Hybrid-interface element for thick laminated composite plates

Computers and Structures 85 (2007) 1484–1499 www.elsevier.com/locate/compstruc

Hybrid-interface element for thick laminated composite plates A.N. Bambole, Y.M. Desai

*

Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India Received 20 June 2006; accepted 22 January 2007 Available online 13 March 2007

Abstract A novel 27-node three-dimensional hexahedral hybrid-interface finite element (FE) model has been presented to analyze laminated composite plates and sandwich plates using the minimum potential energy principle. Fundamental elasticity relationship between components of stress, strain and displacement fields are maintained throughout the elastic continuum as the transverse stress components have been invoked as nodal degrees of freedom. Continuity of the transverse stresses at lamina interface has been maintained. Each lamina is modeled by using hybrid-interface elements at the top and the bottom interfaces and conventional displacement based elements sandwiched between these interfaces. Results obtained from the present formulation have found to be in excellent agreement with the elasticity solutions for thin and thick composite cross-ply, angle-ply laminates, as well as sandwich plates. Additional results have also been presented on the variation of the transverse strains to highlight magnitude of discontinuity in these quantities due to difference in properties of face and core materials of sandwich plates. Present formulation can be used effectively to interface hybrid formulation that uses transverse stresses and displacements as degrees of freedom with conventional purely displacement based formulation for realistic estimates of the transverse stresses.  2007 Elsevier Ltd. All rights reserved. Keywords: Hybrid-interface element; Minimum potential energy; Laminated composite; Cross-ply; Angle-ply; Sandwich plate

1. Introduction Modern composite materials are utilized in a wide range of fields like mechanical, aerospace, marine, automotive, biomedical and MEMS. Inherent complexity of composite materials demands extensive analysis techniques in order to adequately predict response of even simple parts [1]. Finite element (FE) method is widely used to analyze complex system of laminated composite structure. Computationally least demanding two-dimensional elements (plate bending or shell) have been used extensively in the past. Twodimensional FE formulations have been found to yield accurate results for laminates at locations away from boundaries and cutouts. However, two-dimensional analysis fail to give realistic solution near any geometric or material anomaly or near traction-free edges. With the *

Corresponding author. Tel.: +91 22 576 7333/7301; fax: +91 22 576 7302. E-mail address: [email protected] (Y.M. Desai). 0045-7949/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2007.01.022

recent advances in computer technology, researchers are prompted to develop and use computationally demanding three-dimensional (3-D) FE models. Various displacement based layer-wise theories and finite elements have been proposed by Reddy [2], Soldatos [3], Wu and Hsu [4] and others, providing satisfactory results for both the global values (e.g. deflections and flexural stresses) as well as the local values (e.g. transverse stresses) of thin and thick laminates. However, these models could not enforce continuity of the transverse stress components. Hybrid stress (mixed) FE models were developed to overcome lacuna of displacement models in enforcing the continuity of the transverse stresses. Pian [5], Mau et al. [6], for examples, presented hybrid stress FE models which ensured continuity of the transverse shear stress. However, discontinuity of the transverse normal stress and assumption of constant transverse displacement through thickness were major shortcomings. Eight node hybrid stress isoparametric elements were presented by Wen-Jinn and Sun

A.N. Bambole, Y.M. Desai / Computers and Structures 85 (2007) 1484–1499

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Nomenclature 1, 2, 3

local co-ordinate system of lamina (material principal directions) X ; Y ; Z global co-ordinate system of plate x; y; z local co-ordinate system of element 2Lx ; 2Ly ; 2Lz length, width and depth of element, respectively n, g x=Lx ; y=Ly and z/Lz, respectively a, b, t length, width and depth (thickness) of plate, respectively Dij coefficients of constitutive matrix with reference to plate’s as well as element’s reference axes (i.e. X, Z and x; z, respectively) u; v; w displacement components along element’s reference direction x, y, and z, respectively at a point ðx; y; zÞ rx ; ry ; rz ; sxy ; sxz ; syz components of stress at a point with reference to element’s reference axes ex ; ey ; ez ; cyx ; cxz ; cyz components of strain at a point with reference to element’s reference axes

[7] using 55 stress parameters and Spilker [8] using 67 stress parameters per lamina to obtain stress components. Carrera [9,10] presented a mixed FE model with the transverse normal stress component as degrees-of-freedom (DOF) in addition to the transverse shear stress components. In mixed/hybrid finite element models developed using Reissner’s variational principle, stress fields are assumed independent of the displacement fields and fundamental elasticity relation cannot be satisfied. On the other hand, 18-node, 3-dimensional (3-D) mixed FE model based on displacement theory satisfying fundamental elasticity relations has been presented by Ramtekkar et al. [11,12] and Desai et al. [13]. This model has been shown to provide reliable results for stress and displacement of laminated composites, satisfying continuity of displacements as well as the transverse stresses. Present hybrid-interface formulation is aimed to provide a 3-D finite element model, compatible

Pe ; P

total potential energy of an element and plate, respectively [K]e, [K] element and global property matrices, respectively {f}e, [F] element and global influence vectors, respectively {d}, {Q} element and global nodal DOF vectors, respectively ^q; q0 intensity of the transverse sinusoidal load and the maximum amplitude load acting at top surface S aspect ratio (a/t) Z non-dimensionlized thickness of plate (Z/t) u; v; w  non-dimensionlized displacement fields x ; r y ; r z ; sxy ; sxz ; syz non-dimensionlized stress compor nents

with conventional displacement element ensuring through thickness continuity of the transverse stress components as well as displacement components in a laminate. A layer-wise finite element model of laminate with displacement and the transverse stress components as primary variables at interface nodes and only displacement primary variables at non-interface nodes can very well satisfy the requirements of the transverse stress continuity in addition to the continuity of displacement fields (Fig. 1). Unique values of the transverse stress components at interface are directly obtained in this formulation. Thus, integration of equilibrium equations can be advantageously avoided which involve differentiation of in-plane stresses and displacement fields, leading to further approximation in calculation of the transverse stresses. A 27-node 3-D hexahedral hybrid-interface finite element model has been developed in the present work by

(i+1)th layer Interface connected to (i+1)th layer Displacements (u, v, w) u = (u, v, w) l In-plane strains (εx, εy, γxy)u = (εx, εy, γxy)l

Typical laminate

(i+1) layer i layer

In-plane stresses (σx, σy, τxy)u ≠ (σx, σy, τxy)l Transverse stresses (σz , τxz, τyz )u = (σz , τxz, τyz )l Transverse strains (εz , γxz, γyz )u ≠ (εz , γxz, γyz )l

Interface connected to ith layer

ith layer Fig. 1. Global/local continuity requirement of stress and strain (displacement) components at interface of laminated composites.

