Research ArticleAn Edge-Based Smoothed Finite Element for Free VibrationAnalysis of Functionally Graded Porous (FGP) Plates on ElasticFoundation Taking into Mass (EFTIM)

Trung Thanh Tran1 Quoc-Hoa Pham 23 and Trung Nguyen-Thoi23

1Department of Mechanics Le Quy Don Technical University Hanoi Vietnam2Division of Computational Mathematics and Engineering Institute for Computational Science Ton Duc ang UniversityHo Chi Minh City Vietnam3Faculty of Civil Engineering Ton Duc ang University Ho Chi Minh City Vietnam

Correspondence should be addressed to Quoc-Hoa Pham phamquochoatdtueduvn

Received 31 December 2019 Revised 10 March 2020 Accepted 17 March 2020 Published 25 April 2020

Academic Editor Xesus Nogueira

Copyright copy 2020 Trung anh Tran et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this paper free vibration analysis of the functionally graded porous (FGP) plates on the elastic foundation taking into mass(EFTIM) is presented e fundamental equations of the FGP plate are derived using Hamiltonrsquos principle e mixed inter-polation of the tensorial components (MITC) approach and the edge-based smoothed finite element method (ES-FEM) isemployed to avoid the shear locking as well as to improve the accuracy for the triangular element e EFTIM is a foundationmodel based on the two-parameter WinklerndashPasternak model but added a mass parameter of foundation Materials of the plateare FGP with a power-law distribution and maximum porosity distributions in the forms of cosine functions Some numericalexamples are examined to demonstrate the accuracy and reliability of the proposed method in comparison with those available inthe literature

1 Introduction

e plate resting on the elastic foundation (EF) is one of themost common types of structures which have practicalapplications in civil and industrial constructions especiallyin transportation and irrigation In particular the structuresof beams and plates on the EF are subjected to moving loadsof means of transport such as roadbeds affected by vehiclesrailway tracks and aircraft runways In most publicationswhen investigating the mechanical behavior of structures onthe EF the researchers mainly use the one-parameterWinkler foundation model [1] or two-parameter Win-klerndashPasternak foundation model [2 3] e analysis ofplates resting on the WinklerndashPasternak foundation waspreviously addressed by several authors For instanceFazzolari [3] used an analytical method to consider freevibration and buckling of porous FG Sandwich beams

resting on the EF with the WinklerndashPasternak foundationmodel Leissa [4] presented results for the free vibration ofrectangular plates Xiang et al [5] studied free vibration forMindlin plates on the WinklerndashPasternak foundation usingan analytical method Omurtag et al [6] used the finiteelement method (FEM) for the free vibration analysis of theKirchhoff plates resting on the EF Ozccedilelikors et al [7]analyzed the exact solutions of bending buckling and vi-bration problems of a Levy-plate on the two-parameterfoundation Matsunaga [8] used a special higher-order platetheory (HSDT) to analyze vibration and buckling of thickplates on the EF Ayvaz et al [9] developed a modifiedVlasov model to consider the earthquake response of thinplates on the EF Shen et al [10] based on the RayleighndashRitzmethod to study free and forced vibration of the Reiss-nerndashMindlin plates resting on the EF Liew et al [11 12] andHan and Liew [13] analyzed the free vibration of rectangular

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 8278743 17 pageshttpsdoiorg10115520208278743

plates resting on the EF using a differential quadraturemethod Zhou et al [14] considered the vibration of rect-angular plates on the EF using the Ritz method Chu-cheepsakul and Chinnaboon [15] investigated plates by atwo-parameter foundation model using a boundary elementmethod Civalek and Acar [16] investigated the bending ofMindlin plates on the EF by developing a singular convo-lutionmethod Ferreira et al [17] presented bending and freevibration analyses of the FGP plates on the Win-klerndashPasternak foundation by using radial basis functionsShahsavari et al [18] used a new quasi-3D hyperbolic theoryfor the free vibration analysis of the FGP plates resting on theEF Zenkour and Radwan [19] proposed an exact analyticalapproach for free vibration analysis of laminated compositeand Sandwich plates resting on the EF using a four-unknownplate theory Duc et al [20] presented the analysis ofnonlinear thermal dynamic response of shear deformablefunctionally graded plates on the EF Mahmoudi et al [21]developed a refined quasi-three-dimensional shear defor-mation theory to analyze the functionally graded Sandwichplates resting on the two-parameter EF under thermo-mechanical loading Duc et al [22] used an analyticalmethod to calculate static bending and free vibration of FGcarbon nanotube-reinforced composite plate resting onWinklerndashPasternak foundations ang et al [23] consid-ered the effects of variable thickness on buckling andpostbuckling behavior of imperfect sigmoid FGM plates onelastic medium subjected to compressive loading Banh-ien et al [24] developed the isogeometric analysis forbuckling analysis of nonuniform thickness nanoplates in anelastic medium

In recent years the FGP materials have attracted greatinterest frommany researchers over the world due to theirlightness and high strength As a result they are widelyapplied for civil engineering aerospace structures nuclearplants and other applications Kim et al [25] investigatedbending vibration and buckling of the FGP microplatesusing a modified couples stress based on the analyticalmethod Coskun et al [26] presented analytical solutionsto analyze the bending vibration and buckling of the FGmicroplates based on the third-order shear deformationtheory (TSDT) Chen et al [27] investigated the staticbending and buckling of the FGP beams by using theTimoshenko beam theory Rezaei and Saidi [28 29]studied the vibration of rectangular and porous-cellularplates based on an analytical method e vibration of theFGP shallow shells using an improve Fourier method wasexamined by Zhao et al [30] Moreover the dynamics ofthe FGP doubly-curved panels and shells were also in-vestigated in [31] Li et al [32] analyzed the nonlinearvibration and dynamic buckling of the Sandwich FGPplates with graphene platelet reinforcement (GPL) on theEF For nonlinear problems Sahmani et al [33] used thenonlocal method to analyze nonlinear large-amplitudevibrations of the FGP micro-nanoplates with GPL re-inforce Wu et al [34] studied the dynamic of the FGPstructures by using FEM ang et al [35] investigated the

elastic buckling and free vibration of porous cellular platesbased on the first-order shear deformation theory (FSDT)Although the FGP materials have many different types inthis paper the authors only use the distribution of po-rosity as presented in [25 26]

In the other front of the development of numericalmethods for computational mechanics Liu et al [36]have recently proposed an edge-based smoothed FEM(ES-FEM) using triangular elements which show somefollowing excellent properties for the 2D solid mechanicsanalyses such as (1) the numerical results are often foundsuperconvergent and very accurate (2) the method isstable and works well for dynamic analysis and (3) theimplementation of the method is straightforward and nopenalty parameter is used e ES-FEM has been de-veloped for n-sided polygonal elements [37] viscoelas-toplastic analyses [38] 2D piezoelectric [39] primal-dualshakedown analyses [40] fluid structure interaction[41 42] and various applications [43ndash45] Recently in aneffort to improve the accuracy of the plate and shellstructural analyses the classical MITC3 element [46]incorporated with the ES-FEM [36] has been proposedto give the so-called ES-MITC3 element [47ndash51] In theformulation of the ES-MITC3 the system stiffness matrixis employed using strains smoothed over the smoothingdomains associated with the edges of the MITC3 ele-ments e numerical results demonstrated that the ES-MITC3 has the following great properties [47] (1) theES-MITC3 can eliminate transverse shear locking evenwith the ratio of the thickness to the length of thestructures reach 10minus 8 and (2) the ES-MITC3 has betteraccuracy than the existing triangular elements such asMITC3 [46] DSG3 [52] and CS-DSG3 [53] and is a goodcompetitor with the quadrilateral element MITC4 ele-ment [54]

e objective of this research now is to further extendthe ES-MITC3 method for free vibration analyses of theFGP plates resting on the EFTIM e governing equa-tions are derived from the FSDT and the Reiss-nerndashMindlin plate theory due to simplicity andcomputational efficiency Besides the EFTIM is modelledbased on a two-parameter WinklerndashPasternak foundationmodel and added in a mass parameter of foundation eplate is made from the FGP materials with a power-lawdistribution (k) and maximum porosity distributions (Ω)in the forms of cosine functions e accuracy and reli-ability of the present formulation are verified by com-paring with those of other available numerical resultsMoreover the effects of some geometric parameters andmaterial properties on the free vibration of the FGP platesare also examined in detail

2 Functionally Graded Porous MaterialPlates on Elastic Foundation

Let us consider an FGP plate resting on EFTIM as shown inFigure 1 e FGP materials with a variation of two

2 Mathematical Problems in Engineering

constituents and three different distributions of porositythrough-thickness are presented as follows [25 26]

Case 1 Λ(z) Ω cosπz

h1113874 1113875

Case 2 Λ(z) Ω cosπ2

z

h+ 051113874 11138751113876 1113877

Case 3 Λ(z) Ω cosπ2

z

hminus 051113874 11138751113876 1113877

(1)

where Ω is the maximum porosity value A typical materialproperty of the FGP materials can be considered as in thefollowing power-law relations

P(z) Pt minus Pb( 1113857z

h+ 051113874 1113875

k

+ Pb1113890 1113891(1 minus Λ(z)) (2)

where Pt and Pb are the typical material properties at the topand the bottom surfaces respectively and k is the power-lawindex e normalized distributions of porosity through thethickness are shown in Figure 2(a) As shown in Figure 2(a)the porosity distribution of Case 1 is symmetric with respectto the midplane of plates Case 2 and Case 3 are bottom andtop surface-enhanced distributions respectively BesidesFigures 2(b)ndash2(d) show the distributions of a normalizedtypical property associated with three different cases ofporosity distributions with parameters Ω 05 k 1 5 10and PtPb 10

e EFTIM is built based on the WinklerndashPasternakfoundation by adding a mass parameter of foundation

qe k1w(x y t) minus k2z2

zx2 +z2

zy21113888 1113889w(x y t) + mf

z2w(x y t)

zt2

(3)

where w is the displacement of FGP plate k1 and k2 arerespectively Winkler foundation stiffness and shear layerstiffness of the Pasternak foundation In order to mentionthe effectiveness of the foundation mass involved in theoscillation as well as the continuous interaction of the springwith the plate the parameter β with unit kgminus1 is added Itcharacterizes the effective level of the foundation mass in-volved in vibration which is determined based on an ex-perimental basis and the ratio of the density of thefoundation to the density of plate material which is definedas μF ρFρ us the density of mass mf involved vi-bration with the foundation is determined mf βμFρ Fromequation (3) we see that for the static problem the EFTIMmodel and WinklerndashPasternak foundation model are thesame But for the dynamic problems these two models havedifferences and when omitting the influence parameters ofthe foundation mass the EFTIM model is equivalent to theWinklerndashPasternak foundation model In addition thisfoundationmodel also covers theWinkler foundationmodelwhen the influence of shear parameters and foundationmassparameters ignored It was found that the EFTIM model

closely resembles the true feature of the foundation in-cluding the Pasternak and Winkler foundation models

3 The First-Order Shear Deformation Theoryand Weak Form of the FGP Plates

31 First-Order Shear Deformation eory for FGP Platese displacement of the FGP plates in the present workbased on the FSDT model can be expressed as

u (x y z) u0(x y) + zθx(x y)

v (x y z) v0(x y) + zθy(x y)

w (x y z) w0(x y)

⎧⎪⎪⎨

⎪⎪⎩(4)

where u v w θx and θy are five unknown displacements ofthe midsurface of the plate For a bending plate the strainfield can be expressed as follows

ε εm + zκ (5)

where the membrane strain is given as

εm

u0x

v0y

u0y + v0x

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(6)

e bending and transverse shear strains are written as

κ

θxx

θyy

θxy + θyx

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(7)

γ w0x + θx

w0y + θy

1113896 1113897 (8)

FromHookersquos law the linear stress-strain relations of theFGP plates can be expressed as

σx

σy

τxy

τxz

τyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

Q11 Q12 0 0 0

Q21 Q22 0 0 0

0 0 Q66 0 0

0 0 0 Q55 0

0 0 0 0 Q44

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

εx

εy

cxy

cxz

cyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(9)

x

z

y

Spring stiffener

Shear layer

Mass offoundationO

Figure 1 Modeling the FGP plate resting on EFTIM

Mathematical Problems in Engineering 3

where

Q11 Q22 E(z)

1 minus υ2

Q12 Q21 υE(z)

1 minus υ2

Q44 Q55 Q66 E(z)

2(1 + υ)

(10)

where E(z) presents for effective Youngrsquos modulus and υrepresents Poissonrsquos ratio

32 Weak Form Equations To obtain the motion equationsof the FGP plates for the free vibration analysis Hamiltonrsquosprinciple is applied with the following form

1113946t2

t1

(δU minus δK)dt 0 (11)

where U and K are the strain and kinetic energies re-spectively e strain energy is expressed as

U Up

+ Uf

(12)

where Uf is the strain energy

Uf

12

1113946ψ

k1w2

minus k2z2w

zx21113888 1113889

2

+z2w

zy21113888 1113889

2⎡⎣ ⎤⎦⎛⎝ ⎞⎠dψ (13)

and Up is the strain energy

Up

12

1113946ψεTDε + γTCγ1113872 1113873dψ (14)

0 02 04 06 08 1Φ(z) (Ω)

ndash05

ndash03

ndash01

01

03

05z (

h)

Case 1Case 2

Case 3

(a)

k = 1 PtPb = 10 Ω = 05

0 02 04 06 08 1P(z)Pt

ndash05

ndash03

ndash01

01

03

05

z (h)

Case 1Case 2

Case 3(z) = 0

(b)

0 02 04 06 08 1P(z)Pt

ndash05

ndash03

ndash01

01

03

05

z (h)

k = 5 PtPb = 10 Ω = 05

Case 1Case 2

Case 3(z) = 0

(c)

0 02 04 06 08 1P(z)Pt

ndash05

ndash03

ndash01

01

03

05z (

h)

Case 1Case 2

Case 3(z) = 0

k = 10 PtPb = 10 Ω = 05

(d)

Figure 2 Distributions of porosity and typical material property (a) Distribution of porosity along of z-axis (b) distribution materialproperty with k 1 (c) distribution material property with k 5 and (d) distribution material property with k 10

4 Mathematical Problems in Engineering

where ε εm κ1113858 1113859T and

D A B

B F1113890 1113891 (15)

and A B F and C can be given by

(AB F) 1113946h2

minush21 z z

21113872 1113873

Q11 Q12 0

Q21 Q22 0

0 0 Q66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dz (16)

C 1113946h2

minush2

Q55 00 Q44

1113890 1113891dz (17)

e kinetic energy in equation (11) is given by

K Kp

+ Kf

(18)

where Kp is the kinetic energy

Kp

12

1113946ψ

_umpu dψ (19)

where uT u0 v0 w0 θx θy ϕx ϕy1113960 1113961 is the displacementfield and mp is the mass matrix defined by

mp

I1 0 0 I2 0

I1 0 0 I2

I1 0 0

I3 0

I3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

where (I1 I2 I3) 1113938h2minush2 ρ(1 z z2)dz

In equation (11) the kinetic energy of the mass offoundation Kf is defined as

Kf

12

1113946ψ

_wmfw dψ (21)

Substituting equations (12) and (18) into equation (11)the weak formulation for the free vibration of the FGP plateis finally obtained as

1113946ψδεTDε dψ + 1113946

ψδγTCγ dψ + 1113946

ψδwT

middot k1w minus k2z2w

zx2 +z2w

zy21113888 11138891113890 1113891dψ 1113946ψ

_umpu dψ

+ 1113946ψ

_wmfw dψ

(22)

4 Formulation of an ES-MITC3 Method forFGP Plates

41 Formulation of the Finite Element Using the MITC3Element e middle surface of plate ψ is discretized into ne

finite three-node triangular elements with nn nodes such thatψ asymp 1113936

ne

e1ψe and ψi capψj empty ine j en the generalizeddisplacements at any point ue [ue

j vej we

j θexj θ

eyj]

T ofelement ψe can be approximated as

ue(x) 1113944

nne

j1

NI(x) 0 0 0 0

0 NI(x) 0 0 0

0 0 NI(x) 0 0

0 0 0 NI(x) 0

0 0 0 0 NI(x)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dej

1113944nne

j1N(x)de

j

(23)

where nne is the number of nodes of ψe N(x) is the shapefunction matrix and de

j [uej ve

j wej θ

exj θ

eyj]

T is the nodaldegrees of freedom (dof ) associated with the jth node of ψe

e membrane bending strains of MITC3 element canbe expressed in matrix forms as follows

εem Be

m1 Bem2 Be

m31113858 1113859de Be

m de (24)

κe Be

b1 Beb2 Be

b31113858 1113859de Be

b de (25)

where

Bem1

12Ae

b minus c 0 0 0 0

0 d minus a 0 0 0

d minus a b minus c 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (26)

Bem2

12Ae

c 0 0 0 00 minusd 0 0 0

minusd c 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (27)

Bem3

12Ae

minusb 0 0 0 00 a 0 0 0a minusb 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (28)

Beb1

12Ae

0 0 0 b minus c 00 0 0 0 d minus a

0 0 0 d minus a b minus c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (29)

Beb2

12Ae

0 0 0 c 00 0 0 0 minusd

0 0 0 minusd c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (30)

Beb3

12Ae

0 0 0 minusb 00 0 0 0 a

0 0 0 a minusb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)

To eliminate the shear locking phenomenon as thethickness of the plate becomes small the formulation of thetransverse shear strains of the MITC3 element based onFSDT [36] in this study can be written as follows

γe Be

s de (32)

where

Mathematical Problems in Engineering 5

Bes Be

s1 Bes2 Be

s31113858 1113859 (33)

with

Bes1 Jminus 1

0 0 minus1a

3+

d

6b

3+

c

6

0 0 minus1d

3+

a

6c

3+

b

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

Bes2 Jminus 1

0 0 1a

2minus

d

6b

2minus

c

6

0 0 0d

6c

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (35)

Be(0)s3 Jminus 1

0 0 0a

6b

6

0 0 1d

2minus

a

6c

2minus

b

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (36)

where

Jminus 1

12Ae

c minusb

minusd a

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (37)

Here a x2 minus x1 b y2 minus y1 c y3 minus y1 and d x3 minus

x1 are pointed out in and Ae is the area of the three-nodetriangular element as shown in Figure 3

Substituting the discrete displacement field into equation(22) we obtained the discrete system equations for freevibration analysis of FGP plate resting on the EF respec-tively as

K minus ω2M1113872 1113873d 0 (38)

where K and M are the stiffness and mass matricesrespectively

e stiffness matrix in equation (38) can be written as

K 1113944ne

e1Ke

p + Kef1113872 1113873 (39)

where

Kep 1113946

ψe

BTDB dψe + 1113946ψe

BTs CBsdψe (40)

Kef k11113946

ψe

NTwNwdψe + k21113946

ψe

zNw

zx1113888 1113889

TzNw

zx1113888 1113889⎡⎣

+zNw

zy1113888 1113889

TzNw

zy1113888 1113889⎤⎦dψe

(41)

where

Be Be

m Beb1113858 1113859 (42)

Nw 0 0N1 0 0 0 0N2 0 0 0 0N3 0 01113858 1113859 (43)

Next the mass matrix in equation (38) can be defined as

M 1113944ne

e1Me

p + Mef1113872 1113873 (44)

where

Mep 1113946

ψe

NTmpN dψe (45)

Mef mf1113946

ψe

NTwNw dψe (46)

42 Formulation of an ES-MITC3 Method for FGP Platese smoothing domains ψk is constructed based on edges ofthe triangular elements such that ψ cupnk

k1ψk and

ψki cap ψk

j empty for ine j An edge-based smoothing domain ψk

for the inner edge k is formed by connecting two end-nodesof the edge to the centroids of adjacent triangular MITC3elements as shown in Figure 4

Applying the edge-based smooth technique [36] thesmoothed membrane bending and shear strain 1113957εk

m 1113957κk 1113957γk

over the smoothing domain ψk can be created by

1113957εkm 1113946

ψkεmΦ

k(x)dψ (47)

1113957kk 1113946

ψk

κΦk(x)dψ (48)

1113957γk 1113946

ψk

γΦk(x)dψ (49)

where εm κ and γ the compatible membrane bending andthe shear strains respectively Φk(x) is a given smoothingfunction that satisfies at least unity property1113938ψkΦk(x)dψ 1

In this study we use the constant smoothing function

Φk(x)

1Ak

x isin ψk

0 x notin ψk

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(50)

y

d

c

a

bx

3 (x3 y3)

2 (x2 y2)

1 (x1 y1)

Figure 3 ree-node triangular element in the local coordinates

6 Mathematical Problems in Engineering

where Ak is the area of the smoothing domain ψk and isgiven by

Ak

1113946ψk

dψ 13

1113944

nek

i1A

i (51)

where nek is the number of the adjacent triangular elementsin the smoothing domain ψk and Ai is the area of the ithtriangular element attached to the edge k

By substituting equations (47)ndash(49) into equations (24)(25) and (32) the approximation of the smoothed strains onthe smoothing domain ψk can be expressed as follows

1113957εkm 1113944

nnk

j1

1113957Bk

mjdkj

1113957kk 1113944

nnksh

j1

1113957Bk

bjdkj

γk 1113944

nnksh

j1

1113957Bk

sjdkj

(52)

where nnksh is the total number of nodes of the triangular

MITC3 elements attached to edge k (nnksh 3 for boundary

edges and nnksh 4 for inner edges as given in Figure 4 dk

j isthe nodal dof associated with the smoothing domain ψkand 1113957Bk

mj 1113957Bk

bj and 1113957Bk

sj are the smoothed membrane bendingand shear strain gradient matrices respectively at the jthnode of the elements attached to edge k computed by

1113957Bk

mj 1

Ak1113944

nek

i1

13A

iBemj (53)

1113957Bk

bj 1

Ak1113944

nek

i1

13AiB

ebj (54)

1113957Bk

sj 1

Ak1113944

nek

i1

13A

iBesj (55)

e stiffness matrix of the FGP plate using the ES-MITC3 is assembled by

1113957K 1113944

nksh

k1

1113957Kk (56)

where 1113957Kk is the ES-MITC3 stiffness matrix of the smoothingdomain ψk and given by

1113957Kk 1113946

ψk

1113957BKTD1113957Bk+ 1113957Bkt

s C1113957Bk

s1113874 1113875dψ 1113957BKTD1113957BkA

k+ 1113957Bkt

s C1113957Bk

s Ak

(57)

where

1113957BkT 1113957Bk

mj1113957Bk

bj1113876 1113877 (58)

5 Accuracy of the Proposed Method

In this section the various numerical examples are solved toverify the reliability and accuracy of the proposed methodFor convenience the stiffness factors and nondimensionalfrequencies of the plates are defined as the followingequations

K1 k1a

4

H

K2 k2a

2

H

λ ωa2

π2

ρh

H

1113970

withH Eh3

12 1 minus ]2( )

(59)

To demonstrate the performance of numerical resultsthe relative frequency error is defined by

Δ () 100 timesλpr minus λre

11138681113868111386811138681113868

11138681113868111386811138681113868

λre

11138681113868111386811138681113868111386811138681113868

(60)

where λpr and λre are nondimensional frequencies of presentmethod and nondimensional frequencies in [17 18]respectively

e results of the convergence of the first two nondi-mensional frequencies of the plate in the case of fully simplesupport (SSSS) plate and a fully clamped (CCCC) plate withha 01 K1 100 K2 10 respectively are shown in Fig-ure 5 From these results it can be seen that almost allfrequencies corresponding to different cases of boundaryconditions (BC) converge with 18times18 element mesh For18times18 mesh we compare the first three nondimensionalfrequencies of a plate resting on the WinklerndashPasternakfoundation with the published results as shown in Table 1 Itcan be seen that the present results agree well with the resultsof the authors using analytical methods [5 14 17] and aremore accurate than those using the original MITC3 elementand FEM [6] In addition from Table 2 it is obvious that therelative error of the present results compared to [18] is lessthan 2 In [18] they used a new quasi-3D hyperbolic theoryto investigate the free vibration of the FGP plate resting onthe EF ese results are the basis to analyze the free vi-bration of FGP plates on the EFTIM

