thick plate shear deformation

8/9/2019 thick plate shear deformation

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A higher order finite element including transverse normal strain for linear

elastic composite plates with general lamination configurations

Wu Zhen a,b, S.H. Lo b,n, K.Y. Sze c, Chen Wanji a

a Key Laboratory of Liaoning Province for Composite Structural Analysis of Aerocraft and Simulation, Shenyang Aerospace University, Shenyang 110136, Chinab Department of Civil Engineering, University of Hong Kong, Pokfulam Road, Hong Kong, Chinac Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong, China

a r t i c l e i n f o

Article history:Received 5 September 2010

Received in revised form

18 March 2011

Accepted 7 August 2011Available online 31 August 2011

Keywords:

Higher order theory

Finite element

Transverse normal strain

Angle-ply

Laminated composite and sandwich

a b s t r a c t

This paper describes a higher-order global–local theory for thermal/mechanical response of moderatelythick laminated composites with general lamination configurations. In-plane displacement fields are

constructed by superimposing the third-order local displacement field to the global cubic displacement

field. To eliminate layer-dependent variables, interlaminar shear stress compatibility conditions have

been employed, so that the number of variables involved in the proposed model is independent of the

number of layers of laminates. Imposing shear stress free condition at the top and the bottom surfaces,

derivatives of transverse displacement are eliminated from the displacement field, so that C0

interpolation functions are only required for the finite element implementation. To assess the proposed

model, the quadratic six-node C0 triangular element is employed for the interpolation of all the

displacement parameters defined at each nodal point on the composite plate. Comparing to various

existing laminated plate models, it is found that simple C0 finite elements with non-zero normal strain

could produce more accurate displacement and stresses for thick multilayer composite plates subjected

to thermal and mechanical loads. Finally, it is remarked that the proposed model is quite robust, such

that the finite element results are not sensitive to the mesh configuration and can rapidly converge to

3-D elasticity solutions using regular or irregular meshes.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Due to low weight, high strength and rigidity, laminated

composite structures are being widely used in many engineering

fields such as aerospace, automotive and submarines. For safe and

reliable designs, it is necessary to well understand the structural

behavior of laminated composite and sandwich plates. Appro-

priate computational models ought to be developed for accurately

predicting the responses of these laminated structures. Among

the available approaches, the classical laminated plate theory and

the first order shear deformation theory have been proposedto predict thermo-mechanical behavior of laminated plates.

The laminated composites generally possess relatively soft trans-

verse shear modulus. However, the classical laminated plate

theory and the first order shear deformation theory are unable

to adequately model the relatively large transverse shear deforma-

tions, so that they often produce unacceptable errors in predicting

displacement and stresses of thick and moderately thick laminated

composite plates. Although the three-dimensional models are

accurate enough, the three-dimensional analysis requires huge

computational efforts, as the number of unknowns, in general,

depends on the number of layers of the laminate [1].

By adding high-order terms of transverse coordinate z to the

in-plane and transverse displacements, the global higher order

shear deformation theories [2–6] have been widely developed.

Compared to the first-order theory, the global higher-order theory

does not require the shear correction factors and can account for

the warping of the cross-section. In the global higher-order

theory, however, it is assumed that in-plane displacements and

its derivatives are continuous through the thickness of laminates.Owing to this assumption, the continuity conditions of interla-

minar stresses at interfaces will be violated as continuous strains

at the interfaces are multiplied by varying material properties of

different layers. Previous studies [7–11] indicated that the global

higher-order theories overestimate the natural frequencies and

buckling loads of the soft-core sandwiches with a vast difference

in material properties and thickness between the faces and the

core, as these models violate the continuity conditions of trans-

verse shear stresses at interfaces. To determine the natural

frequencies and the critical loads of the soft-core sandwiches,

the mixed layerwise theories [7–10] have been developed.

Although in general the performance of the mixed layerwise

Contents lists available at SciVerse ScienceDirect

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

Finite Elements in Analysis and Design

0168-874X/$ - see front matter & 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.finel.2011.08.003

n Corresponding author.

E-mail address: [email protected] (S.H. Lo).

Finite Elements in Analysis and Design 48 (2012) 1346–1357

http://www.elsevier.com/locate/finelhttp://www.elsevier.com/locate/finelhttp://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.finel.2011.08.003mailto:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.finel.2011.08.003http://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.finel.2011.08.003mailto:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.finel.2011.08.003http://www.elsevier.com/locate/finelhttp://www.elsevier.com/locate/finel


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theories is promising, they are computational expensive and

become intractable when the number of laminates increases, as

the number of unknowns depends on the number of layers.

The zig-zag theories, which can a priori satisfy the continuity

of transverse shear stresses at interfaces, generally provide a

reasonable compromise between accuracy and efficiency. Carrera

[12] introduced the original work and development of the zig-zag

theory in great details. In recent decade, the zig-zag theories have

been developed to study the mechanical, thermal and electricalbehaviors of smart composite plates/shells [13–18]. However, due

to enforcement of the transverse shear stress continuity at the

interfaces, derivatives of transverse displacement are involved in

the displacement field. Hence, C1 continuity of transverse

displacement at element interfaces will be required for the finite

element implementation. Although the continuity conditions of

transverse shear stresses at interfaces can be a priori satisfied, the

zig-zag theories are unable to produce accurate transverse shear

stresses directly from constitutive equations. For zig-zag theories,

accurate transverse shear stresses could only be obtained by

integrating three-dimensional equilibrium equation. However,

third order derivatives of transverse displacement are involved

if transverse shear stresses are to be computed by integrating the

3-D equilibrium equations. Hence, global equilibrium equations

related to the full domain were proposed [19], which is rather

complicated and expensive, and even might fail under irregular

mesh configuration. An eight-node element based on the zig-zag

theory has been proposed by Icardi [20] in which different

degrees of freedom are defined at the nodes of a finite element.

Nonetheless, the proposed element consists of first-order and

second-order derivatives of transverse displacement as nodal

variables, which is rather difficult to handle in a practical analysis.

