A new solution for FGM thin rectangular plate bending with variable thickness

Abstract

This study presents a novel analytical solution for the bending analysis of functionally graded material (FGM) thin rectangular plates with variable thickness. The Levy-type method is extended based on the classical thin-plate theory in order to provide a new exact and practical solution for solving this problem. It is assumed that the material properties varied through the thickness direction in a power-law distribution. Types of loading, thickness variation, plate’s boundary condition, and plate’s material combination were chosen as the state variables which affected on the non-dimensional plate’s deflection. By governing equations, some partial differential equations were appeared which were solved analytically by using generalized Fourier series. Finally, the non-dimensional thin rectangular FGM plate’s deflection under various loading types were calculated. The results demonstrate that the proposed method provides an accurate and computationally efficient tool for analyzing FGM plates with non-uniform thickness, offering significant advantages over conventional numerical approaches in terms of computational cost and solution precision. By increasing the power law index (), non-dimensional deflection values increased through all of the other variables such as plate’s boundary conditions.

Introduction

Functionally graded materials (FGMs) are advanced composites characterized by a continuous variation of material properties, making them suitable for applications in aerospace, automotive, and biomechanical engineering. FGMs are microscopically inhomogeneous in which the mechanical properties vary smoothly and continuously from one surface to the other one. On the other hand, Thin rectangular plates with variable thickness are commonly used in structural components where some special advantages such as weight reduction without sacrificing strength, optimized stress distribution by reducing failure risks, material efficiency for lowering costs, and improved dynamic performance (e.g., vibration, acoustics) are needed. For example, leading edges, morphing wings, and satellite panels in aerospace industries or high-performance brake disc in automotive industries could employ FGM thin rectangular plates with variable thickness to improve their performance. Moreover, turbine blades or heat exchanger in special power plants could increase their thermal resistance by using these kinds of plates. Therefore, in these days, mechanical behavior characterization of FGM thin rectangular plates is highly attended from researchers. One of the most fashionable failure modes for thin rectangular plates with variable thickness is bending. Elastic bending is incorporated in the free energy approach. Analysis of variable thickness plate to be accomplished for a special case by Olsson [1]. He solved the elastic bending problem for these plates by governing the displacement equations. Other researchers completed Olson’s study and improved his solution accuracy for bending of thin rectangular plate with variable thickness by using another approach [2, 3, 4–5]. Mudhaffar et al. [6] applied a four-known quasi-3D shear deformation theory to investigate the bending behavior of a functionally graded plate resting on a viscoelastic foundation and subjected to hygro-thermo-mechanical loading. The results show that the presence of the viscoelastic foundation causes an 18% decrease in the plate deflection and about a 10% increase in transverse shear stresses under both linear and nonlinear loading conditions. Fertis et al. [7, 8–9] developed a convenient and general method to solve variable thickness plate problem with various boundary conditions and different loading type by using closed form solution in the form of an equivalent system of flat plates with both elastic and inelastic analysis. The classical solution was presented as an isotropic plate theory for various boundary conditions by Ferits [10]. An exact solution for the bending of thin rectangular plate with uniform, linear, and quadratic thickness variations was investigated by Znkour [11]. Reddy et al. [12] analyzed the asymmetrical bending of FGM circular plate with uniform thickness. They used the first order shear deformation Mindlin plate theory. They gained the force, moment and displacement equations. that Mechab et al. [13] investigate the bending behavior of FGM plates based on two variables refine plate theory. They derived displacement equations by a close-form solution for simply supported rectangular plate which was subjected to sinusoidal loading. They verified their results by the numerical simulations with a good agreement. Bouguenina et al. [14] analysed thermal buckling condition of FGM simply supported plates, numerically, based on finite difference method. They also performed parameter study to analyze the effect of material and geometery on the thermal buckling of these plates. They found that the thickness variation affects isotropic plates more than FGM plates.Thang et al. [15] presented an analytical solution for buckling and post-buckling behavior of imperfect sigmoid FGM (S-FGM) plates with variable thickness. The results showed that variable thickness plays an important role in the prediction of failure under elastic compression loading. Sayyed et al. [16] developed a simple four-unknown exponential shear deformation theory for the FGM rectangular plate under the non-linear hygrothermomechanical load. Navier solutionwas used to derive the governing equations based on related boundary conditions. The accuracy and efficiency of the presented theory were approved by numerical solution. An exact solution (without any simplifications) for thick rectangular FGM plates under bending load was developed by Vafakhah et al. [17]. They understood that the effect of inhomogeneity parameter is extensive in the thick plates. Yekkalam Tash et al. [18] used an analytical solution for bending of isotropic thick rectangular plates with variable thickness. They compared the behavior materials such as graphite epoxy, E-glass epoxy, zinc, magnesium and steel. Their