Thermal buckling of hybrid laminated composite plates with a hole

Composite Structures 68 (2005) 247–254

www.elsevier.com/locate/compstruct

Thermal buckling of hybrid laminated composite plates with a hole

Ahmet Avci, €Omer Sinan Sahin *, Mesut Uyaner

Department of Mechanical Engineering, Selc�uk University, 42075 Konya, Turkey

Available online 13 May 2004

Abstract

Thermal buckling analysis of symmetric and antisymmetric cross-ply laminated hybrid composite plates with a hole subjected to

a uniform temperature rise for different boundary conditions are presented in this paper. The first-order shear deformation theory in

conjunction with variational energy method is employed in the mathematical formulation. The eight-node Lagrangian finite element

technique is used for finding the thermal buckling temperatures of hybrid laminates. The effects of hole size, lay-up sequences and

boundary conditions on the thermal buckling temperatures are investigated. The results are shown in graphical form for various

conditions.

� 2004 Published by Elsevier Ltd.

Keywords: Hybrid composite plates; Thermal buckling; Finite element method

1. Introduction

The buckling of fiber reinforced plates is an impor-

tant consideration in the design process in a number of

engineering fields. The buckling problem under thermal

loadings is one of the practical importances for struc-

tures work at elevated temperatures.

Gossard et al. [1] was the first to consider the thermal

buckling of plates and in their work simply supported

rectangular isotropic plate subjected to tent-like tem-perature distribution was investigated. Chang and Shiao

[2] investigated the thermal buckling of isotropic and

composite plates with hole by using both closed form

solution and finite elements method. Thermal buckling

of antisymmetric cross-ply composite laminates was

investigated by Mathew et al. [3]. Abramovich[4] in-

vestigated the thermal buckling of cross-ply symmetric

and nonsymmetrical laminated beams by using first-order deformation theory. Murphy and Ferreira [5]

studied theoretical and experimental approaches to

determine the buckling temperature and buckling mode

for flat rectangular plates. Huang and Tauchert [6]

determined thermal buckling of clamped symmetric

angle-ply laminated plates using a Fourier series ap-

proach and the finite element method. Prabhu and

*Corresponding author. Tel.: +90-332-223-18-98; fax: +90-332-241-

06-51.

E-mail address: [email protected] (€O.S. Sahin).

0263-8223/$ - see front matter � 2004 Published by Elsevier Ltd.

doi:10.1016/j.compstruct.2004.03.017

Dhanaraj [7], Chandrashekhara [8], Thangaratnam and

Ramaohandran [9], and Chen et al. [10] also evaluatedthe thermal buckling of the laminates subjected to uni-

form temperature rise or nonuniform temperature fields

using the finite element approach. Mannini [11] inves-

tigated the thermal buckling of cross-ply laminates

by using first-order shear deformation theory and the

Rayleigh–Ritz method. Dawe and Ge [12] presented

results for thermally loaded shear deformable composite

laminates using the spline finite strip method. Yettramand Brown [13] studied the buckling of square perfo-

rated plates under biaxial loadings using a direct matrix

method. Chang and Hsu [14] evaluated the buckling and

vibration of isotropic plates containing circular or a

square opening. The post buckling analysis of isotropic

and composite square plates involving circular holes

were carried out by Vanden Brink and Kamat [15] by

using finite element method. Larson [16] also used a fi-nite element approach and perturbation method for

investigating the buckling of orthotropic compressed

plates with circular holes. Lin and Kuo [17] investigated

the buckling of cross-ply and angle-ply laminated plates

with circular holes under in plane static loadings. In

their study, they utilized the finite elements method in

order to find the critical buckling loads.

The present paper aims to determine the bucklingtemperature and buckling mode shapes for hybrid

composite laminates with different circular hole diame-

ters using the finite element method. The thermal

mail to: [email protected]

248 A. Avci et al. / Composite Structures 68 (2005) 247–254

buckling of symmetric and antisymmetric cross-ply

laminates with holes is investigated, based on the first-

order deformation theory in conjunction with the vari-

ational energy method. The finite element approachis used for finding the thermal buckling temperatures

for aluminum–glass/epoxy, aluminum–boron/epoxy and

aluminum–boron/epoxy–glass/epoxy hybrid laminates.