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using the minimum potential energy principle satisfying the entire requirement discussed above for an N-layered laminated plate. The transverse stress quantities (rz, sxz and syz) are invoked from the assumed displacement fields by using fundamental elasticity equations, giving unique and accurate estimates of these quantities. Difficulty like non-positive definiteness of matrices [14] is advantageously avoided in the present formulation by using fundamental elasticity relations for incorporating the transverse stress variables as nodal DOF in conjunction with the minimum energy principle.

a

3, Z

2

α

1 α

Typical lamina

b

2. 3-D numerical model 2.1. Theoretical formulation An anisotropic composite laminated plate consisting of N-layers of orthotropic lamine is considered for finite element analysis as shown in Fig. 2a–c. The plate has been discretized in to a number of 3-D elements. Each element lies completely within a lamina and no element crosses the interface between any two successive laminae. The plate is modeled by using 27-node hybrid-interface elements at lower interface (Fig. 3b) and upper interface (Fig. 3d) of each lamina, remaining portion of each lamina is modeled using conventional 27-node displacement elements (Fig. 3c) as shown in Fig. 3a.

Z

c

Y

X

t b a

2.2. Kinematics A 27-node 3-D hybrid-interface (lower-interface) finite element model shown in Fig. 3b has been developed by considering displacement fields uðx; y; zÞ; vðx; y; zÞ and wðx; y; zÞ having quadratic variation along the plane of the plate and cubic variation in the transverse direction. Displacement fields are expressed as ! 3 3 X 3 X X m uk ðx; y; zÞ ¼ z gi hj amijk ; k ¼ 1; 2; 3; ð1Þ m¼0

i¼1

j¼1

where n n g1 ¼ ðn  1Þ; g2 ¼ 1  n2 ; g3 ¼ ð1 þ nÞ; 2 2 d d 2 h1 ¼ ðd  1Þ; h2 ¼ 1  d ; h3 ¼ ð1 þ dÞ; 2 2

n ¼ x=Lx ;

ð2Þ

d ¼ y=Ly ;

ð3Þ

and

Fig. 2. Laminate geometry with positive set of lamina/laminate reference axes, (a) (1, 2, 3 ) – lamina reference, (b) ðX ; Y ; ZÞ – laminate reference axes of a general laminated plate, (c) ðX ; Y ; ZÞ – sandwich plate’s reference axes.

typical ith lamina with reference to the fiber-matrix coordinate axes (1, 2, 3) can be presented as 8 9i 2 3i 8 9i C 11 C 12 C 13 0 0 0 e r1 > > > > > > > 1 > > 7> 6 > > >r > >e > > > > C C 0 0 0 2 > 22 23 2 > > > > 7 6 > > > > > > 7> 6 = < 6 C 33 0 0 0 7 e3 = 3 7 6 ¼6 : > s12 > 0 7 c12 > C 44 0 > > > > 7> 6 Sym: > > > > > > > > 7> > 6 > > > > > 4 C 55 0 5 > > > > > > s13 > > c13 > ; ; : : C 66 s23 c23 ð5Þ

Further, amijk are the generalized co-ordinates. Element coordinate axes x, y, z are parallel to the laminate coordinates X, Y, Z.

Here ðr1 ; r2 ; r3 ; s12 ; s13 ; s23 Þ are stress components and ðe1 ; e2 ; e3 ; c12 ; c13 ; c23 Þ are the linear strain components referred to the lamina coordinates (1, 2, 3). Further, C mn ’s ðm; n ¼ 1; . . . ; 6Þ are the elastic constants of the ith lamina. The stress–strain relations for the ith lamina can be written in the laminate coordinates X ; Y ; Z as

2.3. Constitutive equations

frg ¼ ½Dfeg:

u1 ¼ u;

u2 ¼ v;

u3 ¼ w:

ð4Þ

ð6aÞ

Here Each lamina in the laminate has been considered to be in a 3-D state of stress so that the constitutive relation for a

frg ¼ ½ rx

ry

rz

sxy

sxz

syz T ;

ð6bÞ

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Fig. 3. Descretization pattern and elements used: (a) descretization pattern of laminated plate using hybrid-interface and displacement finite elements, (b) 27-node lower-interface element (placed at bottom of each lamina), (c) 27-node displacement element (placed in core of each lamina) (d) 27-node upperinterface element (placed at top of each lamina).

and feg ¼ ½ ex

From Eqs. (7) and (8), ey

ez

cxy

cxz

cyz T

ð6cÞ

are the stress and strain vectors with respect to the laminate axes and   Dij 0 i; j ¼ 1; 2; 3; 4; ½D ¼ ð6dÞ l; m ¼ 5; 6 0 Dlm are the three-dimensional elasticity constants for ith lamina with respect to the laminate reference axes (Fig. 2).

ou 1 ow ¼ ½D66 sxz  D56 syz   ; oz D ox

ð9Þ

ov 1 ow ¼  ½D56 sxz  D55 syz   ; oz D oy

ð10Þ

   ow 1 ou ov ou ov ¼ þ rz  D13  D23  D34 oz D33 ox oy oy ox

ð11Þ

can be obtained, where 2.4. Finite element formulation The transverse stresses can be obtained from the constitutive Eq. (6a) and strain–displacement relations as 9 8 ou ow > 3> > > þ > D55 D56 < oz ox > = 4 5 ; ¼ > ov ow > :s ; > D56 D66 > > > yz ; : þ oz oy   ou ov ow ou ov rz ¼ D13 þ D23 þ D33 þ D34 þ : ox oy oz oy ox 8 9 < sxz =

2

ð7Þ

D ¼ D55  D66  D256 :

Displacement fields uðx; y; zÞ, vðx; y; zÞ and wðx; y; zÞ expressed in Eq. (1) can now be further expressed in terms of nodal variables by using Eqs. (9), (10) and (12), respectively as uk ðx; y; zÞ ¼

27 X n¼1

ð8Þ

ð12Þ

gi hj fq ukn þ

9 X

_

gi hj fp u km :

ð13Þ

m¼1

Here m are nodes with mixed DOF as shown in Fig. 3a and _ _ ukn and u km are the nodal variables, with u km ¼ ouozkm .

