Large deformation flexural behavior of laminated composite skew plates: An analytical approach

Composite Structures 94 (2012) 3722–3735

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Large deformation flexural behavior of laminated composite skew plates: An analytical approach A.K. Upadhyay, K.K. Shukla ⇑ Department of Applied Mechanics, MNNIT, Allahabad 211 004, India

a r t i c l e

i n f o

Article history: Available online 4 July 2012 Keywords: Skew plate TSDT Nonlinear Flexure Analytical Chebyshev polynomials

a b s t r a c t The paper presents the large deformation flexural response of composite laminated skew plates subjected to uniform transverse pressure. Third order shear deformation theory (TSDT) and von-Karman’s nonlinearity is used for the analysis. Skew domain is mapped into a square domain and finite degree double Chebyshev series is used to discretize the space domain. No grid generation is required in the present solution technique. The nonlinear equations are linearized using quadratic extrapolation technique and the behavior of moderately thick laminated composite skew plates is studied. The effects of geometric nonlinearity, transverse shear, boundary conditions, aspect ratio and modular ratio on the behavior of laminated composite skew plates are discussed in detail. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Wide applications of laminated composite plate as structural elements in aerospace, marine, automobile engineering, etc., has drawn the attention of researchers in this area. Circular, rectangular and skew plates are commonly used in these structures. Circular and rectangular plates have been analyzed by many investigators due to simplicity. Skew plates being one of the important members of the quadrilateral plate family has also got the attention of research community. But, the amount of reported work in this area is relatively less due to its complexity. Most of the works till early nineties were limited to isotropic and orthotropic skew plates [1–5]. Liew and Liu [6] obtained the solution for bending of arbitrary shaped Kirchhoff’s plates with different boundary conditions, using differential cubature method. Liew and Han [7] presented the bending analysis of thick and thin skew plates using differential quadrature method. Rajamohan and Raamachandran [8] used charge simulation method for the analysis of isotropic skew plates subjected to uniform transverse loading. Duan and Mahendran [9] analyzed the large deflection behavior of skew plates subjected to point and uniformly distributed transverse loads using hybrid/ mixed shell element in oblique coordinate system. Singh and Elaghabash [10] used a Ritz type formulation for the linear and nonlinear static analysis of isotropic skew plates based on the First order Shear Deformation Theory (FSDT). Muhammad and Singh [11] presented an energy method for the linear static analysis of ⇑ Corresponding author. Tel.: +91 532 2271206; fax: +91 532 2445101. E-mail addresses: [email protected], [email protected] (K.K. Shukla). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.06.015

rectangular, circular, elliptical and skew plates using FSDT. Malekzadeh and Fiouz [12] obtained the nonlinear static response of orthotropic skew plates with rotationally restrained edges using DQM. The analysis of laminated composite skew plates is complex as it not only involves the oblique boundaries but also the coupling among the stiffness coefficients. Kabir [13] developed a shear locking free robust isoparametric three node triangular finite element for the static analysis of laminated composite moderately thick and thin plates. Sheikh et al. [14] used a new high precision triangular element for the linear analysis of laminated composite plates of different shapes. Karami et al. [15] used the differential quadrature method for the static, stability and free vibration analysis of skew and trapezoidal thin composite plates. Nallim et al. [16] presented a general variational approach using a set of beam characteristic orthogonal polynomials for the static and dynamic analyses of composite laminates with different shapes based on classical Kirchhoff’s assumptions. Malekzadeh and Karami [17] utilized the differential quadrature method for the nonlinear static analysis of laminated composite skew plate based on first order shear deformation theory and von-Karman’s nonlinearity. Numerical results are reported for different skew angles, aspect ratio, plate side to thickness ratio and boundary conditions. Most of the reported works relate to the investigation of the flexural response of isotropic and laminated composite skew plates using numerical techniques. To the best of authors’ knowledge, the nonlinear flexural response of laminated composite skew plate employing higher order shear deformation theory and using analytical techniques is scarce. In this work, the authors have made an attempt to present the nonlinear flexural response of laminated

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composite skew plate using third order shear deformation theory and von-Karman’s nonlinearity. Chebyshev series based analytical solution methodology is employed but grid points are not required in the spatial discretization. Chebyshev polynomials have been used for the analysis of circular and rectangular plates. Alwar and Nath [18] obtained the results for thin circular plate. Nath and Sandeep [19] analyzed the nonlinear boundary value problems in rectangular domain. Using FSDT, Shukla and Nath [20] obtained the analytical solutions for moderately thick laminated composite plates undergoing moderately large deformation. Nath et al. [21] analyzed moderately thick sector plates. Pandey et al. [22] used TSDT for the nonlinear flexural response of laminated composite plates. The equations of equilibrium are transformed from physical to computational domain using linear transformation and chain rule of differentiation. The nonlinear terms are linearized using quadratic extrapolation technique. A detailed parametric study is presented for different skew angles, span to thickness ratio, boundary conditions, aspect ratio and modular ratio to observe the flexural behavior of laminated composite skew plates. 2. Mathematical formulation Based on TSDT with cubic variation of in-plane displacements through the thickness and constant transverse displacement, the displacement field at a point in the laminated composite plate is expressed as [22]; 9 8 9 9 9 9 8 8 8 8 > > > = > = = = = > < u0 ðx;yÞ > < wx ðx;yÞ > < u1 ðx;yÞ > < /x ðx;yÞ > < Uðx;y;zÞ > 2 3 Vðx;y;zÞ ¼ v 0 ðx; yÞ þ z wy ðx;yÞ þ z v 1 ðx;yÞ þ z /y ðx;yÞ > > > > > > > > > > ; : ; ; ; ; : : : : w0 ðx;yÞ 0 0 0 Wðx;y;zÞ ð1Þ where u0, v0 are the in-plane displacements and w0 is the transverse displacement of a point (x, y) on the mid plane of the plate. The functions wx and wy are rotations of the normal to the mid plane about y and x axes, respectively. The parameters u1, v1, /x and /y are the higher order terms in the Taylor’s series expansion, representing higher-order transverse cross-sectional deformation modes. Assuming plane stress condition in the lamina, the constitutive stress–strain relations for kth layer in the laminate is written as:

8 rx > > > > > r > < y

9 > > > > > > =

2

Q 11

6 6 Q 12 6 sxy ¼ 6 6 Q 16 > > > > 6 > > s > > 4 0 yz > > > > : ; sxz k 0

Q 12

Q 16

0

Q 22

Q 26

0

Q 26

Q 66

0

0

0

Q 44

0

0

Q 45

9 38 ex > > > > > > 7>e > > > 0 7> y > < = 7 c 0 7 7 > xy > 7> > c > > > Q 45 5 > > > yz > > : ; Q 55 k cxz k 0

9

8

9

8

9

8

9

8

8 9 2 9 38 ½N > ½A ½B ½D ½E > ½e0  > > > > > > > > > < ½M = 6 ½B ½D ½E ½F 7< ½j > = 7 6 ¼ 7 6  1 > ½N  > > 4 ½D ½E ½F ½H 5> > > > > > ½e  > > :  > : ; ; ½M  ½E ½F ½H ½J ½j1 

ð4aÞ

Transverse shear stress resultants are expressed as:

8 9 Qy > > > > > > > > > Q > > > > > x> > > > > < Sy > =

8 9 wy þ w0;y > > > > > > > > wx þ w0;x > > > 2 3> > > > > > > ½A ½B ½D > > 6 7< 2v 1 = 7 ¼6 ½D ½E ½B 4 5> > > Sx > > > > > > 2u1 > > > > > > > > > > ½D ½E ½F > > > > >  > > Qy > > > 3/y > > > > > > > > > > : > : ; ; Qx 3/x

ð4bÞ

where, T T ½N ¼ ½ Nx Ny Nxy  ; ½M ¼ ½ M x My M xy        T    T ½N  ¼ Nx Ny Nxy ; ½M  ¼ M x M y Mxy  T T ½e0  ¼ e0x e0y c0xy ; ½j ¼ ½ jx jy jxy     T T ½e1  ¼ e1x e1y c1xy ; ½j1  ¼ j1x j1y j1xy

½A; ½B; ½D; ½E; ½F; ½H; ½J; ½A; ½B; ½D; ½E; ½F are the plate stiffness coefficients matrices. The plate stiffness coefficients are defined as: n X Rz

ðAij ; Bij ; Dij ; Eij ; F ij ; Hij ; J ij Þ ¼

ðkÞ

k

zk1

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

k¼1

ði; j ¼ 1; 2; 6Þ ðAij ; Bij ; Dij ; Eij ; F ij Þ ¼

n X Rz

ðkÞ

k

zk1

Q ij ð1; z; z2 ; z3 ; z4 Þdz;

ði; j ¼ 4; 5Þ

k¼1

The governing equations of equilibrium are obtained using the Hamilton’s principle and expressed as:

@Nx @N xy þ ¼ 0; @x @y

@Ny @Nxy þ ¼0 @y @x

@Q x @Q y @2w @2w @2w þq¼0 þ þ Nx 2 þ N y 2 þ 2Nxy @x @y @x@y @x @y ð2Þ

@M x @M xy þ  Q x ¼ 0; @x @y @0Nxy

where Q ij for i, j = 1, 2, 4, 5, 6 are transformed reduced stiffness coefficients. Employing von-Karman nonlinear kinematics and using the displacement field in Eq. (1), strain–displacement relations are expressed as:

8 > > > > > > <

Here, comma (,) denotes the differentiation of the term with respect to the variable followed by it. The in-plane stress and moment resultants of the laminated composite plate consisting of n layers are expressed as:



@M x @M xy þ  3Q x ¼ 0; @x @y

@Ny @y

þ

@Nxy @x

ð5a-iÞ

 2Sy ¼ 0

@My @Mxy þ  3Q y ¼ 0 @y @x

If n and s denote the normal and tangential directions at the boundary, then associated admissible boundary conditions obtained are of the form [23]:

9

e1x > j1x > eox > ex > jx > > > > > > > > > > > > > > > > > > > > > > > > > > > > > 1 > 1 > o > > > > > > > > > > ey > > > > < jy > < ey > < jy > < ey > = = > = = = 2 3 o cxy ¼ cxy þ z jxy þ z c1xy þ z j1xy > > > > > > > > > > > > > > > > > > > > > co > > > > > > 2v 1 > > > > > > > > 3/y > 0 > > > cyz > > yz > > > > > > > > > > > > > > > > > > > : : : : ; ; : o ; ; ; cxz cxz 2u1 0 3/x

@Nx þ  2Sx ¼ 0; @x @y

@My @M xy þ  Qy ¼ 0 @y @x

ð3aÞ

un ¼ 0 or Nnn ¼ 0;

wn ¼ 0 or M nn ¼ 0;

us ¼ 0 or Nns ¼ 0;

ws ¼ 0 or M ns ¼ 0;

w0 ¼ 0 or Q n ¼ 0;

u1n ¼ 0 or

N nn

¼ 0;

u1s ¼ 0 or Nns ¼ 0 /n ¼ 0 or M nn ¼ 0 /s ¼ 0 or

Mns

ð6Þ

¼0

where,

eox ¼ uo;x þ 0:5ðwo;x Þ2 ; eoy ¼ v o;y þ 0:5ðwo;y Þ2 ; coxy ¼ uo;y þ v o;x þ wo;x wo;y coyz ¼ wy þ wo;y ; coxz ¼ wx þ wo;x ; jx ¼ wx;x ; jy ¼ wy;y ; jxy ¼ wx;y þ wy;x e1x ¼ u1;x ; e1y ¼ v 1;y ; c1xy ¼ u1;y þ v 1;x ; j1x ¼ /x;x ; j1y ¼ /y;y ; j1xy ¼ /x;y þ /y;x ð3bÞ

3. Transformation of physical domain into computational domain Transformation of a skew plate of sides ‘a’ and ‘b’ and thickness ‘h’ with skew angle h (Fig. 1) into computational domain (1 6 r, s 6 1) is done using the relations;

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Simply supported immovable edge (S):

un ¼ us ¼ w0 ¼ Mnn ¼ ws ¼ u1n ¼ u1s ¼ M nn ¼ /s ¼ 0

ð10bÞ

where

un ¼ u0 nx þ v 0 ny ;

us ¼ u0 ny þ v 0 nx ;

wn ¼ wx nx þ wy ny ;

ws ¼ wx ny þ wy nx u1n ¼ u1 nx þ v 1 ny ;

u1s ¼ u1 ny þ v 1 nx ;

/n ¼ /x nx þ /y ny ;

/s ¼ /x ny þ /y nx M nn ¼ M x n2x þ M y n2y þ 2M xy nx ny

Mnn ¼ Mx n2x þ M y n2y þ 2Mxy nx ny ;

where nx and ny are the direction cosines at respective edges. The boundary conditions considered in the analysis are taken in the following order: r = 1, r = +1, s = +1 and s = 1.