The effects of hole diameter, lay-up sequences and

boundary conditions on the thermal buckling tempera-

tures are numerically solved. The results are presented in

graphical form for various boundary conditions.

2. Mathematical formulation

The laminated orthotropic construction of the plate is

consisted of N layers. Each layer is of thickness tk, sothat h ¼

PNk¼1 tk is the total thickness of the laminate.

The longitudinal and lateral dimensions of the lami-

nate are a and b and subjected to uniform temperature

difference DT between ambient and laminated plate asshown Fig. 1. The linear stress–strain relation for each

layer is expressed with x, y-axes and has the form

rx

ry

sxy

8><>:

9>=>;

k

¼Q11Q12Q16

Q12Q22Q26

Q16Q26Q66

264

375

k

ex � axDT

ey � ay DT

cxy � axy DT

8><>:

9>=>;

k

syzsxz

� �¼ Q44Q45

Q45Q55

" #cyzcxz

� � ð1Þ

where rx, ry , sxy , syz and sxz are the stress components,

Qij are transformed reduced stiffnesses, which can be

expressed in terms of the orientation angle and theengineering constant of the material [20]. DT is tem-

y

d

a

b

Nxy

Ny

Fig. 1. Geometry of the pro

perature increase, ax and ay are the coefficients of ther-

mal expansion in directions of x and y axes, respectively.axy is the apparent coefficient of thermal shear, such as

ax ¼ a1 cos2 hþ a2 sin

2 h

ay ¼ a2 cos2 hþ a1 sin

2 h

axy ¼ 2ða1 � a2Þ sin h � cos hð2Þ

a1 and a2 are the thermal expansion coefficients of the

lamina along the longitudinal and transverse directions

of fibers respectively.

In this study first-order shear deformation theory is

used. The displacements u, v and w can be written as

follows:

uðx; y; zÞ ¼ u0ðx; yÞ þ zwxðx; yÞvðx; y; zÞ ¼ v0ðx; yÞ þ zwyðx; yÞwðx; y; zÞ ¼ wðx; yÞ

ð3Þ

where u0, v0, w are the displacements at any point of themiddle surface, and wx, wy are the bending rotations of

normals to the mid plane about the x and y axes,

respectively. The bending strains ex, ey and transverse

shear strains cxy , cyz, cxz at any point of the laminate are

exeycxy

8<:

9=; ¼

ou0oxov0oy

ou0oy þ

ov0ox

��þ z

owx0oxowy

oy

owxoy þ owy

ox

��

��;

cyzcxz

�� ¼

owoy� wy

owox þ wx

�� ð4Þ

The resultant forces Nx, Ny and Nxy , momentsMx,My and

Mxy and shearing forces Qx, Qy per unit length of theplate are given as

2

xx

y

θ

1

x

z

h/2

h/2

Nx

blem and coordinates.

A. Avci et al. / Composite Structures 68 (2005) 247–254 249

Nx Mx

Ny My

Nxy Mxy

264

375 ¼

Z h=2

�h=2

rx

ry

sxy

8><>:

9>=>;ð1; zÞdz

Qx

Qy

� �¼

Z h=2

�h=2

sxzsyz

� �dz

ð5Þ

The total potential energy P of a laminated plate under

thermal loading is equal to

P ¼ Ub þ Us þ V ð6Þwhere Ub is the strain energy of bending, Us is the strainenergy of shear and V represents the potential energy of

in-plane loadings due to temperature change.

Ub ¼1

2

Z h=2

�h=2

Z ZRðrxex

�þ ryey þ sxycxyÞdA

�dz

Us ¼1

2

Z h=2

�h=2

Z ZRðsxzcxz

�þ syzcyzÞdA

�dz

ð7Þ

V ¼ 1

2

Z ZR½N 1ðow=oxÞ2 þ N 2ðow=oyÞ2

þ 2N 12ðow=oxÞðow=oyÞ�dA�ZoRðNb

nu0n þ N

bsu

0s Þds

Here dA ¼ dxdy, R is the region of a plate excluding the

hole. Nbn and N

bs are in-plane loads applied on the

boundary oR.For the equilibrium, the potential energy P must be

stationary. The equilibrium equations of the cross-ply

laminated plate subjected to temperature change can be

derived from the variational principle through use of

stress–strain and strain–displacement relations. One

may obtain these equations by using dP ¼ 0 [17,18].