Boundary edge m

Inner edge k

Г(k)

ψ(k)

Г(m)

ψ(m)

Field nodeCentroid of triangles

Figure 4 e smoothing domain ψk is formed by triangularelements

Mathematical Problems in Engineering 7

Next we consider an SSSS FGP plate (AlAl2O3)with its material properties as follows metal (Al)Eb 70GPa ρb 2702 kgm3 and ceramic (Al2O3)Et 380GPa ρt 3800 kgm3 Poissonrsquos ratio is fixed atυ 03 e FGP plate with even porosities is expressed asin [18]

P(z) Pb + Pt minus Pb( 1113857z

h+ 051113874 1113875

k

minusξ2

Pt + Pb( 1113857 (61)

where ξ(ξ le 1) presents the porosity volume fraction estiffness factor and nondimensional frequencies of the platesare shown in equation (58) with Hb (Ebh312(1 minus υ2)) and

4 6 8 10 12 14 16 18 20Mesh (nxn)

25

3

35

4

45

SSSSCCCC

λ1

(a)

SSSSCCCC

4 6 8 10 12 14 16 18 20Mesh (nxn)

5

6

7

8

9

10

λ2

(b)

Figure 5 e convergence of element mesh to nondimensional frequency of plate (a)λ1 and (b)λ2

Table 1 Nondimensional frequencies of plates

Plates K1 K2 Author λ1 Δ() λ2 Δ() λ3 Δ()

SSSSυ 03ha 001

100 10

Ferreira et al [17] 26559 55718 85384Zhou et al [14] 26551 003 55717 000 85406 003Xiang et al [5] 26551 003 55718 000 85405 002

MITC3 26604 017 56103 070 86296 107Present 26590 012 55920 037 86017 074

500 10

Ferreira [17] 33406 59285 87754Zhou et al [14] 33398 002 59285 000 87775 002Xiang et al [5] 33400 002 59287 000 87775 002

MITC3 33441 010 59649 061 88642 101Present 33430 007 59477 032 88370 070

SSSSυ 03ha 01

200 10

Ferreira et al [17] 27902 53452 78255Zhou et al [14] 27756 052 52954 093 77279 125Xiang et al [5] 27842 022 53043 077 77287 124

MITC3 27874 010 53258 036 77719 068Present 27887 005 53362 017 77971 036

1000 10

Ferreira et al [17] 39844 60430 83112Zhou et al [14] 39566 070 59757 111 81954 139Xiang et al [5] 39805 010 60078 058 82214 108

MITC3 39827 004 60266 027 82619 059Present 39836 002 60358 012 82856 031

CCCCυ 015ha 0015

13902 16683

Ferreira et al [17] 81669 12821 16842Zhou et al [14] 81675 001 12823 002 16833 005

Omurtag et al [6] 81375 036 12898 060 16932 053MITC3 81842 021 12909 069 17010 100Present 81729 007 12872 040 16939 058

8 Mathematical Problems in Engineering

Table 2 e first nondimensional frequencies of FGP plate according to the WinklerndashPasternak foundation stiffness (k 1)

(K1 K2) haξ 0 ξ 02

Present [18] Δ() Present [18] Δ()

(0 0)

005 9010 9020 011 8485 8370 137010 8823 8818 006 8319 8203 141015 8541 8516 029 8069 7950 150020 8196 8151 055 7762 7641 158

(100 0)

005 9389 9430 043 9020 8917 116010 9207 9231 026 8858 8753 120015 8933 8934 001 8614 8505 128020 8599 8577 026 8315 8203 137

(100 100)

005 15383 15439 036 16338 16320 011010 15213 15245 021 16175 16148 017015 14962 14966 003 15932 15895 023020 14664 14640 016 15639 15595 028

Table 3 Nondimensional frequencies of the FGP plate on EFTIM

λ1 λ2 λ3 λ4 λ5 λ608583 18118 18157 28015 33898 34248

(a) (b)

(c) (d)

(e) (f )

Figure 6 e first six mode sharps the FGP plate on EFTIM

Mathematical Problems in Engineering 9

06

07

08

09

1

11

12

13

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ1

(a)

14

16

18

2

22

24

26

28

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(b)

2

25

3

35

4

45

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(c)

Figure 7 Nondimensional frequencies of the eFGP plate with difference of featured-index β (a) nondimensional frequency λ1 (b)nondimensional frequency λ2 and (c) nondimensional frequency λ3

Table 4 e first three nondimensional frequencies of FGP on EFTIM

Parameter of plate β 0 025 05 075 1Case 1 λ1 11015 09575 08583 07847 07273(K1 100 K2 10) λ2 23078 20156 18118 16595 15400(SSSS) λ3 35651 31154 28015 25665 23822Case 2 λ1 10225 08988 08113 07453 06931(K1 100 K2 10) λ2 20774 18350 16612 15289 14239(SSSS) λ3 32410 28630 25921 23859 22221Case 3 λ1 12537 10735 09538 08669 08001(K1 100 K2 10) λ2 27399 23531 20942 19054 17599(SSSS) λ3 41972 36097 32152 29269 27044

10 Mathematical Problems in Engineering

Table 5 Nondimensional frequencies of the FGP plate with different K1 and K2

Case of Porosity distribution K2K1

100 250 500 750 1000

Case 1(SSSS)

10 08583 08929 09477 09995 1048825 09258 09579 10092 10580 1104750 10285 10575 11042 11490 1192175 11218 11485 11916 12332 12734100 12079 12327 12730 13120 13499

Case 2(SSSS)

10 08113 08452 08988 09493 0997325 08775 09089 09589 10064 1051850 09778 10061 10515 10950 1136975 10687 10947 11365 11769 12160100 11525 11765 12156 12534 12902

Case 3(SSSS)

10 09538 10009 10747 11438 1209025 10452 10883 11566 12211 1282350 11819 12202 12815 13400 1396075 13044 13392 13953 14492 15011100 14163 14484 15004 15507 15993

08100

1

85 100070 850

12

55 700

K1K2

14

55040 40025 25010 100

λ1

(a)

K1

K2

08100

09

85

1

100070

11

850

12

55 700

13

55040 40025 25010 100

λ1

(b)

K1

K2

08100

1

85 1000

12

70 850

14

55 700

16

55040 40025 25010 100

λ1

(c)

Figure 8 Nondimensional frequencies of FGP plate with different K1 and K2 (a) Case 1 (b) Case 2 and (c) Case 3

Mathematical Problems in Engineering 11

Table 6 Nondimensional frequencies of FGP plate with different K1 K2 and β

β K1 (K2 10) K2 (K1 100)Case 1

100 250 500 750 1000 10 25 50 75 1000 11015 11459 12163 12828 13460 11015 11882 13199 14397 15502025 09575 09961 10572 11150 11700 09575 10328 11473 12514 1347505 08583 08929 09477 09995 10488 08583 09258 10285 11218 12079075 07847 08163 08664 09138 09588 07847 08464 09403 10256 110431 07273 07566 08030 08469 08887 07273 07844 08715 09505 10235Case 20 10225 10652 11327 11964 12569 10225 11059 12323 13469 14525025 08988 09363 09957 10517 11049 08988 09721 10833 11840 1276805 08113 08452 08988 09493 09973 08113 08775 09778 10687 11525075 07453 07763 08256 08720 09161 07453 08060 08982 09817 105861 06931 07220 07677 08109 08519 06931 07496 08353 09129 09845Case 30 12537 13156 14127 15035 15891 12537 13739 15536 17146 18617025 10735 11265 12096 12874 13607 10735 11764 13303 14681 1594105 09538 10009 10747 11438 12090 09538 10452 11819 13044 14163075 08669 09097 09768 10396 10988 08669 09500 10742 11855 128721 08001 08395 09015 09595 10141 08001 08767 09914 10942 11880

060

08

025 100

1

25005 400

12

550075 700850

14

1 1000

080

1

025 1025

12

05 4055

14

075 70851 100

λ1

K1

β

λ1

K2

β

(a)

060

08

025 10025005

1

400550075 700

12

8501 1000

0

08

025

1

102505 40

12

55075 70

14

851 100

λ1

K1

β

λ1

K2

β

(b)

Figure 9 Continued

12 Mathematical Problems in Engineering

ωlowast (ωa2h)(ρbEb)

1113968 e first nondimensional frequen-

cies of present work compared with [18] are shown inTable 2

6 Numerical Results and Discussions

For free vibration problems a fully simple support (SSSS)FGP plate is considered whereina b h a10 Ω 05 k 1 Et 10Eb ρt 10ρbυ 038 and μF 05 e first six nondimensional fre-quencies of the FGP plate with porosity distribution of Case1 and stiffener of foundation K1 100 K2 10 are shownin Table 3 and the first six mode shapes are presented inFigure 6 e stiffness factors and nondimensional

frequencies of FGP plate are shown in equation (58) withHb (Ebh312(1 minus υ2))

61 Influence of the Parameters of the EFTIM to the FreeVibration for the FGP Plate Firstly in order to investigatethe effect of the featured-index of themass of foundation β tofree vibration of the FGP plate the featured-index of mass ischanged from 0 to 1 In Figure 7 and Table 4 it is seen that allcases of porosity distribution featured-index of the mass offoundation β significantly influence to free vibration of theFGP plate As β increases the mass of the plate increases andthe frequencies of the plate decrease For all cases of porositydistribution of the FGP plate the porosity distribution of

080

1

025 100

12

25005 400

14

550075 700850

16

1 1000

0

1

025 102505

15

4055075 70

85

2

1 100

λ1

K1

β

λ1

K2

β

(c)

Figure 9 Nondimensional frequencies of FGP plate with different K1 K2 and β (a) Case 1 (b) Case 2 and (c) Case 3

Table 7 Nondimensional frequencies of the FGP plate as a function of k and Ω

ΩSSSS CCCCk k

0 25 5 75 10 0 25 5 75 10Case 10 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06584 08830 09777 10469 10984 10935 12518 13694 14783 1563305 07126 09238 10105 10767 11273 11732 12869 13930 14958 15780075 07820 09677 10446 11070 11564 12711 13154 14102 15061 158481 08745 10105 10776 11357 11836 13924 13226 14130 15021 15766Case 20 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 08604 09563 10288 10833 10977 12011 13156 14273 1516205 07162 08685 09569 10299 10871 11707 11651 12611 13675 14564075 07777 08600 09357 10073 10671 12179 10868 11556 12496 133421 07852 07972 08420 09009 09580 10347 09106 09310 09904 10547Case 30 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 09413 10475 11196 11714 10977 13661 15012 16147 1699405 07162 10785 11908 12621 13122 11707 15836 17243 18317 19084075 07777 13066 14256 14917 15363 12179 19297 20656 21484 220181 07852 18589 19631 19772 19807 10347 24083 24138 23588 23246

Mathematical Problems in Engineering 13

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

constituents and three different distributions of porositythrough-thickness are presented as follows [25 26]

Case 1 Λ(z) Ω cosπz

h1113874 1113875

Case 2 Λ(z) Ω cosπ2

z

h+ 051113874 11138751113876 1113877

Case 3 Λ(z) Ω cosπ2

z

hminus 051113874 11138751113876 1113877

(1)

where Ω is the maximum porosity value A typical materialproperty of the FGP materials can be considered as in thefollowing power-law relations

P(z) Pt minus Pb( 1113857z

h+ 051113874 1113875

k

+ Pb1113890 1113891(1 minus Λ(z)) (2)

where Pt and Pb are the typical material properties at the topand the bottom surfaces respectively and k is the power-lawindex e normalized distributions of porosity through thethickness are shown in Figure 2(a) As shown in Figure 2(a)the porosity distribution of Case 1 is symmetric with respectto the midplane of plates Case 2 and Case 3 are bottom andtop surface-enhanced distributions respectively BesidesFigures 2(b)ndash2(d) show the distributions of a normalizedtypical property associated with three different cases ofporosity distributions with parameters Ω 05 k 1 5 10and PtPb 10

e EFTIM is built based on the WinklerndashPasternakfoundation by adding a mass parameter of foundation

qe k1w(x y t) minus k2z2

zx2 +z2

zy21113888 1113889w(x y t) + mf

z2w(x y t)

zt2

(3)

where w is the displacement of FGP plate k1 and k2 arerespectively Winkler foundation stiffness and shear layerstiffness of the Pasternak foundation In order to mentionthe effectiveness of the foundation mass involved in theoscillation as well as the continuous interaction of the springwith the plate the parameter β with unit kgminus1 is added Itcharacterizes the effective level of the foundation mass in-volved in vibration which is determined based on an ex-perimental basis and the ratio of the density of thefoundation to the density of plate material which is definedas μF ρFρ us the density of mass mf involved vi-bration with the foundation is determined mf βμFρ Fromequation (3) we see that for the static problem the EFTIMmodel and WinklerndashPasternak foundation model are thesame But for the dynamic problems these two models havedifferences and when omitting the influence parameters ofthe foundation mass the EFTIM model is equivalent to theWinklerndashPasternak foundation model In addition thisfoundationmodel also covers theWinkler foundationmodelwhen the influence of shear parameters and foundationmassparameters ignored It was found that the EFTIM model

closely resembles the true feature of the foundation in-cluding the Pasternak and Winkler foundation models

3 The First-Order Shear Deformation Theoryand Weak Form of the FGP Plates

31 First-Order Shear Deformation eory for FGP Platese displacement of the FGP plates in the present workbased on the FSDT model can be expressed as

u (x y z) u0(x y) + zθx(x y)

v (x y z) v0(x y) + zθy(x y)

w (x y z) w0(x y)

⎧⎪⎪⎨

⎪⎪⎩(4)

where u v w θx and θy are five unknown displacements ofthe midsurface of the plate For a bending plate the strainfield can be expressed as follows

ε εm + zκ (5)

where the membrane strain is given as

εm

u0x

v0y

u0y + v0x

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(6)

e bending and transverse shear strains are written as

κ

θxx

θyy

θxy + θyx

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(7)

γ w0x + θx

w0y + θy

1113896 1113897 (8)

FromHookersquos law the linear stress-strain relations of theFGP plates can be expressed as

σx

σy

τxy

τxz

τyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

Q11 Q12 0 0 0

Q21 Q22 0 0 0

0 0 Q66 0 0

0 0 0 Q55 0

0 0 0 0 Q44

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

εx

εy

cxy

cxz

cyz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(9)

x

z

y

Spring stiffener

Shear layer

Mass offoundationO

Figure 1 Modeling the FGP plate resting on EFTIM

Mathematical Problems in Engineering 3

where

Q11 Q22 E(z)

1 minus υ2

Q12 Q21 υE(z)

1 minus υ2

Q44 Q55 Q66 E(z)

2(1 + υ)

(10)

where E(z) presents for effective Youngrsquos modulus and υrepresents Poissonrsquos ratio

32 Weak Form Equations To obtain the motion equationsof the FGP plates for the free vibration analysis Hamiltonrsquosprinciple is applied with the following form

1113946t2

t1

(δU minus δK)dt 0 (11)

where U and K are the strain and kinetic energies re-spectively e strain energy is expressed as

U Up

+ Uf

(12)

where Uf is the strain energy

Uf

12

1113946ψ

k1w2

minus k2z2w

zx21113888 1113889

2

+z2w

zy21113888 1113889

2⎡⎣ ⎤⎦⎛⎝ ⎞⎠dψ (13)

and Up is the strain energy

Up

12

1113946ψεTDε + γTCγ1113872 1113873dψ (14)

0 02 04 06 08 1Φ(z) (Ω)

ndash05

ndash03

ndash01

01

03

05z (

h)

Case 1Case 2

Case 3

(a)

k = 1 PtPb = 10 Ω = 05

0 02 04 06 08 1P(z)Pt

ndash05

ndash03

ndash01

01

03

05

z (h)

Case 1Case 2

Case 3(z) = 0

(b)

0 02 04 06 08 1P(z)Pt

ndash05

ndash03

ndash01

01

03

05

z (h)

k = 5 PtPb = 10 Ω = 05

Case 1Case 2

Case 3(z) = 0

(c)

0 02 04 06 08 1P(z)Pt

ndash05

ndash03

ndash01

01

03

05z (

h)

Case 1Case 2

Case 3(z) = 0

k = 10 PtPb = 10 Ω = 05

(d)

Figure 2 Distributions of porosity and typical material property (a) Distribution of porosity along of z-axis (b) distribution materialproperty with k 1 (c) distribution material property with k 5 and (d) distribution material property with k 10

4 Mathematical Problems in Engineering

where ε εm κ1113858 1113859T and

D A B

B F1113890 1113891 (15)

and A B F and C can be given by

(AB F) 1113946h2

minush21 z z

21113872 1113873

Q11 Q12 0

Q21 Q22 0

0 0 Q66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dz (16)

C 1113946h2

minush2

Q55 00 Q44

1113890 1113891dz (17)

e kinetic energy in equation (11) is given by

K Kp

+ Kf

(18)

where Kp is the kinetic energy

Kp

12

1113946ψ

_umpu dψ (19)

where uT u0 v0 w0 θx θy ϕx ϕy1113960 1113961 is the displacementfield and mp is the mass matrix defined by

mp

I1 0 0 I2 0

I1 0 0 I2

I1 0 0

I3 0

I3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

where (I1 I2 I3) 1113938h2minush2 ρ(1 z z2)dz

In equation (11) the kinetic energy of the mass offoundation Kf is defined as

Kf

12

1113946ψ

_wmfw dψ (21)

Substituting equations (12) and (18) into equation (11)the weak formulation for the free vibration of the FGP plateis finally obtained as

1113946ψδεTDε dψ + 1113946

ψδγTCγ dψ + 1113946

ψδwT

middot k1w minus k2z2w

zx2 +z2w

zy21113888 11138891113890 1113891dψ 1113946ψ

_umpu dψ

+ 1113946ψ

_wmfw dψ

(22)

4 Formulation of an ES-MITC3 Method forFGP Plates

41 Formulation of the Finite Element Using the MITC3Element e middle surface of plate ψ is discretized into ne

finite three-node triangular elements with nn nodes such thatψ asymp 1113936

ne

e1ψe and ψi capψj empty ine j en the generalizeddisplacements at any point ue [ue

j vej we

j θexj θ

eyj]

T ofelement ψe can be approximated as

ue(x) 1113944

nne

j1

NI(x) 0 0 0 0

0 NI(x) 0 0 0

0 0 NI(x) 0 0

0 0 0 NI(x) 0

0 0 0 0 NI(x)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dej

1113944nne

j1N(x)de

j

(23)

where nne is the number of nodes of ψe N(x) is the shapefunction matrix and de

j [uej ve

j wej θ

exj θ

eyj]

T is the nodaldegrees of freedom (dof ) associated with the jth node of ψe

e membrane bending strains of MITC3 element canbe expressed in matrix forms as follows

εem Be

m1 Bem2 Be

m31113858 1113859de Be

m de (24)

κe Be

b1 Beb2 Be

b31113858 1113859de Be

b de (25)

where

Bem1

12Ae

b minus c 0 0 0 0

0 d minus a 0 0 0

d minus a b minus c 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (26)

Bem2

12Ae

c 0 0 0 00 minusd 0 0 0

minusd c 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (27)

Bem3

12Ae

minusb 0 0 0 00 a 0 0 0a minusb 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (28)

Beb1

12Ae

0 0 0 b minus c 00 0 0 0 d minus a

0 0 0 d minus a b minus c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (29)

Beb2

12Ae

0 0 0 c 00 0 0 0 minusd

0 0 0 minusd c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (30)

Beb3

12Ae

0 0 0 minusb 00 0 0 0 a

0 0 0 a minusb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)

To eliminate the shear locking phenomenon as thethickness of the plate becomes small the formulation of thetransverse shear strains of the MITC3 element based onFSDT [36] in this study can be written as follows

γe Be

s de (32)

where

Mathematical Problems in Engineering 5

Bes Be

s1 Bes2 Be

s31113858 1113859 (33)

with

Bes1 Jminus 1

0 0 minus1a

3+

d

6b

3+

c

6

0 0 minus1d

3+

a

6c

3+

b

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

Bes2 Jminus 1

0 0 1a

2minus

d

6b

2minus

c

6

0 0 0d

6c

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (35)

Be(0)s3 Jminus 1

0 0 0a

6b

6

0 0 1d

2minus

a

6c

2minus

b

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (36)

where

Jminus 1

12Ae

c minusb

minusd a

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (37)

Here a x2 minus x1 b y2 minus y1 c y3 minus y1 and d x3 minus

x1 are pointed out in and Ae is the area of the three-nodetriangular element as shown in Figure 3

Substituting the discrete displacement field into equation(22) we obtained the discrete system equations for freevibration analysis of FGP plate resting on the EF respec-tively as

K minus ω2M1113872 1113873d 0 (38)

where K and M are the stiffness and mass matricesrespectively

e stiffness matrix in equation (38) can be written as

K 1113944ne

e1Ke

p + Kef1113872 1113873 (39)

where

Kep 1113946

ψe

BTDB dψe + 1113946ψe

BTs CBsdψe (40)

Kef k11113946

ψe

NTwNwdψe + k21113946

ψe

zNw

zx1113888 1113889

TzNw

zx1113888 1113889⎡⎣

+zNw

zy1113888 1113889

TzNw

zy1113888 1113889⎤⎦dψe

(41)

where

Be Be

m Beb1113858 1113859 (42)

Nw 0 0N1 0 0 0 0N2 0 0 0 0N3 0 01113858 1113859 (43)

Next the mass matrix in equation (38) can be defined as

M 1113944ne

e1Me

p + Mef1113872 1113873 (44)

where

Mep 1113946

ψe

NTmpN dψe (45)

Mef mf1113946

ψe

NTwNw dψe (46)

42 Formulation of an ES-MITC3 Method for FGP Platese smoothing domains ψk is constructed based on edges ofthe triangular elements such that ψ cupnk

k1ψk and

ψki cap ψk

j empty for ine j An edge-based smoothing domain ψk

for the inner edge k is formed by connecting two end-nodesof the edge to the centroids of adjacent triangular MITC3elements as shown in Figure 4

Applying the edge-based smooth technique [36] thesmoothed membrane bending and shear strain 1113957εk

m 1113957κk 1113957γk

over the smoothing domain ψk can be created by

1113957εkm 1113946

ψkεmΦ

k(x)dψ (47)

1113957kk 1113946

ψk

κΦk(x)dψ (48)

1113957γk 1113946

ψk

γΦk(x)dψ (49)

where εm κ and γ the compatible membrane bending andthe shear strains respectively Φk(x) is a given smoothingfunction that satisfies at least unity property1113938ψkΦk(x)dψ 1

In this study we use the constant smoothing function

Φk(x)

1Ak

x isin ψk

0 x notin ψk

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(50)

y

d

c

a

bx

3 (x3 y3)

2 (x2 y2)

1 (x1 y1)

Figure 3 ree-node triangular element in the local coordinates

6 Mathematical Problems in Engineering

where Ak is the area of the smoothing domain ψk and isgiven by

Ak

1113946ψk

dψ 13

1113944

nek

i1A

i (51)

where nek is the number of the adjacent triangular elementsin the smoothing domain ψk and Ai is the area of the ithtriangular element attached to the edge k

By substituting equations (47)ndash(49) into equations (24)(25) and (32) the approximation of the smoothed strains onthe smoothing domain ψk can be expressed as follows

