Available online at www.HighTechJournal.org  

HighTech and Innovation  
Journal 

Vol. 3, No. 3, September, 2022 

 

 

267 

 

ISSN: 2723-9535 

 

Static Elastic Bending Analysis of a Three-Dimensional Clamped 

Thick Rectangular Plate using Energy Method 

F. C. Onyeka 1* , T. E. Okeke 2*, B. O. Mama 2  

1 Department of Civil Engineering, Edo State University, Uzairue, Edo State, Nigeria. 

2 Department of Civil Engineering, University of Nigeria Nsukka, Nigeria. 

Received 03 June 2021; Revised 16 July 2022; Accepted 02 August 2022; Available online 16 August 2022 

Abstract 

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. 

Keywords: Exact Static Theory; Equilibrium Equation; Bending of 3-D Clamped Plate; Trigonometric Model. 

 

1. 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 non-

homogeneous. 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 (𝑎/𝑡), Mahi et al. (2015) [3] and Timoshenko & Woinowsky-Krieger (1959) [4] classified 

rectangular plates with 50 ≤ 𝑎/𝑡 ≤ 100 as thin plate, 20 ≤ 𝑎/𝑡 ≤ 50 as moderately thick and 𝑎/𝑡 ≤ 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 

                                                           
* Corresponding author:  onyeka.festus@edouniversity.edu.ng; edozie.okeke@unn.edu.ng 

 
http://dx.doi.org/10.28991/HIJ-2022-03-03-03 

 This is an open access article under the CC-BY license (https://creativecommons.org/licenses/by/4.0/). 

© Authors retain all copyrights. 

mailto:edozie.okeke@unn.edu.ng
https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0000-0002-2668-9753
https://orcid.org/0000-0002-0175-0060


HighTech and Innovation Journal         Vol. 3, No. 3, September, 2022 

268 

 

due to the impact of forces and moments [9, 10]. As a result of applied load, a structural member is displaced and stresses 

are induced. Consequently, the structure tends to bend to withstand the load. The bending features of plates are strongly 

influenced by their thickness in comparison with their other parameters [10]. To ensure the stability of thick plates for 

resisting design load, bending analysis is needed so as to determine the displacements, moments, and stresses at various 

points of the plate [11, 12]. 

Many researchers have developed and applied several theories to avoid the complexity of analyzing rectangular plates 

as a three-dimensional element. These theories include the classical plate theory (CPT) and the refined plate theory 

(RPT). These theories offer solutions to plate problems in either exact or approximate form. Classical plate theory [13] 

cannot ascertain the proper bending behavior of thick plates as the shear deformation effect is overlooked [14, 15]. The 

introduction of shear deformation effects on the plate displacements distinguishes thick plate theories from thin plate 

theories. This resulted in the formulation of refined plate theories. The refined plate theories (RPT), which can be 

employed for thick plate analysis [16], consists of first-order shear deformation theory (FSDT), also called Reissner-

Mindlin theory [17-19], and higher-order shear deformation theories (HSDTs), that provide zero shear stress conditions 

at the upper and underside of the plates without the shear correction factor [20-22]. 

Refined plate theories, which has been used by scholars such as [23-25], consider five strains, five stress components, 

assuming the normal stress and strain along the z-axis to be zero. Refined plate theories are inadequate to express an 

accurate bending response of a typical 3-D thick plate. In order to overcome the errors of refined plate theories (RPT) in 

analysis of rectangular plates, three-dimensional theory must be employed to ensure that no stress or strain element is 

assumed to be zero. For a typical 3-D thick plate analysis, refined plate theories are indelicate, hence the need for precise 

results through the application of 3-D theory is justified. 

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. 

2. 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. 



HighTech and Innovation Journal         Vol. 3, No. 3, September, 2022 

269 

 

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. 

3. Methodology 

The research methodology of this study is presented by considering a rectangular plate in Figure 1 as a three-

dimensional 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. 



HighTech and Innovation Journal         Vol. 3, No. 3, September, 2022 

270 

 

 

Figure 1. An element of thick rectangular plate showing middle surface 

Figure 2 is a flowchart which indicates the procedures of formulating the potential energy equation in the form of the 

kinematics and three-dimensional constitutive relations for a static elastic theory of plate, thereafter, the governing 

equations were derived and solved to obtain formula for the analysis. 

