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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Nonlinear flexural response of laminated composite plates under hygro-thermo-mechanical loading A.K. Upadhyay a, Ramesh Pandey a, K.K. Shukla b,* a b

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

a r t i c l e

i n f o

Article history: Received 1 July 2009 Received in revised form 7 August 2009 Accepted 20 August 2009 Available online 11 November 2009 Keywords: Analytical Composite plate Nonlinear Hygro-thermo-mechanical Elastic foundation

a b s t r a c t The paper deals with Chebyshev series based analytical solution for the nonlinear flexural response of the elastically supported moderately thick laminated composite rectangular plates subjected to hygro-thermo-mechanical loading. The mathematical formulation is based on higher order shear deformation theory (HSDT) and von-Karman nonlinear kinematics. The elastic foundation is modeled as shear deformable with cubic nonlinearity. The elastic and hygrothermal properties of the fiber reinforced composite material are considered to be dependent on temperature and moisture concentration and have been evaluated utilizing micromechanics model. The quadratic extrapolation technique is used for linearization and fast converging finite double Chebyshev series is used for spatial discretization of the governing nonlinear equations of equilibrium. The effects of Winkler and Pasternak foundation parameters, temperature and moisture concentration on nonlinear flexural response of the laminated composite rectangular plate with different lamination scheme and boundary conditions are presented. Ó 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 major structural elements in aerospace, naval, automobile, etc. are often subjected to hostile environmental conditions during their operational life. The development of solid propellant rocket motors and increased use of soft filaments in aerospace structures etc. have intensified the need for the solutions of various plate/panel problems supported by elastic medium. Also, the sandwich plates/panels may be viewed as problem of plates/panels supported with elastic medium. In addition to mechanical loading, these structures are often subjected to hygroscopic as well as destabilizing thermal loadings also. The structural components of high-speed aircrafts, spacecrafts and re-entry space vehicle encounter hygrothermal loading conditions. The adsorbed moisture and induced temperature adversely affects the material properties, which in turn reduces the stiffness and strength of the structure thus affecting the performance of the structure. Hence, the degradation in performance of the structure due to moisture concentration and high temperature has become increasingly more important with the prolonged 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 been the 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 significant effect on strength and stiffness of the composites. Therefore, there is a need to understand the behavior of composite * Corresponding author. Tel.: +91 532 2271206; fax: +91 532 2445101. E-mail addresses: [email protected] (A.K. Upadhyay), [email protected] (R. Pandey), [email protected], [email protected] (K.K. Shukla). 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.08.026

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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 deformation. Sai Ram and Sinha [5] presented static analysis of laminated composites plates using First Order Shear Deformation Theory (FSDT) and employing finite element method. The effects of moisture and temperature on the deflections and stress resultants are presented for simply supported and clamped anti-symmetric cross-ply and angle-ply laminates using reduced lamina properties at elevated moisture concentration and temperature. Lee et al. [6] studied the influence of hygrothermal effects on the cylindrical bending of symmetric angle-ply laminated plates subjected to uniform transverse load for different boundary conditions via classical laminated plate theory and von-Karman’s large deflection theory. The material properties of the composite are assumed to be independent of temperature and moisture variation. It has been observed that the classical laminated plate theory may not be adequate for the analysis of composite laminates even in the small deflection range. Shen [7] studied the influence of hygrothermal effects on the nonlinear bending of shear deformable laminated plates using a micro-to-macro-mechanical analytical model and Reddy’s higher order shear deformation plate theory. A perturbation technique is employed to determine the load-deflection and load-bending moment curves. Patel et al. [8] used a higherorder theory to study the static and dynamic characteristics of thick composite laminates exposed to hygrothermal environment. The formulation accounts for the nonlinear variation of the in-plane and transverse displacements through the thickness, and abrupt discontinuity in slope of the in-plane displacements at any interface. Rao and Sinha [9] studied the effects of moisture and temperature on the bending characteristics of thick multidirectional fibrous composite plates. The finite element analysis accounts for the hygrothermal strains and reduced elastic properties of multidirectional composites at an elevated moisture concentration and temperature. Deflections and stresses are evaluated for thick multidirectional composite plates under uniform and linearly varying through-the-thickness moisture concentration and temperature. Results reveal the 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 supported cross-ply and angle-ply laminated composite plates under hygrothermal environment. Higher order shear deformation theory (HSDT), von-Karman nonlinear kinematics, finite double Chebyshev series and quadratic extrapolation technique are 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 type elastic foundation as shown in Fig. 1. Based on the global higher order shear deformation theory with cubic variation of inplane displacements through the thickness and constant transverse displacement, the displacement field at a point in the laminated plate is expressed as (Kant and Swaminathan [10])

9 9 9 9 9 8 8 8 8 8 > > > > = = = = = > < u0 ðx; yÞ > < wx ðx; yÞ > < u1 ðx; yÞ > < /x ðx; yÞ > < Uðx; y; zÞ > ¼ v 0 ðx; yÞ þ z wy ðx; yÞ þ z2 v 1 ðx; yÞ þ z3 /y ðx; yÞ Vðx; y; zÞ > > > > > > > > > ; ; ; ; ; > : : : : : w0 ðx; yÞ 0 0 0 Wðx; y; zÞ

ð1Þ

where, the parameters u0 ; v 0 and w0 are the in-plane and transverse displacements of a point ðx; yÞ on the middle plane of the plate, respectively. The functions wx and wy are rotations of the normal to the middle plane about y- and x-axes, respectively. The parameters u1 ; v 1 ; /x and /y are the higher order terms representing higher-order transverse cross-sectional deformation modes.

a h/2 h/2

b θ

x, u0

Winkler and nonlinear foundation (k1, k2) y, v 0 Shear layer (k3) z, w0 Fig. 1. Geometry of the laminated composite rectangular plate resting on nonlinear Pasternak type elastic foundation.