1113957εkm 1113944

nnk

j1

1113957Bk

mjdkj

1113957kk 1113944

nnksh

j1

1113957Bk

bjdkj

γk 1113944

nnksh

j1

1113957Bk

sjdkj

(52)

where nnksh is the total number of nodes of the triangular

MITC3 elements attached to edge k (nnksh 3 for boundary

edges and nnksh 4 for inner edges as given in Figure 4 dk

j isthe nodal dof associated with the smoothing domain ψkand 1113957Bk

mj 1113957Bk

bj and 1113957Bk

sj are the smoothed membrane bendingand shear strain gradient matrices respectively at the jthnode of the elements attached to edge k computed by

1113957Bk

mj 1

Ak1113944

nek

i1

13A

iBemj (53)

1113957Bk

bj 1

Ak1113944

nek

i1

13AiB

ebj (54)

1113957Bk

sj 1

Ak1113944

nek

i1

13A

iBesj (55)

e stiffness matrix of the FGP plate using the ES-MITC3 is assembled by

1113957K 1113944

nksh

k1

1113957Kk (56)

where 1113957Kk is the ES-MITC3 stiffness matrix of the smoothingdomain ψk and given by

1113957Kk 1113946

ψk

1113957BKTD1113957Bk+ 1113957Bkt

s C1113957Bk

s1113874 1113875dψ 1113957BKTD1113957BkA

k+ 1113957Bkt

s C1113957Bk

s Ak

(57)

where

1113957BkT 1113957Bk

mj1113957Bk

bj1113876 1113877 (58)

5 Accuracy of the Proposed Method

In this section the various numerical examples are solved toverify the reliability and accuracy of the proposed methodFor convenience the stiffness factors and nondimensionalfrequencies of the plates are defined as the followingequations

K1 k1a

4

H

K2 k2a

2

H

λ ωa2

π2

ρh

H

1113970

withH Eh3

12 1 minus ]2( )

(59)

To demonstrate the performance of numerical resultsthe relative frequency error is defined by

Δ () 100 timesλpr minus λre

11138681113868111386811138681113868

11138681113868111386811138681113868

λre

11138681113868111386811138681113868111386811138681113868

(60)

where λpr and λre are nondimensional frequencies of presentmethod and nondimensional frequencies in [17 18]respectively

e results of the convergence of the first two nondi-mensional frequencies of the plate in the case of fully simplesupport (SSSS) plate and a fully clamped (CCCC) plate withha 01 K1 100 K2 10 respectively are shown in Fig-ure 5 From these results it can be seen that almost allfrequencies corresponding to different cases of boundaryconditions (BC) converge with 18times18 element mesh For18times18 mesh we compare the first three nondimensionalfrequencies of a plate resting on the WinklerndashPasternakfoundation with the published results as shown in Table 1 Itcan be seen that the present results agree well with the resultsof the authors using analytical methods [5 14 17] and aremore accurate than those using the original MITC3 elementand FEM [6] In addition from Table 2 it is obvious that therelative error of the present results compared to [18] is lessthan 2 In [18] they used a new quasi-3D hyperbolic theoryto investigate the free vibration of the FGP plate resting onthe EF ese results are the basis to analyze the free vi-bration of FGP plates on the EFTIM

Boundary edge m

Inner edge k

Г(k)

ψ(k)

Г(m)

ψ(m)

Field nodeCentroid of triangles

Figure 4 e smoothing domain ψk is formed by triangularelements

Mathematical Problems in Engineering 7

Next we consider an SSSS FGP plate (AlAl2O3)with its material properties as follows metal (Al)Eb 70GPa ρb 2702 kgm3 and ceramic (Al2O3)Et 380GPa ρt 3800 kgm3 Poissonrsquos ratio is fixed atυ 03 e FGP plate with even porosities is expressed asin [18]

P(z) Pb + Pt minus Pb( 1113857z

h+ 051113874 1113875

k

minusξ2

Pt + Pb( 1113857 (61)

where ξ(ξ le 1) presents the porosity volume fraction estiffness factor and nondimensional frequencies of the platesare shown in equation (58) with Hb (Ebh312(1 minus υ2)) and

4 6 8 10 12 14 16 18 20Mesh (nxn)

25

3

35

4

45

SSSSCCCC

λ1

(a)

SSSSCCCC

4 6 8 10 12 14 16 18 20Mesh (nxn)

5

6

7

8

9

10

λ2

(b)

Figure 5 e convergence of element mesh to nondimensional frequency of plate (a)λ1 and (b)λ2

Table 1 Nondimensional frequencies of plates

Plates K1 K2 Author λ1 Δ() λ2 Δ() λ3 Δ()

SSSSυ 03ha 001

100 10

Ferreira et al [17] 26559 55718 85384Zhou et al [14] 26551 003 55717 000 85406 003Xiang et al [5] 26551 003 55718 000 85405 002

MITC3 26604 017 56103 070 86296 107Present 26590 012 55920 037 86017 074

500 10

Ferreira [17] 33406 59285 87754Zhou et al [14] 33398 002 59285 000 87775 002Xiang et al [5] 33400 002 59287 000 87775 002

MITC3 33441 010 59649 061 88642 101Present 33430 007 59477 032 88370 070

SSSSυ 03ha 01

200 10

Ferreira et al [17] 27902 53452 78255Zhou et al [14] 27756 052 52954 093 77279 125Xiang et al [5] 27842 022 53043 077 77287 124

MITC3 27874 010 53258 036 77719 068Present 27887 005 53362 017 77971 036

1000 10

Ferreira et al [17] 39844 60430 83112Zhou et al [14] 39566 070 59757 111 81954 139Xiang et al [5] 39805 010 60078 058 82214 108

MITC3 39827 004 60266 027 82619 059Present 39836 002 60358 012 82856 031

CCCCυ 015ha 0015

13902 16683

Ferreira et al [17] 81669 12821 16842Zhou et al [14] 81675 001 12823 002 16833 005

Omurtag et al [6] 81375 036 12898 060 16932 053MITC3 81842 021 12909 069 17010 100Present 81729 007 12872 040 16939 058

8 Mathematical Problems in Engineering

Table 2 e first nondimensional frequencies of FGP plate according to the WinklerndashPasternak foundation stiffness (k 1)

(K1 K2) haξ 0 ξ 02

Present [18] Δ() Present [18] Δ()

(0 0)

005 9010 9020 011 8485 8370 137010 8823 8818 006 8319 8203 141015 8541 8516 029 8069 7950 150020 8196 8151 055 7762 7641 158

(100 0)

005 9389 9430 043 9020 8917 116010 9207 9231 026 8858 8753 120015 8933 8934 001 8614 8505 128020 8599 8577 026 8315 8203 137

(100 100)

005 15383 15439 036 16338 16320 011010 15213 15245 021 16175 16148 017015 14962 14966 003 15932 15895 023020 14664 14640 016 15639 15595 028

Table 3 Nondimensional frequencies of the FGP plate on EFTIM

λ1 λ2 λ3 λ4 λ5 λ608583 18118 18157 28015 33898 34248

(a) (b)

(c) (d)

(e) (f )

Figure 6 e first six mode sharps the FGP plate on EFTIM

Mathematical Problems in Engineering 9

06

07

08

09

1

11

12

13

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ1

(a)

14

16

18

2

22

24

26

28

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(b)

2

25

3

35

4

45

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(c)

Figure 7 Nondimensional frequencies of the eFGP plate with difference of featured-index β (a) nondimensional frequency λ1 (b)nondimensional frequency λ2 and (c) nondimensional frequency λ3

Table 4 e first three nondimensional frequencies of FGP on EFTIM

Parameter of plate β 0 025 05 075 1Case 1 λ1 11015 09575 08583 07847 07273(K1 100 K2 10) λ2 23078 20156 18118 16595 15400(SSSS) λ3 35651 31154 28015 25665 23822Case 2 λ1 10225 08988 08113 07453 06931(K1 100 K2 10) λ2 20774 18350 16612 15289 14239(SSSS) λ3 32410 28630 25921 23859 22221Case 3 λ1 12537 10735 09538 08669 08001(K1 100 K2 10) λ2 27399 23531 20942 19054 17599(SSSS) λ3 41972 36097 32152 29269 27044

10 Mathematical Problems in Engineering

Table 5 Nondimensional frequencies of the FGP plate with different K1 and K2

Case of Porosity distribution K2K1

100 250 500 750 1000

Case 1(SSSS)

10 08583 08929 09477 09995 1048825 09258 09579 10092 10580 1104750 10285 10575 11042 11490 1192175 11218 11485 11916 12332 12734100 12079 12327 12730 13120 13499

Case 2(SSSS)

10 08113 08452 08988 09493 0997325 08775 09089 09589 10064 1051850 09778 10061 10515 10950 1136975 10687 10947 11365 11769 12160100 11525 11765 12156 12534 12902

Case 3(SSSS)

10 09538 10009 10747 11438 1209025 10452 10883 11566 12211 1282350 11819 12202 12815 13400 1396075 13044 13392 13953 14492 15011100 14163 14484 15004 15507 15993

08100

1

85 100070 850

12

55 700

K1K2

14

55040 40025 25010 100

λ1

(a)

K1

K2

08100

09

85

1

100070

11

850

12

55 700

13

55040 40025 25010 100

λ1

(b)

K1

K2

08100

1

85 1000

12

70 850

14

55 700

16

55040 40025 25010 100

λ1

(c)

Figure 8 Nondimensional frequencies of FGP plate with different K1 and K2 (a) Case 1 (b) Case 2 and (c) Case 3

Mathematical Problems in Engineering 11

Table 6 Nondimensional frequencies of FGP plate with different K1 K2 and β

β K1 (K2 10) K2 (K1 100)Case 1

100 250 500 750 1000 10 25 50 75 1000 11015 11459 12163 12828 13460 11015 11882 13199 14397 15502025 09575 09961 10572 11150 11700 09575 10328 11473 12514 1347505 08583 08929 09477 09995 10488 08583 09258 10285 11218 12079075 07847 08163 08664 09138 09588 07847 08464 09403 10256 110431 07273 07566 08030 08469 08887 07273 07844 08715 09505 10235Case 20 10225 10652 11327 11964 12569 10225 11059 12323 13469 14525025 08988 09363 09957 10517 11049 08988 09721 10833 11840 1276805 08113 08452 08988 09493 09973 08113 08775 09778 10687 11525075 07453 07763 08256 08720 09161 07453 08060 08982 09817 105861 06931 07220 07677 08109 08519 06931 07496 08353 09129 09845Case 30 12537 13156 14127 15035 15891 12537 13739 15536 17146 18617025 10735 11265 12096 12874 13607 10735 11764 13303 14681 1594105 09538 10009 10747 11438 12090 09538 10452 11819 13044 14163075 08669 09097 09768 10396 10988 08669 09500 10742 11855 128721 08001 08395 09015 09595 10141 08001 08767 09914 10942 11880

060

08

025 100

1

25005 400

12

550075 700850

14

1 1000

080

1

025 1025

12

05 4055

14

075 70851 100

λ1

K1

β

λ1

K2

β

(a)

060

08

025 10025005

1

400550075 700

12

8501 1000

0

08

025

1

102505 40

12

55075 70

14

851 100

λ1

K1

β

λ1

K2

β

(b)

Figure 9 Continued

12 Mathematical Problems in Engineering

ωlowast (ωa2h)(ρbEb)

1113968 e first nondimensional frequen-

cies of present work compared with [18] are shown inTable 2

6 Numerical Results and Discussions

For free vibration problems a fully simple support (SSSS)FGP plate is considered whereina b h a10 Ω 05 k 1 Et 10Eb ρt 10ρbυ 038 and μF 05 e first six nondimensional fre-quencies of the FGP plate with porosity distribution of Case1 and stiffener of foundation K1 100 K2 10 are shownin Table 3 and the first six mode shapes are presented inFigure 6 e stiffness factors and nondimensional

frequencies of FGP plate are shown in equation (58) withHb (Ebh312(1 minus υ2))

61 Influence of the Parameters of the EFTIM to the FreeVibration for the FGP Plate Firstly in order to investigatethe effect of the featured-index of themass of foundation β tofree vibration of the FGP plate the featured-index of mass ischanged from 0 to 1 In Figure 7 and Table 4 it is seen that allcases of porosity distribution featured-index of the mass offoundation β significantly influence to free vibration of theFGP plate As β increases the mass of the plate increases andthe frequencies of the plate decrease For all cases of porositydistribution of the FGP plate the porosity distribution of

080

1

025 100

12

25005 400

14

550075 700850

16

1 1000

0

1

025 102505

15

4055075 70

85

2

1 100

λ1

K1

β

λ1

K2

β

(c)

Figure 9 Nondimensional frequencies of FGP plate with different K1 K2 and β (a) Case 1 (b) Case 2 and (c) Case 3

Table 7 Nondimensional frequencies of the FGP plate as a function of k and Ω

ΩSSSS CCCCk k

0 25 5 75 10 0 25 5 75 10Case 10 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06584 08830 09777 10469 10984 10935 12518 13694 14783 1563305 07126 09238 10105 10767 11273 11732 12869 13930 14958 15780075 07820 09677 10446 11070 11564 12711 13154 14102 15061 158481 08745 10105 10776 11357 11836 13924 13226 14130 15021 15766Case 20 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 08604 09563 10288 10833 10977 12011 13156 14273 1516205 07162 08685 09569 10299 10871 11707 11651 12611 13675 14564075 07777 08600 09357 10073 10671 12179 10868 11556 12496 133421 07852 07972 08420 09009 09580 10347 09106 09310 09904 10547Case 30 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 09413 10475 11196 11714 10977 13661 15012 16147 1699405 07162 10785 11908 12621 13122 11707 15836 17243 18317 19084075 07777 13066 14256 14917 15363 12179 19297 20656 21484 220181 07852 18589 19631 19772 19807 10347 24083 24138 23588 23246

Mathematical Problems in Engineering 13

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

where

Q11 Q22 E(z)

1 minus υ2

Q12 Q21 υE(z)

1 minus υ2

Q44 Q55 Q66 E(z)

2(1 + υ)

(10)

where E(z) presents for effective Youngrsquos modulus and υrepresents Poissonrsquos ratio

32 Weak Form Equations To obtain the motion equationsof the FGP plates for the free vibration analysis Hamiltonrsquosprinciple is applied with the following form

1113946t2

t1

(δU minus δK)dt 0 (11)

where U and K are the strain and kinetic energies re-spectively e strain energy is expressed as

U Up

+ Uf

(12)

where Uf is the strain energy

Uf

12

1113946ψ

k1w2

minus k2z2w

zx21113888 1113889

2

+z2w

zy21113888 1113889

2⎡⎣ ⎤⎦⎛⎝ ⎞⎠dψ (13)

and Up is the strain energy

Up

12

1113946ψεTDε + γTCγ1113872 1113873dψ (14)

0 02 04 06 08 1Φ(z) (Ω)

ndash05

ndash03

ndash01

01

03

05z (

h)

Case 1Case 2

Case 3

(a)

k = 1 PtPb = 10 Ω = 05

0 02 04 06 08 1P(z)Pt

ndash05

ndash03

ndash01

01

03

05

z (h)

Case 1Case 2

Case 3(z) = 0

(b)

0 02 04 06 08 1P(z)Pt

ndash05

ndash03

ndash01

01

03

05

z (h)

k = 5 PtPb = 10 Ω = 05

Case 1Case 2

Case 3(z) = 0

(c)

0 02 04 06 08 1P(z)Pt

ndash05

ndash03

ndash01

01

03

05z (

h)

Case 1Case 2

Case 3(z) = 0

k = 10 PtPb = 10 Ω = 05

(d)

Figure 2 Distributions of porosity and typical material property (a) Distribution of porosity along of z-axis (b) distribution materialproperty with k 1 (c) distribution material property with k 5 and (d) distribution material property with k 10

4 Mathematical Problems in Engineering

where ε εm κ1113858 1113859T and

D A B

B F1113890 1113891 (15)

and A B F and C can be given by

(AB F) 1113946h2

minush21 z z

21113872 1113873

Q11 Q12 0

Q21 Q22 0

0 0 Q66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dz (16)

C 1113946h2

minush2

Q55 00 Q44

1113890 1113891dz (17)

e kinetic energy in equation (11) is given by

K Kp

+ Kf

(18)

where Kp is the kinetic energy

Kp

12

1113946ψ

_umpu dψ (19)

where uT u0 v0 w0 θx θy ϕx ϕy1113960 1113961 is the displacementfield and mp is the mass matrix defined by

mp

I1 0 0 I2 0

I1 0 0 I2

I1 0 0

I3 0

I3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

where (I1 I2 I3) 1113938h2minush2 ρ(1 z z2)dz

In equation (11) the kinetic energy of the mass offoundation Kf is defined as

Kf

12

1113946ψ

_wmfw dψ (21)

Substituting equations (12) and (18) into equation (11)the weak formulation for the free vibration of the FGP plateis finally obtained as

1113946ψδεTDε dψ + 1113946

ψδγTCγ dψ + 1113946

ψδwT

middot k1w minus k2z2w

zx2 +z2w

zy21113888 11138891113890 1113891dψ 1113946ψ

_umpu dψ

+ 1113946ψ

_wmfw dψ

(22)

4 Formulation of an ES-MITC3 Method forFGP Plates

41 Formulation of the Finite Element Using the MITC3Element e middle surface of plate ψ is discretized into ne

finite three-node triangular elements with nn nodes such thatψ asymp 1113936

ne

e1ψe and ψi capψj empty ine j en the generalizeddisplacements at any point ue [ue

j vej we

j θexj θ

eyj]

T ofelement ψe can be approximated as

ue(x) 1113944

nne

j1

NI(x) 0 0 0 0

0 NI(x) 0 0 0

0 0 NI(x) 0 0

0 0 0 NI(x) 0

0 0 0 0 NI(x)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dej

1113944nne

j1N(x)de

j

(23)

where nne is the number of nodes of ψe N(x) is the shapefunction matrix and de

j [uej ve

j wej θ

exj θ

eyj]

T is the nodaldegrees of freedom (dof ) associated with the jth node of ψe

e membrane bending strains of MITC3 element canbe expressed in matrix forms as follows

εem Be

m1 Bem2 Be

m31113858 1113859de Be

m de (24)

κe Be

b1 Beb2 Be

b31113858 1113859de Be

b de (25)

where

Bem1

12Ae

b minus c 0 0 0 0

0 d minus a 0 0 0

d minus a b minus c 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (26)

Bem2

12Ae

c 0 0 0 00 minusd 0 0 0

minusd c 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (27)

Bem3

12Ae

minusb 0 0 0 00 a 0 0 0a minusb 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (28)

Beb1

12Ae

0 0 0 b minus c 00 0 0 0 d minus a

0 0 0 d minus a b minus c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (29)

Beb2

12Ae

0 0 0 c 00 0 0 0 minusd

0 0 0 minusd c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (30)

Beb3

12Ae

0 0 0 minusb 00 0 0 0 a

0 0 0 a minusb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)

To eliminate the shear locking phenomenon as thethickness of the plate becomes small the formulation of thetransverse shear strains of the MITC3 element based onFSDT [36] in this study can be written as follows

γe Be

s de (32)

where

Mathematical Problems in Engineering 5

Bes Be

s1 Bes2 Be

s31113858 1113859 (33)

with

Bes1 Jminus 1

0 0 minus1a

3+

d

6b

3+

c

6

0 0 minus1d

3+

a

6c

3+

b

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

Bes2 Jminus 1

0 0 1a

2minus

d

6b

2minus

c

6

0 0 0d

6c

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (35)

Be(0)s3 Jminus 1

0 0 0a

6b

6

0 0 1d

2minus

a

6c

2minus

b

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (36)

where

Jminus 1

12Ae

c minusb

minusd a

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (37)

Here a x2 minus x1 b y2 minus y1 c y3 minus y1 and d x3 minus

x1 are pointed out in and Ae is the area of the three-nodetriangular element as shown in Figure 3

Substituting the discrete displacement field into equation(22) we obtained the discrete system equations for freevibration analysis of FGP plate resting on the EF respec-tively as

K minus ω2M1113872 1113873d 0 (38)

where K and M are the stiffness and mass matricesrespectively

e stiffness matrix in equation (38) can be written as

K 1113944ne

e1Ke

p + Kef1113872 1113873 (39)

where

Kep 1113946

ψe

BTDB dψe + 1113946ψe

BTs CBsdψe (40)

Kef k11113946

ψe

NTwNwdψe + k21113946

ψe

zNw

zx1113888 1113889

TzNw

zx1113888 1113889⎡⎣

+zNw

zy1113888 1113889

TzNw

zy1113888 1113889⎤⎦dψe

(41)

where

Be Be

m Beb1113858 1113859 (42)

Nw 0 0N1 0 0 0 0N2 0 0 0 0N3 0 01113858 1113859 (43)

Next the mass matrix in equation (38) can be defined as

M 1113944ne

e1Me

p + Mef1113872 1113873 (44)

where

Mep 1113946

ψe

NTmpN dψe (45)

Mef mf1113946

ψe

NTwNw dψe (46)

42 Formulation of an ES-MITC3 Method for FGP Platese smoothing domains ψk is constructed based on edges ofthe triangular elements such that ψ cupnk

k1ψk and

ψki cap ψk

j empty for ine j An edge-based smoothing domain ψk

for the inner edge k is formed by connecting two end-nodesof the edge to the centroids of adjacent triangular MITC3elements as shown in Figure 4

Applying the edge-based smooth technique [36] thesmoothed membrane bending and shear strain 1113957εk

m 1113957κk 1113957γk

over the smoothing domain ψk can be created by

1113957εkm 1113946

ψkεmΦ

k(x)dψ (47)

1113957kk 1113946

ψk

κΦk(x)dψ (48)

1113957γk 1113946

ψk

γΦk(x)dψ (49)

where εm κ and γ the compatible membrane bending andthe shear strains respectively Φk(x) is a given smoothingfunction that satisfies at least unity property1113938ψkΦk(x)dψ 1

In this study we use the constant smoothing function

Φk(x)

1Ak

x isin ψk

0 x notin ψk

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(50)

y

d

c

a

bx

3 (x3 y3)

2 (x2 y2)

1 (x1 y1)

Figure 3 ree-node triangular element in the local coordinates

6 Mathematical Problems in Engineering

where Ak is the area of the smoothing domain ψk and isgiven by

Ak

1113946ψk

dψ 13

1113944

nek

i1A

i (51)

where nek is the number of the adjacent triangular elementsin the smoothing domain ψk and Ai is the area of the ithtriangular element attached to the edge k

By substituting equations (47)ndash(49) into equations (24)(25) and (32) the approximation of the smoothed strains onthe smoothing domain ψk can be expressed as follows

1113957εkm 1113944

nnk

j1

1113957Bk

mjdkj

1113957kk 1113944

nnksh

j1

1113957Bk

bjdkj

γk 1113944

nnksh

j1

1113957Bk

sjdkj

(52)

where nnksh is the total number of nodes of the triangular

MITC3 elements attached to edge k (nnksh 3 for boundary

edges and nnksh 4 for inner edges as given in Figure 4 dk

j isthe nodal dof associated with the smoothing domain ψkand 1113957Bk

mj 1113957Bk

bj and 1113957Bk

sj are the smoothed membrane bendingand shear strain gradient matrices respectively at the jthnode of the elements attached to edge k computed by

1113957Bk

mj 1

Ak1113944

nek

i1

13A

iBemj (53)

1113957Bk

bj 1

Ak1113944

nek

i1

13AiB

ebj (54)

1113957Bk

sj 1

Ak1113944

nek

i1

13A

iBesj (55)

e stiffness matrix of the FGP plate using the ES-MITC3 is assembled by

1113957K 1113944

nksh

k1

1113957Kk (56)

where 1113957Kk is the ES-MITC3 stiffness matrix of the smoothingdomain ψk and given by

1113957Kk 1113946

ψk

1113957BKTD1113957Bk+ 1113957Bkt

s C1113957Bk

s1113874 1113875dψ 1113957BKTD1113957BkA

k+ 1113957Bkt

s C1113957Bk

s Ak

(57)

where

1113957BkT 1113957Bk

mj1113957Bk

bj1113876 1113877 (58)

