Composite Structures 108 (2014) 848–855

Contents lists available at ScienceDirect

Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Technical Note

Hygrothermal effects on the free vibration and buckling of laminatedcomposites with cutouts

0263-8223/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2013.10.009

⇑ Corresponding author. Address: School of Civil & Environmental Engineering,The University of New South Wales, Sydney, NSW 2052, Australia. Tel.: +61 293855030.

E-mail address: sundararajan.natarajan@gmail.com (S. Natarajan).

Sundararajan Natarajan a,⇑, Pratik S. Deogekar b, Ganapathi Manickam c, Salim Belouettar d

a School of Civil & Environmental Engineering, The University of New South Wales, Sydney, Australiab Department of Civil Engineering, Indian Institute of Technology, Bombay, Indiac Stress & DTA, IES-Aerospace, Mahindra Satyam Computer Services Ltd., Bangalore, Indiad Henri Tudor Research Center, 29JFK, Avenue John F Kennedy, L-195 Luxembourg, Luxembourg

a r t i c l e i n f o

Article history:Available online 19 October 2013

Keywords:VibrationBucklingReissner–Mindlin plateExtended finite element methodHygrothermal effectsLaminated composites.

a b s t r a c t

The effect of moisture concentration and the thermal gradient on the free flexural vibration and bucklingof laminated composite plates are investigated. The effect of a centrally located cutout on the globalresponse is also studied. The analysis is carried out within the framework of the extended finite elementmethod. A Heaviside function is used to capture the jump in the displacement and an enriched shear flex-ible 4-noded quadrilateral element is used for the spatial discretization. The formulation takes intoaccount the transverse shear deformation and accounts for the lamina material properties at elevatedmoisture concentrations and temperature. The influence of the plate geometry, the geometry of the cut-out, the moisture concentration, the thermal gradient and the boundary conditions on the free flexuralvibration is numerically studied.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Fiber reinforced laminated composites belong to a class of engi-neered materials that has found increased utilization as structuralelements in the construction of aeronautical and aerospace vechi-cles, sports, as well as civil and mechanical structures. This is be-cause of their excellent strength-to and stiffness-to-weight ratiosand the possibility to tailor their properties to optimize the struc-tural response. However, the analysis of such structures is verydemanding due to coupling between membrane, torsion and bend-ing strains; weak transverse rigidities; and discontinuity of themechanical characteristics through the thickness of the laminates.The application of analytical/numerical methods based on various2D theories have attracted the attention of the research commu-nity. In general, three different approaches have been used to studylaminated composite structures: single layer theories, discretelayer theories and mixed plate theory. In the single layer theory ap-proach, layers in laminated composites are assumed to be oneequivalent single layer (ESL), whereas in the discrete layer theoryapproach, each layer is considered in the analysis. Although thediscrete layer theories provide very accurate prediction of thedisplacements and the stresses, increasing the number of layers

increases the number of unknowns. This can be prohibitively costlyand significantly increase the computational time [34]. To over-come the above limitation, zig-zag models developed by Muruka-mi [16] can satisfy the transverse shear stresses continuityconditions at the interfaces. Moreover, the number of unknownsare independent of the number of layers. Carrera and Demasi[6,17] and Carrera [5] derived a series of axiomatic approaches,coined as ‘Carrera Unified Formulation’ (CUF) for the generaldescription of two-dimensional formulations for multilayeredplates and shells. With this unified formulation it is possible toimplement in a single software a series of hierarchical formula-tions, thus affording a systematic assessment of different theories,ranging from simple ESL models up to higher order layerwisedescriptions. This formulation is a valuable tool for gaining a deepinsight into the complex mechanics of laminated structures. Theinvestigation of the static and the dynamic characteristics of lam-inated composites is fairly well covered in the literature. Existingapproaches employ finite element based on Lagrange basis func-tions [8], meshfree methods [7,4], smoothed finite element meth-ods [22,25] and very recently iso-geometric analysis [32,12]. Theabove mentioned list is no way complete, for a detailed overviewinterested readers are referred to [13] and references therein.

Plates with cutouts are extensively used in transport vehiclestructures. Cutouts are made to lighten the structure, for ventila-tion, to provide accessibility to other parts of the structures andfor altering the resonant frequency. Therefore, the natural frequen-cies of plates with cutouts are of considerable interest to designers

Fig. 1. Coordinate system of a rectangular laminated plate.