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

9 > > > > > > =

8 ex > > > > > > < ey

c

xy > > > > > cyz > > > > > > > ; :

¼

cxz

8 o e > > > xo > > > < ey

co

9 > > > > > > =

xy > > > > > coyz > > > > > > > ; :

þz

coxz

8 jx > > > > > > < jy

j

9 > > > > > > =

xy > > > > > 2v 1 > > > > > > > ; : 2u1

þ z2

8 1 e > > > x1 > > > < ey

c1

9 > > > > > > =

xy > > > > > > > > 3/ > y> > > ; : 3/x

þ z3

8 1 j > > > x1 > > > < jy

j1

9 > > > > > > =

xy > > > > > > > > 0 > > > > ; : 0

ð2Þ

where,

9 > > > > > > =

8 > > > > > > <

8 > > > > > > > > > > <

@u0 @x

þ 12

@w 2 0

@x

eox  2 @v 0 0 þ 12 @w eoy @y @y  coxy ¼ @u0 þ @v 0 þ @w0  @w0 > > > @y @x @x @y > o > > > > > > > > > > cyz > @w0 > > > wy þ @y : o ; > > > cxz > : 8 > <

9 > =

8 > > <

8 > <

9

8

8 > <

9

@wx @x @wy @y

9 > > > > > > > > > > =

0 wx þ @w @x 9 > > =

ð3Þ

> > > > > > > > > > ;

jx jy ¼ > > > > : ; : @wx @wy > jxy ; > þ @y

@x

ð4Þ

9

1 > > @u e1x > > @x = = > < @v 1 1 ey ¼ @y > > ; > > : 1 > ; : @u1 þ @ v 1 > cxy @y @x

8

ð5Þ

9

@/

x > > j1x > @x > = = > < @/ y j1y ¼ @y > > > : 1 > ; : @/x @/y > jxy ; > þ

@y

ð6Þ

@x

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

8 rx > > > > > > < ry

9 > > > > > > =

sxy > > > > > syz > > > > > > > : sxz ;k

2

Q 11

6 6 Q 12 6 ¼6 6 Q 16 6 40 0

Q 12

Q 16

0

Q 22

Q 26

0

Q 26 0

Q 66 0

0 Q 44

0

0

Q 45

9 38 ex  ax DT  bx DC > > > > > > 7 > e  a DT  b DC > > > > 0 7> y y y = 7< 7 0 7 cxy  axy DT  bxy DC > > > 7> > > cyz > Q 45 5 > > > > > ; : cxz Q 55 k k 0

ð7Þ

where, Q ij 0 s are transformed reduced stiffness coefficients. DT ¼ ðT  T 0 Þ = Applied temperature  reference temperature. DC ¼ ðC  C 0 Þ = 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 directions and can be expressed as

9 2 3 bx > m2 n2   = 6 7 a11 ; b11 by ¼ 4 n2 m2 5 > a22 ; b22 axy ; bxy ; 2mn 2mn

8 > < ax ; ay ; > : where,

m ¼ Cos h; n ¼ Sin h;

h ¼ fibre orientation angle:

ð8Þ

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

2637

The elastic and hygrothermal properties of the composite material are dependent on the temperature and moisture concentration. It becomes important to consider the temperature and moisture dependent properties of the polymer based fibre reinforced composite material in order to predict the response of the laminated composite plate in hygrothermal environment, accurately. The material properties are evaluated utilizing micro-mechanics model. Since, the effect of temperature and moisture is dominant in polymer based matrix material; the degradation of the composite material properties is estimated by degrading the matrix property only. The matrix mechanical property retention ratio is expressed as (Chamis and Sinclair [11])

 Fm ¼

T gw  T T go  T 0

12 ð9Þ

where T ¼ T 0 þ DT and T is the temperature at which material property is to be predicted, T 0 is the reference temperature, DT is the increase in temperature from reference temperature, T gw and T go are the glass transition temperatures for wet and reference dry conditions, respectively. The glass transition temperature for wet material is determined as (Chamis [12])

T gw ¼ ð0:005C 2  0:10C þ 1:0ÞT go

ð10Þ

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

E11 ¼ Ef 1 V f þ F m Em V m E22

ð11Þ

qffiffiffiffiffiffi  ¼ 1:0  V f F m Em þ

pffiffiffiffiffiffi F m Em V f   pffiffiffiffiffiffi 1:0  V f 1:0  F mE Em

ð12Þ

pffiffiffiffiffiffi F m Gm V f   pffiffiffiffiffiffi 1:0  V f 1:0  FGm Gm

ð13Þ

f2

G12

qffiffiffiffiffiffi ¼ 1:0  V f F m Gm þ 

f 12

m12 ¼ mf 12 V f þ mm V m

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

F h ¼ 1=F m

ð15Þ

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

a11 ¼

Ef 1 V f af 1 þ F m Em V m F h am Ef 1 V f þ F m Em V m

a22 ¼ af 2 V f þ V m F h am þ

V f V m ðmf 12 F m Em  mm Ef 1 Þ ðaf 1  F h am Þ Ef 1 V f þ F m Em V m

ð16Þ

ð17Þ

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

b11 ¼

Ef 1 V f bf 1 þ F m Em V m F h bm Ef 1 V f þ F m Em V m

ð18Þ

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

b22 ¼

 V m F h bm ð1 þ mm ÞðEf 1 V f þ F m Em V m Þ  ðmf 12 V f þ mm V m ÞEm F m Ef 1 V f þ F m Em V m

ð19Þ

Eqs. (9)–(19) presented herein are used to evaluate the stiffness coefficients in Eq. (7), thermal expansion coefficients and hygroscopic coefficients in Eq. (8). The nonlinear elastic foundation is considered as Pasternak type with foundation nonlinearity. It can be modeled as a nonlinear spring and a shear layer. The up-thrust due to nonlinear elastic foundation (Pasternak type) can be expressed as (Nath et al. [16])

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R ¼ K1W þ K2W 3  K3

@2W @2W þ @x2 @y2

! ð20Þ

where, K 1 ; K 2 and K 3 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 @Nxy þ ¼0 @x @y