2.1. Finite element formulations

In general, a closed form solution is difficult to obtain

for buckling problems [17,18]. Therefore numerical

methods are usually used for finding an approximate

solution.

In order to study the buckling of the plate, an eight-

node Lagrangian finite element analysis is applied in this

study. The stiffness matrix of the plate is obtained byusing the minimum potential energy principle. Bending

stiffness [Kb], shear stiffness [Ks] and geometric stiffness

[Kg] matrices can be expressed as

½Kb� ¼ZA½Bb�T½Db�½Bb�dA ð8Þ

½Ks� ¼ZA½Bs�T½Ds�½Bs�dA ð9Þ

and

½Kg� ¼ZA½Bg�T½Dg�½Bg�dA ð10Þ

where

½Db� ¼Aij Bij

Bij Dij

� �½Ds� ¼

k21A44 0

0 k22A55

" #

½Dg� ¼N 1 N 12

N 12 N 2

" #ð11Þ

ðAij;Bij;DijÞ ¼Z h=2

�h=2Qijð1; z; z2Þdz ði; j ¼ 1; 2; 6Þ ð12Þ

ðA44;A55Þ ¼Z h=2

�h=2ðQ44;Q55Þdz ð13Þ

A44 and A55, are the shear correction factors for rectan-

gular cross-section are given by k21 ¼ k22 ¼ 5=6 [19].The total potential energy principle for the plate

satisfies the assembly of the element equations.

The element stiffness and the geometric stiffness

matrices are assembled. The corresponding eigenvalue

problem can be solved using any standard eigenvalue

extraction procedures [10,18]

½½K0� � kb½K0g��uiviwi

8<:

9=; ¼ 0 ð14Þ

where

½K0� ¼ ½Kb� þ ½Ks�;�kb½K0g� ¼ ½Kg� ð15ÞThe product of kb and the initial guest value DT is thecritical buckling temperature Tcr, that is

Tcr ¼ kbDT ð16Þ

3. Numerical result and discussion

There are many techniques to solve eigenvalue

problems. In this study the Newton Raphson method is

applied to obtain numerical solutions of the problem.

For thermal buckling due to a DT temperature change inthe plate, the uniaxial or biaxial in-plane loads are

developed along the rectangular edges, while the circular

hole edge is free.

The cross-ply laminated hybrid composite plates used

here have several thicknesses and bonded symmetrically

and antisymmetrically.

For the computations thermo-elastic properties con-

sidered for the aluminum, E-glass/epoxy, and boron/epoxy composites are given in Table 1. Here, E1 and E2

are elastic moduli in 1 and 2 directions respectively, m12 isPoisson’s ratio and a1 and a2 are thermal expansion

coefficients of the materials used in the solution. The

effect of a12 is neglected.

Stacking sequence of hybrid composite plates have

been taken both symmetric and antisymmetric. Some of

Table 1

Material properties

Material E1 (GPa) E2 (GPa) G12 (GPa) m12 a1 (�C�1) a2 (�C�1)

Aluminum 70 70 52 0.33 23.6· 10�6 23.6· 10�6

E-glass/epoxy 15 6 3 0.3 7.0 · 10�6 2.30· 10�5

Boron/epoxy 207 19 4.8 0.21 4.14· 10�6 1.91· 10�5

Fig. 2. Typical mesh for four edges clamped (CCCC) plate.