 
Figure 2. Flowchart to the article analysis procedure as presented in the research methodology 

3.1. 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: 

Static theory of elasticity was used to get strain 
and stress relationship

Potential energy equation 
formulation

Equilibrium and governing equation of 
3-D plate were developed

Solving equilibrium equation 
to obtain rotation function

Getting deflection function and substitute its 
value into energy equation to obtain 

deflection coefficients

Deflection coefficient was used to obtain 
the expression for displacement and 

stress of the plate

x 

a 

b 

z 

y 

t 

Middle surface 
 

Bottom fiber 

 

Top fiber 
 

Total Deformation line  

𝜃𝑥  𝑜𝑟 𝜃𝑦 

𝑡 

 



HighTech and Innovation Journal         Vol. 3, No. 3, September, 2022 

271 

 

𝑥 =
𝜕𝑢

𝜕𝑧
  (1) 

𝑦 =
𝜕𝑣

𝜕𝑧
  (2) 

where, 𝜃𝑥  and 𝜃𝑦 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: 

𝑥 =
1

t
.
𝜕𝑢

𝜕𝑠
  (3) 

𝑦 =
1

t
.
𝜕𝑣

𝜕𝑠
  (4) 

Where: 

𝑧 = 𝑡𝑠  (5) 

Re-arranging Equation 3 and 4 gives: 

𝑢 = 𝑡𝑠. 𝑠𝑥  (6) 

𝑣 = 𝑡𝑠. 𝑠𝑦   (7) 

where, 𝑢 and 𝑣 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: 

𝑥 = 
1

a
.
𝜕𝑢

𝜕𝑅
  (8) 

𝑦 =
1

aβ
.
𝜕𝑣

𝜕𝑄
  (9) 

𝑧 =
1

t
.
𝜕𝑤

𝜕𝑠
  (10) 


𝑥𝑦

=
1

aβ
.
𝜕𝑢

𝜕𝑄
+

1

a
.
𝜕𝑣

𝜕𝑅
  (11) 


𝑥𝑧

=
1

t
.
𝜕𝑢

𝜕𝑠
+ 

1

a
.
𝜕𝑤

𝜕𝑅
  (12) 


𝑦𝑧

=
1

t
.
𝜕𝑣

𝜕𝑠
+

1

aβ
.
𝜕𝑤

𝜕𝑄
  (13) 

where, 𝜀𝑥, 𝜀𝑦 and 𝜀𝑧 are normal strain along x axis, y axis and z axis respectively, 𝛾𝑥𝑦 , 𝛾𝑥𝑧 𝑎𝑛𝑑 𝛾𝑦𝑧 represents the shear 

strain in the plane parallel to the x-y, x-z and y-z plane. 

3.2. Constitutive Relations 

The three dimensional constitutive relation is determined using a generalized Hooke’s principle as: 

[
 
 
 
 
 
𝜎𝑥

𝜎𝑦

𝜎𝑧
τxz

τyz

τxy]
 
 
 
 
 

=
E

(1+𝜇)(1−2𝜇)

[
 
 
 
 
 
 
 
(1 − 𝜇) 𝜇 𝜇 0 0 0

𝜇 (1 − 𝜇) 𝜇 0 0 0

𝜇 𝜇 (1 − 𝜇) 0 0 0

0 0 0 (
1−2𝜇

2
) 0 0

0 0 0 0 (
1−2𝜇

2
) 0

0 0 0 0 0 (
1−2𝜇

2
)]
 
 
 
 
 
 
 

[
 
 
 
 
 
 
 
 
𝜀𝑥

𝜀𝑦

𝜀𝑧

γxz

γyz

γxy]
 
 
 
 
 
 
 
 

   (14) 

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: 

𝑥 =
E

(1+μ)(1−2μ)
[ (1 − μ)  

ts

a
.
𝜕𝑠𝑥

𝜕𝑅
+ 

ts

aβ
.
𝜕𝑠𝑦

𝜕𝑄
+ 

1

t
.
∂w

∂S
]  (15) 

𝑦 =
E

(1+μ)(1−2μ)
[ ts .