ð21Þ

@Ny @Nxy þ ¼0 @y @x

ð22Þ

@Q x @Q y @ 2 w0 @ 2 w0 @ 2 w0 þ Ny þ 2N xy þ þ Nx þqR¼0 2 2 @x @y @x @y @x@y

ð23Þ

@Mx @Mxy þ  Qx ¼ 0 @x @y

ð24Þ

@My @Mxy þ  Qy ¼ 0 @y @x

ð25Þ



@Nx @Nxy þ  2Sx ¼ 0 @x @y

ð26Þ

@Ny @Nxy þ  2Sy ¼ 0 @y @x

ð27Þ



@Mx @Mxy þ  3Q x ¼ 0 @x @y

ð28Þ

@My @Mxy þ  3Q y ¼ 0 @y @x

ð29Þ

The associated admissible boundary conditions obtained are of the form at x ¼  2a

u0 ¼ 0 or Nx ¼ 0;

v 0 ¼ 0 or Nxy ¼ 0;

wx ¼ 0 or M x ¼ 0 wy ¼ 0 or M xy 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;

ð30Þ

/y ¼ 0 or M xy ¼ 0

v 1 ¼ 0 or Nxy ¼ 0; at x ¼  2b

u0 ¼ 0 or Nxy ¼ 0;

wx ¼ 0 or M xy ¼ 0

v 0 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

v 1 ¼ 0 or

N xy ¼ 0; Ny ¼ 0;

/y ¼ 0 or

M y

ð31Þ

¼0

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

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

9 8 Nx > > > > > > > > > Ny > > > > > > > > > > > N > xy > > > > > > > > Mx > > > > > > > > 2 > > My > > > A > > > > > = 6B < M xy > 6 ¼6  > > 4D > Nx > > > >  > > > > E > > Ny > > > >  > > > > > N > > xy > > > >  > > > > > Mx > > > > > >  > > > > My > > > > ; :  > M xy

B D D E E

F

F

H

8 T 9 8 m 9 Nx > 8 0 9 > > > Nx > > > > ex > > > > > T > > > > > > > > > > > Ny > > > > > > > e0 > Nm > > > > > > y > > > > > > y > > > > > T > > > > > > m > > > > > > > N > > > 0 > > > xy > N > > > > > c xy > > > > > xy > > > > > > > T > > > > > > m > > > > > > M > > > > > > M x j > x > > > > x > > > > > > > > > > > > > T m > > > 3> > > M > > > > > > M j y > > > y > E > y > > > > > > > > > > > > > > > > > T > m > = = < 7 < < Mxy M xy = F 7 jxy   7 1 > > NT > > H 5> N m > > > ex > x > x > > > > > > > > > > > > 1 > > > > > > > m > T e > > > > > J > N > > > Ny > > > y > y > > > > > > > > > > > > > > > > 1 > m > > > > > > c > > > > > > NT > > xy > N > > > xy > > > > > > xy > > > > > > > > > > > 1 m > > > > > > T > >M > > jx > > > x > > > > > > M > > > > > > > > > > > > > xT > > M m > > j1y > > > > > > > > > > > > > y > > > My > > > ; > > > : 1 > > ; > : m > > jxy M xy : T ; Mxy

2639

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 ; F ij ; Hij ; J ij Þ ¼

n Z X k¼1

zk

zk1

ðkÞ

Q ij ð1; z; z2 ; z3 ; z4 ; z5 ; z6 Þdz;

ðfor i; j ¼ 1; 2; 6Þ

ð33Þ

Transverse shear stress resultants are expressed as

9 8 8 9 0 wy þ @w > Qy > > > > @y > > > > > > > > > > > > @w0 > > > > 2 3 Q x > wx þ @x > > > > > > > > A B D > > > > = = < < Sy 6 7 2v 1 ¼ 4B D E 5 > > > > > 2u1 > > Sx > > > > D E F > > > > > > > > > > > > > Q > > > > y> 3/ > > > y > > : ; ; : Qx 3/x

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 ; F ij Þ ¼

n Z X k¼1

zk zk1

ðkÞ

Q ij ð1; z; z2 ; z3 ; z4 Þdz;

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 obtained and expressed as

9 8 T 2 > > N ; M Tx ; NT MT > Q 11 x ; x > = X < x n Z zk T T T T 6 Ny ; My ; Ny ; My ¼ 4 Q 12 > k¼1 zk1 > > ; : NT ; M T ; N T ; MT > Q 16 xy xy xy xy

Q 12 Q 22 Q 26

9 38 > ax > Q 16 < = 7 2 3 Q 26 5 ay DTð1; z; z ; z Þdz > > ; : axy Q 66

ð36Þ

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

8 m 2 m m m 9 > Q 11 = X < Nx ; Mx ; Nx ; Mx > n Z zk m m m m 6 Ny ; My ; Ny ; My ¼ 4 Q 12 > > ; k¼1 zk1 : m Nxy ; M m N m Mm Q 16 xy ; xy ; xy

Q 12 Q 22 Q 26

9 38 Q 16 > = < bx > 7 2 3 Q 26 5 by DCð1; z; z ; z Þdz > > ; : b Q 66 xy

ð37Þ

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

ðLa þ Lb þ Lc Þd þ Q  R ¼ 0 where,

La ¼ La1

@2 @2 @2 @ @ þ La2 2 þ La3 þ La6 þ La4 þ La5 2 @x @y @x @y @x@y

Lb ¼ Lb1

@2 @2 @2 þ Lb2 2 þ Lb3 2 @x @y @x@y

ð38Þ

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Lc ¼ Lc1

@2 @2 @2 þ Lc2 2 þ Lc3 2 @x @y @x@y

 d¼ u

v

w wx

wy

v 1

1 u

Q ¼ ½0 0 Q 0 0 0 0 0 0T ;