Fig. 1. Geometry of the skew plate.

5. Solution methodology Table 1 Convergence of symmetric and anti symmetric cross ply laminated composite skew plate (skew angle = 600, a/b = 1, a/h = 20). M=N

6 7 8 9 10

r¼

[0/90/90/0]

[0/90/0/90]

Q = 200

Q = 500

Q = 200

Q = 500

0.5407 0.5180 0.5481 0.5589 0.5635

0.8881 0.8992 0.9512 0.9743 0.9827

0.5232 0.4916 0.5213 0.5319 0.5368

0.8917 0.8796 0.9295 0.9556 0.9620

Fox and parker [24] have discussed the properties of Chebyshev polynomials in detail. The ith term in a Chebyshev polynomial is given by:

T i ðrÞ ¼ CosðivÞ; Cos v ¼ r; 1 6 r 6 1 The recurrence relation can be written as:

T nþ1 ðrÞ þ T n1 ðrÞ ¼ 2rT n ðrÞ

2x  ðy cot h þ a þ y cot hÞ 2x  ð2y cot h þ aÞ ¼ ; r¼ ðy cot h þ a  y cot hÞ a 2y  ðb sin h þ 0Þ 2y s¼ ¼ 1 ðb sin h  0Þ b sin h

ð8Þ

Once the transformation Eq. (8) is known, the equations of equilibrium (5a-i) can be transformed to the computational domain using chain rule of differentiation:

@u @u @r @u @s ¼ þ @x @r @x @s @x @2u @r @ 2 u @r @s @ 2 u @s @ 2 u @ 2 r @u @ 2 s @u þ ð Þ2 2 þ 2 þ ¼ ð Þ2 2 þ 2 @x2 @x @r @x @x @r@s @x @s @x @r @x2 @s

NS XX @ RS gðr; sÞ MR @ RS g ¼ d ij @rR sS @r R sS i¼0 j¼0

ð13Þ

! T i ðrÞT j ðsÞ;

1 6 r; s 6 1

ð14Þ

ij

where R and S are the orders of derivatives with respect to r and s, respectively. The function dij used in Eqs. (13) and (14) takes the following values [20]:

d ¼ 0:25 if i ¼ 0 & j ¼ 0 d ¼ 0:5 if i ¼ 0 & j – 0 or i – 0 & j ¼ 0

ð15Þ

d ¼ 1:0 otherwise The derivative function ð@ RS g=@r R sS Þij is evaluated using the recurrence relation (Eq. (12)). The derivative with respect to ‘r/s’ can be expressed as:

! @Rg @r R

¼ ði1Þj

¼ @ RS g @r R @sS

4. Boundary conditions

! ¼ ði1Þðj1Þ

! @Rg þ 2i @r R ðiþ1Þj ! @Sg þ 2j @sS iðjþ1Þ ! @ RS g @r R @sS

ðiþ1Þðjþ1Þ

! @ ðR1Þ g ; @rðR1Þ ij ! @ ðS1Þ g @sðS1Þ

! @Sg @sS

ij

@ ðR1ÞðS1Þ g þ 4ij @rðR1Þ @sðS1Þ

iðj1Þ

! ij

ð16Þ

The skew plate and rectangular plate are different at their boundaries, and hence the transformation of the boundary conditions is necessary. Following boundary conditions and there combinations are used in the present work [23]: Clamped immovable edge (C):

un ¼ us ¼ w0 ¼ wn ¼ ws ¼ u1n ¼ u1s ¼ /n ¼ /s ¼ 0

1 6 r; s 6 1;

where M and N are the number of terms in finite degree double Chebyshev series. The spatial derivatives of the function for e.g. g (r, s) are expressed as:

ð9Þ Similarly, other derivatives can also be obtained. These values are substituted in the Eqs. (5a-i) to obtain the required governing equations of equilibrium (Appendix A).

M X N X dij gi;j T i ðrÞT j ðsÞ; i¼0 j¼0

ð7Þ

where x1 and x2 are the lower and upper limits of x, y1 and y2 are the lower and upper limits of y. These limits are either constant numerical values (as in the case of a rectangular plate) or the functions of y, i.e. (x1 = f1(y), x2 = f2(y)) and the functions of x, i.e. (y1 = f1(x), y2 = f2(x)) according to the geometry of plate (Fig. 1). Using Eq. (7), the coordinates can be computed as:

ð12Þ

The displacement functions and the loadings are approximated in space domain and expressed as:

gðr; sÞ ¼

2x  ðx2 þ x1 Þ 2y  ðy2 þ y1 Þ ;s ¼ x2  x1 y2  y1

ð11Þ

ð10aÞ

The nonlinear terms are linearized at any step of marching variable (load) using quadratic extrapolation technique. A typical nonlinear function G at step J is expressed as,

"

MR N XX

#

ðgij ÞR T i ðrÞT j ðsÞ

GJ ¼ d

i¼0 j¼0

" 

J

d

# M X NS X ðgij ÞS T i ðrÞT j ðsÞ i¼0 j¼0

J

ð17Þ

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A.K. Upadhyay, K.K. Shukla / Composite Structures 94 (2012) 3722–3735

5

0.35

90 Linear

Lamination Scheme, M=N

0.3

A ,6 A,7 A,8 A,9 A,10 B,6 B,7 B,8 B,9 B,10

0.2 0.15

75 Linear

4 60 Linear

3

W*

0.25

W_c

CCCC, a/b = 1, a/h =20, Skew Angle = 30

45 Linear

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

0.1

A = [45/-45/-45/45] B = [45/-45/45/-45]

1

30 Linear

0.05 90

Skew Angle

75

60

45

30

0

0 0

100

200

300

400

0

500

100

200

400

500

Q

Q Fig. 2. Convergence behavior of symmetric [45/45/45/45] and anti symmetric [45/45/45/45] angle ply moderately thick (a/h = 20) clamped laminated composite skew (h = 300) skew plate.

300

Fig. 5. Linear and nonlinear displacements of clamped [0/90/90/0] laminated cross ply skew plate (a/h = 20).

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

1.8

1.2 1

90 75 60 45 30

8

Mx

1.4

W_c

10

Present 90 Ref.[10], 90 Ref.[5], 90 Present 60 Ref.[10], 60 Ref.[5], 60 Present 45 Ref.[10], 45 Ref.[5], 45

1.6

Skew Angle

6 4

0.8 0.6

2

0.4 0 0.2 0

0

100

200

300

400

500

Q 0

50

100

150

200

250

300

Q

Fig. 6. Effect of skew angle on the non dimensional bending moment (Mx) of clamped [0/90/90/0] laminated cross ply skew plate (a/h = 20).

Fig. 3. Comparison of non dimensional transverse central deflection of isotropic, clamped skew plates.

0.9 CCCC, a/b = 1, [0/90/90/0]

1.2 Skew Angle, a/h

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

90,10

1

Skew Angle

W_c

0.6

90,100 60,100

W_c

90 75 60 45 L 45 30 L 30

0.8

60,10

0.6

30,10 30,100

0.3

0.4 0.2 0 0 0

100

200

300

400

500

0

50

100

150

200

Q

Q Fig. 4. Effect of skew angle on the non dimensional center deflection of clamped [0/90/90/0] laminated cross ply skew plate (a/h = 20).

Fig. 7. Effect of span to thickness ratio (a/h), on the non dimensional center deflection of laminated [0/90/90/0] cross ply clamped skew plate with different skew angles.

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A.K. Upadhyay, K.K. Shukla / Composite Structures 94 (2012) 3722–3735

1.4

10

CCCC, a/h = 20, [0/90/90/0]

CCCC, a/b = 1, [0/90/90/0] Skew Angle, a/h

1.2

90,10 90,100 60,10 60,100 30,10 30,100

Mx

6

Skew Angle, b/a 90,2 60,2 90,1 60,1 30,2 30,1

1 0.8

W_c

8

0.6

4

0.4 2

0.2 0

0

50

100

150

0

200

0

100

200

Q Fig. 8. Effect of span to thickness ratio (a/h), on the non dimensional bending moment (Mx) of laminated [0/90/90/0] cross ply clamped skew plate with different skew angles.

Skew Angle, E 1/E2 90,5 60,5 90,10 60,10 90,20 60,20 90,40 60,40 30,5 30,10 30,20 30,40

0.9 1.2

0.45

W_c

0.6

W_c

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

SSSS, 90 SSSS, 60 CCSS, 90 CCCC, 90 CCSS, 60 CCCC, 60 SSSS, 30 CCSS, 30 CCCC, 30

0.75

400

500

Fig. 11. Effect of b/a ratio on the non dimensional center deflection of laminated [0/ 90/90/0] cross ply clamped skew plate (a/h = 20) with different skew angles.

1.6

Boundary Condition, Skew Angle

300

Q

0.8

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

0.4

0.3 0

0.15

0

100

200

300

400

500

Q 0 0

25

50

75

100

125

150

Q Fig. 9. Effect of boundary conditions on the non dimensional center deflection of laminated [0/90/90/0] cross ply skew plate (a/h = 20) with different skew angles.

Fig. 12. Effect of E1/E2 ratio on the nondimensional center deflection of laminated [0/90/90/0] cross ply clamped skew plate (a/h = 20) with different skew angles.

The variable at any step ‘J’ is predicted by using the values of three consecutive previous steps and expressed as:

ðgij ÞJ ¼ Aðgij ÞJ1 þ Bðgij ÞJ2 þ Cðgij ÞJ3

During initial steps of marching variables, the coefficients A, B, C of the quadratic extrapolation scheme of linearization takes the following values [20]:

8 Boundary Condition, Skew Angle

Mx

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

SSSS, 90 SSSS, 60 CCSS, 90 CCCC, 90 CCSS, 60 CCCC, 60 CCSS, 30 SSSS, 30 CCCC, 30

6

4

ð18Þ

1; 0; 0ðJ ¼ 1Þ;

2; 1; 0ðJ ¼ 2Þ;

3; 3; 1ðJ P 3Þ

Using the procedure described above, the differential equations are discretized in the space domain. Collocating the zeroes of the Chebyshev polynomial, the governing equations are reduced to a set of linear simultaneous equations and finally expressed as: M2 N2 XX

2

F k ðu0ij ; v 0ij ; w0ij ; wxij ; wyij ; u1ij ; v 1ij ; /xij ; /yij ; Q ij ÞT i ðrÞT j ðsÞ ¼ 0;

i¼0 j¼0

ðk ¼ 1 to 9Þ 0 0

25

50

75

100

125

150

Q Fig. 10. Effect of boundary conditions on the non dimensional bending moment (Mx) of clamped laminated [0/90/90/0] cross ply skew plate (a/h = 20) with different skew angles.