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Shape functions in the z-direction are expressed as 1 f1 ¼ ð8g þ 2g2 þ 6g3 Þ; 4 1 1 f 2 ¼ ð4 þ 8g  4g  8g3 Þ; f 3 ¼ ð2g2 þ 2g3 Þ; 4 4 h f 4 ¼ ð4g þ 4g3 Þ; g ¼ z=Lz : 4

gi hj fq 6 6 0 6 ! ½N n ð36Þ ¼ 6 6  QQ13 g0i hj fp 33 4 Q34   Q g i h j fp 33

fd n g ¼ ½ un

vn

wn

g0i ¼

for n ¼ 1–9; fd n g ¼ ½ un vn

wn T

6 6 0 60 1 6 6  Q13 g0 h _ f 6 B Q33 i j p C 6@ _ A 6 6  QQ34 gi hj f p 6 33 6 6 6 gi hj fq 6 6 6 1 ½Bn  ¼ 6 0 _ 6 66 6 B g i hj f p f q C 6B Q 6 B  13 g00 h f C 6 @ Q33 i j p C A 6 6  Q34 g0 h f 6 Q33 i j p 6 6 6 6  Q13 g0 h f ! 6 Q33 i j p 6 4  Q34 g h f i j p Q 33

0

Z

T

feg frg dV 

V

0

g0i hj fp _

g i hj f p

T

fqg fpb g dV 

Z

fqgT fpt g ds;

ð17Þ

R

V

ð18aÞ

frg ¼ ½D½Bfqg;

ð18bÞ

where B2

   Bn

   B27 ;

0

0

2g0i hj fp

Q66 g h f r i j p  Qr56 g0i hj fp

Q55 r

ð19Þ

0

gi hj fp

_

g0i hj f p þg0i hj fq

! Q66 r

_

g i hj f p

!

 Qr56 gi hj fp  Qr55 g0i hj fp _

 Qr56 gi hj f p

1

B C   _ B Q34 0  C   B  Q33 gi hj fp C gi hj f p þ gi hj fq @ A Q23   Q gi hj fp

_

 Qr56 gi hj f p

Q55 r

_

g i hj f p

3

7 7 7 7 7 _ 7 1 gh f f 7 Q33 i j p p 7 7 7 7 7 7 0 7 7 7 7 7 7 7 7 1 0 g h f j p 7 Q33 i 7 7 7 7 7 7 7 7  1 7 g h f p Q33 i j 5 0

_

33

 QQ23 33

Z

gi hj f p fq

gi h j fp

!

ð16Þ

 Qr56 g0i hj fp

gi hj fq

33

ohj : oy

Q66 0 ghf r i j p Q56  r gi hj fp

g00i hj fp

 QQ34 g00i hj fp

gi hj fq

and

0

g0i hj fq

0

7 5 for n ¼ 10–27;

feg ¼ ½Bfqg;

ð15cÞ ð15dÞ ð15eÞ

1 _ Q34 0  g h f B Q33 i j p C @ _ A  QQ23 gi hj f p

0 0

hj ¼

and

½B ¼ ½ B1

g0i hj fq

3

0 gi hj fq

where {pb} is the body force vector per unit volume and {pt} is traction load vector acting on any surface of the laminated plate. Here ‘R’ is a surface of the element subjected to the traction forces. The strain vector {e} and the stress vector {r} can be expressed as

and 2

1 Q33

33

2

ogi ox

1 P¼ 2

ðrz Þn 

for n ¼ 10–27;

7 7 7 7 7 g i h j fp 5 0

The total potential energy P of the laminate can be obtained from

T

ðsyz Þn

3

and

ð15bÞ ðsxz Þn

0

0

Q55 ghf r i j p

ð15gÞ

where ½N n ð36Þ ; n ¼ 1–9 and ½N n ð33Þ ; n ¼ 10–27;

0

 QQ23 gi hj fp

0

ð15aÞ w T ;

gi hj fq

!

 QQ34 g 0 h j fp 33 i

g i hj f q 6 ¼4 0

½N n ð33Þ

Finally, Eq. (13) yields displacement fields uðx; y; zÞ, vðx; y; zÞ and wðx; y; zÞ in terms of the nodal degrees of freedom as

v

gi hj fq

Q66 ghf r i j p  Qr56 gi hj fp

 Qr56 gi hj fp

gi hj fp

ð15fÞ

Further,

ð14Þ

i ¼ 1; 2; 3 for the nodes with n ¼ 1, n ¼ 0 and n ¼ 1, respectively, j ¼ 1; 2; 3 for the nodes with d ¼ 1, d ¼ 0 and d ¼ 1, respectively, q ¼ 1; 2; 3 and for the nodes with g ¼ 1, g ¼ 0 and g ¼ 1, respectively, and p ¼ 4 for the nodes with g ¼ 1:

fug ¼ ½ u

g0i hj fp

for n ¼ 1–9:

Further, shape functions gi ðnÞ, hj ðdÞ and fq=p ðgÞ are calculated considering

fug ¼ ½N fdg;

0

!

33

for n ¼ 1–9; ð20aÞ

A.N. Bambole, Y.M. Desai / Computers and Structures 85 (2007) 1484–1499

and

2

g0i hj fq

6 0 6 6 6 0  ½Bn  ¼ 6 6 gi hj fq 63 6 _ 6g h f 4 i j p 0

0 gi hj fq 0 g0i hj fq 0

_

g i hj f p

where [K], {Q} and {F} are, respectively, the global property matrix, the global degrees of freedom vector and the global influence vector.

3

0 0

7 7 7 g i hj f p 7 0 7 7 7 0 gi hj fq 7 5  gi hj fq _

for n ¼ 10–27:

3. Numerical results and discussion

ð20bÞ Further, g00i ¼

o2 g i ; ox2

h j ¼

o2 hj ; oy 2

_

fp ¼

ofq ; oz

_

and

fp ¼

ofp : oz ð21Þ

Total potential energy functional given in Eq. (17) subjected to extremum principle (Principle of minimum potential energy) will yield the element property matrix [K]e and the element influence vector {f}e as Z Lx Z LY Z Lz ½Ke ¼ ½BT ½D½B dx dy dz; ð22Þ Lx Ly Lz Z Lx Z Ly Z Lz Z Z e T T ff g ¼ ½N  fpb g dx dy dz þ ½N  fpt g ds: Lx

Ly

1489

Lz

A computer program incorporating conventional 27node 3-D displacement element with compatible present 3-D 27-node, hybrid-interface elements which has been developed in FORTRAN-90 to analyze laminated plates for various boundary and loading conditions as shown in Fig. 4. Boundary conditions are tabulated under Table 1 and material properties used are presented under Table 2. Numerical investigations have been undertaken for a couple of representative square and rectangular symmetric cross-ply, angle-ply laminates and sandwich plates under idealistic simple and clamped support conditions at all four sides and subjected to bi-directional sinusoidal and uniform transverse load on the top surface as shown in Fig. 5(a and c) respectively. Intensity of bi-directional sinusoidal loading can be expressed as ^qðX ; Y Þ ¼ q0 sin

R

ð23Þ The global equation can be obtained in the following form after assembly ½KfQg ¼ fF g;

ð24Þ

pX pY sin ; a b

ð25Þ

where q0 represents the intensity of distributed load. Also, a three layer cross-ply (0/90/0) laminated square plate, subjected to bi-directional sinusoidal loading (Eq. (25)) on its top surface have been investigated under a realistic simple support conditions for comparing the differ-

Simple

Free

Y=b

b

Free

a

a

X=0

X=a Clamped

d

Free

Clamped

Clamped

Clamped)

Y=b

X=a

b

Y=b Clamped

c

Y=0

b

Y=0

Clamped

free

a

a

X=0

b

Y=0

Y=0 Simple

X=0

Simple

Simple

Simple

Simple

Y=b

X=a

X=0

X=a

Fig. 4. Different boundary conditions: (a) plate with simple supports (b) plate simply supported on two edges (cylindrical bending) (c) plate clamped on all four edges (d) plate clamped on two edges.

