Document Type : Research Paper
Authors
^{1} Department of Mechanical Engineering DDU, Gorakhpur, India
^{2} Department of Mechanical Engineering, GLA University, Mathura, India
^{3} Department of Civil Engineering MMMUT, Gorakhpur, India
^{4} Oil and gas, HEPIPL, Adani Group, Ahmedabad, India
^{5} Department of Mechanical Engineering IIT, BHU, India
^{6} Department of Mechanical Engineering MMMUT, Gorakhpur, India
^{7} Mechanical Engineering SGI, Kukas, Jaipur, India
Abstract
Keywords
Main Subjects
^{a }Department of Mechanical Engineering DDU, Gorakhpur, India
^{b }Department of Mechanical Engineering, GLA University, Mathura, India
^{c }Department of Civil Engineering MMMUT, Gorakhpur, India
^{d }Oil and gas, HEPIPL, Adani Group, Ahmedabad, India
^{e }Department of Mechanical Engineering IIT, BHU, India
^{f }Department of Mechanical Engineering MMMUT, Gorakhpur, India
^{g }Mechanical Engineering SGI, Kukas, Jaipur, India
^{h} Department of Mechanical Engineering DDU, Gorakhpur, India
KEYWORDS 

ABSTRACT 
FGM plate; Spline RBF; Meshfree; Buckling; Foundation; Aspect ratio. 
In the present paper, the effect of aspect ratio on the initial buckling response of an elastically supported rectangular FGM plate, the thin plate spline radial basis function (RBF) method can be used. This method involves the use of a thin plate spline to interpolate the displacement field of the plate, allowing for accurate analysis of the plate's behavior. The higherorder shear deformation theory (HSDT) is used for the analysis of the FGM rectangular plate resting on two parameters of elastic foundation. The displacement field consists of five unknown variables and approximately parabolic distribution of the transverse stress profile through the plate thickness and tangential stressfree boundary conditions on the plate surface. The governing differential equations (GDEs) of the plate are developed in the framework of Hamilton’s principle. The thin plate spline radial basic functionbased Meshfree method is used for discretizing the GDEs. To demonstrate the accuracy and efficacy of the present approach, the results obtained by the present approach are compared with the results given in the literature. The effect of various aspect ratios, grading index, spantothickness ratio, and two parameters of elastic foundations on the normalized buckling load is proposed. Some new results are also presented, which may be beneficial for future research works. The novelty of the present is the effects of aspect ratio on the elastically supported FGM plates. 
Functionally graded materials (FGMs) are a class of advanced engineering materials that have a continuously varying composition and structure over their volume. The composition and structure of FGMs are designed to change gradually from one end to the other, resulting in a material with unique properties that can be tailored for specific applications. FGMs are typically made by combining two or more materials with different properties, such as metals and ceramics, and then gradually varying the composition of each material along a specified direction. The resulting material can have a wide range of properties, such as high strength, toughness, and wear resistance, as well as the ability to withstand high temperatures and harsh environments. FGMs have a wide range of applications in various fields, such as aerospace, automotive, biomedical, and energy. For example, FGMs can be used in aircraft engines to improve their thermal and mechanical properties, or in dental implants to promote bone growth and integration with the surrounding tissue.
Various numerical techniques are being widely implemented for buckling analysis of FGMs rectangular plates so for by researchers. In this direction, Feldman and Aboudi [1] investigated the analytical technique for the buckling analysis of FGM plates subjected to a uniaxial compression load using classical plate theory (CPT). Mohammadi et al. [2] carried out a Levy solution for the buckling response of thin FGMs rectangular plates via the CPT model. Bodaghi and Saidi [3] studied analytical techniques for the buckling response of thick FGM plates based on HSDT. Thai and Uy [4] carried out analytical solutions for the buckling response of FGMs plate via refined plate theory. Singh and Harsha [5] examined the buckling response of FGM plates subjected to various types of inplane loads in the framework of the HSDT model. However, these analytical solutions are not pertinent for general cases, and they are not yet feasible for complex geometry and loading conditions. Subsequently, various numerical methods are being utilized, such as the finite elements method (FEM) [6], meshless method, spline finite strip method, and so for. Among various numerical methods, the meshfree technique has long been utilized as one of the most effective numerical methods for the buckling response of plates. Bateni et al. [7] examined the initial buckling response of FGM plates under thermal and mechanical loading via the Galerkin method. Bui et al. [8] studied the stability response of the plate using the moving Kriging interpolation technique. Liew et al. [9] studied the buckling and vibration response of plates using the reproducing kernel particle approximate technique and firstorder shear deformation theory (FSDT). Kiarasi et al.,[10] investigated buckling analysis of sandwich plates via analytical approach and classical plate theory. Shahani and Kiarasi, [11] studied the postbuckling response of shells with rectangular cutouts under axial loading via the finite element method (FEM). Kumari et al. [12] carried out a stress analysis of a single hole on an infinite plate via Airy’s stress function. Rahul and Singh [13] examined the buckling analysis of the FGM plate via the RBF technique.Tan et al. [14] investigated 3D isogeometric analysis and meshfree coupling approach to examine the structural responses for FGM structures. Kiarasi et al.[15] investigated the buckling response of porous FGM rectangular plates subjected to different loading conditions. Asemi et al. [16] developed a 3D FEM formulation for the buckling response of anisotropic FGMs plate with arbitrary orthotropy directions. Shahsavari et al. [17] investigated thermal postbuckling analysis of FGM annular sector plates subjected to uniform temperature rise via FEM approach. Singh et al. [18] used the molecular dynamics approach for the mechanical properties of multiwalled carbon nanotubes. Khatoonabadi et al.,[19] studied shear buckling analysis of FGM annular sector plate reinforced with graphene nanoplatelets (GPLs) under porous medium via FEM pproach. Kumar et al.