Nonlinear flexural response of laminated composite plates under hygro-thermo-mechanical loading

Commun Nonlinear Sci Numer Simulat 15 (2010) 2634–2650

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Commun Nonlinear Sci Numer Simulat

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

Nonlinear flexural response of laminated composite plates underhygro-thermo-mechanical loading

A.K. Upadhyay a, Ramesh Pandey a, K.K. Shukla b,*

a Department of Applied Mechanics, MNNIT Allahabad, UP 211004, Indiab Civil Engineering Department, MNNIT Allahabad, UP 211004, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 July 2009Received in revised form 7 August 2009Accepted 20 August 2009Available online 11 November 2009

Keywords:AnalyticalComposite plateNonlinearHygro-thermo-mechanicalElastic foundation

1007-5704/$ - see front matter � 2009 Elsevier B.Vdoi:10.1016/j.cnsns.2009.08.026

* Corresponding author. Tel.: +91 532 2271206; fE-mail addresses: [email protected] (A.K. Upa

The paper deals with Chebyshev series based analytical solution for the nonlinear flex-ural response of the elastically supported moderately thick laminated composite rect-angular plates subjected to hygro-thermo-mechanical loading. The mathematicalformulation is based on higher order shear deformation theory (HSDT) and von-Kar-man nonlinear kinematics. The elastic foundation is modeled as shear deformable withcubic nonlinearity. The elastic and hygrothermal properties of the fiber reinforced com-posite material are considered to be dependent on temperature and moisture concen-tration and have been evaluated utilizing micromechanics model. The quadraticextrapolation technique is used for linearization and fast converging finite doubleChebyshev series is used for spatial discretization of the governing nonlinear equationsof equilibrium. The effects of Winkler and Pasternak foundation parameters, tempera-ture and moisture concentration on nonlinear flexural response of the laminated com-posite rectangular plate with different lamination scheme and boundary conditions arepresented.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

The plates/panels made up of the polymer based fibre reinforced composite materials, primarily used as one of the majorstructural elements in aerospace, naval, automobile, etc. are often subjected to hostile environmental conditions during theiroperational life. The development of solid propellant rocket motors and increased use of soft filaments in aerospace struc-tures etc. have intensified the need for the solutions of various plate/panel problems supported by elastic medium. Also, thesandwich plates/panels may be viewed as problem of plates/panels supported with elastic medium. In addition to mechan-ical loading, these structures are often subjected to hygroscopic as well as destabilizing thermal loadings also. The structuralcomponents of high-speed aircrafts, spacecrafts and re-entry space vehicle encounter hygrothermal loading conditions. Theadsorbed moisture and induced temperature adversely affects the material properties, which in turn reduces the stiffnessand strength of the structure thus affecting the performance of the structure. Hence, the degradation in performance ofthe structure due to moisture concentration and high temperature has become increasingly more important with the pro-longed use of fiber-reinforced polymer composite material in many structural applications.

The deformation and stress analysis of the laminated composite plates subjected to moisture and temperature has beenthe subject of research interest of many investigators. Adams and Miller [1], Ishikawa et al. [2] and Strife and Prewo [3]studied the effect of environment on the material properties of composite materials and observed that it has significanteffect on strength and stiffness of the composites. Therefore, there is a need to understand the behavior of composite

. All rights reserved.

ax: +91 532 2445101.dhyay), [email protected] (R. Pandey), [email protected], [email protected] (K.K. Shukla).

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A.K. Upadhyay et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 2634–2650 2635

structures subjected to hygrothermal conditions. Whiteny and Ashton [4] studied the hygrothermal effects on bending,buckling and vibration of composite laminated plates using the Ritz method and neglecting the transverse shear deforma-tion. Sai Ram and Sinha [5] presented static analysis of laminated composites plates using First Order Shear DeformationTheory (FSDT) and employing finite element method. The effects of moisture and temperature on the deflections andstress resultants are presented for simply supported and clamped anti-symmetric cross-ply and angle-ply laminates usingreduced lamina properties at elevated moisture concentration and temperature. Lee et al. [6] studied the influence ofhygrothermal effects on the cylindrical bending of symmetric angle-ply laminated plates subjected to uniform transverseload for different boundary conditions via classical laminated plate theory and von-Karman’s large deflection theory. Thematerial properties of the composite are assumed to be independent of temperature and moisture variation. It has beenobserved that the classical laminated plate theory may not be adequate for the analysis of composite laminates even in thesmall deflection range.

Shen [7] studied the influence of hygrothermal effects on the nonlinear bending of shear deformable laminated platesusing a micro-to-macro-mechanical analytical model and Reddy’s higher order shear deformation plate theory. A perturba-tion technique is employed to determine the load-deflection and load-bending moment curves. Patel et al. [8] used a higher-order theory to study the static and dynamic characteristics of thick composite laminates exposed to hygrothermal environ-ment. The formulation accounts for the nonlinear variation of the in-plane and transverse displacements through the thick-ness, and abrupt discontinuity in slope of the in-plane displacements at any interface. Rao and Sinha [9] studied the effects ofmoisture and temperature on the bending characteristics of thick multidirectional fibrous composite plates. The finite ele-ment analysis accounts for the hygrothermal strains and reduced elastic properties of multidirectional composites at an ele-vated moisture concentration and temperature. Deflections and stresses are evaluated for thick multidirectional compositeplates under uniform and linearly varying through-the-thickness moisture concentration and temperature. Results revealthe effects of fiber directionality on deflection and stresses.

In the present study an attempt is made to present analytical solution of nonlinear flexural response of elastically sup-ported cross-ply and angle-ply laminated composite plates under hygrothermal environment. Higher order shear deforma-tion theory (HSDT), von-Karman nonlinear kinematics, finite double Chebyshev series and quadratic extrapolation techniqueare utilized in the formulation and solution methodology.

2. Problem formulation

It is assumed that perfect bonding exists between the layers of the laminated composite plate resting on Pasternak typeelastic foundation as shown in Fig. 1. Based on the global higher order shear deformation theory with cubic variation of in-plane displacements through the thickness and constant transverse displacement, the displacement field at a point in thelaminated plate is expressed as (Kant and Swaminathan [10])

Uðx; y; zÞVðx; y; zÞWðx; y; zÞ

8><>:

9>=>; ¼

u0ðx; yÞv0ðx; yÞw0ðx; yÞ

8><>:

9>=>;þ z

wxðx; yÞwyðx; yÞ0

8><>:

9>=>;þ z2

u1ðx; yÞv1ðx; yÞ0

8><>:

9>=>;þ z3

/xðx; yÞ/yðx; yÞ0

8><>:

9>=>; ð1Þ

where, the parameters u0; v0 and w0 are the in-plane and transverse displacements of a point ðx; yÞ on the middle plane ofthe plate, respectively. The functions wx and wy are rotations of the normal to the middle plane about y- and x-axes, respec-tively. The parameters u1;v1;/x and /y are the higher order terms representing higher-order transverse cross-sectionaldeformation modes.

h/2 h/2

a

b

y, v0

x, u0 θ

z, w0

Shear layer (k3)

Winkler and nonlinear foundation (k1, k2)

Fig. 1. Geometry of the laminated composite rectangular plate resting on nonlinear Pasternak type elastic foundation.