ð19Þ

Similarly, the appropriate sets of boundary conditions are also discretized and expressed in form of linear simultaneous equations. The set of generating Eq. (19) give rise to 9(M  1)  (N  1) algebraic equations. All edges clamped (CCCC) boundary condition, gives (18 M + 18 N + 36) algebraic equations. Thus total number of equations obtained in this case are 9(M + 1) (N + 1) + 36. The total

A.K. Upadhyay, K.K. Shukla / Composite Structures 94 (2012) 3722–3735

number of unknown coefficients is 9(M + 1) (N + 1) i.e. total number of equations is more than the total number of unknowns which is true for other boundary conditions also. In order to have a unique and compatible solution, multiple regression analysis based on least-square error norms is used. The set of linear equations are expressed in the matrix form as:

Aa ¼ Q

ð20Þ

where the error function is

SðaÞ ¼ ðQ  AaÞT ðQ  AaÞ where A is (p  q) coefficient matrix, a is (q  1) displacement coefficient vector, Q is (p  1) load vector. Multiple regression analysis gives

a ¼ ðAT AÞ1 AT Q a ¼ BQ

ð21Þ

The matrix B is evaluated once and retained for subsequent usage. The values of the displacement vector a obtained from Eq. (21) is put into the Eq. (13) to evaluate the displacement at the desired location on the mid-plane of the plate. 6. Results and discussions Detailed convergence and parametric studies are carried out and discussed in this section. The material properties used in the present analysis, unless otherwise stated are:

E1 =E2 ¼ 10;

m12 ¼ 0:22; G12 ¼ 0:33E2 ; G23 ¼ 0:2E2 ; G13 ¼ G12 :

Following non dimensional parameters are used in the analysis: 4

Q ¼ qa4 =E2 h ; W c ¼ w0 =h; M x ¼ Mx hb2 =D11 ðBar sign over Mx is omitted here onwardsÞ: Table 1 shows the convergence of symmetric and anti-symmetric cross-ply clamped laminated composite 600 skew plate (a/ h = 20, a/b = 1). It can be seen that good convergence (within 1%) is achieved at 9 terms. The convergence of symmetric and antisymmetric angle-ply clamped laminated composite 300 skew plate (a/h = 20, a/b = 1) is shown in Fig. 2. A very good convergence of the deflection is achieved at 8 terms. In the present study, nine terms expansion of the variables in the Chebyshev series is used. In order to validate the present solution methodology, the nonlinear non-dimensional transverse deflections of isotropic clamped plate for different skew angles is obtained and compared with the results due to Singh and Elaghabash [10] and Pica et al. [5]. The comparison of results is shown in Fig. 3 and it can be seen that results are in good agreement. Nonlinear transverse central displacements of symmetric crossply [0/90/90/0] laminated composite clamped skew plate (a/h = 20, a/b = 1) for different skew angles are obtained and shown in Fig. 4. The linear (L) results for skew angles 450 and 300 are also obtained and shown in the figure along with nonlinear results. It is observed that the rate of decrease of deflection increases with decrease in skew angle. There is decrease in deflection of skew plates in comparison to square plate. This decrease is 3.91%, 17.62%, 41.94%, and 73.87% for skew plates with skew angles 750, 600, 450 and 300, respectively at transverse load at Q = 500. The results depict the increase in stiffening behavior with decrease in skew angle. Fig. 5 depicts the variation of transverse deflection per unit transverse load (i.e. slope of curves in Fig. 4, W⁄ = 1000  W_c/Q) with transverse load. It is observed that for high skew angles, the difference between linear and nonlinear results is appreciable even at very small loads. In case of 300 skew plates, this difference becomes negligible even at higher loads. The large deflection behavior in

3727

the skew plates with high and moderate skew angles (>450) is observed because w0/h  1 is attainable in these plates at higher loads. This clearly shows that for small skew angles, linear analysis is sufficient to make good estimates even at higher loads but for high skew angles, nonlinear analysis is essentially required. The variations of bending moment Mx with transverse load for skew plates with different skew angles are shown in Fig. 6. The bending moment of skew plates in comparison to square plate decreases by 6.2%, 25.93%, 48.72% and 77.38% corresponding to skew angles 750, 600, 450 and 300, respectively at Q = 500. The effect of span to thickness ratio of moderately thick (a/ h = 10) and thin (a/h = 100) plates on non-dimensional transverse central displacement and bending moment of a clamped laminated composite [0/90/90/0], skew plate (a/b = 1) for different skew angles are shown in Figs. 7 and 8, respectively. The difference in non dimensional deflection of moderately thick and thin square plate is 46.96% while for 600 and 300 skew plates, this difference is 51.85% and 90.74%, respectively at Q = 200. This comparison shows the dominance of transverse shear deformation in highly skewed plates. In the case of bending moment, moderately thick and thin skew plates with skew angle 300 have almost same bending moment at lower loads but a sharp decrease in bending moment for moderately thick plate is observed at higher loads. It may be concluded that the higher order shear deformation theories (Cubic or higher) can predict the response for highly skewed plates more accurately. The effect of boundary conditions on non-dimensional transverse central displacement and bending moment of a laminated composite [0/90/90/0], skew plate (a/h = 20, a/b = 1) for different skew angles is shown in Figs. 9 and 10, respectively. The decrease in deflection of clamped skew plates with respect to square plates is found to be maximum and it is minimum in case of simply supported skew plates. Moreover, the nonlinear behavior in simply supported skew plates is observed clearly whereas in case of clamped skew plates with skew angle 300, the response is almost linear. This clearly indicates that linear or nonlinear analysis in case of skew plates (especially highly skewed plates) is dependent on boundary conditions also. The nonlinear non-dimensional transverse central displacements of symmetric cross-ply [0/90/90/0] laminated composite clamped skew plate (a/h = 20) for different skew angles and aspect ratio (b/a) are obtained and shown in Fig. 11. It is observed that skew plates with high and moderate skew angles (>450) show almost similar behavior irrespective of load and aspect ratio. In case of plates (skew angle = 900), the deflection of the square plate (b/ a = 1) decreases by 13.5% and 16.59% in comparison to that of rectangular plate (b/a = 2) corresponding to non-dimensional loads Q = 200 and Q = 500, respectively. But, in the case of skew plates with skew angle 300, the difference in the deflections of skew plate with aspect ratio 1 and 2 is 57.1% and 43.68% corresponding to transverse loads Q = 200 and Q = 500, respectively. It indicates that the effect of aspect ratio on transverse deflection of highly skewed plates is appreciable even at lower loads. The effect of modular ratio E1/E2 on the non dimensional transverse central displacement of a laminated composite [0/90/90/0], skew plate (a/h = 20, a/b = 1) for different skew angles is shown in Fig. 12. It is observed that transverse central displacement decreases with increase in orthotropy ratio for rectangular as well as skew plates.

7. Conclusions Large deformation flexural response of laminated composite skew plates is obtained using Chebyshev polynomials. One of the advantages of the current methodology is that grid points are not

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required at all. The effect of skew angle on the flexural response of laminated composite skew plate is studied in detail. It is observed that for moderate and high skew angles (>450), the nonlinear effects are dominating. But for low skew angles i.e. for highly skewed plates, the difference between linear and nonlinear results is relatively small due to stiffening of the plate. Also, the effect of transverse shear deformation is more pronounced in highly skewed plates as compared to rectangular and moderately skewed plates. For highly skewed plates, linear analysis based on HSDT (third order or even higher) may give reasonably good results and thus saving computational efforts. It is also observed that the nonlinear effects are also dependent on boundary conditions. Hence, proper selection of displacement field along with boundary conditions depending on plate thickness is essential for the analysis of moderately thick laminated composite skew plates.

A21 ¼ 2r x sx E11 þ 2r y sy E66 þ 2ðrx sy þ r y sx ÞE16 ;

Appendix A

A28 ¼ ðA11 sx þ A16 sy Þr 2x þ ðA66 sx þ A26 sy Þr2y

AD22 ¼ E16 r 2x þ E26 r 2y þ ðE12 þ E66 Þrx r y A23 ¼ E16 s2x þ E26 s2y þ ðE12 þ E66 Þsx sy ; A24 ¼ 2r x sx E16 þ 2r y sy E26 þ ðr x sy þ r y sx ÞðE12 þ E66 Þ A25 ¼ ðA11 rx þ A16 r y Þr2x þ ðA66 rx þ A26 ry Þr 2y þ ð2A16 rx þ ðA12 þ A66 Þry Þr x ry

AD26 ¼ ðA11 r x þ A16 r y Þs2x þ ðA66 r x þ A26 r y Þs2y þ ð2A16 rx þ ðA12 þ A66 Þry Þsx sy A27 ¼ ðA11 r x þ A16 r y Þ2r x sx þ ðA66 r x þ A26 r y Þ2ry sy þ ð2A16 r x þ ðA12 þ A66 Þr y Þðrx sy þ ry sx Þ

þ ð2A16 sx þ ðA12 þ A66 Þsy Þr x r y

A.1. Eq. (5a)

A29 ¼ ðA11 sx þ A16 sy Þs2x þ ðA66 sx þ A26 sy Þs2y þ ð2A16 sx þ ðA12 þ A66 Þsy Þsx sy

A1

@ 2 u0 @ 2 u0 @ 2 u0 @2v 0 @2v 0 @2v 0 @ 2 wx þ A2 2 þ A3 þ A5 þ A6 þ A4 þ A7 2 2 2 @r @s @r@s @r @s @r@s @r 2 2 2 2 2 2 @ wy @ wy @ wy @ wx @ wx þ A8 þ A9 þ AD11 þ AD12 þ A10 @s2 @r@s @r 2 @s2 @r@s @ 2 u1 @ 2 u1 @ 2 u1 @2v 1 @2v 1 þ A17 þ AD16 þ A13 2 þ A14 2 þ A15 @r @s @r@s @r2 @s2 2 2 2 2 @ 2 /y @ v1 @ /x @ /x @ /x þ A18 þ A20 þ A21 þ A19 þ AD22 @r@s @r 2 @s2 @r@s @r 2 ! 2 2 2 2 @ /y @ /y @ w0 @ w0 @ 2 w0 @w0 þ A23 þ A24 þ AD26 þ A27 þ A25 @s2 @r@s @r2 @s2 @r@s @r ! @ 2 w0 @ 2 w0 @ 2 w0 @w0 þ A28 þ A29 þ A30 ¼0 @r2 @s2 @r@s @s

A1 ¼ A11 r 2x þ A66 r 2y þ 2A16 r x r y ;

A2 ¼ A11 s2x þ A66 s2y þ 2A16 sx sy

A4 ¼ A16 r 2x þ A26 r 2y þ ðA12 þ A66 Þr x ry

A8 ¼ B11 s2x þ B66 s2y þ 2B16 sx sy

A9 ¼ 2r x sx B11 þ 2r y sy B66 þ 2ðr x sy þ ry sx ÞB16 ;

þ

A13 ¼

þ

D66 r2y

þ ðB12 þ B66 Þsx sy ;

þ 2D16 r x r y ;

A14 ¼

D11 s2x

þ

D66 s2y

A17 ¼

þ

@s ; @x

sy ¼

@s @y

A.2. Eq. (5b)

B1

@ 2 u0 @ 2 u0 @ 2 u0 @2v 0 @2v 0 @2v 0 þ B2 2 þ B3 þ B5 þ B6 þ B4 2 2 2 @r @s @r@s @r @s @r@s þ B7

