Document Type : Research Paper
Authors
^{1} Department of Mechanical Engineering, MLR Institute of Technology, Hyderabad, India
^{2} Department of Mechanical Engineering, Rajeev Gandhi Memorial College of Engineering &Technology, Nandyal, India
^{3} Department of Mechanical Engineering, JNTU Hyderabad, Hyderabad, India
Abstract
Keywords
Main Subjects
Comparison of Deflection in TwoDirectional Functionally Graded Tapered Beam
^{a} Department of Mechanical Engineering, MLR Institute of Technology, Hyderabad, India
^{b }Department of Mechanical Engineering, Rajeev Gandhi Memorial College of Engineering &Technology, Nandyal, India.
^{c} Department of Mechanical Engineering, JNTU Hyderabad, Hyderabad, India.
KEYWORDS 

ABSTRACT 
Comparison of deflection; Reddy’s higher order shear Deformation theory; TwoDimensional functionally graded taper beam; Hamilton's principle; Static responses. 
Traditional engineering materials lack the necessary properties that are needed in aerospace as well as other modern industries. In order to address the aforementioned problem, a number of different materials are used in concert. With the help of functionally graded material, all the necessary characteristics can be achieved. A fast transition between two different materials can lead to debonding, thermal stresses, residual stresses, and stress concentrations; a gradual change in material properties might mitigate these problems. This work provides a comparison of the analytical solutions for the deflections in Two Dimensional Functionally Graded Taper Beam (2DFGTB) under a uniformly distributed load, adapting Reddy’s higherorder shear deformation theory. All the material properties of the beam are graded along the thickness and length dimensions using the powerlaw formula. The thickness of the beam is assumed to change linearly along its length. Equations of motion are derived based on Hamilton's principle and Navier’s solutions. A parametric investigation is conducted to explore the effects of various material and geometrical parameters on the mechanics of 2DFGTB. These parameters are found to be very significant in studying the static responses of 2DFGTB. 
Functionally Graded Material (FGM) [1] was developed in Japan in the mid1980s for structural and functional applications requiring hightemperature resistance and strength. FGM is an advanced composite material with dimensiondependent properties. Hightemperature FGMs are inhomogeneous composites made of ceramic and metal phases. FGM effortlessly changes mechanical and physical properties by mixing metal and ceramic at suitable volumes. Ceramic resists high temperatures due to low thermal conductivity [2]. Ductile metal can withstand hightemperature gradient stresses for a short time [3]. Thus, low and hightemperature FGM structural components can use metals and ceramics. FGM is mostly used in structural applications that integrate refractoriness and toughness. Rocket engine combustion chambers, highperformance tools, etc. Heat shields in spacecraft, tubes in a heat exchanger, implants in biomedical applications, and fusion reactor plasma facings can use FGMs [4, 5]. Chemical facilities, nuclear reactors, aircraft, and other incompatible applications use FGM [6]. Hightemperature thermal barrier coatings attach ceramic layers to metallic structures.
In hightemperature applications, a beam's end surfaces often have different temperatures (top and bottom). Consequently, the member is subjected to thermal loading because steadystate conditions facilitate heat conduction across the beam thickness [7].
Beam theories describe beamtype structures. Euler–Bernoulli beam theory is the oldest and most famous beam theory is Classical Beam Theory (CBT). Top and bottom shear stress violates free boundary conditions. Firstorder shear deformation theory's shear force discrepancy requires a shear correction factor (k) [8]. Researchers developed Higher Order Shear Deformation Theory (HSDT) to overcome this problem. Shear deformation is widely noticeable in the thick beam. Beam theory ignores these effects. HSDT must accurately describe thick beam bending behavior, including crosssectional warping. Appropriate kinematics and constitutive models enable this. The higherorder theory is accounted for transverse shear deformation by incorporating the function f(x) into the displacement field [910].