2636 A.K. Upadhyay et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 2634–2650

Employing von-Karman nonlinear kinematics and using the displacement field in Eq. (1), strain–displacement relationsare expressed as

ex

ey

cxy

cyz

cxz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

eox

eoy

coxy

coyz

coxz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;þ z

jx

jy

jxy

2v1

2u1

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;þ z2

e1x

e1y

c1xy

3/y

3/x

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;þ z3

j1x

j1y

j1xy

00

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð2Þ

where,

eox

eoy

coxy

coyz

coxz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

@u0@x þ 1

2@w0@x

� �2

@v0@y þ 1

2@w0@y

� �2

@u0@y þ

@v0@x þ

@w0@x

� �@w0@y

� �wy þ @w0

@y

wx þ @w0@x

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;

ð3Þ

jx

jy

jxy

8><>:

9>=>; ¼

@wx@x@wy

@y

@wx@y þ

@wy

@x

8>><>>:

9>>=>>; ð4Þ

e1x

e1y

c1xy

8><>:

9>=>; ¼

@u1@x@v1@y

@u1@y þ

@v1@x

8>><>>:

9>>=>>; ð5Þ

j1x

j1y

j1xy

8><>:

9>=>; ¼

@/x@x@/y

@y

@/x@y þ

@/y

@x

8>><>>:

9>>=>>; ð6Þ

Assuming plane stress condition in the lamina, the constitutive stress–strain relations for kth layer in the laminate underhygrothermal environment can be written as

rx

ry

sxy

syz

sxz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

k

¼

Q 11 Q 12 Q 16 0 0Q 12 Q 22 Q 26 0 0Q 16 Q 26 Q 66 0 00 0 0 Q44 Q 45

0 0 0 Q45 Q 55

26666664

37777775

k

ex � axDT � bxDC

ey � ayDT � byDC

cxy � axyDT � bxyDC

cyz

cxz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

k

ð7Þ

where,

Qij0s are transformed reduced stiffness coefficients.

DT ¼ ðT � T0Þ = Applied temperature � reference temperature.DC ¼ ðC � C0Þ = Moisture concentration � reference moisture concentration.ax; ay; axy = transformed thermal expansion or contraction coefficients due to temperature.bx; by; bxy= transformed swelling or contraction coefficients due to moisture.

The coefficients ax;ay;axy; bx; by; bxy are obtained by transformation from a11;a22; b11; b22 in the principal material direc-tions and can be expressed as

ax; bx

ay; by

axy; bxy

8><>:

9>=>; ¼

m2 n2

n2 m2

2mn �2mn

264

375 a11; b11

a22; b22

� �ð8Þ

where,

m ¼ Cos h; n ¼ Sin h; h ¼ fibre orientation angle:


The elastic and hygrothermal properties of the composite material are dependent on the temperature and moisture con-centration. It becomes important to consider the temperature and moisture dependent properties of the polymer based fibrereinforced composite material in order to predict the response of the laminated composite plate in hygrothermal environ-ment, accurately.

The material properties are evaluated utilizing micro-mechanics model. Since, the effect of temperature and moisture isdominant in polymer based matrix material; the degradation of the composite material properties is estimated by degradingthe matrix property only. The matrix mechanical property retention ratio is expressed as (Chamis and Sinclair [11])

Fm ¼Tgw � TTgo � T0

� 12

ð9Þ

where T ¼ T0 þ DT and T is the temperature at which material property is to be predicted, T0 is the reference temperature, DTis the increase in temperature from reference temperature, Tgw and Tgo are the glass transition temperatures for wet and ref-erence dry conditions, respectively.

The glass transition temperature for wet material is determined as (Chamis [12])

Tgw ¼ ð0:005C2 � 0:10C þ 1:0ÞTgo ð10Þ

where, C ¼ C0 þ DC is the weight percent of moisture in the matrix material. C0 ¼ 0 weight % and DC is the increase in mois-ture concentration. The elastic constants are evaluated utilizing the following equations (Gibson [13])

E11 ¼ Ef 1Vf þ FmEmVm ð11Þ

E22 ¼ 1:0�ffiffiffiffiffiffiVf

q� �FmEm þ

FmEmffiffiffiffiffiffiVf

p1:0�

ffiffiffiffiffiffiVf

p1:0� FmEm

Ef 2

� � ð12Þ

G12 ¼ 1:0�ffiffiffiffiffiffiVf

q� �FmGm þ

FmGmffiffiffiffiffiffiVf

p1:0�

ffiffiffiffiffiffiVf

p1:0� FmGm

Gf 12

� � ð13Þ

m12 ¼ mf 12Vf þ mmVm ð14Þ

where, ‘V’ is volume fraction, subscripts ‘f’ and ‘m’ is used for fiber and matrix, respectively.The effect of increased temperature and moisture concentration on the coefficients of thermal expansion ðaÞ and hygro-

scopic expansion ðbÞ is opposite from the corresponding effect on strength and stiffness. Hygroscopic expansion coefficientsfor fibers are taken as zero ignoring the effect of moisture on the fiber. The matrix hygrothermal property retention ratio isapproximated as

Fh ¼ 1=Fm ð15Þ

Coefficients of thermal expansion are expressed as (Lee [14])

a11 ¼Ef 1Vf af 1 þ FmEmVmFham

Ef 1Vf þ FmEmVmð16Þ

a22 ¼ af 2Vf þ VmFham þVf Vmðmf 12FmEm � mmEf 1Þ

Ef 1Vf þ FmEmVmðaf 1 � FhamÞ ð17Þ

The longitudinal coefficient of hygroscopic expansion in a composite with isotropic matrix constituent can be expressed as(Gibson [13])

b11 ¼Ef 1Vf bf 1 þ FmEmVmFhbm

Ef 1Vf þ FmEmVmð18Þ

The moisture absorbed by fibers is generally negligible in comparison with the moisture absorbed by matrix. The transversecoefficient of hygroscopic expansion in a composite with isotropic matrix constituent can be expressed as (Lee [15])

b22 ¼VmFhbm ð1þ mmÞðEf 1Vf þ FmEmVmÞ � ðmf 12Vf þ mmVmÞEmFm

� �Ef 1Vf þ FmEmVm

ð19Þ

Eqs. (9)–(19) presented herein are used to evaluate the stiffness coefficients in Eq. (7), thermal expansion coefficients andhygroscopic coefficients in Eq. (8).