S. Natarajan et al. / Composite Structures 108 (2014) 848–855 849

of such structures. Most of the earlier investigations on plates withcutouts have been confined to isotropic plates [23,1,11] and lami-nated composites [29,30]. Moreover, the laminated compositesmay be subjected to moisture and temperature environment dur-ing its service life. The moisture concentration and thermal envi-ronment can have significant impact on the response of suchlaminated structures. Whitney and Ashton [33] employed Ritzmethod to analyze the effect of environment on the free vibrationof symmetric laminates. Patel et al. [24] employed shear flexibleQ8 quadrilateral element to study the hygrothermal effects onthe structural behavior of thick composite laminates. Patel et al.employed higher order accurate theory and studied the importanceof retaining higher order terms in the displacement approximation.The effect of cutouts on the buckling behavior of laminated com-posites were studied in [21,28]. And more recently, Aydin Komuret al. [14] and Ghannadpour et al. [10] studied the buckling behav-ior of laminated composites with circular and elliptical cutoutsusing the finite element method and first order shear deformationtheory. Their study was restricted to a limited number of configu-rations, because the mesh has to conform to the geometry. More-over, to the author’s knowledge the effect of cutout on the freevibration and buckling behavior of laminated composites in hygro-thermal environment has not been studied earlier or was limited tosimple configurations. In this study, we present a framework thatprovides flexibility to handle internal discontinuities. Without lossof generality, we only present the results for standard cutouts.Interactions of cutouts and cracks emanating from cutouts can behandled within this framework. For more details, interested read-ers are referred to [26] and references therein.

In this paper, we study the influence of a centrally located cutouton the fundamental frequency and the critical load of multilayeredcomposite laminated plates in hygrothermal environment. Bothcircular and elliptical cutouts are considered for the study. A struc-tured quadrilateral mesh is used and the cutouts are modelled inde-pendent of the mesh within the extended finite element (XFEM)framework. A systematic parametric study is carried out to bringthe effect of the boundary conditions, the thermal gradient DT, thechange in moisture concentration DC, the geometry of the cutouton the free flexural vibration and buckling of laminated composites.

2. Theoretical formulation

Using the Mindlin formulation, the displacements u, v, w at apoint (x, y, z) in the plate (see Fig. 1) from the medium surfaceare expressed as functions of the mid-plane displacements uo, vo,wo and independent rotations bx, by of the normal in yz and xzplanes, respectively, as

uðx; y; z; tÞ ¼ uoðx; y; tÞ þ zbxðx; y; tÞvðx; y; z; tÞ ¼ voðx; y; tÞ þ zbyðx; y; tÞwðx; y; z; tÞ ¼ woðx; y; tÞ ð1Þ

where t is the time. The strains in terms of mid-plane deformationcan be written as:

e ¼ep

0

� �þ

zeb

es

� �� feog ð2Þ

The midplane strains ep, the bending strains eb and the shearstrains es in Eq. (2) are written as:

ep ¼uo;x

vo;y

uo;y þ vo;x

8><>:

9>=>;; eb ¼

bx;x

by;y

bx;y þ by;x

8><>:

9>=>;; es ¼

bx þwo;x

by þwo;y

( )

ð3Þ

where the subscript ‘comma’ represents the partial derivative withrespect to the spatial coordinate succeeding it. The strain vectorfeog due to temperature and moisture is represented as:

eo ¼exx

eyy

exy

8><>:

9>=>; ¼ DT

ax

ay

axy

8><>:

9>=>;þ DC

cx

cy

cxy

8><>:

9>=>; ð4Þ

where DT and DC are the rise in temperature and the moisture con-centration, respectively. ax, ay and axy are the thermal expansioncoefficients in the plate coordinates and can be related to the ther-mal coefficients (a1, a2 and a3) in the material principal directionsand cx, cy and cxy are the moisture expansion coefficients similarto thermal expansion coefficients in the plate coordinates. The con-stitutive relations for an arbitrary layer k in the laminate (x, y, z)coordinate system can be expressed as:

r ¼rxx

ryy

rxy

8><>:

9>=>; ¼ ½Q k�

ep

0

� �þ

zeb

es

� �� feog

� �ð5Þ

where the terms of ½Qk� matrix of kth ply are referred to the lami-nate axes and can be obtained from the [Qk] corresponding to thefiber directions with the appropriate transformations. The govern-ing equations are obtained by applying Lagrangian equations ofmotion:

ddt

@ðT � UÞ@ _di

� �� @ðT � UÞ

@di

� �¼ 0; i ¼ 1; . . . ;n ð6Þ

where T is the kinetic energy, given by:

TðdÞ ¼ 12

ZXfpð _u2

o þ _v2o þ _w2

oÞ þ Ið _b2x þ _b2

yÞgdX ð7Þ

where p ¼R h=2�h=2 q dz, I ¼

R h=2�h=2 z2q dz and q is the mass density of the

plate. The strain energy function U is given by:

UðdÞ ¼ 12

ZZ Xn

k¼1

Z hkþ1

hk

rTe dz

" #dx dy ð8Þ

where d = {u, v, w, bx, by} is the vector of the degrees of freedomassociated to the displacement field in a finite element discretiza-tion. Substituting Eqs. (7) and (8) in Lagrange’s equations of motionwith the constitutive equations and following the procedure givenin [27], the following discretized equation is obtained:

M€dþ ½Kþ KR þ KG�d ¼ fT ð9Þ

where K is the global linear stiffness matrix, KR and KG are the glo-bal geometric stiffness due to the residual stresses and the appliedin-plane mechanical loads, respectively, M is the global mass matrixand fT is the global hygrothermal load vector. After substituting the

Table 2Elastic moduli of graphite/epoxy lamina at different moisture concentrations,G13 = G12, G23 = 0.5G12, m12 = 0.3, c1 = 0 and c2 = 0.44.

Elastic moduli (GPa) Moisture concentration C (%)

0.0 0.25 0.50 0.75 1.00 1.25 1.50

E1 130 130 130 130 130 130 130E2 9.50 9.25 9.00 8.75 8.50 8.50 8.50G12 6.0 6.0 6.0 6.0 6.0 6.0 6.0

Table 3

850 S. Natarajan et al. / Composite Structures 108 (2014) 848–855

characteristic of the time function [9] €d ¼ �x2d, the following alge-braic equation is obtained:

Static bending: Kd = fT

Free vibration: [(K + KR) �x2M]d = 0Buckling: [(K + KR) � kKG]d = 0

where x is the natural frequency and k is the buckling load. Theresidual stress state depends on the ply lay-up. Hence, to evaluatethe stress state, pre-buckling displacement field for the assumedhygro-thermal–mechanical load is obtained by solving static bend-ing. The displacement field is then used to calculate the stressesand in turn, KR and KG in matrices.

3. Spatial discretization

The plate element employed here is a Co continuous shear flex-ible field consistent element with five degrees of freedom (uo, vo,wo, bx, by) at four nodes in a 4-noded quadrilateral (QUAD-4) ele-ment. The displacement field within the element is approximatedby:

fueo;v

eo;w

eo; b

ex; b

eyg ¼

X4

J¼1

NJfuoJ;voJ;woJ ;bxJ ;byJg ð10Þ

where uoJ, voJ, woJ, bxJ, byJ are the nodal variables and NJ are the shapefunctions for the bi-linear QUAD-4 element. If the interpolationfunctions for a QUAD-4 are used directly to interpolate the five vari-ables (uo, vo, wo, bx, by) in deriving the shear strains and the mem-brane strains, the element will lock and show oscillations in theshear and the membrane stresses. The oscillations are due to thefact that the derivative functions of the out-of plate displacement,wo do not match that of the rotations (bx, by) in the shear strain def-inition, given by Eq. (3). To alleviate the locking phenomenon, theterms corresponding to the derivative of the out-of plate displace-ment, wo must be consistent with the rotation terms, bx and by.

Table 1Integration rules for enriched and non-enriched ele-ments in the presence of a cutout.

Element type Gauß points

Non-enriched element 4Split element 3 per triangleSplit blending element 4

(a)

Fig. 2. Plate with a centrally located circular and an elliptical cutout. r is the radius of

The present formulation, when applied to thin plates, exhibits shearlocking. In this study, field redistributed shape functions are used toalleviate the shear locking [31,18]. The field consistency requiresthat the transverse shear strains and the membrane strains mustbe interpolated in a consistent manner. Thus, the bx and by termsin the expressions for the shear strain es have to be consistent withthe derivative of the field functions, wo,x and wo,y.