Ebrahimi et al. [11] using a Naviertype solution, this work investigates the free vibration characteristics of Functionally Graded (FG) nanobeams based on the thirdorder shear deformation beam theory. Material properties of FG nanobeam should vary continuously along the thickness according to the powerlaw model. The nonlocal governing equations are developed using Hamilton's principle and thirdorder shear deformation beam theory, then solved using an analytic solution. Cukanovi et al. [12] describe the bending analysis of thick and moderately thick FG square and rectangular plates on a WinklerPasternak elastic basis. The plates are assumed to have an isotropic, twocomponent material distribution throughout their thickness, with the modulus of elasticity increasing in terms of the volume percentages of the constituents according to a powerlaw distribution. Sayyad [13] the purpose of their work was to create a unified theory of shear deformation based on displacements to study advanced composite beams and plates that are susceptible to shear deformation. In order to model transverse shear deformation, a variety of coordinate shape functions based on trigonometry, hyperbola, parabola, and exponential were created. Bending and shear rotations affected inplane displacements. This theory's parabolic shape function produces Reddy's theory. Prashik Malhari Ramteke et al. [14] in this study, the nonlinear Eigen frequency responses of an FGM panel in a temperature environment are computed using finite elements. Consideration is given to the elements of the FGM as a function of temperature. The material properties of the FGM panel are evaluated using Voigt's micromechanical model in conjunction with three unique material distribution patterns. Thom et al. [15] TwoDimensional Functionally Graded Plates (2DFGP) were numerically analyzed for buckling and bending using a finite element model described plate kinematics without a shear correction factor or shear locking treatment.
Coskun et al. [16] by applying general thirdorder plate theory studied the bending, free vibration, and critical buckling behavior of FG porous microplates. Analytical results were obtained using the Navier method. In the context of thirdorder plate theory, analyzed the effects of three different porosity distributions on the resulting porosity variations. Kim et al. [17] examined the FG porous microplates for bending, free vibration, as well as buckling responses with classical and firstorder shear deformation plate theories. Analysis using Navier’s solution of supported rectangular plates was executed. Two material constituents were modeled using powerlaw distribution across plate thickness. Singh et al. [18] an analytical model for FG thinwalled beam flexural response was developed using firstorder shear deformation theory and Vlasov's theory. The power law distribution of mild steel and alumina volume fractions varied material properties along the depth. FG thin beams under uniform vertical loading were numerically analyzed. Mergen et al. [19] established the dynamic and static responses in a nonlinearly forced axially FG tapered beam. Hareram et al. [20], axial FG thin taper beams with different taper profiles and material gradations were studied for free vibration. The mathematical formulation uses energy and that could be resolved in two halves, static as well as dynamic.
Prashik Malhari Ramteke et al. [21] in this study, nonlinear thermal frequency characteristics of multidirectional, doublecurved, FG panels are investigated numerically using finite elements. The material properties of the FG panel are estimated utilizing the temperaturedependent individual properties of the FG constituents, Voigt's micromechanical model, and various forms of material distribution patterns. Rajasekarana et al. [22] discussed the mechanical behavior of nonuniform nanobeams. Smallscale effects are described according to the nonlocal strain gradient theory, the beam is represented according to the EulerBernoulli beam theory, and nonuniformity with a general change in cross section is postulated. Nguyen et al. [23] beams that are either transversely or axially tapered in FG are analyzed statically. The numerical results show that a TFG beam behaves differently than an isotropic beam. On the contrary, an AFG beam behaves exactly like an isotropic beam in terms of axial displacement and stress. Ketabdari et al. [24] using their elastic basis, Winkler and Pasternak analyzed the free vibration in homogenous and FG plates. The elastic foundation was an electric support by Pasternak and Winkler, whose stiffness coefficients varied both parabolically and linearly in each direction. Khoram et al. [25] there were tests conducted on the magnetic and mechanical deflection of FG nanobeams. The assumption was made that the nanobeam would be supported by the WinklerPasternak basis. The mechanical behavior of nanobeams was described by the Timoshenko beam model and the nonlocal elasticity theory of Eringen.