/x

/y

T

R ¼ ½0 0 R 0 0 0 0 0 0T

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

R ¼ k1 w þ k2 w3  k3

@2w @X 2

þ

! @2w @Y 2

ð39Þ

where k1 ; k2 and k3 are non-dimensional Winkler, nonlinear and shear foundation parameters. La1  La6 ; Lb1  Lb3 and Lc1  Lc3 used in Eq. (38) are defined in Appendix A. The non-dimensional parameters used in the above formulation are described in Appendix B. The admissible boundary conditions obtained from Eqs. (30) and (31) are expressed in non-dimensional form as Simply supported immovable edge (S)

y ¼ u  y ¼ Mx ¼ M  ¼ 0 at X ¼ 1  1 ¼ v 1 ¼ / u¼v ¼w¼w x

ð40Þ

x ¼ u  x ¼ My ¼ M  ¼ 0 at Y ¼ 1  1 ¼ v 1 ¼ / u¼v ¼w¼w y

ð41Þ

Clamped immovable edge (C)

x ¼ w y ¼ u x ¼ /  y ¼ 0 at X; Y ¼ 1  1 ¼ v 1 ¼ / u¼v ¼w¼w

ð42Þ

Free edge (F)

Nx ¼ Nxy ¼ Mx ¼ M xy ¼ Nx ¼ Nxy ¼ M x ¼ M xy ¼ Q x ¼ 0 at X ¼ 1

ð43Þ

Ny ¼ Nxy ¼ My ¼ M xy ¼ Ny ¼ Nxy 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 analytical technique. The coupled nonlinear equations are linearized utilizing total linearization scheme based on quadratic extrapolation 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 loading Q is approximated in space domain by finite degree Chebyshev polynomial. A typical displacement/loading function nðx; yÞ is expressed, using finite degree Chebyshev polynomial (Fox and Parker [17]) as

nðx; yÞ ¼

M X N X i¼0

dij nij T i ðxÞT j ðyÞ :

ð45Þ

j¼0

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

nrs ij ¼

Mr X Ns X i¼0

dij nrs ij T i ðxÞT j ðyÞ;

ð46Þ

j¼0

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

8 > < 0:25; i ¼ 0; j ¼ 0 dij ¼ 0:50; i ¼ 0; j – 0 or i – 0; j ¼ 0 > : 1:0; i – 0; j – 0 The derivative function nrs ij is evaluated using the recurrence relations (Fox and Parker [17]) ðr1Þ;s

rs nrs ði1Þ;j ¼ nðiþ1Þ;j þ 2ini;j

r;ðs1Þ

rs nrs i;ðj1Þ ¼ ni;ðjþ1Þ þ 2jni;j

ð47Þ

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

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

" GJ ¼

Mr X N X i¼0

#" M X Ns X

dij nrij T i ðxÞT j ðyÞ

j¼0

J

i¼0

# dij nsij T i ðxÞT j ðyÞ

j¼0

ð48Þ J

where,

ðnij ÞJ ¼ g1 ðnij ÞJ1 þ g2 ðnij ÞJ2 þ g3 ðnij ÞJ3 During initial steps of marching variables, the coefficients g1 ; g2 and g3 of the quadratic extrapolation scheme of linearization take 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

T i ðxÞT j ðyÞT k ðxÞT l ðyÞ ¼ ½T ðiþkÞ ðxÞT ðjþ1Þ ðyÞ þ T ðiþkÞ ðxÞT ðj1Þ ðyÞ þ T ðikÞ ðxÞT ðjþ1Þ ðyÞ þ T ðikÞ ðxÞT ðj1Þ ðyÞ=4

ð49Þ

Using the procedures described herein, the set of governing nonlinear equilibrium equations (38) is linearized and discretized in space domain and finally expressed in form of a set of linear simultaneous equations as M2 N2 XX i¼0

 xij ; w  yij ; u  xij ; /  yij ; Q ij ÞT i ðxÞT j ðyÞ ¼ 0;  1ij ; v 1ij ; / F k ðuij ; v ij ; wij ; w

k ¼ 1; 9

ð50Þ

j¼0

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 (loading) and transferred to the right side so that the left side matrix remains invariant with respect to the loading. Thus, the load vector 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 regression analysis gives

d ¼ ðAT AÞ1 AT P d ¼ BP and the values of the coefficients of the displacement vector ‘d’ obtained are put into Eq. (45) to evaluate the displacements at the desired location on the mid-plane of the plate.

0.6 SSSS, a/b=1, a/h=10 ΔC=1%, [0/90/0/90] 0.4 Wc

M=N 5 6 7 8 9 10 11

0.2

0 0

10

20

30 Q

40

50

60

Fig. 2. Convergence of transverse central deflection of simply supported, square [0/90/0/90] anti-symmetric cross-ply laminated composite (a/h = 10) plate subjected to uniform transverse pressure in hygroscopic environment ðDC ¼ 1%Þ.

2642

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

4. Results and discussions In order to show the accuracy and efficiency of the present solution methodology, the results of convergence study performed on a simply supported moderately thick (a/h = 10), square laminated composite plate in hygroscopic environment ðDC ¼ 1%Þ are obtained and presented in Fig. 2. It is observed that convergence is achieved beyond 7–8 terms expansion of variables in Chebyshev series. In the present study results are obtained using 9 terms expansion of the variables in Chebyshev 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 composite plate subjected to hygro-thermo-mechanical loading is obtained for different hygrothermal conditions and shown along with the results due to Shen [7] in Fig. 3. It is observed that the results are in very good agreement. Table 1 shows the comparisons of 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 composite plate subjected to uniform transverse pressure and supported on elastic foundation ðk1 ¼ k2 ¼ k3 ¼ 50Þ with the results due to Malekzadeh and Setoodeh [19]. It is noticed that the results are in reasonably good agreement. It can be seen that the results due to present analytical technique agree well with the results available in open literature and the present solution methodology may be efficiently used for the nonlinear flexural analysis of the laminated composite plates subjected to different loading conditions. The material properties are taken directly from the reference papers and are not mentioned here for the sake of brevity.