3.1. Representation of discontinuity surface

The finite element framework requires the underlying finite ele-ment mesh to conform to the discontinuity surface. The recent intro-duction of implicit boundary definition-based methods, viz., theextended/generalized FEM (XFEM/GFEM), alleviates the shortcom-ings associated with the meshing of the discontinuity surface. In thisstudy, the partition of unity framework is employed to represent thediscontinuity surface independent of the underlying mesh.

ðuh;vh;wh; bhx ; b

hyÞðxÞ ¼

XI2N fem NIðxÞðus

I ;vsI ;w

sI ;b

sxI;b

syIÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

FEM

þX

J2N c NJðxÞHðxÞðbuJ ; b

vJ ; b

wJ ; b

bxJ ; b

byJ Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Enriched part

ð11Þ

(b)

the circular cutout, 2d and 2e are the major and minor axes defining the ellipse.

Elastic moduli of graphite/epoxy lamina at different temperatures, G13 = G12,G23 = 0.5G12, m12 = 0.3, a1 = � 0.3 � 10�6/K and a2 = 28.1 � 10�6/K.

Elastic moduli (GPa) Temperature T (K)

300 325 350 375 400 425

E1 130 130 130 130 130 130E2 9.50 8.50 8.00 7.50 7.00 6.75G12 6.0 6.0 5.5 5.0 4.75 4.50

40

50

60

70

80

90

100

naliz

ed m

ode

1 fr

eque

ncy

S. Natarajan et al. / Composite Structures 108 (2014) 848–855 851

where N fem is a set of all the nodes in the finite element mesh andN c is a set of nodes that are enriched with the Heaviside function. InEq. (11), ðus

I ; vsI ;w

sI ;b

sxI; bs

yIÞ are the nodal unknown vectors associ-

ated with the continuous part of the finite element solution, bJ isthe nodal enriched degree of freedom vector associated with theHeaviside (discontinuous) function. In this study, a level set ap-proach is followed to model the cutouts. The geometric interface(for example, the boundary of the cutout) is represented by the zerolevel curve / � /(x, t) = 0. The interface is located from the value ofthe level set information stored at the nodes. The standard FE shapefunctions can be used to interpolate / at any point x in the domainas:

/ðxÞ ¼X

I

NIðxÞ/I ð12Þ

where the summation is over all the nodes in the connectivity of theelements that contact x and /I are the nodal values of the level setfunction. For circular cutout, the level set function is given by:

/I ¼ jjxI � xcjj � rc ð13Þ

where xc and rc are the center and the radius of the cutout. For anelliptical cutout oriented at an angle h, measured from the x-axisthe level set function is given by:

/I ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1ðxI � xcÞ2 � a2ðxI � xcÞðyI � ycÞ þ a3ðyI � ycÞ

2q

� 1 ð14Þ

where

Table 4Comparison of natural frequency and critical load for four layered cross-ply laminateswith a/h = 100.

Mesh Frequency, X ¼ xa2ffiffiffiffiffiffiffiffiffiffi

qðE2 h2Þ

qNcr

xx ¼ kcr=Kþcr

C = 0.1% T = 325 K C = 0.1% T = 325 K

10 � 10 9.6133 8.2604 0.6158 0.457120 � 20 9.4596 8.0926 0.6100 0.448830 � 30 9.4345 8.0651 0.6090 0.447540 � 40 9.4260 8.0559 0.6087 0.4393Ref. [33] 9.4110 8.0680 0.6091 0.4477Ref. [24] 9.3993 8.0531 0.6084 0.4466

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.410

20

30

40

50

60

70

80

90

100

a/b

Non

-dim

ensi

onal

ized

mod

e 1

freq

uenc

y

a/h=10

a/h=20

Fig. 3. Influence of the plate aspect ratio on the fundamental frequency of a simplysupported laminated plate with (0�/90�/90�/0�) and a centrally located circularcutout, ro/a = 0.2. The laminated plate is exposed to a moisture concentration, C = 1%and T = 300 K.

a1 ¼cos h

d

� 2

; a2 ¼ 2 cos h sin h1

d2 �1e2

� ;

a3 ¼sin h

d

� 2

þ cos he

� 2

ð15Þ

where d and e are the major and minor axes of the ellipse and (xc, yc)is the center of the ellipse.