5 Accuracy of the Proposed Method

In this section the various numerical examples are solved toverify the reliability and accuracy of the proposed methodFor convenience the stiffness factors and nondimensionalfrequencies of the plates are defined as the followingequations

K1 k1a

4

H

K2 k2a

2

H

λ ωa2

π2

ρh

H

1113970

withH Eh3

12 1 minus ]2( )

(59)

To demonstrate the performance of numerical resultsthe relative frequency error is defined by

Δ () 100 timesλpr minus λre

11138681113868111386811138681113868

11138681113868111386811138681113868

λre

11138681113868111386811138681113868111386811138681113868

(60)

where λpr and λre are nondimensional frequencies of presentmethod and nondimensional frequencies in [17 18]respectively

e results of the convergence of the first two nondi-mensional frequencies of the plate in the case of fully simplesupport (SSSS) plate and a fully clamped (CCCC) plate withha 01 K1 100 K2 10 respectively are shown in Fig-ure 5 From these results it can be seen that almost allfrequencies corresponding to different cases of boundaryconditions (BC) converge with 18times18 element mesh For18times18 mesh we compare the first three nondimensionalfrequencies of a plate resting on the WinklerndashPasternakfoundation with the published results as shown in Table 1 Itcan be seen that the present results agree well with the resultsof the authors using analytical methods [5 14 17] and aremore accurate than those using the original MITC3 elementand FEM [6] In addition from Table 2 it is obvious that therelative error of the present results compared to [18] is lessthan 2 In [18] they used a new quasi-3D hyperbolic theoryto investigate the free vibration of the FGP plate resting onthe EF ese results are the basis to analyze the free vi-bration of FGP plates on the EFTIM

Boundary edge m

Inner edge k

Г(k)

ψ(k)

Г(m)

ψ(m)

Field nodeCentroid of triangles

Figure 4 e smoothing domain ψk is formed by triangularelements

Mathematical Problems in Engineering 7

Next we consider an SSSS FGP plate (AlAl2O3)with its material properties as follows metal (Al)Eb 70GPa ρb 2702 kgm3 and ceramic (Al2O3)Et 380GPa ρt 3800 kgm3 Poissonrsquos ratio is fixed atυ 03 e FGP plate with even porosities is expressed asin [18]

P(z) Pb + Pt minus Pb( 1113857z

h+ 051113874 1113875

k

minusξ2

Pt + Pb( 1113857 (61)

where ξ(ξ le 1) presents the porosity volume fraction estiffness factor and nondimensional frequencies of the platesare shown in equation (58) with Hb (Ebh312(1 minus υ2)) and

4 6 8 10 12 14 16 18 20Mesh (nxn)

25

3

35

4

45

SSSSCCCC

λ1

(a)

SSSSCCCC

4 6 8 10 12 14 16 18 20Mesh (nxn)

5

6

7

8

9

10

λ2

(b)

Figure 5 e convergence of element mesh to nondimensional frequency of plate (a)λ1 and (b)λ2

Table 1 Nondimensional frequencies of plates

Plates K1 K2 Author λ1 Δ() λ2 Δ() λ3 Δ()

SSSSυ 03ha 001

100 10

Ferreira et al [17] 26559 55718 85384Zhou et al [14] 26551 003 55717 000 85406 003Xiang et al [5] 26551 003 55718 000 85405 002

MITC3 26604 017 56103 070 86296 107Present 26590 012 55920 037 86017 074

500 10

Ferreira [17] 33406 59285 87754Zhou et al [14] 33398 002 59285 000 87775 002Xiang et al [5] 33400 002 59287 000 87775 002

MITC3 33441 010 59649 061 88642 101Present 33430 007 59477 032 88370 070

SSSSυ 03ha 01

200 10

Ferreira et al [17] 27902 53452 78255Zhou et al [14] 27756 052 52954 093 77279 125Xiang et al [5] 27842 022 53043 077 77287 124

MITC3 27874 010 53258 036 77719 068Present 27887 005 53362 017 77971 036

1000 10

Ferreira et al [17] 39844 60430 83112Zhou et al [14] 39566 070 59757 111 81954 139Xiang et al [5] 39805 010 60078 058 82214 108

MITC3 39827 004 60266 027 82619 059Present 39836 002 60358 012 82856 031

CCCCυ 015ha 0015

13902 16683

Ferreira et al [17] 81669 12821 16842Zhou et al [14] 81675 001 12823 002 16833 005

Omurtag et al [6] 81375 036 12898 060 16932 053MITC3 81842 021 12909 069 17010 100Present 81729 007 12872 040 16939 058

8 Mathematical Problems in Engineering

Table 2 e first nondimensional frequencies of FGP plate according to the WinklerndashPasternak foundation stiffness (k 1)

(K1 K2) haξ 0 ξ 02

Present [18] Δ() Present [18] Δ()

(0 0)

005 9010 9020 011 8485 8370 137010 8823 8818 006 8319 8203 141015 8541 8516 029 8069 7950 150020 8196 8151 055 7762 7641 158

(100 0)

005 9389 9430 043 9020 8917 116010 9207 9231 026 8858 8753 120015 8933 8934 001 8614 8505 128020 8599 8577 026 8315 8203 137

(100 100)

005 15383 15439 036 16338 16320 011010 15213 15245 021 16175 16148 017015 14962 14966 003 15932 15895 023020 14664 14640 016 15639 15595 028

Table 3 Nondimensional frequencies of the FGP plate on EFTIM

λ1 λ2 λ3 λ4 λ5 λ608583 18118 18157 28015 33898 34248

(a) (b)

(c) (d)

(e) (f )

Figure 6 e first six mode sharps the FGP plate on EFTIM

Mathematical Problems in Engineering 9

06

07

08

09

1

11

12

13

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ1

(a)

14

16

18

2

22

24

26

28

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(b)

2

25

3

35

4

45

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(c)

Figure 7 Nondimensional frequencies of the eFGP plate with difference of featured-index β (a) nondimensional frequency λ1 (b)nondimensional frequency λ2 and (c) nondimensional frequency λ3

Table 4 e first three nondimensional frequencies of FGP on EFTIM

Parameter of plate β 0 025 05 075 1Case 1 λ1 11015 09575 08583 07847 07273(K1 100 K2 10) λ2 23078 20156 18118 16595 15400(SSSS) λ3 35651 31154 28015 25665 23822Case 2 λ1 10225 08988 08113 07453 06931(K1 100 K2 10) λ2 20774 18350 16612 15289 14239(SSSS) λ3 32410 28630 25921 23859 22221Case 3 λ1 12537 10735 09538 08669 08001(K1 100 K2 10) λ2 27399 23531 20942 19054 17599(SSSS) λ3 41972 36097 32152 29269 27044

10 Mathematical Problems in Engineering

Table 5 Nondimensional frequencies of the FGP plate with different K1 and K2

Case of Porosity distribution K2K1

100 250 500 750 1000

Case 1(SSSS)

10 08583 08929 09477 09995 1048825 09258 09579 10092 10580 1104750 10285 10575 11042 11490 1192175 11218 11485 11916 12332 12734100 12079 12327 12730 13120 13499

Case 2(SSSS)

10 08113 08452 08988 09493 0997325 08775 09089 09589 10064 1051850 09778 10061 10515 10950 1136975 10687 10947 11365 11769 12160100 11525 11765 12156 12534 12902

Case 3(SSSS)

10 09538 10009 10747 11438 1209025 10452 10883 11566 12211 1282350 11819 12202 12815 13400 1396075 13044 13392 13953 14492 15011100 14163 14484 15004 15507 15993

08100

1

85 100070 850

12

55 700

K1K2

14

55040 40025 25010 100

λ1

(a)

K1

K2

08100

09

85

1

100070

11

850

12

55 700

13

55040 40025 25010 100

λ1

(b)

K1

K2

08100

1

85 1000

12

70 850

14

55 700

16

55040 40025 25010 100

λ1

(c)

Figure 8 Nondimensional frequencies of FGP plate with different K1 and K2 (a) Case 1 (b) Case 2 and (c) Case 3

Mathematical Problems in Engineering 11

Table 6 Nondimensional frequencies of FGP plate with different K1 K2 and β

β K1 (K2 10) K2 (K1 100)Case 1

100 250 500 750 1000 10 25 50 75 1000 11015 11459 12163 12828 13460 11015 11882 13199 14397 15502025 09575 09961 10572 11150 11700 09575 10328 11473 12514 1347505 08583 08929 09477 09995 10488 08583 09258 10285 11218 12079075 07847 08163 08664 09138 09588 07847 08464 09403 10256 110431 07273 07566 08030 08469 08887 07273 07844 08715 09505 10235Case 20 10225 10652 11327 11964 12569 10225 11059 12323 13469 14525025 08988 09363 09957 10517 11049 08988 09721 10833 11840 1276805 08113 08452 08988 09493 09973 08113 08775 09778 10687 11525075 07453 07763 08256 08720 09161 07453 08060 08982 09817 105861 06931 07220 07677 08109 08519 06931 07496 08353 09129 09845Case 30 12537 13156 14127 15035 15891 12537 13739 15536 17146 18617025 10735 11265 12096 12874 13607 10735 11764 13303 14681 1594105 09538 10009 10747 11438 12090 09538 10452 11819 13044 14163075 08669 09097 09768 10396 10988 08669 09500 10742 11855 128721 08001 08395 09015 09595 10141 08001 08767 09914 10942 11880

060

08

025 100

1

25005 400

12

550075 700850

14

1 1000

080

1

025 1025

12

05 4055

14

075 70851 100

λ1

K1

β

λ1

K2

β

(a)

060

08

025 10025005

1

400550075 700

12

8501 1000

0

08

025

1

102505 40

12

55075 70

14

851 100

λ1

K1

β

λ1

K2

β

(b)

Figure 9 Continued

12 Mathematical Problems in Engineering

ωlowast (ωa2h)(ρbEb)

1113968 e first nondimensional frequen-

cies of present work compared with [18] are shown inTable 2

6 Numerical Results and Discussions

For free vibration problems a fully simple support (SSSS)FGP plate is considered whereina b h a10 Ω 05 k 1 Et 10Eb ρt 10ρbυ 038 and μF 05 e first six nondimensional fre-quencies of the FGP plate with porosity distribution of Case1 and stiffener of foundation K1 100 K2 10 are shownin Table 3 and the first six mode shapes are presented inFigure 6 e stiffness factors and nondimensional

frequencies of FGP plate are shown in equation (58) withHb (Ebh312(1 minus υ2))

61 Influence of the Parameters of the EFTIM to the FreeVibration for the FGP Plate Firstly in order to investigatethe effect of the featured-index of themass of foundation β tofree vibration of the FGP plate the featured-index of mass ischanged from 0 to 1 In Figure 7 and Table 4 it is seen that allcases of porosity distribution featured-index of the mass offoundation β significantly influence to free vibration of theFGP plate As β increases the mass of the plate increases andthe frequencies of the plate decrease For all cases of porositydistribution of the FGP plate the porosity distribution of

080

1

025 100

12

25005 400

14

550075 700850

16

1 1000

0

1

025 102505

15

4055075 70

85

2

1 100

λ1

K1

β

λ1

K2

β

(c)

Figure 9 Nondimensional frequencies of FGP plate with different K1 K2 and β (a) Case 1 (b) Case 2 and (c) Case 3

Table 7 Nondimensional frequencies of the FGP plate as a function of k and Ω

ΩSSSS CCCCk k

0 25 5 75 10 0 25 5 75 10Case 10 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06584 08830 09777 10469 10984 10935 12518 13694 14783 1563305 07126 09238 10105 10767 11273 11732 12869 13930 14958 15780075 07820 09677 10446 11070 11564 12711 13154 14102 15061 158481 08745 10105 10776 11357 11836 13924 13226 14130 15021 15766Case 20 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 08604 09563 10288 10833 10977 12011 13156 14273 1516205 07162 08685 09569 10299 10871 11707 11651 12611 13675 14564075 07777 08600 09357 10073 10671 12179 10868 11556 12496 133421 07852 07972 08420 09009 09580 10347 09106 09310 09904 10547Case 30 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 09413 10475 11196 11714 10977 13661 15012 16147 1699405 07162 10785 11908 12621 13122 11707 15836 17243 18317 19084075 07777 13066 14256 14917 15363 12179 19297 20656 21484 220181 07852 18589 19631 19772 19807 10347 24083 24138 23588 23246

Mathematical Problems in Engineering 13

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

where ε εm κ1113858 1113859T and

D A B

B F1113890 1113891 (15)

and A B F and C can be given by

(AB F) 1113946h2

minush21 z z

21113872 1113873

Q11 Q12 0

Q21 Q22 0

0 0 Q66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dz (16)

C 1113946h2

minush2

Q55 00 Q44

1113890 1113891dz (17)

e kinetic energy in equation (11) is given by

K Kp

+ Kf

(18)

where Kp is the kinetic energy

Kp

12

1113946ψ

_umpu dψ (19)

where uT u0 v0 w0 θx θy ϕx ϕy1113960 1113961 is the displacementfield and mp is the mass matrix defined by

mp

I1 0 0 I2 0

I1 0 0 I2

I1 0 0

I3 0

I3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

where (I1 I2 I3) 1113938h2minush2 ρ(1 z z2)dz

In equation (11) the kinetic energy of the mass offoundation Kf is defined as

Kf

12

1113946ψ

_wmfw dψ (21)

Substituting equations (12) and (18) into equation (11)the weak formulation for the free vibration of the FGP plateis finally obtained as

1113946ψδεTDε dψ + 1113946

ψδγTCγ dψ + 1113946

ψδwT

middot k1w minus k2z2w

zx2 +z2w

zy21113888 11138891113890 1113891dψ 1113946ψ

_umpu dψ

+ 1113946ψ

_wmfw dψ

(22)

4 Formulation of an ES-MITC3 Method forFGP Plates

41 Formulation of the Finite Element Using the MITC3Element e middle surface of plate ψ is discretized into ne

finite three-node triangular elements with nn nodes such thatψ asymp 1113936

ne

e1ψe and ψi capψj empty ine j en the generalizeddisplacements at any point ue [ue

j vej we

j θexj θ

eyj]

T ofelement ψe can be approximated as

ue(x) 1113944

nne

j1

NI(x) 0 0 0 0

0 NI(x) 0 0 0

0 0 NI(x) 0 0

0 0 0 NI(x) 0

0 0 0 0 NI(x)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dej

1113944nne

j1N(x)de

j

(23)

where nne is the number of nodes of ψe N(x) is the shapefunction matrix and de

j [uej ve

j wej θ

exj θ

eyj]

T is the nodaldegrees of freedom (dof ) associated with the jth node of ψe

e membrane bending strains of MITC3 element canbe expressed in matrix forms as follows

εem Be

m1 Bem2 Be

m31113858 1113859de Be

m de (24)

κe Be

b1 Beb2 Be

b31113858 1113859de Be

b de (25)

where

Bem1

12Ae

b minus c 0 0 0 0

0 d minus a 0 0 0

d minus a b minus c 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (26)

Bem2

12Ae

c 0 0 0 00 minusd 0 0 0

minusd c 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (27)

Bem3

12Ae

minusb 0 0 0 00 a 0 0 0a minusb 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (28)

Beb1

12Ae

0 0 0 b minus c 00 0 0 0 d minus a

0 0 0 d minus a b minus c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (29)

Beb2

12Ae

0 0 0 c 00 0 0 0 minusd

0 0 0 minusd c

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (30)

Beb3

12Ae

0 0 0 minusb 00 0 0 0 a

0 0 0 a minusb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)

To eliminate the shear locking phenomenon as thethickness of the plate becomes small the formulation of thetransverse shear strains of the MITC3 element based onFSDT [36] in this study can be written as follows

γe Be

s de (32)

where

Mathematical Problems in Engineering 5

Bes Be

s1 Bes2 Be

s31113858 1113859 (33)

with

Bes1 Jminus 1

0 0 minus1a

3+

d

6b

3+

c

6

0 0 minus1d

3+

a

6c

3+

b

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

Bes2 Jminus 1

0 0 1a

2minus

d

6b

2minus

c

6

0 0 0d

6c

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (35)

Be(0)s3 Jminus 1

0 0 0a

6b

6

0 0 1d

2minus

a

6c

2minus

b

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (36)

where

Jminus 1

12Ae

c minusb

minusd a

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (37)

Here a x2 minus x1 b y2 minus y1 c y3 minus y1 and d x3 minus

x1 are pointed out in and Ae is the area of the three-nodetriangular element as shown in Figure 3

Substituting the discrete displacement field into equation(22) we obtained the discrete system equations for freevibration analysis of FGP plate resting on the EF respec-tively as

K minus ω2M1113872 1113873d 0 (38)

where K and M are the stiffness and mass matricesrespectively

e stiffness matrix in equation (38) can be written as

K 1113944ne

e1Ke

p + Kef1113872 1113873 (39)

where

Kep 1113946

ψe

BTDB dψe + 1113946ψe

BTs CBsdψe (40)

Kef k11113946

ψe

NTwNwdψe + k21113946

ψe

zNw

zx1113888 1113889

TzNw

zx1113888 1113889⎡⎣

+zNw

zy1113888 1113889

TzNw

zy1113888 1113889⎤⎦dψe

(41)

where

Be Be

m Beb1113858 1113859 (42)

Nw 0 0N1 0 0 0 0N2 0 0 0 0N3 0 01113858 1113859 (43)

Next the mass matrix in equation (38) can be defined as

M 1113944ne

e1Me

p + Mef1113872 1113873 (44)

where

Mep 1113946

ψe

NTmpN dψe (45)

Mef mf1113946

ψe

NTwNw dψe (46)

42 Formulation of an ES-MITC3 Method for FGP Platese smoothing domains ψk is constructed based on edges ofthe triangular elements such that ψ cupnk

k1ψk and

ψki cap ψk

j empty for ine j An edge-based smoothing domain ψk

for the inner edge k is formed by connecting two end-nodesof the edge to the centroids of adjacent triangular MITC3elements as shown in Figure 4

Applying the edge-based smooth technique [36] thesmoothed membrane bending and shear strain 1113957εk

m 1113957κk 1113957γk

over the smoothing domain ψk can be created by

1113957εkm 1113946

ψkεmΦ

k(x)dψ (47)

1113957kk 1113946

ψk

κΦk(x)dψ (48)

1113957γk 1113946

ψk

γΦk(x)dψ (49)

where εm κ and γ the compatible membrane bending andthe shear strains respectively Φk(x) is a given smoothingfunction that satisfies at least unity property1113938ψkΦk(x)dψ 1

In this study we use the constant smoothing function

Φk(x)

1Ak

x isin ψk

0 x notin ψk

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(50)

y

d

c

a

bx

3 (x3 y3)

2 (x2 y2)

1 (x1 y1)

Figure 3 ree-node triangular element in the local coordinates

6 Mathematical Problems in Engineering

where Ak is the area of the smoothing domain ψk and isgiven by

Ak

1113946ψk

dψ 13

1113944

nek

i1A

i (51)

where nek is the number of the adjacent triangular elementsin the smoothing domain ψk and Ai is the area of the ithtriangular element attached to the edge k

By substituting equations (47)ndash(49) into equations (24)(25) and (32) the approximation of the smoothed strains onthe smoothing domain ψk can be expressed as follows

1113957εkm 1113944

nnk

j1

1113957Bk

mjdkj

1113957kk 1113944

nnksh

j1

1113957Bk

bjdkj

γk 1113944

nnksh

j1

1113957Bk

sjdkj

(52)

where nnksh is the total number of nodes of the triangular

MITC3 elements attached to edge k (nnksh 3 for boundary

edges and nnksh 4 for inner edges as given in Figure 4 dk

j isthe nodal dof associated with the smoothing domain ψkand 1113957Bk

mj 1113957Bk

bj and 1113957Bk

sj are the smoothed membrane bendingand shear strain gradient matrices respectively at the jthnode of the elements attached to edge k computed by

1113957Bk

mj 1

Ak1113944

nek

i1

13A

iBemj (53)

1113957Bk

bj 1

Ak1113944

nek

i1

13AiB

ebj (54)

1113957Bk

sj 1

Ak1113944

nek

i1

13A

iBesj (55)

e stiffness matrix of the FGP plate using the ES-MITC3 is assembled by

1113957K 1113944

nksh

k1

1113957Kk (56)

where 1113957Kk is the ES-MITC3 stiffness matrix of the smoothingdomain ψk and given by

1113957Kk 1113946

ψk

1113957BKTD1113957Bk+ 1113957Bkt

s C1113957Bk

s1113874 1113875dψ 1113957BKTD1113957BkA

k+ 1113957Bkt

s C1113957Bk

s Ak

(57)

where

1113957BkT 1113957Bk

mj1113957Bk

bj1113876 1113877 (58)

5 Accuracy of the Proposed Method

In this section the various numerical examples are solved toverify the reliability and accuracy of the proposed methodFor convenience the stiffness factors and nondimensionalfrequencies of the plates are defined as the followingequations

K1 k1a

4

H

K2 k2a

2

H

λ ωa2

π2

ρh

H

1113970

withH Eh3

12 1 minus ]2( )

(59)

To demonstrate the performance of numerical resultsthe relative frequency error is defined by

Δ () 100 timesλpr minus λre

11138681113868111386811138681113868

11138681113868111386811138681113868

λre

11138681113868111386811138681113868111386811138681113868

(60)

where λpr and λre are nondimensional frequencies of presentmethod and nondimensional frequencies in [17 18]respectively

e results of the convergence of the first two nondi-mensional frequencies of the plate in the case of fully simplesupport (SSSS) plate and a fully clamped (CCCC) plate withha 01 K1 100 K2 10 respectively are shown in Fig-ure 5 From these results it can be seen that almost allfrequencies corresponding to different cases of boundaryconditions (BC) converge with 18times18 element mesh For18times18 mesh we compare the first three nondimensionalfrequencies of a plate resting on the WinklerndashPasternakfoundation with the published results as shown in Table 1 Itcan be seen that the present results agree well with the resultsof the authors using analytical methods [5 14 17] and aremore accurate than those using the original MITC3 elementand FEM [6] In addition from Table 2 it is obvious that therelative error of the present results compared to [18] is lessthan 2 In [18] they used a new quasi-3D hyperbolic theoryto investigate the free vibration of the FGP plate resting onthe EF ese results are the basis to analyze the free vi-bration of FGP plates on the EFTIM

Boundary edge m

Inner edge k

Г(k)

ψ(k)

Г(m)

ψ(m)

Field nodeCentroid of triangles

Figure 4 e smoothing domain ψk is formed by triangularelements

Mathematical Problems in Engineering 7

Next we consider an SSSS FGP plate (AlAl2O3)with its material properties as follows metal (Al)Eb 70GPa ρb 2702 kgm3 and ceramic (Al2O3)Et 380GPa ρt 3800 kgm3 Poissonrsquos ratio is fixed atυ 03 e FGP plate with even porosities is expressed asin [18]

P(z) Pb + Pt minus Pb( 1113857z

h+ 051113874 1113875

k

minusξ2

Pt + Pb( 1113857 (61)

where ξ(ξ le 1) presents the porosity volume fraction estiffness factor and nondimensional frequencies of the platesare shown in equation (58) with Hb (Ebh312(1 minus υ2)) and

4 6 8 10 12 14 16 18 20Mesh (nxn)

25

3

35

4

45

SSSSCCCC

λ1

(a)

SSSSCCCC

4 6 8 10 12 14 16 18 20Mesh (nxn)

5

6

7

8

9

10

λ2

(b)

Figure 5 e convergence of element mesh to nondimensional frequency of plate (a)λ1 and (b)λ2

Table 1 Nondimensional frequencies of plates

Plates K1 K2 Author λ1 Δ() λ2 Δ() λ3 Δ()