results demonstrated that thickness variation shifted the deflection to the thinner edge.Khakpour komarsofla et al. [19] optimized bending specifications of FGM rectangular plates subjected to thermomechanical loads. They employed a full layer-wise theory and Navier trigonometric series for finding the displacement distribution. Finally, they verified the results by numerical finite element (FE) solutions. Chahardoli et al. [20] investigated the mechanical response of light-weighted sandwich panel which was used in vehicle’s radiator under bending loads. They showed that aluminum addition could increase the energy absorption in these sandwich panels up to 268%. Also, their results demonstrated that the strength of these structures was related to loading plane. Addou et al. [21] solved the elastic bending problem for a porous FG plate by employing a novel higher quasi-3D hyperbolic shear deformation theory. The impact of the porosity parameter, aspect ratio, and thickness variation were shown through several numerical results. The deflection, stress components, the resultant forces and bending moments of a thick functionally graded material plate were expressed analytically in terms of the deflection of the reference homogenous Kirchhoff plate with the same geometry, loadings and boundary constraints by Li et al. [22]. They believed that, their approach could be used as a benchmar; to check numerical solutions of static bending of FGM plates based on different higher-order shear deformation theories. The effect of material gradient index on the FGM rectangular plate bending was studied by Noorial. [23]. They utilized FE analysis to examine the deflection and stress responses of FGM rectangular plates with different material gradient profiles and various boundary conditions. The results indicated the variations in deflection and stresses for different material gradients, and boundary conditions. Hadji et al. [24] investigated the bending behavior of FGM sandwich plates. They reduced the number of unknowns and equations of motion by dividing the transverse displacement into the bending and shear parts, based on shear deformation theory. They concluded that the proposed theory was accurate, and simple in solving the problem of the bending behavior of functionally graded plates. Lafi et al. [25] studied the thermodynamically bending behavior of functionally graded plates, laying on the Winkler/Pasternak/Kerr foundation with different boundary conditions, subjected to harmonic thermal load through plate thickness. They found that the sandwich plate's non-dimensional deflection increased as the aspect ratio raised for all conditions. Chitour et al. [26] used a novel quasi-3D high shear deformation theory to investigate the influence of porosity on the stability behavior of thick functionally graded sandwich plates subjected to mechanical loads. They showed the impact of aspect ratio, material index, loading type, porosity, and various foam shapes on critical buckling behavior. Lakhdar et al. [27] developed a new first-order Timoshenko’s theory to evaluate the vibrational behavior of power-law, sigmoid, and exponential functionally graded beams. They employed a parameter study to analyze the impact of the effects of the material exponent parameter, slenderness ratio, and porosity index on the dynamic behavior of imperfect FG beams. Driz et al. [28] promoted a new approach to improve structural designs based on the dynamic behavior of functionally graded plates on viscoelastic foundations. Their findings demonstrated that enhancing the damping coefficient of the viscoelastic foundation could improve the free-vibrational response of functionally graded material plates. Kaddari et al. [29] used a quasi-3D shear deformation plate theory to investigate the buckling response of functionally graded porous sandwich plates. They claimed that, transverse normal deformations determined the plate stiffness index. Benchohra et al. [30] evaluated bending behavior and free vibration characteristics of imperfect functionally graded beams. They presented a comprehensive discussion on the effects of span-to-depth ratio, porosity volume percentage, and viscoelastic foundations on the bending behavior and free vibration responces of FG beams. Baid et al. [31] introduced a new computational solution for buckling of thin plates with nonlinearity geometry. In this study, partial differential equations were formulated based on Kirchhoff’s theory and discretized by Galerkin method. Finally, they improved Hermite-type point interpolation method (HPIM) by radial basis functions to ensure a well-conditioned moment matrix. Influence of porosity on the thermal free vibration of rotating FGM plate with bi-directional thickness variation was studied by andal et al. [32]. The dynamic characteristics of S-FGM blades idealized as cantilever thin plates were studied using the FEM based on the first-order shear deformation theory (FSDT). The parametric study demonstrated that the frequencies of porous S-FGM cantilever plate decrease with the increase in gradient index and pre-twist angle. Zahari et al. [33] used a high order shear deformation theory in a unified general formulation for FGM sandwich plates. Their study extended a fifth order shear deformation theory (FOSDT) for kinematic modeling of the free vibration of various types of functionally graded material. A numerical comparative study of FOSDT with other kinematic theories showed as well as with the three-dimensional (3D) elasticity model was carried out when evaluating the non-dimensional frequency. Based on literature review, this paper extended a new approach to solve the bending analysis of FGM thin rectangular plates with thickness variations based on the Levy-type method, providing a more realistic model for practical applications. The two important advantages of this method are precision and solving time. This method could predict the plate deflection under various loading condition by more than four decimal places. On the other hand, minimizing the huge mathematical calculations compare to the traditional solutions for bending problem is another advantage of this method.