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Table 1 Boundary conditions Surface/edges

BC on displacement field

BC on stress field

(a) Cross-ply laminate and sandwich plate with cross-ply face sheets under idealistic simply supports (considering symmetry) (i) Face X ¼ 0 v¼w¼0 – (ii) Face X ¼ a=2 u¼0 sxz ¼ 0 (iii) Face Y ¼ 0 u¼w¼0 – (iv) Face Y ¼ b=2 v¼0 syz ¼ 0 (v) Face Z ¼ t=2 – rz ¼ ^qðX ; Y Þ; sxz ¼ syz ¼ 0 (vi) Face Z ¼ t=2 – rz ¼ sxz ¼ syz ¼ 0 (b) Cross-ply laminates with realistic simply supports (i) Edges X ¼ 0; Z ¼ t=2 (ii) Edges X ¼ a; Z ¼ t=2 (iii) Faces X ¼ 0; a; Z 6¼ t=2 (iv) Edges Y ¼ 0; Z ¼ t=2 (v) Edges Y ¼ b; Z ¼ t=2 (vi) Faces Y ¼ 0; b; Z 6¼ t=2 (vii) Top face Z ¼ t=2 (viii) Bottom face Z ¼ t=2

u¼v¼w¼0 v¼w¼0 v¼w¼0 u¼v¼w¼0 u¼w¼0 u¼w¼0 – –

– – sxz ¼ 0 – – syz ¼ 0 rz ¼ ^qðX ; Y Þ; sxz ¼ syz ¼ 0 rz ¼ sxz ¼ syz ¼ 0

(c) Angle-ply laminate under cylindrical bending (i) Edge X ¼ 0; Z ¼ 0 (ii) Edge X ¼ a; Z ¼ 0 (iii) Faces X ¼ 0 and X ¼ a (iv) Top face Z ¼ t=2 (v) Bottom face Z ¼ t=2

u¼v¼w¼0 v¼w¼0 w¼0 – –

– – – rz ¼ ^qðX Þ; sxz ¼ syz ¼ 0 rz ¼ sxz ¼ syz ¼ 0

(d) Angle-ply laminate with clamped–clamped support (i) Faces X ¼ 0and X ¼ a (ii) Top face Z ¼ t=2 (iii) Bottom face Z ¼ t=2

u¼v¼w¼0 – –

– rz ¼ ^qðX Þ; sxz ¼ syz ¼ 0 rz ¼ sxz ¼ syz ¼ 0

(e) Angle-ply laminates and sandwich plate with angle-ply face sheets under simple supports on all four sides (i) Edges X ¼ 0; Z ¼ 0 and X ¼ a; Z ¼ 0 u¼w¼0 (ii) Edges Y ¼ 0; Z ¼ 0 and Y ¼ b; Z ¼ 0 v¼w¼0 (iii) Faces X ¼ 0; X ¼ a and Y ¼ 0; Y ¼ b w¼0 (iv) Top face Z ¼ t=2 – (iv) Bottom face Z ¼ t=2 –

– – – rz ¼ ^qðX ; Y Þ; sxz ¼ syz ¼ 0 rz ¼ sxz ¼ syz ¼ 0

(f) Cross ply Laminate/ Sandwich with cross ply face sheets with clamped supports on all four edges (considering symmetry) (i) Face X ¼ 0 u¼v¼w¼0 – (ii) Face X ¼ a=2 u¼0 sxz ¼ 0 (iii) Face Y ¼ 0 u¼v¼w¼0 – (iv) Face Y ¼ b=2 v¼0 syz ¼ 0 (v) Face Z ¼ t=2 – rz ¼ ^qðX ; Y Þ; sxz ¼ syz ¼ 0 (vi) Face Z ¼ t=2 – rz ¼ sxz ¼ syz ¼ 0 Note: ‘–’ indicates no boundary condition imposed on that degree-of-freedom.

Table 2 Material properties Material properties of graphite/epoxy composite for cross-ply laminate, angle-ply laminate and face sheet of sandwich plate E1 ¼ 172:4 GPa (25 · 106 psi) E2 ¼ E3 ¼ 6:89 GPa (106 psi) G23 ¼ 1:378 GPa (0.2 · 106 psi) G12 ¼ G13 ¼ 3:45 GPa (0.5 · 106 psi) m12 ¼ m13 ¼ m23 ¼ 0:25 Properties of core material for sandwich plate E1 ¼ E2 ¼ 0:276 GPa (0.04 · 106 psi) G13 ¼ G23 ¼ 0:414 GPa (0.06 · 106 psi) m12 ¼ m13 ¼ m23 ¼ 0:25

ence between the idealistic and the realistic boundary conditions. Angle-ply laminates under cylindrical bending due to sinusoidal loading on the top surface have also been con-

E3 ¼ 3:45 GPa (0.5 · 106 psi) G12 ¼ 0:1104 GPa (0.016 · 106 psi)

sidered in the investigation as shown in Fig. 5b. Expression for the loading has been considered to be pX ^qðX ; Y Þ ¼ q0 sin : ð26Þ a

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Fig. 5. (a–c) Loading on laminated/sandwich plate: (a) plate subjected to bi-directional sinusoidal loads, (b) plate subjected to uni-directional sinusoidal loads, (c) plate subjected to uniformly distributed loads.