@ 2 wy @ 2 wy @ 2 wx @ 2 wx @ 2 wx þ B8 þ B9 þ BD11 þ B10 2 2 2 @r @s @r@s @r @s2 @ 2 wy @ 2 u1 @ 2 u1 @ 2 u1 @2v 1 þ B þ B þ B13 þ BD 14 15 16 @r@s @r 2 @s2 @r@s @r 2

@2v 1 @2v 1 @ 2 /x @ 2 /x @ 2 /x þ B þ B þ B þ B 18 19 20 21 @s2 @r@s @r 2 @s2 @r@s

@ 2 /y @ 2 /y @ 2 /y @ 2 w0 @ 2 w0 þ B23 þ B24 þ BD26 þ B25 2 2 2 @r @s @r@s @r @s2 ! ! @ 2 w0 @w0 @ 2 w0 @ 2 w0 @ 2 w0 @w0 þ B27 þ B þ B þ B28 ¼0 29 30 @r@s @r @r 2 @s2 @r@s @s

B4 ¼ A66 r 2x þ A22 r 2y þ 2A26 r x r y þ 2D16 sx sy

AD16 ¼ D16 r 2x þ D26 r 2y þ ðD12 þ D66 Þrx r y D26 s2y

sx ¼

B3 ¼ 2r x sx A16 þ 2r y sy A26 þ ðA12 þ A66 Þðr x sy þ r y sx Þ;

A15 ¼ 2rx sx D11 þ 2ry sy D66 þ 2ðr x sy þ r y sx ÞD16 ;

D16 s2x

@r ; @y

B2 ¼ A16 s2x þ A26 s2y þ ðA12 þ A66 Þsx sy

AD12 ¼ 2r x sx B16 þ 2ry sy B26 þ ðr x sy þ r y sx ÞðB12 þ B66 Þ D11 r 2x

ry ¼

B1 ¼ A16 r 2x þ A26 r 2y þ ðA12 þ A66 Þr x ry ;

A10 ¼ B16 r 2x þ B26 r 2y þ ðB12 þ B66 Þrx r y AD11 ¼

@r ; @x

þ BD22

A6 ¼ 2r x sx A16 þ 2r y sy A26 þ ðr x sy þ r y sx ÞðA12 þ A66 Þ

B26 s2y

rx ¼

þ B17

A5 ¼ A16 s2x þ A26 s2y þ ðA12 þ A66 Þsx sy ;

B16 s2x

þ ð2A16 sx þ ðA12 þ A66 Þsy Þðr x sy þ ry sx Þ where

þ BD12

A3 ¼ 2r x sx A11 þ 2r y sy A66 þ 2ðrx sy þ r y sx ÞA16 ;

A7 ¼ B11 r2x þ B66 r 2y þ 2B16 r x r y ;

A30 ¼ ðA11 sx þ A16 sy Þ2r x sx þ ðA66 sx þ A26 sy Þ2ry sy

þ ðD12 þ D66 Þsx sy ;

B5 ¼ A66 s2x þ A22 s2y þ 2A26 sx sy ; B6 ¼ 2r x sx A66 þ 2r y sy A22 þ ðr x sy þ r y sx Þ2A26 B7 ¼ B16 r 2x þ B26 r 2y þ ðB12 þ B66 Þr x r y ; B8 ¼ B16 s2x þ B26 s2y þ ðB12 þ B66 Þsx sy

A18 ¼ 2rx sx D16 þ 2ry sy D26 þ ðr x sy þ r y sx ÞðD12 þ D66 Þ B9 ¼ 2r x sx B16 þ 2ry sy B26 þ ðB12 þ B66 Þðrx sy þ r y sx Þ; A19 ¼ E11 r 2x þ E66 r2y þ 2E16 r x r y ;

A20 ¼ E11 s2x þ E66 s2y þ 2E16 sx sy

B10 ¼ B66 r2x þ B26 r 2y þ 2B26 r x r y

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BD11 ¼ B66 s2x þ B26 s2y þ 2B26 sx sy ; BD12 ¼ 2r x sx B66 þ 2r y sy B22 þ ðrx sy þ ry sx Þ2B26 B13 ¼ D16 r 2x þ D26 r 2y þ ðD12 þ D66 Þrx r y ; B14 ¼ D16 s2x þ D26 s2y þ ðD12 þ D66 Þsx sy B15 ¼ 2r x sx D16 þ 2r y sy D26 þ ðD12 þ D66 Þðr x sy þ ry sx Þ; BD16 ¼ D66 r 2x þ D26 r2y þ 2D26 r x r y B17 ¼ D66 s2x þ D26 s2y þ 2D26 sx sy ; B18 ¼ 2r x sx D66 þ 2r y sy D22 þ ðrx sy þ r y sx Þ2D26 B19 ¼ E16 r 2x þ E26 r 2y þ ðE12 þ E66 Þrx r y ; B20 ¼ E16 s2x þ E26 s2y þ ðE12 þ E66 Þsx sy B21 ¼ 2r x sx E16 þ 2r y sy E26 þ ðE12 þ E66 Þðr x sy þ ry sx Þ; BD22 ¼ E66 r 2x þ E26 r2y þ 2E26 r x ry B23 ¼ E66 s2x þ E26 s2y þ 2E26 sx sy ; B24 ¼ 2r x sx E66 þ 2r y sy E22 þ ðrx sy þ r y sx ÞE26 B25 ¼ ðA16 rx þ A66 ry Þr2x þ ðA26 rx þ A22 ry Þr2y þ ð2A26 ry þ ðA12 þ A66 Þrx Þrx ry BD26 ¼ ðA16 r x þ A66 r y Þs2x þ ðA26 rx þ A22 ry Þs2y þ ð2A16 ry þ ðA12 þ A66 Þrx Þsx sy

B27 ¼ ðA16 r x þ A66 r y Þ2r x sx þ ðA26 rx þ A22 ry Þ2r y sy þ ð2A26 ry þ ðA12 þ A66 Þrx Þðr x sy þ ry sx Þ B28 ¼ ðA16 rx þ A66 ry Þr2x þ ðA26 rx þ A22 ry Þr2y þ ð2A26 ry þ ðA12 þ A66 Þrx Þrx ry B29 ¼ ðA16 rx þ A66 ry Þs2x þ ðA26 rx þ A22 ry Þs2y þ ð2A26 ry þ ðA12 þ A66 Þrx Þsx sy

B30 ¼ ðA16 sx þ A66 sy Þ2r x sx þ ðA26 sx þ A22 sy Þ2r y sy þ ð2A26 sy þ ðA12 þ A66 Þsx Þðr x sy þ ry sx Þ A.3. Eq. (5c)

C1

@wy @wy @ 2 w0 @ 2 w0 @ 2 w0 @w @w þ C2 þ C3 þ C4 x þ C5 x þ C6 þ C7 2 2 @r @s @r@s @r @s @r @s @u1 @u1 @v 1 @v 1 @/x @/x þ C8 þ C9 þ C 10 þ C 11 þ C 12 þ C 13 @r @s @r @s @r @s ! @/y @/y @ 2 w0 @ 2 w0 @ 2 w0 @u0 þ C 15 þ C 16 þ C þ C þ C 14 17 18 @r @s @r2 @s2 @r@s @r ! @ 2 w0 @ 2 w0 @ 2 w0 @u0 þ C 19 þ C 20 þ C 21 @r2 @s2 @r@s @s ! @ 2 w0 @ 2 w0 @ 2 w0 @ v 0 þ C 22 þ C þ C 23 24 @r2 @s2 @r@s @r ! 2 2 @ w0 @ w0 @ 2 w0 @ v 0 þ C 25 þ C 26 þ C 27 @r2 @s2 @r@s @s ! @ 2 w0 @ 2 w0 @ 2 w0 @wx þ C 28 þ C þ C 29 30 @r2 @s2 @r@s @r ! 2 2 @ w0 @ w0 @ 2 w0 @wx þ C 31 þ C 32 þ C 33 @r2 @s2 @r@s @s

!

@ 2 w0 @ 2 w0 @ 2 w0 @wy þ C þ C 35 36 @r 2 @s2 @r@s @r ! @ 2 w0 @ 2 w0 @ 2 w0 @wy þ C þ C þ C 37 38 39 @r 2 @s2 @r@s @s ! 2 2 @ w0 @ w0 @ 2 w0 @u1 þ C 41 þ C 42 þ C 40 @r 2 @s2 @r@s @r ! @ 2 w0 @ 2 w0 @ 2 w0 @u1 þ C 43 þ C þ C 44 45 @r 2 @s2 @r@s @s ! 2 2 @ w0 @ w0 @ 2 w0 @ v 1 þ C 47 þ C 48 þ C 46 @r 2 @s2 @r@s @r ! @ 2 w0 @ 2 w0 @ 2 w0 @ v 1 þ C þ C þ C 49 50 51 @r 2 @s2 @r@s @s ! 2 2 @ w0 @ w0 @ 2 w0 @/x þ C 53 þ C 54 þ C 52 @r 2 @s2 @r@s @r ! @ 2 w0 @ 2 w0 @ 2 w0 @/x þ C þ C þ C 55 56 57 @r 2 @s2 @r@s @s ! 2 2 @ w0 @ w0 @ 2 w0 @/y þ C 58 þ C 59 þ C 60 @r 2 @s2 @r@s @r ! @ 2 w0 @ 2 w0 @ 2 w0 @/y þ C þ C þ C 61 62 63 @r 2 @s2 @r@s @s !  2 2 2 1 @w0 @ w0 @ w0 @ 2 w0 C 64 þ C 65 þ C 66 þ 2 @r @r 2 @s2 @r@s !  2 1 @w0 @ 2 w0 @ 2 w0 @ 2 w0 C 67 þ C þ C þ 68 69 2 @s @r 2 @s2 @r@s !   2 2 @w0 @w0 @ w0 @ w0 @ 2 w0 þ C 73 q ¼ 0 þ C 71 þ C 72 C 70 þ @r @s @r 2 @s2 @r@s

þ C 34

C 1 ¼ A55 r 2x þ A44 r 2y þ 2A45 r x r y ;

C 2 ¼ A55 s2x þ A44 s2y þ 2A45 sx sy ;

C 3 ¼ 2A55 r x sx þ 2A44 r y sy þ 2A45 ðsy rx þ sx r y Þ C 4 ¼ A55 r x þ A45 r y ;

C 5 ¼ A55 sx þ A45 sy ;

C 7 ¼ A45 sx þ A44 sy ;

C 8 ¼ 2ðB55 r x þ B45 r y Þ

C 9 ¼ 2ðB55 sx þ B45 sy Þ;

C 6 ¼ A45 r x þ A44 r y ;

C 10 ¼ 2ðB45 rx þ B44 r y Þ;

C 11 ¼ 2ðB45 sx þ B44 sy Þ C 12 ¼ 3ðD55 r x þ D45 r y Þ;

C 13 ¼ 3ðD55 sx þ D45 sy Þ;

C 14 ¼ 3ðD45 r x þ D44 r y Þ;

