HighTech Static Elastic Bending Analysis of a Three-Dimensional Clamped Thick Rectangular Plate using Energy Method

Analytical formulations and solutions for the thick rectangular plate static analysis with clamped support based on a three-dimensional (3-D) elasticity theory is developed using the energy method. The theoretical model, whose formulation is based on the static elastic principle as already reported in the literature, is presented herein to obviate the shear correction coefficients while considering shear deformation effect and transverse normal strain/stress in the analysis. The equilibrium equations are obtained using 3-D kinematic and constitutive relations. The deflection and rotation functions, which are the solutions of the equilibrium equation, are obtained in closed form using a general variational technique for solving the boundary value problem. The minimization energy equation yields the general equation which was used to obtain the theoretical model for the deflection and stresses of the plate. The results are compared with the available literature and the results-computed trigonometric displacement function shows that this 3-D predicts the vertical displacement and the stresses more accurately than previous studies considered in this paper. The result showed that the percentage difference between the present work and those of 2-D Mindlin FSDT, 2-D numeric analysis, and 2-D HSDT of polynomial shape functions was about 3.02%, 0.62%, and 0.33%, respectively. It is concluded that the 3-D trigonometric model gives an exact solution, unlike other 2-D theories, and can be used for clamped-supported thick plate analysis.


Introduction
Plates are three-dimensional structural elements with spatial dimensions along x, y, and z axes, whose applications are prevalent in different aspects of engineering, such as marine, naval, aerospace, mechanical, and structural engineering. Plates can be classified in terms of shapes such as: quadrilateral, square, circular, or rectangular. Depending on their constituent materials, they may also be classified as isotropic, anisotropic, orthotropic, homogeneous, or nonhomogeneous. They can also be defined based on thickness as thin, thick, or moderately thick plates [1,2]. As regards to its span-to-depth ratio ( / ), Mahi et al. (2015) [3] and Timoshenko & Woinowsky-Krieger (1959) [4] classified rectangular plates with 50 ≤ / ≤ 100 as thin plate, 20 ≤ / ≤ 50 as moderately thick and / ≤ 20 as thick plate [5]. The use of thick plates has greatly increased in structural engineering as a result of its cost benefits and other advantages such as its light weight, high strength and load resistance ability [6,7].
In general, plate research consists of buckling, deflection, and vibration analysis [8]. The bending of the thick rectangular plate is considered in this paper. Bending is the deformation of the plate at right angles to the plate surface The purpose of this research is to apply 3-D theory in solving the problem of deflection for a clamped isotropic rectangular thick plate, investigating the impact of aspect ratio and displacement of the moment, shear force, stresses, and stress resultant of the plate using the Energy method. This study was undertaken with the following objectives in mind:  To create the internal energy of a three-dimensional rectangular thick plate;  To generate the compatibility and overall governing equations of the plate and derive equations of displacements and shear deformation slope coefficient for x, y and z coordinates;  To obtain the exact expressions of the displacements, bending moment, shear force and stresses for the thick rectangular plate.