3.2. Numerical integration over enriched elements

A consequence of adding custom tailored enrichment functionsto the FE approximation basis, which are not necessarily smoothfunctions is that, special care has to be taken in numerically

0 0.2 0.4 0.6 0.8 1 1.2 1.40

10

20

30

Moisture Concentration, C (%)

Non

-dim

ensi

o

a/h=10 (SSSS)a/h=100 (SSSS)a/h=10 (CCCC)

a/h=100 (CCCC)

Fig. 4. Normalized fundamental frequency as a function of moisture concentration,C (%) for a simply supported square laminated plate with (0�/90�/90�/0�) and acentrally located circular cutout, ro/a = 0.2 and for various plate thickness.

0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.410

20

30

40

50

60

70

80

Cutout radius ro/a

Non

-dim

ensi

onal

ized

mod

e 1

freq

uenc

y

T=300K

T=400K

Fig. 5. Non-dimensionalized mode 1 frequency as a function of cutout radius ro/afor a simply supported square laminated plate with a/h = 10 and (0�/90�/90�/0�).The plate is exposed to different thermal environment and moisture concentrationC = 0%.

852 S. Natarajan et al. / Composite Structures 108 (2014) 848–855

integrating over the elements that are intersected by the discon-tinuity surface. The standard Gauß quadrature cannot be appliedin elements enriched by discontinuous terms, because Gaußquadrature implicitly assumes a polynomial approximation.One potential solution for the purpose of numerical integrationis by partitioning the elements into subcells (to triangles forexample) aligned to the discontinuous surface in which the inte-grands are continuous and differentiable [2]. The other tech-niques that can be employed are Schwarz Christoffel Mapping[19,20], Generalized quadrature [15] and Smoothed eXtendedFEM [3]. In the present study, a triangular quadrature withsub-division is employed along with the integration rules de-scribed in Table 1. For the elements that are not enriched, astandard 2 � 2 Gaussian quadrature rule is used.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 225

30

35

40

45

50

55

Size of the ellipse, d/e

=0o

=45o

0 20 40 60 8038

40

42

44

46

48

50

52

Orientation of the ellipse,

Non

-dim

ensi

onal

ized

mod

e1 f

requ

ency

a/h=10, C=0%, T=300K

Non

-dim

ensi

onal

ized

mod

e1 f

requ

ency

Fig. 6. Influence of the geometry and the orientation of the cutout on the linearfrequency for a square (0�/90�/90�/0�) plate with a/h = 10. The plate is exposed totemperature T = 300 K and moisture concentration, C = 0%.

4. Numerical examples

In this section, we study the influence of a centrally located cut-out on the fundamental frequencies of FGM plates. We considerboth square and rectangular plates with simply supported andclamped boundary conditions. Two different cutout shapes, viz.,circular and elliptical cutouts (see Fig. (2)) are considered in thisstudy. Although the formulation presented here is general, theanalysis is carried out for cross-ply laminates subjected to uniformdistributions of moisture and/or temperature. The lamina proper-ties at the elevated moisture concentration and temperature is gi-ven in Tables 2 and 3. The effect of the plate slenderness ratio a/h,the plate aspect ratio b/a, the cutout radius r/a, the cutout geome-try d/e and the boundary condition on the natural frequencies andthe critical buckling load are numerically studied. The boundaryconditions for simply supported and clamped cases are:

Simply supported boundary condition:

uo ¼ wo ¼ by ¼ 0 on x ¼ 0; a; vo ¼ wo ¼ bx ¼ 0 on y ¼ 0; b ð16Þ

Clamped boundary condition:

uo ¼ wo ¼ by ¼ vo ¼ bx ¼ 0 on x ¼ 0; a & y ¼ 0; b ð17Þ

4.1. Validation

Before proceeding with a detailed study on the effect of differ-ent parameters on the natural frequency and the critical load, theformulation developed herein is validated against available closedform/analytical solutions. The critical load and the natural fre-quency of a cross-ply laminate exposed to moisture and tempera-ture are presented in Table 4, along with the Ritz solutions [33] andwith Q8 element [24]. It can be seen that the results from the pres-ent formulation compare very well with the available solutions andbased on a progressive mesh refinement, a structured quadrilateralmesh of 40 � 40 is found to be adequate to model the full laminatefor the present analysis. From the Table 4, it is seen that the per-centage difference between a structured 30 � 30 and 40 � 40 meshis less than 0.1%, hence, a structured mesh of 30 � 30 is used forthe analysis. Next, through the present formulation, the influence

0 10 20 30 40 50 60 70 80 9032

34

36

38

40

42

44

46

Orientation of the ellipse,

Non

-dim

ensi

onal

ized

mod

e1 f

requ

ency

θ=0o θ=45o θ=90o

Fig. 7. Non-dimensionalized mode 1 frequency as a function of orientation of thecutout w for a simply supported square laminated plate with a/h = 10 and forvarious ply orientations. In this case, only one single lamina is considered and theplate is exposed to temperature T = 300 K and moisture concentration C = 0%.