The nonlinear elastic foundation is considered as Pasternak type with foundation nonlinearity. It can be modeled as anonlinear spring and a shear layer. The up-thrust due to nonlinear elastic foundation (Pasternak type) can be expressedas (Nath et al. [16])


R ¼ K1W þ K2W3 � K3@2W@x2 þ

@2W@y2

!ð20Þ

where, K1;K2 and K3 are Winkler, nonlinear and shear foundation parameters, respectively.The governing equations of equilibrium and appropriate boundary conditions are derived using the Variational principle

and expressed as

@Nx

@xþ @Nxy

@y¼ 0 ð21Þ

@Ny

@yþ @Nxy

@x¼ 0 ð22Þ

@Qx

@xþ@Q y

@yþ Nx

@2w0

@x2 þ Ny@2w0

@y2 þ 2Nxy@2w0

@x@yþ q� R ¼ 0 ð23Þ

@Mx

@xþ @Mxy

@y� Q x ¼ 0 ð24Þ

@My

@yþ @Mxy

@x� Q y ¼ 0 ð25Þ

@N�x@xþ@N�xy

@y� 2Sx ¼ 0 ð26Þ

@N�y@yþ@N�xy

@x� 2Sy ¼ 0 ð27Þ

@M�x

@xþ@M�

xy

@y� 3Q �x ¼ 0 ð28Þ

@M�y

@yþ@M�

xy

@x� 3Q �y ¼ 0 ð29Þ

The associated admissible boundary conditions obtained are of the form at x ¼ � a2

u0 ¼ 0 or Nx ¼ 0; wx ¼ 0 or Mx ¼ 0v0 ¼ 0 or Nxy ¼ 0; wy ¼ 0 or Mxy ¼ 0

w0 ¼ 0 or Q x ¼ 0 /x ¼ 0 or M�x ¼ 0

u1 ¼ 0 or N�x ¼ 0; /y ¼ 0 or M�xy ¼ 0

v1 ¼ 0 or N�xy ¼ 0;

ð30Þ

at x ¼ � b2

u0 ¼ 0 or Nxy ¼ 0; wx ¼ 0 or Mxy ¼ 0v0 ¼ 0 or Ny ¼ 0; wy ¼ 0 or My ¼ 0

w0 ¼ 0 or Q y ¼ 0 /x ¼ 0 or M�xy ¼ 0

u1 ¼ 0 or N�xy ¼ 0; /y ¼ 0 or M�y ¼ 0

v1 ¼ 0 or N�y ¼ 0;

ð31Þ

where, the in-plane stress and moment resultants of the laminated composite plate consisting of n layers and subjected tohygro-thermo-mechanical loading can be expressed as


Nx

Ny

Nxy

Mx

My

Mxy

N�xN�yN�xy

M�x

M�y

M�xy

8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;

¼

A B D EB D E F

D E F H

E F H J

26664

37775

e0x

e0y

c0xy

jx

jy

jxy

e1x

e1y

c1xy

j1x

j1y

j1xy

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>;

�

NTx

NTy

NTxy

MTx

MTy

MTxy

N�Tx

N�Ty

N�Txy

M�Tx

M�Ty

M�Txy

8>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

�

Nmx

Nmy

Nmxy

Mmx

Mmy

Mmxy

N�mx

N�my

N�mxy

M�mx

M�my

M�mxy

8>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>;

ð32Þ

where, A;B;D; E; F;H; J are the plate stiffness coefficients matrices and the elements of these are defined as

ðAij;Bij;Dij; Eij; Fij;Hij; JijÞ ¼Xn

k¼1

Z zk

zk�1

Q ðkÞij ð1; z; z2; z3; z4; z5; z6Þdz; ðfor i; j ¼ 1;2;6Þ ð33Þ

Transverse shear stress resultants are expressed as

Q y

Q x

Sy

Sx

Q �yQ �x

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;¼

A B D

B D E

D E F

264

375

wy þ @w0@y

wx þ @w0@x

2v1

2u1

3/y

3/x

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

ð34Þ

where, A;B;D; E; F are the plate stiffness coefficients matrices and the elements of these are defined as

ðAij;Bij;Dij; Eij; FijÞ ¼Xn

k¼1

Z zk

zk�1

Q ðkÞij ð1; z; z2; z3; z4Þdz; ðfor i; j ¼ 4;5Þ ð35Þ

The thermal stress and moment resultants of the plates due to uniform temperature over the surface of the plate are ob-tained and expressed as

NTx ; MT

x ; N�Tx ; M�Tx

NTy ; MT

y ; N�Ty ; M�Ty

NTxy; MT

xy; N�Txy ; M�Txy

8>><>>:

9>>=>>; ¼

Xn

k¼1

Z zk

zk�1

Q 11 Q 12 Q16

Q 12 Q 22 Q26

Q 16 Q 26 Q66

264

375

ax

ay

axy

8><>:

9>=>;DTð1; z; z2; z3Þdz ð36Þ

Similarly, the hygroscopic stress and moment resultants of the plates due to uniform moisture concentration over the sur-face of the plate are expressed as

Nmx ; Mm

x ; N�mx ; M�mx

Nmy ; Mm

y ; N�my ; M�my

Nmxy; Mm

xy; N�mxy ; M�mxy

8><>:

9>=>; ¼

Xn

k¼1

Z zk

zk�1

Q 11 Q 12 Q 16

Q 12 Q 22 Q 26

Q 16 Q 26 Q 66

264

375

bx

by

bxy

8><>:

9>=>;DCð1; z; z2; z3Þdz ð37Þ

The governing differential equations of equilibrium (21)–(29) are finally expressed in terms of displacement componentsand further these equations are cast in compact non-dimensional form as

ðLa þ Lb þ LcÞdþ Q � R ¼ 0 ð38Þ

where,

La ¼ La1@2

@x2 þ La2@2

@y2 þ La3@2

@x@yþ La4

@

@xþ La5

@

@yþ La6

Lb ¼ Lb1@2

@x2 þ Lb2@2

@y2 þ Lb3@2

@x@y


Lc ¼ Lc1@2

@x2 þ Lc2@2

@y2 þ Lc3@2

@x@y

d ¼ u v w wx wy �u1 �v1 /x /y� �T

Q ¼ 0 0 Q 0 0 0 0 0 0½ �T ; R ¼ 0 0 R 0 0 0 0 0 0½ �T

where, Q represents non-dimensional transverse pressure and R is the non-dimensional elastic foundation parameter (Pas-ternak type with foundation nonlinearity) and expressed as

R ¼ k1wþ k2w3 � k3@2w

@X2 þ@2w

@Y2

!ð39Þ

where k1; k2 and k3 are non-dimensional Winkler, nonlinear and shear foundation parameters. La1 � La6; Lb1 � Lb3 andLc1 � Lc3 used in Eq. (38) are defined in Appendix A. The non-dimensional parameters used in the above formulation are de-scribed in Appendix B.