Literature Review
For a rectangular SSSS Kirchhoff plate, the Ritz method was used by Nwoji et al. (2018) [26] to analyze the plate bending problem. The method used by the authors yielded exact identical solutions as the exact results obtained by those who employed the Navier double Fourier sine series method. Using the exact deflection shape function, the authors obtained an exact solution. Ike (2017) [27] applied the Kantorovich-Galerkin method in studying the bending of CSSS plates with an assumed displacement function. The author formulated the equation of equilibrium in line with the work of Euler-Lagrange and solved to obtain the deflection and bending moment coefficients for deflection at the center of the plates under the uniform. The author did not consider the stresses in the direction of thickness axis neither was plates the CCCC boundary condition taken into account. The author did not apply the general variational method in the derivation of the displacement function and shape function used was assumed, which made the result not a close-form solution.
Using an analytical method, Onyeka et al. (2019) [2] employed third-order refined theory for solving the bending of a thick rectangular plate that is simply supported on all the edges. To determine displacement coefficients, the equation of total stored energy of a thick plate that was generated from elastic. Integral direct integration method of the exact analytical solution approach was used to determine the work, stresses, displacement and the shear deformation equation and the values obtained from their study conformed to the values from previous studies. However, the authors did not consider a full 3-D analogy for a typical three-dimensional plate with all round clamped edges using the energy method. Ibearugbulem et al. (2018) [28] applied shear deformation theory with a polynomial shape function to analyze the bending of CCCC rectangular thick plates. As with other higher-order theories, the condition of zero shear stress on the surfaces of the plate were met with the transverse shear stress derived from the constitutive relation of the theory. The authors did not consider a trigonometric shape function. Even though the result of their displacements and stresses, a 3-D theory was not applied. Onyeka & Edozie (2021) [29] analyzed the displacements and stresses of thick rectangular CCFC plate applying the higher order polynomial which was derived from the governing equation using the general variation method. The results of their study agreed well with those of refined plate theory, but varied more with the value of the classical plate theory. The considerations of authors will not yield a good result for a 3-D plate because it is limited to a 2-D plate theory. The trigonometric shape function and CCCC boundary condition was not considered.
Analyzing the problem of displacement-stresses in thick plates with simply supported edges, Sayyad & Ghugal, (2012) [23] used the refined theory of shear deformation and exponential functions. The shear transverse distortion and rotary inertia were found using the theory and the functions in thickness coordinate form. Compared with other refined plate theories, the displacements and stresses achieved in their result were satisfactory. The authors did not consider trigonometric displacement function in an energy method using the 3-D theory. Also, their analysis did not cover for thick plates with all-round clamped boundary conditions. Onyeka et al. (2020) [30] analyzed the bending behavior of rectangular thick CSCS and SCFS plates based on fourth-order polynomial shear deformation function. The authors developed a new approach to achieve the critical load of the plate from the established equation. The deflection and stresses obtained in their study were identical with the other order theories, but they did not analyze the in-plane displacement and moment that induce mending in the plate. Also, the author neither analyzes the plate as a typical 3-D element nor did they consider a thick plate of all round clamped edges.
Applying the numerical method on account of the three-dimensional theory of elasticity, the study of bending solutions of thick plates with clamped edge conditions, was carried out Grigorenko et al. (2013) [31]. The authors employed two coordinate directions of spline collocation and the resultant displacements and stresses in the clamped thick rectangular plates were satisfactory. The result of their study were not exactly because they did not consider the analytical approach neither did they use the energy method that is more simplified.
Onyeka & Ibearugbulem (2020) [32], used the direct variation energy method to obtain closed form solutions for bending analysis of CCCC and CCFC thick rectangular plates, applying the nonlinear strain-displacement polynomial shape function of fourth order shear deformation theory. From the principle of variational calculus, the authors obtained the governing equations which were used to solve the deflection problem of the plates. They also developed formulas for calculating actual and maximum lateral loads imposed on the plate before deformation gets to the specified maximum specified limit and elastic yield respectively. Their result confirmed that the actual load that causes the bending problem can be predicted using this theory. The 3-D theory was also not employed, nor did the authors consider the use of the trigonometric shape function. The authors investigated only the aspect ratio effects on the critical thickness of the plate without considering the displacement and stresses. A three-dimensional analysis of a thick SSSS plate was presented analytically by Fu et al. (2022) [33].
To obtain a total potential energy function, strain and stress with six components each, were used by the shear deformation theory of third order. The rotation and deflection expressions were derived from the solutions of compatibility equations that were obtained by minimizing the function with respect to shear deformation rotations. The deflection equation was found by solving the governing equation derived from further minimization of the function with respect to deflect. The values of the calculated deflections and stresses obtained from the 3-D analysis were coarse compared with those of refined plate theories. The work is limited as there is no application of trigonometric displacement functions which produces an exact solution.
Ibearugbulem & Onyeka (2020) [34] employed a direct variation energy method to solve the bending problem of clamped rectangular plates using third order plate theory. The method used did not require shearing correction factors and the results obtained revealed its precision by numerical comparison. The authors did not analyze for the critical lateral load and the solutions of their study were not exactly as a result of the assumed shape function and non-application of the general variational method. The authors did not consider the use of trigonometric displacement function and did not apply the 3D theory. Most of these reviewed studies are mostly based on refined plate theories. Aside from the work by Ibearugbulem & Onyeka et al. (2020) [34], one can hardly see work on the bending behavior of thick plates based on 3-D theory. The need for this current research work cannot be neglected, as it is worthwhile to fill this gap in the literature. The peculiarity of this study with the various previous respective works resides in the type of plate theory, method of analysis, the displacement functions, and the plate supports. In this study, the general variation of the total potential energy was performed in order to get an exact trigonometric shape function from the elastic principle without assumption. Investigating the bending features for a CCCC rectangular, this work also went ahead to determine the displacements and stresses of the plate using 3-D plate theory.

Methodology
The research methodology of this study is presented by considering a rectangular plate in Figure 1 as a threedimensional element in which the deformation exists in the three axis: length (a), width (b) and thickness (t). The analytical approach of the energy method was used to obtain formulas for the analysis.