1.2 a/h=10, a/b=1, SSSS, [45/- 45/45/- 45],V f =0.6

1.0

0.8 Wc

ΔT( 0C), ΔC(%)

0.6

0, 0, Ref. [ 7 ]

0.4

100, 1, Ref. [ 7 ]

0, 0, Present 100, 1, Present 200, 2, Ref. [ 7 ]

0.2

200, 2, Present

0.0 0

50

100

150 Q

200

250

300

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

Table 1 Comparison of nonlinear central deflection ðW c Þ, stress resultant ðN x Þ 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]

900 900 1500 1500

0.5656 0.5702 0.8435 0.8491

0.0895 0.0887 0.2006 0.1981

2.5992 2.3310 3.8544 3.4528

‘a’ ‘b’ ‘a’ ‘b’

2

100

[0/90/90/0]

900 900 1500 1500

0.4981 0.5085 0.7638 0.7795

0.0007 0.0006 0.0015 0.0014

5.1370 5.6380 7.7284 8.1858

‘a’ ‘b’ ‘a’ ‘b’

3

10

[0/90/0/90]

900 900 1500 1500

0.7640 0.7600 1.0550 1.0379

0.1359 0.1440 0.2757 0.2842

2.0309 1.8965 1.5992 1.1947

‘a’ ‘b’ ‘a’ ‘b’

4

100

[0/90/0/90]

900 900 1500 1500

0.7193 0.7281 1.0130 1.0170

0.0010 0.0009 0.0023 0.0022

4.3294 4.9912 4.5825 5.0380

‘a’ ‘b’ ‘a’ ‘b’

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

2643

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

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

Ef 1 ¼ 220 GPa; Ef 2 ¼ 13:79 GPa; Em ¼ 3:45 GPa; Gf 12 ¼ 8:97 GPa; 6 

6 

mf 12 ¼ 0:20; mm ¼ 0:35;

6 

af 1 ¼ 0:99  10 = C; af 2 ¼ 10:08  10 = C; am ¼ 72:0  10 = C; bm ¼ 0:33; T go ¼ 216  C The effects of temperature, moisture concentration and their combination on the non-dimensional displacement response of the anti-symmetric cross ply [0/90/0/90] plate are shown in Fig. 4. It is observed that central deflection increases with increase in moisture concentration, temperature and increase in both simultaneously. The increase is highest when hygrothermal condition is taken and it is least when only effect of moisture is considered. At Q ¼ 200, the increase in central deflection 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% corresponding to DC ¼ 1% and DT ¼ 100  C.

0.8

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

Wc

0.6

0.4

ΔT( 0C), ΔC(%) 0, 0.0 0, 1.0 100, 0.0 100,1.0

0.2

0.0 0

50

100

150

200

250

Q Fig. 4. Effect of temperature, moisture concentration and their combination on the non-dimensional central deflection of clamped, square, anti-symmetric, cross- ply laminated composite plate (a/h = 10) subjected to uniform transverse loading.

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

1.2

Wc

0.9

ΔT( 0C), ΔC(%)

0.6

150, 1.5 100, 1.0 50, 0.5 0, 0.0

0.3

0.0 0

150

300

450

600

750

Q Fig. 5. Effect of hygrothermal environment on transverse central deflection of clamped, square [0/90/0/90] laminated composite plate (a/h = 20) subjected to uniform transverse pressure.

2644

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

1.0 a/h=10, a/b=1, CCCC, V f = 0.6, [45/-45/45/-45]

0.8

Wc

0.6

ΔT(0C), ΔC(%)

0.4

100, 1.0 50, 0.5

0.2 0,

0

0.0 0

50

100

150 Q

200

250

300

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

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

0.8 0.6 Wc

Vf 0.4 0.5 0.6

0.4 0.2 0.0 0

50

100

150 Q

200

250

300

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.

0.8 a/h=10, a/b=1, V f = 0.6 CCCC,[45/-45/-45/45] ΔT=50 C,ΔC=0.5% 0

Wc

0.6

k1 , k 2 , k 3

0.4

0,0,0 75,0,0 75,0,30 75,100,30 75,200,30

0.2

0.0 0

50

100

150 Q

200

250

300

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.

2645

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

0.8 a/h=10, a/b=1 CCCC, V f = 0.6, [45/-45/-45/45] ΔT=500C, ΔC=0.5%

Wc

0.6

0.4

k1,-k2, k3 0.2

75, 0, 30 75, 100, 30, 75, 200, 30

0.0 0

50

100

150

200

250

Q 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.

0.3 a/h=10, a/b=1 V f = 0.6, [0/90/90/0] ΔT = 500C, ΔC = 0.5% k1=75, k2=100, k3=30

Wc

0.2

CFCC SSSS CSSS CCSS CSCC CCCC

0.1

0 0

50

100

150

200

250

Q Fig. 10. Effect of boundary conditions on the transverse central deflection of elastically supported square, symmetric cross-ply laminated composite plate subjected 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] laminated composite plate subjected to hygro-thermo-mechanical loading. Appreciable increase in deflection is observed at temperature closer to glass transition temperature, indicating the reduction in stiffness of the plate at the increased temperature and moisture. The effect of temperature and moisture concentration on transverse central deflection of clamped, anti-symmetric angleply [45/45/45/45] laminate at fiber volume fraction 0.6 is shown in Fig. 6. It is observed that with increase in temperature and moisture concentration, deflection increases as expected. Fig. 7 shows the effect of fiber volume fraction on transverse central deflection of clamped, moderately thick, anti-symmetric cross-ply [0/90/0/90] laminate subjected to hygro-thermo-mechanical loading ðDT ¼ 100  C and DC ¼ 1%Þ and it is observed that with increase in fiber volume fraction, transverse central deflection decreases. It is due to the fact that with increase in fiber volume fraction, the stiffness of the plate increases.