Theoretical formulation

Governing equation

Figure 1 shows a thin rectangular plate of length a, width b and made of functionally graded material. The plate is assumed to be functionally graded through the thickness direction. The constituent materials are assumed to be ceramic and metal corresponding to the power law are expressed as [34].

1

where is the thickness coordinate which varied , is the plate thickness, and is the power law index that takes values greater than or equal to zero (K ≥ 0) [33]. The value of equal to zero represents a fully ceramic plate and for K ≥ 0 a functionally graded material is demonstrated. In this paper, it is assumed that the variation of the composition of ceramic and metal is linear. The mechanical properties of FGMs are determined from the volume fraction of the material constituents. It assumed that non homogeneous material properties such as the elasticity modulus changed in the thickness direction based on Voigt’s rule over the whole range of volume fraction [35]:

2

,

[See PDF for image]

Fig. 1

Geometry and coordinate system of thin rectangular plate with variable thickness

[See PDF for image]

Fig. 2

Various type of transverse loads

Where subscripts m and c refer to the metal and ceramic constituents, respectively. By substituting volume fraction ratio from Equations. (1) in Equations. (2), material properties of FGMs plate are determined, which are the same as the equations proposed by [34]:

3

where

4

If the plate thickness h varies in the y direction only, that is, . The governing partial differential equation of the plate subjected to a transverse distributor load on its upper surface may be written in this case as the follows:

5

where D represents the flexural rigidity of the FGM variable thickness plate:

6

If , equation of isotropic plate bending with variable thickness would be obtained.

The differential Eq. (5), together with the boundary conditions defines the deflections of variable thickness plate, is derived by neglecting the effect of normal stress in z direction , and shearing stresses and . Therefore, it could be written as the follows:

7

where is the constant reference thickness value located in Fig. 1, describes the thickness variation in which n is the degree of non-uniformity, and is a small parameter. The values of could be chosen so the function could be constrained such as bellows:

8

The Eq. (8) keeps the plate, thin enough to fall within the range of thin-plate theory. Using Equations. (6) and (7), the plate flexural rigidity is given by:

9

where the constant is called the flexural rigidity of the FGM flat plate and given by:

10

The appropriate solution to the linear differential Eq. (5) could be written with the help of the small parameter method as:

11

where the is considered to be smaller than 0.5. Substituting Eq. (11) into Eq. (5). It could be re-written same as follows:

12

Note that, represents a series of solutions corresponding to the series of appearing on the right side of the equation. Equating the coefficients for both sides of Eq. (12), it could be easily obtained a set of differential equations representing equivalent flat plates with constant rigidity and uniform thickness . The solution of each differential equation may be carried out by using appropriate known methods for flat plates. Numerical and approximate methods could be used for this purpose. For practical applications, an accurate solution may be obtained by using only the first three or four equations from the set of equations given by Eq. (12).

Exact solution for bending

A generalized Levy-type solution is used to determine the thin plate deflections. all of the plate edges are considered simply supported along (see Fig. 1). Therefore, the solution of the present problem could be represented as:

13

where , and the function must fulfill the support conditions at . The above representation satisfies the simply supported boundary conditions at and x = a. Similarly, the load in a unitary from could be presented in terms of a single Fourier series as:

14

where

15

Three types of transverse load distributions would be considered in the analysis, namely uniform (UN), triangular with maximum intensity of at (TX), and triangular with maximum intensity of at (TY) as represented in Fig. 2. Therefore, Eq. (16) should be shown all of the loading types:

16

Introducing the dimensionless variables and , and after that substitution of Equations. (13) & (14) in to the Eq. (12) and performing the required different equations, the following series of differential equations would be obtained:

17

The first one of the preceding set of equations, Eq. (16) with , represents a FGM flat plate with constant rigidity D₀ and loading that is identical to the load applied on the original variable thickness plate. The remaining equations in this set, represent flat plates with different loads and so on. These loads could be determined once the displacements from the preceding equations. Thus, Eq. (17) is the set of equations that is describing an equivalent system, which replace the original variable – thickness plate.