Kinematics
The 3-D displacement kinematics along x, y and z axis (u, v and w) shown in the Figure 3 are obtained assuming that the x-z section and y-z section, is no longer normal to x-y plane after bending.

Figure 3. Rotation of x-z (or y-z) section after bending
Resolving the deformation diagram in Figure 3 using trigonometric relations, the algebraic relationship between the displacement and slope along the x axis and y becomes: where, and is the shear deformation rotation along x axis and y axis.
Taking into account, the thick plate assumption as stated in this section, the non-dimensional form of the Equations 1 and 2 gives: ( Where: Re-arranging Equation 3 and 4 gives: where, and is the in-plane displacement along x-axis and y axis respectively, thus, the six non-dimensional coordinates strain components were derived using strain-displacement expression according to Hooke's law and presented in Equations 8 to 13: where, , and are normal strain along x axis, y axis and z axis respectively, , represents the shear strain in the plane parallel to the x-y, x-z and y-z plane.

Constitutive Relations
The three dimensional constitutive relation is determined using a generalized Hooke's principle as: where, E and µ are the modulus of elasticity and Poisson's ratio.
The six stress components were obtained by substituting Equations 8 to 13 into Equation 14 and simplifying the outcome as: . [

Strain Energy
The strain energy ( ) is mathematically defined as: Substituting the values of stresses (Equations 8 to 13) and strain (Equations 15 to 20) into Equation 21, and integrate the dot product with respect to gives:

Energy Equation Formulation
The total potential energy is mathematically expressed as: where, V, q, 1 and h are the external work, uniformly distributed load, coefficient of deflection and shape function of the plate respectively, a and b is the length and breadth of the plate.

Substituting Equations 22 and 25 into Equation 24
gives:

Governing Equation
The solution of the governing equation is presented as the result of energy functional minimization with respect to deflection to give exact plate's shape function: Let; where; 2 and 3 are the coefficient of shear deformation along x axis and coefficient of shear deformation along y axis respectively. Substituting Where; = ∫ ∫ (

Numerical Analysis
The numerical analysis of a rectangular thick plate whose Poisson's ratio is 0.3 under CCCC boundary conditions as shown in the Figure 4 and carrying uniformly distributed load (including self-weight) is presented. An exact trigonometric functions as was obtained in the Equation 27 and applied here to get the actual values of the shape functions, coefficients of deflection and shear deformation rotations at x and y axis of the plate.
where the coefficient of the deflection, while the shape function

Exact Displacement and Stress Expression
By substituting the value of 1 , 2 and 3 in Equations 49, 38 and 39 into Equation 15 to 20 and substitute appropriately, the in-plane displacement along x-axis becomes: The in-plane displacement along y-axis becomes: The deflection equation of the plate as: The six stress elements are presented in Equations 15 to 20 as:

Results and Discussion
The parametric data for the trigonometric stiffness coefficient, kx, kxy, ky, kz, k2z and kq for CCCC shape functions are presented in Table 1. This data was obtained by substituting Equation 58 into Equations 30, 31, 32, 33, 34 and 35 as presented in the Figure 5. This stiffness coefficients were used to obtain the value of the shape functions and displacement and rotation of the plate material when subjected to a uniformly distributed transverse load under the same boundary conditions. The graph in Figure 5 showed that kx and ky have the highest coefficient followed by kxy while kz, k2z and kq contains the lowest amount of stiffness coefficient.

Figure 5. Stiffness coefficient for the CCCC plate boundary condition
The numerical results of the non-dimensional displacements (u, v & w) and the stresses characteristics of a 3-D clamped rectangular plate which was subjected to uniform distributed load was obtained using the established exact trigonometric displacement function. Figures 6 and 7 contains the result of the non-dimensional value of displacements and stresses at different span-thickness aspect ratio in a rectangular thick plate aspect ratio of 1 and 2 respectively.