2646

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

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 deflection of the elastically supported plate is lower than the plate without elastic foundation. It is also clear from the figure that effect of shear layer foundation parameter ðk3 Þ is more pronounced than the effect of Winkler foundation parameter ðk1 Þ. The effect of foundation nonlinearity ðk2 Þ is appreciable only at higher loads. With the increase in hardening type of nonlinearity ðþk2 Þ, deflection of the laminated composite plate decreases. Fig. 9 presents the effect of softening type of foundation nonlinearity ðk2 Þ on the displacement response of symmetric angle-ply [45/45/45/45] plate. It can be observed that with increase in ðk2 Þ value, the deflection increases and at higher values 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 moderately thick symmetric cross-ply laminate under hygro-thermo-mechanical loading. The deflection of clamped plate is least and the plate with three edges clamped and one free is highest. It is observed that increase in degree of fixity, decreases the deflection 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 to hygro-thermo-mechanical loading is obtained using fast converging finite double Chebyshev series. It is observed that hygrothermal dependent mechanical and thermal properties greatly affect the flexural behavior of the laminated composite plates. The flexural response of the laminated composite plate deteriorates considerably with the increase in temperature and moisture concentration and this hygrothermal environment becomes more detrimental as the working temperature reaches closer to the glass transition temperature. The deflection of the elastically supported laminated composite plate is smaller and the presence of shear layer in the foundation (Pasternak type) is relatively more predominant and has significant effect on the displacement response of the plate. The effects of various boundary conditions are also discussed, showing the applicability of the present solution methodology. Appendix A

2

La1

1

6 6 A16 6 A22 6 6 0 6 6 hB11 6D 6 11 6 hB16 6 ¼ 6 D22 6 6 1 6 6D 6 16 6 D22 6 6 E11 6 hD11 4 E16 hD22

A16 A11

0

B11 A11 h

B16 A11 h

A66 A22

0

B16 A22 h

B66 A22 h

0

La1 ð3; 3Þ 0

0 1

hB66 D22

0

D16 D11

hB16 D11

D11 A11 h2 D16 A22 h2

D16 A11 h2 D66 A22 h2

E11 A11 h3 E16 A22 h3

0

0

0

0

D16 D11

E11 hD11

E16 hD11

D16 D22

D66 D22

E16 hD22

E66 hD22

F 11 h2 D11 F 16 h2 D22 H11 h3 D11 H16 h3 D22

F 16 h2 D11 F 66 h2 D22 H16 h3 D11 H66 h3 D22

F 11 h D11 F 16 h2 D22 H11 h3 D11 H16 h3 D22 J 11 h4 D11 J 16 h4 D22

0

E11 hD11

E16 hD11

D66 D22

0

E16 hD22

E66 hD22

E16 hD11

0

E66 hD22

0

F 11 h2 D11 F 16 h2 D22

F 16 h2 D11 F 66 h2 D22

2

E16 A11 h3 E66 A22 h3

3

7 7 7 7 0 7 7 F 16 7 7 2 h D11 7 F 66 7 7 h2 D22 7 7 H16 7 h3 D11 7 7 H66 7 h3 D22 7 7 J 16 7 h4 D11 7 5 J 66 h4 D22

where,

La1 ð3; 3Þ ¼

2

La2

  bA55  A11 NTx þ N m x bA22

A66 A11

6 6 A26 6 A22 6 6 0 6 6 hB 6 66 6 D11 6 6 hB 2 6 26 ¼ k 6 D22 6 D66 6D 6 11 6D 6 26 6 D22 6 6 E66 6 hD11 4 E26 hD22

A26 A11

0

B66 A11 h

B26 A11 h

D66 A11 h2

D26 A11 h2

E66 A11 h3

1

0

B26 A22 h

B22 A22 h

D26 A22 h2

D22 A22 h2

E26 A22 h3

0

La2 ð3; 3Þ

0

0

0

0

0

hB26 D11

0

D66 D11

D26 D11

E66 hD11

E26 hD11

F 66 h2 D11

hB22 D22

0

D26 D22

1

E26 hD22

E22 hD22

F 26 h2 D22

D26 D11

0

E66 hD11

E26 hD11

F 66 h2 D11

F 26 h2 D11

H66 h3 D11

1

0

E26 hD22

E22 hD22

F 26 h2 D22

F 22 h2 D22

H26 h3 D22

E26 hD11

0

F 66 h2 D11

F 26 h2 D11

H66 h3 D11

H26 h3 D11

J 66 h4 D11

E22 hD22

0

F 26 h2 D22

F 22 h2 D22

H26 h3 D22

H22 h3 D22

J 26 h4 D22

E26 A11 h3

3

7 7 7 7 0 7 7 7 F 26 7 h2 D11 7 7 F 22 7 7 h2 D22 7 H26 7 7 h3 D11 7 7 H22 7 h3 D22 7 7 J 26 7 7 h4 D11 5 E22 A22 h3

J 22 h4 D22

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

where,

La2 ð3; 3Þ ¼

2

La3

  bA44  A22 NTy þ Nm y bA22

2A16

6 A11 6 A12 þA66 6 A 22 6 6 6 0 6 6 2hB16 6 D11 6 6 hðB12 þB66 Þ 6 ¼ k6 D22 6 6 2D16 6 D11 6 6 D12 þD66 6 D22 6 6 2E16 6 6 hD11 4