The boundary conditions for simply supported , clamped and free at the edges would be considered as:

18

The total solution of Eq. (17) consists the complementary solution and particular solution Is independent of the loading. A single expression could be derived for that is valid for all rectangular plates having particular boundary conditions on the two opposite sides. Clearly, for each specific loading , a solution must be obtained. Generally, the complementary and particular solutions are given as the follows:

19

20

21

where denotes a dummy variable associated and which are weighting coefficients given by applying the boundary conditions in Eq. (17). It should be emphasized that all equations are formulated at all points and various boundary conditions have been applied. Therefore, results could be obtained by solving the corresponding equation directly.

It could be seen that the formulation in this approach is very convenient and easily coded in to a general computer program. The solution for FGM plates with various boundary conditions and thickness variations could be presented based on the same procedure by entering different weighting coefficients and different types of thickness function .

Numerical results and discussion

Numerical validation

In order to verify the method accuracy and results precision, the non-dimensional deflection of isotropic thin rectangular plate with different plate thickness variation (constant thickness (n = 0), linear variation (n = 1), and quadratic variation (n = 2)) under bending load and SS plate boundary condition is compared by Zenkour [11] in Table 1.

Table 1. Comparing non-dimensional deflection of an SS isotropic rectangular plate with various thickness variation under UN loading (Point 2, ) with Ref [11]

	n = 0		n = 1		n = 2
Aspect Ratio	This Study	Ref [11]	This Study	Ref [11]	This Study	Ref [11]
1.0	0.4062	0.4062	0.4100	0.4100	0.3494	0.3494
1.1	0.4869	0.4869	0.4913	0.4913	0.4195	0.4195
1.2	0.5651	0.5651	0.5701	0.5701	0.4884	0.4884
1.3	0.6392	0.6392	0.6449	0.6449	0.5547	0.5547
1.4	0.7085	0.7085	0.7149	0.7149	0.6176	0.6176
1.5	0.7724	0.7724	0.7795	0.7795	0.6766	0.6766
1.6	0.8308	0.8308	0.8386	0.8386	0.7315	0.7315
1.7	0.8838	0.8838	0.8923	0.8923	0.7821	0.7821
1.8	0.9316	0.9316	0.9406	0.9406	0.8286	0.8286
1.9	0.9745	0.9745	0.9840	0.9840	0.8710	0.8710
2.0	1.0129	1.0129	1.0229	1.0229	0.9097	0.9097
3.0	1.2233	1.2233	1.2340	1.2340	1.1415	1.1415
4.0	1.2819	1.2819	1.2904	1.2904	1.2247	1.2247
5.0	1.2971	1.2971	1.3033	1.3033	1.2572	1.2572

As seen in Table 1 the non-dimensional deflection of mentioned plate has high compliance with four decimal places with Zenkour [11]. Thus, the validity of the mentioned method is approved. Therefore, other results for the FGM thin rectangular could be trusted.

Discussion

The bending of FGM thin rectangular plates with variable thickness were considered. The combination of materials consists of aluminum and alumina. The young’s modulus for an aluminum was assumed , and for alumina . The Poisson’s ratio was chosen 0.3 for aluminum and alumina. As well as for the non-uniform thickness variation (h), the function may be expressed as follows:

22

and all of the results herein are in the following dimensionless forms:

23

Tables 2 and 3 summarized the non-dimensional numerical results for the deflection of isotropic and FGM plates for SS boundary condition and subjected UN, TX and TY loaded versus the various values of power law index (k). As it seen, by increasing plate aspect ratio (b/a) in a constant k, non-dimensional deflection of plate was increased for every loading type. This is due to the greater induced moment for long plate compared to the short one. Moreover, the moment values caused smaller non-dimensional deflection in the TX loading type compared to the other loading types.

Table 2. Non–dimensional deflection of an SS rectangular isotropic and FGM plate with uniform, linear, and quadratic thickness variations and subjected to UN load (Point 2, )