Figure 6. The result of displacements and stresses of a clamped square plates
The result covered the 3-D bending and stress analysis of rectangular plate at varying thickness. The span to thickness ratio considered is ranged between 4, 5, 10, 15, 20, 50, 100 and CPT, which is obviously seen to span from the thick plate, moderately thick plate and thin plate [22]. The present work obtained non-dimensional result of stresses and displacements of the plate by expressing the deflection and rotation functions in the form of trigonometry to analyze the bending characteristics of the plate.
The non-dimensional result in the Figure 6 shows that as the span-thickness ratio of the plate increase, the in-plane displacement along x and y axis (u and v) increases too, whereas, the deflection (w) which occurs at the plate due to the applied load decrease with increases in the value of the span-thickness ratio of the plate. On the other hand, the stress perpendicular to the x, y and z axis ( , & ) decreases as the span-depth ratio of the plate increases. Meanwhile, the increase at the span-thickness ratio of the plate increases the value shear stress along the x-y ( ) while the span -depth ratio causes a decrease in the value shear stress along the x-z and y-z plane ( & ). These decrease continue until failure occurs in the plate structure. Figure 6 shows that, at a span-thickness ratio between 4 and 20, the value of out of plane displacement varies between 0.0026 and 0.0137. These values maintain a constant value of 0.0132 at the span -thickness 50 till 100 which is the same as the CPT. A variation in deflection is discovered more when the plate is thicker and less when the span-thickness increase (thinner plate) under the same loading capacity/condition. This deflection becomes constant and equal to the value of the CPT at span-thickness ratio of 50 and above under the same loading capacity/condition. These decrease continue until the plate structure deflects beyond the elastic yield stress, hence, failure occurs. Thus, it can be said that at spanthickness ratio between 4 and 20 the plate is regarded as thick. The spanthickness ratio beyond 20 till 50 the plate is regarded as moderately thick while the thin plate is regarded as those beyond spanthickness ratio beyond 50.

Figure 7, Displacement and stresses of a CCCC plate aspect ratio of 2
The non-dimensional result in the Figure 7 shows that as the span-thickness ratio of the plate increase, the in-plane displacement along x and y axis (u and v) increases too, whereas, the deflection (w) which occurs at the plate due to the applied load decrease with increases in the value of the span-thickness ratio of the plate. On the other hand, the stress perpendicular to the x, y and z axis( , & ) decreases as the span-depth ratio of the plate increases. Meanwhile, the increase at the span-thickness ratio of the plate increases the value shear stress along the x-y ( ) while the span-depth ratio causes a decrease in the value shear stress along the x-z and y-z plane ( & ). Figure 7 shows that, at a span-thickness ratio between 4 and 20, the value of out of plane displacement varies between 0.0509 and 0.0291. These values maintain a constant value of 0.0283 at the span -thickness 50 till 100 which is equal to the value of the CPT. A variation in deflection is discovered more when the plate is thicker and less when the spanthickness increase (thinner plate) under the same loading capacity/condition. This deflection becomes constant and the same as the CPT at span-thickness ratio of 50 and above under the same loading capacity/condition. These decrease continue until the plate structure deflects beyond the elastic yield stress, hence, failure occurs. Thus, it can be said that at spanthickness ratio between 4 and 20 the plate is regarded as thick. The spanthickness ratio beyond 20 till 50 the plate is regarded as moderately thick while the thin plate is regarded as those beyond spanthickness ratio beyond 50. Figure 6 and 7 shows that as the aspect ratio of the plate increase, the in-plane displacement along x and y axis (u and v) decrease whereas, the deflection (w) which occurs at the plate due to the applied load increase with increases in the value of the span-thickness ratio of the plate. On the other hand, the stress perpendicular to the x, y and z axis ( , & ) increases as the span-depth ratio of the plate increases. This means that, if the plate material is stretched beyond the elastic limit, the failure in a plate structure is bound to occur as the more stresses are induced within the plate element which affects the performance in terms of the serviceability of the plate. Thus, caution must be taken when selecting the depth and other dimensions along the x and y co-ordinate of the plate to ensure accuracy of the analysis and safety in the construction.