A12 þA66 A11

0

2B16 A11 h

B12 þB66 A11 h

2D16 A11 h2

D12 þD66 A11 h2

2E16 A11 h3

2A26 A22

0

B12 þB66 A22 h

2B26 A22 h

D12 þD66 A22 h2

2D26 A22 h2

E12 þE66 A22 h3

0

La3 ð3; 3Þ

0

0

0

0

0

hðB12 þB66 Þ D11

0

2D16 D11

D12 þD66 D11

2E16 hD11

E12 þE66 hD11

2F 16 h2 D11

2hB26 D22

0

D12 þD66 D22

2D26 D22

E12 þE66 hD22

2E26 hD22

F 12 þF 66 h2 D22

D12 þD66 D11

0

2E16 hD11

E12 þE66 hD11

2F 16 h2 D11

F 12 þF 66 h2 D11

2H16 h3 D11

2D26 D22

0

E12 þE66 hD22

2E26 hD22

F 12 þF 66 h2 D22

2F 26 h2 D22

H12 þH66 h3 D22

E12 þE66 hD11

0

2F 16 h2 D11

F 12 þF 66 h2 D11

2H16 h3 D11

H12 þH66 h3 D11

2J 16 h4 D11

2E26 hD22

0

F 12 þF 66 h2 D22

2F 26 h2 D22

H12 þH66 h3 D22

2H26 h3 D22

J 12 þJ 66 h4 D22

E12 þE66 hD22

where,

La3 ð3; 3Þ ¼

2

La4

0 60 6 6 60 6 6 60 6 6 6 0 ¼6 6 6 60 6 6 60 6 6 60 4

2

La5

  2bA45  2A66 NTxy þ Nm xy bA22

0

0

0

0

0

0

0

0

0

0

0

A55 b 2A22

A45 b 2A22

B55 b A22 h

B45 b A22 h

3D55 b 2A22 h2

0

A55 bh2 2D11

0

0

0

0

0

0

A45 bh2 2D22

0

0

0

0

0

0

B55 bh D11

0

0

0

0

0

0

B45 bh D22

0

0

0

0

0

0

3D55 b 2D11

0

0

0

0

0

0 0

3D45 b 2D22

0

0

0

0

0

0 60 6 6 60 6 6 60 6 6 6 0 ¼6 6 6 60 6 6 60 6 6 60 4 0

0

0

0

0

0

0 7 7 7 7 7 7 0 7 7 7 7 0 7 7 7 0 7 7 7 0 7 7 7 0 7 5

3D45 b 2A22 h2

0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

A45 kb 2A22

A44 kb 2A22

B45 kb A22 h

B44 kb A22 h

3D45 kb 2A22 h2

0

A45 kbh2 2D11

0

0

0

0

0

2

3

0

0

A44 kbh 2D22

0

0

0

0

0

0

B45 kbh D11

0

0

0

0

0

0

B44 kbh D22

0

0

0

0

0

0

3D45 kb 2D11

0

0

0

0

0

0

3D44 kb 2D22

0

0

0

0

0

3 0 0 7 7 3D44 kb 7 2 7 2A22 h 7 7 0 7 7 7 7 0 7 7 7 0 7 7 7 0 7 7 7 0 7 5 0

E12 þE66 A11 h3

3

7 7 7 7 7 0 7 7 F 12 þF 66 7 7 h2 D11 7 2F 26 7 7 h2 D22 7 7 H12 þH66 7 h3 D11 7 7 2H26 7 7 3 h D22 7 J 12 þJ 66 7 7 h4 D11 7 5 2E26 A22 h3

2J 26 h4 D22

2647

2648

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

2

La6

0 60 6 60 6 6 60 6 6 60 6 ¼6 60 6 6 60 6 6 60 6 4 0

2

Lb1

0 6 6 60 6 6 60 6 60 6 6 ¼6 60 6 60 6 6 60 6 6 60 4

0 0 0 0

0 0 0 0 0 0 2 2 0  A554Db11h 2 2

0 0  A454Db22h 2

45 b h  B2D 22

0 0

 3D4D5511b

2

 3D4D4522b

2

0 0

0 0 0 0 0 0 0

0 0

2 b



@w @X

16 þ kA A11

0 0 0

0 0 0

55 b h 45 b h  B2D  B2D  3D4D5511b 11 11

2 2

45 b h 44 b h  B2D  B2D  3D4D4522b 22 22

2

45 b h  B2D 11 2

44 b h  B2D 22

 3D4D4511b

2

 3D4D4422b

2

@w @Y

0 0 0

2 2

 A444Db22h

2

0 0

0 0 0

 A454Db11h

55 b h  B2D 11

0 0

0

0 0 0



2

2

2

2

2

2

 DD5511b

2

 DD4511b

2

55 b  3E 2D11 h

 DD4522b

2

 DD4422b

2

45 b  3E 2D22 h

55 b  3E 2D11 h

2

45 b  3E 2D11 h

2

9F 55 b  4D h2

2

44 b  3E 2D22 h

2

9F 45 b  4D h2

45 b  3E 2D22 h

2

2

2

11

0 0 0 0 0 0

2

22

3

7 7 7 7 3D45 b2 7  4D11 7 7 2 7  3D4D4422b 7 7 2 7 45 b 7  3E 2D11 h 7 2 7 44 b 7  3E 2D22 h 7 7 9F 45 b2 7  4D 2 7 11 h 5 9F 44 b2  4D h2 22

3

7 7 0 0 0 0 0 07 7 7 Lb1 ð3; 3Þ 0 0 0 0 0 07 7   2h 0 0 0 0 0 07 B11 @w þ kB16 @w 7 bD11 @X @Y 7   2h @w @w 7 0 0 0 0 0 0 B þ kB 16 @X 66 @Y bD22 7   7 2 @w @w 7 0 0 0 0 0 0 D þ kD 11 16 bD11 @X @Y 7   7 2 @w @w 0 0 0 0 0 07 D16 @X þ kD66 @Y bD22 7   7 2 0 0 0 0 0 07 E11 @w þ kE16 @w hbD11 @X @Y 5   2 @w @w 0 0 0 0 0 0 E þ kE 16 66 hbD22 @X @Y 2 bA22



A16 @w þ kA66 @w @X @Y



       x x 1 1 @w @w u u w w @u @u 2 where, Lb1 ð3; 3Þ ¼ bA222 A11 @X þ kA16 @Y þ A16 @@Xv þ kA12 @@Yv þ hbA2 22 B11 @@X þ kB16 @@Y þ B16 @Xy þ kB12 @Yy þ h2 bA D11 @@X þ kD16 @@Y þ 22         @ / @ / v1 v1 /x /x 2 D16 @@X þ kD12 @@Y Þ þ h3 bA E11 @@X þ kE16 @@Y þ E16 @Xy þ kE12 @Xy 22