By virtue of the zig-zag theory, Chakrabarti and Sheikh [21]

developed a six-node nonconforming triangular element for the

analysis of sandwich plates, which violates the normal slope

continuity requirement. Recent study [22] indicated that the

six-node nonconforming triangular element [21] was unable to

produce the accurate dynamic response of the sandwich plates, as

it violates the continuity conditions of the normal slope at the

interelement boundary. Pandit et al. [23] proposed an improved

zig-zag theory in which transverse displacement is assumed to be

quadratic over the core thickness and constant over the face

sheets. Nevertheless, the derivatives of transverse displacement

are involved in the displacement fields. To avoid the use of C1

interpolation functions in the finite element implementation,

artificial constraints have to be imposed using a penalty approach.

Subsequently, the improved zig-zag theory was extended to study

the static behaviors of sandwich plate with random material

properties [24].

To avoid using the C1 interpolation function in the finite

element formulation, a C0-type global–local higher-order theory

[25] was proposed for the static analysis of laminated composites.

Compared to the previous global–local higher-order theories[26,27], C0 interpolation functions are only required for the finite

element implementation, as derivatives of transverse displace-

ment are eliminated from displacement field based on known

shear stress conditions. Numerical results indicated that the

C0-type global–local higher-order theory neglecting transverse

normal strain [25] can be employed to study the deformations

and stresses of laminated composites subjected to mechanical

loads. However, for thick multilayer composite plates under

thermal loads, transverse normal strain plays an important role.

In view of this situation, this paper aims to develop a C0-type

global–local theory including transverse normal strain for the

analysis of thick multilayer plates under thermal/mechanical

loads. Furthermore, the proposed model is applicable not only

to cross-ply but also to angle-ply laminated composite plates.

Based on the proposed model, the six-node C0 triangular element

is formulated for the thermal/mechanical analysis of thick multi-

layer plates with general lamination configurations.

2. Higher-order global–local theory

In order to model the zig-zag shape distribution of in-plane

displacement through the thickness of laminates, a higher-orderin-plane displacement field is defined by superimposing third-

order local displacements to the global cubic displacement fields.

To include the transverse normal strain effect, the transverse

displacement is assumed to be linear across the thickness for

thermo-mechanical problems of multilayered plates. The initial

displacement fields are given by

ukð x, y, z Þ ¼ uGð x, y, z ÞþukLð x, y, z Þþû

kLð x, y, z Þ,

vkð x, y, z Þ ¼ vGð x, y, z ÞþvkLð x, y, z Þþ v̂

kLð x, y, z Þ,

wkð x, y, z Þ ¼ wGð x, y, z Þ: ð1Þ

The global displacement components in Eq. (1) are given by

uGð x, y, z Þ ¼ u0ð x, yÞþX3i ¼ 1

z iuið x, yÞ,

vGð x, y, z Þ ¼ v0ð x, yÞþX3i ¼ 1

z ivið x, yÞ,

wGð x, y, z Þ ¼ w0ð x, yÞþ zw1ð x, yÞ: ð2Þ

The local displacement components of the kth ply can be

written as follows:

ukLð x, y, z Þ ¼ zkuk1ð x, yÞþz

2k u

k2ð x, yÞ,

vkLð x, y, z Þ ¼ zkvk1ð x, yÞþz

2k v

k2ð x, yÞ,

ûkLð x, y, z Þ ¼ z

3k u

k3ð x, yÞ,

v̂kLð x, y, z Þ ¼ z3k vk3ð x, yÞ, ð3Þ

in which, zk¼ak z bk, ak ¼ ð2=ð z kþ1 z kÞÞ, bk ¼ ð z kþ1þ z kÞ=ð z kþ1

z kÞÞ, and x, y and z are the global coordinates of the plate. The

reference plane ( z ¼0) is taken at the mid-surface of the laminate.

The local coordinates for a layer are denoted by x, y and zk, where

zkA[1,1]. The relationships between the global coordinates and

the local coordinates are depicted in Fig. 1.

In Eq. (1), there are 6nþ10 variables in the displacement fields,

where n denotes the number of layers of the composite plate. In

order to reduce the layer-dependent variables, the interlaminar

continuity compatibility conditions of in-plane displacements

[26] and transverse shear stresses will be employed. The inter-

laminar continuity conditions of in-plane displacements can be

Fig. 1. Schematic diagram for the laminated plate and coordinates.

W. Zhen et al. / Finite Elements in Analysis and Design 48 (2012) 1346–1357 1347


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expressed as

ukLð x, y, z kÞ ¼ uk1L ð x, y, z kÞ

ûkLð x, y, z kÞ ¼ û

k1L ð x, y, z kÞ

vkLð x, y, z kÞ ¼ vk1L ð x, y, z kÞ

v̂kLð x, y, z kÞ ¼ v̂

k1L ð x, y, z kÞ

, where k ¼ 2,3,4,. . .,n: ð4Þ

Using the interlaminar continuity conditions of in-plane dis-

placement, 4(n1) variables are eliminated from the initial

displacement field. The interlaminar continuity conditions and

free surface conditions of transverse shear stresses can be used

to further reduce the layer-dependent variables. For angle-ply

composite plate, the transverse shear stresses for the kth ply are

given by

tk xz ð z Þ ¼ Q 44kek

xz ð z ÞþQ 45kek

yz ð z Þ,

tk yz ð z Þ ¼ Q 45kek

xz ð z ÞþQ 55kek

yz ð z Þ, ð5Þ

where Q ijk are the transformed material constants with respect to

the global coordinates for the kth layer and transverse shear

strains are given by

ek xz ð z Þ ¼ @w0@ x

þ z @w1@ x

þu1þ2 zu2þ3 z 2u3þaku

k1þ2akzku

k2þ3akz

2k u

k3,

ek yz ð z Þ ¼ @w0@ y

þ z @w1@ y

þv1þ2 zv2þ3 z 2v3þakv

k1þ2akzkv

k2þ3akz

2k v

k3,

ð6Þ

In Eq. (6), it can be found that derivatives of transverse

displacements are present in the strain components. The deriva-

tives of transverse displacements will be involved in the in-plane

displacement fields, as the interlaminar continuity conditions

of transverse shear stresses are used. As the derivatives of

transverse displacements are required in the in-plane displace-

ment fields, C1 interpolation functions would be required for the

finite element implementation. To avoid using C1 interpolation

functions, the derivatives of lateral displacement have to be

eliminated by employing the stress free conditions at the bound-

ary surfaces.