0

50

100

150

200

250

0.01 0.015 0.020 0.025 0.01 0.015 0.020 0.025

Tc

o C

CCCC

SSSS

SFSF

CFCF

0

30

60

90

120

150

180

h/bh/b

Tc

oC

CCCC

SSSS

SFSF

CFCF

(a) (b)

Fig. 3. Effect of stacking on buckling temperature for (a) antisym-

metric and (b) symmetric boron–epoxy/glass–epoxy hybrid composites

without hole.


stacking sequences have been represented below. The

letters A, B and G represent, aluminum, boron/epoxy

and glass/epoxy composites respectively.

The sequence of 10 layers symmetric lay up of boron/

epoxy–glass/epoxy is 0�G/90�B/90�B/0�G/90�B/90�B/0�G/90�B/90�B/0�G.

The sequence of 6 layers antisymmetric lay up of

Aluminum–boron/epoxy–glass/epoxy is 0�G/90�B/0�A/0�G/90�B/0�A.

And sequence of 6 layers symmetric lay up of alu-

minum–boron/epoxy–glass/epoxy is 0�G/90�B/0�A/0�A/90�B/0�G.

Each layer has 0.25 mm thickness and the length of

one edge of plate is 100 mm. h=b ratio, represents the

total thickness of composite plate to length of one side

of composite plate and d=b ratio, represents the hole sizeto length of one side of composite plate.

A wide range of boundary conditions can be

accommodated, but only four kinds of boundary con-

ditions are chosen as defined below.

1. Four edges simply supported (SSSS):

At x ¼ � a2;a2; u ¼ w ¼ wy ¼ 0

At y ¼ � b2;b2; v ¼ w ¼ wx ¼ 0

2. Two edges simply supported and two edges are free

(SFSF):

At x ¼ � a2;a2; u ¼ w ¼ wy ¼ 0

3. Four edges clamped (CCCC):

At x ¼ � a2;a2; u ¼ w ¼ wy ¼ wx ¼ 0

At y ¼ � b2;b2; v ¼ w ¼ wx ¼ wy ¼ 0

4. Two edges clamped and two edges are free (CFCF):

At x ¼ � a2;a2; u ¼ w ¼ wy ¼ wx ¼ 0

Fig. 2 shows the meshed plate. Four edges of plate

have divided into 10 parts disregarding the hole size.

The variation of critical buckling temperature Tc withh=b (plate thickness to length of the one edge of the

plate), for boron–epoxy/glass–epoxy hybrid composites

consisting of 4, 6, 8 and 10 layers is shown in Figs. 3

and 4.

For this material, the buckling temperatures for both

symmetric and antisymmetric plates increase as the h=bratio increases. Both plates with hole and plates without

hole show the same behavior for four different boundary

conditions. It is observed that the critical buckling

temperature reaches its maximum value for four edgeclamped (CCCC) plate. This behavior is same for all

hole sizes except for the case d=b ¼ 0:3. The critical

0

100

200

300

400

0.01 0.015 0.02 0.025

h/b

0.01 0.015 0.02 0.025

h/b

Tc

oC

CCCC

SSSS

SFSF

CFCF

0

100

200

300

400

Tc

oC

CCCC

SSSS

SFSF

CFCF

(a) (b)

Fig. 4. Effect of stacking on buckling temperature for (a) antisym-


for d=b ¼ 0:3.

01020304050607080

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

d/b d/b

CCCC

SSSS

SFSF

CFCF

0

10

20

30

40

50

Tc

o C

Tc

o C

CCCC

SSSS

SFSF

CFCF

(a) (b)

Fig. 5. Effect of hole size on buckling temperature for (a) antisym-


composed of four layers.


buckling temperature for the case of d=b ¼ 0:3 reaches

its maximum value for two edge clamped (CFCF) plate.

In Figs. 3 and 4, the curves present a monotonically

increasing Tcr as the size of hole increases. It can be seen

that the critical temperatures of clamped cases are al-ways higher than those of the simply supported plates.

Fig. 5 represents the variation of critical buckling

temperature Tc with d=b for boron/epoxy–glass/epoxy

hybrid composites. For symmetric and antisymmetric

lay up, it is concluded that, there is no change at critical

00 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

20406080

100120140160180

d/b d/b

Tco C

CCCC

SSSS

SFSF

CFCF

020

4060

80100

120

140

Tc

o C

CCCC

SSSS

SFSF

CFCF

(a) (b)


metric and (b) symmetric aluminum/boron–epoxy/glass–epoxy hybrid

composites composed of six layers.

buckling temperature with the variation of hole size for

SFSF and SSSS plates. For four edges clamped (CCCC)

plate, critical temperature gradually increases as the d=bincreases.