A quarter plate model has been used for cross-ply laminate and sandwich plate with cross ply face sheets under idealistic simple support condition, due to the bi-axial symmetry. On the other hand, full plate model has been considered for angle-ply laminates, sandwich plate with angle-ply face sheets and also for cross-ply laminate under realistic simple support conditions. For thick to moderately thick plates, present formulation has been found to yield converging results by considering 4 to 6 elements along X and Y directions of the plate along with the descretization of the thickness (i.e. along Z-direction) in such way that smaller of Lx =Lz and Ly =Lz ratios in an element was between 8 and 12. Typical convergence trend for simply supported cross-ply laminated (0/90/0) square thick plate ðS ¼ 4Þ

is presented in Table 3. Convergence of the normal stress rx ða=2; b=2; þh=2Þ and the transverse shear stress, on the other hand, are shown in Fig. 6a and b, respectively. Mesh schemes used for numerical examples are presented in Table 4. On the basis of numerical investigation of thin plates, it has been found that the present 3D FE formulation does not pose ill-conditioning. Results have been found to match very well with the available elasticity solution however they have been omitted here for brevity. Patch test [15] has also been performed by using two layers (lower-interface and upper-interface) of hexahedral elements in thickness direction for a range of element aspect ratios ðLx =Lz Þ ¼ 10–100. Present interface elements have been

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Table 3 Convergence of stresses of a simply supported cross-ply laminated (0/90/0) square plate for S ¼ 4 Lx =Lz or Ly=Lz

x ða2 ; b2 ;  h2Þ r

Z

Total no. of element

6 9 12 15 18 21 24 27 30

96 144 192 240 288 336 384 432 480

6 9 12 15 18 21 24 27 30

+0.820 +0.810 +0.809 +0.809 +0.809 +0.809 +0.809 +0.809 +0.809 +0.801

No. of elements along X

Y

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Exact Solution [17]

found to satisfy the patch test even for a thin patch. To check the locking behavior of elements, a square clamped steel plate with (L ¼ 100 mm t ¼ 0:01 mm) subjected to uniformly distributed load (UDL) of intensity 1:454 107 with 10 · 10 · 1 elements (upper-interface element layer) descretization in X ; Y and Z direction was analysed

y ða2 ; b2 ;  h6Þ r

0.780 0.766 0.764 0.763 0.763 0.763 0.763 0.763 0.763 0.755

Element Aspect Ratio Lx /Lz 6

9

12

15

18

21

24

27

30

33

36

0.84

⎯σx

syz ða2 ; 0; 0Þ

0.0522 0.0514 0.0513 0.0512 0.0512 0.0512 0.0512 0.0512 0.0512 0.0505

0.266 0.260 0.259 0.259 0.259 0.258 0.258 0.258 0.258 0.256

0.224 0.223 0.223 0.223 0.222 0.222 0.222 0.222 0.222 0.217

+0.0526 +0.0519 +0.0518 +0.0518 +0.0518 +0.0518 +0.0518 +0.0518 +0.0518 +0.0511

a E2 u 100E2 w ¼ s ¼ ; u ¼ ; w ; 3 t q0 ts q0 ts4 1 y ; sxy Þ ¼ ð rx ; r ðrx ; ry ; sxy Þ; q0 s 2 1 rz z ¼ : ðsxz ; syz Þ; r ðsxz ; syz Þ ¼ q0 s q0 Z ; t

ð27Þ

Here ‘E2’ is modulus of elasticity of graphite/epoxy composite presented in Table 2, along the lamina coordinate axis 2 (Fig. 2a). The percentage difference in results has been calculated as

0.82

0.80

Percentage difference ðPDÞ   approximate value-true value ¼  100: true value

0.78

100

200

300

400

500

No. of elements

3

6

9

12

15

18

21

24

27

ð28Þ

Illustrative numerical examples considered in the present work have been discussed next.

Element Aspect Ratio Lx /Lz 0.29

0.575 0.564 0.563 0.563 0.563 0.563 0.563 0.563 0.563 0.556

sxz ð0; b2 ; 0Þ

[16]. The present hybrid-interface elements did not exhibit any locking. Following normalization factors have been used in the numerical examples considered in the work for a proper comparison of results Z¼

3

+0.552 +0.542 +0.541 +0.540 +0.540 +0.540 +0.540 +0.540 +0.540 +0.534

sxy ð0; 0;  h2Þ

30

33

36

3.1. Cross-ply laminated plates

0.28

⎯τxz 0.27 0.26

0.25

0.24 100

200

300

400

500

No. of elements

Fig. 6. (a and b) Convergence of stresses for a simply supported cross-ply laminated (0/90/0) square plate ðS ¼ 4Þ: (a) convergence of the normalized in-plane stress, (b) convergence of the normalized transverse shear stress.

(i) A three-layer cross-ply (0/90/0) laminated square plate with aspect ratio S ¼ 4 subjected to bi-directional sinusoidal loads (Eq. (25), Fig. 5a) under idealistic (Table 1a) and realistic (Table 1b) simple support condition (Fig. 4a) was considered in study. Variations of the in-plane normal stress ð rx Þ, shear stress ðsxy Þ, the transverse shear stress ðsxz Þ and displacement ðuÞ through the laminate thickness are shown in Fig. 7a–d. Results are compared with elasticity solution by Pagano [17] and FE model of Ramtekkar et al. [11]. Results obtained for plate with idealistic boundary condition are in close agreement with corresponding elasticity and mixed FE solution. Close matching of result is observed for the in-plane normal stress ð rx Þ and the in-plane displacement ðuÞ for plate with idealistic and realistic support conditions whereas significant difference

A.N. Bambole, Y.M. Desai / Computers and Structures 85 (2007) 1484–1499

1493

Table 4 Mesh scheme used in numerical examples Plate description

Descretization X-dir

Y-dir

Z-dir

Simply supported cross-ply plate (0/90/0) ðS ¼ 4; a ¼ bÞ Simply supported cross-ply plate (0/90/0) ðS ¼ 2; b ¼ 3aÞ Simply supported cross-ply plate (0/90/0) ðS ¼ 4; b ¼ 3aÞ Simply supported cross-ply plate (0/90/0) ðS ¼ 10; b ¼ 3aÞ Simply supported cross-ply plate (0/90/0) ðS ¼ 20; b ¼ 3aÞ Angle-ply plate simply supported on two opposite edges (cylindrical bending and plane strain condition) (+30/30/30/+30) ðS ¼ 10Þ Angle-ply plate clamped on two opposite edges (plane-strain condition) (+a/-a)s with a ¼ 15 ; 30 ; 45 ; 60 and 75 ðS ¼ 4Þ Simply supported angle-ply plate (30/+30/30/+30) ðS ¼ 4; a ¼ b) Simply supported sandwich plate ð45 =core=45 Þ ðS ¼ 4; a ¼ bÞ Clamped sandwich plate (0/90/core/0/90) ðS ¼ 5; a ¼ bÞ

4 3 4 6 6 6

4 4 6 8 8 1

12 12 12 12 12 12

4

1

12

4 4 4

4 4 4

12 12 15

Present (Idealistic BC)

a 0. 5

Elasticity [17]

b

Z

0.5

a b 2 2

-0. 5

0. 0 0 .0

0. 5

-0.05

-0.5

0.00

-0.5

d

Z 0. 5

τ xy (0,0, Z )