Using the conditions of zero transverse shear stresses at the

lower surface, the following equations can be obtained:

@w0@ x

¼ H 1u11þH 2u

12þH 3u

13þH 4u1þH 5u2þH 6u3þH 7

@w1@ x

,

@w0@ y

¼ N 1v11þN 2v

12þN 3v

13þN 4v1þN 5v2þN 6v3þN 7

@w1@ y

, ð7Þ

where coefficients H i and N i (i¼17) can be found in

Appendix A. Recall the transverse shear stress continuity conditions

at interfaces,

tkþ1 xz ð z kÞ ¼ tk

xz ð z kÞ,

tkþ1 yz ð z kÞ ¼ tk

yz ð z kÞ: ð8Þ

Applying the interlaminar continuity condition and free con-

dition at the bottom surface of transverse shear stresses, local

displacement variables for the kth ply can be expressed as

where coefficients F ki , Gki , H

ki , L

ki , M

ki and N

ki (i¼114) are given in

Appendix.

Employing the stress conditions of zero transverse shear

stresses at the top surface, @w1=@ x and @w1=@ y can be, respec-

tively, expressed as

@w1@ x

¼ r 1u11þr 2u

12þr 3u

13þs1u1þs2u2þs3u3þc 1v

11þc 2v

12þc 3v

13

þd1v1þd2v2þd3v3,

@w1@ y

¼ e1u11þe2u

12þe3u

13þ f 1u1þ f 2u2þ f 3u3 þ g 1v

11þ g 2v

12

þ g 3v13þh1v1þh2v2þh3v3, ð10Þ

where the coefficients in Eq. (10) can be found in the Appendix A.

Substituting Eqs. (9) and (10) into Eq. (1), the final displacement

fields can be written as

uk ¼ u0þX3i ¼ 1

Fki ð z Þu1i þ

X3 j ¼ 1

Fk jþ3ð z Þu jþX3r ¼ 1

Fkr þ6ð z Þv1r þ

X3s ¼ 1

Fksþ9ð z Þvs,

vk ¼ v0þX3i ¼ 1

Cki ð z Þu1i þ

X3 j ¼ 1

Ck jþ3ð z Þu j þX3r ¼ 1

Ckr þ6ð z Þv1r þ

X3s ¼ 1

Cksþ9ð z Þvs,

wk ¼w0þ zw1, ð11Þ

where Fki and Cki are functions of material constants and the

thickness of the laminated plate, respectively, which are alsoshown in Appendix A. Based on known traction conditions at the

upper surface, the derivatives of transverse displacement @wi=@ x

and @wi=@ y (i¼0, 1) are eliminated. This is a major breakthrough

in the finite element formulation of composite plates, in which

simple planar C0 finite elements could be used for the interpola-

tion of displacement parameters.

3. Finite element formulation

As the first derivatives of transverse displacement are excluded

from the in-plane displacement fields, the classical six-node C0

quadratic triangular element can be used in the finite element

analysis. The interpolations for the displacement parameters used

u1

u2

u3

v1

v2

v3

26666666664

37777777775

k

¼

F 1 F 2 F 3 F 4 F 5 F 6 F 7 F 8 F 9 F 10 F 11 F 12 F 13 F 14

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14

H 1 H 2 H 3 H 4 H 5 H 6 H 7 H 8 H 9 H 10 H 11 H 12 H 13 H 14

L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14

M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 M 9 M 10 M 11 M 12 M 13 M 14

N 1 N 2 N 3 N 4 N 5 N 6 N 7 N 8 N 9 N 10 N 11 N 12 N 13 N 14

26666666664

37777777775

k

u11u12u13u1

u2

u3@w1@ x

v11v12v13v1

v2

v3@w1@ y

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

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

, ð9Þ



4/12

in the generalized displacement field are given by

u0 ¼X6r ¼ 1

N r u0r , v0 ¼X6r ¼ 1

N r v0r , w0 ¼X6r ¼ 1

N r w0r , w1 ¼X6r ¼ 1

N r w1r ,

u11 ¼X6r ¼ 1

N r u11r , u

12 ¼

X6r ¼ 1

N r u12r , u j ¼

X6r ¼ 1

N r u jr , ð j ¼ 123Þ,

v11 ¼X6r ¼ 1

N r v11r , v

12 ¼

X6r ¼ 1

N r v12r , v j ¼

X6r ¼ 1

N r v jr , ð j ¼ 123Þ, ð12Þ

where N i¼(2Li1)Li, N 4¼4L1L2, N 5¼4L2L3, N 6¼4L3L1; Li are area

coordinates, (i¼1,2,3).

3.1. The strain and the stiffness matrix

By virtue of linear strain–displacement relationships, the

strain for the kth layer can be written as

ek ¼ @uk ¼ B1 B2 B3 B4 B5 B6

de, ð13Þ

where de ¼ ½ de1 d

e2 d

e3 d

e4 d

e5 d

e6

T and dei (i¼1–6) are displa-

cement parameters at the ith node. The sixteen global displace-ment parameters at each node are given by

@¼

@@ x 0 0

@@ z 0

@@ y

0 @@ y 0 0 @

@ z @@ x

0 0 @@ z @@ x

@@ y 0

2664

3775

T

,

Bi ¼

@N i@ x

0 0 0 0 @N i@ y

0 @N i@ y 0 0 0 @N i@ x

0 0 0 @N i@ x@N i@ y

0

0 0 N i z @N i@ x z

@N i@ y 0

Fk1@N i@ x C

k1

@N i@ y 0

@Fk1@ z N i

@Ck1@ z N i F

k1

@N i@ y þC

k1

@N i@ x

^ ^ ^ ^ ^ ^

Fk6@N i@ x C

k6

@N i@ y

0 @Fk6

@ z N i@Ck6@ z N i F

k6

@N i@ y þC

k6

@N i@ x

Fk7@N i@ x C

k7

@N i@ y

0 @Fk7


k7

@N i@ y þC

k7

@N i@ x

^ ^ ^ ^ ^ ^

Fk12@N i@ x C

k12

@N i@ y

0 @Fk12


k12

@N i@ y þC

k12

@N i@ x

266666666666666666666666664

377777777777777777777777775

T

, ði ¼ 126Þ:

By means of the strain matrix B , the element stiffness matrix

Ke can be readily evaluated using the following equation:

Ke ¼X

n

k ¼ 1

Z hk

ZZ BT Q kB dxdy

dz , ð14Þ

where Q k is the transformed material constant matrix of the

kth ply.