b/a	Flat plate				n = 1				n = 2
	k = 0	k = 1	k = 5	k = 10	k = 0	k = 1	K = 5	k = 10	k = 0	k = 1	K = 5	k = 10
1.0	0.4062	0.6861	0.9735	1.1811	0.4100	0.6925	0.9826	1.1921	0.3494	0.5901	0.8373	1.0159
1.1	0.4869	0.8223	1.1668	1.4156	0.4913	0.8297	1.1773	1.4284	0.4195	0.7086	1.0054	1.2198
1.2	0.5651	0.9543	1.3541	1.6229	0.5701	0.9628	1.3662	1.6575	0.4884	0.8248	1.1704	1.4200
1.3	0.6392	1.0796	1.5318	1.8585	0.6449	1.0892	1.5455	1.8751	0.5547	0.9368	1.3293	1.6127
1.4	0.7085	1.1966	1.6978	2.0599	0.7149	1.2074	1.7132	2.0785	0.6176	1.0430	1.4800	1.7957
1.5	0.7724	1.3045	1.8510	2.2457	0.7795	1.3166	1.8681	2.2665	0.6766	1.1427	1.6215	1.9672
1.6	0.8308	1.4032	1.9910	2.4155	0.8386	1.4164	2.0097	2.4383	0.7315	1.2354	1.7529	2.1267
1.7	0.8838	1.4926	2.1179	2.5696	0.8923	1.5069	2.1382	2.5942	0.7821	1.3209	1.8742	2.2739
1.8	0.9316	1.5734	2.2325	2.7086	0.9406	1.5886	2.2541	2.7348	0.8286	1.3994	1.9856	2.4090
1.9	0.9745	1.6458	2.3353	2.8333	0.9840	1.6619	2.3582	2.8611	0.8710	1.4711	2.0873	2.5325
2.0	1.0129	1.7106	2.4272	2.9449	1.0229	1.7275	2.4512	2.9739	0.9097	1.5364	2.1800	2.6449
3.0	1.2233	2.0660	2.9315	3.5566	1.2340	2.0842	2.9573	3.5880	1.1415	1.9279	2.7356	3.3190
4.0	1.2819	2.1649	3.0719	3.7269	1.2904	2.1793	3.0922	3.7516	1.2247	2.0684	2.9349	3.5608
5.0	1.2971	2.1906	3.1083	3.7712	1.3033	2.2011	3.1231	3.7891	1.2572	2.1233	3.0127	3.6552

Table 3. Non–dimensional deflection of a rectangular isotropic and FGM plate with linear thickness variations () and subjected to TX and TY load

b/a	TX load				TY load
	k = 0	k = 1	k = 5	k = 10	k = 0	k = 1	k = 5	k = 10
1.0	0.2050	0.3462	0.4913	0.5961	0.3902	0.6591	0.9352	1.1346
1.1	0.2456	0.4149	0.5886	0.7142	0.4698	0.7934	1.1257	1.3658
1.2	0.2850	0.4814	0.6831	0.8287	0.5472	0.9242	1.3113	1.5910
1.3	0.3225	0.5446	0.7721	0.9376	0.6211	1.0489	1.4883	1.8058
1.4	0.3575	0.6037	0.8566	1.0393	0.6904	1.1660	1.6544	2.0072
1.5	0.3898	0.6583	0.9340	1.1332	0.7546	1.2744	1.8083	2.1939
1.6	0.4193	0.7082	1.0048	1.2191	0.8135	1.3739	1.9494	2.3651
1.7	0.4461	0.7535	1.0691	1.2971	0.8671	1.4644	2.0778	2.5210
1.8	0.4703	0.7943	1.1271	1.3674	0.9156	1.5463	2.1941	2.6620
1.9	0.4920	0.8310	1.1791	1.4305	0.9592	1.6200	2.2986	2.7888
2.0	0.5114	0.8637	1.2256	1.4869	0.9983	1.6861	2.3924	2.9026
3.0	0.6170	1.0421	1.4786	1.7940	1.2142	2.0507	2.9097	3.5302
4.0	0.6452	1.0896	1.5461	1.8758	1.2745	2.1527	3.0545	3.7059
5.0	0.6516	1.1005	1.5616	1.8946	1.2905	2.1794	3.0925	3.7519

The effect of power law index () on the non-dimensional deflection of FGM plate versus y distance for different loading types and boundary conditions was shown in Figs. 3, 4, 5, 6, 7, 8, 9, and 10. As observed, these figures reveal that by increasing the power law index () the values of non-dimensional deflection increase for every loading type and boundary conditions. But, at a constant power law index the non-dimensional deflection of FGM plate varied in a different manner. For example, in Fig. 6, for a plate by uniform thickness under UN loading type and FF plate’s boundary condition the plate non-dimensional deflection has a constant value, however, in the same condition, just by changing plate thickness by a linear variation, the plate non-dimensional deflection decreased significantly. The material gradient variation is the main reason of this observation. On the other hand, in Figs. 4, 5, and 7 the plate non-dimensional deflection has a sinusoidal behavior in comparison to the Figs. 8, 9, and 10. This is due to the plate’s boundary condition which is the most important factor to deal with the plate deflection under various loading types. It could be claimed that, the symmetric plate’s boundary conditions cause a sinusoidal plate deflection, however one free edge could disturb these symmetry results.