Ammar Melaibari et al. [26] present the first mathematical definition of the dynamical behavior of BiDirectional Functionally Graded Porous Plates (BDFGPP) resting on a Winkler–Pasternak foundation based on unified higherorder plate theories. Using the kinematic displacement fields, the null shear strain/stress at the bottom and top surfaces of the plate is satisfied without the requirement for a shear factor correction. Mohammadi et al. [27] FG was used to create a core for a composite nanoplate with lipid face sheets to study nonlinear vibrations. There are three distinct types of FG core materials. The viscoelasticity of lipid membranes was studied using the KelvinVoigt model. Adapting the VonKarman hypotheses, a nonlinear differential equation was derived for analyzing vibration in composite nanoplates. Mohammadi et al. [28] used FG material with a lipid core and two layers for vibration analysis. Nonlocal elasticity theory yields nonlinear differential governing equations. KelvinVoigt equation modeling lipid layer viscoelasticity. Nejad et al. [29] The phenomenon of buckling in a nanoplate constructed from a BDFG material of arbitrary composition was investigated. Eringen studied micro buckling loads with his nonlocal theory. Using minimum potential energy, the governing equations were derived. TrungKien Nguyen et al. [30] new HSDT was proposed for buckling, and free vibration analysis of isotropic and FG sandwich beams. The transverse shear stress distribution was hyperbolic and boundary conditions were satisfied.
Rahmani et al. [31] Timoshenko's beam theory examined FG material size dependence. FG nanobeam material properties varied power lawlike along the thickness. Ammar Melaibari et al. [32] offer a novel mathematical framework to represent the static deflection of a BiDirectionally Functionally Graded (BDFG) porous plate lying on an elastic base. Detailing the validity of the static response produced by middle surface versus neutral surface formulations and the location of the boundary conditions. Şimşek [33] the effects of a shifting load on a Timoshenko beam which is actually made of bidirectional FGM were investigated. Both the width and length of the beam expanded and contracted at an exponential rate. Slimane et al. [34] using HSDT, performed a vibration behavior of a porous, simply supported FG plate. The thickness of the FG porous plate had an effect on the material's behavior.
Ahmad Reshad Noori et al. [35] an effective numerical method for solving the dynamic response of functionally graded porous (FGP) beams is introduced. Along the thickness direction, the elastic modulus and mass density of porous materials are believed to have nonuniform distributions. It is assumed that the standard opencell metal foam governs the material constitutive law. Thai et al. [36] vibration analysis in an FG beam was determined by HSD theories. Adjustments to the shear strain boundary produce parallel beam surfaces on top and bottom. Vo, et al. [37] FGB static bending and vibration were examined using an enhanced shear deformation theory. Simsek [38] investigated the behavior of the Timoshenko beam made of bidirectional FGM while it was subjected to a varying load. Both the width and length of the beam expanded and contracted at an exponential rate. Chen et al. [39] Timoshenko's beam theory analyzed porous FG beam buckling and statics. Porous composite materials had different thicknessdependent elasticity modules and mass densities.
A displacementbased unified shear deformation theory was developed by Sayyad et al. [40] for advanced composite plates and beams. Transverse shear deformation affected shape function in this theory. Zenkour et al. [41] temperature and humidity impact FG porous beam bending. Normal and shear strains affect this investigation.
Menasria et al. using revised shear deformation theory, the current work present a dynamic analysis of an FGsandwich plate sat on an elastic foundation with various types of support. The current analytical model is simplified by reducing the number of unknowns. Without inserting correction factors, the free surfaces of the FGsandwich plate are guaranteed to have zero shear stresses. The study of Prashik Malhari Ramteke et al. [42] describes the massive geometric deformationinduced deformation and stress analysis of a bidirectional (2D) functionally graded material (FGM) shell panel subjected to changing mechanical loads (static and dynamic). The reactions of graded structures are computed numerically using an inhouse finite element model with displacement functions of higher order.