3.2. Transverse shear stresses from 3-D equilibrium equation

Apart from using constitutive equations, more accurate trans-

verse shear stresses could be obtained by integrating three-

dimensional local stress equilibrium equations through the thick-

ness of laminated plates, i.e.

t xz ð z Þ ¼Z z

h

2

@s x

@ x þ

@t xy

@ y

dz ,

t yz ð z Þ ¼Z z h2

@t xy@ x

þ@s y@ y

dz , ð15Þ

where s x, s y and t xy are in-plane stresses of the element.

4. Numerical examples

In this section, the finite element results of laminated compo-sites based on the proposed model are presented. The perfor-

mance of finite element is assessed by comparing with the three-

dimensional elasticity solutions and other published results. The

effects of transverse normal strain and stacking sequence are also

studied. The finite element meshes for the static analysis of the

composite plates are shown in Fig. 2. The material constants used

in the examples are given as follows:

Material (1) for laminated composite plates [28]

E 1 ¼ 172:5GPa, E 2 ¼ E 3 ¼ 6:9GPa, G12 ¼ G13 ¼ 3:45GPa,

G23 ¼ 1:38GPa, v12 ¼ v13 ¼ v23 ¼ 0:25:


C 11 ¼ 1:0025E 2, C 12 ¼ 0:25E 2, C 22 ¼ 25:0625E 2, C 33 ¼ C 11,

C 44 ¼ 0:5E 2, C 55 ¼ 0:2E 2, C 66 ¼ C 44:


C 11 ¼ 32:0625E 2, C 12 ¼ 0:2495E 2, C 22 ¼ 1:00195E 2, C 33 ¼ C 22,

C 44 ¼ 0:2E 2, C 55 ¼ 0:8E 2, C 66 ¼ C 55:


C 11 ¼ 25:0625E 2, C 12 ¼ 0:25E 2, C 22 ¼ 1:0025E 2, C 33 ¼ C 22,

C 44 ¼ 0:2E 2, C 55 ¼ 0:5E 2, C 66 ¼ C 55:

Material (5) for sandwich plates [30]

Face sheets (h/52):

E 1 ¼ 200GPa, E 2 ¼ E 3 ¼ 8GPa, G12 ¼ G13 ¼ 5GPa,

G23 ¼ 2:2GPa,v12 ¼ v13 ¼ 0:25, v23 ¼ 0:35, a1 ¼2 106=K ,

a2 ¼ a3 ¼ 50 106=K :

Core material (3h/5):

E c 1 ¼ E c 2 ¼ 1GPa, E

c 3 ¼ 2GPa, G

c 12 ¼ 3:7GPa, G

c 13 ¼ G

c 23 ¼ 0:8GPa,

vc 12 ¼ 0:35, vc 13 ¼ v

c 23 ¼ 0:25, a

c 1 ¼a

c 2 ¼ a

c 3 ¼ 30 10

6=K :

where 1 and 2 denote the in-plane directions and 3 denotes

transverse direction of laminates.

Example 1. Cylindrical bending of laminated composite plate

strips subjected to a sinusoidal loading q ¼ q0 sinðp x=aÞ.

The normalized displacements and stresses are given by

s x ¼ s xða=2,b=2, z Þh2

q0a2 , t xz ¼

t xz ð0,b=2, z Þh

q0a ,

~u ¼ E 2uð0,b=2, z Þ

q0h

, ~s x ¼ s xða=2,b=2, z Þ

q0,

dei ¼ u0i v0i w0i w1i u

11i u

12i u

13i u1i u2i u3i v

11i v

12i v

13i v1i v2i v3i

h iT :



5/12

~t xy ¼ t xyða=2,b=2, z Þ

q0, ~t xz ¼

t xz ð0,b=2, z Þq0

:

The following boundary conditions proposed by Bogdanovich

and Yushanov [31] are used.

u0 ¼ v0 ¼ w0 ¼w1 ¼ 0, at x ¼ 0; u0 ¼ w0 ¼ w1 ¼ 0, at x ¼ a:

Material (1) is used in this example for various models

considered, and the entire plate is modeled in the analysis of

the static response of angle-ply laminated plates. In the Figures,

acronyms have been used. GLHTI represents the results obtained

Fig. 2. Finite element meshes of the entire plate. (a) Regular mesh configuration (Mesh sh 1, m m), (b) irregular mesh configuration (Mesh 2, 24 elements), (c) irregular

mesh configuration (Mesh 3, 24 elements) and (d) irregular mesh configuration (Mesh 4, 24 elements).

Table 1

Convergence rate of in-plane stress s xða=2,b=2, z Þ for [151/151] plate (Mesh 1,a/h¼4).

z /h Present 3-D [31]

44 88 1212 1414

0.5 1.0292 0.9992 0.9925 0.9910 0.9960

0.2 0.0371 0.0356 0.0353 0.0353 0.0373

0.0 0.4719 0.4582 0.4552 0.4546 0.4526

0.0þ 0.4926 0.4771 0.4738 0.4732 0.4719

0.2 0.0240 0.0235 0.0234 0.0234 0.0260

1.0 1.0812 1.0481 1.0411 1.0396 1.0446

Table 2

Convergence rate of in-plane stress s xða=2,b=2, z Þ for [151/151] plate (Mesh 2,a/h¼4).

z /h Present 3-D [31]

96 elements 384 elements 600 elements

0.5 1.005 0.9975 0.9966 0.9960

0.2 0.0367 0.0356 0.0355 0.0373

0.0 0.4595 0.4570 0.4566 0.4526

0.0þ 0.4785 0.4755 0.4751 0.4719

0.2 0.0240 0.0236 0.0237 0.0260

1.0 1.0532 1.0456 1.0448 1.0446

Table 3

Comparison of nondimensional stresses at strategic points of laminated composite

plate [01/901/01].

a/h s x a2 ,b2 ,

h2

t xz 0,b2,0

4

Present (1212) 1.1763 0.3531

Exact [28] 1.1678 0.3576

10

Present (1212) 0.7343 0.4196

Exact [28] 0.7369 0.4238

20

Present (1212) 0.6540 0.4375

Exact [28] 0.6580 0.437450

Present (1212) 0.6299 0.4854

Exact [28] 0.6348 0.4415



6/12

from the proposed six-node triangular element based on the

C0-type Global Local Higher-order Theory, and 16 variables are

defined at each node. GLHTN denotes the results obtained from

the six-node triangular element based on the C0-type Global Local

Higher-order Theory neglecting transverse normal strain [25], and

13 variables are defined at each node. FSDT represents the results

obtained from the six-node triangular element based on the First

order Shear Deformation Theory, and there are 5 variables in the

displacement field of this model. A suffix has been introduced tothe acronyms: C and –E denote, respectively, transverse shear

stresses obtained directly from constitutive equations and by

integrating 3-D equilibrium equation; –M1 and M2 denote,

respectively, results obtained from Mesh 1 and Mesh 2. It is

noted that results of GLHTN and FSDT are produced by the

authors based on their own computer codes.