The admissible boundary conditions obtained from Eqs. (30) and (31) are expressed in non-dimensional form asSimply supported immovable edge (S)

u ¼ v ¼ w ¼ �wy ¼ �u1 ¼ �v1 ¼ �/y ¼ Mx ¼ M�x ¼ 0 at X ¼ �1 ð40Þ

u ¼ v ¼ w ¼ �wx ¼ �u1 ¼ �v1 ¼ �/x ¼ My ¼ M�y ¼ 0 at Y ¼ �1 ð41Þ

Clamped immovable edge (C)

u ¼ v ¼ w ¼ �wx ¼ �wy ¼ �u1 ¼ �v1 ¼ �/x ¼ �/y ¼ 0 at X;Y ¼ �1 ð42Þ

Free edge (F)

Nx ¼ Nxy ¼ Mx ¼ Mxy ¼ N�x ¼ N�xy ¼ M�x ¼ M�

xy ¼ Q x ¼ 0 at X ¼ �1 ð43Þ

Ny ¼ Nxy ¼ My ¼ Mxy ¼ N�y ¼ N�xy ¼ M�y ¼ M�

xy ¼ Q y ¼ 0 at Y ¼ �1 ð44Þ

3. Solution methodology

The governing nonlinear equations of equilibrium along with appropriate boundary conditions are solved using an ana-lytical technique. The coupled nonlinear equations are linearized utilizing total linearization scheme based on quadraticextrapolation technique. The fast converging, orthogonal, double Chebyshev polynomial in the range of �1 6 X 6 1 and�1 6 Y 6 1 is used for spatial discretization of the linear differential equations. The displacement functions d and loadingQ is approximated in space domain by finite degree Chebyshev polynomial. A typical displacement/loading functionnðx; yÞ is expressed, using finite degree Chebyshev polynomial (Fox and Parker [17]) as

nðx; yÞ ¼XM

i¼0

XN

j¼0

dijnijTiðxÞTjðyÞ : ð45Þ

The spatial derivative of the function nðx; yÞ can be expressed as

nrsij ¼

XM�r

i¼0

XN�s

j¼0

dijnrsij TiðxÞTjðyÞ; ð46Þ

where, r and s are the order of differentiation of the function with respect to X and Y , respectively. The function dij used inEqs. (45) and (46) is expressed as (Shukla and Nath [18])

dij ¼0:25; i ¼ 0; j ¼ 00:50; i ¼ 0; j – 0 or i – 0; j ¼ 01:0; i – 0; j – 0

8><>:

The derivative function nrsij is evaluated using the recurrence relations (Fox and Parker [17])

nrsði�1Þ;j ¼ nrs

ðiþ1Þ;j þ 2inðr�1Þ;si;j

nrsi;ðj�1Þ ¼ nrs

i;ðjþ1Þ þ 2jnr;ðs�1Þi;j

ð47Þ


The nonlinear terms appearing in the set of governing Eq. (38) are linearized at any step of marching variable (loading) usingquadratic extrapolation technique. A typical nonlinear function G at a step J is expressed as

Fig. 2.subject

GJ ¼XM�r

i¼0

XN

j¼0

dijnrijTiðxÞTjðyÞ

" #J

XM

i¼0

XN�s

j¼0

dijnsijTiðxÞTjðyÞ

" #J

ð48Þ

where,

ðnijÞJ ¼ g1ðnijÞJ�1 þ g2ðnijÞJ�2 þ g3ðnijÞJ�3

During initial steps of marching variables, the coefficients g1;g2 and g3 of the quadratic extrapolation scheme of linearizationtake the following values (Shukla and Nath [18])

1; 0;0 ðJ ¼ 1Þ; 2;�1;0 ðJ ¼ 2Þ; 3;�3;1 ðJ P 3Þ

The product of two Chebyshev polynomials is expressed as

TiðxÞTjðyÞTkðxÞTlðyÞ ¼ ½TðiþkÞðxÞTðjþ1ÞðyÞ þ TðiþkÞðxÞTðj�1ÞðyÞ þ Tði�kÞðxÞTðjþ1ÞðyÞ þ Tði�kÞðxÞTðj�1ÞðyÞ�=4 ð49Þ

Using the procedures described herein, the set of governing nonlinear equilibrium equations (38) is linearized and discret-ized in space domain and finally expressed in form of a set of linear simultaneous equations as

XM�2

i¼0

XN�2

j¼0

Fkðuij; v ij;wij; �wxij; �wyij; �u1ij; �v1ij; �/xij; �/yij;Q ijÞTiðxÞTjðyÞ ¼ 0; k ¼ 1;9 ð50Þ

Similarly, the appropriate sets of boundary conditions are also discretized.The loads are incremented in small steps and the nonlinear terms are computed at each step of marching variable (load-

ing) and transferred to the right side so that the left side matrix remains invariant with respect to the loading. Thus, the loadvector gets updated at every iteration of each step. The set of linear equations are expressed in the matrix form as

Ad ¼ P ð51Þ

where A is ði� jÞ coefficient matrix, d is ðj� 1Þ displacement coefficient vector, P is ði� 1Þ load vector. Multiple regressionanalysis gives

d ¼ ðATAÞ�1 ATPd ¼ BP

and the values of the coefficients of the displacement vector ‘d’ obtained are put into Eq. (45) to evaluate the displacementsat the desired location on the mid-plane of the plate.

0

0.2

0.4

0.6

0 10 20 30 40 50 60

567891011

SSSS, a/b=1, a/h=10

ΔC=1%, [0/90/0/90]

M=N

Q

Wc

Convergence of transverse central deflection of simply supported, square [0/90/0/90] anti-symmetric cross-ply laminated composite (a/h = 10) plateed to uniform transverse pressure in hygroscopic environment ðDC ¼ 1%Þ.