C 15 ¼ 3ðD45 sx þ D44 sy Þ

C 16 ¼ ðA11 r x þ A16 r y Þr 2x þ ðA12 r x þ A26 r y Þr 2y þ 2rx r y ðA16 r x þ A66 r y Þ C 17 ¼ ðA11 r x þ A16 r y Þs2x þ ðA12 r x þ A26 r y Þs2y þ 2sx sy ðA16 rx þ A66 ry Þ C 18 ¼ 2rx sx ðA11 rx þ A16 r y Þ þ 2r y sy ðA12 r x þ A26 r y Þ þ 2ðr x sy þ r y sx ÞðA16 r x þ A66 r y Þ C 19 ¼ r 2x ðA11 sx þ A16 sy Þ þ r 2y ðA12 sx þ A26 sy Þ þ 2rx r y ðA16 sx þ A66 sy Þ C 20 ¼ s2x ðA11 sx þ A16 sy Þ þ s2y ðA12 sx þ A26 sy Þ þ 2sx sy ðA16 sx þ A66 sy Þ C 21 ¼ 2rx sx ðA11 sx þ A16 sy Þ þ 2r y sy ðA12 sx þ A26 sy Þ þ 2ðr x sy þ r y sx ÞðA16 sx þ A66 sy Þ

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A.K. Upadhyay, K.K. Shukla / Composite Structures 94 (2012) 3722–3735

C 22 ¼ r 2x ðA16 r x þ A12 r y Þ þ r2y ðA26 r x þ A22 r y Þ þ 2r x r y ðA66 r x þ A26 r y Þ

C 48 ¼ 2r x sx ðD16 r x þ D12 r y Þ þ 2ry sy ðD26 rx þ D22 r y Þ þ 2ðr x sy þ r y sx ÞðD66 rx þ D26 ry Þ

C 23 ¼ s2x ðA16 r x þ A12 r y Þ þ s2y ðA26 r x þ A22 r y Þ þ 2sx sy ðA66 r x þ A26 r y Þ C 49 ¼ r2x ðD16 sx þ D12 sy Þ þ r2y ðD26 sx þ D22 sy Þ þ 2rx r y ðD66 sx þ D26 sy Þ C 24 ¼ 2rx sx ðA16 r x þ A12 r y Þ þ 2r y sy ðA26 r x þ A22 ry Þ þ 2ðr x sy þ r y sx ÞðA66 r x þ A26 r y Þ C 25 ¼ r 2x ðA16 sx þ A12 sy Þ þ r 2y ðA26 sx þ A22 sy Þ þ 2rx r y ðA66 sx þ A26 sy Þ

C 50 ¼ s2x ðD16 sx þ D12 sy Þ þ s2y ðD26 sx þ D22 sy Þ þ 2sx sy ðD66 sx þ D26 sy Þ C 51 ¼ 2r x sx ðD16 sx þ D12 sy Þ þ 2ry sy ðD26 sx þ D22 sy Þ þ 2ðr x sy þ r y sx ÞðD66 sx þ D26 sy Þ

C 26 ¼ s2x ðA16 sx þ A12 sy Þ þ s2y ðA26 sx þ A22 sy Þ þ 2sx sy ðA66 sx þ A26 sy Þ C 52 ¼ r2x ðE11 r x þ E16 r y Þ þ r 2y ðE12 rx þ E26 r y Þ þ 2r x r y ðE16 r x þ E66 ry Þ C 27 ¼ 2rx sx ðA16 sx þ A12 sy Þ þ 2r y sy ðA26 sx þ A22 sy Þ þ 2ðr x sy þ r y sx ÞðA66 sx þ A26 sy Þ C 28 ¼ r 2x ðB11 r x þ B16 ry Þ þ r 2y ðB12 rx þ B26 r y Þ þ 2r x ry ðB16 r x þ B66 r y Þ

C 53 ¼ s2x ðE11 r x þ E16 r y Þ þ s2y ðE12 rx þ E26 r y Þ þ 2sx sy ðE16 rx þ E66 ry Þ C 54 ¼ 2r x ss ðE11 rx þ E16 r y Þ þ 2ry sy ðE12 rx þ E26 ry Þ þ 2ðr x sy þ r y sx ÞðE16 r x þ E66 ry Þ

C 29 ¼ s2x ðB11 r x þ B16 ry Þ þ s2y ðB12 rx þ B26 r y Þ þ 2sx sy ðB16 r x þ B66 r y Þ C 55 ¼ ðE11 sx þ E16 sy Þr 2x þ ðE12 sx þ E26 sy Þr 2y þ 2rx r y ðE16 sx þ E66 sy Þ C 30 ¼ 2rx sx ðB11 r x þ B16 r y Þ þ 2r y sy ðB12 rx þ B26 r y Þ þ 2ðr x sy þ r y sx Þ  ðB16 r x þ B66 r y Þ C 31 ¼ r 2x ðB11 sx þ B16 sy Þ þ r 2y ðB12 sx þ B26 sy Þ þ 2r x ry ðB16 sx þ B66 sy Þ

C 56 ¼ ðE11 sx þ E16 sy Þs2x þ ðE12 sx þ E26 sy Þs2y þ 2sx sy ðE16 sx þ E66 sy Þ C 57 ¼ ðE11 sx þ E16 sy Þ2rx sx þ ðE12 sx þ E26 sy Þ2r y sy þ 2ðr x sy þ r y sx ÞðE16 sx þ E66 sy Þ

C 32 ¼ s2x ðB11 sx þ B16 sy Þ þ s2y ðB12 sx þ B26 sy Þ þ 2sx sy ðB16 sx þ B66 sy Þ

C 58 ¼ ðE16 r x þ E12 r y Þr 2x þ ðE26 r x þ E22 r y Þr 2y þ 2r x r y ðE66 r x þ E26 ry Þ

C 33 ¼ 2rx sx ðB11 sx þ B16 sy Þ þ 2r y sy ðB12 sx þ B26 sy Þ þ 2ðr x sy þ r y sx ÞðB16 sx þ B66 sy Þ C 34 ¼ r 2x ðB16 r x þ B12 ry Þ þ r 2y ðB26 rx þ B22 r y Þ þ 2r x ry ðB66 r x þ B26 r y Þ

C 59 ¼ ðE16 r x þ E12 r y Þs2x þ ðE26 r x þ E22 r y Þs2y þ 2sx sy ðE66 rx þ E26 ry Þ C 60 ¼ ðE16 r x þ E12 r y Þ2rx sx þ ðE26 r x þ E22 r y Þ2r y sy þ 2ðr x sy þ r y sx ÞðE66 r x þ E26 ry Þ

C 35 ¼ s2x ðB16 r x þ B12 ry Þ þ s2y ðB26 rx þ B22 r y Þ þ 2sx sy ðB66 r x þ B26 r y Þ

C 61 ¼ ðE16 sx þ E12 sy Þr 2x þ ðE26 sx þ E22 sy Þr 2y þ 2rx r y ðE66 sx þ E26 sy Þ

C 36 ¼ 2rx sx ðB16 r x þ B12 r y Þ þ 2r y sy ðB26 rx þ B22 r y Þ þ 2ðr x sy þ r y sx ÞðB66 rx þ B26 r y Þ C 37 ¼ r 2x ðB16 sx þ B12 sy Þ þ r 2y ðB26 sx þ B22 sy Þ þ 2r x ry ðB66 sx þ B26 sy Þ

C 62 ¼ ðE16 sx þ E12 sy Þs2x þ ðE26 sx þ E22 sy Þs2y þ 2sx sy ðE66 sx þ E26 sy Þ C 63 ¼ ðE16 sx þ E12 sy Þ2rx sx þ ðE26 sx þ E22 sy Þ2r y sy þ 2ðr x sy þ r y sx ÞðE66 sx þ E26 sy Þ

C 38 ¼ s2x ðB16 sx þ B12 sy Þ þ s2y ðB26 sx þ B22 sy Þ þ 2sx sy ðB66 sx þ B26 sy Þ C 39 ¼ 2rx sx ðB16 sx þ B12 sy Þ þ 2r y sy ðB26 sx þ B22 sy Þ þ 2ðr x sy þ r y sx ÞðB66 sx þ B26 sy Þ C 40 ¼ r 2x ðD11 rx þ D16 r y Þ þ r 2y ðD12 r x þ D26 ry Þ þ 2r x r y ðD16 r x þ D66 ry Þ C 41 ¼ s2x ðD11 rx þ D16 r y Þ þ s2y ðD12 rx þ D26 ry Þ þ 2sx sy ðD16 r x þ D66 ry Þ C 42 ¼ 2rx sx ðD11 rx þ D16 ry Þ þ 2r y sy ðD12 r x þ D26 ry Þ þ 2ðr x sy þ r y sx ÞðD16 r x þ D66 r y Þ C 43 ¼ r 2x ðD11 sx þ D16 sy Þ þ r 2y ðD12 sx þ D26 sy Þ þ 2r x r y ðD16 sx þ D66 sy Þ C 44 ¼ s2x ðD11 sx þ D16 sy Þ þ s2y ðD12 sx þ D26 sy Þ þ 2sx sy ðD16 sx þ D66 sy Þ C 45 ¼ 2rx sx ðD11 sx þ D16 sy Þ þ 2r y sy ðD12 sx þ D26 sy Þ þ 2ðr x sy þ r y sx ÞðD16 sx þ D66 sy Þ C 46 ¼ r 2x ðD16 rx þ D12 r y Þ þ r 2y ðD26 r x þ D22 ry Þ þ 2r x r y ðD66 r x þ D26 ry Þ C 47 ¼ s2x ðD16 rx þ D12 r y Þ þ s2y ðD26 rx þ D22 ry Þ þ 2sx sy ðD66 r x þ D26 ry Þ

C 64 ¼ ðA11 r 2x þ A12 r 2y þ 2r x r y A16 Þr 2x þ ðA12 r 2x þ A22 r 2y þ 2rx r y A26 Þr2y   þ 2r x ry A16 r 2x þ A26 r 2y þ 2r x r y A66 C 65 ¼ ðA11 s2x þ A12 s2y þ 2sx sy A16 Þr 2x þ ðA12 s2x þ A22 s2y þ 2sx sy A26 Þr2y   þ 2r x ry A16 s2x þ A26 s2y þ 2sx sy A66 C 66 ¼ 2ðrx sx A11 þ r y sy A12 þ ðrx sy þ r y sx ÞA16 Þr2x þ 2ðr x sx A12 þ r y sy A22 þ ðrx sy þ r y sx ÞA26 Þr2y þ 4r x ry ðr x sx A16 þ ry sy A26 þ ðr x sy þ ry sx ÞA66 Þ   C 67 ¼ A11 r 2x þ A12 r 2y þ 2rx r y A16 s2x þ ðA12 r 2x þ A22 r 2y þ 2rx r y A26 Þs2y   þ 2sx sy A16 r 2x þ A26 r 2y þ 2r x r y A66     C 68 ¼ A11 s2x þ A12 s2y þ 2sx sy A16 s2x þ A12 s2x þ A22 s2y þ 2sx sy A26 s2y   þ 2sx sy A16 s2x þ A26 s2y þ 2sx sy A66 C 69 ¼ 2ðrx sx A11 þ r y sy A12 þ ðrx sy þ r y sx ÞA16 Þs2x þ 2ðr x sx A12 þ r y sy A22 þ ðrx sy þ r y sx ÞA26 Þs2y þ 4sx sy ðr x sx A16 þ ry sy A26 þ ðr x sy þ ry sx ÞA66 Þ

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    C 70 ¼ A11 r 2x þ A12 r2y þ 2r x r y A16 rx sx þ A12 r 2x þ A22 r2y þ 2r x r y A26 ry sy   þ ðr x sy þ r y sx Þ A16 r2x þ A26 r 2y þ 2rx r y A66     C 71 ¼ A11 s2x þ A12 s2y þ 2sx sy A16 r x sx þ A12 s2x þ A22 s2y þ 2sx sy A26 r y sy   þ ðr x sy þ r y sx Þ A16 s2x þ A26 s2y þ 2sx sy A66 C 72 ¼ 2ðr x sx A11 þ ry sy A12 þ ðrx sy þ r y sx ÞA16 Þrx sx