0.0 1 .0

c

Z

σx ( , , Z) -1 .0

Present (Realistic BC)

Ramtekkar et al [11]

0.05

-0.015

-0.010

0. 0 0.000

-0.005

b u (0, , Z ) 2 0.005

0.010

-0.5

Z 0. 5

τ xz

0. 0 0. 0

0 .1

0. 2

0 .3

Present at (0, b/2, Z) Elasticity [17] at (0, b/2, Z) Ramtekkar et al [11] at (0, b/2, Z) Present at (a/8, b/2, Z)(Realistic BC) Present at (a/8, b/2, Z)(Idealistic BC)

-0 .5

Fig. 7. Variation of the normalized (a) in-plane normal stress ( rx ); (b) in-plane shear stress ðsxy Þ; (c) transverse shear stress (sxz ); and (d) in-plane displacement ð uÞ through the thickness of a simply supported cross-ply (0/90/0) square laminate with S ¼ 4, subjected to the transverse bi-directional sinusoidal loading.

in values of the transverse shear stress ðsxz Þ is observed. The maximum value of the transverse stress ðsxz Þ of plate with realistic simple supports is greater by 9.32% of that of plate

with idealistic simple supports. Therefore, appropriate modeling of support condition seems to be important for reliable estimate of the transverse stresses.

1494

A.N. Bambole, Y.M. Desai / Computers and Structures 85 (2007) 1484–1499

(ii) A three-layer cross-ply (0/90/0) simply supported (Table 1a, Fig. 4a) laminated rectangular plates ðb ¼ 3aÞ subjected to bi-directional sinusoidal loads (Eq. (25), Fig. 5a), aspect ratios S ¼ 2; 4; 10 and 20 have been considered. Results of the normalized in-plane normal and shear stresses, transverse shear stresses and transverse displacement are presented in Table 5. Variations of the transverse shear stress ðsyz Þ through the laminate thickness are also presented in Fig. 8 for S ¼ 4. Results are compared with elasticity solution [17] and higher order analysis by Reddy [19]. Excellent agreement of present solution with elasticity solution shows the ability of formulation in predicting response of square as well as rectangular laminated composite plates.

0.5

Z

Present Elasticity [17]

0.0 0.00

b 2

τ xz (0, , Z ) 0.01

0.02

0.03

-0.5

3.2. Angle-ply laminates (i) Four-layer symmetric angle-ply laminated plate with ply orientation of (+30/30/30/+30) and aspect ratios S ¼ 10 under cylindrical bending (Table 1c, Fig. 4b) subjected to unidirectional sinusoidal loads (Eq. (26), Fig. 5b) have been considered for investigation. The plane-strain condition assumed by Pagano [18] has been simulated in the computer program. Variations of the inplane and transverse stresses through the thickness have been plotted in Fig. 9. Results have been found to be in excellent agreement with the analytical solution by Pagano [18] and Mixed FE formulation of Desai et al. [13]. Fig. 9e shows variation of the transverse normal and shear stresses at the layer interface (+30/30) along longitudinal axis X. The figure shows presence of high magnitude of the transverse shear stress components at interfaces. Example shows capability of present formulation to estimate reliable values of interlaminar transverse stress components in angle-ply laminates. (ii) An angle-ply laminated plate with aspect ratio of S ¼ 4, and lamination scheme +a/a/a/+a with

Fig. 8. Variation of the normalized transverse shear stress (syz Þ through the thickness of a simply supported cross-ply (0/90/0) rectangular laminate ðb ¼ 3a; S ¼ 4Þ subjected to the transverse bi-directional sinusoidal loading.

a ¼ 15 ; 30 ; 45 ; 60 and 75 have been considered under clamped-clamped support condition (Table 1d, Fig. 4d) subjected to unidirectional sinusoidal loading (Eq. (26), Fig. 5b). The plane-strain condition assumed by Pagano [18] has been simulated in the computer program. Variation of the transverse shear stresses and in plane normal and shear stresses through the thickness have been presented in Fig. 10. Variation of both the transverse stresses ðsxz and syz Þ shows the presence of extremely high stress gradient at the top and bottom surfaces. Variation of syz shows presence of such high stress gradient at the layer interfaces (+a/a and a/+a) also. Only a very refined theoretical model can handle the presence of such a high stress gradient. The present formulation has shown its capability of handling such situations. (iii) An angle-ply laminated square plate with four layers of equal thickness having ply orientation of (30/+30/

Table 5 Maximum stresses and deflection in symmetric simply supported cross-ply (0/90/0) rectangular ðb ¼ 3aÞ laminated plate under bi-directional sinusoidal transverse load Source

r X ða2 ; b2 ;  h2Þ

2

i. Elasticity [17] ii. Present iii. Reddy [19]

2.130 2.170 –

1.620 1.660 –

0.230 0.231 –

4

i. Elasticity [17] ii. Present iii. Reddy [19]

1.140 1.158 –

1.100 1.112 1.030

10

i. Elasticity [17] ii. Present iii. Reddy [19]

0.726 0.737 –

0.725 0.737 0.692

20

i. Elasticity [17] ii. Present iii. Reddy [19]

±0.650 ±0.657 ±0.641

S

r Y ða2 ; b2 ;  h6Þ

sXY ð0; 0;  h2Þ

 ða ; b ; 0Þ W 2 2

0.0564 0.0575 –

0.0548 0.0560 –

8.17 8.17 –

0.0334 0.0342 0.0348

0.0269 0.0273 –

0.0281 0.0285 0.0263

2.82 2.82 2.64

0.0152 0.0165 0.0170

0.0120 0.0121 –

0.0123 0.0124 0.0115

0.919 0.919 0.862

0.0119 0.0142 0.0139

0:0093 0:0094 0.0091

sXZ ð0; b2 ; 0Þ

sYZ ða2 ; 0; 0Þ

0.268 0.229 –

0.257 0.262 –

0.455(.30) 0.472(.30) –

0.0668 0.0680 –

0.109 0.111 –

0.119 0.119 0.103

0.351 0.357 

0.387(.27) 0.388(.27) 0.272

0.0418 0.0420 –

0.0435 0.0436 0.0398

0.420 0.433 –

0.420(.03) 0.438(.03) 0.286

0.0294 0.0298 –

0.0299 0.0298 0.0289

0.434 0.458 0.288

0.0673(.03) 0.0683(.03) –

Note: (i) ‘–’ Result not available. (ii) Number within ‘( )’ indicates the position of the point on thickness dimension where the stress is maximum.