2

Lb2

0 6 60 6 6 60 6 6 60 6 6 6 ¼ 60 6 60 6 6 6 60 6 6 60 4 0

0 0 0 0 0 0 0 0 0

2k2 bA11



A66 @w þ kA26 @w @X @Y



0 0 0 0 0 0

3

7 0 0 0 0 0 07 7 7 0 0 0 0 0 07 Lb2 ð3; 3Þ 7   7 2k2 h B66 @w þ kB26 @w 0 0 0 0 0 07 bD11 @X @Y 7 7   2k2 h @w @w 7 0 0 0 0 0 0 A þ kA 26 @X 22 @Y 7 bD22 7  2  7 2k @w @w 0 0 0 0 0 0 D þ kD 66 @X 26 @Y 7 bD11 7   7 2k2 @w @w 0 0 0 0 0 0 D þ kD 7 26 @X 22 @Y bD22 7  2  7 2k E66 @w þ kE26 @w 0 0 0 0 0 07 @X @Y bhD11 5   2k2 @w @w E þ kE 0 0 0 0 0 0 26 @X 22 @Y bhD22 2k2 bA22



A26 @w þ kA22 @w @X @Y



       2 x x 1 1 @w @w u u w w 2k2 @u @u 2k2 where, Lb2 ð3; 3Þ ¼ bA A12 @X þ kA26 @Y þ A26 @@Xv þ kA22 @@Yv þ hbA B12 @@X þ kB26 @@Y þ B26 @Xy þ kB22 @Yy þ h22k D12 @@X þ kD26 @@Y þ 22 22 bA22   y y x x 1 1 @/ @/ @v @v @/ @/ 2k2 D26 @X þ kD22 @Y Þ þ h3 bA E12 @X þ kE26 @Y þ E26 @X þ kE22 @Y

2

Lb3

22

0 0

6 60 6 6 60 6 60 6 6 6 ¼ 60 6 60 6 6 60 6 6 60 4 0

0 0 0 0 0 0 0 0

2k bA11

 

2A16 @w þ kðA12 þ A66 Þ @w @X @Y



 @w

0 0 0 0 0 0

3

7 0 0 0 0 0 07 7 7 Lb3 ð3; 3Þ 0 0 0 0 0 07 7   2kh 0 0 0 0 0 07 2B16 @w þ kðB12 þ B66 Þ @w 7 bD11 @X @Y 7   2kh @w @w 7 0 0 0 0 0 0 ðB þ B Þ þ 2kB 12 66 @X 26 @Y 7 bD22 7   2k @w @w 7 0 0 0 0 0 0 2D þ kðD þ D Þ 16 12 66 7 bD11 @X @Y   7 2k 0 0 0 0 0 07 ðD12 þ D66 Þ @w þ 2kD26 @w bD22 @X @Y 7   7 2k @w @w 7 0 0 0 0 0 0 2E þ kðE þ E Þ 16 @X 12 66 @Y bhD11 5   2k @w @w 0 0 0 0 0 0 ðE þ E Þ þ 2kE 12 66 26 bhD22 @X @Y 2k bA22

ðA12 þ A66 Þ @w þ 2kA26 @Y @X

2649

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

      x x @w @w w w @u @u @v 4k where, Lb3 ð3; 3Þ ¼ bA4k22 A16 @X þ kA66 @Y þ A66 @@Xv þ kA26 @Y B16 @@X þ kB66 @@Y þ B66 @Xy þ kB26 @Yy þ þ hbA 22     x x @/ @/ v 1 v 1 / / D66 @@X þ kD26 @@Y Þ þ h3 4k E16 @@X þ kE66 @@Y þ E66 @Xy þ kE26 @Yy bA 22

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

Lc1 ð3; 3Þ ¼

Lc2 ð3; 3Þ ¼

Lc3 ð3; 3Þ ¼

2 2

b A22

2k2 b2 A22 4k b2 A22

A16

 2  2   @w @w @w @w þ k2 A12 þ 2kA16 @X @Y @X @Y

A12

 2  2   @w @w @w @w þ k2 A22 þ 2kA26 @X @Y @X @Y

A16

 2  2   @w @w @w @w þ k2 A26 þ 2kA66 @X @Y @X @Y

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

X¼

2x ; a

q ¼

qa2 ; 4A22 h

Q¼

 m

Nx ; NTx ; N x

¼

a k¼ ; b

a b¼ ; h

qa4

u0 ; h

4E2 h

y ¼ w ; w y

x ¼ w ; w x 

2y ; b

Y¼

4

;u ¼

A11

;

v0 h

v 1 ¼ v 1 h;

 1 ¼ u1 h; u

  Nx ; NTx ; Nm x b

v¼

;

w¼

w0 ; h

 x ¼ / h2 ; / x

 y ¼ / h2 ; / y

    Ny ; NTy ; Nm y b Ny ; NTy ; Nm ; ¼ y A22

    N xy ; NTxy ; Nm xy b T m Nxy ; Nxy ; N xy ¼ ; A66



  m   b N x ; N T y ; Ny  T m Ny ; Ny ; Ny ¼ ; 2 A22 h

  m   Nxy ; NT xy ; N xy b  T m Nxy ; Nxy ; Nxy ¼ ; 2 A66 h



 m

Mx ; M Tx ; Mx



 2 Mx ; M Tx ; Mm x hb

m Nx ; NT x ; Nx





 ¼



 m b Nx ; NT x ; Nx 2

A11 h

;

  2 M y ; MTy ; M m y hb

; My ; M Ty ; Mm ; ¼ y D D  11   22  m 2   b2 M xy ; MTxy ; M m M x ; M T   xy hb x ; Mx m ¼ M xy ; M Txy ; Mm ; M x ; MT ; xy ¼ x ; Mx D66 D11 h ¼

  m   b2 M x ; MT y ; My m ¼ M y ; MT ; M ; y y D22 h

Qx ¼

Q xb ; A55

k1 ¼

k1 a4 ; D11

Qy ¼

Q yb ; A44

k2 ¼

k2 a4 h ; D11











Sx ¼

Sx b ; A55 h

2

  m 2   Mxy ; M T xy ; M xy b m ¼ Mxy ; M T ; M ; xy xy D66 h

Sy ¼

Sy b ; A44 h

Q x ¼

Q x b A55 h

2

;

Q y ¼

Q y b A44 h

2

;



k3 ¼

k3 a2 ; D11

where, k1 ; k2 and k3 are Winkler, nonlinear and shear foundation parameters, respectively.

4k h2 bA22

 1 1 u u D16 @@X þ kD66 @@Y þ

2650

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

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