𝜕𝑥

𝑎𝜕𝑅
+

(1−𝜇)ts

𝑎𝛽
.
𝜕𝑦

𝜕𝑄
+



𝑡
.
𝜕𝑤

𝜕𝑆
]  (16) 

𝑧 =
E

(1+μ)(1−2μ)
[ ts .

𝜕𝑥

𝑎𝜕𝑅
+

ts

𝑎𝛽
.
𝜕𝑦

𝜕𝑄
+

(1−𝜇)

𝑡
.
𝜕𝑤

𝜕𝑆
]  (17) 



HighTech and Innovation Journal         Vol. 3, No. 3, September, 2022 

272 

 

𝑥𝑦 =
𝐸(1−2)

(1+𝜇)(1−2𝜇)
. [

ts

2𝑎𝛽

𝜕𝑥

𝜕𝑄
+

ts𝜕𝑦

2𝑎𝜕𝑅
]  (18) 

𝑥𝑧 =
(1−2)𝐸

(1+𝜇)(1−2𝜇)
. [
𝑥

2
+

1

2𝑎

𝜕𝑤

𝜕𝑅
]  (19) 

𝑦𝑧 =
(1−2)𝐸

(1+𝜇)(1−2𝜇)
. [
𝑦

2
+

1

2𝑎𝛽

𝜕𝑤

𝜕𝑄
]  (20) 

3.3. Strain Energy 

The strain energy (𝑈) is mathematically defined as: 

𝑈 =
𝑎𝑏𝑡

2
∫ ∫ ∫ (𝑥𝑥 + 𝑦𝑦 + 𝑧𝑧 + 𝜏𝑥𝑦𝑥𝑦

+ 𝜏𝑥𝑧𝑥𝑧
+ 𝜏𝑦𝑧𝑦𝑧

)
0.5

−0.5

1

0

1

0
𝑑𝑅 𝑑𝑄 𝑑𝑆  (21) 

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: 

U =
Et3𝑎𝑏

24(1+μ)(1−2μ)a2 ∫ ∫ [(1 − μ) (
𝜕𝑠𝑥

𝜕𝑅
)

2

 +
1

𝛽

𝜕𝑠𝑥

𝜕𝑅
.
𝜕𝑠𝑦

𝜕𝑄
+

(1−μ)

𝛽2 (
𝜕𝑠𝑦

𝜕𝑄
)

2

+
(1−2)

2β2 (
𝜕𝑠𝑥

𝜕𝑄
)

2

+
(1−2)

2
(

𝜕𝑠𝑦

𝜕𝑅
)

2

+
1

0

1

0

12.(1−2)

2t2
(a2𝑠𝑥

2 + a2𝑠𝑦
2 + (

𝜕w

𝜕𝑅
)

2

+
1

β2 (
𝜕w

𝜕𝑄
)

2

+ 2a. 𝑠𝑥
𝜕w

𝜕𝑅
+

2a.𝑠𝑦

𝛽

𝜕w

𝜕𝑄
) + 0 ∗ 2

μa

t2
. ( 

𝜕𝑠𝑥

𝜕𝑅
.
𝜕w

𝜕𝑆
+

1

β
.
𝜕𝑠𝑦

𝜕𝑄
.
𝜕w

𝜕𝑆
) +

(1−μ)a2

𝑡4 (
𝜕w

𝜕𝑆
)

2

] dR dQ  

(22) 

Where; 

𝐷∗ =
𝐸𝑡3

12(1+𝜇)(1−2𝜇)
  (23) 

3.4. Energy Equation Formulation 

The total potential energy is mathematically expressed as: 

 = U − V  (24) 

𝑉 = 𝑎𝑏𝑞𝐴1 ∫ ∫ ℎ
1

0

1

0
𝑑𝑅 𝑑𝑄  (25) 

where, V, q, 𝐴1 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: 

 =
𝐷∗𝑎𝑏

2𝑎2 ∫ ∫ [(1 − 𝜇) (
𝜕𝑠𝑥

𝜕𝑅
)

2

 +
1

𝛽

𝜕𝑠𝑥

𝜕𝑅
.
𝜕𝑠𝑦

𝜕𝑄
+

(1−𝜇)