SSSSυ 03ha 001

100 10

Ferreira et al [17] 26559 55718 85384Zhou et al [14] 26551 003 55717 000 85406 003Xiang et al [5] 26551 003 55718 000 85405 002

MITC3 26604 017 56103 070 86296 107Present 26590 012 55920 037 86017 074

500 10

Ferreira [17] 33406 59285 87754Zhou et al [14] 33398 002 59285 000 87775 002Xiang et al [5] 33400 002 59287 000 87775 002

MITC3 33441 010 59649 061 88642 101Present 33430 007 59477 032 88370 070

SSSSυ 03ha 01

200 10

Ferreira et al [17] 27902 53452 78255Zhou et al [14] 27756 052 52954 093 77279 125Xiang et al [5] 27842 022 53043 077 77287 124

MITC3 27874 010 53258 036 77719 068Present 27887 005 53362 017 77971 036

1000 10

Ferreira et al [17] 39844 60430 83112Zhou et al [14] 39566 070 59757 111 81954 139Xiang et al [5] 39805 010 60078 058 82214 108

MITC3 39827 004 60266 027 82619 059Present 39836 002 60358 012 82856 031

CCCCυ 015ha 0015

13902 16683

Ferreira et al [17] 81669 12821 16842Zhou et al [14] 81675 001 12823 002 16833 005

Omurtag et al [6] 81375 036 12898 060 16932 053MITC3 81842 021 12909 069 17010 100Present 81729 007 12872 040 16939 058

8 Mathematical Problems in Engineering

Table 2 e first nondimensional frequencies of FGP plate according to the WinklerndashPasternak foundation stiffness (k 1)

(K1 K2) haξ 0 ξ 02

Present [18] Δ() Present [18] Δ()

(0 0)

005 9010 9020 011 8485 8370 137010 8823 8818 006 8319 8203 141015 8541 8516 029 8069 7950 150020 8196 8151 055 7762 7641 158

(100 0)

005 9389 9430 043 9020 8917 116010 9207 9231 026 8858 8753 120015 8933 8934 001 8614 8505 128020 8599 8577 026 8315 8203 137

(100 100)

005 15383 15439 036 16338 16320 011010 15213 15245 021 16175 16148 017015 14962 14966 003 15932 15895 023020 14664 14640 016 15639 15595 028

Table 3 Nondimensional frequencies of the FGP plate on EFTIM

λ1 λ2 λ3 λ4 λ5 λ608583 18118 18157 28015 33898 34248

(a) (b)

(c) (d)

(e) (f )

Figure 6 e first six mode sharps the FGP plate on EFTIM

Mathematical Problems in Engineering 9

06

07

08

09

1

11

12

13

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ1

(a)

14

16

18

2

22

24

26

28

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(b)

2

25

3

35

4

45

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(c)

Figure 7 Nondimensional frequencies of the eFGP plate with difference of featured-index β (a) nondimensional frequency λ1 (b)nondimensional frequency λ2 and (c) nondimensional frequency λ3

Table 4 e first three nondimensional frequencies of FGP on EFTIM

Parameter of plate β 0 025 05 075 1Case 1 λ1 11015 09575 08583 07847 07273(K1 100 K2 10) λ2 23078 20156 18118 16595 15400(SSSS) λ3 35651 31154 28015 25665 23822Case 2 λ1 10225 08988 08113 07453 06931(K1 100 K2 10) λ2 20774 18350 16612 15289 14239(SSSS) λ3 32410 28630 25921 23859 22221Case 3 λ1 12537 10735 09538 08669 08001(K1 100 K2 10) λ2 27399 23531 20942 19054 17599(SSSS) λ3 41972 36097 32152 29269 27044

10 Mathematical Problems in Engineering

Table 5 Nondimensional frequencies of the FGP plate with different K1 and K2

Case of Porosity distribution K2K1

100 250 500 750 1000

Case 1(SSSS)

10 08583 08929 09477 09995 1048825 09258 09579 10092 10580 1104750 10285 10575 11042 11490 1192175 11218 11485 11916 12332 12734100 12079 12327 12730 13120 13499

Case 2(SSSS)

10 08113 08452 08988 09493 0997325 08775 09089 09589 10064 1051850 09778 10061 10515 10950 1136975 10687 10947 11365 11769 12160100 11525 11765 12156 12534 12902

Case 3(SSSS)

10 09538 10009 10747 11438 1209025 10452 10883 11566 12211 1282350 11819 12202 12815 13400 1396075 13044 13392 13953 14492 15011100 14163 14484 15004 15507 15993

08100

1

85 100070 850

12

55 700

K1K2

14

55040 40025 25010 100

λ1

(a)

K1

K2

08100

09

85

1

100070

11

850

12

55 700

13

55040 40025 25010 100

λ1

(b)

K1

K2

08100

1

85 1000

12

70 850

14

55 700

16

55040 40025 25010 100

λ1

(c)

Figure 8 Nondimensional frequencies of FGP plate with different K1 and K2 (a) Case 1 (b) Case 2 and (c) Case 3

Mathematical Problems in Engineering 11

Table 6 Nondimensional frequencies of FGP plate with different K1 K2 and β

β K1 (K2 10) K2 (K1 100)Case 1

100 250 500 750 1000 10 25 50 75 1000 11015 11459 12163 12828 13460 11015 11882 13199 14397 15502025 09575 09961 10572 11150 11700 09575 10328 11473 12514 1347505 08583 08929 09477 09995 10488 08583 09258 10285 11218 12079075 07847 08163 08664 09138 09588 07847 08464 09403 10256 110431 07273 07566 08030 08469 08887 07273 07844 08715 09505 10235Case 20 10225 10652 11327 11964 12569 10225 11059 12323 13469 14525025 08988 09363 09957 10517 11049 08988 09721 10833 11840 1276805 08113 08452 08988 09493 09973 08113 08775 09778 10687 11525075 07453 07763 08256 08720 09161 07453 08060 08982 09817 105861 06931 07220 07677 08109 08519 06931 07496 08353 09129 09845Case 30 12537 13156 14127 15035 15891 12537 13739 15536 17146 18617025 10735 11265 12096 12874 13607 10735 11764 13303 14681 1594105 09538 10009 10747 11438 12090 09538 10452 11819 13044 14163075 08669 09097 09768 10396 10988 08669 09500 10742 11855 128721 08001 08395 09015 09595 10141 08001 08767 09914 10942 11880

060

08

025 100

1

25005 400

12

550075 700850

14

1 1000

080

1

025 1025

12

05 4055

14

075 70851 100

λ1

K1

β

λ1

K2

β

(a)

060

08

025 10025005

1

400550075 700

12

8501 1000

0

08

025

1

102505 40

12

55075 70

14

851 100

λ1

K1

β

λ1

K2

β

(b)

Figure 9 Continued

12 Mathematical Problems in Engineering

ωlowast (ωa2h)(ρbEb)

1113968 e first nondimensional frequen-

cies of present work compared with [18] are shown inTable 2

6 Numerical Results and Discussions

For free vibration problems a fully simple support (SSSS)FGP plate is considered whereina b h a10 Ω 05 k 1 Et 10Eb ρt 10ρbυ 038 and μF 05 e first six nondimensional fre-quencies of the FGP plate with porosity distribution of Case1 and stiffener of foundation K1 100 K2 10 are shownin Table 3 and the first six mode shapes are presented inFigure 6 e stiffness factors and nondimensional

frequencies of FGP plate are shown in equation (58) withHb (Ebh312(1 minus υ2))

61 Influence of the Parameters of the EFTIM to the FreeVibration for the FGP Plate Firstly in order to investigatethe effect of the featured-index of themass of foundation β tofree vibration of the FGP plate the featured-index of mass ischanged from 0 to 1 In Figure 7 and Table 4 it is seen that allcases of porosity distribution featured-index of the mass offoundation β significantly influence to free vibration of theFGP plate As β increases the mass of the plate increases andthe frequencies of the plate decrease For all cases of porositydistribution of the FGP plate the porosity distribution of

080

1

025 100

12

25005 400

14

550075 700850

16

1 1000

0

1

025 102505

15

4055075 70

85

2

1 100

λ1

K1

β

λ1

K2

β

(c)

Figure 9 Nondimensional frequencies of FGP plate with different K1 K2 and β (a) Case 1 (b) Case 2 and (c) Case 3

Table 7 Nondimensional frequencies of the FGP plate as a function of k and Ω

ΩSSSS CCCCk k

0 25 5 75 10 0 25 5 75 10Case 10 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06584 08830 09777 10469 10984 10935 12518 13694 14783 1563305 07126 09238 10105 10767 11273 11732 12869 13930 14958 15780075 07820 09677 10446 11070 11564 12711 13154 14102 15061 158481 08745 10105 10776 11357 11836 13924 13226 14130 15021 15766Case 20 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 08604 09563 10288 10833 10977 12011 13156 14273 1516205 07162 08685 09569 10299 10871 11707 11651 12611 13675 14564075 07777 08600 09357 10073 10671 12179 10868 11556 12496 133421 07852 07972 08420 09009 09580 10347 09106 09310 09904 10547Case 30 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 09413 10475 11196 11714 10977 13661 15012 16147 1699405 07162 10785 11908 12621 13122 11707 15836 17243 18317 19084075 07777 13066 14256 14917 15363 12179 19297 20656 21484 220181 07852 18589 19631 19772 19807 10347 24083 24138 23588 23246

Mathematical Problems in Engineering 13

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

Bes Be

s1 Bes2 Be

s31113858 1113859 (33)

with

Bes1 Jminus 1

0 0 minus1a

3+

d

6b

3+

c

6

0 0 minus1d

3+

a

6c

3+

b

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)

Bes2 Jminus 1

0 0 1a

2minus

d

6b

2minus

c

6

0 0 0d

6c

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (35)

Be(0)s3 Jminus 1

0 0 0a

6b

6

0 0 1d

2minus

a

6c

2minus

b

6

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (36)

where

Jminus 1

12Ae

c minusb

minusd a

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (37)

Here a x2 minus x1 b y2 minus y1 c y3 minus y1 and d x3 minus

x1 are pointed out in and Ae is the area of the three-nodetriangular element as shown in Figure 3

Substituting the discrete displacement field into equation(22) we obtained the discrete system equations for freevibration analysis of FGP plate resting on the EF respec-tively as

K minus ω2M1113872 1113873d 0 (38)

where K and M are the stiffness and mass matricesrespectively

e stiffness matrix in equation (38) can be written as

K 1113944ne

e1Ke

p + Kef1113872 1113873 (39)

where

Kep 1113946

ψe

BTDB dψe + 1113946ψe

BTs CBsdψe (40)

Kef k11113946

ψe

NTwNwdψe + k21113946

ψe

zNw

zx1113888 1113889

TzNw

zx1113888 1113889⎡⎣

+zNw

zy1113888 1113889

TzNw

zy1113888 1113889⎤⎦dψe

(41)

where

Be Be

m Beb1113858 1113859 (42)

Nw 0 0N1 0 0 0 0N2 0 0 0 0N3 0 01113858 1113859 (43)

Next the mass matrix in equation (38) can be defined as

M 1113944ne

e1Me

p + Mef1113872 1113873 (44)

where

Mep 1113946

ψe

NTmpN dψe (45)

Mef mf1113946

ψe

NTwNw dψe (46)

42 Formulation of an ES-MITC3 Method for FGP Platese smoothing domains ψk is constructed based on edges ofthe triangular elements such that ψ cupnk

k1ψk and

ψki cap ψk

j empty for ine j An edge-based smoothing domain ψk

for the inner edge k is formed by connecting two end-nodesof the edge to the centroids of adjacent triangular MITC3elements as shown in Figure 4

Applying the edge-based smooth technique [36] thesmoothed membrane bending and shear strain 1113957εk

m 1113957κk 1113957γk

over the smoothing domain ψk can be created by

1113957εkm 1113946

ψkεmΦ

k(x)dψ (47)

1113957kk 1113946

ψk

κΦk(x)dψ (48)

1113957γk 1113946

ψk

γΦk(x)dψ (49)

where εm κ and γ the compatible membrane bending andthe shear strains respectively Φk(x) is a given smoothingfunction that satisfies at least unity property1113938ψkΦk(x)dψ 1

In this study we use the constant smoothing function

Φk(x)

1Ak

x isin ψk

0 x notin ψk

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(50)

y

d

c

a

bx

3 (x3 y3)

2 (x2 y2)

1 (x1 y1)

Figure 3 ree-node triangular element in the local coordinates

6 Mathematical Problems in Engineering

where Ak is the area of the smoothing domain ψk and isgiven by

Ak

1113946ψk

dψ 13

1113944

nek

i1A

i (51)

where nek is the number of the adjacent triangular elementsin the smoothing domain ψk and Ai is the area of the ithtriangular element attached to the edge k

By substituting equations (47)ndash(49) into equations (24)(25) and (32) the approximation of the smoothed strains onthe smoothing domain ψk can be expressed as follows

1113957εkm 1113944

nnk

j1

1113957Bk

mjdkj

1113957kk 1113944

nnksh

j1

1113957Bk

bjdkj

γk 1113944

nnksh

j1

1113957Bk

sjdkj

(52)

where nnksh is the total number of nodes of the triangular

MITC3 elements attached to edge k (nnksh 3 for boundary

edges and nnksh 4 for inner edges as given in Figure 4 dk

j isthe nodal dof associated with the smoothing domain ψkand 1113957Bk

mj 1113957Bk

bj and 1113957Bk

sj are the smoothed membrane bendingand shear strain gradient matrices respectively at the jthnode of the elements attached to edge k computed by

1113957Bk

mj 1

Ak1113944

nek

i1

13A

iBemj (53)

1113957Bk

bj 1

Ak1113944

nek

i1

13AiB

ebj (54)

1113957Bk

sj 1

Ak1113944

nek

i1

13A

iBesj (55)

e stiffness matrix of the FGP plate using the ES-MITC3 is assembled by

1113957K 1113944

nksh

k1

1113957Kk (56)

where 1113957Kk is the ES-MITC3 stiffness matrix of the smoothingdomain ψk and given by

1113957Kk 1113946

ψk

1113957BKTD1113957Bk+ 1113957Bkt

s C1113957Bk

s1113874 1113875dψ 1113957BKTD1113957BkA

k+ 1113957Bkt

s C1113957Bk

s Ak

(57)

where

1113957BkT 1113957Bk

mj1113957Bk

bj1113876 1113877 (58)

5 Accuracy of the Proposed Method

In this section the various numerical examples are solved toverify the reliability and accuracy of the proposed methodFor convenience the stiffness factors and nondimensionalfrequencies of the plates are defined as the followingequations

K1 k1a

4

H

K2 k2a

2

H

λ ωa2

π2

ρh

H

1113970

withH Eh3

12 1 minus ]2( )

(59)

To demonstrate the performance of numerical resultsthe relative frequency error is defined by

Δ () 100 timesλpr minus λre

11138681113868111386811138681113868

11138681113868111386811138681113868

λre

11138681113868111386811138681113868111386811138681113868

(60)

where λpr and λre are nondimensional frequencies of presentmethod and nondimensional frequencies in [17 18]respectively

e results of the convergence of the first two nondi-mensional frequencies of the plate in the case of fully simplesupport (SSSS) plate and a fully clamped (CCCC) plate withha 01 K1 100 K2 10 respectively are shown in Fig-ure 5 From these results it can be seen that almost allfrequencies corresponding to different cases of boundaryconditions (BC) converge with 18times18 element mesh For18times18 mesh we compare the first three nondimensionalfrequencies of a plate resting on the WinklerndashPasternakfoundation with the published results as shown in Table 1 Itcan be seen that the present results agree well with the resultsof the authors using analytical methods [5 14 17] and aremore accurate than those using the original MITC3 elementand FEM [6] In addition from Table 2 it is obvious that therelative error of the present results compared to [18] is lessthan 2 In [18] they used a new quasi-3D hyperbolic theoryto investigate the free vibration of the FGP plate resting onthe EF ese results are the basis to analyze the free vi-bration of FGP plates on the EFTIM

Boundary edge m

Inner edge k

Г(k)

ψ(k)

Г(m)

ψ(m)

Field nodeCentroid of triangles

Figure 4 e smoothing domain ψk is formed by triangularelements

Mathematical Problems in Engineering 7

Next we consider an SSSS FGP plate (AlAl2O3)with its material properties as follows metal (Al)Eb 70GPa ρb 2702 kgm3 and ceramic (Al2O3)Et 380GPa ρt 3800 kgm3 Poissonrsquos ratio is fixed atυ 03 e FGP plate with even porosities is expressed asin [18]

P(z) Pb + Pt minus Pb( 1113857z

h+ 051113874 1113875

k

minusξ2

Pt + Pb( 1113857 (61)

where ξ(ξ le 1) presents the porosity volume fraction estiffness factor and nondimensional frequencies of the platesare shown in equation (58) with Hb (Ebh312(1 minus υ2)) and

4 6 8 10 12 14 16 18 20Mesh (nxn)

25

3

35

4

45

SSSSCCCC

λ1

(a)

SSSSCCCC

4 6 8 10 12 14 16 18 20Mesh (nxn)

5

6

7

8

9

10

λ2

(b)

Figure 5 e convergence of element mesh to nondimensional frequency of plate (a)λ1 and (b)λ2

Table 1 Nondimensional frequencies of plates

Plates K1 K2 Author λ1 Δ() λ2 Δ() λ3 Δ()

SSSSυ 03ha 001

100 10

Ferreira et al [17] 26559 55718 85384Zhou et al [14] 26551 003 55717 000 85406 003Xiang et al [5] 26551 003 55718 000 85405 002

MITC3 26604 017 56103 070 86296 107Present 26590 012 55920 037 86017 074

500 10

Ferreira [17] 33406 59285 87754Zhou et al [14] 33398 002 59285 000 87775 002Xiang et al [5] 33400 002 59287 000 87775 002

MITC3 33441 010 59649 061 88642 101Present 33430 007 59477 032 88370 070

SSSSυ 03ha 01

200 10

Ferreira et al [17] 27902 53452 78255Zhou et al [14] 27756 052 52954 093 77279 125Xiang et al [5] 27842 022 53043 077 77287 124

MITC3 27874 010 53258 036 77719 068Present 27887 005 53362 017 77971 036

1000 10

Ferreira et al [17] 39844 60430 83112Zhou et al [14] 39566 070 59757 111 81954 139Xiang et al [5] 39805 010 60078 058 82214 108

MITC3 39827 004 60266 027 82619 059Present 39836 002 60358 012 82856 031

CCCCυ 015ha 0015

13902 16683

Ferreira et al [17] 81669 12821 16842Zhou et al [14] 81675 001 12823 002 16833 005

Omurtag et al [6] 81375 036 12898 060 16932 053MITC3 81842 021 12909 069 17010 100Present 81729 007 12872 040 16939 058

8 Mathematical Problems in Engineering

Table 2 e first nondimensional frequencies of FGP plate according to the WinklerndashPasternak foundation stiffness (k 1)

(K1 K2) haξ 0 ξ 02

Present [18] Δ() Present [18] Δ()

(0 0)

005 9010 9020 011 8485 8370 137010 8823 8818 006 8319 8203 141015 8541 8516 029 8069 7950 150020 8196 8151 055 7762 7641 158

(100 0)

005 9389 9430 043 9020 8917 116010 9207 9231 026 8858 8753 120015 8933 8934 001 8614 8505 128020 8599 8577 026 8315 8203 137

(100 100)

005 15383 15439 036 16338 16320 011010 15213 15245 021 16175 16148 017015 14962 14966 003 15932 15895 023020 14664 14640 016 15639 15595 028

Table 3 Nondimensional frequencies of the FGP plate on EFTIM

λ1 λ2 λ3 λ4 λ5 λ608583 18118 18157 28015 33898 34248

(a) (b)

(c) (d)

(e) (f )

Figure 6 e first six mode sharps the FGP plate on EFTIM

Mathematical Problems in Engineering 9

06

07

08

09

1

11

12

13

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ1

(a)

14

16

18

2

22

24

26

28

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(b)

2

25

3

35

4

45

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(c)

Figure 7 Nondimensional frequencies of the eFGP plate with difference of featured-index β (a) nondimensional frequency λ1 (b)nondimensional frequency λ2 and (c) nondimensional frequency λ3

Table 4 e first three nondimensional frequencies of FGP on EFTIM

Parameter of plate β 0 025 05 075 1Case 1 λ1 11015 09575 08583 07847 07273(K1 100 K2 10) λ2 23078 20156 18118 16595 15400(SSSS) λ3 35651 31154 28015 25665 23822Case 2 λ1 10225 08988 08113 07453 06931(K1 100 K2 10) λ2 20774 18350 16612 15289 14239(SSSS) λ3 32410 28630 25921 23859 22221Case 3 λ1 12537 10735 09538 08669 08001(K1 100 K2 10) λ2 27399 23531 20942 19054 17599(SSSS) λ3 41972 36097 32152 29269 27044

10 Mathematical Problems in Engineering

Table 5 Nondimensional frequencies of the FGP plate with different K1 and K2

Case of Porosity distribution K2K1

100 250 500 750 1000

Case 1(SSSS)

10 08583 08929 09477 09995 1048825 09258 09579 10092 10580 1104750 10285 10575 11042 11490 1192175 11218 11485 11916 12332 12734100 12079 12327 12730 13120 13499

Case 2(SSSS)

10 08113 08452 08988 09493 0997325 08775 09089 09589 10064 1051850 09778 10061 10515 10950 1136975 10687 10947 11365 11769 12160100 11525 11765 12156 12534 12902

Case 3(SSSS)

10 09538 10009 10747 11438 1209025 10452 10883 11566 12211 1282350 11819 12202 12815 13400 1396075 13044 13392 13953 14492 15011100 14163 14484 15004 15507 15993

08100

1

85 100070 850

12

55 700

K1K2

14

55040 40025 25010 100

λ1

(a)

K1

K2

08100

09

85

1

100070

11

850

12

55 700

13

55040 40025 25010 100

λ1

(b)

K1

K2

08100

1

85 1000

12

70 850

14

55 700

16

55040 40025 25010 100

λ1

(c)

Figure 8 Nondimensional frequencies of FGP plate with different K1 and K2 (a) Case 1 (b) Case 2 and (c) Case 3

Mathematical Problems in Engineering 11

Table 6 Nondimensional frequencies of FGP plate with different K1 K2 and β

β K1 (K2 10) K2 (K1 100)Case 1

100 250 500 750 1000 10 25 50 75 1000 11015 11459 12163 12828 13460 11015 11882 13199 14397 15502025 09575 09961 10572 11150 11700 09575 10328 11473 12514 1347505 08583 08929 09477 09995 10488 08583 09258 10285 11218 12079075 07847 08163 08664 09138 09588 07847 08464 09403 10256 110431 07273 07566 08030 08469 08887 07273 07844 08715 09505 10235Case 20 10225 10652 11327 11964 12569 10225 11059 12323 13469 14525025 08988 09363 09957 10517 11049 08988 09721 10833 11840 1276805 08113 08452 08988 09493 09973 08113 08775 09778 10687 11525075 07453 07763 08256 08720 09161 07453 08060 08982 09817 105861 06931 07220 07677 08109 08519 06931 07496 08353 09129 09845Case 30 12537 13156 14127 15035 15891 12537 13739 15536 17146 18617025 10735 11265 12096 12874 13607 10735 11764 13303 14681 1594105 09538 10009 10747 11438 12090 09538 10452 11819 13044 14163075 08669 09097 09768 10396 10988 08669 09500 10742 11855 128721 08001 08395 09015 09595 10141 08001 08767 09914 10942 11880

060

08

025 100

1

25005 400

12

550075 700850

14

1 1000

080

1

025 1025

12

05 4055

14

075 70851 100

λ1

K1

β

λ1

K2

β

(a)

060

08

025 10025005

1

400550075 700

12

8501 1000

0

08

025

1

102505 40

12

55075 70

14

851 100

λ1

K1

β

λ1

K2

β

(b)