Study in the
In summary, there are three categories of rectangular plates. The plates whose deflection and vertical shear stress do not vary much with CPT is categorized as thin plate. Hence, the plate whose deflection and transverse shear stress varies very much from zero is categorized as thick plates. Thus, the span-thickness ratio for these categories of rectangular plates are: Thick plate is categorized as the plate with the span to thickness ratio∶ / ≤ 20 while the thin plate is categorized as the plate with the span-thickness ratio: / ≥ 100. In between the thick and thin plate exist, the moderately thick plate. Thick plate is categorized as the plate with the span-thickness ratio∶ / > 20 < 50. Meanwhile, the present theory stress prediction shows that the result of the displacement and stress of thin and moderately thick plate using the 3-D theory is the same for the bending analysis of rectangular plate under the CCCC boundary condition.
The comparative analysis was performed in this study as presented in the Table 2 and Table 3 to show the disparity between different theories used in the plate analysis. This theory includes the analytical process ranging from Double integration, according to Levi, Mindlin theory, FSDT, HSDT and 3-D elasticity. Numerical and approximate approaches were also adopted to compare and show the validity of the derived relationships. The present study was also validated with the past works using different shape or mathematical functions such as polynomial, exponential, hyperbolic and trigonometric displacement functions. The result of the percentage difference evaluation showed that the plate with the largest thickness (a/t of 4) gives a percentage difference of 1.74, 0.55, 0.37, 0.37, 1.29, 1.01 and 3.12% of the work of Ibearugbulem et al. (2018) [28],   [34], Li et al. (2015) [35], Liu & Liew (1998) [36], Lok & Cheng (2001) [37], Shen & He (1995) [38] and Zhong & Xu (2017) [39] respectively, when compared with the present study. On the other hand, the thick plate at a/t of 10 gives a percentage difference of 0. 33 [35], Liu & Liew (1998) [36] and Shen & He (1995) [38] respectively, when compared with the present study. The result of the work of Lok & Cheng (2001) [37] and Zhong & Xu (2017) [39] at a/t of 20 is not available in the literature in consideration. Table 3 shows that, the difference with past works in consideration percentagewise decreases and converges as the plate is getting thinner. It can be deduced that, the difference with past works in consideration percentagewise at a/t of 10 and 20 gives a constant value of 0.15% and 0.13% respectively, a value which could be the same difference when compared with the value of the CPT.  Table 3, it is found that the average the difference with 3-D elasticity trigonometric theory percentagewise and those of the 2-D HSDT with assumed polynomial shape function [30] and 2-HSDT with exact shape function [34] is 0.36% and 0.29% respectively. The average the difference percentagewise with 2-D Mindlin FSDT [37,39] is about 2.41% and 3.62%, while the average difference percentagewise with the 2-D thick plate numeric analysis [35] and moderately thick [38] is 53% and 61% respectively. The overall average difference percentage wise with 2-D Mindlin FSDT [37,39] is about 3.02% while the overall average difference percentagewise with the 2-D numeric analysis [36,39] is about 0.62%. The present study overall average difference values of deflection percentagewise with those using 2-D HSDT shape functions [30,34] is about 0.33%. This negligible difference showed that HSDT is preferable to the Levi, Mindlin theory and numerical method in the thick plate analysis. Consequently, the smaller value of percentage between the present study and those of 2-D HSDT (0.33%) showed that HSDT using derived polynomial displacement function is better compared to those of HSDT with an assumed shape function as it predicted an exact deflection which proved more reliable in the analysis of thick plate under the same boundary condition. Despite the fact that both Ibearugbulem et al. (2018) [28],   [34], Li et al. (2015) [35], Liu & Liew (1998) [36], Lok & Cheng (2001) [37], Shen & He (1995) [38] and Zhong & Xu (2017) [39] used shear deformation theory their work differs more when compared with the present study. This shows that HSDT derived shape function, enhanced close form solution in plate analysis. However, the overall average difference values of deflection percentagewise with Ibearugbulem et al. (2018) [28], Ibearugbulem & Onyeka (2020) [34], Li et al. (2015) [35], Liu & Liew (1998) [36], Lok & Cheng (2001) [37], Shen & He (1995) [38] and Zhong & Xu (2017) [39] is 1. 19%. This showed that at the 98 % confidence level, both theory and methods are the same for a thick plate analysis.
It is worth noting that the 2-D RPT with exact deflection gives a closer result when compared with exact 3-D plate theory than those 2-D RPTs with an assumed deflection and other RPT and CPT in the thick plate analysis. Hence, an exact 3-D theory is required to achieve efficiency. Thus, the present model uses the six stress elements to yield the exact solution for the analysis of a thick plate that is clamped and supported on all the edges (CCCC). Hence, the result of the present analysis, which contains all the stress elements with an exact deflection function, ensures that the variation of the stresses through the thickness of the plate which induces stresses can be used with confidence for bending analysis of the plate.

Conclusions
The 3-D bending and stress analysis of thick rectangular plates using 3-D elasticity theory has been investigated, and the following conclusion has been drawn:  A closer-form solution is predicted by the trigonometric shape function than by the polynomial displacement function.
 The present theory of stress prediction shows that the result of the displacement and stress of thin and moderately thick plates using the 3-D theory is the same at a span-thickness ratio beyond 50% for the bending analysis of rectangular plates under the CCCC boundary condition.
 Classical theory is good for thin plates but over-predicts buckling loads in relatively thick plates.
 Plate analysis requires 3-D analogy for a true solution, but the 2-D shear deformation theory gives an approximate solution which is practically unrealistic.
 The 3-D exact plate model developed in this study can be used in the analysis of any category of the plate.

Data Availability Statement
Data sharing is not applicable to this article.

Funding
The authors received no financial support for the research, authorship, and/or publication of this article.

Ethical Approval
Not applicable.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.