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Int. J Sci. Emerging Tech. Vol-2 No. 2 November, 2011

53

Thermal analysis of Composite Laminated

Plates using Higher-order shear deformation

theory with Zig-Zag FunctionT.Dharma Raju1 and J. Suresh Kumar2

1Dept of Mechanical Engg, Godavari Institute of Engineering &Technology, Rajahmundry-533294, India

2Department of Mechanical Engineering, JNTU college of engineering, Hyderabad-500085,India

[email protected] and [email protected]

Abstract - In this paper an analytical procedure is

developed to investigate the Thermal characteristics of

laminated composite plates under thermal loadingbased on higher-order displacement model with zig-zag

function, with out enforcing zero transverse shear

stresses on the top and bottom faces of the laminated

plates. This function improves slope discontinuities atthe interfaces of laminated composite plates. The

related functions are obtained using the dynamicversion of principle of virtual work or Hamiltons

principle. The solutions are obtained using Naviers andnumerical methods for anti-symmetric cross-ply and

angle-ply laminates with a specific type of simply

supported boundary conditions SS-1 and SS-2. The

Numerical results are presented for anti-symmetriccross-ply and angle-ply laminated plates. All the

solutions presented are close agreement with the theory

of elasticity and available literature.

Keywords: zig-zag function, Thermal analysis, Hamiltons

Principle, cross-ply and angle-ply.

1 IntroductionThe composite laminate plates are straight, planesurface structures whose thickness is slight compare

to other dimensions geometrically, they are boundeither by straight or curved lines. As the prediction ofthe response characteristics of laminated compositestructure is a challenging task depending on their

intrinsic anisotropy, heterogeneity, and low ratio ofthe transverse shear modulus to the in-plane Youngsmodulus. So, it is necessary to analyze the thermal

characteristics of laminated composite plates. Many

plate theories have been developed to analyzelaminated composite plates. Padovon J [1]

investigated the effects of mechanical and thermalloads on the local stationary fields of generallylaminated plates, based on three-dimensional thermo

elasticity theory. Tungikar V.B et al., [2] presented athree dimensional exact solutions for temperaturedistribution and thermal stresses in simply supportedfinite rectangular orthotropic laminates and is

subjected to prescribed boundary conditions undercombined thermal and mechanical loading, whichwill be used to check the accuracy of more

generalized numerical tools. Savoia M and Reddy

J.N. [3] discussed the stress analysis of multilayeredplates subjected to thermal and mechanical loads inthe context of the three-dimensional quasi-statictheory of thermo elasticity. Ali J.S.M et al., [4]

formulated a new displacement based higher-ordertheory. The theory employs realistic displacementvariations and is shown to be accurate for even thick

laminates and for any combination of mechanical andthermal loading. Dafedar J.B. et al., [5] investigatedbuckling response of laminated composite platessubjected to mechanical and hygrothermal loads. Forthis they used analytical mixed theory based on thepotential energy principle. Maenghyo Cho et al., [6]

developed a higher order zig-zag plate theory torefine the predictions of the mechanical, thermal, andelectric behaviors partially coupled. The in-planedisplacement fields are constructed by superimposing

linear zig-zag field to the smooth globally cubicvarying field through the thickness. E.Carrera [7]discussed the use of the Murakamis zig-zag function

in the modeling of layered plates and shells. In this,ZZF modeling was discussed with various loads andcompared with other theories. Kamran Daneshjo et

al., [8] proposed A new mixed finite elementformulation to analyze transient coupledthermoelastic problems. Coupled model of dynamicthermoelasticity is selected for a laminated compositeand a homogeneous isotropic plate. Jinho Oh et al.,[9] developed a higher order zig-zag shell theory

based on general tensor formulation to refine thepredictions of the mechanical, thermal, and electricbehaviors. S.H. Lo et al ., [10] proposed a four-node

quadrilateral plate element based on the globallocal

higher order theory to study the response oflaminated composite plates due to a variation intemperature and moisture concentrations.

2 The Higher-Order ShearsDeformation Theory with ZIG-ZAG

Function:

Consider a rectangular plate of 0 x a; 0 y

b and2

h z

2

h.

The higher-order shear deformation theory withzig-zag function is assumed to be

__________________________________________________________________________

International Journal of Science & Emerging Technologies

IJSET, E-ISSN: 2048 - 8688

Copyright ExcelingTech, Pub, UK (http://excelingtech.co.uk/ )
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54

),(),,(

),(),(

),(),(),(),,(

),(),(

),(),(),(),,(

2

*3

*2

1

*3

*2

yxwzyxw

yxsyxz

yxvzyxzyxvzyxv

yxsyxz

yxuzyxzyxuzyxu

o

ky

oyo

kx

oxo

(1)Where u0, v0 , wo, s1and s2 denote the

displacements of a point (x, y) on the mid-plane.

k is the Zig-Zag function, defined as:

k

kk

kh

Z)1(2

Zkis the local transverse coordinate with itsorigin at the center of the k

thlayer.

hkis the corresponding layer thickness.

x , y are rotations of the normal to the

midplane about y and xaxes

u0*, v0

*,

*

x ,*

y are the higherorder

deformation terms defined at the mid-plane.