4. Results and discussions

In order to show the accuracy and efficiency of the present solution methodology, the results of convergence study per-formed on a simply supported moderately thick (a/h = 10), square laminated composite plate in hygroscopic environmentðDC ¼ 1%Þ are obtained and presented in Fig. 2. It is observed that convergence is achieved beyond 7–8 terms expansionof variables in Chebyshev series. In the present study results are obtained using 9 terms expansion of the variables in Cheby-shev series to obtain reasonably accurate results at relatively low computational cost.

The transverse central displacement response of simply supported, square, [45/�45/45/�45] anti-symmetric laminated com-posite plate subjected to hygro-thermo-mechanical loading is obtained for different hygrothermal conditions and shown alongwith the results due to Shen [7] in Fig. 3. It is observed that the results are in very good agreement. Table 1 shows the comparisonsof the central deflection (Wc), in-plane stress and moment resultants (Nx and Mx) of symmetric and anti-symmetric cross-ply,moderately thick (a/h=10) and thin (a/h=100), CSSS (one edge clamped and other three simply supported) laminated compositeplate subjected to uniform transverse pressure and supported on elastic foundation ðk1 ¼ k2 ¼ k3 ¼ 50Þwith the results due toMalekzadeh and Setoodeh [19]. It is noticed that the results are in reasonably good agreement. It can be seen that the resultsdue to present analytical technique agree well with the results available in open literature and the present solution methodologymay be efficiently used for the nonlinear flexural analysis of the laminated composite plates subjected to different loading con-ditions. The material properties are taken directly from the reference papers and are not mentioned here for the sake of brevity.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 50 100 150 200 250 300Q

W 0, 0, Ref. [ 7 ]

0, 0, Present

100, 1, Ref. [ 7 ]

100, 1, Present

200, 2, Ref. [ 7 ]

200, 2, Present

a/h=10, a/b=1, SSSS, [45/- 45/45/- 45],Vf =0.6

ΔT( 0C), ΔC(%)

c

Fig. 3. Comparison of the transverse central displacement of [45/�45/45/�45] anti-symmetric angle-ply, simply supported, square laminated compositeplate subjected to hygro-thermo-mechanical loading.

Table 1Comparison of nonlinear central deflection ðWcÞ, stress resultant ðNxÞ and moment resultant (Mx) at centre of an elastically supported ðk1 ¼ k2 ¼ k3 ¼ 50Þ, CSSS,laminated composite plate subjected to uniform transverse pressure.

S. No. (a/h) Lamination scheme Q Wc Nx Mx Reference*

1 10 [0/90/90/0] 900 0.5656 0.0895 2.5992 ‘a’900 0.5702 0.0887 2.3310 ‘b’

1500 0.8435 0.2006 3.8544 ‘a’1500 0.8491 0.1981 3.4528 ‘b’

2 100 [0/90/90/0] 900 0.4981 0.0007 5.1370 ‘a’900 0.5085 0.0006 5.6380 ‘b’

1500 0.7638 0.0015 7.7284 ‘a’1500 0.7795 0.0014 8.1858 ‘b’

3 10 [0/90/0/90] 900 0.7640 0.1359 2.0309 ‘a’900 0.7600 0.1440 1.8965 ‘b’

1500 1.0550 0.2757 1.5992 ‘a’1500 1.0379 0.2842 1.1947 ‘b’

4 100 [0/90/0/90] 900 0.7193 0.0010 4.3294 ‘a’900 0.7281 0.0009 4.9912 ‘b’

1500 1.0130 0.0023 4.5825 ‘a’1500 1.0170 0.0022 5.0380 ‘b’

* Malekzadeh and Setoodeh [19] – ‘a’, Present – ‘b’.


The nonlinear flexural response of laminated composite plates subjected to uniform transverse pressure in hygrothermalenvironment is studied and the results are depicted in graphical form in Figs. 4–10. The material properties of the compositematerial are considered to be dependent on temperature and moisture. The material properties are taken in the analysis atthe reference temperature 21�C and moisture concentration 0% as given below

Fig. 4.cross- p

Fig. 5.to unifo

Ef 1 ¼ 220 GPa; Ef 2 ¼ 13:79 GPa; Em ¼ 3:45 GPa; Gf 12 ¼ 8:97 GPa; mf 12 ¼ 0:20; mm ¼ 0:35;

af 1 ¼ �0:99� 10�6=�C; af 2 ¼ 10:08� 10�6=

�C; am ¼ 72:0� 10�6=�C; bm ¼ 0:33; Tgo ¼ 216 �C

The effects of temperature, moisture concentration and their combination on the non-dimensional displacement re-sponse of the anti-symmetric cross ply [0/90/0/90] plate are shown in Fig. 4. It is observed that central deflection increaseswith increase in moisture concentration, temperature and increase in both simultaneously. The increase is highest whenhygrothermal condition is taken and it is least when only effect of moisture is considered. At Q ¼ 200, the increase in centraldeflection is 0.89% corresponding to DC ¼ 1%; DT ¼ 0 �C; 5:2% corresponding to DC ¼ 0%; DT ¼ 100 �C and 7.14% corre-sponding to DC ¼ 1% and DT ¼ 100 �C.

0.0

0.2

0.4

0.6

0.8

0 50 100 150 200 250Q

Wc

0, 0.00, 1.0100, 0.0100,1.0

a/h=10, a/b=1, CCCC,Vf = 0.6, [0/90/0/90]

ΔT( 0C), ΔC(%)

Effect of temperature, moisture concentration and their combination on the non-dimensional central deflection of clamped, square, anti-symmetric,ly laminated composite plate (a/h = 10) subjected to uniform transverse loading.

0.0

0.3

0.6

0.9

1.2

1.5

0 150 300 450 600 750

150, 1.5100, 1.0 50, 0.5 0, 0.0

a/h=20, a/b=1, CCCC,Vf = 0.6, [0/90/0/90]

ΔT( 0C), ΔC(%)

Q

Wc

Effect of hygrothermal environment on transverse central deflection of clamped, square [0/90/0/90] laminated composite plate (a/h = 20) subjectedrm transverse pressure.

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250 300

100, 1.0

50, 0.5

0, 0

a/h=10, a/b=1, CCCC, Vf = 0.6, [45/-45/45/-45]

ΔT(0C), ΔC(%)

Q

Wc

Fig. 6. Effect of hygrothermal environment on transverse central deflection of clamped, square [45/�45/45/�45] laminated composite (a/h = 10) platesubjected to uniform transverse pressure.