[See PDF for image]

Fig. 3

Comparison of the dimensionless deflection () along the centerline of isotropic and FGM, square plates subjected to UN load for SS boundary condition () with: (a) linear thickness variations (); and (b) quadratic thickness variations ()

[See PDF for image]

Fig. 4

Comparison of the dimensionless deflection () along the centerline of isotropic and FGM, square plates subjected for SS boundary condition with linear thickness variations(, ) for: (a) TX load; and (b) TY load

[See PDF for image]

Fig. 5

Comparison of the dimensionless deflection () along the centerline of isotropic and FGM, square plates subjected to UN load for CC boundary condition with: (a) uniform thickness variations (); and (b) linear thickness variations (, )

[See PDF for image]

Fig. 6

Comparison of the dimensionless deflection () along the centerline of isotropic and FGM, square plates subjected to UN load for FF boundary condition with: (a) uniform thickness variations (); and (b) linear thickness variations (, )

[See PDF for image]

Fig. 7

Comparison of the dimensionless deflection () along the centerline of isotropic and FGM, square plates subjected to UN load for SC boundary condition with: (a) uniform thickness variations (); and (b) linear thickness variations (, )

[See PDF for image]

Fig. 8

Comparison of the dimensionless deflection () along the centerline of isotropic and FGM, square plates subjected to UN load for SF boundary condition with: (a) uniform thickness variations (); and (b) linear thickness variations (, )

[See PDF for image]

Fig. 9

Comparison of the dimensionless deflection () along the centerline of isotropic and FGM, square plates subjected to UN load for CF boundary condition with: (a) uniform thickness variations (); and (b) linear thickness variations (, )

[See PDF for image]

Fig. 10

Comparison of the dimensionless deflection () versus the thickness parameter () of isotropic and FGM, square plates subjected to UN load for SS boundary condition with: (a) linear thickness variations (); and (b) quadratic thickness variations ()

Also, when the non- dimensional deflection of FGM plates variation compared with the isotropic plate () which is made from ceramic, the various of elasticity modulus between ceramic and metal would cause sharp difference in plate’s deflection values.

Conclusions

In this paper, a new solution for the elastic bending problem for the FGM thin rectangular plates with variable thickness was presented based on the Levy-type method. The plates deflection was calculated for plate’s different boundary conditions, different material property, various aspect ratio, and diverse thickness parameters. Some of the results are listed bellows:

1. This new method (extended Levy-type method) reduced the solving time and increased the answer precision for the elastic bending problem of FGM thin rectangular plates with variable thickness.

2. This method is very useful for designing any high-performance components which are affected by elevated thermal loads and bending loads such as rotary reactors.

3. Non-dimensional deflection of isotropic and FGM thin rectangular plate with variable thickness under UN loading condition gained higher values in comparison to the TX and TY loading conditions.

4. As the plate boundary conditions tended to clamp type, the plate non-dimensional deflection was reduced under the same loading type. However, As the plate boundary conditions tended to simply supported type, plate non-dimensional deflection increased significantly up to 115% compare to the clamp boundary condition.

5. By rising the elasticity modulus variation in FGM plate (K parameter) the non-dimensional deflection of FGM thin rectangular under the same loading condition was increased sharply up to 200%.

6. By increasing the power law index (), non-dimensional deflection values increased through all of the plate’s boundary conditions.

7. In a constant k, non-dimensional deflection values have a diverse behavior through y direction based on the plates’ boundary conditions.

8. Some experimental works on the three-point bending test of FGM plates could approve the method accuracy as a future scope.

Author contributions

S. Ghorbanhosseini: Methodology, Data Analysis, Review. M. H. Jowkar: Writing, Formulations, Data Analysis, Review. M. M. Najafizadeh: Methodology, Supervision, Review.

Funding

No funding was received for this study.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Ethical approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. Olsson, RG. Biegung der Rechteckplatte bei linear veränderlicher Biegungssteifigkeit. Ing-Arch; 1934; 5, 5 pp. 363-373. [DOI: https://dx.doi.org/10.1007/BF02092076]

2. Reissner, ME. Remark on the theory of bending of plates of variable thickness. J Math Phys; 1937; 16, 1–4 pp. 43-45. [DOI: https://dx.doi.org/10.1002/sapm193716143]

3. Conway, HD. Axially symmetrical plates with linearly varying thickness. J Appl Mech; 1951; 47489 [DOI: https://dx.doi.org/10.1115/1.4010267]

4. Conway, HD. Closed-form solutions for plates of variable thickness. J Appl Mech; 1953; [DOI: https://dx.doi.org/10.1115/1.4010767]

5. Ohga, M; Shigematsu, T. Bending analysis of plates with variable thickness by boundary element-transfer matrix method. Comput Struct; 1988; 28, 5 pp. 635-640. [DOI: https://dx.doi.org/10.1016/0045-7949(88)90008-9]

6. Mudhaffar, IM; Chikh, A; Tounsi, A; Al-Osta, MA; Al-Zahrani, MM; Al-Dulaijan, SU. Impact of viscoelastic foundation on bending behavior of FG plate subjected to hygro-thermo-mechanical loads. Structural Engineering and Mechanics, An Int'l Journal; 2023; 86, 2 pp. 167-180.