Fig. 6 shows the variation of critical buckling tem-

perature Tc versus d=b for aluminum–boron/epoxy–

glass/epoxy six layered hybrid composite plates. It is

concluded that, there are no change at the critical tem-

peratures for four edge simple supported (SSSS) and

two edge simply supported (SFSF) plates with the var-

iation of hole size for both symmetric and antisymmetric

plates. The greatest buckling temperature is reached forfour edge clamped plate at antisymmetric lay up, though

it can be seen a little decrease at small hole sizes. The

behavior of symmetric and antisymmetric single edge

supported plates is different. Symmetric cross-ply lami-

nates does not yield the highest buckling resistance as

usually expected, because of absence of bending–exten-

sion coupling.

The buckling temperature of four edges and twoedges clamped antisymmetric plate remains constant

though at small hole sizes, but, buckling temperatures of

four edges and two edges clamped symmetric plate have

a gradual increment by d=b ratio.

The variation of critical buckling temperature Tc withd=b for aluminum/boron–epoxy hybrid composites is

shown in Fig. 7. The critical temperatures for four edges

simple supported and two edge simply supported platesare not affected by hole size. A gradual increment is seen

at four edge clamped plate. For two edge clamped plate

with hole size of d=b ¼ 0–0.1 the buckling temperature

increases, but buckling temperature is almost constant

for plate with bigger hole sizes. Both symmetric and

antisymmetric plates show almost same behavior.

In Fig. 8, the variation of critical buckling tempera-

ture Tc versus d=b for aluminum/glass–epoxy hybridcomposites are shown. It can be observed that the crit-

ical temperatures for all boundary conditions except for

four edge clamped plates are not affected by hole size.

The critical temperature Tc for four edges clamped plate

rapidly increases as d=b ratio increases.

0

10

20

30

40

50

d/b d/b

Tc

o C

Tc

o C

CCCC

SSSS

SFSF

CFCF

0

10

20

30

40

50

CCCC

SSSS

SFSF

CFCF

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5(a) (b)


metric and (b) symmetric aluminum/boron–epoxy hybrid composites

composed of four layers.

0

10

20

30

40 CCCC

SSSS

SFSF

CFCF

0

10

20

30

40

50

o

CCCC

SSSS

SFSF

CFCF

0 0.1 0.2 0.3 0.4 0.5

d/b

0 0.1 0.2 0.3 0.4 0.5

d/b

Tco C

T co C

(a) (b)


metric and (b) symmetric aluminum/glass–epoxy hybrid composites

with four layers.

0

10

20

30

40

50

60

70Antisym

Sym.

Antisym

Sym.

0

2

4

6

8

10

12

14

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

d/b d/b

Tco C

T co C

(a) (b)

Fig. 9. Relationship between Buckling temperature and hole size of

boron–epoxy/glass–epoxy composite plates composed of four layers

for (a) two edge clamped and (b) two edge simply supported condi-

tions.

050

100150200250300350400450

4 Layer 4 Layer

6 Layer

8 Layer

10 Layer

050

100150200250300350400

6 Layer

8 Layer

10 Layer

0 0.1 0.2 0.3 0.4 0.5d/b

0 0.1 0.2 0.3 0.4 0.5d/b

Tco C

T co C

(a) (b)

Fig. 10. Relationship between buckling temperature and hole size of

boron/epoxy–glass/epoxy composite plates for four edge clamped

(a) antisymmetric, (b) symmetric plates.

020406080

100120140160180

A-G

A-B

B-G

A-B-G

A-G

A-B

B-G

A-B-G

0

30

60

90

120

150

0 0.1 0.2 0.3 0.4 0.5

d/b

0 0.1 0.2 0.3 0.4 0.5

d/b

Tc

o C

Tc

o C

(a) (b)

Fig. 11. Relationship between buckling temperature and hole size of

various composite plates for four edge clamped plates consisting of six

layers (a) antisymmetrical, (b) symmetrical plate.