𝛽2 (
𝜕𝑠𝑦

𝜕𝑄
)

2

+
(1−2)

2𝛽2 (
𝜕𝑠𝑥

𝜕𝑄
)

2

+
(1−2)

2
(

𝜕𝑠𝑦

𝜕𝑅
)

2

+
1

0

1

0

6(1−2)

𝑡2 (𝑎2𝑠𝑥
2 + 𝑎2𝑠𝑦

2 + (
𝜕𝑤

𝜕𝑅
)

2

+
1

𝛽2 (
𝜕𝑤

𝜕𝑄
)

2

+ 2𝑎. 𝑠𝑥
𝜕𝑤

𝜕𝑅
+

2𝑎.𝑠𝑦

𝛽

𝜕𝑤

𝜕𝑄
) +

(1−𝜇)𝑎2

𝑡4 (
𝜕𝑤

𝜕𝑆
)

2

] 𝑑𝑅 𝑑𝑄 −

∫ ∫ 𝑎𝑏𝑞ℎ𝐴1𝜕𝑅𝜕𝑄 
1

0

1

0
  

(26) 

3.5. 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: 

ℎ = [1   𝑅   𝐶𝑜𝑠(𝑐1𝑅)  𝑆𝑖𝑛(𝑐1𝑅)] [

𝑎0

𝑎1
𝑎2

𝑎3

] . [1   𝑄   𝐶𝑜𝑠(𝑐1𝑄)  𝑆𝑖𝑛(𝑐1𝑄)] [

𝑏0

𝑏1

𝑏2

𝑏3

] /𝐴1  (27) 

𝜃𝑥 =
𝑐

𝑎
. ∆0. [1   𝑐1𝑆𝑖𝑛(𝑐1𝑅)  𝑐1𝐶𝑜𝑠(𝑐1𝑅)] [

𝑎1

𝑎2

𝑎3

] . [1   𝑄   𝐶𝑜𝑠 (𝑐1𝑄)  𝑆𝑖𝑛 (𝑐1𝑄)] [

𝑏0

𝑏1

𝑏2

𝑏3

]  (28) 

𝑦 =
𝑐

𝑎β
. ∆0. [1   𝑅   𝐶𝑜𝑠(𝑐1𝑅)  𝑆𝑖𝑛(𝑐1𝑅)] [

𝑎0

𝑎1
𝑎2

𝑎3

] . [1  𝑐1𝑆𝑖𝑛(𝑐1𝑄)  𝑐1𝐶𝑜𝑠(𝑐1𝑄)] [

𝑏1

𝑏2

𝑏3

]  (29) 

Let; 



HighTech and Innovation Journal         Vol. 3, No. 3, September, 2022 

273 

 

𝑤 = 𝐴1. ℎ  (30) 

𝑥 =
𝐴2

𝑎
.
𝜕ℎ

𝜕𝑅
  (31) 

𝑦 =
𝐴3

𝑎𝛽
.
𝜕ℎ

𝜕𝑄
  (32) 

where; 𝐴2 and 𝐴3 are the coefficient of shear deformation along x axis and coefficient of shear deformation along y axis 

respectively. 

Substituting Equations 30, 31 and 32 into 26, gives: 

 =
𝐷∗𝑎𝑏

2𝑎4 [(1 − 𝜇)𝐴2
2𝑘𝑥  +

1

𝛽2 [𝐴2. 𝐴3 +
(1−2)𝐴2

2

2
+

(1−2)𝐴3
2

2
] 𝑘𝑥𝑦 +

(1−𝜇)𝐴3
2

𝛽4 𝑘𝑦 + 6(1 − 2) (
𝑎

𝑡
)

2

([𝐴2
2 +

𝐴1
2 + 2𝐴1𝐴2]. 𝑘𝑧 +

1

𝛽2 . [𝐴3
2 + 𝐴1

2 + 2𝐴1𝐴3]. 𝑘2𝑧) −
2𝑞𝑎4𝑘ℎ𝐴1

𝐷∗ ]  
(33) 

Where; 

𝑘𝑥 = ∫ ∫ (
𝜕2ℎ

𝜕𝑅2)
2

1

0

1

0
𝑑𝑅𝑑𝑄  (34) 