D21 ¼ 2r x sx F 11 þ 2ry sy F 66 þ 2ðrx sy þ ry sx ÞF 16 ; DD22 ¼ F 16 r 2x þ F 26 r2y þ ðF 12 þ F 66 Þr x ry D23 ¼ F 16 s2x þ F 26 s2y þ ðF 12 þ F 66 Þsx sy ; D24 ¼ 2r x sx F 16 þ 2ry sy F 26 þ ðr x sy þ r y sx ÞðF 12 þ F 66 Þ D25 ¼ ðB11 r x þ B16 r y Þr 2x þ ðB11 rx þ B16 r y Þr 2y þ ð2B16 r x þ ðB12 þ B66 Þr y Þr x r y DD26 ¼ ðB11 rx þ B16 ry Þs2x þ ðB11 rx þ B16 ry Þs2y þ ð2B16 r x þ ðB12 þ B66 Þry Þsx sy

þ 2ðr x sx A12 þ ry sy A22 þ ðr x sy þ ry sx ÞA26 Þry sy þ 2ðr x sy þ ry sx Þðr x sx A16 þ ry sy A26 þ ðrx sy þ ry sx ÞA66 Þ

D27 ¼ ðB11 rx þ B16 r y Þ2rx sx þ ðB66 r x þ B26 r y Þ2r y sy C 73 ¼ qa2 =A22 h

þ ð2B16 r x þ ðB12 þ B66 Þr y Þðrx sy þ r y sx Þ D28 ¼ ðB11 sx þ B16 sy Þr 2x þ ðB11 sx þ B16 sy Þr 2y þ ð2B16 sx

A.4. Eq. (5d)

þ ðB12 þ B66 Þsy Þr x r y 2

D1

2

2

2

2

2

@ u0 @ u0 @ u0 @ v0 @ v0 @ v0 þ D2 2 þ D3 þ D5 þ D6 þ D4 @r 2 @s @r@s @r 2 @s2 @r@s @ 2 wy @ 2 wy @ 2 wx @ 2 wx @ 2 wx þ D8 þ D9 þ DD11 þ D7 þ D10 @r 2 @s2 @r@s @r 2 @s2 2 2 2 2 @ wy @ u1 @ u1 @ u1 @2v 1 þ DD12 þ D14 2 þ D15 þ D13 þ DD16 2 @r@s @r @s @r@s @r2 2 2 2 2 2 @ v1 @ v1 @ /x @ /x @ /x þ D17 þ D18 þ D20 þ D21 þ D19 @s2 @r@s @r 2 @s2 @r@s @ 2 /y @ 2 /y @ 2 /y @ 2 w0 @ 2 w0 þ DD22 þ D þ D þ DD þ ðD 23 24 25 26 @r 2 @s2 @r@s @r2 @s2 ! 2 2 2 2 @ w0 @w0 @ w0 @ w0 @ w0 @w0 þ D27 þ D29 þ D30 Þ þ D28 @r@s @r @r 2 @s2 @r@s @s @w0 @w0 þ D32 þ D33 wx þ D34 wy þ D35 u1 þ D36 v 1 @r @s þ D37 /x þ D38 /y ¼ 0 þ D31

D1 ¼ B11 r2x þ B66 r 2y þ 2B16 r x r y ;

D2 ¼ B11 s2x þ B66 s2y þ 2B16 sx sy

D3 ¼ 2r x sx B11 þ 2r y sy B66 þ 2ðr x sy þ ry sx ÞB16 ; D4 ¼ B16 r2x þ B26 r 2y þ ðB12 þ B66 Þrx r y D5 ¼ B16 s2x þ B26 s2y þ ðB12 þ B66 Þsx sy ; D6 ¼ 2r x sx B16 þ 2r y sy B26 þ ðrx sy þ r y sx ÞðB12 þ B66 Þ D7 ¼ D11 r 2x þ D66 r2y þ 2D16 r x r y ;

D8 ¼ D11 s2x þ D66 s2y þ 2D16 sx sy

D9 ¼ 2r x sx D11 þ 2ry sy D66 þ 2ðr x sy þ r y sx ÞD16 ; D10 ¼ D16 r2x þ D26 r 2y þ ðD12 þ D66 Þr x r y DD11 ¼ D16 s2x þ D26 s2y þ ðD12 þ D66 Þsx sy ; DD12 ¼ 2r x sx D16 þ 2r y sy D26 þ ðrx sy þ r y sx ÞðD12 þ D66 Þ

D29 ¼ ðB11 sx þ B16 sy Þs2x þ ðB11 sx þ B16 sy Þs2y þ ð2B16 sx þ ðB12 þ B66 Þsy Þsx sy

D30 ¼ ðB11 sx þ B16 sy Þ2rx sx þ ðB66 sx þ B26 sy Þ2r y sy þ ð2B16 sx þ ðB12 þ B66 Þsy Þðrx sy þ r y sx Þ D31 ¼ ðA55 rx þ A45 ry Þ; D33 ¼ A55 ; D35 ¼ 2B55 ;

D32 ¼ ðA55 sx þ A45 sy Þ;

D34 ¼ A45 D36 ¼ 2B45 ;

D37 ¼ 3D55 ;

D38 ¼ 3D45

A.5. Eq. (5e)

@ 2 u0 @ 2 u0 @ 2 u0 @2v 0 @2v 0 @2v 0 þ ED2 2 þ ED3 þ E5 þ E6 þ E4 2 2 2 @r @s @r@s @r @s @r@s @ 2 wy @ 2 wy @ 2 wy @ 2 wx @ 2 wx @ 2 wx þ E7 þ E8 þ E9 þ ED11 þ ED12 þ E10 2 2 2 2 @r @s @r@s @r @s @r@s @ 2 u1 @ 2 u1 @ 2 u1 @2v 1 @2v 1 þ E13 2 þ E14 2 þ E15 þ E17 þ ED16 @r @s @r@s @r 2 @s2 2 2 2 2 @ 2 /y @ v1 @ /x @ /x @ /x þ E18 þ E20 þ E21 þ E19 þ ED22 2 2 @r@s @r @s @r@s @r 2 ! @ 2 /y @ 2 /y @ 2 w0 @ 2 w0 @ 2 w0 @w0 þ E23 þ E24 þ ED26 þ E27 þ E25 @s2 @r@s @r2 @s2 @r@s @r ! 2 2 2 @ w0 @ w0 @ w0 @w0 @w0 @w0 þ E29 þ E30 þ E28 þ E31 þ E32 @r 2 @s2 @r@s @s @r @s

ED1

þ E33 wx þ E34 wy þ E35 u1 þ E36 v 1 þ E37 /x þ E38 /y ¼ 0

ED1 ¼ B16 r 2x þ B26 r 2y þ ðB12 þ B66 Þr x r y ; ED2 ¼ B16 s2x þ B26 s2y þ ðB12 þ B66 Þsx sy ED3 ¼ 2r x sx B16 þ 2ry sy B26 þ ðr x sy þ r y sx ÞðB12 þ B66 Þ; E4 ¼ B66 r 2x þ B22 r2y þ 2B26 r x r y

D13 ¼ E11 r2x þ E66 r2y þ 2E16 r x r y ;

D14 ¼ E11 s2x þ E66 s2y þ 2E16 sx sy

D15 ¼ 2rx sx E11 þ 2r y sy E66 þ 2ðr x sy þ r y sx ÞE16 ; DD16 ¼

E16 r 2x

þ

E26 r 2y

þ ðE12 þ E66 Þrx r y

D17 ¼ E16 s2x þ E26 s2y þ ðE12 þ E66 Þsx sy ; D18 ¼ 2rx sx E16 þ 2r y sy E26 þ ðr x sy þ r y sx ÞðE12 þ E66 Þ D19 ¼ F 11 r 2x þ F 66 r 2y þ 2F 16 r x ry ;

D20 ¼ F 11 s2x þ F 66 s2y þ 2F 16 sx sy

E5 ¼ B66 s2x þ B22 s2y þ 2B26 sx sy ; E6 ¼ 2rx sx B66 þ 2r y sy B22 þ 2ðr x sy þ r y sx ÞB26 E7 ¼ D16 r2x þ D26 r 2y þ ðD12 þ D66 Þr x r y ; E8 ¼ D16 s2x þ D26 s2y þ ðD12 þ D66 Þsx sy E9 ¼ 2rx sx D16 þ 2r y sy D26 þ ðr x sy þ r y sx ÞðD12 þ D66 Þ; E10 ¼ D66 r 2x þ D22 r 2y þ 2D26 rx r y

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A.K. Upadhyay, K.K. Shukla / Composite Structures 94 (2012) 3722–3735

ED11 ¼ D66 s2x þ D22 s2y þ 2D26 sx sy ;

F 1 ¼ D11 r 2x þ D66 r 2y þ 2D16 r x ry ;

ED12 ¼ 2r x sx D66 þ 2r y sy D22 þ 2ðr x sy þ ry sx ÞD26

F 2 ¼ D11 s2x þ D66 s2y þ 2D16 sx sy

F 3 ¼ 2r x sx D11 þ 2r y sy D66 þ 2ðr x sy þ r y sx ÞD16 ;

E13 ¼ E16 r2x þ E26 r 2y þ ðE12 þ E66 Þr x r y ;

F 4 ¼ D16 r 2x þ D26 r 2y þ ðD12 þ D66 Þr x ry

E14 ¼ E16 s2x þ E26 s2y þ ðE12 þ E66 Þsx sy

F 5 ¼ D16 s2x þ D26 s2y þ ðD12 þ D66 Þsx sy ;

E15 ¼ 2rx sx E16 þ 2r y sy E26 þ ðr x sy þ r y sx ÞðE12 þ E66 Þ; ED16 ¼ E66 r 2x þ E22 r 2y þ 2E26 rx r y

F 6 ¼ 2r x sx D16 þ 2r y sy D26 þ ðr x sy þ ry sx ÞðD12 þ D66 Þ

E17 ¼ E66 s2x þ E22 s2y þ 2E26 sx sy ;

F 7 ¼ E11 r 2x þ E66 r 2y þ 2E16 r x ry ;

E18 ¼ 2rx sx E66 þ 2r y sy E22 þ 2ðr x sy þ r y sx ÞE26

F 9 ¼ 2r x sx E11 þ 2r y sy E66 þ 2ðr x sy þ r y sx ÞE16 ;

E19 ¼ F 16 r 2x þ F 26 r 2y þ ðF 12 þ F 66 Þr x r y ; E20 ¼

F 16 s2x

þ

F 26 s2y

F 10 ¼ E16 r 2x þ E26 r2y þ ðE12 þ E66 Þr x ry

þ ðF 12 þ F 66 Þsx sy

FD11 ¼ E16 s2x þ E26 s2y þ ðE12 þ E66 Þsx sy ;

E21 ¼ 2rx sx F 16 þ 2r y sy F 26 þ ðr x sy þ r y sx ÞðF 12 þ F 66 Þ;

FD12 ¼ 2r x sx E16 þ 2r y sy E26 þ ðrx sy þ r y sx ÞðE12 þ E66 Þ

ED22 ¼ F 66 r2x þ F 22 r 2y þ 2F 26 r x r y E23 ¼

F 66 s2x

þ

F 22 s2y

F 8 ¼ E11 s2x þ E66 s2y þ 2E16 sx sy

F 13 ¼ F 11 r 2x þ F 66 r2y þ 2F 16 rx r y ;

F 14 ¼ F 11 s2x þ F 66 s2y þ 2F 16 sx sy

þ 2F 26 sx sy ; F 15 ¼ 2rx sx F 11 þ 2r y sy F 66 þ 2ðr x sy þ r y sx ÞF 16 ;