S. Natarajan et al. / Composite Structures 108 (2014) 848–855 853

of various parameters on the natural frequency and the criticalload is studied.

4.2. Vibration

Consider a plate with side lengths a and b and thickness h. Alaminated plate with ply sequence (0�/90�/90�/0�) is consideredfor the analysis. In all cases, we present the non-dimensionalizedfree flexural frequencies as, unless specified otherwise:

X ¼ xa2

h

� ffiffiffiffiffiqE2

rð18Þ

Fig. 3 shows the influence of plate aspect ratio a/b on the naturalfrequency for a simply supported laminated plate (0�/90�/90�/0�)with a centrally located circular cutout ro/a = 0.2 and exposed tomoisture concentration C = 1% and temperature T = 300 K. The effectof plate thickness is also shown in Fig. 3. It is seen that increasingthe plate aspect ratio and decreasing the thickness of the plate, in-creases the natural frequency. The effect of moisture concentrationC, the plate thickness a/h and the boundary conditions is shown inFig. 4. Increasing the moisture concentration has greater impact for

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0.2

0.4

0.6

0.8

1

Moisture Concentration C %

Cri

tical

buck

lingl

oad

λ cr/

Λ+ cr

a/ h =10a/ h =20a/ h =30a/ h =40

(a) a/b =1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0.75

0.8

0.85

0.9

0.95

1

Moisture Concentration C %

Cri

tical

buck

lingl

oad

λ cr/

Λ+ cr

a/ h =10a/ h =20a/ h =30a/ h =40

(b) a/b =2

Fig. 8. Influence of moisture concentration (C %) on the normalized critical bucklingload kcr=K

þcr for a square and a rectangular laminated plate (0�/90�/90�/0�). Kþcr is the

critical buckling load of a composite laminate without a cutout.

a plate with smaller thickness and the natural frequency of the lam-inate plate with clamped boundary conditions is greater than thesimply supported plate as expected. The influence of circular cutoutradius and temperature on the natural frequency is shown in Fig. 5.It is seen that increasing the temperature, decreases the fundamen-tal frequency while increasing the cutout radius, the fundamentalfrequency increases. The effect of the geometry of the cutout d/eand the orientation of the cutout w is shown in Fig. 6. It can be seenthat the orientation w of the cutout and the size of the cutout d/ehas strong influence on the fundamental frequencies. The funda-mental frequency decreases with increasing cutout size irrespectiveof the orientation of the cutout. With increasing orientation from 0�to 90�, the frequency decreases and reaches minimum when thecutout is oriented at 60� and with further increase in the orienta-tion, the frequency increases. Next, the effect of the cutout orienta-tion and ply orientation on the fundamental frequency is studied. Inthis case, only one lamina is considered and the results are depictedin Fig. 7. It is seen that for the ply orientation h = 45�, the frequencyis symmetric with respect to cutout orientation w = 45�.

Fig. 9. Influence of moisture concentration (C %), Temperature and a centrallylocated circular cutout on the normalized critical buckling load kcr=K

þcr for a square

laminated plate with (0�/90�/90�/0�). Kþcr is the critical buckling load of a compositelaminate without a cutout.

Fig. 10. Effect of various boundary conditions and the size of a centrally locatedcircular cutout on the normalized critical buckling load kcr=K

þcr for a square

composite laminate with (0�/90�/90�/0�). Kþcr is the critical buckling load of acomposite laminate without a cutout.