The convergence characteristics of the GLHTI for an angle-ply

moderately thick [151/15] plate (a/h¼4) are shown in

Tables 1 and 2. It is found that stresses can rapidly converge to

within 1% of the 3-D elasticity solutions [31] by refining regular or

irregular meshes. In Table 3, comparison of nondimensional

stresses at strategic points of laminated composite plate [01/901/01]

with different length-to-thickness ratios has been presented. It is

found that the proposed model is applicable not only to thick

plates but also to thin composite plates. In Fig. 3, convergence rate

of in-plane stress by different meshes for three-layer [301/301/301]

plate (a/h¼4) is also presented. It is found that converged results

close to the 3-D elasticity solution [31] could be obtained from

regular 1212 meshes and irregular meshes of 384 elements, for

which mesh configurations had little influence on the accuracy of

results. Effect of transverse normal strain on transverse shear

stresses has been shown in Fig. 4. Numerical results show thatGLHTI including transverse normal strain agree well with the 3-D

elasticity solution [31]. However, GLHTN neglecting transverse

normal strain is less accurate compared to 3-D elasticity solution.

For the five-layer [151/451/01/301/151] plate, the results are,

respectively, presented in Figs. 5–8. It is noted that results of

GLHTI are in good agreement with the 3-D elasticity solutions

[32]. However, the results (FSDT) obtained from the first order

theory seem to be less accurate in comparison with the 3-D

elasticity solutions.

Example 2. Simply-supported laminated composite plate sub-

jected to a doubly sinusoidal transverse loading q ¼ q0 sinðp x=aÞsinðp y=bÞ.

-1 -0.5 0 0.5 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [31]

GLHTI (4x4)

GLHTI (8x8)

GLHTI (12x12)

-1 -0.5 0 0.5 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.53-D [31]

GLHTI (96 elements)

GLHTI (384 elements)


hz

hz

-1 -0.5 0 0.5 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [31]

GLHTI (24 elements)

GLHTI (96 elements)



-1 -0.5 0 0.5 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.53-D [31]

GLHTI (24 elements)

GLHTI (96 elements)



( )z baσ x ,2/,2/( )z baσ x ,2/,2/

( )z baσ x ,2/,2/ ( )z baσ x ,2/,2/

hz

hz

Fig. 3. In-plane stress by different meshes for three-layer [301/30

1/30

1] plate (a/h¼4). (a) Mesh 1, (b) mesh 2, (c) mesh 3 and (d) mesh 4.



7/12

-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [31]

2-D [31]

GLHTI-C (12x12)

GLHTN-C (12x12)

( )z byz ,2/,0τ

( )z byz ,2/,0τ

-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [31]

2-D [31]

GLHTI-C (384 elements)

GLHTI-E (384 elements)

hz

hz

Fig. 4. Transverse shear stress for three-layer [301/301/301] plate (a/h¼4). (a)

Mesh 1 and (b) mesh 2.

-1.5 -1 -0.5 0 0.5 1 1.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [32]

GLHTI-M1 (12x12)

GLHTI-M2 (384 elements)

FSDT-M1 (12x12)

( )z bu ,2/,0~

hz

Fig. 5. In-plane displacement for five-layer [151/45

1/0

1/30

1/15

1] plate (a/h¼4).

-20 -15 -10 -5 0 5 10 15 20-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.53-D [32]

GLHTI-M1 (12x12)


FSDT-M1 (12x12)

( )z ba x ,2/,2/~σ

hz

Fig. 6. In-plane stress for five-layer [151/451/01/301/151] plate (a/h¼4).

-6 -4 -2 0 2 4-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [32]

GLHTI-M1 (12x12)


FSDT-M1 (12x12)

( )z ba xy ,2/,2/~τ

h

z

Fig. 7. In-plane stress for five-layer [151/451/01/301/151] plate (a/h¼4).

0 0.5 1 1.5 2-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [32]

GLHTI-M1 (12x12)


( )z b xz ,2/,0~τ

hz

Fig. 8. Transverse shear five-layer [151/45

1/0

1/30

1/15

1] plate (a/h¼4).



8/12


u ¼ E 2h2u=q0a

3, ðs y,t xyÞ ¼ ðs y,t xyÞh

2=q0a2

, t xz ¼ t xz h=q0a:

Owing to symmetry in geometry, material properties and

loading conditions, one-quarter of the laminated plate is consid-

ered in this example. The boundary conditions used are given in

Table 4.

To further assess the performance of the finite element based

on the proposed model, a five-layer plate with different thickness

and material properties at each ply is studied. The plies of the

composite plate are of thickness 0.3h/0.2h/0.15h/0.25h/0.1h and

of materials 4/2/4/3/2. Distributions of displacement and stresses

through the thickness are plotted in Figs. 9–12. Due to the rapid

changes of material properties through the thickness direction,

the postprocessing method from the zig-zag theory proposed by

Cho and Choi [29] did not produce accurate results of in-plane

displacement and transverse shear stresses. However, the present

finite element based on the proposed model could produce more

accurate displacement and stress distributions.

Table 4

Boundary conditions for examples 2 and 3.