Figure 9 Continued

12 Mathematical Problems in Engineering

ωlowast (ωa2h)(ρbEb)

1113968 e first nondimensional frequen-

cies of present work compared with [18] are shown inTable 2

6 Numerical Results and Discussions

For free vibration problems a fully simple support (SSSS)FGP plate is considered whereina b h a10 Ω 05 k 1 Et 10Eb ρt 10ρbυ 038 and μF 05 e first six nondimensional fre-quencies of the FGP plate with porosity distribution of Case1 and stiffener of foundation K1 100 K2 10 are shownin Table 3 and the first six mode shapes are presented inFigure 6 e stiffness factors and nondimensional

frequencies of FGP plate are shown in equation (58) withHb (Ebh312(1 minus υ2))

61 Influence of the Parameters of the EFTIM to the FreeVibration for the FGP Plate Firstly in order to investigatethe effect of the featured-index of themass of foundation β tofree vibration of the FGP plate the featured-index of mass ischanged from 0 to 1 In Figure 7 and Table 4 it is seen that allcases of porosity distribution featured-index of the mass offoundation β significantly influence to free vibration of theFGP plate As β increases the mass of the plate increases andthe frequencies of the plate decrease For all cases of porositydistribution of the FGP plate the porosity distribution of

080

1

025 100

12

25005 400

14

550075 700850

16

1 1000

0

1

025 102505

15

4055075 70

85

2

1 100

λ1

K1

β

λ1

K2

β

(c)

Figure 9 Nondimensional frequencies of FGP plate with different K1 K2 and β (a) Case 1 (b) Case 2 and (c) Case 3

Table 7 Nondimensional frequencies of the FGP plate as a function of k and Ω

ΩSSSS CCCCk k

0 25 5 75 10 0 25 5 75 10Case 10 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06584 08830 09777 10469 10984 10935 12518 13694 14783 1563305 07126 09238 10105 10767 11273 11732 12869 13930 14958 15780075 07820 09677 10446 11070 11564 12711 13154 14102 15061 158481 08745 10105 10776 11357 11836 13924 13226 14130 15021 15766Case 20 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 08604 09563 10288 10833 10977 12011 13156 14273 1516205 07162 08685 09569 10299 10871 11707 11651 12611 13675 14564075 07777 08600 09357 10073 10671 12179 10868 11556 12496 133421 07852 07972 08420 09009 09580 10347 09106 09310 09904 10547Case 30 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 09413 10475 11196 11714 10977 13661 15012 16147 1699405 07162 10785 11908 12621 13122 11707 15836 17243 18317 19084075 07777 13066 14256 14917 15363 12179 19297 20656 21484 220181 07852 18589 19631 19772 19807 10347 24083 24138 23588 23246

Mathematical Problems in Engineering 13

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

where Ak is the area of the smoothing domain ψk and isgiven by

Ak

1113946ψk

dψ 13

1113944

nek

i1A

i (51)

where nek is the number of the adjacent triangular elementsin the smoothing domain ψk and Ai is the area of the ithtriangular element attached to the edge k

By substituting equations (47)ndash(49) into equations (24)(25) and (32) the approximation of the smoothed strains onthe smoothing domain ψk can be expressed as follows

1113957εkm 1113944

nnk

j1

1113957Bk

mjdkj

1113957kk 1113944

nnksh

j1

1113957Bk

bjdkj

γk 1113944

nnksh

j1

1113957Bk

sjdkj

(52)

where nnksh is the total number of nodes of the triangular

MITC3 elements attached to edge k (nnksh 3 for boundary

edges and nnksh 4 for inner edges as given in Figure 4 dk

j isthe nodal dof associated with the smoothing domain ψkand 1113957Bk

mj 1113957Bk

bj and 1113957Bk

sj are the smoothed membrane bendingand shear strain gradient matrices respectively at the jthnode of the elements attached to edge k computed by

1113957Bk

mj 1

Ak1113944

nek

i1

13A

iBemj (53)

1113957Bk

bj 1

Ak1113944

nek

i1

13AiB

ebj (54)

1113957Bk

sj 1

Ak1113944

nek

i1

13A

iBesj (55)

e stiffness matrix of the FGP plate using the ES-MITC3 is assembled by

1113957K 1113944

nksh

k1

1113957Kk (56)

where 1113957Kk is the ES-MITC3 stiffness matrix of the smoothingdomain ψk and given by

1113957Kk 1113946

ψk

1113957BKTD1113957Bk+ 1113957Bkt

s C1113957Bk

s1113874 1113875dψ 1113957BKTD1113957BkA

k+ 1113957Bkt

s C1113957Bk

s Ak

(57)

where

1113957BkT 1113957Bk

mj1113957Bk

bj1113876 1113877 (58)

5 Accuracy of the Proposed Method

In this section the various numerical examples are solved toverify the reliability and accuracy of the proposed methodFor convenience the stiffness factors and nondimensionalfrequencies of the plates are defined as the followingequations

K1 k1a

4

H

K2 k2a

2

H

λ ωa2

π2

ρh

H

1113970

withH Eh3

12 1 minus ]2( )

(59)

To demonstrate the performance of numerical resultsthe relative frequency error is defined by

Δ () 100 timesλpr minus λre

11138681113868111386811138681113868

11138681113868111386811138681113868

λre

11138681113868111386811138681113868111386811138681113868

(60)

where λpr and λre are nondimensional frequencies of presentmethod and nondimensional frequencies in [17 18]respectively

e results of the convergence of the first two nondi-mensional frequencies of the plate in the case of fully simplesupport (SSSS) plate and a fully clamped (CCCC) plate withha 01 K1 100 K2 10 respectively are shown in Fig-ure 5 From these results it can be seen that almost allfrequencies corresponding to different cases of boundaryconditions (BC) converge with 18times18 element mesh For18times18 mesh we compare the first three nondimensionalfrequencies of a plate resting on the WinklerndashPasternakfoundation with the published results as shown in Table 1 Itcan be seen that the present results agree well with the resultsof the authors using analytical methods [5 14 17] and aremore accurate than those using the original MITC3 elementand FEM [6] In addition from Table 2 it is obvious that therelative error of the present results compared to [18] is lessthan 2 In [18] they used a new quasi-3D hyperbolic theoryto investigate the free vibration of the FGP plate resting onthe EF ese results are the basis to analyze the free vi-bration of FGP plates on the EFTIM

Boundary edge m

Inner edge k

Г(k)

ψ(k)

Г(m)

ψ(m)

Field nodeCentroid of triangles

Figure 4 e smoothing domain ψk is formed by triangularelements

Mathematical Problems in Engineering 7

Next we consider an SSSS FGP plate (AlAl2O3)with its material properties as follows metal (Al)Eb 70GPa ρb 2702 kgm3 and ceramic (Al2O3)Et 380GPa ρt 3800 kgm3 Poissonrsquos ratio is fixed atυ 03 e FGP plate with even porosities is expressed asin [18]

P(z) Pb + Pt minus Pb( 1113857z

h+ 051113874 1113875

k

minusξ2

Pt + Pb( 1113857 (61)

where ξ(ξ le 1) presents the porosity volume fraction estiffness factor and nondimensional frequencies of the platesare shown in equation (58) with Hb (Ebh312(1 minus υ2)) and

4 6 8 10 12 14 16 18 20Mesh (nxn)

25

3

35

4

45

SSSSCCCC

λ1

(a)

SSSSCCCC

4 6 8 10 12 14 16 18 20Mesh (nxn)

5

6

7

8

9

10

λ2

(b)

Figure 5 e convergence of element mesh to nondimensional frequency of plate (a)λ1 and (b)λ2

Table 1 Nondimensional frequencies of plates

Plates K1 K2 Author λ1 Δ() λ2 Δ() λ3 Δ()

SSSSυ 03ha 001

100 10

Ferreira et al [17] 26559 55718 85384Zhou et al [14] 26551 003 55717 000 85406 003Xiang et al [5] 26551 003 55718 000 85405 002

MITC3 26604 017 56103 070 86296 107Present 26590 012 55920 037 86017 074

500 10

Ferreira [17] 33406 59285 87754Zhou et al [14] 33398 002 59285 000 87775 002Xiang et al [5] 33400 002 59287 000 87775 002

MITC3 33441 010 59649 061 88642 101Present 33430 007 59477 032 88370 070

SSSSυ 03ha 01

200 10

Ferreira et al [17] 27902 53452 78255Zhou et al [14] 27756 052 52954 093 77279 125Xiang et al [5] 27842 022 53043 077 77287 124

MITC3 27874 010 53258 036 77719 068Present 27887 005 53362 017 77971 036

1000 10

Ferreira et al [17] 39844 60430 83112Zhou et al [14] 39566 070 59757 111 81954 139Xiang et al [5] 39805 010 60078 058 82214 108

MITC3 39827 004 60266 027 82619 059Present 39836 002 60358 012 82856 031

CCCCυ 015ha 0015

13902 16683

Ferreira et al [17] 81669 12821 16842Zhou et al [14] 81675 001 12823 002 16833 005

Omurtag et al [6] 81375 036 12898 060 16932 053MITC3 81842 021 12909 069 17010 100Present 81729 007 12872 040 16939 058

8 Mathematical Problems in Engineering

Table 2 e first nondimensional frequencies of FGP plate according to the WinklerndashPasternak foundation stiffness (k 1)

(K1 K2) haξ 0 ξ 02

Present [18] Δ() Present [18] Δ()

(0 0)

005 9010 9020 011 8485 8370 137010 8823 8818 006 8319 8203 141015 8541 8516 029 8069 7950 150020 8196 8151 055 7762 7641 158

(100 0)

005 9389 9430 043 9020 8917 116010 9207 9231 026 8858 8753 120015 8933 8934 001 8614 8505 128020 8599 8577 026 8315 8203 137

(100 100)

005 15383 15439 036 16338 16320 011010 15213 15245 021 16175 16148 017015 14962 14966 003 15932 15895 023020 14664 14640 016 15639 15595 028

Table 3 Nondimensional frequencies of the FGP plate on EFTIM

λ1 λ2 λ3 λ4 λ5 λ608583 18118 18157 28015 33898 34248

(a) (b)

(c) (d)

(e) (f )

Figure 6 e first six mode sharps the FGP plate on EFTIM

Mathematical Problems in Engineering 9

06

07

08

09

1

11

12

13

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ1

(a)

14

16

18

2

22

24

26

28

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(b)

2

25

3

35

4

45

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(c)

Figure 7 Nondimensional frequencies of the eFGP plate with difference of featured-index β (a) nondimensional frequency λ1 (b)nondimensional frequency λ2 and (c) nondimensional frequency λ3

Table 4 e first three nondimensional frequencies of FGP on EFTIM

Parameter of plate β 0 025 05 075 1Case 1 λ1 11015 09575 08583 07847 07273(K1 100 K2 10) λ2 23078 20156 18118 16595 15400(SSSS) λ3 35651 31154 28015 25665 23822Case 2 λ1 10225 08988 08113 07453 06931(K1 100 K2 10) λ2 20774 18350 16612 15289 14239(SSSS) λ3 32410 28630 25921 23859 22221Case 3 λ1 12537 10735 09538 08669 08001(K1 100 K2 10) λ2 27399 23531 20942 19054 17599(SSSS) λ3 41972 36097 32152 29269 27044

10 Mathematical Problems in Engineering

Table 5 Nondimensional frequencies of the FGP plate with different K1 and K2

Case of Porosity distribution K2K1

100 250 500 750 1000

Case 1(SSSS)

10 08583 08929 09477 09995 1048825 09258 09579 10092 10580 1104750 10285 10575 11042 11490 1192175 11218 11485 11916 12332 12734100 12079 12327 12730 13120 13499

Case 2(SSSS)

10 08113 08452 08988 09493 0997325 08775 09089 09589 10064 1051850 09778 10061 10515 10950 1136975 10687 10947 11365 11769 12160100 11525 11765 12156 12534 12902

Case 3(SSSS)

10 09538 10009 10747 11438 1209025 10452 10883 11566 12211 1282350 11819 12202 12815 13400 1396075 13044 13392 13953 14492 15011100 14163 14484 15004 15507 15993

08100

1

85 100070 850

12

55 700

K1K2

14

55040 40025 25010 100

λ1

(a)

K1

K2

08100

09

85

1

100070

11

850

12

55 700

13

55040 40025 25010 100

λ1

(b)

K1

K2

08100

1

85 1000

12

70 850

14

55 700

16

55040 40025 25010 100

λ1

(c)

Figure 8 Nondimensional frequencies of FGP plate with different K1 and K2 (a) Case 1 (b) Case 2 and (c) Case 3

Mathematical Problems in Engineering 11

Table 6 Nondimensional frequencies of FGP plate with different K1 K2 and β

β K1 (K2 10) K2 (K1 100)Case 1

100 250 500 750 1000 10 25 50 75 1000 11015 11459 12163 12828 13460 11015 11882 13199 14397 15502025 09575 09961 10572 11150 11700 09575 10328 11473 12514 1347505 08583 08929 09477 09995 10488 08583 09258 10285 11218 12079075 07847 08163 08664 09138 09588 07847 08464 09403 10256 110431 07273 07566 08030 08469 08887 07273 07844 08715 09505 10235Case 20 10225 10652 11327 11964 12569 10225 11059 12323 13469 14525025 08988 09363 09957 10517 11049 08988 09721 10833 11840 1276805 08113 08452 08988 09493 09973 08113 08775 09778 10687 11525075 07453 07763 08256 08720 09161 07453 08060 08982 09817 105861 06931 07220 07677 08109 08519 06931 07496 08353 09129 09845Case 30 12537 13156 14127 15035 15891 12537 13739 15536 17146 18617025 10735 11265 12096 12874 13607 10735 11764 13303 14681 1594105 09538 10009 10747 11438 12090 09538 10452 11819 13044 14163075 08669 09097 09768 10396 10988 08669 09500 10742 11855 128721 08001 08395 09015 09595 10141 08001 08767 09914 10942 11880

060

08

025 100

1

25005 400

12

550075 700850

14

1 1000

080

1

025 1025

12

05 4055

14

075 70851 100

λ1

K1

β

λ1

K2

β

(a)

060

08

025 10025005

1

400550075 700

12

8501 1000

0

08

025

1

102505 40

12

55075 70

14

851 100

λ1

K1

β

λ1

K2

β

(b)

Figure 9 Continued

12 Mathematical Problems in Engineering

ωlowast (ωa2h)(ρbEb)

1113968 e first nondimensional frequen-

cies of present work compared with [18] are shown inTable 2

6 Numerical Results and Discussions

For free vibration problems a fully simple support (SSSS)FGP plate is considered whereina b h a10 Ω 05 k 1 Et 10Eb ρt 10ρbυ 038 and μF 05 e first six nondimensional fre-quencies of the FGP plate with porosity distribution of Case1 and stiffener of foundation K1 100 K2 10 are shownin Table 3 and the first six mode shapes are presented inFigure 6 e stiffness factors and nondimensional

frequencies of FGP plate are shown in equation (58) withHb (Ebh312(1 minus υ2))

61 Influence of the Parameters of the EFTIM to the FreeVibration for the FGP Plate Firstly in order to investigatethe effect of the featured-index of themass of foundation β tofree vibration of the FGP plate the featured-index of mass ischanged from 0 to 1 In Figure 7 and Table 4 it is seen that allcases of porosity distribution featured-index of the mass offoundation β significantly influence to free vibration of theFGP plate As β increases the mass of the plate increases andthe frequencies of the plate decrease For all cases of porositydistribution of the FGP plate the porosity distribution of

080

1

025 100

12

25005 400

14

550075 700850

16

1 1000

0

1

025 102505

15

4055075 70

85

2

1 100

λ1

K1

β

λ1

K2

β

(c)

Figure 9 Nondimensional frequencies of FGP plate with different K1 K2 and β (a) Case 1 (b) Case 2 and (c) Case 3

Table 7 Nondimensional frequencies of the FGP plate as a function of k and Ω

ΩSSSS CCCCk k

0 25 5 75 10 0 25 5 75 10Case 10 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06584 08830 09777 10469 10984 10935 12518 13694 14783 1563305 07126 09238 10105 10767 11273 11732 12869 13930 14958 15780075 07820 09677 10446 11070 11564 12711 13154 14102 15061 158481 08745 10105 10776 11357 11836 13924 13226 14130 15021 15766Case 20 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 08604 09563 10288 10833 10977 12011 13156 14273 1516205 07162 08685 09569 10299 10871 11707 11651 12611 13675 14564075 07777 08600 09357 10073 10671 12179 10868 11556 12496 133421 07852 07972 08420 09009 09580 10347 09106 09310 09904 10547Case 30 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 09413 10475 11196 11714 10977 13661 15012 16147 1699405 07162 10785 11908 12621 13122 11707 15836 17243 18317 19084075 07777 13066 14256 14917 15363 12179 19297 20656 21484 220181 07852 18589 19631 19772 19807 10347 24083 24138 23588 23246

Mathematical Problems in Engineering 13

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

Next we consider an SSSS FGP plate (AlAl2O3)with its material properties as follows metal (Al)Eb 70GPa ρb 2702 kgm3 and ceramic (Al2O3)Et 380GPa ρt 3800 kgm3 Poissonrsquos ratio is fixed atυ 03 e FGP plate with even porosities is expressed asin [18]

P(z) Pb + Pt minus Pb( 1113857z

h+ 051113874 1113875

k

minusξ2

Pt + Pb( 1113857 (61)

where ξ(ξ le 1) presents the porosity volume fraction estiffness factor and nondimensional frequencies of the platesare shown in equation (58) with Hb (Ebh312(1 minus υ2)) and

4 6 8 10 12 14 16 18 20Mesh (nxn)

25

3

35

4

45

SSSSCCCC

λ1

(a)

SSSSCCCC

4 6 8 10 12 14 16 18 20Mesh (nxn)

5

6

7

8

9

10

λ2

(b)

Figure 5 e convergence of element mesh to nondimensional frequency of plate (a)λ1 and (b)λ2

Table 1 Nondimensional frequencies of plates

Plates K1 K2 Author λ1 Δ() λ2 Δ() λ3 Δ()

SSSSυ 03ha 001

100 10

Ferreira et al [17] 26559 55718 85384Zhou et al [14] 26551 003 55717 000 85406 003Xiang et al [5] 26551 003 55718 000 85405 002

MITC3 26604 017 56103 070 86296 107Present 26590 012 55920 037 86017 074

500 10

Ferreira [17] 33406 59285 87754Zhou et al [14] 33398 002 59285 000 87775 002Xiang et al [5] 33400 002 59287 000 87775 002

MITC3 33441 010 59649 061 88642 101Present 33430 007 59477 032 88370 070

SSSSυ 03ha 01

200 10

Ferreira et al [17] 27902 53452 78255Zhou et al [14] 27756 052 52954 093 77279 125Xiang et al [5] 27842 022 53043 077 77287 124

MITC3 27874 010 53258 036 77719 068Present 27887 005 53362 017 77971 036

1000 10

Ferreira et al [17] 39844 60430 83112Zhou et al [14] 39566 070 59757 111 81954 139Xiang et al [5] 39805 010 60078 058 82214 108

MITC3 39827 004 60266 027 82619 059Present 39836 002 60358 012 82856 031

CCCCυ 015ha 0015

13902 16683

Ferreira et al [17] 81669 12821 16842Zhou et al [14] 81675 001 12823 002 16833 005

Omurtag et al [6] 81375 036 12898 060 16932 053MITC3 81842 021 12909 069 17010 100Present 81729 007 12872 040 16939 058

8 Mathematical Problems in Engineering

Table 2 e first nondimensional frequencies of FGP plate according to the WinklerndashPasternak foundation stiffness (k 1)

(K1 K2) haξ 0 ξ 02

Present [18] Δ() Present [18] Δ()

(0 0)

005 9010 9020 011 8485 8370 137010 8823 8818 006 8319 8203 141015 8541 8516 029 8069 7950 150020 8196 8151 055 7762 7641 158

(100 0)

005 9389 9430 043 9020 8917 116010 9207 9231 026 8858 8753 120015 8933 8934 001 8614 8505 128020 8599 8577 026 8315 8203 137

(100 100)

005 15383 15439 036 16338 16320 011010 15213 15245 021 16175 16148 017015 14962 14966 003 15932 15895 023020 14664 14640 016 15639 15595 028

Table 3 Nondimensional frequencies of the FGP plate on EFTIM

λ1 λ2 λ3 λ4 λ5 λ608583 18118 18157 28015 33898 34248

(a) (b)

(c) (d)

(e) (f )

Figure 6 e first six mode sharps the FGP plate on EFTIM

Mathematical Problems in Engineering 9

06

07

08

09

1

11

12

13

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ1

(a)

14

16

18

2

22

24

26

28

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(b)

2

25

3

35

4

45

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(c)

Figure 7 Nondimensional frequencies of the eFGP plate with difference of featured-index β (a) nondimensional frequency λ1 (b)nondimensional frequency λ2 and (c) nondimensional frequency λ3

Table 4 e first three nondimensional frequencies of FGP on EFTIM

Parameter of plate β 0 025 05 075 1Case 1 λ1 11015 09575 08583 07847 07273(K1 100 K2 10) λ2 23078 20156 18118 16595 15400(SSSS) λ3 35651 31154 28015 25665 23822Case 2 λ1 10225 08988 08113 07453 06931(K1 100 K2 10) λ2 20774 18350 16612 15289 14239(SSSS) λ3 32410 28630 25921 23859 22221Case 3 λ1 12537 10735 09538 08669 08001(K1 100 K2 10) λ2 27399 23531 20942 19054 17599(SSSS) λ3 41972 36097 32152 29269 27044

10 Mathematical Problems in Engineering

Table 5 Nondimensional frequencies of the FGP plate with different K1 and K2

Case of Porosity distribution K2K1

100 250 500 750 1000

Case 1(SSSS)

10 08583 08929 09477 09995 1048825 09258 09579 10092 10580 1104750 10285 10575 11042 11490 1192175 11218 11485 11916 12332 12734100 12079 12327 12730 13120 13499

Case 2(SSSS)

10 08113 08452 08988 09493 0997325 08775 09089 09589 10064 1051850 09778 10061 10515 10950 1136975 10687 10947 11365 11769 12160100 11525 11765 12156 12534 12902

Case 3(SSSS)

10 09538 10009 10747 11438 1209025 10452 10883 11566 12211 1282350 11819 12202 12815 13400 1396075 13044 13392 13953 14492 15011100 14163 14484 15004 15507 15993

08100

1

85 100070 850

12

55 700

K1K2

14

55040 40025 25010 100

λ1

(a)

K1

K2

08100

09

85

1

100070

11

850

12

55 700

13

55040 40025 25010 100

λ1

(b)

K1

K2

08100

1

85 1000

12

70 850

14

55 700

16

55040 40025 25010 100

λ1

(c)

Figure 8 Nondimensional frequencies of FGP plate with different K1 and K2 (a) Case 1 (b) Case 2 and (c) Case 3

Mathematical Problems in Engineering 11

Table 6 Nondimensional frequencies of FGP plate with different K1 K2 and β

β K1 (K2 10) K2 (K1 100)Case 1

100 250 500 750 1000 10 25 50 75 1000 11015 11459 12163 12828 13460 11015 11882 13199 14397 15502025 09575 09961 10572 11150 11700 09575 10328 11473 12514 1347505 08583 08929 09477 09995 10488 08583 09258 10285 11218 12079075 07847 08163 08664 09138 09588 07847 08464 09403 10256 110431 07273 07566 08030 08469 08887 07273 07844 08715 09505 10235Case 20 10225 10652 11327 11964 12569 10225 11059 12323 13469 14525025 08988 09363 09957 10517 11049 08988 09721 10833 11840 1276805 08113 08452 08988 09493 09973 08113 08775 09778 10687 11525075 07453 07763 08256 08720 09161 07453 08060 08982 09817 105861 06931 07220 07677 08109 08519 06931 07496 08353 09129 09845Case 30 12537 13156 14127 15035 15891 12537 13739 15536 17146 18617025 10735 11265 12096 12874 13607 10735 11764 13303 14681 1594105 09538 10009 10747 11438 12090 09538 10452 11819 13044 14163075 08669 09097 09768 10396 10988 08669 09500 10742 11855 128721 08001 08395 09015 09595 10141 08001 08767 09914 10942 11880