The strain components are

**

**2

32

3

ykzyozsyzkyoy

xkzxozsxzkxox

0z

*

*2

**

2

32

xzxzozsxxz

yzyzozsyyz

xykzxyozsxyzkxyoxy

( 2)

The stressstrain relationships in the global x-y-z

coordinate system can be written as

xz

xyxy

yy

xx

L

xz

yz

xy

y

x

T

T

T

Q

Q

QQQ

QQQ

QQQ

yz

L

55

44

332313

232212

131211

0000

0000

00

00

00

(3)The governing equations of displacement model will

be derived using the principle of virtualwork as

0)(0

dtKVUT

(4)The virtual work statement shown in Eq. (4) ,

integrating through the thickness of laminate, the in-plane and transverse force and moment resultantrelations in the form of matrix obtained as:

0

0

|0|0

0||

0||

*

*

*

*

*

0

0

*

*

*

T

T

T

T

s

s

b

t

M

M

N

N

K

K

D

DB

BA

Q

Q

M

M

N

N

(5)

Equating the coefficients of each of virtual

displacements uo v0, w0, x , y , u0*, v0

*,

*

x ,*

y , s1, s2 to zero, the equations of motion

are obtained. These Equations are expressed in terms

of displacements uo, v0, w0,

x ,

y ,u0

*,v0

*,

*

x ,*

y ,s1,s2 by substituting for the force

and moment resultants

2.1 The Naviers Solutions of Simply SupportedAnti Symmetric Cross Ply Laminated Plates:

2.1.1The ss-1 Boundary Conditions for the Anti-

Symmetric Cross Ply Laminated Plates are:

At edges x = 0 and x = a

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v0 = 0, wo = 0, y = 0, Mx = 0, v0*

= 0,*

y =

0, Mx*

= 0, Nx = 0, Nx*

= 0, 02 s 6 (a)

At edges y = 0 and y = b

u0= 0, wo = 0, x = 0, My = 0, u0

*= 0,

*

x = 0,

My*

= 0, Ny = 0, Ny*

= 0, 01 s 6(b)

2.1.2 The SS-2 Boundary Conditions for the anti-

symmetric Angle Ply Laminated Plates are:

At edges x = 0 and x = a

u0= 0, wo = 0, y = 0, Nxy= 0, Mx = 0, u0

*=

0,*

y = 0, Mx*

= 0, Nxy*

= 0 , 01 s 7(a)

At edges y = 0 and y = b

v0 = 0, wo = 0, x = 0, Nxy = 0, My = 0, v0*

=

0,*

x = 0, My*

= 0, Nxy*

= 0, 02 s

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

thickness co ordinate(z/h)

N.trasverse

shea

rstress(

t

xz)

HSDTWZF

HSDT

Fig.2 Non-dimensionalized max.transverse shear stress (t xz) Vs thickness co-ordinate (z/h) for

simply supported anti-symmetric angle -ply laminated square plate

(-45/45)8 ss2

-0.09

-0.06

-0.03

0

0.03

0.06

0.09

0.12

0.15

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

N.T

ransverse

shearst

ress

(xz)

HSDTWZF

HSDT

(0/90)

ss1 a/h=5

Thickness coordinate (z/h)

Fig.1 Non-dimensionalized max.transverse shear stress (txz) Vs thickness co-ordinate (z/h) f or simply supported anti-symmetric cross-ply laminated square

plate

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The displacements at the mid plane will be defined tosatisfy the boundary conditions in Eq.(6) &(7).Thesedisplacements will be substituted in governingequations to obtain the equations in terms of A,B,Dparameters. The obtained equations will be solved to

find the behavior of the laminated composite plates.

3. Results and DiscussionThe simply supported boundary conditions

(SS-1) shown in Eq. (6) are considered for solutions

of anti-symmetric cross-ply laminates and Eq. (7) forsolutions of anti-symmetric angle-ply laminates usinga higher order shear deformation theory with zig- zag

function.

The material properties of graphite epoxy used foreach lamina of the laminated composite plate are:

E1 / E2 = 25, G12 / E2 = 0.5, G23 / E2 = 0.2, 12 /13 =

23 = 0.25, 2 /1 = 1125

The deflections and stresses are presented here innon-dimensional form using the following

multipliers:

M1 = w/h1T S2

M2 = (u, v)/h1T S M3 =

(i, ij) / E21 T

Fig. 1 through 3 shows the plots of the normal andtransverse stresses through the thickness of anti-

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

thickness co ordinate (z/h)N.transverse

shearstre

ss(

t

yz)

HSDTWZF

HSDT

Fig.3 Non-dimensionalized max.transverse shear stress (tyz) Vs thickness co-ordinate (z/h) for

simply supported anti-symmetric angle -ply laminated square plate .

(-45/45)4 ss2

0

0.02

0.04

0.06

0.08

0.1

0.12

0 5 10 15 20 25 30

side to thickness ratio(a/h)

N.t

ransverse

shearstress(

t

xz)

HSDTWZF

HSDT

(0/90) ss1

Fig. 4 Non-dimensionalized max.transverse shear s tress (t xz) Vs side to thickness

ratio(a/h)for simply supported anti-symmetric cross-ply laminated square plate

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symmetric cross-ply and angle-ply laminated plates.

From the figures it is observed that the slopediscontinuities in HSDT at the interfaces are achievedwith the inclusion of zig-zag function in the theories.

From fig.2 it is noted that the maximum shear stressobtain from HSDTWZF is 1.2 times than that ofHSDT. Fig.4 contain plots of non-dimensionalized

transverse stresses as a function of side to thicknessratio (a/h) for anti-symmetric angle-ply and cross- plylaminated plates. The effect of transverse sheardeformation and coupling is negligible for all values

of a/h is > 10 and it is quite significant for all thevalues of a/h is

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