Boundary

conditions

x¼constant y¼constant

Simply-

supported

boundary

w0 ¼ w1 ¼ v11 ¼ v

12 ¼ v

13 ¼ v1 ¼ v2 ¼ v3 ¼ 0 w0 ¼ w1 ¼ u

11 ¼ u

12 ¼ u

13 ¼ u1 ¼ u2 ¼ u3 ¼ 0

Symmetric

axis

u0 ¼ u11 ¼ u

12 ¼ u

13 ¼ u1 ¼ u2 ¼ u3 ¼ 0 v0 ¼ v

11 ¼ v

12 ¼ v

13 ¼ v1 ¼ v2 ¼ v3 ¼ 0

-0.015 -0.01 -0.005 0 0.005 0.01-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.53-D [29]

GLHTI (12x12)

ZZT [29]

( )z au ,0,2/

h

z

Fig. 9. In-plane displacement through thickness of five-layer plate with different

thickness at each ply (a/h¼4).

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [29]

GLHTI (12x12)

ZZT [29]

( )z y ,0,0σ

h

z

Fig. 10. In-plane stresses through thickness of five-layer plate with different


-0.04 -0.02 0 0.02 0.04 0.06-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [29]

GLHTI (12x12)

ZZT [29]

( )z ba xy ,2/,2/τ

hz

Fig. 11. In-plane shear stresses through thickness of five-layer plate with different


-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [29]

GLHTI (12x12)

ZZT [29]

( )z a xz ,0,2/τ

h

z

Fig. 12. Transverse shear stresses through thickness of five-layer plate with

different thickness at each ply (a/h¼4).



9/12

Example 3. A square sandwich plate (01/core/01) subjected to

thermal load DT ð x, y, z Þ ¼ T 0 sinðp x=aÞsinðp y=bÞ has been analyzed.


w ¼ w

a0T 0, ðs x,t xy,t xz ,Þ ¼

ðs x,t xy,t xz Þ

ða0T 0E T Þ , a0 ¼ 10

6=K :

Taking into account the symmetry in geometry, material

properties and loading conditions, one-quarter of the sandwich

plate is considered in this example. The boundary conditions used

are given in Table 4.

In order to study the effects of thermal stress and displace-

ment of sandwich plates due to transverse normal strain,

three-layer sandwich plate with material (5) is considered. Dis-

tributions of transverse displacement and stresses through the

thickness are shown in Figs. 13–16. It is found that the proposedmodel GLHTI can produce much better lateral displacement

distribution through the thickness. However, lateral displacement

obtained from GLHTN is zero, which is in contradiction with the

exact solution [30]. Due to neglect of transverse normal strain, the

model GLHTN fails to produce accurate in-plane stresses.

The distributions of transverse shear stresses along the thickness

are shown in Fig. 16. Numerical results show that the transverse

shear stresses computed by the proposed model (GLHTI) are in

excellent agreement with the exact solutions. However, trans-

verse shear stress obtained from the model GLHTN is less reliable.

5. Conclusions

By introducing transverse normal strain, a C0-type global-localhigher order theory is developed to enhance the analysis of thermal/

mechanical response of thick multilayer plates with general lamina-

tion configurations. The proposed model is applicable not only to

cross-ply but also to angle-ply laminated composite plates.

By employing transverse shear free conditions at upper and lower

surfaces and interlaminar continuity conditions of transverse shear

stresses, the layer-dependent displacement variables can all be

eliminated. As a result, the number of variables involved in the

proposed model is independent of the number of layers of laminates.

One major advantage of the proposed model is that C0 interpolation

functions are only required for the finite element implementation, as

the derivatives of transverse displacement have been eliminated from

the general displacement field. The six-node quadratic C0 triangular

element can be conveniently applied to laminated composite and

-30 -20 -10 0 10 20 30-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [30]

GLHTI (12x12)

GLHTN (12x12)

( )z w ,0,0

h

z

Fig. 13. Transverse displacement for sandwich plate under thermal loads (a/h¼4).

-60 -50 -40 -30 -20 -10 0 10 20-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [30]

GLHTI (12x12)

GLHTN (12x12)

( )z x ,0,0σ

h

z

Fig. 14. In-plane stress for sandwich plate under thermal loads (a/h¼4).

-18 -16 -14 -12 -10 -8 -6-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [30]

GLHTI (12x12)

GLHTN (12x12)

( )z ba xy ,2/,2/τ

hz

Fig. 15. In-plane stress for sandwich plate under thermal loads (a/h¼4).

-1.5 -1 -0.5 0 0.5 1 1.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3-D [30]

GLHTI (12x12)

GLHTN (12x12)

( )z a xz ,0,2/τ

h

z

Fig. 16. Transverse shear stress for sandwich plate under thermal loads ( a/h¼4).



10/12

sandwich plates of various material characteristics under different

load conditions. It is remarked that the finite element results are not

affected by the mesh configuration, which could rapidly converge to

the 3-D elasticity solution using regular or irregular meshes. Further-

more, effects of transverse normal strain on thermal/mechanical

behaviors of thick multilayer plates have been studied. It is found

that transverse normal strain has a crucial effect on the lateral

displacement and in-plane stresses of thick multilayer composite

and sandwich plates under thermal loads.

Acknowledgement

The work described in this paper was supported by the

University Development Fund (2009) on Computational Science

and Engineering at the University of Hong Kong, the National

Natural Sciences Foundation of China (Nos. 10802052, 11072156),

the Program for Liaoning Excellent Talents in University

(LR201033), and the Program for Science and Technology of

Shenyang (F10-205-1-16).

Appendix

The coefficients H i and N i (i¼1–7) are given by

H 1 ¼a1, H 2 ¼ 2a1, H 3 ¼3a1, H 4 ¼1, H 5 ¼2 z 1,

H 6 ¼3 z 21, H 7 ¼ z 1,

N 1 ¼a1, N 2 ¼ 2a1, N 3 ¼3a1, N 4 ¼1, N 5 ¼2 z 1,

N 6 ¼3 z 21, N 7 ¼ z 1:

For k¼1, the coefficients are written as

F 11 ¼ 1, F 12 ¼ F

13 ¼ F

14 ¼ . . .F

114 ¼ 0, G

12 ¼ 1, G

11 ¼

G13 ¼G14 ¼ . . .G

114 ¼ 0,

H 13 ¼ 1, H 11 ¼ H

12 ¼ H

14 ¼ . . .H

114 ¼ 0;