Nonclassical Timoshenko beam model [43] is created using a variational formulation based on the modified couple stress theory and GurtinMurdoch surface elasticity theory to investigate the mechanics of 2DFG nonuniform micro/nanobeams. The Hamilton principle is used to simultaneously derive the equations of motion and boundary conditions that account for bending and axial deformations, Poisson's effect, and the physical neutral axis concept.
Nguyen et al. [44] analytical solutions for the static analysis of functionally graded transversely or axially tapered crosssection beams are given. Depending on the power form, the beam's elastic modulus changes continuously in the thickness or axial direction.
Samaniego et al. [45] Using DNNs to tackle boundary value problems has been investigated. Many extremely relevant computational mechanics cases have been solved using DNNs to construct the approximation space, demonstrating that it is possible to approach the solution of extremely relevant BVPs using concepts and methods derived from deep learning.
Hongwei et al. [46] Deep Collocation Method (DCM) for thin plate bending problems is proposed in this work. Deep learning employs computational graphs and backpropagation algorithms, which are utilized in this method. In addition, the planned DCM is based on a feedforward deep neural network (DNN) and varies from the majority of previous deep learning applications for mechanical challenges.
The majority of the research is concentrated on static, dynamic, and buckling studies of conventional, FG onedimensional, and twodimensional beams. As a result of this, we are motivated to apply higherorder theory in this research on crosssection (tapered) in FGM. The author is only vaguely aware of a small handful of studies that have been conducted and presented on the subject of the deflection of twodirectional tapered FGBs that are predicated on a thirdorder shear deformation theory
In this study, a comparison is made between the analytical solutions of beam deflection under evenly distributed loads in 2DFGTB. It is presumable that the material properties in the beam will vary according to a power law distribution in thickness as well as length directions. Simply supported (SS) and Clampedclamped (CC) beams are subjected to homogenous load with varying lengthtothickness ratios (L/h) and gradient exponents so that the deflection of each type of beam can be compared. Hamilton's principle and RHSDT are used to generate governing equations of motion.
2DFGTBs are favored in this study. The composition of a material can vary along the length or thickness axes, as well as simultaneously along the length (xaxis) and thickness (zaxis) axes. Material parameters such as Young's modulus ' ,' Poisson's ratio ' μ,' and mass density ( ) vary (zaxis). The thicknesses at the left and right ends of the nonuniform beams are h_{2} and h_{1} (<h_{2}), respectively. At any axial location, the thickness h(x) in the tapered beam is given by h(x) =h_{2} [1  n(x/L)]. Where n is the taperness parameter given by n= [1h_{1}/h_{2}]. In view of this, the present theory is having following important assumptions.
Fig. 1. Geometry of FG Tapered Beam
Due to variations in the volume fraction of constituent materials, the FG beam's properties vary continuously. Assume a functional relationship between thickness coordinate and material property. The volume fraction of metal (V_{m}) could be expressed based on power law as [43],
Material properties of TDFG tapered beam can be defined as [43]:
Metal phases are denoted by “m” and ceramic phases by “c”. 'P_{x}' and 'P_{z}' stands for power law exponents that indicate the material variation profile through the axial and transverse directions of the beam. The power law exponents are (p_{x }= p_{z }= 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1) as shown in Figure 2.
Fig. 2. Assessing metal volume fractions
in the z/h and x/L directions.
Modulus of elasticity "E" Poisson's ratio "μ” and density "ρ" of FG tapered beam are listed below [43].
For the purpose of presenting the displacement equations, Reddy's HSDT with a shear strain function is taken into account [30].
(5)
(6)
The axial displacement, transverse displacement shear slope at a given point on the neutral axis is represented by u_{0}, w_{0}, Determining transverse shear deformation distribution using the inverse elastoplastic function (shape function) i.e. f (z).