854 S. Natarajan et al. / Composite Structures 108 (2014) 848–855

4.3. Buckling

Next, the effect of moisture concentration, the cutout and tem-perature on the critical load is numerically investigated. A lami-nated plate with ply sequence (0�/90�/90�/0�) is considered forthe analysis. In all cases, we present the non-dimensionalized crit-ical load as, unless specified otherwise:

Ncrxx ¼ kcr=K

þcr ð19Þ

where Kþcr is the critical load of the laminate plate without a cutoutand with moisture concentration C = 0% and temperature T = 300 K.Fig. 8 shows the effect of moisture concentration, the plate thick-ness and the aspect ratio on the critical buckling load for a simplysupported laminated plate. It is seen that with increasing moistureconcentration and a/h, the critical buckling load decreases, whilstwith increasing plate aspect ratio, the critical load increases. The ef-fect of cutout radius ro/a, the moisture concentration C and the ther-mal gradient on the critical buckling load is shown in Fig. 9. It isseen that with increasing the cutout radius, the moisture concentra-tion and thermal gradient, the critical buckling load decreases. Thiscan be attributed to the stiffness degradation. Fig. 10 shows theinfluence of various boundary conditions and the size of a centrallylocated circular cutout on the normalized critical buckling loadkcr=K

þcr for a square composite laminate with (0�/90�/90�/0�). It is

seen that with increasing cutout radius, the critical buckling loaddecreases for a simply supported plate, while for a clamped plate,the critical buckling load first decreases and then increases.

5. Conclusion

The extended finite element framework was adopted to studythe hygrothermal effects on the free vibration and buckling of mul-tilayered laminated composites with a centrally located cutout.The formulation developed is general in nature and can handlenon-uniform distributions of moisture and temperature, althoughonly uniform distribution is considered in the present analysis.The broad conclusion that can be made from this parametric studyis that with the increase in the uniform moisture concentrationand the temperature, the reduction in the fundamental natural fre-quency and the critical load need not be linear and could lead toinstability depending on the value of the moisture content,

temperature and side-to-thickness ratio and aspect ratio. It canalso be concluded that the presence of moisture content has negli-gible effect on the fundamental frequency of thick laminated plate.

Acknowledgement

Sundararajan Natarajan would like to acknowledge the finan-cial support of the School of Civil and Environmental Engineering,The University of New South Wales for his research fellowshipsince September 2012.

References

[1] Ali R, Atwal SJ. Prediction of natural frequencies of vibration of rectangularplates with rectangular cutouts. Comput Struct 1980;12:819–23.

[2] Belytschko Ted, Gracie Robert, Ventura Giulio. A review of extended/generalized finite element methods for material model. Model Simul MaterSci Eng 2009;17(4):1–24.

[3] Bordas S, Natarajan S, Kerfriden P, Augarde CE, Mahapatra DR, Rabczuk T, et al.On the performance of strain smoothing for quadratic and enriched finiteelement approximations (XFEM/GFEM/PUFEM). Int J Numer Methods Eng2011;86(4–5):637–66.

[4] Bui Tinh Quoc, Nguyen Minh Ngoc, Zhang Chuanzeng. An efficient meshfreemethod for vibration analysis of laminated composite plates. Comput Mech2011;48:175–93.

[5] Carrera E. Theories and finite elements for multilayered plates and shells: aunified compact formulation with numerical assessment and benchmarking.Arch Comput Meth Eng 2003;10:215–96.

[6] Carrera E, Demasi L. Classical and advanced multilayered plate elements basedupon PVD and RMVT. Part 1: derivation of finite element matrices. Int J NumerMethods Eng 2002;55:191–231.

[7] Ferreira AJM, Roque CMC, Neves AMA, Jorge RMN, Soares CMM. Buckling andvibration analysis of isotropic and laminated plates by radial basis functions.Compos Part B: Eng 2011;42:592–606.

[8] Ganapathi M, Kalyani Amit, Mondal B, Prakash T. Free vibration analysis ofsimply supported composite laminated panels. Compos Struct 2009;90:100–3.

[9] Ganapathi M, Varadan TK, Sarma BS. Nonlinear flexural vibrations of laminatedorthotropic plates. Comput Struct 1991;39:685–8.

[10] Ghannadpour SAM, Najafi A, Mohammadi B. On the buckling behaviour ofcross-ply laminated composite plates due to circular/elliptical cutouts.Compos Struct 2006;75:3–6.

[11] Huang M, Sakiyama T. Free vibration analysis of rectangular plates withvariously-shaped holes. J Sound Vib 1999;226(4):769–86.