7. Fertis, DG. Advanced mechanics of structures; 2024; CRC Press: [DOI: https://dx.doi.org/10.1201/9781003574019]

8. Fertis, DG; Mijatov, MM. Equivalent systems for variable thickness plates. J Eng Mech; 1989; 115, 10 pp. 2287-2300.

9. Fertis, DG; Lee, CT. Elastic and inelastic analysis of variable thickness plates, using equivalent systems. J Struct Mech; 1993; 21, 2 pp. 201-236.

10. Fertis, DG. Nonlinear mechanics; 1998; CRC Press:

11. Zenkour, AM. An exact solution for the bending of thin rectangular plates with uniform, linear, and quadratic thickness variations. Int J Mech Sci; 2003; 45, 2 pp. 295-315. [DOI: https://dx.doi.org/10.1016/S0020-7403(03)00050-X]

12. Reddy, JN; Wang, CM; Kitipornchai, S. Axisymmetric bending of functionally graded circular and annular plates. European Journal of Mechanics-A/Solids; 1999; 18, 2 pp. 185-199. [DOI: https://dx.doi.org/10.1016/S0997-7538(99)80011-4]

13. Mechab, I; Atmane, HA; Tounsi, A; Belhadj, HA; Bedia, EAA. A two variable refined plate theory for the bending analysis of functionally graded plates. Acta Mech Sin; 2010; 26, pp. 941-949.2763914 [DOI: https://dx.doi.org/10.1007/s10409-010-0372-1]

14. Bouguenina, O; Belakhdar, K; Tounsi, A; Bedia, EA. Numerical analysis of FGM plates with variable thickness subjected to thermal buckling. Steel Compos Struct; 2015; 19, 3 pp. 679-695. [DOI: https://dx.doi.org/10.12989/scs.2015.19.3.679]

15. Thang, PT; Nguyen-Thoi, T; Lee, J. Closed-form expression for nonlinear analysis of imperfect sigmoid-FGM plates with variable thickness resting on elastic medium. Compos Struct; 2016; 143, pp. 143-150. [DOI: https://dx.doi.org/10.1016/j.compstruct.2016.02.002]

16. Sayyad, AS; Ghugal, YM. Effects of nonlinear hygrothermomechanical loading on bending of FGM rectangular plates resting on two-parameter elastic foundation using four-unknown plate theory. J Therm Stress; 2019; 42, 2 pp. 213-232. [DOI: https://dx.doi.org/10.1080/01495739.2018.1469962]

17. Vafakhah, Z; Neya, BN. An exact three-dimensional solution for bending of thick rectangular FGM plate. Compos Part B Eng; 2019; 156, pp. 72-87. [DOI: https://dx.doi.org/10.1016/j.compositesb.2018.08.036]

18. Tash, FY; Neya, BN. An analytical solution for bending of transversely isotropic thick rectangular plates with variable thickness. Appl Math Model; 2020; 77, pp. 1582-1602.4029036 [DOI: https://dx.doi.org/10.1016/j.apm.2019.08.017]

19. Komarsofla, MK; Salami, SJ; Shakeri, M; Komarsofla, AK. Optimization of three-dimensional up to yield bending behavior using a full layer-wise theory for FGM rectangular plate subjected to thermo-mechanical loads. Compos Struct; 2021; 257, [DOI: https://dx.doi.org/10.1016/j.compstruct.2020.113172] 113172.

20. Chahardoli, S; Akhavan Attar, A; Ghorbanhosseini, S; Hossaeini Marashi, SM. Investigation of the bending and crushing for the light-weight structures used in vehicle's radiator. Mech Based Des Struct Mach; 2023; 51, 8 pp. 4491-4507. [DOI: https://dx.doi.org/10.1080/15397734.2021.1967165]

21. Addou, FY; Bourada, F; Meradjah, M; Bousahla, AA; Tounsi, A; Ghazwani, MH; Alnujaie, A. Impact of porosity distribution on static behavior of functionally graded plates using a simple quasi-3D HSDT. Comput Concr; 2023; 32, 1 87.