(7) 

(8) 

(9) 

(10) 

(11) 

(12) 
where
=
=
The beam in functionally graded materials obeys Hooke’s law, so the behavioral relations can be given as follows:

(12a) 

(13) 

(14) 
Governing equations derived from the principle of virtual displacements yields, [30].

(15) 
where U is variation strain energy, k is variation in kinetic energy, V is variation in work done and t is the time.

(16) 

(17) 
The bending stress of the tapered beam in terms of virtual strain energy and work done can be shown by:

(18) 

(19) 



(20) 



(21) 

(22) 

(22a) 

(23) 
where, is axial resultant force, is the resultant bending moment, is resultant moment owing to shear deformation and are resultant shear forces. By integrating displacement gradations, governing equations of equilibrium can be derived.

(24) 

(25) 

(26) 
(z) dz 
(27) 
(z) zdz 
(28) 
(z) dz 
(29) 
z) dz 
(30) 
z)dz 
(31) 
Z)]^{2}dz 
(32) 
The motion equations admit Navier’s solutions applied to get analytical results for the various boundary conditions. The coefficients can be written assuming the following variations:
= 
(33) 
where , and are unknown coefficients, and complex number and .
The transverse load q is also expressed by double series of Fourier sine as follows:

(34) 
Substitute the expressions of Eq. (33) in the equations of motion (24), (25), and (26). The analytical solutions are given in the following form:

(35) 
(z) dz ( 
(36) 
(z) zdz ( 
(37) 
z) dz ( 
(38) 
z) dz ( 
(39) 
z) dz ( 
(40) 
z)]^{2}dz ( 
(41) 
To foretell the static analysis of the 2DFGTB subject to several boundary conditions, such as for SS and CC, the HSDTbased numerical studies presented in Table 1 are carried out.
Deflection analysis is presented and validates the accuracy of the current theory. 2DFGTB is considered for numerical results made of Alumina and Aluminum with material properties as follows.
Alumina: 
E_{c}=380 Gpa, 

μ_{c}_{ }= 0.3 
Aluminum: 
E_{m}=70 Gpa, 

μ_{m}_{ }= 0.3 
Table 1. Kinematic boundary conditions can be
applied to numerical calculations.
Boundary condition 
X= L/2 
X= L/2 
Simply Supported 
u=0, w=0 
w=0 
ClampedCamped 
u=0, w=0, 𝜙=0, w’=0 
u=0,w=0, 𝜙=0, w’=0 
Transverse deflection ( )

(42) 
= transverse deflection
= transverse deflection at any point in the beam
= Young's modulus of the metal
= Height of taper beam
= load
= Length of beam
A methodical and exhaustive validation of the 2DFGTB with higherorder shear deformation theory was conducted. Eq. (42) represents the nondimensional deflection. The results presented in Table 2 are based on idealized beams with different boundary conditions. The results from the current code differ marginally from those reported by Rabab et al. [43]. The deflection results of Table 3 are also identical for the CC beam.
Table 2. Comparison of dimensionless transverse deflection ( ) values of SS 2DFGTB with different aspect ratios (L/h=5, 20), taper ratio (n=0.1), and
gradation exponents(p).
Theory 
P=0 
P=0.5 
P=1 
P=2 
P=5 
P=10 
TBT [43] 
L/h=5 

1.1306 
0.6536 
0.5671 
0.5133 
0.4831 
0.4777 

HSDT 
1.0714 
0.6466 
0.5541 
0.5025 
0.4495 
0.4180 
% Error 
5.23% 
1.07% 
2.29% 
2.1% 
6.9% 
12.4% 
 