060

08

025 100

1

25005 400

12

550075 700850

14

1 1000

080

1

025 1025

12

05 4055

14

075 70851 100

λ1

K1

β

λ1

K2

β

(a)

060

08

025 10025005

1

400550075 700

12

8501 1000

0

08

025

1

102505 40

12

55075 70

14

851 100

λ1

K1

β

λ1

K2

β

(b)

Figure 9 Continued

12 Mathematical Problems in Engineering

ωlowast (ωa2h)(ρbEb)

1113968 e first nondimensional frequen-

cies of present work compared with [18] are shown inTable 2

6 Numerical Results and Discussions

For free vibration problems a fully simple support (SSSS)FGP plate is considered whereina b h a10 Ω 05 k 1 Et 10Eb ρt 10ρbυ 038 and μF 05 e first six nondimensional fre-quencies of the FGP plate with porosity distribution of Case1 and stiffener of foundation K1 100 K2 10 are shownin Table 3 and the first six mode shapes are presented inFigure 6 e stiffness factors and nondimensional

frequencies of FGP plate are shown in equation (58) withHb (Ebh312(1 minus υ2))

61 Influence of the Parameters of the EFTIM to the FreeVibration for the FGP Plate Firstly in order to investigatethe effect of the featured-index of themass of foundation β tofree vibration of the FGP plate the featured-index of mass ischanged from 0 to 1 In Figure 7 and Table 4 it is seen that allcases of porosity distribution featured-index of the mass offoundation β significantly influence to free vibration of theFGP plate As β increases the mass of the plate increases andthe frequencies of the plate decrease For all cases of porositydistribution of the FGP plate the porosity distribution of

080

1

025 100

12

25005 400

14

550075 700850

16

1 1000

0

1

025 102505

15

4055075 70

85

2

1 100

λ1

K1

β

λ1

K2

β

(c)

Figure 9 Nondimensional frequencies of FGP plate with different K1 K2 and β (a) Case 1 (b) Case 2 and (c) Case 3

Table 7 Nondimensional frequencies of the FGP plate as a function of k and Ω

ΩSSSS CCCCk k

0 25 5 75 10 0 25 5 75 10Case 10 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06584 08830 09777 10469 10984 10935 12518 13694 14783 1563305 07126 09238 10105 10767 11273 11732 12869 13930 14958 15780075 07820 09677 10446 11070 11564 12711 13154 14102 15061 158481 08745 10105 10776 11357 11836 13924 13226 14130 15021 15766Case 20 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 08604 09563 10288 10833 10977 12011 13156 14273 1516205 07162 08685 09569 10299 10871 11707 11651 12611 13675 14564075 07777 08600 09357 10073 10671 12179 10868 11556 12496 133421 07852 07972 08420 09009 09580 10347 09106 09310 09904 10547Case 30 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 09413 10475 11196 11714 10977 13661 15012 16147 1699405 07162 10785 11908 12621 13122 11707 15836 17243 18317 19084075 07777 13066 14256 14917 15363 12179 19297 20656 21484 220181 07852 18589 19631 19772 19807 10347 24083 24138 23588 23246

Mathematical Problems in Engineering 13

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

Table 2 e first nondimensional frequencies of FGP plate according to the WinklerndashPasternak foundation stiffness (k 1)

(K1 K2) haξ 0 ξ 02

Present [18] Δ() Present [18] Δ()

(0 0)

005 9010 9020 011 8485 8370 137010 8823 8818 006 8319 8203 141015 8541 8516 029 8069 7950 150020 8196 8151 055 7762 7641 158

(100 0)

005 9389 9430 043 9020 8917 116010 9207 9231 026 8858 8753 120015 8933 8934 001 8614 8505 128020 8599 8577 026 8315 8203 137

(100 100)

005 15383 15439 036 16338 16320 011010 15213 15245 021 16175 16148 017015 14962 14966 003 15932 15895 023020 14664 14640 016 15639 15595 028

Table 3 Nondimensional frequencies of the FGP plate on EFTIM

λ1 λ2 λ3 λ4 λ5 λ608583 18118 18157 28015 33898 34248

(a) (b)

(c) (d)

(e) (f )

Figure 6 e first six mode sharps the FGP plate on EFTIM

Mathematical Problems in Engineering 9

06

07

08

09

1

11

12

13

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ1

(a)

14

16

18

2

22

24

26

28

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(b)

2

25

3

35

4

45

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(c)

Figure 7 Nondimensional frequencies of the eFGP plate with difference of featured-index β (a) nondimensional frequency λ1 (b)nondimensional frequency λ2 and (c) nondimensional frequency λ3

Table 4 e first three nondimensional frequencies of FGP on EFTIM

Parameter of plate β 0 025 05 075 1Case 1 λ1 11015 09575 08583 07847 07273(K1 100 K2 10) λ2 23078 20156 18118 16595 15400(SSSS) λ3 35651 31154 28015 25665 23822Case 2 λ1 10225 08988 08113 07453 06931(K1 100 K2 10) λ2 20774 18350 16612 15289 14239(SSSS) λ3 32410 28630 25921 23859 22221Case 3 λ1 12537 10735 09538 08669 08001(K1 100 K2 10) λ2 27399 23531 20942 19054 17599(SSSS) λ3 41972 36097 32152 29269 27044

10 Mathematical Problems in Engineering

Table 5 Nondimensional frequencies of the FGP plate with different K1 and K2

Case of Porosity distribution K2K1

100 250 500 750 1000

Case 1(SSSS)

10 08583 08929 09477 09995 1048825 09258 09579 10092 10580 1104750 10285 10575 11042 11490 1192175 11218 11485 11916 12332 12734100 12079 12327 12730 13120 13499

Case 2(SSSS)

10 08113 08452 08988 09493 0997325 08775 09089 09589 10064 1051850 09778 10061 10515 10950 1136975 10687 10947 11365 11769 12160100 11525 11765 12156 12534 12902

Case 3(SSSS)

10 09538 10009 10747 11438 1209025 10452 10883 11566 12211 1282350 11819 12202 12815 13400 1396075 13044 13392 13953 14492 15011100 14163 14484 15004 15507 15993

08100

1

85 100070 850

12

55 700

K1K2

14

55040 40025 25010 100

λ1

(a)

K1

K2

08100

09

85

1

100070

11

850

12

55 700

13

55040 40025 25010 100

λ1

(b)

K1

K2

08100

1

85 1000

12

70 850

14

55 700

16

55040 40025 25010 100

λ1

(c)

Figure 8 Nondimensional frequencies of FGP plate with different K1 and K2 (a) Case 1 (b) Case 2 and (c) Case 3

Mathematical Problems in Engineering 11

Table 6 Nondimensional frequencies of FGP plate with different K1 K2 and β

β K1 (K2 10) K2 (K1 100)Case 1

100 250 500 750 1000 10 25 50 75 1000 11015 11459 12163 12828 13460 11015 11882 13199 14397 15502025 09575 09961 10572 11150 11700 09575 10328 11473 12514 1347505 08583 08929 09477 09995 10488 08583 09258 10285 11218 12079075 07847 08163 08664 09138 09588 07847 08464 09403 10256 110431 07273 07566 08030 08469 08887 07273 07844 08715 09505 10235Case 20 10225 10652 11327 11964 12569 10225 11059 12323 13469 14525025 08988 09363 09957 10517 11049 08988 09721 10833 11840 1276805 08113 08452 08988 09493 09973 08113 08775 09778 10687 11525075 07453 07763 08256 08720 09161 07453 08060 08982 09817 105861 06931 07220 07677 08109 08519 06931 07496 08353 09129 09845Case 30 12537 13156 14127 15035 15891 12537 13739 15536 17146 18617025 10735 11265 12096 12874 13607 10735 11764 13303 14681 1594105 09538 10009 10747 11438 12090 09538 10452 11819 13044 14163075 08669 09097 09768 10396 10988 08669 09500 10742 11855 128721 08001 08395 09015 09595 10141 08001 08767 09914 10942 11880

060

08

025 100

1

25005 400

12

550075 700850

14

1 1000

080

1

025 1025

12

05 4055

14

075 70851 100

λ1

K1

β

λ1

K2

β

(a)

060

08

025 10025005

1

400550075 700

12

8501 1000

0

08

025

1

102505 40

12

55075 70

14

851 100

λ1

K1

β

λ1

K2

β

(b)

Figure 9 Continued

12 Mathematical Problems in Engineering

ωlowast (ωa2h)(ρbEb)

1113968 e first nondimensional frequen-

cies of present work compared with [18] are shown inTable 2

6 Numerical Results and Discussions

For free vibration problems a fully simple support (SSSS)FGP plate is considered whereina b h a10 Ω 05 k 1 Et 10Eb ρt 10ρbυ 038 and μF 05 e first six nondimensional fre-quencies of the FGP plate with porosity distribution of Case1 and stiffener of foundation K1 100 K2 10 are shownin Table 3 and the first six mode shapes are presented inFigure 6 e stiffness factors and nondimensional

frequencies of FGP plate are shown in equation (58) withHb (Ebh312(1 minus υ2))

61 Influence of the Parameters of the EFTIM to the FreeVibration for the FGP Plate Firstly in order to investigatethe effect of the featured-index of themass of foundation β tofree vibration of the FGP plate the featured-index of mass ischanged from 0 to 1 In Figure 7 and Table 4 it is seen that allcases of porosity distribution featured-index of the mass offoundation β significantly influence to free vibration of theFGP plate As β increases the mass of the plate increases andthe frequencies of the plate decrease For all cases of porositydistribution of the FGP plate the porosity distribution of

080

1

025 100

12

25005 400

14

550075 700850

16

1 1000

0

1

025 102505

15

4055075 70

85

2

1 100

λ1

K1

β

λ1

K2

β

(c)

Figure 9 Nondimensional frequencies of FGP plate with different K1 K2 and β (a) Case 1 (b) Case 2 and (c) Case 3

Table 7 Nondimensional frequencies of the FGP plate as a function of k and Ω

ΩSSSS CCCCk k

0 25 5 75 10 0 25 5 75 10Case 10 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06584 08830 09777 10469 10984 10935 12518 13694 14783 1563305 07126 09238 10105 10767 11273 11732 12869 13930 14958 15780075 07820 09677 10446 11070 11564 12711 13154 14102 15061 158481 08745 10105 10776 11357 11836 13924 13226 14130 15021 15766Case 20 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 08604 09563 10288 10833 10977 12011 13156 14273 1516205 07162 08685 09569 10299 10871 11707 11651 12611 13675 14564075 07777 08600 09357 10073 10671 12179 10868 11556 12496 133421 07852 07972 08420 09009 09580 10347 09106 09310 09904 10547Case 30 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 09413 10475 11196 11714 10977 13661 15012 16147 1699405 07162 10785 11908 12621 13122 11707 15836 17243 18317 19084075 07777 13066 14256 14917 15363 12179 19297 20656 21484 220181 07852 18589 19631 19772 19807 10347 24083 24138 23588 23246

Mathematical Problems in Engineering 13

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

06

07

08

09

1

11

12

13

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ1

(a)

14

16

18

2

22

24

26

28

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(b)

2

25

3

35

4

45

0 01β

02 03 04 05 06 07 08 09 1

Case 1Case 2Case 3

λ2

(c)

Figure 7 Nondimensional frequencies of the eFGP plate with difference of featured-index β (a) nondimensional frequency λ1 (b)nondimensional frequency λ2 and (c) nondimensional frequency λ3

Table 4 e first three nondimensional frequencies of FGP on EFTIM

Parameter of plate β 0 025 05 075 1Case 1 λ1 11015 09575 08583 07847 07273(K1 100 K2 10) λ2 23078 20156 18118 16595 15400(SSSS) λ3 35651 31154 28015 25665 23822Case 2 λ1 10225 08988 08113 07453 06931(K1 100 K2 10) λ2 20774 18350 16612 15289 14239(SSSS) λ3 32410 28630 25921 23859 22221Case 3 λ1 12537 10735 09538 08669 08001(K1 100 K2 10) λ2 27399 23531 20942 19054 17599(SSSS) λ3 41972 36097 32152 29269 27044

10 Mathematical Problems in Engineering

Table 5 Nondimensional frequencies of the FGP plate with different K1 and K2

Case of Porosity distribution K2K1

100 250 500 750 1000

Case 1(SSSS)

10 08583 08929 09477 09995 1048825 09258 09579 10092 10580 1104750 10285 10575 11042 11490 1192175 11218 11485 11916 12332 12734100 12079 12327 12730 13120 13499

Case 2(SSSS)

10 08113 08452 08988 09493 0997325 08775 09089 09589 10064 1051850 09778 10061 10515 10950 1136975 10687 10947 11365 11769 12160100 11525 11765 12156 12534 12902

Case 3(SSSS)

10 09538 10009 10747 11438 1209025 10452 10883 11566 12211 1282350 11819 12202 12815 13400 1396075 13044 13392 13953 14492 15011100 14163 14484 15004 15507 15993

08100

1

85 100070 850

12

55 700

K1K2

14

55040 40025 25010 100

λ1

(a)

K1

K2

08100

09

85

1

100070

11

850

12

55 700

13

55040 40025 25010 100

λ1

(b)

K1

K2

08100

1

85 1000

12

70 850

14

55 700

16

55040 40025 25010 100

λ1

(c)

Figure 8 Nondimensional frequencies of FGP plate with different K1 and K2 (a) Case 1 (b) Case 2 and (c) Case 3

Mathematical Problems in Engineering 11

Table 6 Nondimensional frequencies of FGP plate with different K1 K2 and β

β K1 (K2 10) K2 (K1 100)Case 1

100 250 500 750 1000 10 25 50 75 1000 11015 11459 12163 12828 13460 11015 11882 13199 14397 15502025 09575 09961 10572 11150 11700 09575 10328 11473 12514 1347505 08583 08929 09477 09995 10488 08583 09258 10285 11218 12079075 07847 08163 08664 09138 09588 07847 08464 09403 10256 110431 07273 07566 08030 08469 08887 07273 07844 08715 09505 10235Case 20 10225 10652 11327 11964 12569 10225 11059 12323 13469 14525025 08988 09363 09957 10517 11049 08988 09721 10833 11840 1276805 08113 08452 08988 09493 09973 08113 08775 09778 10687 11525075 07453 07763 08256 08720 09161 07453 08060 08982 09817 105861 06931 07220 07677 08109 08519 06931 07496 08353 09129 09845Case 30 12537 13156 14127 15035 15891 12537 13739 15536 17146 18617025 10735 11265 12096 12874 13607 10735 11764 13303 14681 1594105 09538 10009 10747 11438 12090 09538 10452 11819 13044 14163075 08669 09097 09768 10396 10988 08669 09500 10742 11855 128721 08001 08395 09015 09595 10141 08001 08767 09914 10942 11880

060

08

025 100

1

25005 400

12

550075 700850

14

1 1000

080

1

025 1025

12

05 4055

14

075 70851 100

λ1

K1

β

λ1

K2

β

(a)

060

08

025 10025005

1

400550075 700

12

8501 1000

0

08

025

1

102505 40

12

55075 70

14

851 100

λ1

K1

β

λ1

K2

β

(b)

Figure 9 Continued

12 Mathematical Problems in Engineering

ωlowast (ωa2h)(ρbEb)

1113968 e first nondimensional frequen-

cies of present work compared with [18] are shown inTable 2

6 Numerical Results and Discussions

For free vibration problems a fully simple support (SSSS)FGP plate is considered whereina b h a10 Ω 05 k 1 Et 10Eb ρt 10ρbυ 038 and μF 05 e first six nondimensional fre-quencies of the FGP plate with porosity distribution of Case1 and stiffener of foundation K1 100 K2 10 are shownin Table 3 and the first six mode shapes are presented inFigure 6 e stiffness factors and nondimensional

frequencies of FGP plate are shown in equation (58) withHb (Ebh312(1 minus υ2))

61 Influence of the Parameters of the EFTIM to the FreeVibration for the FGP Plate Firstly in order to investigatethe effect of the featured-index of themass of foundation β tofree vibration of the FGP plate the featured-index of mass ischanged from 0 to 1 In Figure 7 and Table 4 it is seen that allcases of porosity distribution featured-index of the mass offoundation β significantly influence to free vibration of theFGP plate As β increases the mass of the plate increases andthe frequencies of the plate decrease For all cases of porositydistribution of the FGP plate the porosity distribution of

080

1

025 100

12

25005 400

14

550075 700850

16

1 1000

0

1

025 102505

15

4055075 70

85

2

1 100

λ1

K1

β

λ1

K2

β

(c)

Figure 9 Nondimensional frequencies of FGP plate with different K1 K2 and β (a) Case 1 (b) Case 2 and (c) Case 3

Table 7 Nondimensional frequencies of the FGP plate as a function of k and Ω

ΩSSSS CCCCk k

0 25 5 75 10 0 25 5 75 10Case 10 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06584 08830 09777 10469 10984 10935 12518 13694 14783 1563305 07126 09238 10105 10767 11273 11732 12869 13930 14958 15780075 07820 09677 10446 11070 11564 12711 13154 14102 15061 158481 08745 10105 10776 11357 11836 13924 13226 14130 15021 15766Case 20 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 08604 09563 10288 10833 10977 12011 13156 14273 1516205 07162 08685 09569 10299 10871 11707 11651 12611 13675 14564075 07777 08600 09357 10073 10671 12179 10868 11556 12496 133421 07852 07972 08420 09009 09580 10347 09106 09310 09904 10547Case 30 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 09413 10475 11196 11714 10977 13661 15012 16147 1699405 07162 10785 11908 12621 13122 11707 15836 17243 18317 19084075 07777 13066 14256 14917 15363 12179 19297 20656 21484 220181 07852 18589 19631 19772 19807 10347 24083 24138 23588 23246

Mathematical Problems in Engineering 13

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

Table 5 Nondimensional frequencies of the FGP plate with different K1 and K2

Case of Porosity distribution K2K1

100 250 500 750 1000

Case 1(SSSS)

10 08583 08929 09477 09995 1048825 09258 09579 10092 10580 1104750 10285 10575 11042 11490 1192175 11218 11485 11916 12332 12734100 12079 12327 12730 13120 13499

Case 2(SSSS)

10 08113 08452 08988 09493 0997325 08775 09089 09589 10064 1051850 09778 10061 10515 10950 1136975 10687 10947 11365 11769 12160100 11525 11765 12156 12534 12902

Case 3(SSSS)

10 09538 10009 10747 11438 1209025 10452 10883 11566 12211 1282350 11819 12202 12815 13400 1396075 13044 13392 13953 14492 15011100 14163 14484 15004 15507 15993

08100

1

85 100070 850

12

55 700

K1K2

14

55040 40025 25010 100

λ1

(a)

K1

K2

08100

09

85

1

100070

11

850

12

55 700

13

55040 40025 25010 100

λ1

(b)

K1

K2

08100

1

85 1000

12

70 850

14

55 700

16

55040 40025 25010 100

λ1

(c)

Figure 8 Nondimensional frequencies of FGP plate with different K1 and K2 (a) Case 1 (b) Case 2 and (c) Case 3

Mathematical Problems in Engineering 11

Table 6 Nondimensional frequencies of FGP plate with different K1 K2 and β

β K1 (K2 10) K2 (K1 100)Case 1

100 250 500 750 1000 10 25 50 75 1000 11015 11459 12163 12828 13460 11015 11882 13199 14397 15502025 09575 09961 10572 11150 11700 09575 10328 11473 12514 1347505 08583 08929 09477 09995 10488 08583 09258 10285 11218 12079075 07847 08163 08664 09138 09588 07847 08464 09403 10256 110431 07273 07566 08030 08469 08887 07273 07844 08715 09505 10235Case 20 10225 10652 11327 11964 12569 10225 11059 12323 13469 14525025 08988 09363 09957 10517 11049 08988 09721 10833 11840 1276805 08113 08452 08988 09493 09973 08113 08775 09778 10687 11525075 07453 07763 08256 08720 09161 07453 08060 08982 09817 105861 06931 07220 07677 08109 08519 06931 07496 08353 09129 09845Case 30 12537 13156 14127 15035 15891 12537 13739 15536 17146 18617025 10735 11265 12096 12874 13607 10735 11764 13303 14681 1594105 09538 10009 10747 11438 12090 09538 10452 11819 13044 14163075 08669 09097 09768 10396 10988 08669 09500 10742 11855 128721 08001 08395 09015 09595 10141 08001 08767 09914 10942 11880

060

08

025 100

1

25005 400

12

550075 700850

14

1 1000

080

1

025 1025

12

05 4055

14

075 70851 100

λ1

K1

β

λ1

K2

β

(a)

060

08

025 10025005

1

400550075 700

12

8501 1000

0

08

025

1

102505 40

12

55075 70

14

851 100

λ1

K1

β

λ1

K2

β

(b)

Figure 9 Continued

12 Mathematical Problems in Engineering

ωlowast (ωa2h)(ρbEb)

1113968 e first nondimensional frequen-

cies of present work compared with [18] are shown inTable 2

6 Numerical Results and Discussions

For free vibration problems a fully simple support (SSSS)FGP plate is considered whereina b h a10 Ω 05 k 1 Et 10Eb ρt 10ρbυ 038 and μF 05 e first six nondimensional fre-quencies of the FGP plate with porosity distribution of Case1 and stiffener of foundation K1 100 K2 10 are shownin Table 3 and the first six mode shapes are presented inFigure 6 e stiffness factors and nondimensional

frequencies of FGP plate are shown in equation (58) withHb (Ebh312(1 minus υ2))

61 Influence of the Parameters of the EFTIM to the FreeVibration for the FGP Plate Firstly in order to investigatethe effect of the featured-index of themass of foundation β tofree vibration of the FGP plate the featured-index of mass ischanged from 0 to 1 In Figure 7 and Table 4 it is seen that allcases of porosity distribution featured-index of the mass offoundation β significantly influence to free vibration of theFGP plate As β increases the mass of the plate increases andthe frequencies of the plate decrease For all cases of porositydistribution of the FGP plate the porosity distribution of

080

1

025 100

12

25005 400

14

550075 700850

16

1 1000

0

1

025 102505

15

4055075 70

85

2

1 100

λ1

K1

β

λ1

K2

β

(c)

Figure 9 Nondimensional frequencies of FGP plate with different K1 K2 and β (a) Case 1 (b) Case 2 and (c) Case 3

Table 7 Nondimensional frequencies of the FGP plate as a function of k and Ω

ΩSSSS CCCCk k

0 25 5 75 10 0 25 5 75 10Case 10 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06584 08830 09777 10469 10984 10935 12518 13694 14783 1563305 07126 09238 10105 10767 11273 11732 12869 13930 14958 15780075 07820 09677 10446 11070 11564 12711 13154 14102 15061 158481 08745 10105 10776 11357 11836 13924 13226 14130 15021 15766Case 20 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 08604 09563 10288 10833 10977 12011 13156 14273 1516205 07162 08685 09569 10299 10871 11707 11651 12611 13675 14564075 07777 08600 09357 10073 10671 12179 10868 11556 12496 133421 07852 07972 08420 09009 09580 10347 09106 09310 09904 10547Case 30 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 09413 10475 11196 11714 10977 13661 15012 16147 1699405 07162 10785 11908 12621 13122 11707 15836 17243 18317 19084075 07777 13066 14256 14917 15363 12179 19297 20656 21484 220181 07852 18589 19631 19772 19807 10347 24083 24138 23588 23246