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250 300

0.4

0.5

0.6

a/h=10, a/b=1, CCCC, [0/90/0/90], ΔT=1000C, ΔC=1%

Vf

Q

Wc

Fig. 7. Effect of fibre volume fraction on transverse central deflection of clamped, square, anti-symmetric cross-ply laminated composite plate (a/h = 10)subjected to uniform transverse pressure in hygrothermal environment.

0.0

0.2

0.4

0.6

0.8

0 50 100 150 200 250 300

0,0,075,0,075,0,3075,100,3075,200,30

a/h=10, a/b=1, Vf = 0.6

CCCC,[45/-45/-45/45]

ΔT=500C,ΔC=0.5%

k1, k2, k3

Q

Wc

Fig. 8. Effect of elastic foundation parameters on transverse central deflection of clamped, square, [45/-45/-45/45] laminated composite plate (a/h = 10)subjected to uniform transverse pressure in hygrothermal environment.


0.0

0.2

0.4

0.6

0.8

0 50 100 150 200 250

75, 0, 30

75, 100, 30,

75, 200, 30

a/h=10, a/b=1 CCCC, Vf = 0.6, [45/-45/-45/45]

ΔT=500C, ΔC=0.5%

k1,-k2, k3

Q

Wc

Fig. 9. Effect of softening type foundation nonlinearity on transverse central deflection of clamped, square, [45/-45/-45/45] laminated composite plate (a/h = 10) subjected to uniform transverse pressure in hygrothermal environment.

0

0.1

0.2

0.3

0 50 100 150 200 250

CFCCSSSSCSSSCCSSCSCCCCCC

a/h=10, a/b=1 Vf = 0.6, [0/90/90/0]

ΔT = 500C, ΔC = 0.5% k1=75, k2=100, k3=30

Q

Wc

Fig. 10. Effect of boundary conditions on the transverse central deflection of elastically supported square, symmetric cross-ply laminated composite platesubjected to uniform transverse pressure in hygrothermal environment.


Fig. 5 represents the response of clamped, moderately thick (a/h = 20), square anti-symmetric cross-ply [0/90/0/90] lam-inated composite plate subjected to hygro-thermo-mechanical loading. Appreciable increase in deflection is observed attemperature closer to glass transition temperature, indicating the reduction in stiffness of the plate at the increased temper-ature and moisture.

The effect of temperature and moisture concentration on transverse central deflection of clamped, anti-symmetric angle-ply [45/�45/45/�45] laminate at fiber volume fraction 0.6 is shown in Fig. 6. It is observed that with increase in temperatureand moisture concentration, deflection increases as expected.

Fig. 7 shows the effect of fiber volume fraction on transverse central deflection of clamped, moderately thick, anti-sym-metric cross-ply [0/90/0/90] laminate subjected to hygro-thermo-mechanical loading ðDT ¼ 100 �C and DC ¼ 1%Þ and it isobserved that with increase in fiber volume fraction, transverse central deflection decreases. It is due to the fact that withincrease in fiber volume fraction, the stiffness of the plate increases.


Fig. 8 shows the effect of foundation parameters on transverse central deflection of symmetric angle-ply [45/�45/�45/45] laminated composite plate subjected to hygro-thermo-mechanical loading. It is observed that transverse central deflec-tion of the elastically supported plate is lower than the plate without elastic foundation. It is also clear from the figure thateffect of shear layer foundation parameter ðk3Þ is more pronounced than the effect of Winkler foundation parameter ðk1Þ. Theeffect of foundation nonlinearity ðk2Þ is appreciable only at higher loads. With the increase in hardening type of nonlinearityðþk2Þ, deflection of the laminated composite plate decreases.

Fig. 9 presents the effect of softening type of foundation nonlinearity ð�k2Þ on the displacement response of symmetricangle-ply [45/�45/�45/45] plate. It can be observed that with increase in ð�k2Þ value, the deflection increases and at highervalues of k2 (softening type), the plate shows the softening type of nonlinear behavior.

Fig. 10 represents the effect of boundary conditions on the transverse central deflection of elastically supported moder-ately thick symmetric cross-ply laminate under hygro-thermo-mechanical loading. The deflection of clamped plate is leastand the plate with three edges clamped and one free is highest. It is observed that increase in degree of fixity, decreases thedeflection of the elastically supported plate in hygrothermal environment.

5. Conclusions

Analytical solutions to the nonlinear flexural response of the moderately thick laminated composite plate subjected tohygro-thermo-mechanical loading is obtained using fast converging finite double Chebyshev series. It is observed thathygrothermal dependent mechanical and thermal properties greatly affect the flexural behavior of the laminated compositeplates. The flexural response of the laminated composite plate deteriorates considerably with the increase in temperatureand moisture concentration and this hygrothermal environment becomes more detrimental as the working temperaturereaches closer to the glass transition temperature. The deflection of the elastically supported laminated composite plateis smaller and the presence of shear layer in the foundation (Pasternak type) is relatively more predominant and has signif-icant effect on the displacement response of the plate. The effects of various boundary conditions are also discussed, showingthe applicability of the present solution methodology.

