1284
−0.0083
−0.0619
−0.7495
0.5984
0.7703
−0.0496
0.3714As shown in Tables4 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.


## 5. Conclusions

The 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. The 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. The dimensionless deflection of the CCCF rectangular thin plate is smaller than the dimensionless deflection of the SCSF rectangular thin plate. The 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 ratioλ and Poisson’s ratio μ. The results obtained by this paper can provide the references for the similar thin plates and plane gates in hydraulic engineering.



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*Source: 1560171-2019-03-07.xml*