Mathematical Problems in Engineering 13

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

Table 6 Nondimensional frequencies of FGP plate with different K1 K2 and β

β K1 (K2 10) K2 (K1 100)Case 1

100 250 500 750 1000 10 25 50 75 1000 11015 11459 12163 12828 13460 11015 11882 13199 14397 15502025 09575 09961 10572 11150 11700 09575 10328 11473 12514 1347505 08583 08929 09477 09995 10488 08583 09258 10285 11218 12079075 07847 08163 08664 09138 09588 07847 08464 09403 10256 110431 07273 07566 08030 08469 08887 07273 07844 08715 09505 10235Case 20 10225 10652 11327 11964 12569 10225 11059 12323 13469 14525025 08988 09363 09957 10517 11049 08988 09721 10833 11840 1276805 08113 08452 08988 09493 09973 08113 08775 09778 10687 11525075 07453 07763 08256 08720 09161 07453 08060 08982 09817 105861 06931 07220 07677 08109 08519 06931 07496 08353 09129 09845Case 30 12537 13156 14127 15035 15891 12537 13739 15536 17146 18617025 10735 11265 12096 12874 13607 10735 11764 13303 14681 1594105 09538 10009 10747 11438 12090 09538 10452 11819 13044 14163075 08669 09097 09768 10396 10988 08669 09500 10742 11855 128721 08001 08395 09015 09595 10141 08001 08767 09914 10942 11880

060

08

025 100

1

25005 400

12

550075 700850

14

1 1000

080

1

025 1025

12

05 4055

14

075 70851 100

λ1

K1

β

λ1

K2

β

(a)

060

08

025 10025005

1

400550075 700

12

8501 1000

0

08

025

1

102505 40

12

55075 70

14

851 100

λ1

K1

β

λ1

K2

β

(b)

Figure 9 Continued

12 Mathematical Problems in Engineering

ωlowast (ωa2h)(ρbEb)

1113968 e first nondimensional frequen-

cies of present work compared with [18] are shown inTable 2

6 Numerical Results and Discussions

For free vibration problems a fully simple support (SSSS)FGP plate is considered whereina b h a10 Ω 05 k 1 Et 10Eb ρt 10ρbυ 038 and μF 05 e first six nondimensional fre-quencies of the FGP plate with porosity distribution of Case1 and stiffener of foundation K1 100 K2 10 are shownin Table 3 and the first six mode shapes are presented inFigure 6 e stiffness factors and nondimensional

frequencies of FGP plate are shown in equation (58) withHb (Ebh312(1 minus υ2))

61 Influence of the Parameters of the EFTIM to the FreeVibration for the FGP Plate Firstly in order to investigatethe effect of the featured-index of themass of foundation β tofree vibration of the FGP plate the featured-index of mass ischanged from 0 to 1 In Figure 7 and Table 4 it is seen that allcases of porosity distribution featured-index of the mass offoundation β significantly influence to free vibration of theFGP plate As β increases the mass of the plate increases andthe frequencies of the plate decrease For all cases of porositydistribution of the FGP plate the porosity distribution of

080

1

025 100

12

25005 400

14

550075 700850

16

1 1000

0

1

025 102505

15

4055075 70

85

2

1 100

λ1

K1

β

λ1

K2

β

(c)

Figure 9 Nondimensional frequencies of FGP plate with different K1 K2 and β (a) Case 1 (b) Case 2 and (c) Case 3

Table 7 Nondimensional frequencies of the FGP plate as a function of k and Ω

ΩSSSS CCCCk k

0 25 5 75 10 0 25 5 75 10Case 10 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06584 08830 09777 10469 10984 10935 12518 13694 14783 1563305 07126 09238 10105 10767 11273 11732 12869 13930 14958 15780075 07820 09677 10446 11070 11564 12711 13154 14102 15061 158481 08745 10105 10776 11357 11836 13924 13226 14130 15021 15766Case 20 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 08604 09563 10288 10833 10977 12011 13156 14273 1516205 07162 08685 09569 10299 10871 11707 11651 12611 13675 14564075 07777 08600 09357 10073 10671 12179 10868 11556 12496 133421 07852 07972 08420 09009 09580 10347 09106 09310 09904 10547Case 30 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 09413 10475 11196 11714 10977 13661 15012 16147 1699405 07162 10785 11908 12621 13122 11707 15836 17243 18317 19084075 07777 13066 14256 14917 15363 12179 19297 20656 21484 220181 07852 18589 19631 19772 19807 10347 24083 24138 23588 23246

Mathematical Problems in Engineering 13

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

ωlowast (ωa2h)(ρbEb)

1113968 e first nondimensional frequen-

cies of present work compared with [18] are shown inTable 2

6 Numerical Results and Discussions

For free vibration problems a fully simple support (SSSS)FGP plate is considered whereina b h a10 Ω 05 k 1 Et 10Eb ρt 10ρbυ 038 and μF 05 e first six nondimensional fre-quencies of the FGP plate with porosity distribution of Case1 and stiffener of foundation K1 100 K2 10 are shownin Table 3 and the first six mode shapes are presented inFigure 6 e stiffness factors and nondimensional

frequencies of FGP plate are shown in equation (58) withHb (Ebh312(1 minus υ2))

61 Influence of the Parameters of the EFTIM to the FreeVibration for the FGP Plate Firstly in order to investigatethe effect of the featured-index of themass of foundation β tofree vibration of the FGP plate the featured-index of mass ischanged from 0 to 1 In Figure 7 and Table 4 it is seen that allcases of porosity distribution featured-index of the mass offoundation β significantly influence to free vibration of theFGP plate As β increases the mass of the plate increases andthe frequencies of the plate decrease For all cases of porositydistribution of the FGP plate the porosity distribution of

080

1

025 100

12

25005 400

14

550075 700850

16

1 1000

0

1

025 102505

15

4055075 70

85

2

1 100

λ1

K1

β

λ1

K2

β

(c)

Figure 9 Nondimensional frequencies of FGP plate with different K1 K2 and β (a) Case 1 (b) Case 2 and (c) Case 3

Table 7 Nondimensional frequencies of the FGP plate as a function of k and Ω

ΩSSSS CCCCk k

0 25 5 75 10 0 25 5 75 10Case 10 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06584 08830 09777 10469 10984 10935 12518 13694 14783 1563305 07126 09238 10105 10767 11273 11732 12869 13930 14958 15780075 07820 09677 10446 11070 11564 12711 13154 14102 15061 158481 08745 10105 10776 11357 11836 13924 13226 14130 15021 15766Case 20 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 08604 09563 10288 10833 10977 12011 13156 14273 1516205 07162 08685 09569 10299 10871 11707 11651 12611 13675 14564075 07777 08600 09357 10073 10671 12179 10868 11556 12496 133421 07852 07972 08420 09009 09580 10347 09106 09310 09904 10547Case 30 06146 08460 09468 10184 10706 10275 12153 13432 14571 15444025 06609 09413 10475 11196 11714 10977 13661 15012 16147 1699405 07162 10785 11908 12621 13122 11707 15836 17243 18317 19084075 07777 13066 14256 14917 15363 12179 19297 20656 21484 220181 07852 18589 19631 19772 19807 10347 24083 24138 23588 23246

Mathematical Problems in Engineering 13

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

Case 3 leads to the maximum values of frequencies of theplate while the porosity distribution of Case 2 leads to theminimum values It can be observed that the FGP plate withporosity distribution Case 3 is stiffer than plates with otherporosity distributions

Next the influence of nondimensional parameters offoundation stiffness K1 and K2 is investigated We change K1from 100 to 1000 and K2 from 10 to 100 with respect toβ 05 and μF 05 e first nondimensional frequency ofthe FGP plate with three cases of porosity distribution is

06010

08

8 02046

k

1

4 06

12

2 080 1

SSSS

1010

12

8 02046

k

14

064

16

2 0810

CCCC

λ1λ1

ΩΩ

(a)

0610 0

07

8 02

08

6 04

09

k4 06

1

11

0820 1

SSSS

08010

1

8 02

12

046

k

14

064

16

0820 1

CCCC

λ1λ1

ΩΩ

(b)

0510 0

1

8 026 04

k

15

064

2

2 0810

SSSS

1010

15

8 02046

k

2

4 06

25

0820 1

CCCC

λ1λ1

ΩΩ

(c)

Figure 10 Nondimensional frequencies vibration of the FGP plate as a function of k (a) Case 1 (b) Case 2 and (c) Case 3

14 Mathematical Problems in Engineering

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

presented in Table 5 and shown in Figure 8 As shown inthese figure and table when K1 and K2 increase the non-dimensional frequency of plate also increases We also ex-amine the effect of (β K1) and (β K2) parameters tonondimensional frequency e numerical result is pre-sented in Table 6 and Figure 9 Consequently Winklerfoundation stiffness K1 and shear layer stiffness of Pasternakfoundation K2 make stiffness of plate become greater and themass of the EF involved in the platersquos vibrationmakes reducefrequencies

62 Influence of the Parameters-FGP to Free Vibration of thePlate on EFTIM Let us consider the effect of materialsproperty to free vibration of the FGP plate e power-lawindex k is changed from 0 to 10 and maximum porositydistributions Ω has the value from 0 to 1 We examine theSSSS FGP plate and fully clamped (CCCC) plate resting onEFTIM e parameters of EFTIM are given by β 05μF 05 K1 100 and K1 10 e first nondimensionalfrequencies of plate with three cases of porosity distributionis shown in Table 7 and Figure 10

As shown in these figures and tables when k and Ωchange the values of nondimensional frequency change withno rule It is understandable because with each case ofchange in porosity distributions k and Ω the stiffness andthe weight of the plate changes From Figure 10 in the caseof the CCCC plate nondimensional frequency depending onk andΩ value varies more complex than the case of the SSSSplate If k andΩ values are same the frequency of the CCCCplate is larger than that of the SSSS platee results are quitereasonable because the SSSS boundary condition inherentlyoffers more flexible boundary conditions than the CCCCboundary condition

7 Conclusions

In this paper new numerical results of free vibration of theFGP plate resting on EFTIM are studied We used the ES-MITC3 to establish the fundamental equation of the FGPplate e computed results obtained by this approach are inexcellent agreement with others published Our work has thefollowing advantages

e novel ES-MITC3 which computes the free vibrationof the plate on EF takes into account the mass of foundation

e numerical results obtained by ES-MITC3 show goodagreement with the reference solutions and are more ac-curate than those obtained by the original MITC3

e elastic foundation of Pasternak with three-param-eters is developed by adding the featured-index of mass β toaccurately describe the actual elastic foundation

e mass of the elastic foundation involved in the vi-bration of the plate reduces the frequency of vibration whiletwo parameters K1 and K2 effect the stiffness of the plate

e material parameters k Ω and the case of po-rosity distribution effect of vibration of the plate Nu-merical results are useful for calculation design andtesting of material parameters in engineering andtechnologies

is study suggests some further works on the dynamicresponse and heat transfer problems of the FGP plate restingon EF using different plate theories

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Vietnam National Foun-dation for Science and Technology Development(NAFOSTED) under Grant no 10702-2019330

References

[1] E Winkler Die Lehre von der Elasticitaet und Festigkeit PragDominicus 1867

[2] P L Pasternak On a New Method of Analysis of an ElasticFoundation by Means of Two Foundation Constants Gosu-darstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhi-tekture Moscow Russia in Russian 1954

[3] F A Fazzolari ldquoGeneralized exponential polynomial andtrigonometric theories for vibration and stability analysis ofporous FG sandwich beams resting on elastic foundationsrdquoComposites Part B Engineering vol 136 pp 254ndash271 2018

[4] A W Leissa ldquoe free vibration of rectangular platesrdquoJournal of Sound and Vibration vol 31 no 3 pp 257ndash2931973

[5] Y Xiang C M Wang and S Kitipornchai ldquoExact vibrationsolution for initially stressed mindlin plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 36 no 4 pp 311ndash316 1994

[6] M H Omurtag A Ozutok A Y Akoz and Y OzccedilelikorsldquoFree vibration analysis of Kirchhoff plates resting on elasticfoundation by mixed finite element formulation based onGateaux differentialrdquo International Journal for NumericalMethods in Engineering vol 40 no 2 pp 295ndash317 1997

[7] K Y Ozccedilelikors C M Wang and X Q He ldquoCanonical exactsolutions for Levy-plates on two-parameter foundation usingGreenrsquos functionsrdquo Engineering Structures vol 22 no 4pp 364ndash378 2000

[8] H Matsunaga ldquoVibration and stability of thick plates onelastic foundationsrdquo Journal of Engineering Mechanicsvol 126 no 1 pp 27ndash34 2000

[9] Y Ayvaz A Daloglu and A Dogangun ldquoApplication of amodified Vlasovmodel to earthquake analysis of plates restingon elastic foundationsrdquo Journal of Sound and Vibrationvol 212 no 3 pp 499ndash509 1998

[10] H-S Shen J Yang and L Zhang ldquoFree and forced vibrationof Reissner-mindlin plates with free edges resting on elasticfoundationsrdquo Journal of Sound and Vibration vol 244 no 2pp 299ndash320 2001

[11] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996

Mathematical Problems in Engineering 15

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

[12] K M Liew and T M Teo ldquoDifferential cubature method foranalysis of shear deformable rectangular plates on Pasternakfoundationsrdquo International Journal of Mechanical Sciencesvol 44 no 6 pp 1179ndash1194 2002

[13] J-B Han and K M Liew ldquoNumerical differential quadraturemethod for Reissnermindlin plates on two-parameterfoundationsrdquo International Journal of Mechanical Sciencesvol 39 no 9 pp 977ndash989 1997

[14] D Zhou Y K Cheung S H Lo and F T K Au ldquoree-dimensional vibration analysis of rectangular thick plates onpasternak foundationrdquo International Journal for NumericalMethods in Engineering vol 59 no 10 pp 1313ndash1334 2004

[15] S Chucheepsakul and B Chinnaboon ldquoAn alternative do-mainboundary element technique for analyzing plates ontwo-parameter elastic foundationsrdquo Engineering Analysis withBoundary Elements vol 26 no 6 pp 547ndash555 2002

[16] O Civalek and M H Acar ldquoDiscrete singular convolutionmethod for the analysis of mindlin plates on elastic foun-dationsrdquo International Journal of Pressure Vessels and Pipingvol 84 no 9 pp 527ndash535 2007

[17] A J M Ferreira C M C Roque A M A NevesR M N Jorge and C M M Soares ldquoAnalysis of plates onPasternak foundations by radial basis functionsrdquo Computa-tional Mechanics vol 46 no 6 pp 791ndash803 2010

[18] D Shahsavari M Shahsavari Li Li and B Karami ldquoA novelquasi-3D hyperbolic theory for free vibration of FG plateswith porosities resting on WinklerPasternakKerr founda-tionrdquo Aerospace Science and Technology vol 72 pp 134ndash1492018

[19] A Zenkour and A Radwan ldquoFree vibration analysis ofmultilayered composite and soft core sandwich plates restingon Winkler-Pasternak foundationsrdquo Journal of SandwichStructures amp Materials vol 20 no 2 pp 169ndash190 2018

[20] N D Duc D H Bich and P H Cong ldquoNonlinear thermaldynamic response of shear deformable FGM plates on elasticfoundationsrdquo Journal of ermal Stresses vol 39 no 3pp 278ndash297 2016

[21] A Mahmoudi S Benyoucef A Tounsi A BenachourE A Adda Bedia and SMahmoud ldquoA refined quasi-3D sheardeformation theory for thermo-mechanical behavior offunctionally graded sandwich plates on elastic foundationsrdquoJournal of Sandwich Structures amp Materials vol 21 no 6pp 1906ndash1929 2019

[22] N D Duc J Lee T Nguyen-oi and P T ang ldquoStaticresponse and free vibration of functionally graded carbonnanotube-reinforced composite rectangular plates resting onWinkler-Pasternak elastic foundationsrdquo Aerospace Scienceand Technology vol 68 pp 391ndash402 2017

[23] P-T ang T Nguyen-oi and J Lee ldquoClosed-form ex-pression for nonlinear analysis of imperfect sigmoid-FGMplates with variable thickness resting on elastic mediumrdquoComposite Structures vol 143 pp 143ndash150 2016

[24] T Banh-ien H Dang-Trung L Le-Anh V Ho-Huu andT Nguyen-oi ldquoBuckling analysis of non-uniform thicknessnanoplates in an elastic medium using the isogeometricanalysisrdquo Composite Structures vol 162 pp 182ndash193 2017

[25] J Kim K K Zur and J N Reddy ldquoBending free vibrationand buckling of modified couples stress-based functionallygraded porous micro-platesrdquo Composite Structures vol 209pp 879ndash888 2019

[26] S Coskun J Kim and H Toutanji ldquoBending free vibrationand buckling analysis of functionally graded porous micro-plates using a general third-order plate theoryrdquo Journal ofComposites Science vol 3 no 1 p 15 2019

[27] D Chen J Yang and S Kitipornchai ldquoElastic buckling andstatic bending of shear deformable functionally graded porousbeamrdquo Composite Structures vol 133 pp 54ndash61 2015

[28] A S Rezaei and A R Saidi ldquoApplication of Carrera UnifiedFormulation to study the effect of porosity on natural fre-quencies of thick porous-cellular platesrdquo Composites Part BEngineering vol 91 pp 361ndash370 2016

[29] A S Rezaei and A R Saidi ldquoExact solution for free vibrationof thick rectangular plates made of porous materialsrdquoComposite Structures vol 134 pp 1051ndash1060 2015

[30] J Zhao F Xie A Wang C Shuai J Tang and Q Wang ldquoAunified solution for the vibration analysis of functionallygraded porous (FGP) shallow shells with general boundaryconditionsrdquo Composites Part B Engineering vol 156pp 406ndash424 2019

[31] J Zhao F Xie A Wang C Shuai J Tang and Q WangldquoVibration behavior of the functionally graded porous (FGP)doubly-curved panels and shells of revolution by using a semi-analytical methodrdquo Composites Part B Engineering vol 157pp 219ndash238 2019

[32] Q Li D Wu X Chen L Liu Y Yu and W Gao ldquoNonlinearvibration and dynamic buckling analyses of sandwich func-tionally graded porous plate with graphene platelet rein-forcement resting on Winkler-Pasternak elastic foundationrdquoInternational Journal of Mechanical Sciences vol 148pp 596ndash610 2018

[33] S Sahmani M M Aghdam and T Rabczuk ldquoNonlocal straingradient plate model for nonlinear large-amplitude vibrationsof functionally graded porous micronano-plates reinforcedwith GPLsrdquo Composite Structures vol 198 pp 51ndash62 2018

[34] D Wu A Liu Y Huang Y Huang Y Pi and W GaoldquoDynamic analysis of functionally graded porous structuresthrough finite element analysisrdquo Engineering Structuresvol 165 pp 287ndash301 2018

[35] P T ang T Nguyen-oi D Lee J Kang and J LeeldquoElastic buckling and free vibration analyses of porous-cel-lular plates with uniform and non-uniform porosity distri-butionsrdquo Aerospace Science and Technology vol 79pp 278ndash287 2018

[36] G R Liu T Nguyen-oi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free andforced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[37] T Nguyen-oi G R Liu and H Nguyen-Xuan ldquoAn n-sidedpolygonal edge-based smoothed finite element method (nES-FEM) for solid mechanicsrdquo International Journal for Nu-merical Methods in Biomedical Engineering vol 27 no 9pp 1446ndash1472 2011

[38] T Nguyen-oi G R Liu H C Vu-Do and H Nguyen-Xuan ldquoAn edge-based smoothed finite element method forvisco-elastoplastic analyses of 2D solids using triangularmeshrdquo Computational Mechanics vol 45 no 1 pp 23ndash442009

[39] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method (ES-FEM) for analysis of two-dimensional piezoelectric struc-turesrdquo Smart Materials and Structures vol 18 p 12 2009

[40] T N anh G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoAn edge-based smoothed finite element method forprimal-dual shakedown analysis of structuresrdquo InternationalJournal for Numerical Methods in Engineering vol 82 no 7pp 917ndash938 2010

[41] T Nguyen-oi P Phung-Van T Rabczuk H Nguyen-Xuan and C Le-Van ldquoAn application of the ES-FEM in solid

16 Mathematical Problems in Engineering

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17

domain for dynamic analysis of 2D fluid-solid interactionproblemsrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340003 2013

[42] T Nguyen-oi P Phung-Van V Ho-Huu and L Le-AnhldquoAn edge-based smoothed finite element method (ES-FEM)for dynamic analysis of 2D Fluid-Solid interaction problemsrdquoKSCE Journal of Civil Engineering vol 19 no 3 pp 641ndash6502015

[43] C V Le H Nguyen-Xuan H Askes T Rabczuk andT Nguyen-oi ldquoComputation of limit load using edge-basedsmoothed finite element method and second-order coneprogrammingrdquo International Journal of ComputationalMethods vol 10 no 1 Article ID 1340004 2013

[44] H Nguyen-Xuan L V Tran T Nguyen-oi and H C Vu-Do ldquoAnalysis of functionally graded plates using an edge-based smoothed finite element methodrdquo Composite Struc-tures vol 93 no 11 pp 3019ndash3039 2011

[45] H H Phan-Dao H Nguyen-Xuan C ai-HoangT Nguyen-oi and T Rabczuk ldquoAn edge-based smoothedfinite element method for analysis of laminated compositeplatesrdquo International Journal of Computational Methodsvol 10 no 1 Article ID 1340005 2013

[46] P-S Lee and K-J Bathe ldquoDevelopment of MITC isotropictriangular shell finite elementsrdquo Computers amp Structuresvol 82 no 11-12 pp 945ndash962 2004

[47] T Chau-Dinh Q Nguyen-Duy and H Nguyen-XuanldquoImprovement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysisrdquo ActaMechanica vol 228 no 6 pp 2141ndash2163 2017

[48] T-K Nguyen V-H Nguyen T Chau-Dinh T P Vo andH Nguyen-Xuan ldquoStatic and vibration analysis of isotropicand functionally graded sandwich plates using an edge-basedMITC3 finite elementsrdquo Composites Part B Engineeringvol 107 pp 162ndash173 2016

[49] Q-H Pham T-V Tran T-D Pham and D-H Phan ldquoAnedge-based smoothed MITC3 (ES-MITC3) shell finite ele-ment in laminated composite shell structures analysisrdquo In-ternational Journal of Computational Methods vol 15 no 7Article ID 1850060 2018

[50] Q-H Pham T-D Pham V T Quoc and D-H PhanldquoGeometrically nonlinear analysis of functionally gradedshells using an edge-based smoothed MITC3 (ES-MITC3)finite elementsrdquo Engineering with Computers vol 33 pp 1ndash14 2019

[51] D Pham-Tien H Pham-Quoc T Vu-Khac and N Nguyen-Van ldquoTransient analysis of laminated composite shells usingan edge-based smoothed finite element methodrdquo in Pro-ceedings of the International Conference on Advances inComputational Mechanics 2017 pp 1075ndash1094 SpringerBerlin Germany 2018

[52] K-U Bletzinger M Bischoff and E Ramm ldquoA unified ap-proach for shear-locking-free triangular and rectangular shellfinite elementsrdquo Computers amp Structures vol 75 no 3pp 321ndash334 2000

[53] T Nguyen-oi P Phung-Van H Nguyen-Xuan andC ai-Hoang ldquoA cell-based smoothed discrete shear gapmethod using triangular elements for static and free vibrationanalyses of Reissner-Mindlin platesrdquo International Journal forNumerical Methods in Engineering vol 91 no 7 pp 705ndash7412012

[54] K-J Bathe and E N Dvorkin ldquoA formulation of general shellelements-the use of mixed interpolation of tensorial com-ponentsrdquo International Journal for Numerical Methods inEngineering vol 22 no 3 pp 697ndash722 1986

Mathematical Problems in Engineering 17