[20] studied the wear response of FGM under the influences of centrifugal casting processing. Neves et al. [21] examined the structural response of sandwich plates via quasi3D HSDT and meshfree methods. Singh et al. [18] investigated the effect of functionalization on the mechanical properties of multiwalled carbon nanotubes. Rahimi et al.[22] studied the exact thermoelastic response of an FGM piezoelectrical rotating cylinder with the combination of electrical, thermal, and mechanical loads simultaneously. Arefi and Rahimi [23] studied the 2D electroelastic response of an FGM piezoelectric cylinder under internal pressure using the FSDT model.Arefi and Rahimi [24] investigated different boundary conditions for the thermoelastic response of an FG cylinder via the FSDT model.Arefi and Zenkour [25] studied the nonlocal thermomagnetoelectromechanical bending response of a threelayered nanoplate via sinusoidal sheardeformation plate theoryFerreira et al. [26] studied the buckling and vibration analysis of laminated plates via the RBF technique and FSDT model. Rodrigues et al. [27] applied Murakami’s zigzag theory with RBF finite difference collocation method for the structural analysis of composite plate. Rahimi et al. [28] examined the electroelastic response of FGM piezoelectric cylinder via FSDT model.Khoshgoftar et al.[29] examined the elastic response of a FGM thickwalled cylindrical via FSDT model and analytical approach.Arefi and Rahimi [30] investigated nonlinear analysis of a FG square plate with two smart layers as a sensor and actuator under pressure using VonKarman assumption.Ferreira et al. [31] investigated the stability response of isotropic and laminated plates via the FSDT model and RBF with wavelets. Chilakala et al. [32] used Element Free Galerkin Method for the Cracks analysis in Automotive Coatings Under Mechanical and Thermal Loading. Kumar et al. [33] investigated the thermomechanical buckling response of a bidirectional porous FGM plate using HSDT displacement field. Arefi and Rahimi [34] studied the nonlinear response of the FGM piezoelectric annular plate in the framework of VonKarman assumption.Arefi and Rahimi [35] introduced the 3D multifield formulation of a FGM piezoelectric thick shell of revolution via tensor analysis
To represent the correlation between the plate and foundation, different types of foundation models have been considered so for. The plate supported by the elastic foundation is increasingly used in many structural applications such as the foundation of runways of large aircraft, foundation deep wells, bridges and buildings, etc. The easiest one is the Winkler model, which considers the establishment as a progression of isolated springs without coupling impacts among one another [36]. Pasternak alters this model by considering a shear spring among the isolated springs in Winkler's model, known as the twoparameters model [37]. Thai et al. [38] examined the structural response of elastically supported plates using a simple, refined theory. Malekzadeh and Karami [39] applied mixed differential quadrature and FEM for vibration and stability response of elastically supported thick beams.
Sobhy [40] examined the stability and vibration analysis of elastically supported FGM sandwich plates. In the abovementioned literature, little work has been done for buckling analysis of thin and thick rectangular FGM plates supported by the elastic foundation to the best of the author’s knowledge. Modified thin plate spline RBFbased meshfree method capable of producing a result for rectangles without changing the shape parameter.
In the present work, the influence of the aspect ratio of an elastically supported rectangular FGM plate is carried out for the initial buckling response. The displacement field is based on five variables HSDT model. The governing differential equations and boundary conditions are derived using Hamilton’s principle. The thin plate spline RBFbased Meshfree method is implemented for discretizing the GDE. The accuracy of the present code is verified by convergence study and the results obtained by the present method are compared with reported results available in the literature. The aspect ratio of a structure significantly affects its buckling behavior. Design considerations for buckling stability should take into account the aspect ratio to ensure the structural integrity and safety of the system. The influences of grading index, foundation parameters, spantothickness ratio, and plate aspect ratio on the initial buckling of rectangular FGMs are discussed.
A FGM plate with dimensions a, b, and h indicated as length, width, and thickness, respectively, in the coordinates xy and zaxis are shown in Fig. 1. The reference plane is considered from the midplane of the plate.
The volume fraction of the metal side is expressed as [41]
(1)
where n is the volume fraction of metal. The material property (Young's modulus, E where subscripts m and c represent the metallic and ceramic) varying through the thickness of the plate is expressed as [42],[43],[44]
(2)
where
The effect of variation of Poisson’s ratio (v) is assumed as constant.
Fig. 1. The elastically supported FGM plate
The relationship between the FGM plate and supporting foundation is expressed as [45],[46]
(3)
where the parameters k_{w} and k_{s} are the normal and shear stiffness of the foundation, respectively.
2.1. Displacement Field
Assuming the displacement components u and v are the inplane displacements in the x and y directions respectively and w is the transverse displacement in the zdirection. These displacements are small in comparison with the plate thickness. The inplane displacement u in the xdirection and v in the ydirection each consist of two parts: (a) a displacement component analogous to displacement in classical plate theory of bending; (b) a displacement component due to shear deformation which is assumed to be parabolic in nature with respect to thickness coordinate.
The present displacement field is assumed to be the HSDT model with five unknown displacement variables expressed as [47]:

(4) 
where, is a transverse shear function where p=0.9
2.2. StressStrain Relations
The Generalise Hook’s law for the isotropic FGM plate is represented as [48];

(5) 
2.3. Governing Differential Equations
The GDEs of the elastically supported FGM plate are produced by the energy principle and collecting the coefficients of variables and as [49], [54],[55].

(6) 

(7) 

(8) 

(9) 

(10) 
where, and are the inplane axial forces along x and y directions, respectively. Uniaxial compression along the xaxis is known as Uniaxial A type and uniaxial compression along the ydirection is known as UniaxialB type and the rest is biaxial compression along the xand yaxes as shown in Fig. 2.
(a) Uniaxial A 
(b) Uniaxial B 
(c) Biaxial 
Fig. 2. Types of inplane forces of FGM rectangular plates
The boundary condition for simply supported plate is expressed as [2526]

(11) 
The RBF is a mathematical function that depends only on the distance between the input points and some fixed center points in a multidimensional space. In other words, the value of an RBF at any point in the space is determined by the distance between that point and a set of predefined center points. However, the method's accuracy and stability are extremely dependent on an appropriate choice of a shape parameter ‘c’. The solution of the linear GDEs is assumed in terms of modified thin plate spline RBF expressed as for nodes 1: N, as;
where a and b are the length and breadth of the rectangular plate and 'c' is the shape parameter. First, the convergence of results is checked by shifting the shape parameter. When convergence is achieved within 1% and is stable, the accuracy is verified. The unknown field variables and appearing in governing differential equations are assumed in terms of radial basis function as [51],[52],[53]:

(12) 

(13) 

(14) 

(15) 

(16) 
where, N is the total number of nodes, which is equal to the summation of boundary nodes NB and domain interior nodes NI, is an unknown coefficient, is radial basis function, is the distance between the nodes and c is the shape parameter.
The discretized GDEs for buckling analysis are written as:
(17) 
(18) 
(19) 
(20) 
(21) 
The final matrix in the form of unknown variables is written as above. For calculating the unknown variables, a generalized MATLAB code is developed as it is easy for matrix operations to find out the unknown variables.
Several numerical examples are presented for buckling analysis of FGM rectangular plates with different plate aspect ratios and thickness ratios for uniaxial and biaxial compressions, along with varying parameters of foundation. FGM plate is assumed as simply supported, and the material properties used in computations are as Ec=360 GPa, Em=70 GPa, and v=0.3.
The following nondimensional critical buckling is expressed as where N_{cr} is the critical dimensional buckling load.
The Winkler (K_{w}) and shear foundation parameters (K_{s}) are expressed as
and
where K_{s} and K_{w} are the input shear and spring foundation parameters.
4.1. Convergence and Validation
The convergence and comparison study for the normalized buckling with respect to the number of nodes is presented in Table 1 for a/h=10, a/b=0.5, and k_{s}=0, k_{w}=0. The present results are compared with results available in literature by Kulkarni et al. [56] using the 2D analytical solution based on HSDT and Sekkal et al. [57] using quasi3D HSDT. It is noted that the present results are in perfect agreement with the results available in the literature, and they converged within 1% and became closer to the results of Kulkarni et al. [56] and Sekkal et al. [57] at 15X 15. It is clearly observed that the present result produced in 15X15 nodes showed 0.04 % average difference the results of Kulkarni et al. [56] and 2.40 % average difference the results of Sekkal et al. [57] under UniaxialB loading and 0.04 % average difference the results of Kulkarni et al. [56] and 2.40 % average difference the results of Sekkal et al. [57] under Biaxial loading Hence 15X15 nodes are taken for further analysis of the numerical study. With the increase of power index law, the dimensionless buckling load decreases due to more content of metal portion being added. In between the uniaxial and biaxial loading, biaxial loading provides better lower buckling resistance as compared to uniaxial loading due to a higher amount of inplane loading.