Appendix A

A B B D D E E2 3

La1 ¼

1 16A11

0 11A11h

16A11h

11

A11h216

A11h211

A11h316

A11h3

A16A22

A66A22

0 B16A22h

B66A22h

D16

A22h2D66

A22h2E16

A22h3E66

A22h3

0 0 La1ð3;3Þ 0 0 0 0 0 0hB11D11

hB16D11

0 1 D16D11

E11hD11

E16hD11

F11

h2D11

F16

h2D11

hB16D22

hB66D22

0 D16D22

D66D22

E16hD22

E66hD22

F16

h2D22

F66

h2D22

1 D16D11

0 E11hD11

E16hD11

F11

h2D11

F16

h2D11

H11

h3D11

H16

h3D11D16D22

D66D22

0 E16hD22

E66hD22

F16

h2D22

F66

h2D22

H16

h3D22

H66

h3D22

E11hD11

E16hD11

0 F11

h2D11

F16

h2D11

H11

h3D11

H16

h3D11

J11

h4D11

J16

h4D11

E16hD22

E66hD22

0 F16

h2D22

F66

h2D22

H16

h3D22

H66

h3D22

J16

h4D22

J66

h4D22

6666666666666666666664

7777777777777777777775
where,
La1ð3;3Þ ¼bA55 � A11 NT

x þ Nmx

� �bA22

La2 ¼ k2

A66A11

A26A11

0 B66A11h

B26A11h

D66

A11h2D26

A11h2E66

A11h3E26

A11h3

A26A22

1 0 B26A22h

B22A22h

D26

A22h2D22

A22h2E26

A22h3E22

A22h3

0 0 La2ð3;3Þ 0 0 0 0 0 0hB66D11

hB26D11

0 D66D11

D26D11

E66hD11

E26hD11

F66

h2D11

F26

h2D11

hB26D22

hB22D22

0 D26D22

1 E26hD22

E22hD22

F26

h2D22

F22

h2D22

D66D11

D26D11

0 E66hD11

E26hD11

F66

h2D11

F26

h2D11

H66

h3D11

H26

h3D11

D26D22

1 0 E26hD22

E22hD22

F26

h2D22

F22

h2D22

H26

h3D22

H22

h3D22

E66hD11

E26hD11

0 F66

h2D11

F26

h2D11

H66

h3D11

H26

h3D11

J66

h4D11

J26

h4D11

E26hD22

E22hD22

0 F26

h2D22

F22

h2D22

H26

h3D22

H22

h3D22

J26

h4D22

J22

h4D22

266666666666666666666664

377777777777777777777775


where,

La2ð3;3Þ ¼bA44 � A22 NT

y þ Nmy

� �bA22

La3 ¼ k

2A16A11

A12þA66A11

0 2B16A11h

B12þB66A11h

2D16

A11h2D12þD66

A11h22E16

A11h3E12þE66

A11h3

A12þA66A22

2A26A22

0 B12þB66A22h

2B26A22h

D12þD66

A22h22D26

A22h2E12þE66

A22h32E26

A22h3

0 0 La3ð3;3Þ 0 0 0 0 0 02hB16

D11

hðB12þB66ÞD11

0 2D16D11

D12þD66D11

2E16hD11

E12þE66hD11

2F16

h2D11

F12þF66

h2D11

hðB12þB66ÞD22

2hB26D22

0 D12þD66D22

2D26D22

E12þE66hD22

2E26hD22

F12þF66

h2D22

2F26

h2D22

2D16D11

D12þD66D11

0 2E16hD11

E12þE66hD11

2F16

h2D11

F12þF66

h2D11

2H16

h3D11

H12þH66

h3D11

D12þD66D22

2D26D22

0 E12þE66hD22

2E26hD22

F12þF66

h2D22

2F26

h2D22

H12þH66

h3D22

2H26

h3D22

2E16hD11

E12þE66hD11

0 2F16

h2D11

F12þF66

h2D11

2H16

h3D11

H12þH66

h3D11

2J16

h4D11

J12þJ66

h4D11

E12þE66hD22

2E26hD22

0 F12þF66

h2D22

2F26

h2D22

H12þH66

h3D22

2H26

h3D22

J12þJ66

h4D22

2J26

h4D22

2666666666666666666666664

3777777777777777777777775

where,

La3ð3;3Þ ¼2bA45 � 2A66 NT

xy þ Nmxy

� �bA22

La4 ¼

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 A55b

2A22

A45b2A22

B55bA22h

B45bA22h

3D55b

2A22h23D45b

2A22h2

0 0 �A55bh2

2D110 0 0 0 0 0

0 0 �A45bh2

2D220 0 0 0 0 0

0 0 �B55bhD11

0 0 0 0 0 0

0 0 �B45bhD22

0 0 0 0 0 0

0 0 �3D55b2D11

0 0 0 0 0 0

0 0 �3D45b2D22

0 0 0 0 0 0

2666666666666666666664

3777777777777777777775

La5 ¼

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 A45kb

2A22

A44kb2A22

B45kbA22h

B44kbA22h

3D45kb

2A22h23D44kb

2A22h2

0 0 �A45kbh2

2D110 0 0 0 0 0

0 0 �A44kbh2

2D220 0 0 0 0 0

0 0 �B45kbhD11

0 0 0 0 0 0

0 0 �B44kbhD22

0 0 0 0 0 0

0 0 �3D45kb2D11

0 0 0 0 0 0

0 0 �3D44kb2D22

0 0 0 0 0 0

2666666666666666666664

3777777777777777777775


La6 ¼

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 � A55b2h2

4D11� A45b2h2

4D11� B55b2h

2D11� B45b2h

2D11� 3D55b2

4D11� 3D45b2

4D11

0 0 0 � A45b2h2

4D22� A44b2h2

4D22� B45b2h

2D22� B44b2h

2D22� 3D45b2

4D22� 3D44b2

4D22

0 0 0 � B55b2h2D11

� B45b2h2D11

� D55b2

D11� D45b2

D11� 3E55b2

2D11h � 3E45b2

2D11h

0 0 0 � B45b2h2D22

� B44b2h2D22

� D45b2

D22� D44b2

D22� 3E45b2

2D22h � 3E44b2

2D22h

0 0 0 � 3D55b2

4D11� 3D45b2

4D11� 3E55b2

2D11h � 3E45b2

2D11h � 9F55b2

4D11h2 � 9F45b2

4D11h2

0 0 0 � 3D45b2

4D22� 3D44b2

4D22� 3E45b2

2D22h � 3E44b2

2D22h � 9F45b2

4D22h2 � 9F44b2

4D22h2

266666666666666666664

377777777777777777775

Lb1 ¼

0 0 2b