Fig. 9 depicts the relationship between buckling

temperature and hole size of boron–epoxy/glass–epoxy

composite plates for two edge clamped (a) and two edge

simply supported conditions. As shown in Fig. 9, for

both two edge clamped and two edge simply supported

plates, higher temperature difference is needed for

buckling of antisymmetrical stacked plate than sym-

metric one.Expected results are seen in Fig. 10. As number of

layers increases, the temperature difference for buckling

Fig. 12. First buckling mode shapes for (a) four edge simply supported and

layers stacked antisymmetrically.

increases. This behavior is same for both antisymmetric

and symmetric lay up. But, temperature difference forbuckling of antisymmetrical stacked plate is higher than

symmetric stacked plate.

Fig. 11 shows the relationship between buckling

temperature and hole size of various composite plates

for four edges clamped plates consisting of six layers. As

seen in Fig. 11, for all materials used in the solution, as

the hole size increases, the buckling temperature in-

creases rapidly. For these four composite materials, the

(b) two edge simply supported aluminum–glass/epoxy plates with four

Fig. 13. First buckling mode shapes for (a) two edge clamped and (b) four edge clamped aluminum–glass/epoxy plates with four layers stacked

antisymmetrically.

Fig. 14. (a) First and (b) second buckling mode shapes for two edge clamped aluminum–glass/epoxy plates with four layers stacked antisymmet-

rically.


highest buckling temperature is reached for aluminum–

boron/epoxy–glass/epoxy composites.The first and second buckled mode shapes generated

aluminum–glass/epoxy cross-ply laminated plates with

four boundary conditions are shown in Figs. 12–14. It is

found that critical temperature for four edge clamped

plate is 16.33 �C, for two edge clamped is 14.32 �C, forfour edge simply supported plate is 5.86 �C and for two

edge simply supported plate is 4.15 �C, respectively. As

shown in Fig. 11 critical temperatures are 12.02 formode1 and 18.26 for mode 2 for two edge clamped

aluminum–glass/epoxy plates with four layers stacked

antisymmetrically. The mode shapes presented in Figs.

12–14 show considerable skewing for the laminated

plates.

4. Conclusions

Thermal buckling behaviors of cross-ply laminated

hybrid plates with holes have been examined by em-ploying the first-order shear deformation theory and

finite element technique. Both symmetric and antisy-

metric lay-up sequence are considered and various

boundary conditions are taken in to account.

For boron/epoxy–glass/epoxy hybrid composites,

critical buckling temperature increases as h=b ratio in-

creases.

The boundary condition has a strong impact on the

critical buckling temperature. It is evident that, clampedplates have a much higher buckling temperature because

of stiffer constraints. Four edges clamped plates have the

highest buckling temperature and for this boundary

condition, critical buckling temperature is strongly de-

pend on hole size. It is concluded that there are no

significant change at the critical buckling temperature of

four edges simply supported and two edges simply

supported plates. It can be seen that the critical bucklingtemperatures of clamped cases are always higher than

those of the simply supported plates. For this material,

there is a little difference between symmetric and anti-

symmetric lay up. Because of absence of bending–

extension coupling, symmetric cross-ply laminates does

not yield the highest buckling resistance as usually

expected.

It is also observed that there are no significant vari-ations at the critical buckling temperatures of alumi-

num/boron–epoxy/glass–epoxy, aluminum/glass–epoxy

and aluminum/boron–epoxy hybrid composites for all

boundary conditions used in the solution except for four

edges clamped plates.

It can be seen that the critical buckling temperatures

goes up as the number of layers increases. This behavior

is same for both antisymmetric and symmetric lay up.But, buckling temperatures of antisymmetric stacked

plates are higher than those of symmetric stacked plates.


The highest buckling temperature is obtained for

aluminum–boron/epoxy–glass/epoxy composites.

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Documents

Thermal buckling of hybrid laminated composite plates with a hole