4.2. Parametric study
In this section, the influence of aspect ratio on the critical buckling loads of FGM rectangular plate with different spantothickness ratios for a/h =5,10 and 20 under UniaxialA load and n=1, and k_{s}=0, k_{w}=0 is investigated and shown in Fig. 3. For the graph with thickness ratio a/h=5, it is noticed that by increasing the aspect ratio (a/b) from 0.5 to 0.8, the buckling load start decreases and further increases from 0.9 to 1.2. A similar nature is observed in aspect ratios from 1.3 to 2.3. In addition, with an increase in aspect ratio, the peck of normalized buckling shifted toward the right for all spantothickness ratios. The same nature is followed by all the spantothickness ratios. For the given aspect ratio, the buckling load increases with the increase of span spantothickness ratio.
The influences of aspect ratio with grading index on normalized critical buckling of rectangular FGM plate under uniaxialA loading is investigated for a/h=10, and Ks=0, Kw=0 and shown in Fig. 4. It is interesting that with the increase of the gradation index, the buckling load decreases. For the same gradation index, with the increase of aspect ratio, the buckling load decreases from 0.5 to 0.8 and then increases from 0.9 to 1.2, as discussed in the previous example of 2.1.1. For the gradation index n=0, the buckling load is highest due to the addition of a fully ceramic portion in the plate.
Further, the effect of aspect ratio on normalized critical buckling of rectangular FGM plate under various loading conditions for a/h=10, n=2, Ks=0, Kw=0 is shown in Fig. 5. In addition to the above discussion, it is worth noting that the nature of changing the critical buckling is repeated for UniaxialA loading, as discussed in section 4.2.2. For uniaxialB and biaxial loading, buckling load decreases continuously with the increase of aspect ratio.
Fig. 3. Effect of plate length to thickness ratio and aspect ratio on normalized critical buckling
under UniaxialA loading.
Fig. 4. Effect of aspect ratio and grading index under UniaxialA load.
Table 2 represents the contour shape of different aspect ratios (b/a=0.5,1,2, and3) on normalized buckling of FGM rectangular plate under UniaxialA, UniaxialB, and biaxial loading for a/h=10, n=1, and Ks=0, Kw=0. It is very interesting that the first mode is observed for a small change in aspect ratios(b/a) from 0.5 and 1.0 for uniaxialA, while the second and third mode is observed for further increase of aspect ratio from 2.0 to onward. For uniaxialB, the second and first mode is observed for aspect ratio 0.5 and 1.0, and the first mode is observed for further increasing the aspect ratio from 1 to 3. For biaxial loading, with the increase of aspect ratio, the first mode is observed so far. The reason for changing the shape contour is due to local deformation in the shape of the structure, and more than one apex may be created for the given specific arrangement of uniaxial compression. Among the given different arrangements of UniaxialA and UniaxialB and biaxial inplane loading, with the increase of aspect ratio, buckling load of UniaxialA arrangement decreases very sharply from 0.1 to 1 and then very little change is observed for further increase the aspect ratio from 1 to onward. For the UniaxialB arrangement, with the increase of aspect ratio, the buckling load decreases very sharply, and a similar trend is observed in biaxial compression also.
Table 1. Convergence and validation study of FGM rectangular plate under uniaxialB and biaxial loading.
Loading 
Methods 
Grading Index 
Avg.% 
Avg. % 3D[52] 