L/h=20 

HSDT 
1.0562 
0.6057 
0.5114 
0.4657 
0.4126 
0.3663 
Table 3. Comparison of dimensionless transverse deflection ( ) values of CC 2DFGTB with different aspect ratios (L/h=5, 20), taper ratio (n=0.1), and
gradation exponents(p).
Theory 
P=0 
P=0.5 
P=1 
P=2 
P=5 
P=10 
L/h=5 

HSDT 
0.2143 
0.1268 
0.1131 
0.1013 
0.0862 
0.0836 
L/h=20 

HSDT 
0.2071 
0.1151 
0.1003 
0.0862 
0.0750 
0.0643 
Table 4. Dimensionless transverse deflection ( ) values of 2DFGTB with different aspect ratio (L/h=5, 20), taper
ratio (n=0.1), and gradation exponents (p_{x}, p_{z}).
n 
pz 
Px 

0 
0.5 
1 
2 
5 
10 

L/h=5 

0.1 
0 
0.2143 
0.1402 
0.1341 
0.1233 
0.1093 
0.0986 
0.5 
0.2115 
0.1268 
0.1236 
0.1144 
0.1034 
0.0976 

1 
0.2087 
0.1265 
0.1131 
0.1156 
0.0974 
0.0902 

2 
0.2059 
0.1206 
0.1046 
0.1013 
0.0915 
0.0888 

5 
0.2031 
0.1148 
0.0999 
0.0920 
0.0862 
0.0846 

10 
0.2004 
0.1104 
0.0926 
0.0879 
0.085 
0.0836 

L/h=20 

0.1 
0 
0.2071 
0.1384 
0.1185 
0.1089 
0.0946 
0.0812 
0.5 
0.2043 
0.1325 
0.1094 
0.1014 
0.0897 
0.0771 

1 
0.2016 
0.1265 
0.1003 
0.0938 
0.0848 
0.0711 

2 
0.1988 
0.1206 
0.0911 
0.0862 
0.0799 
0.0674 

5 
0.1960 
0.1147 
0.0888 
0.0787 
0.0750 
0.0658 

10 
0.1932 
0.1087 
0.0847 
0.0742 
0.0701 
0.0643 
Fig. 3. Dimensionless transverse deflection ( ) values of CC TDFG tapered beam at aspect ratios (L/h=5),
taper ratio (n=0.1).
Fig. 4. Dimensionless transverse deflection ( ) values of CC 2DFGTB at aspect ratios (L/h=5), taper ratio (n=0.5).
Fig. 5. Dimensionless transverse deflection ( ) values of CC TDFG tapered beam at aspect ratios (L/h=20),
taper ratio (n=0.1).
Fig. 6. Dimensionless transverse deflection ( ) values of CC 2DFGTB at aspect ratios (L/h=20), taper ratio (n=0.5).
Table 5. Dimensionless transverse deflection ( ) values of SS 2DFGTB with different aspect ratio (L/h=5, 20),
taper ratio (n=0.5).
n 
pz 
px 

0 
0.5 
1 
2 
5 
10 

0.1 
L/h=5 

0 
1.0714 
0.7151 
0.6570 
0.6114 
0.5443 
0.4928 

0.5 
1.0575 
0.6466 
0.6055 
0.5675 
0.5147 
0.4882 

1 
1.0436 
0.6452 
0.5541 
0.5236 
0.4851 
0.4512 

2 
1.0296 
0.6152 
0.5125 
0.5025 
0.4654 
0.4435 

5 
1.0157 
0.5853 
0.4896 
0.4565 
0.4495 
0.4229 

10 
1.0018 
0.5631 
0.4537 
0.4362 
0.4237 
0.4180 

L/h=20 

0.1 
0 
1.0562 
0.6850 
0.6046 
0.5883 
0.5205 
0.4628 
0.5 
1.0421 
0.6557 
0.5580 0.5114 
0.5474 
0.4935 
0.4396 