[12] Kapoor Hitesh, Kapania Rakesh K, Soni Som R. Interlaminar stress calculationin composite and sandwich plates in nurbs isogeometric finite elementanalysis. Compos Struct 2013;106:537–48.

[13] Khandan Rasoul, Noroozi Siamak, Sewell Philip, Vinney John. The developmentof laminated compoiste plate theories: a review. J Mater Sci 2012;47:5901–10.

[14] Aydin Komur M, Sen Faruk, Atas� Akin, Arslan Nurettin. Buckling analysis oflaminated composite plates with an elliptical/circular cutout using fem. AdvEng Softw 2010;41:161–4.

[15] Mousavi SE, Sukumar N. Numerical integration of polynomials anddiscontinuous functions on irregular convex polygons and polyhedrons.Comput Mech 2011;47(5):535–54.

[16] Murukami H. Laminated composite plate theory with improved in-planeresponses. J Appl Mech 1986;53:661–6.

[17] Carrera E, Demasi L. Classical and advanced multilayered plate elements basedupon PVD and RMVT. Part 2: Numerical implementations. Int J NumerMethods Eng 2002;55:253–91.

[18] Natarajan S, Baiz PM, Bordas SPA, Kerfriden P, Rabczuk T. Natural frequenciesof cracked functionally graded material plates by the extended finite elementmethod. Compos Struct 2011;93:3082–92.

[19] Natarajan S, Bordas S, Mahapatra DR. Numerical integration over arbitrarypolygonal domains based on Schwarz Christoffel conformal mapping. Int JNumer Methods Eng 2009;80(1):103–34.

[20] Natarajan S, Mahapatra DR, Bordas S. Integrating strongand weak discontinuities without integration subcells and exampleapplications in an XFEM/GFEM framework. Int J Numer Methods Eng 2010;83(3):269–94.

[21] Nemeth Michael P. Buckling behaviour of compression-loaded symmetricallylaminated angle-ply plates with holes. AIAA J 1988;26:330–6.

[22] Nguyen-Xuan H, Rabczuk T, Bordas S, Debongnie JF. A smoothed finite elementmethod for plate analysis. Comput Methods Appl Mech Eng 2008;197:1184–203.

[23] Paramasivam P. Free vibration of square plates with square openings. J SoundVib 1973;30:173–8.

[24] Patel BP, Ganapathi M, Makhecha DP. Hygrothermal effects on the structuralbehaviour of thick composite laminates using higher order theory. ComposStruct 2002;56:25–34.

[25] Phan-Dao H, Nguyen-Xuan H, Thai-Hoang C, Nguyen-Thoi T, Rabczuk T. Anedge based smoothed finite element method for analysis of laminatedcomposite plates. Int J Comput Methods 2013;10:340005–1340031.

S. Natarajan et al. / Composite Structures 108 (2014) 848–855 855

[26] Ahmad Akbari R, Natarajan Sundararajan, Bordas Stéphane. Vibration offunctionally graded material plates with cutouts and cracks in thermalenvironment. Key Eng Mater 2013;560:157–80.

[27] Rajasekaran S, Murray DW. Incremental finite element matrices. ASCE J StructDiv 1973;99:423–2438.

[28] Sai Ram KS, Sinha PK. Vibration and buckling of laminated plates with cutoutin hygrothermal environment. AIAA J 1992;30:2353–5.

[29] Reddy JN. Large amplitude flexural vibration of layered composite plates withcutouts. J Sound Vib 1982;83(1):1–10.

[30] Sivakumar K, Iyengar NGR, Deb Kalyanmoy. Optimum design of laminatedcomposite plates with cutouts using a genetic algorithm. Compos Struct 1998;42:265–79.

[31] Somashekar BR, Prathap G, Ramesh Babu C. A field-consistent four-noded laminated anisotropic plate/shell element. Comput Struct 1987;25:345–53.

[32] Thai CH, Ngyuen-Xuan H, Nguyen-Thanh N, Le TH, Rabczuk T. Static freevibration and buckling analysis of laminated composites Reissner–Mindlinplates using NURBS based isogemetric approach. Int J Numer Methods Eng2012;91:571–603.

[33] Whitney JM, Ashton JE. Effect of environment on the elastic response oflayered composite plates. AIAA 1971;9:1708–13.

[34] Wu Z, Chen R, Chen W. Refined laminated composite plate element based onglobal–local higher order shear deformation theory. Compos Struct 2005;70:135–52.