0 
0.5 
1 
2 
5 
10 

UniaxialB 
9X9 
8.9862 
5.364 
3.7543 
2.6877 
2.3787 
2.3974 
8.59 
9.32 
11X11 
7.4907 
4.8546 
3.7215 
2.8874 
2.4262 
2.2127 
0.59 
2.19 

13X13 
7.4187 
4.8254 
3.7123 
2.889 
2.4182 
2.1938 
0.09 
2.35 

15X15 
7.4079 
4.8213 
3.711 
2.8892 
2.4168 
2.1906 
0.04 
2.40 

2D [51] 
7.4102 
4.8234 
3.7131 
2.8907 
2.4152 
2.1904 
 
 

3D[52] 
7.4126 
4.8904 
3.8221 
3.0168 
2.509 
2.2374 
 
 

Biaxial 
9X9 
7.6392 
4.441 
3.0673 
2.1808 
1.9386 
1.9807 
11.05 
11.06 
11X11 
5.9935 
3.8852 
2.9792 
2.3121 
1.9427 
1.7713 
0.62 
2.13 

13X13 
5.9349 
3.8604 
2.97 
2.3115 
1.9348 
1.7551 
0.09 
2.34 

15X15 
5.9263 
3.857 
2.9688 
2.3114 
1.9334 
1.7525 
0.04 
2.40 

2D [51] 
5.9281 
3.8587 
2.9705 
2.3125 
1.9322 
1.7523 
 
 