0.610 0.609 0.594

A.N. Bambole, Y.M. Desai / Computers and Structures 85 (2007) 1484–1499

Z

Z

0.5

0.5

0.5

τ xz(0, Z )

0.0 0.0

τ yz(0, Z ) -0.15

0.5

-0.10

0.0 0.00

-0.05

-0.5

-0.4

-0.2

-0.5

0.5

-0.4

τ xy(a/2,Z)

0.0 0.0

0.2

0.4

-0.5

0.4 0.3

Present Pagano [18] Desai et al [13]

-0.8

Z

τ xz;τ yz;σz ;

Z

0.0 0.0

1495

τ xz( X, −t / 4)

0.2

σ x (a / 2, Z ) 0.4

0.1

σ z ( X, −t / 4)

0.0 0

0.8 -0.1

1

2

3

4

5

6

7

8

9

10

X

τ yz ( X , −t / 4)

-0.2 -0.3

-0.5

-0.4

Fig. 9. Variation of the normalized (a) transverse shear stress (sxz Þ; (b) transverse shear stress (syz Þ; (c) in-plane shear stress (sxy Þ; and (d) in-plane normal z along the length of a four-layer angle-ply laminate (30/30/30/30) with S ¼ 10 under stress ( rx Þ through the thickness and (e) sxz , syz and r cylindrical bending due to the transverse sinusoidal loading.

30/+30) for S ¼ 4, has been considered for investigation. Plate has been kept under simple support conditions (Table 1e, Fig. 4a) on all the four sides and subjected to bi-directional sinusoidal loads (Eq. (25), Fig. 5a). Variations of the various transverse stresses through the thickness have been plotted in Fig. 11a–c. Excellent match of results of the transverse stress components with elasticity solution [20] is observed. Example shows the capability of present formulation to predict reliable response for angleply plates. 3.3. Sandwich plates (i) A thick square sandwich plate (aspect ratio, S ¼ 4) consisting of angle-ply face sheets and flexible core (45/core/45) with simple supports (Table 1e, Fig. 4a) subjected to bi-directional sinusoidal loads (Eq. (25), Fig. 5a) have been considered. The top and bottom face sheets have thickness equal to 0.1 times the total thickness of the plate. The ratio of shear modulus of the face sheet material with that of the core material is 86.20. Through thickness variation of the normalized transverse shear stress ðsyz Þ at three different locations of the plate have been presented in Fig. 12a–c. It has been observed that though

there is excellent match in the variation of the transverse shear stress in the core region with those given by Rao and Meyer-Piening [21], results do not match in the face sheet regions. The reason for this discrepancy may be attributed to the fact that Rao and Meyer-Piening [21] have made two basic assumptions in their analysis: (i) the variation of in-plane displacement field is linear within each layer; and (ii) the transverse displacement remains constant throughout the thickness, which implies that the transverse normal strain, ez ¼ 0. It has been shown through investigations presented in Fig. 13 that both of these assumptions made by Rao and Meyer-Piening [21] may lead to erroneous results. (ii) A square sandwich plate with lamination 0/90/ core/0/90, clamped on all the four sides (Table 1f, Fig. 4c) with aspect ratio S ¼ 5 subjected to UDL (Fig. 5c) has been considered for investigation. The individual thickness of the top and the bottom face sheets is onetenth of the total thickness of the sandwich plate. Each face sheet is equally divided into two layers having 0/90 ply orientation. The through thickness variation of the transverse shear and the normal stresses are presented in Fig. 14a–c and those of the transverse shear and the normal strains are presented in Fig. 15a–c. A very high stress

1496

A.N. Bambole, Y.M. Desai / Computers and Structures 85 (2007) 1484–1499

Z

Z

0.5

0.5

τ xz ( 0.0 0.0

0.1

0 .2

0.3

a , Z) 20

0 .4

τ yz ( 0.0 -0.2

-0.1

0 .0

-0.5

0.1

a , Z) 20

0 .2

-0.5

15º

45º

30º

75º

60º

Z

Z

0.5

0. 5

σ x (a / 2, Z )

0. 0 0. 0

-0 .5

-0.3

0 .5

τ xy (a / 2, Z )

0.0 -0.2

-0.1

0.0

0.1

0.2

0.3

-0.5

-0 .5

Fig. 10. Variation of the normalized (a) transverse shear stress (sxz Þ; (b) transverse shear stress (syz Þ; (c) in-plane normal stress ( rx Þ; and (d) in-plane shear stress ðsxy Þ through the thickness of four layer symmetric (a/a/a/a) angle-ply with S ¼ 4 with clamped-clamped support on the two parallel edges and subjected to the transverse sinusoidal loading, for a ¼ 15 , 30 ; 45 ; 60 ; 75 .

Present Analysis Z

Z

0.5

0.5

ab 22

σz ( , , Z ) 0.0 0.0

-0.5

Elasticity [20]

0.5

1.0

0.5

a 2

Z

a 2

τ xz ( ,0, Z ) -0.1 2

-0.06

0.0 0.00

-0.5

0.06

0.12

0.0 0.0

τ yz ( ,0, Z ) 0 .1

0.2

-0.5

Fig. 11. Variation of the normalized (a) transverse normal stress ( rz Þ; (b) transverse shear stress (sxz Þ; and (c) transverse shear stress (syz Þ through the thickness of a idealistic simply supported angle-ply laminate (30/30/30/30), for S ¼ 4, subjected to the transverse bi-directional sinusoidal loading and with simple supports.

A.N. Bambole, Y.M. Desai / Computers and Structures 85 (2007) 1484–1499

Present

Rao and Mayer-Piening [21]

Z

Z 0.5

a b

a 2

0.0

1.0

τyz(0, , Z )

4 4

0.0

0.5

0.0

b 2

τ yz ( , , Z )

τ yz ( , 0, Z ) 0.0

Ramtekkar et al [12]

Z

0.5

0.5

1497

0.2

0.4

0.0

0.6

0.8

-0.8

-0.4

0.4

0.8

-0.5

-0.5

-0.5

0.0

Fig. 12. Variation of the normalized transverse shear stress (syz Þ at various locations, through the thickness of square sandwich laminate (45/core/45) with S ¼ 4 and d ft ¼ d fb ¼ 0:1d; d c ¼ 0:8d with simple supports and subjected to the transverse bi-directional sinusoidal loading.

Present 0.5

Z

Z

0.5

a b 2 2

Ramtekkar et al [12] 0.5

a b 2 2

σz ( , , Z ) 0.0 0.0

0.5

0.0 0.0

-0.5

5.0x10

a 2

γ yz ( ,0, Z )

εz ( , , Z )

1.0

Z

0.0 0.0

-5

1.0x10

-4

-0.5

-0.5

Fig. 13. Variation of the (a) transverse normal stress r z ; (b) transverse normal strain ez ; and (c) transverse shear strain cxz, through the thickness of a square sandwich plate (45/core/45) with S ¼ 4, subjected to the transverse bi-directional sinusoidal loading with simple supports.