L18 ¼ 1, L11 ¼ L12 ¼ L13 ¼ . . .L114 ¼ 0,

M 19 ¼ 1,M 11 ¼ M

12 ¼ M

13 ¼ . . .M

114 ¼ 0

N 110 ¼ 1, N 11 ¼N

12 ¼N

13 ¼ . . .N

114 ¼ 0:

The coefficients for k41 can be determined from the following

recursive equations:

F ki ¼ð2þwkÞF k1i 2ð1þwkÞG

k1i 3ð1þwkÞH

k1i

þgkðLk1i þ2M

k1i þ3N

k1i ÞþS i;

Lki ¼ ð2þBkÞLk1i 2ð1þBkÞM

k1i 3ð1þBkÞN

k1i

þykðF k1i þ2G

k1i þ3H

k1i ÞþS i;

Gk

i

¼ F k

i

þF k1

i

þGk1

i

, H k

i

¼H k1

i

;

M ki ¼ Lki þ L

k1i þM

k1i , N

ki ¼N

k1i

where

S 1 ¼ H 1Wk, S 2 ¼ H 2Wk, S 3 ¼H 3Wk, S 4 ¼ ðH 4þ1ÞWk,

S 5 ¼ ðH 5þ2 z kÞWk,

S 6 ¼ ðH 6þ3 z 2k ÞWk, S 7 ¼ ðH 7þ z kÞWk, S 8 ¼ N 1rk,

S 9 ¼ N 2rk, S 10 ¼ N 3rk,

S 11 ¼ ðN 4þ1Þrk, S 12 ¼ ðN 5þ2 z kÞrk,

S 13 ¼ ðN 6þ3 z 2kÞrk, S 14 ¼ ðN 7þ z kÞrk;

S 1 ¼ H 1ck, S 2 ¼H 2ck, S 3 ¼H 3ck,

S 4 ¼ ðH 4þ1Þck, S 5 ¼ ðH 5þ2 z kÞck

S 6 ¼ ðH 6þ3 z 2kÞck, S 7 ¼ ðH 7þ z kÞck,

S 8 ¼ N 1Zk,

S 9 ¼ N 2Zk,

S 10 ¼ N 3Zk,

S 11 ¼ ðN 4þ1ÞZk, S 12 ¼ ðN 5þ2 z kÞZk,

S 13 ¼ ðN 6þ3 z 2k ÞZk, S 14 ¼ ðN 7þ z kÞZk:

k ¼ 2,3,:::n:

wk ¼ Q 44k1Q 55kQ 45k1Q 45k

Q 44kQ 55kQ 245k

!ak1

ak;

Wk¼ 1þ

Q 45k1Q 45kQ 44k1Q 55k

Q 44kQ 55kQ 245k ! 1

ak;

gk ¼ Q 55k1Q 45kQ 45k1Q 55k

Q 44kQ 55kQ 245k

!ak1

ak;

rk ¼ Q 55k1Q 45kQ 45k1Q 55k

Q 44kQ 55kQ 245k

! 1

ak;

Bk ¼ Q 55k1Q 44kQ 45k1Q 45k

Q 44kQ 55kQ 245k

!ak1

ak;

Zk ¼ 1þQ 45k1Q 45kQ 55k1Q 44k

Q 44kQ 55kQ 245k

! 1

ak;

yk ¼ Q 44k1Q 45kQ 45k1Q 44k

Q 44kQ 55kQ 245k

!ak1

ak;

ck ¼ Q 44k1Q 45kQ 45k1Q 44k

Q 44kQ 55kQ 245k

! 1

ak:

Employing the stress free conditions of transverse shear

stresses at the upper surface, @w1=@ x and @w1=@ y can be respec-

tively expressed as

@w1@ x

¼ r 1u11þr 2u

12þr 3u

13þs1u1þs2u2þs3u3þc 1v

11

þc 2v12þc 3v

13þd1v1þd2v2þd3v3,

@w1@ y

¼ e1u11þe2u

12þe3u

13þ f 1u1þ f 2u2þ f 3u3þ g 1v

11

þ g 2v12þ g 3v

13þh1v1þh2v2þh3v3,

where

r 1 ¼ b1a14ða1þH 1Þðb14þN 7þ z nþ1Þ

D ,


D ,


D ,

s1 ¼ b4a14ða4þH 4þ1Þðb14þN 7þ z nþ1Þ

D ,

s2 ¼ b5a14ða5þH 5þ2 z nþ1Þðb14þN 7þ z nþ1Þ

D ,

s3 ¼b6a14ða6þH 6þ3 z

2nþ1Þðb14þN 7þ z nþ1Þ

D ,

c 1 ¼

ðb8þN 1Þa14a8ðb14þN 7þ z nþ1Þ

D ,

c 2 ¼ ðb9þN 2Þa14a9ðb14þN 7þ z nþ1Þ

D ,

c 3 ¼ ðb10þN 3Þa14a10ðb14þN 7þ z nþ1Þ

D ,

d1 ¼ ðb11þN 4þ1Þa14a11ðb14þN 7þ z nþ1Þ

D ,

d2 ¼ ðb12þN 5þ2 z nþ1Þa14a12ðb14þN 7þ z nþ1Þ

D ,

d3 ¼ðb13þN 6þ3 z

2nþ1Þa14a13ðb14þN 7þ z nþ1Þ

D

e1 ¼ ða1þH 1Þb7b1ða7þH 7þ z nþ1Þ

D ,


D ,



11/12


D ,

f 1 ¼ ða4þH 4þ1Þb7b4ða7þH 7þ z nþ1Þ

D ,

f 2 ¼ ða5þH 5þ2 z nþ1Þb7b5ða7þH 7þ z nþ1Þ

D ,

f 3 ¼ða6þH 6þ3 z

2nþ1Þb7b6ða7þH 7þ z nþ1Þ

D ,

g 1 ¼ a8b7ðb8þN 1Þða7þH 7þ z nþ1Þ

D ,


D ,


D ,

h1 ¼ a11b7ðb11þN 4þ1Þða7þH 7þ z nþ1Þ

D ,

h2 ¼ a12b7ðb12þN 5þ2 z nþ1Þða7þH 7þ z nþ1Þ

D ,

h3 ¼a13b7ðb13þN 6þ3 z

2nþ1Þða7þH 7þ z nþ1Þ

D :