3D[52] 
5.9301 
3.9123 
3.0577 
2.4134 
2.0072 
1.7899 
 
 
Table 2. Contour shapes of normalized critical buckling for different aspect ratios under various inplane loadings.
b/a 
UniaxialA 
UniaxialB 
Biaxial 
0.5 
13.6801 
32.8877 
10.9443 
1 
9.3453 
9.3453 
4.6727 
2 
9.3360 
3.7110 
2.9688 
3 
9.3441 
2.9402 
2.6462 
Table 3. Contour shape with the effect of foundation on normalized buckling loads of rectangular FGM plate under inplane loading.
Kw 
Ks 
Uniaxial A 
Uniaxial B 
Biaxial 
0 
0 
7.5353 
2.9558 
2.3647 
10 
0 
7.6281 
3.0485 
2.4389 
100 
0 
8.4634 
3.8832 
3.1066 
0 
10 
9.3670 
4.1004 
3.2804 
0 
100 
22.0088 
14.4024 
11.5222 
Table 4. Contour shape of two modes on normalized buckling of FGM plate under uniaxial loading. (n=2, a/h=20).
Loading 
b/a=1 
b/a=2 
Uniaxial A 
Mode 1= 7.5360 Mode 2= 11.5477 
Mode 1= 7.5353 Mode 2= 8.7809 
Uniaxial B 
Mode 1= 7.5360 Mode 2= 11.5477 
Mode 1= 2.9558 Mode 2= 7.5339 
Fig. 5. Effect of aspect ratio under various inplane loading
The effect of foundation parameters (K_{w} and K_{s}) with plate aspect ratios on normalized buckling loads of FGM plate under UniaxialB loading is shown in Fig. 6. It is observed that at a constant aspect ratio, buckling load increases by increasing Winkler and Pasternak stiffness foundation. This is because the foundation stiffness increases the overall stiffness of the plate. The effect of the Pasternak stiffness foundation is more sensitive than the Winkler stiffness foundation. It is also observed that the nature followed is similar to the impact of plate aspect ratio in section 4.2.1. It is interesting that for the higher value of shear foundation parameters, the peak of normalized critical buckling is shifted toward the left due to becoming more stronger plate by the action of the foundation.
Fig. 6. Effect of elastic foundation with aspect ratios on normalized buckling loads under UniaxialB loading. (a/h=10, n=1)
The effect of foundation parameters with contour shape on normalized buckling loads of FGM plate under different inplane loading for a/h=20, n=2, and b/a=2shown in Table 3. It is observed that with the increase of foundation parameters, normalized critical buckling increases due to an increase in the stiffness matrix. The effect of foundation parameters is more severe in UniaxialA type arrangement due to higher length along width direction as compared to other UniaxialB and biaxial compression. Due to the higher length along the width direction, there would be more than one peak, as shown in the first column of the given Table. It is discussed earlier; the effect of the shear foundation parameter is more crucial on critical buckling load as compared to the Winkler foundation parameter.
Table 4 shows the first two modes shape of a rectangular FGM plate for a/h=20, n=2, and b/a=1 and 2 for uniaxialA and uniaxialB loading. In between the two loading conditions, with the increase of aspect ratio, there would be more than one peak in uniaxialA types of arrangement while there would not be any change in uniaxialB types of arrangement due to the load acting along xdirection with constant length.
In this study, initial buckling analysis of elastically supported rectangular FGM plate using a modified RBFbased meshfree method for different inplan loading is carried out. The result produced by modified RBF for rectangular plates is in good agreement with the published results. The displacement field with five unknown variables is based on higherorder shear deformation theory. The need for shear correction factor is not required. The GDEs are obtained by deriving the energy principle. The following conclusions are noted from the limited study:
Overall, FGMs offer a unique and versatile approach to material design and have the potential to revolutionize many industries in the future. Buckling analysis of these plates can help in optimizing their design and improving their performance under different loading and support conditions. This can ensure the structural integrity and safety of the engineering structures.
Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this manuscript.