Z

Z

0.5

0.5

0.5

a 2 2

-0.5

0.5

1.0

a b ,Z) 2 20

b τ xz ( a , , Z )

σ z ( , b, Z ) 0.0 0.0

Z

20 2

0.0

0.0

-0.5

0.5

1.0

1.5

τ yz ( , 0.0

2.0

0.0

0.5

1.0

1.5

2.0

-0.5

Fig. 14. Variation of the normalized (a) transverse normal stress ( rz Þ; (b) transverse shear stress (sxz Þ; and (c) transverse shear stress (syz Þ through the thickness of a clamped square sandwich laminate (0/90/core/0/90), with S ¼ 5, subjected to the transverse uniformly distributed loading.

1498

A.N. Bambole, Y.M. Desai / Computers and Structures 85 (2007) 1484–1499 Z

Z

0.5

Z

0.5

a b 22

0.5

εz ( , , Z ) 0.0 0.0

-0.5

-6

1.0x10

γ xz ( -6

2.0x10

0.0

0.0

1.0x10

-5

2.0x10

-5

a b , ,Z ) 20 2 3.0x10

-0.5

-5

a b , Z) 2 20

γ yz ( , 0.0 0.0

1.0x10

-5

2.0x10

-5

3.0x10

-5

-0.5

Fig. 15. Variation of the (a) transverse normal strain (z); (b) transverse shear strain (cxz); and (c) transverse shear strain (cyz) through the thickness of a clamped square sandwich laminate (0/90/core/0/90), with S ¼ 5, subjected to the transverse uniformly distributed loading.

gradient at the top and bottom surfaces as well as at the interface of the face sheet and the core along with a vast difference in the magnitude of the transverse stress in the face sheet region and that at the core region has been observed. The transverse strain variations presented in Fig. 15a–c also show a steep gradient in the variation of the transverse shear strains along with a discontinuity of large magnitude at the layer interface because of the large difference in the shear modulus of the face and the core material. It clearly indicates the necessity of a refined three-dimensional FE analysis to predict reliable response for such problems.

metric cases. Present formulation do not exhibit illconditioning and elements developed do not lock when used for thin plates with S ¼ 1000. However the present formulation is better suited for analysis of thick-moderately thick plates.

4. Conclusions

[1] Jones RM. Mechanics of composite materials. New York: Int. Student edition; 1990. [2] Reddy JN. A generalization of two-dimensional theories of laminated composite plates. Commun Appl Numer Meth 1987;3:173–80. [3] Soldatos KPA. A general laminated plate theory accounting for continuity of displacements and transverse shear stresses at material interfaces. Compos Struct 1992;20:195–211. [4] Wu CP, Hsu HC. A new local higher-order laminate theory. Compos Struct 1993;25:439–48. [5] Pian THH. Derivation of element stiffness matrices by assumed stress distributions. AIAA J 1964;2:1333–6. [6] Mau ST, Tong P, Pian THH. Finite element solutions for laminated thick plates. J Compos Mater 1972;6:304–11. [7] Wen-Jinn L, Sun CT. A three-dimensional hybrid stress isoparametric element for analysis of laminated composite plates. Comput Struct 1987;25(2):241–9. [8] Spilker RL. Hybrid-stress eight-node elements for thin and thick multilayer laminated plates. Int J Numer Meth Eng 1982;18:801–28. [9] Carrera E. Mixed layer-wise models for multilayered plates analysis. Compos Struct 1998;43:57–70. [10] Carrera E. Transverse normal stress effects in multilayered plates. ASME J Appl Mech 1999;66:1004–12. [11] Ramtekkar GS, Desai YM, Shah AH. Mixed finite-element model for thick composite laminated plates. Mech Adv Mater Struct 2002;9:133–56. [12] Ramtekkar GS, Desai YM, Shah AH. Application of a threedimensional mixed finite element to the flexure of sandwich plate. Comput Struct 2003;81:2183–98. [13] Desai YM, Ramtekkar GS, Shah AH. A novel 3D mixed finiteelement model for statics of angle-ply laminates. Int J Numer Meth Eng 2003;57:1695–716.

A novel 3-D 27-node hybrid-interface element based on the minimum potential energy principle and compatible with conventional displacement element is presented for analysis of laminated composite plates. Present formulation ensures the fundamental elasticity relationship between stress, strain and displacement fields within elastic continuum. The transverse stress components considered as primary variables at interfaces ensures its through thickness continuity in addition to continuity of displacement components. Thus, the present formulation can be implemented for realistic estimate of failure loads, as well as initiation of delamination. Formulation enables specification of the transverse stress boundary condition in addition to displacement boundary conditions in the delaminated portion of lamina interfaces. Such a feature can be advantageously used to predict near life response of the laminates with built-in or induced delamination. Capability of the present formulation to deal with high stress gradient has been evidently brought to notice vide examples of plates with clamped supports. Excellent agreement of results with the elasticity solutions suggest that the formulation is capable of dealing with laminated composite plates (cross-ply as well as angle ply) and sandwich plates for a variety of support and loading conditions for symmetric as well as asym-

Acknowledgement Constructive criticisms of the reviewers are gratefully acknowledged. References

A.N. Bambole, Y.M. Desai / Computers and Structures 85 (2007) 1484–1499 [14] Noor AK. Multifield (mixed and hybrid) finite element models. In: Noor AK, Pilkey WD, editors. State of the art surveys on finite element technology. New York: ASME; 1983. p. 127–62. [15] Bathe KJ. Finite element procedures. New Jersey: Prentice Hall; 1996. [16] Iosilevich A, Bathe KJ, Brezzi F. On evaluating the inf-sup condition for plate bending elements. Int J Numer Meth Eng 1997;40:3639–63. [17] Pagano NJ. Exact solutions for rectangular bi-directional composites and sandwich plates. J Compos Mater 1970;4:20–34.

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[18] Pagano NJ. Influence of shear coupling in cylindrical bending of anisotropic laminates. J Compos Mater 1970;4:330–43. [19] Reddy JN. A simple higher order theory of laminated composite plates. ASME J Appl Mech 1984;51:742–52. [20] Savio M, Reddy JN. A variational approach to three-dimensional elasticity solutions of laminated composite plates. J Appl Mech (ASME) 1992;59:S116–75. [21] Rao KM, Meyer-Piening HR. Analysis of sandwich plates using a hybrid-stress finite element. AIAA J 1991;29(9):1498–506.