𝑘𝑥𝑦 = ∫ ∫ (
𝜕2ℎ

𝜕𝑅𝜕𝑄
)

2
1

0

1

0
𝑑𝑅𝑑𝑄  (35) 

𝑘𝑦 = ∫ ∫ (
𝜕2ℎ

𝜕𝑄2)
2

1

0

1

0
𝑑𝑅𝑑𝑄  (36) 

𝑘𝑧 = ∫ ∫ (
𝜕ℎ

𝜕𝑅
)

21

0

1

0
𝑑𝑅𝑑𝑄  (37) 

𝑘2𝑧 = ∫ ∫ (
𝜕ℎ

𝜕𝑄
)

21

0

1

0
𝑑𝑅𝑑𝑄  (38) 

𝑘ℎ = ∫ ∫ ℎ
1

0
.

1

0
𝑑𝑅𝑑𝑄  (39) 

Minimizing Equation 33 with respect to 𝐴2 gives: 

𝜕

𝜕𝐴2
= (1 − 𝜇)𝐴2𝑘𝑥 +

1

2𝛽2
[𝐴3 + 𝐴2(1 − 2)]𝑘𝑥𝑦 + 6(1 − 2) (

𝑎

𝑡
)

2
[𝐴2 + 𝐴1]. 𝑘𝑧 = 0  (40) 

Minimizing Equation 33 with respect to 𝐴3 gives: 

𝜕

𝜕𝐴2
=

(1−𝜇)𝐴3

𝛽4 𝑘𝑦 +
1

2𝛽2
[𝐴2 + 𝐴3(1 − 2)]𝑘𝑥𝑦 +

6

𝛽2
(1 − 2) (

𝑎

𝑡
)

2
([𝐴3 + 𝐴1]. 𝑘2𝑧) = 0  (41) 

Rewriting Equations 34 and 35 gives: 

[(1 − 𝜇)𝑘𝑥 +
1

2𝛽2
(1 − 2)𝑘𝑥𝑦 + 6(1 − 2) (

𝑎

𝑡
)

2

𝑘𝑧] 𝐴2 + [
1

2𝛽2 𝑘𝑥𝑦] 𝐴3 = [−6(1 − 2) (
𝑎

𝑡
)

2

𝑘𝑧] 𝐴1  (42) 

[
1

2𝛽2 𝑘𝑥𝑦] 𝐴2 + [
(1−𝜇)

𝛽4 𝑘𝑦 +
1

2𝛽2
(1 − 2)𝑘𝑥𝑦 +

6

𝛽2
(1 − 2) (

𝑎

𝑡
)

2

𝑘2𝑧] 𝐴3 = [−
6

𝛽2
(1 − 2) (

𝑎

𝑡
)

2

𝑘𝑄] 𝐴1  (43) 

Solving Equations 42 and 43 simultaneously gives: 

𝐴2 = 𝑀𝐴1  (44) 

𝐴3 = 𝑁𝐴1  (45) 

Let: 

𝑀 =
(𝑟12𝑟23−𝑟13𝑟22)

(𝑟12𝑟12−𝑟11𝑟22)
  (46) 

𝑁 =
(𝑟12𝑟13−𝑟11𝑟23)

(𝑟12𝑟12−𝑟11𝑟22)
  (47) 

Where; 

𝑟11 = (1 − 𝜇)𝑘𝑥 +
1

2𝛽2
(1 − 2)𝑘𝑥𝑦 + 6(1 − 2) (

𝑎

𝑡
)

2

𝑘𝑧  (48) 

𝑟22 =
(1−𝜇)

𝛽4 𝑘𝑦 +
1

2𝛽2
(1 − 2)𝑘𝑥𝑦 +

6

𝛽2
(1 − 2) (

𝑎

𝑡
)

2

𝑘2𝑧  (49) 



HighTech and Innovation Journal         Vol. 3, No. 3, September, 2022 

274 

 

𝑟12 = 𝑟21 =
1

2𝛽2 𝑘𝑥𝑦;  𝑟13 = −6(1 − 2) (
𝑎

𝑡
)

2

𝑘𝑧;  𝑟23 = 𝑟32 = −
6

𝛽2
(1 − 2) (

𝑎

𝑡
)