22. Li, SR; Zhang, F; Liu, RG. Classical and homogenized expressions for the bending solutions of FGM plates based on the four variable plate theories. Mech Adv Mater Struct; 2024; 31, 15 pp. 3413-3424. [DOI: https://dx.doi.org/10.1080/15376494.2023.2177909]

23. Noori, M; Bilgin, A; Diallo, H; Al Rousan, MO; Noorı, AR. A study on the influence of material gradient index on bending and stress responses of FGM rectangular plates using the finite element method. J Sustain Constr Mater Technol; 2024; 9, 3 pp. 239-254. [DOI: https://dx.doi.org/10.47481/jscmt.1555157]

24. Hadji, L; Plevris, V; Papazafeiropoulos, G. Investigation of the static bending response of FGM sandwich plates. J Appl Comput Mech; 2024; 10, 1 pp. 26-37.

25. Lafi, DE; Bouhadra, A; Mamen, B; Menasria, A; Bourada, M; Bousahla, AA; Bourada, F; Tounsi, A; Tounsi, A; Yaylacı, M. Combined influence of variable distribution models and boundary conditions on the thermodynamic behavior of FG sandwich plates lying on various elastic foundations. Structural Engineering and Mechanics, An Int'l Journal; 2024; 89, 2 pp. 103-119.

26. Chitour, M., Bouhadra, A., Bourada, F., Mamen, B., Bousahla, A.A., Tounsi, A., Tounsi, A., Salem, M.A. and Khedher, K.M., 2024, March. Stability analysis of imperfect FG sandwich plates containing metallic foam cores under various boundary conditions. In Structures (Vol. 61, p. 106021). Elsevier.

27. Lakhdar, K; Sadoun, M; Addou, FY; Bourada, F; Bousahla, AA; Tounsi, A; Khedher, KM; Tounsi, A. Free vibrational characteristics of various imperfect FG beam via a novel integral Timoshenko’s theory. Acta Mech; 2024; 235, 10 pp. 6287-6304.4800661 [DOI: https://dx.doi.org/10.1007/s00707-024-04046-2]

28. Driz, H; Attia, A; Bousahla, AA; Addou, FY; Bourada, M; Tounsi, A; Tounsi, A; Balubaid, M; Mahmoud, SR. Dynamic response of imperfect functionally graded plates: impact of graded patterns and viscoelastic foundation. Structural Engineering and Mechanics, An Int'l Journal; 2024; 91, 6 pp. 551-565.

29. Kaddari, M; Balubaid, M; Mahmoud, SR; Kaci, A; Bousahla, AA; Tounsi, A; Al-Osta, MA; Tounsi, A. Influence of porosity distribution on buckling behavior of FG sandwich plates using a quasi-3D refined plate theory. Mech Based Des Struct Mach; 2025; 53, 5 pp. 3635-3662. [DOI: https://dx.doi.org/10.1080/15397734.2024.2434552]

30. Benchohra, M; Laoufi, I; Attia, A; Bousahla, AA; Tounsi, A; Tounsi, A; Yaylaci, M; Alluqmani, AE; Mahmoud, SR. Analytical solution for imperfect functionally graded beams resting on viscoelastic foundation. Struct Eng Mech; 2025; 95, 2 155.

31. Baid, S; Hilali, Y; Mesmoudi, S; Bourihane, O. Buckling analysis of functionally graded sandwich thin plates using a meshfree Hermite Radial Point Interpolation Method. Eng Comput; 2025; 41, 1 pp. 627-643. [DOI: https://dx.doi.org/10.1007/s00366-024-02011-0]

32. Mandal, R; Singha, TD; Bandyopadhyay, T; Karmakar, A. Influence of porosity on thermal free vibration of rotating porous sigmoid functionally graded plate with bi-directional thickness variation. J Vib Eng Technol; 2025; 13, 1 109. [DOI: https://dx.doi.org/10.1007/s42417-024-01639-y]

33. Zahari, K; Hilali, Y; Mesmoudi, S; Bourihane, O. Fundamental frequency of FGM sandwich plates using a fifth order shear deformation theory in a unified general formulation. J Vib Eng Technol; 2025; 13, 1 pp. 1-20. [DOI: https://dx.doi.org/10.1007/s42417-024-01647-y]

34. Praveen, GN; Reddy, JN. Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Int J Solids Struct; 1998; 35, 33 pp. 4457-4476. [DOI: https://dx.doi.org/10.1016/S0020-7683(97)00253-9]

35. Sugano, Y. An analytical solution for a plane thermal stress problem in nonhomogeneous multiply connected regions: unaxisymmetric steady-state thermal stresses in a nonhomogeneous hollow circular plate. JSME international journal Ser 1, Solid mechanics, strength of materials; 1990; 33, 2 pp. 136-144. [DOI: https://dx.doi.org/10.1299/jsmea1988.33.2_136]