@w@X þ

kA16A11

@w@Y

� �0 0 0 0 0 0

0 0 2bA22

A16@w@X þ kA66

@w@Y

� �0 0 0 0 0 0

0 0 Lb1ð3;3Þ 0 0 0 0 0 00 0 2h

bD11B11

@w@X þ kB16

@w@Y

� �0 0 0 0 0 0

0 0 2hbD22

B16@w@X þ kB66

@w@Y

� �0 0 0 0 0 0

0 0 2bD11

D11@w@X þ kD16

@w@Y

� �0 0 0 0 0 0

0 0 2bD22

D16@w@X þ kD66

@w@Y

� �0 0 0 0 0 0

0 0 2hbD11

E11@w@X þ kE16

@w@Y

� �0 0 0 0 0 0

0 0 2hbD22

E16@w@X þ kE66

@w@Y

� �0 0 0 0 0 0

2666666666666666666664

3777777777777777777775

where, Lb1ð3;3Þ ¼ 2bA22

A11@u@X þ kA16

@u@Y þ A16

@v@X þ kA12

@v@Y

� �þ 2

hbA22B11

@�wx@X þ kB16

@�wx@Y þ B16

@�wy

@X þ kB12@�wy

@Y

� �þ 2

h2bA22D11

@�u1@X þ kD16

@�u1@Y þ

�D16

@�v1@X þ kD12

@�v1@Y Þ þ 2

h3bA22E11

@�/x@X þ kE16

@�/x@Y þ E16

@�/y

@X þ kE12@�/y

@X

� �

Lb2 ¼

0 0 2k2

bA11A66

@w@X þ kA26

@w@Y

� �0 0 0 0 0 0

0 0 2k2

bA22A26

@w@X þ kA22

@w@Y

� �0 0 0 0 0 0

0 0 Lb2ð3;3Þ 0 0 0 0 0 0

0 0 2k2hbD11

B66@w@X þ kB26

@w@Y

� �0 0 0 0 0 0

0 0 2k2hbD22

A26@w@X þ kA22

@w@Y

� �0 0 0 0 0 0

0 0 2k2

bD11D66

@w@X þ kD26

@w@Y

� �0 0 0 0 0 0

0 0 2k2

bD22D26

@w@X þ kD22

@w@Y

� �0 0 0 0 0 0

0 0 2k2

bhD11E66

@w@X þ kE26

@w@Y

� �0 0 0 0 0 0

0 0 2k2

bhD22E26

@w@X þ kE22

@w@Y

� �0 0 0 0 0 0

26666666666666666666664

37777777777777777777775

where, Lb2ð3;3Þ ¼ 2k2

bA22A12

@u@X þ kA26

@u@Y þ A26

@v@X þ kA22

@v@Y

� �þ 2k2

hbA22B12

@�wx@X þ kB26

@�wx@Y þ B26

@�wy

@X þ kB22@�wy

@Y

� �þ 2k2

h2bA22D12

@�u1@X þ kD26

@�u1@Y þ

�D26

@�v1@X þ kD22

@�v1@Y Þ þ 2k2

h3bA22E12

@�/x@X þ kE26

@�/x@Y þ E26

@�/y

@X þ kE22@�/y

@Y

� �

Lb3 ¼

0 0 2kbA11

2A16@w@X þ kðA12 þ A66Þ @w

@Y

� �0 0 0 0 0 0

0 0 2kbA22

ðA12 þ A66Þ @w@X þ 2kA26

@w@Y

� �0 0 0 0 0 0

0 0 Lb3ð3;3Þ 0 0 0 0 0 00 0 2kh

bD112B16

@w@X þ kðB12 þ B66Þ @w

@Y

� �0 0 0 0 0 0

0 0 2khbD22

ðB12 þ B66Þ @w@X þ 2kB26

@w@Y

� �0 0 0 0 0 0

0 0 2kbD11

2D16@w@X þ kðD12 þ D66Þ @w

@Y

� �0 0 0 0 0 0

0 0 2kbD22

ðD12 þ D66Þ @w@X þ 2kD26

@w@Y

� �0 0 0 0 0 0

0 0 2kbhD11

2E16@w@X þ kðE12 þ E66Þ @w

@Y

� �0 0 0 0 0 0

0 0 2kbhD22

ðE12 þ E66Þ @w@X þ 2kE26

@w@Y

� �0 0 0 0 0 0

266666666666666666664

377777777777777777775


where, Lb3ð3;3Þ ¼ 4kbA22

A16@u@X þ kA66

@u@Y þ A66

@v@X þ kA26

@v@Y

� �þ 4k

hbA22B16

@�wx@X þ kB66

@�wx@Y þ B66

@�wy

@X þ kB26@�wy

@Y

� �þ 4k

h2bA22D16

@�u1@X þ kD66

@�u1@Y þ

�D66

@�v1@X þ kD26

@�v1@Y Þ þ 4k

h3bA22E16

@�/x@X þ kE66

@�/x@Y þ E66

@�/y

@X þ kE26@�/y

@Y

� �The non-zero terms of [9 � 9] matrices Lc1; Lc2; and Lc3 are as following;

Lc1ð3;3Þ ¼2

b2A22A16

@w@X

� 2

þ k2A12@w@Y

� 2

þ 2kA16@w@X

� @w@Y

�

Lc2ð3;3Þ ¼2k2

b2A22A12

@w@X

� 2

þ k2A22@w@Y

� 2

þ 2kA26@w@X

� @w@Y

�

Lc3ð3;3Þ ¼4k

b2A22A16

@w@X

� 2

þ k2A26@w@Y

� 2

þ 2kA66@w@X

� @w@Y

�

Appendix B

Non-dimensional parameters used in the problem formulation are defined as

X ¼ 2xa; Y ¼ 2y

b; k ¼ a

b; b ¼ a

h;

q� ¼ qa2

4A22h; Q ¼ qa4

4E2h4 ;u ¼u0

h; v ¼ v0

h; w ¼ w0

h;

�wx ¼ wx;�wy ¼ wy; �u1 ¼ u1h; �v1 ¼ v1h; �/x ¼ /xh2

; �/y ¼ /yh2;

Nx;NTx ;N

mx

� �¼

Nx;NTx ;N

mx

� �b

A11; Ny;NT

y ;Nmy

� �¼

Ny;NTy ;N

my

� �b

A22;

Nxy;NTxy;N

mxy

� �¼

Nxy;NTxy;N

mxy

� �b

A66; N�x;N

�Tx ;N

�mx

� �¼

N�x;N�Tx ;N

�mx

� �b

A11h2 ;

N�y;N�Ty ;N

�my

� �¼

N�x;N�Ty ;N

�my

� �b

A22h2 ; N�xy;N�Txy ;N

�mxy

� �¼

N�xy;N�Txy ;N

�mxy

� �b

A66h2 ;

Mx;MTx ;M

mx

� �¼

Mx;MTx ;M

mx

� �hb2

D11; My;MT

y ;Mmy

� �¼

My;MTy ;M

my

� �hb2

D22;

Mxy;MTxy;M

mxy

� �¼

Mxy;MTxy;M

mxy

� �hb2

D66; M�

x;M�Tx ;M

�mx

� �¼

M�x;M

�Tx ;M

�mx

� �b2

D11h;

M�y;M

�Ty ;M

�my

� �¼

M�x;M

�Ty ;M

�my

� �b2

D22h; M�

xy;M�Txy ;M

�mxy

� �¼

M�xy;M

�Txy ;M

�mxy

� �b2

D66h;

Q x ¼Q xbA55

; Qy ¼Q yb

A44; Sx ¼

SxbA55h

; Sy ¼Syb

A44h; Q �x ¼

Q �xb

A55h2 ; Q �y ¼Q �yb

A44h2 ;

k1 ¼k�1a4

D11; k2 ¼

k�2a4h2

D11; k3 ¼

k�3a2

D11;

where, k�1; k�2 and k�3 are Winkler, nonlinear and shear foundation parameters, respectively.


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