2

𝑘2𝑧  (50) 

Minimizing Equation 33 with respect to A1 gives: 

𝜕𝛱

𝜕𝐴1
=

𝐷∗𝑎𝑏

2𝑎4 [6(1 − 2) (
𝑎

𝑡
)

2

([2𝐴1 + 2𝐴2]. 𝑘𝑧 +
1

𝛽2 . [2𝐴1 + 2𝐴3]. 𝑘2𝑧) −
2𝑞𝑎4𝑘ℎ

𝐷∗ ] = 0  (51) 

That is: 

6(1 − 2) (
𝑎

𝑡
)

2

([𝐴1 + 𝑈𝐴1]. 𝑘𝑧 +
1

𝛽2 . [𝐴1 + 𝑉𝐴1]. 𝑘2𝑧) −
𝑞𝑎4𝑘ℎ

𝐷∗ = 0  (52) 

Factorizing Equations 52 and simplifying gives: 

6(1 − 2) (
𝑎

𝑡
)

2

𝐴1 ([1 + 𝑈]. 𝑘𝑧 +
1

𝛽2 . [1 + 𝑉]. 𝑘2𝑧) =
𝑞𝑎4𝑘ℎ

𝐷∗   (53) 

𝑇𝐴1 =
𝑞𝑎4𝑘ℎ

𝐷∗   (54) 

𝐴1 =
𝑞𝑎4

𝐷∗ ( 
𝑘ℎ

𝑇
)  (55) 

Where; 

𝑇 = 6(1 − 2) (
𝑎

𝑡
)

2

∗ ([1 + 𝑈]. 𝑘𝑧 +
1

𝛽2 . [1 + 𝑉]. 𝑘2𝑧)  (56) 

3.6. 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. 

 

Figure 4. CCCC Rectangular Plate 

The boundary conditions of the plate in Figure 4 are as follows: 

At 𝑅 =  𝑄 =  0;  𝑤 = 0 (57) 

At 𝑅 =  𝑄 =  0; 
𝑑𝑤

𝑑𝑅
=

𝑑𝑤

𝑑𝑄
= 0 (58) 

At 𝑅 =  𝑄 =  1;  𝑤 = 0 (59) 

At 𝑅 =  𝑄 =  1;
𝑑𝑤

𝑑𝑅
=

𝑑𝑤

𝑑𝑄
= 0 (60) 

The derived trigonometric deflection 𝑤 (𝑥, 𝑦) functions is subjected to a CCCC boundary condition to get the 

particular solution of the deflection. Hence, the analytical solution of the deflection of the plate in trigonometric form 

after satisfying the boundary conditions for all edges clamped rectangular plate presented in the Equation 61: 

a 

b 

𝑄 

𝑅 O 

C 

C 

C 

C 



HighTech and Innovation Journal         Vol. 3, No. 3, September, 2022 

275 

 

𝑤 = 𝑎2 × 𝑏2( 𝐶𝑜𝑠2𝜋𝑅 − 1). (𝐶𝑜𝑠2𝜋𝑄 − 1)  (61) 

where the coefficient of the deflection, 

𝐴1 = 𝑎2 × 𝑏2  (62) 

while the shape function 

ℎ = ( 𝐶𝑜𝑠2𝜋𝑅 − 1). (𝐶𝑜𝑠2𝜋𝑄 − 1)  (63) 

3.7. Exact Displacement and Stress Expression 

By substituting the value of 𝐴1, 𝐴2 and 𝐴3 in Equations 49, 38 and 39 into Equation 15 to 20 and substitute 

appropriately, the in-plane displacement along x-axis becomes: 

𝑢 = 𝑡𝑠.
𝑀

𝑎
.
𝑞𝑎4

𝐷∗ (
𝑘ℎ

𝑇
)

𝜕ℎ

𝜕𝑅
  (64) 

The in-plane displacement along y-axis becomes: 

𝑣 = 𝑡𝑠.
𝑁

𝑎𝛽
.
𝑞𝑎4

𝐷∗ (
𝑘ℎ

𝑇
)

𝜕ℎ

𝜕𝑄
  (65) 

The deflection equation of the plate as: 