E24 ¼ 2rx sx F 66 þ 2r y sy F 22 þ 2ðrx sy þ r y sx ÞF 26

FD16 ¼ F 16 r2x þ F 26 r 2y þ ðF 12 þ F 66 Þrx r y

E25 ¼ ðB16 r x þ A66 r y Þr 2x þ ðB26 r x þ A22 r y Þr2y þ ð2B26 r y þ ðB12 þ B66 Þr x Þr x r y

F 17 ¼ F 16 s2x þ F 26 s2y þ ðF 12 þ F 66 Þsx sy ;

ED26 ¼ ðB16 rx þ A66 r y Þs2x þ ðB26 r x þ B22 r y Þs2y þ ð2B26 ry þ ðB12 þ B66 Þr x Þsx sy

F 18 ¼ 2rx sx F 16 þ 2r y sy F 26 þ ðr x sy þ r y sx ÞðF 12 þ F 66 Þ F 19 ¼ H11 r 2x þ H66 r2y þ 2H16 r x r y ;

E27 ¼ ðB16 r x þ B66 r y Þ2r x sx þ ðB26 r x þ B22 ry Þ2r y sy þ ð2B26 r y þ ðB12 þ B66 Þr x Þðr x sy þ r y sx Þ

F 20 ¼ H11 s2x þ H66 s2y þ 2H16 sx sy

F 21 ¼ 2rx sx H11 þ 2ry sy H66 þ 2ðr x sy þ r y sx ÞH16 ;

E428 ¼ ðB16 sx þ B66 sy Þr2x þ ðB26 sx þ B22 sy Þr2y þ ð2B26 sy þ ðB12 þ B66 Þsx Þrx ry

FD22 ¼ H16 r 2x þ H26 r 2y þ ðH12 þ H66 Þrx r y F 23 ¼ H16 s2x þ H26 s2y þ ðH12 þ H66 Þsx sy ;

E29 ¼ ðB16 sx þ B66 sy Þs2x þ ðB26 sx þ B22 sy Þs2y þ ð2B26 sy

F 24 ¼ 2rx sx H16 þ 2ry sy H26 þ ðr x sy þ ry sx ÞðH12 þ H66 Þ

þ ðB12 þ B66 Þsx Þsx sy

F 25 ¼ ðD11 r x þ D16 r y Þr 2x þ ðD11 rx þ D16 r y Þr 2y

E30 ¼ ðB16 sx þ B66 sy Þ2r x sx þ ðB26 sx þ B22 sy Þ2r y sy

þ ð2D16 r x þ ðD12 þ D66 Þry Þrx r y

þ ð2B26 sy þ ðB12 þ B66 Þsx Þðr x sy þ r y sx Þ E31 ¼ ðA45 r x þ A44 r y Þ; E32 ¼ ðA45 sx þ A44 sy Þ; E33 ¼ A45 ; E34 ¼ A44

FD26 ¼ ðD11 r x þ D16 ry Þs2x þ ðD11 r x þ D16 ry Þs2y

E35 ¼ 2B45 ;

F 27 ¼ ðD11 r x þ D16 r y Þ2r x sx þ ðD66 rx þ D26 r y Þ2r y sy

E36 ¼ 2B44 ;

E37 ¼ 3D45 ;

þ ð2D16 r x þ ðD12 þ D66 Þr y Þsx sy

E38 ¼ 3D44

þ ð2D16 r x þ ðD12 þ D66 Þry Þðr x sy þ r y sx Þ A.6. Eq. (5f)

F 28 ¼ ðD11 sx þ D16 sy Þr 2x þ ðD11 sx þ D16 sy Þr 2y 2

F1

2

2

2

2

2

2

@ u0 @ u0 @ u0 @ v0 @ v0 @ v0 @ w þ F2 2 þ F3 þ F5 2 þ F6 þ F4 þ F 7 2x @r 2 @s @r@s @r 2 @s @r@s @r @ 2 wy @ 2 wy @ 2 wy @ 2 wx @ 2 wx @ 2 u1 þ F8 2 þ F9 þ F 10 2 þ FD11 2 þ FD12 þ F 13 2 @s @r@s @r @s @r@s @r @ 2 u1 @ 2 u1 @2v 1 @2v 1 @2v 1 @ 2 /x þ F 14 2 þ F 15 þ F 17 2 þ F 18 þ FD16 þ F 19 2 @s @r@s @r 2 @s @r@s @r @ 2 /y @ 2 /y @ 2 /y @ 2 /x @ 2 /x þ F 20 2 þ F 21 þ FD22 2 þ F 23 2 þ F 24 @s @r@s @r @r@s ! @s 2 2 @ w0 @ w0 @ 2 w0 @w0 þ FD26 þ F 27 þ F 25 @r2 @s2 @r@s @r ! 2 2 2 @ w0 @ w0 @ w0 @w0 @w0 @w0 þ F 29 þ F 30 þ F 31 þ F 32 þ F 28 @r2 @s2 @r@s @s @r @s þ F 33 wx þ F 34 wy þ F 35 u1 þ F 36 v 1 þ F 37 /x þ F 38 /y ¼ 0

þ ð2D16 sx þ ðD12 þ D66 Þsy Þr x ry F 29 ¼ ðD11 sx þ D16 sy Þs2x þ ðD11 sx þ D16 sy Þs2y þ ð2D16 sx þ ðD12 þ D66 Þsy Þsx sy F 30 ¼ ðD11 sx þ D16 sy Þ2r x sx þ ðD66 sx þ D26 sy Þ2r y sy þ ð2D16 sx þ ðD12 þ D66 Þsy Þðr x sy þ r y sx Þ F 31 ¼ 2ðB55 r x þ B45 ry Þ;

F 32 ¼ 2ðB55 sx þ B45 sy Þ;

F 33 ¼ 2B55 ;

F 34 ¼ 2B45

F 35 ¼ 4D55 ;

F 36 ¼ 4D45 ;

F 37 ¼ 6E55 ;

F 38 ¼ 6E45

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A.K. Upadhyay, K.K. Shukla / Composite Structures 94 (2012) 3722–3735

A.7. Eq. (5g)

G1

@ 2 u0 @ 2 u0 @ 2 u0 @2v 0 @2v 0 @2v 0 þ G2 2 þ G3 þ G5 þ G6 þ G4 2 2 2 @r @s @r@s @r @s @r@s @ 2 wy @ 2 wy @ 2 wx @ 2 wx @ 2 wx þ G8 þ G9 þ G11 þ G10 þ G7 2 2 2 @r @s @r@s @r @s2 þ GD12

@ 2 wy @ 2 u1 @ 2 u1 @ 2 u1 @2v 1 þ GD13 2 þ G14 2 þ G15 þ G16 @r@s @r @s @r@s @r 2

@2v 1 @2v 1 @ 2 /x @ 2 /x @ 2 /x þ G17 þ G þ G þ G þ G 18 19 20 21 @s2 @r@s @r 2 @s2 @r@s @ 2 /y @ 2 /y @ 2 /y @ 2 w0 @ 2 w0 þ GD23 þ G24 þ G26 þ G22 þ ðG25 2 2 2 @r @s @r@s @r @s2 ! 2 2 2 2 @ w0 @w0 @ w0 @ w0 @ w0 @w0 þ G27 þ G29 þ G30 Þ þ G28 @r@s @r @r 2 @s2 @r@s @s @w0 @w0 þ G31 þ G32 þ G33 wx þ G34 wy þ G35 u1 þ G36 v 1 @r @s þ G37 /x þ G38 /y ¼ 0 G1 ¼ D16 r 2x þ D26 r2y þ ðD12 þ D66 Þr x r y ;

G26 ¼ ðD16 r x þ D66 ry Þs2x þ ðD26 rx þ D22 r y Þs2y þ ð2D26 ry þ ðD12 þ D66 Þr x Þsx sy

G27 ¼ ðD16 r x þ D66 ry Þ2r x sx þ ðD26 r x þ D22 r y Þ2r y sy þ ð2D26 r y þ ðD12 þ D66 Þr x Þðrx sy þ r y sx Þ G28 ¼ ðD16 sx þ D66 sy Þr2x þ ðD26 sx þ D22 sy Þr2y þ ð2D26 sy þ ðD12 þ D66 Þsx Þr x r y G29 ¼ ðD16 sx þ D66 sy Þs2x þ ðD26 sx þ D22 sy Þs2y þ ð2D26 sy þ ðD12 þ D66 Þsx Þsx sy G30 ¼ ðD16 sx þ D66 sy Þ2r x sx þ ðD26 sx þ D22 sy Þ2r y sy þ ð2D26 sy þ ðD12 þ D66 Þsx Þðrx sy þ r y sx Þ G31 ¼ 2ðB45 rx þ B44 r y Þ;

G32 ¼ 2ðB45 sx þ B44 sy Þ;

G33 ¼ 2B45 ;

G34 ¼ 2B44

G35 ¼ 4D45 ;

G36 ¼ 4D44 ;

G37 ¼ 6E45 ;

G38 ¼ 6E44

A.8. Eq. (5h)

G2 ¼ D16 s2x þ D26 s2y þ ðD12 þ D66 Þsx sy G3 ¼ 2rx sx D16 þ 2ry sy D26 þ ðr x sy þ r y sx ÞðD12 þ D66 Þ; G4 ¼ D66 r 2x þ D22 r2y þ 2D26 r x r y G5 ¼ D66 s2x þ D22 s2y þ 2D26 sx sy ; G6 ¼ 2rx sx D66 þ 2ry sy D22 þ 2ðr x sy þ r y sx ÞD26 G7 ¼ E16 r 2x þ E26 r2y þ ðE12 þ E66 Þr x ry ; G8 ¼ E16 s2x þ E26 s2y þ ðE12 þ E66 Þsx sy G9 ¼ 2rx sx E16 þ 2ry sy E26 þ ðr x sy þ ry sx ÞðE12 þ E66 Þ; G10 ¼ E66 r2x þ E22 r 2y þ 2E26 r x r y G11 ¼ E66 s2x þ E22 s2y þ 2E26 sx sy ; GD12 ¼ 2r x sx E66 þ 2r y sy E22 þ 2ðr x sy þ ry sx ÞE26 GD13 ¼ F 16 r2x þ F 26 r 2y þ ðF 12 þ F 66 Þr x r y ; G14 ¼ F 16 s2x þ F 26 s2y þ ðF 12 þ F 66 Þsx sy G15 ¼ 2r x sx F 16 þ 2r y sy F 26 þ ðr x sy þ r y sx ÞðF 12 þ F 66 Þ; G16 ¼ F 66 r 2x þ F 22 r 2y þ 2F 26 r x ry G17 ¼ F 66 s2x þ F 22 s2y þ 2F 26 sx sy ; G18 ¼ 2r x sx F 66 þ 2r y sy F 22 þ 2ðrx sy þ r y sx ÞF 26 G19 ¼ H16 r2x þ H26 r 2y þ ðH12 þ H66 Þr x r y ; G20 ¼ H16 s2x þ H26 s2y þ ðH12 þ H66 Þsx sy G21 ¼ 2r x sx H16 þ 2r y sy H26 þ ðr x sy þ r y sx ÞðH12 þ H66 Þ; G22 ¼ H66 r2x þ H22 r 2y þ 2H26 r x r y GD23 ¼ H66 s2x þ H22 s2y þ 2H26 sx sy ; G24 ¼ 2r x sx H66 þ 2r y sy H22 þ 2ðr x sy þ r y sx ÞH26 G25 ¼ ðD16 r x þ D66 r y Þr 2x þ ðD26 rx þ D22 r y Þr 2y þ ð2D26 r y þ ðD12 þ D66 Þr x Þr x ry