ai ¼ anF ni þ2anG

ni þ3anH

ni ; bi ¼ anL

ni þ2anM

ni þ3anN

ni

D¼ ðanF n

7 þ2anGn

7þ3anH n

7 þH 7þ z nþ1ÞðanLn

14þ2anM n

14

þ3anN n14þN 7þ z nþ1ÞðanF

n14þ2anG

n14þ3anH

n14Þ

ðanLn7þ2anM

n7 þ3anN

n7Þ

Finally, coefficients Fki and Cki are given by

Fki ¼ Rki zkþS

ki z

2k þT

ki z

3k þ Z i,

Cki ¼ Oki zkþP

ki z

2k þQ

ki z

3k þ Z i ,

where

Z 4 ¼ z , Z 5 ¼ z 2

, Z 6 ¼ z 3

, Z i ¼ 0 ðia4,5,6Þ,

Z 10 ¼ z , Z 11 ¼ z 2

, Z 12 ¼ z 3

, Z i ¼ 0 ðia10,11,12Þ,

Rk1 ¼ F k1 þF

k7r 1þF

k14e1, S

k1 ¼G

k1þG

k7r 1þG

k14e1,

T k1 ¼H

k1þH

k7r 1þH

k14e1

,

Rk2 ¼ F k2 þF

k7r 2þF

k14e2, S

k2 ¼G

k2þG

k7r 2þG

k14e2,

T k2 ¼H k2þH

k7r 2þH

k14e2,

Rk3 ¼ F k3 þF

k7r 3þF

k14e3, S

k3 ¼G

k3þG

k7r 3þG

k14e3,

T k3 ¼H k3þH

k7r 3þH

k14e3,

Rk4 ¼ F k4 þF

k7s1þF

k14 f 1, S

k4 ¼ G

k4þG

k7s1þG

k14 f 1,

T k4 ¼H k4þH

k7s1þH

k14 f 1,

Rk5 ¼ F k5 þF

k7s2þF

k14 f 2, S

k5 ¼ G

k5þG

k7s2þG

k14 f 2,

T k5 ¼H k5þH

k7s2þH

k14 f 2,

Rk6 ¼ F k6 þF

k7s3þF

k14 f 3, S

k6 ¼ G

k6þG

k7s3þG

k14 f 3,

T k6 ¼H k6þH

k7s3þH

k14 f 3,

Rk7 ¼ F

k8 þF

k7c 1þF

k14 g 1

,

S k7 ¼G

k8þG

k7c 1þG

k14 g 1

,

T k7 ¼H k8þH

k7c 1þH

k14 g 1,

Rk8 ¼ F k9 þF

k7c 2þF

k14 g 2, S

k8 ¼G

k9þG

k7c 2þG

k14 g 2,

T k8 ¼H k9þH

k7c 2þH

k14 g 2,

Rk9 ¼ F k10þF

k7c 3þF

k14 g 3, S

k9 ¼G

k10þG

k7c 3þG

k14 g 3,

T k9 ¼H k10þH

k7c 3þH

k14 g 3,

Rk10 ¼ F k11þF

k7d1þF

k14h1, S

k10 ¼ G

k11þG

k7d1þG

k14h1,

T k10 ¼H k11þH

k7d1þH

k14h1,

Rk11 ¼ F k12þF

k7d2þF

k14h2, S

k11 ¼ G

k12þG

k7d2þG

k14h2,

T k11 ¼ H k12þH

k7d2þH

k14h2,

Rk12 ¼ F k13þF

k7d3þF

k14h3, S

k12 ¼ G

k13þG

k7d3þG

k14h3,

T k

12 ¼ H k

13þH k

7d3þH k

14h3;

Ok1 ¼ Lk1þL

k7r 1þL

k14e1, P

k1 ¼ M

k1þM

k7r 1þM

k14e1,

Q k1 ¼ N k1þN

k7r 1þN

k14e1,

Ok2 ¼ Lk2þL

k7r 2þL

k14e2, P

k2 ¼ M

k2þM

k7r 2þM

k14e2,

Q k2 ¼ N k2þN

k7r 2þN

k14e2,

Ok3 ¼ Lk3þL

k7r 3þL

k14e3, P

k3 ¼ M

k3þM

k7r 3þM

k14e3,

Q k3 ¼ N k3þN

k7r 3þN

k14e3,

Ok4 ¼ Lk4þL

k7s1þL

k14 f 1, P

k4 ¼ M

k4þM

k7s1þM

k14 f 1,

Q k4 ¼ N k4þN

k7s1þN

k14 f 1,

Ok5 ¼ Lk5þL

k7s2þL

k14 f 2, P

k5 ¼ M

k5þM

k7s2þM

k14 f 2,

Q k5 ¼ N k5þN

k7s2þN

k14 f 2,

Ok6 ¼ Lk6þL

k7s3þL

k14 f 3, P

k6 ¼ M

k6þM

k7s3þM

k14 f 3,

Q k6 ¼ N k6þN

k7s3þN

k14 f 3,

Ok7 ¼ Lk8þL

k7c 1þL

k14 g 1, P

k7 ¼M

k8þM

k7c 1þM

k14 g 1,

Q k7 ¼ N k8þN

k7c 1þN

k14 g 1

Ok8 ¼ Lk9þL

k7c 2þL

k14 g 2, P

k8 ¼M

k9þM

k7c 2þM

k14 g 2,

Q k8 ¼ N k9þN

k7c 2þN

k14 g 2,

Ok9 ¼ Lk10þL

k7c 3þL

k14 g 3, P

k9 ¼M

k10þM

k7c 3þM

k14 g 3,

Q k9 ¼ N k10þN

k7c 3þN

k14 g 3,

Ok10 ¼ Lk11þLk7d1þLk14h1, P k10 ¼ M k11þM k7d1þM k14h1,

Q k10 ¼N k11þN

k7d1þN

k14h1,

Ok11 ¼ Lk12þL

k7d2þL

k14h2, P

k11 ¼ M

k12þM

k7d2þM

k14h2,

Q k11 ¼ N k12þN

k7d2þN

k14h2,

Ok12 ¼ Lk13þL

k7d3þL

k14h3, P

k12 ¼ M

k13þM

k7d3þM

k14h3,

Q k12 ¼ N k13þN

k7d3þN

k14h3:

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thick plate shear deformation

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