𝑤 = ( 𝐶𝑜𝑠2𝜋𝑅 − 1). (𝐶𝑜𝑠2𝜋𝑄 − 1).
𝑞𝑎4

𝐷∗ (
𝑘ℎ

𝑇
)  (66) 

The six stress elements are presented in Equations 15 to 20 as: 

𝑥 =
E

(1+μ)(1−2μ)
[ (1 − μ)  

ts

a
.
∂2ℎ

∂𝑅2 + 
ts

aβ
.
∂2ℎ

∂𝑄2 + 
1

t
.
𝑞𝑎4

𝐷∗ (
𝑘ℎ

𝑇
)

∂h

∂S
]  (67) 

𝑦 =
E

(1+μ)(1−2μ)
[
ts

𝑎
 .

∂2ℎ

∂𝑅2 +
(1−𝜇)ts

𝑎𝛽
.
∂2ℎ

∂𝑄2 +


𝑡
.
𝑞𝑎4

𝐷∗ (
𝑘ℎ

𝑇
)

∂h

∂S
]  (68) 

𝑧 =
E

(1+μ)(1−2μ)
[
ts

𝑎
 .

∂2ℎ

∂𝑅2 +
ts

𝑎𝛽
.
∂2ℎ

∂𝑄2 +
(1−𝜇)

𝑡
.
𝑞𝑎4

𝐷∗ (
𝑘ℎ

𝑇
)

∂h

∂S
]  (69) 

𝑥𝑦 =
𝐸(1−2)

(1+𝜇)(1−2𝜇)
. [

ts

2𝑎𝛽
.

∂2𝜕ℎ

∂𝑅 ∂𝑄
+

ts

2𝑎
.

∂2𝜕ℎ

∂𝑅 ∂𝑄
]  (70) 

𝑥𝑧 =
(1−2)𝐸

(1+𝜇)(1−2𝜇)
. [

1

2

∂h

∂𝑅
+

1

2𝑎
.
𝑞𝑎4

𝐷∗ (
𝑘ℎ

𝑇
)

𝜕ℎ

𝜕𝑅
]  (71) 

𝑦𝑧 =
(1−2)𝐸

(1+𝜇)(1−2𝜇)
. [

1

2

∂h

∂𝑄
+

1

2𝑎𝛽
.
𝑞𝑎4

𝐷∗ (
𝑘ℎ

𝑇
)

𝜕ℎ

𝜕𝑄
]  (72) 

Thus, the stiffness coefficients of CCCC rectangular plate is obtained from Equations 34 to 39 and presented in the 

Table 1. 

Table 1. Trigonometric form of stiffness coefficients of CCCC rectangular plate 

Deflection form 𝒌𝒙 𝒌𝒙𝒚 𝒌𝒚 𝒌𝒛 𝒌𝟐𝒛 𝒌𝒉 

Trigonometry 12𝜋4 4𝜋4 12𝜋4 3𝜋2 3𝜋2 1.0 

4. 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. 



HighTech and Innovation Journal         Vol. 3, No. 3, September, 2022 

276 

 

 

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 

0

200

400

600

800

1000

1200

1400

kx kxy ky kz k2z kq

B
o
u

n
d

a
ry

 c
o
n

d
it

io
n

Stiffness Coefficient

Boundary condition

-1

-0.5

0

0.5

1

1.5

2

2.5

4 5 10 15 20 50 100 CPT

D
is

p
la

c
e
m

e
n

t 
(w

, 
u

) 
a

n
d

 S
tr

e
ss

 (
σ

x
, 

σ
z
, 
τ
x
y

, 
τ
y

z
) 

Span-thicknes ratio (a/t)

w u Qx Qz Txy Tyz



HighTech and Innovation Journal         Vol. 3, No. 3, September, 2022 

277 

 

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 (𝜎𝑥, 𝜎𝑦 & 𝜎𝑧) 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 (𝜏𝑥𝑦) while the span - depth 

ratio causes a decrease in the value shear stress along the x-z and y-z plane ( 𝜏𝑥𝑧 & 𝜏𝑦𝑧). These decrease continue until 

failure occurs in the plate structure. 