H1

@ 2 u0 @ 2 u0 @ 2 u0 @2v 0 @2v 0 @2v 0 þ H2 2 þ H3 þ H5 þ H6 þ H4 2 2 2 @r @s @r@s @r @s @r@s @ 2 wy @ 2 wy @ 2 wx @ 2 wx @ 2 wx þ H8 þ H9 þ HD11 þ H7 þ H10 @r 2 @s2 @r@s @r2 @s2 2 2 2 @ 2 wy @ u1 @ u1 @ u1 @2v 1 þ HD12 þ H14 2 þ H15 þ H13 þ HD16 2 @r@s @r @s @r@s @r2 @2v 1 @2v 1 @ 2 /x @ 2 /x @ 2 /x þ H17 þ H18 þ H20 þ H21 þ H19 @s2 @r@s @r 2 @s2 @r@s 2 2 2 2 @ /y @ /y @ /y @ w0 @ 2 w0 þ HD22 þ H þ H þ HD þ ðH 23 24 25 26 2 @r2 @s2 @r@s @r2 ! @s @ 2 w0 @w0 @ 2 w0 @ 2 w0 @ 2 w0 @w0 þ H27 þ H29 þ H30 Þ þ H28 @r@s @r @r 2 @s2 @r@s @s @w0 @w0 þ H32 þ H33 wx þ H34 wy þ H35 u1 þ H36 v 1 @r @s þ H37 /x þ H38 /y ¼ 0 þ H31

H1 ¼ E11 r2x þ E66 r2y þ 2r x r y E16 ;

H2 ¼ E11 s2x þ E66 s2y þ 2sx sy E16

H3 ¼ 2rx sx E66 þ 2r y sy E66 þ 2ðr x sy þ r y sx ÞE16 ; H4 ¼ E16 r2x þ E26 r2y þ r x ry ðE12 þ E66 Þ H5 ¼ E16 s2x þ E26 s2y þ sx sy ðE12 þ E66 Þ; H6 ¼ 2rx sx E16 þ 2r y sy E26 þ ðr x sy þ r y sx ÞðE12 þ E66 Þ H7 ¼ F 11 r 2x þ F 66 r 2y þ 2r x r y F 16 ; H8 ¼ F 11 s2x þ F 66 s2y þ 2sx sy F 16 H9 ¼ 2rx sx F 11 þ 2r y sy F 66 þ 2ðr x sy þ r y sx ÞF 16 ; H10 ¼ F 16 r 2x þ F 26 r 2y þ r x r y ðF 12 þ F 66 Þ HD11 ¼ F 16 s2x þ F 26 s2y þ sx sy ðF 12 þ F 66 Þ; HD12 ¼ 2r x sx F 16 þ 2r y sy F 26 þ ðr x sy þ ry sx ÞðF 12 þ F 66 Þ H13 ¼ H11 r 2x þ H66 r 2y þ 2r x r y H16 ; H14 ¼ H11 s2x þ H66 s2y þ 2sx sy H16

3734

A.K. Upadhyay, K.K. Shukla / Composite Structures 94 (2012) 3722–3735

P5 ¼ E66 s2x þ E22 s2y þ 2sx sy E26 ;

H15 ¼ 2rx sx H11 þ 2r y sy H66 þ 2ðr x sy þ r y sx ÞH16 ; H16 r 2x

HD16 ¼

þ

H26 r 2y

þ r x r y ðH12 þ H66 Þ

P6 ¼ 2r x sx E66 þ 2r y sy E22 þ 2ðr x sy þ ry sx ÞE26

H17 ¼ H16 s2x þ H26 s2y þ sx sy ðH12 þ H66 Þ;

P7 ¼ F 16 r2x þ F 26 r 2y þ r x r y ðF 12 þ F 66 Þ;

H18 ¼ 2rx sx H16 þ 2r y sy H26 þ ðr x sy þ r y sx ÞðH12 þ H66 Þ

P8 ¼ F 16 s2x þ F 26 s2y þ sx sy ðF 12 þ F 66 Þ

H19 ¼ J 11 r 2x þ J 66 r 2y þ 2r x r y J 16 ;

P9 ¼ 2r x sx F 16 þ 2r y sy F 26 þ ðrx sy þ ry sx ÞðF 12 þ F 66 Þ;

H20 ¼

J 11 s2x

þ

J 66 s2y

P10 ¼ F 66 r 2x þ F 22 r2y þ 2r x ry F 26

þ 2sx sy J 16

P11 ¼ F 66 s2x þ F 22 s2y þ 2sx sy F 26 ;

H21 ¼ 2rx sx J 11 þ 2ry sy J 66 þ 2ðrx sy þ ry sx ÞJ 16 ; J 16 r2x

HD22 ¼

þ

J 26 r2y

þ rx r y ðJ 12 þ J 66 Þ

P12 ¼ 2r x sx F 66 þ 2r y sy F 22 þ 2ðr x sy þ r y sx ÞF 26

H23 ¼ J 16 s2x þ J 26 s2y þ sx sy ðJ 12 þ J 66 Þ;

P13 ¼ H16 r 2x þ H26 r2y þ rx r y ðH12 þ H66 Þ;

H24 ¼ 2rx sx J 16 þ 2ry sy J 26 þ ðr x sy þ r y sx ÞðJ 12 þ J 66 Þ

P14 ¼ H16 s2x þ H26 s2y þ sx sy ðH12 þ H66 Þ

H25 ¼ ðE16 r x þ E66 r y Þr2x þ ðE26 rx þ E22 ry Þr2y þ r x r y ððE12 þ E66 Þrx þ 2E26 r y Þ

P15 ¼ 2r x sx H16 þ 2ry sy H26 þ ðr x sy þ ry sx ÞðH12 þ H66 Þ; P16 ¼ H66 r 2x þ H22 r2y þ 2r x ry H26

HD26 ¼ ðE16 rx þ E66 ry Þs2x þ ðE26 rx þ E22 r y Þs2y þ sx sy ððE12 þ E66 Þrx þ 2E26 ry Þ

H27 ¼ ðE16 r x þ E66 r y Þ2r x sx þ ðE26 rx þ E22 r y Þ2r y sy

P17 ¼ H66 s2x þ H22 s2y þ 2sx sy H26 ; P18 ¼ 2r x sx H66 þ 2ry sy H22 þ 2ðr x sy þ r y sx ÞH26

þ ðr x sy þ sx r y ÞððE12 þ E66 Þr x þ 2E26 ry Þ H28 ¼ ðE16 sx þ E66 sy Þr2x

þ ðE26 sx þ E22 sy Þr 2y

þ rx ry ððE12 þ E66 Þsx þ 2E26 sy Þ

H29 ¼ ðE16 sx þ E66 sy Þs2x þ ðE26 sx þ E22 sy Þs2y þ sx sy ððE12 þ E66 Þsx þ 2E26 sy Þ

P19 ¼ J 16 r 2x þ J 26 r 2y þ r x ry ðJ12 þ J 66 Þ; P20 ¼ J 16 s2x þ J 26 s2y þ sx sy ðJ 12 þ J 66 Þ P21 ¼ 2r x sx J 16 þ 2r y sy J 26 þ ðr x sy þ r y sx ÞðJ 12 þ J 66 Þ; P22 ¼ J 66 r 2x þ J 22 r 2y þ 2r x r y J 26

H30 ¼ ðE16 sx þ E66 sy Þ2rx sx þ ðE26 sx þ E22 sy Þ2r y sy P23 ¼ J 66 s2x þ J 22 s2y þ 2sx sy J26 ;

þ ðr x sy þ sx r y ÞððE12 þ E66 Þsx þ 2E26 sy Þ

P24 ¼ 2r x sx J 66 þ 2r y sy J 22 þ 2ðrx sy þ r y sx ÞJ 26 H31 ¼ 3ðD55 r x þ D45 r y Þ;

H32 ¼ 3ðD55 sx þ D45 sy Þ;

H33 ¼ 3D55 ;

H34 ¼ 3D45

H35 ¼ 6E55 ;

H36 ¼ 6E45 ;

P25 ¼ ðE16 rx þ E66 r y Þr 2x þ ðE26 rx þ E22 ry Þr 2y þ r x ry ððE12 þ E66 Þr x H37 ¼ 9F 55 ;

H38 ¼ 9F 45

þ 2E26 ry Þ P26 ¼ ðE16 rx þ E66 r y Þs2x þ ðE26 rx þ E22 ry Þs2y þ sx sy ððE12 þ E66 Þr x

A.9. Eq. (5i)

P1

@ 2 u0 @ 2 u0 @ 2 u0 @2v 0 @2v 0 @2v 0 @ 2 wx þ P2 2 þ P3 þ P5 þ P6 þ P4 þ P7 @r2 @s @r@s @r 2 @s2 @r@s @r2 þ P8

@ 2 wy @ 2 wy @ 2 wy @ 2 wx @ 2 wx @ 2 u1 þ P9 þ P11 þ P12 þ P10 þ P13 2 2 2 @s @r@s @r @s @r@s @r 2

þ P14

@ 2 u1 @ 2 u1 @2v 1 @2v 1 @2v 1 þ P15 þ P17 þ P18 þ P16 2 2 2 @s @r@s @r @s @r@s

@ 2 /y @ 2 /y @ 2 /x @ 2 /x @ 2 /x þ P þ P þ P þ P þ P19 20 21 22 23 @r 2 @s2 @r@s @r 2 @s2 @ 2 /y @ 2 w0 @ 2 w0 @ 2 w0 @w0 þ P26 þ P27 þ ðP25 Þ 2 2 @r@s @r @s @r@s @r ! @ 2 w0 @ 2 w0 @ 2 w0 @w0 @w0 @w0 þ P þ P þ P31 þ P32 þ P28 29 30 @r 2 @s2 @r@s @s @r @s

þ P24

þ P33 wx þ P 34 wy þ P35 u1 þ P36 v 1 þ P37 /x þ P38 /y ¼ 0

þ 2E26 ry Þ P27 ¼ ðE16 rx þ E66 r y Þ2r x sx þ ðE26 rx þ E22 ry Þ2r y sy þ ðrx sy þ sx r y ÞððE12 þ E66 Þr x þ 2E26 r y Þ P28 ¼ ðE16 sx þ E66 sy Þr 2x þ ðE26 sx þ E22 sy Þr 2y þ r x r y ððE12 þ E66 Þsx þ 2E26 sy Þ P29 ¼ ðE16 sx þ E66 sy Þs2x þ ðE26 sx þ E22 sy Þs2y þ sx sy ððE12 þ E66 Þsx þ 2E26 sy Þ P30 ¼ ðE16 sx þ E66 sy Þ2r x sx þ ðE26 sx þ E22 sy Þ2r y sy þ ðrx sy þ sx r y ÞððE12 þ E66 Þsx þ 2E26 sy Þ P31 ¼ 3ðD45 rx þ D44 r y Þ; P32 ¼ 3ðD45 sx þ D44 sy Þ; P33 ¼ 3D45 ; P 34 ¼ 3D44 P35 ¼ 6E45 ;

P 36 ¼ 6E44 ;

P37 ¼ 9F 45 ;

P38 ¼ 9F 44

P1 ¼ E16 r 2x þ E26 r 2y þ r x r y ðE12 þ E66 Þ; P2 ¼ E16 s2x þ E26 s2y þ sx sy ðE12 þ E66 Þ P3 ¼ 2r x sx E16 þ 2r y sy E26 þ ðr x sy þ r y sx ÞðE12 þ E66 Þ; P4 ¼ E66 r 2x þ E22 r 2y þ 2r x r y E26

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