1 
1.0279 
0.6263 
0.5114 
0.5066 
0.4665 
0.4054 

2 
1.0137 
0.5969 
0.4648 
0.4657 
0.4396 
0.3839 

5 
0.9996 
0.5676 
0.4527 
0.4249 
0.4126 
0.3751 

10 
0.9854 
0.5382 
0.4321 
0.4009 
0.3857 
0.3663 
Fig. 7. Dimensionless transverse deflection ( ) values of SS 2DFGTB at aspect ratios (L/h=5), taper ratio (n=0.1).
Fig. 8. Dimensionless transverse deflection ( ) values of SS 2DFGTB at aspect ratios (L/h=5), taper ratio (n=0.5).
Fig. 9. Dimensionless transverse deflection ( ) values of SS 2DFGTB at aspect ratios (L/h= 20), taper ratio (n=0.1).
Fig. 10. Dimensionless transverse deflection ( ) values of SS 2DFGTB at aspect ratios (L/h= 20), taper ratio (n=0.5).
The results of transverse deflections of the present theory are compared with the results of a taper nano beam [43] for SS boundary conditions. It is found that the results are agreeable. The remaining values are also acceptable for CC as well.
The effects of nonuniform parameters on thickness variation, h(x), and on the maximum dimensionless deflection at different powerlaw exponents and supporting types are presented ahead.
In this scenario, the width of the beam is constant; therefore, the applied distribution load is constant along the beam, but the moment of inertia and elastic modulus change along the length of the beam. The variation of the moment of inertia is due to the thickness variation along the length of the beam. It is obvious that, for all supported types, the maximum dimensionless deflection decreases as shown in Tables 4, and 5.
while the nonuniformity parameter decreases ‘n’ and the rate of increase depends on the supporting types as shown in Figures 36 for the CC beam and Figures 710 for the SS beam.
The maximum rate of dimensionless deflection appears in the SS beam as shown in Table 5, in comparison with other supports like CC as shown in Table 4. The main factors that affected the maximum dimensionless deflection and its rate of change are the distribution of the young modulus, the position of the clamped end, and the change in moment of inertia.
Generally, the width variation effects are not significant on the deflection in the beam in comparison with the thickness variation effects and the combination of both width and thickness variations, where it has the greater effects. In other words, the effects of variations mean that the dimensionless deflection increases as the nonuniformity parameters are reduced. For SS beam the effects of width variation, thickness variation, and both width and thickness variation cause to increase in the dimensionless deflection, in addition to changing the position of maximum dimensionless deflection. Generally, the midspan is the position of the maximum deflection in a SS beam (i.e., x = l/2). It seems that the position of dimensionless deflection is changed due to the variation in the dimensions of the beam and the distribution of elastic modulus.
The effects of aspect ratio (L/h) on the maximum dimensionless deflection, when the aspect ratio increases the maximum dimensionless deflection decrease as shown in Figure 5, Figure 6 as compared with Figure 3, Figure 4 for CC beam, and Figure 9, Figure 10 as compared with Figure 7, Figure 8 for SS beam due to the structure becomes relatively narrower or shorter in one direction compared to the other, which reduces its bending stiffness and makes it more susceptible to deflection under load.
Using a variational formulation based on Reddy's higherorder shear deformation theory, a new 2DFGTB model is developed in this article. In order to derive the equation of motion and corresponding boundary conditions that account for deflection and the neutral axis concept, the Hamilton principle is utilized simultaneously. In addition to the material properties, the model includes a material length scale parameter and surface elasticity constants that vary along the axial and transverse directions of the beam according to a power law.
Navier's method is applied to develop a 2DFGTB model to derive an analytical solution for the deflection responses of simply supported and clampedclamped 2DFGTB. A comprehensive parametric study is performed to investigate the effect of various material and geometrical parameters on the deflection responses. The principal findings can be summed up as follows:
Boundary conditions: Reddy's HSDT assumes that the plate's edges are simply supported or clamped. However, in practice, plates may have more complex boundary conditions, such as free or partially clamped edges, which can affect the analysis accuracy.
References