 

Figure 7, Displacement and stresses of a CCCC plate aspect ratio of 2 

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 span – thickness ratio between 4 and 20 the plate is regarded as thick. The span – thickness ratio beyond 20 till 50 the 

plate is regarded as moderately thick while the thin plate is regarded as those beyond span – thickness ratio beyond 50. 

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(𝜎𝑥, 𝜎𝑦 & 𝜎𝑧) 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 (𝜏𝑥𝑦) while the span-depth 

ratio causes a decrease in the value shear stress along the x-z and y-z plane (𝜏𝑥𝑧 & 𝜏𝑦𝑧). 

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 span-

thickness 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 span – thickness ratio between 4 and 20 the plate is regarded as thick. The span – thickness ratio beyond 20 till 50 the 

plate is regarded as moderately thick while the thin plate is regarded as those beyond span – thickness ratio beyond 50. 

Study in the 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 (𝜎𝑥 , 𝜎𝑦 & 𝜎𝑧) 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 

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4 5 10 15 20 50 100 CPTD
is

p
la

ce
m

en
t 

(w
, 
u

) 
a

n
d

 s
tr

e
ss

 (
σ

x
, 

σ
z,

 τ
x
y

, 
τ
y

z)

Span-thicknes ratio (a/t)

w u Qx Qz Txy Tyz



HighTech and Innovation Journal         Vol. 3, No. 3, September, 2022 

278 

 

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. 

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∶ 𝑎/𝑡 ≤  20 while the thin plate is 

categorized as the plate with the span-thickness ratio: 𝑎/𝑡 ≥ 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∶ 𝑎/𝑡 >  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], 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] 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, 0.98, 0.98, 0.98, 1.64, 

0.98% and 2.95% of the work of 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] 

respectively, when compared with the present study. More so, the thick plate at a/t of 20 gives a percentage difference 

of 0.80, 0.80, 2.85, 2.85 and 2.89% of the work of Ibearugbulem et al. (2018) [28], Ibearugbulem & Onyeka (2020) [34], 

Li et al. (2015) [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 2. Comparative deflection analysis for square plate at varying span-thickness ratio (β = a/t) between present study 

and past studies 

a/t Present [28]  [34] [35] [36] [37] [38] [39] 

5 0.2178 0.214 0.219 0.217 0.217 0.215 0.220 0.211 

10 0.1525 0.153 0.154 0.151 0.151 0.150 0.151 0.148 

20 0.1369 0.138 0.138 0.133 0.133 - 0.133 - 

From 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. 

Table 3. Percentage difference between the present study and past studies 

 %𝐃𝐢𝐟𝐟 =  
𝐀𝐛𝐬𝐨𝐥𝐮𝐭𝐞 𝐝𝐢𝐟𝐟𝐞𝐫𝐞𝐧𝐜𝐞 𝐛𝐞𝐭𝐰𝐞𝐞 𝐩𝐫𝐞𝐬𝐞𝐧𝐭 𝐚𝐧𝐝 𝐩𝐚𝐬𝐫 𝐯𝐚𝐥𝐮𝐞

𝐏𝐚𝐬𝐭 𝐯𝐚𝐥𝐮𝐞
 

Span-to depth ratio (a/t) [28] [34] [35] [36] [37] [38] [39] 

5 1.745 0.551 0.367 0.367 1.286 1.010 3.122 

10 0.328 0.984 0.984 0.984 1.639 0.984 2.951 

20 0.804 0.804 2.849 2.849 - 2.849 - 

Average % Difference 0.36 0.29 0.53 0.52 2.41 0.61 3.62 

Total Ave.% Difference 1.19 



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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], 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] 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. 

5. 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. 

6. Declarations 

6.1. Author Contributions 

Conceptualization, F.C.O., T.E.O. and B.O.M.; methodology, F.C.O., T.E.O. and B.O.M.; software, F.C.O., T.E.O. 

and B.O.M.; formal analysis, F.C.O., T.E.O. and B.O.M.; writing—original draft preparation, F.C.O., T.E.O. and 

B.O.M.; writing—review and editing, F.C.O., T.E.O. and B.O.M. All authors have read and agreed to the published 

version of the manuscript. 

6.2. Data Availability Statement 

Data sharing is not applicable to this article. 

6.3. Funding 

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

6.4. Ethical Approval 

Not applicable. 

6.5. 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. 



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