A higher order finite element including transverse normal strain for linear elastic composite plates with general lamination configurations

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

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Finite Elements in Analysis and Design

0168-87

doi:10.1

n Corr

E-m

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

A higher order finite element including transverse normal strain for linearelastic 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

4X/$ - see front matter & 2011 Elsevier B.V.

016/j.finel.2011.08.003

esponding author.

ail address: [email protected] (S.H. Lo).

a b s t r a c t

This paper describes a higher-order global–local theory for thermal/mechanical response of moderately

thick 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, laminatedcomposite structures are being widely used in many engineeringfields such as aerospace, automotive and submarines. For safe andreliable designs, it is necessary to well understand the structuralbehavior of laminated composite and sandwich plates. Appro-priate computational models ought to be developed for accuratelypredicting the responses of these laminated structures. Amongthe available approaches, the classical laminated plate theory andthe 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 platetheory and the first order shear deformation theory are unableto adequately model the relatively large transverse shear deforma-tions, so that they often produce unacceptable errors in predictingdisplacement and stresses of thick and moderately thick laminatedcomposite plates. Although the three-dimensional models are

All rights reserved.

accurate enough, the three-dimensional analysis requires hugecomputational 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 thein-plane and transverse displacements, the global higher ordershear deformation theories [2–6] have been widely developed.Compared to the first-order theory, the global higher-order theorydoes not require the shear correction factors and can account forthe warping of the cross-section. In the global higher-ordertheory, however, it is assumed that in-plane displacements andits 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 strainsat the interfaces are multiplied by varying material properties ofdifferent layers. Previous studies [7–11] indicated that the globalhigher-order theories overestimate the natural frequencies andbuckling loads of the soft-core sandwiches with a vast differencein material properties and thickness between the faces and thecore, as these models violate the continuity conditions of trans-verse shear stresses at interfaces. To determine the naturalfrequencies 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

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dx.doi.org/10.1016/j.finel.2011.08.003

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dx.doi.org/10.1016/j.finel.2011.08.003

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

theories is promising, they are computational expensive andbecome intractable when the number of laminates increases, asthe number of unknowns depends on the number of layers.

The zig-zag theories, which can a priori satisfy the continuityof transverse shear stresses at interfaces, generally provide areasonable compromise between accuracy and efficiency. Carrera[12] introduced the original work and development of the zig-zagtheory in great details. In recent decade, the zig-zag theories havebeen developed to study the mechanical, thermal and electricalbehaviors of smart composite plates/shells [13–18]. However, dueto enforcement of the transverse shear stress continuity at theinterfaces, derivatives of transverse displacement are involved inthe displacement field. Hence, C1 continuity of transversedisplacement at element interfaces will be required for the finiteelement implementation. Although the continuity conditions oftransverse shear stresses at interfaces can be a priori satisfied, thezig-zag theories are unable to produce accurate transverse shearstresses directly from constitutive equations. For zig-zag theories,accurate transverse shear stresses could only be obtained byintegrating three-dimensional equilibrium equation. However,third order derivatives of transverse displacement are involvedif transverse shear stresses are to be computed by integrating the3-D equilibrium equations. Hence, global equilibrium equationsrelated to the full domain were proposed [19], which is rathercomplicated and expensive, and even might fail under irregularmesh configuration. An eight-node element based on the zig-zagtheory has been proposed by Icardi [20] in which differentdegrees of freedom are defined at the nodes of a finite element.Nonetheless, the proposed element consists of first-order andsecond-order derivatives of transverse displacement as nodalvariables, 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 theanalysis of sandwich plates, which violates the normal slopecontinuity requirement. Recent study [22] indicated that thesix-node nonconforming triangular element [21] was unable toproduce the accurate dynamic response of the sandwich plates, asit violates the continuity conditions of the normal slope at theinterelement boundary. Pandit et al. [23] proposed an improvedzig-zag theory in which transverse displacement is assumed to bequadratic over the core thickness and constant over the facesheets. Nevertheless, the derivatives of transverse displacementare 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 studythe static behaviors of sandwich plate with random materialproperties [24].

To avoid using the C1 interpolation function in the finiteelement 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 finiteelement implementation, as derivatives of transverse displace-ment are eliminated from displacement field based on knownshear stress conditions. Numerical results indicated that theC0-type global–local higher-order theory neglecting transversenormal strain [25] can be employed to study the deformationsand stresses of laminated composites subjected to mechanicalloads. However, for thick multilayer composite plates underthermal loads, transverse normal strain plays an important role.In view of this situation, this paper aims to develop a C0-typeglobal–local theory including transverse normal strain for theanalysis of thick multilayer plates under thermal/mechanicalloads. Furthermore, the proposed model is applicable not onlyto cross-ply but also to angle-ply laminated composite plates.

Based on the proposed model, the six-node C0 triangular elementis 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-planedisplacement 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 transversedisplacement is assumed to be linear across the thickness forthermo-mechanical problems of multilayered plates. The initialdisplacement fields are given by

ukðx,y,zÞ ¼ uGðx,y,zÞþukLðx,y,zÞþ u

kLðx,y,zÞ,

vkðx,y,zÞ ¼ vGðx,y,zÞþvkLðx,y,zÞþ v

kLðx,y,zÞ,

wkðx,y,zÞ ¼wGðx,y,zÞ: ð1Þ

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

uGðx,y,zÞ ¼ u0ðx,yÞþX3

i ¼ 1

ziuiðx,yÞ,

vGðx,y,zÞ ¼ v0ðx,yÞþX3

i ¼ 1

ziviðx,yÞ,

wGðx,y,zÞ ¼w0ðx,yÞþzw1ðx,yÞ: ð2Þ

The local displacement components of the kth ply can bewritten as follows:

ukLðx,y,zÞ ¼ zkuk

1ðx,yÞþz2k uk

2ðx,yÞ,

vkLðx,y,zÞ ¼ zkvk

1ðx,yÞþz2k vk

2ðx,yÞ,

ukLðx,y,zÞ ¼ z3

k uk3ðx,yÞ,

vkLðx,y,zÞ ¼ z3

kvk3ðx,yÞ, ð3Þ

in which, zk¼akz�bk, ak ¼ ð2=ðzkþ1�zkÞÞ, bk ¼ ðzkþ1þzkÞ=ðzkþ1�

zkÞÞ, and x, y and z are the global coordinates of the plate. Thereference plane (z¼0) is taken at the mid-surface of the laminate.The local coordinates for a layer are denoted by x, y and zk, wherezkA[�1,1]. The relationships between the global coordinates andthe 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. Inorder to reduce the layer-dependent variables, the interlaminarcontinuity 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

W. Zhen et al. / Finite Elements in Analysis and Design 48 (2012) 1346–13571348

expressed as

ukLðx,y,zkÞ ¼ uk�1

L ðx,y,zkÞ

ukLðx,y,zkÞ ¼ u

k�1L ðx,y,zkÞ

vkLðx,y,zkÞ ¼ vk�1

L ðx,y,zkÞ

vkLðx,y,zkÞ ¼ v

k�1L ðx,y,zkÞ

, where k¼ 2,3,4,. . .,n: ð4Þ

u1

u2

u3

v1

v2

v3

26666666664

37777777775

k

¼

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14

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

H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14

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

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14

N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N13 N14

26666666664

37777777775

k

u11

u12

u13

u1

u2

u3@w1@x

v11

v12

v13

v1

v2

v3@w1@y

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

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

, ð9Þ

Using the interlaminar continuity conditions of in-plane dis-placement, 4(n�1) variables are eliminated from the initialdisplacement field. The interlaminar continuity conditions andfree surface conditions of transverse shear stresses can be usedto further reduce the layer-dependent variables. For angle-plycomposite plate, the transverse shear stresses for the kth ply aregiven by

tkxzðzÞ ¼Q44kek

xzðzÞþQ45kekyzðzÞ,

tkyzðzÞ ¼Q45kek

xzðzÞþQ55kekyzðzÞ, ð5Þ

where Qijk are the transformed material constants with respect tothe global coordinates for the kth layer and transverse shearstrains are given by

ekxzðzÞ ¼

@w0

@xþz

@w1

@xþu1þ2zu2þ3z2u3þakuk

1þ2akzkuk2þ3akz

2k uk

3,

ekyzðzÞ ¼

@w0

@yþz

@w1

@yþv1þ2zv2þ3z2v3þakvk

1þ2akzkvk2þ3akz

2kvk

3,

ð6Þ

In Eq. (6), it can be found that derivatives of transversedisplacements are present in the strain components. The deriva-tives of transverse displacements will be involved in the in-planedisplacement fields, as the interlaminar continuity conditionsof transverse shear stresses are used. As the derivatives oftransverse displacements are required in the in-plane displace-ment fields, C1 interpolation functions would be required for thefinite element implementation. To avoid using C1 interpolationfunctions, the derivatives of lateral displacement have to beeliminated by employing the stress free conditions at the bound-ary surfaces.

Using the conditions of zero transverse shear stresses at thelower surface, the following equations can be obtained:

@w0

@x¼H1u1

1þH2u12þH3u1

3þH4u1þH5u2þH6u3þH7@w1

@x,

@w0

@y¼N1v1

1þN2v12þN3v1

3þN4v1þN5v2þN6v3þN7@w1

@y, ð7Þ

where coefficients Hi and Ni (i¼1�7) can be found inAppendix A. Recall the transverse shear stress continuity conditions

at interfaces,

tkþ1xz ðzkÞ ¼ tk

xzðzkÞ,

tkþ1yz ðzkÞ ¼ tk

yzðzkÞ: ð8Þ

Applying the interlaminar continuity condition and free con-dition at the bottom surface of transverse shear stresses, localdisplacement variables for the kth ply can be expressed as

where coefficients Fki , Gk

i , Hki , Lk

i , Mki and Nk

i (i¼1�14) are given inAppendix.

Employing the stress conditions of zero transverse shearstresses at the top surface, @w1=@x and @w1=@y can be, respec-tively, expressed as

@w1

@x¼ r1u1

1þr2u12þr3u1

3þs1u1þs2u2þs3u3þc1v11þc2v1

2þc3v13

þd1v1þd2v2þd3v3,

@w1

@y¼ e1u1

1þe2u12þe3u1

3þ f1u1þ f2u2þ f3u3þg1v11þg2v1

2

þg3v13þh1v1þh2v2þh3v3, ð10Þ

where the coefficients in Eq. (10) can be found in the Appendix A.Substituting Eqs. (9) and (10) into Eq. (1), the final displacementfields can be written as

uk ¼ u0þX3

i ¼ 1

Fki ðzÞu

1i þ

X3

j ¼ 1

Fkjþ3ðzÞujþ

X3

r ¼ 1

Fkrþ6ðzÞv

1r þ

X3

s ¼ 1

Fksþ9ðzÞvs,

vk ¼ v0þX3

i ¼ 1

Cki ðzÞu

1i þ

X3

j ¼ 1

Ckjþ3ðzÞujþ

X3

r ¼ 1

Ckrþ6ðzÞv

1r þ

X3

s ¼ 1

Cksþ9ðzÞvs,

wk ¼w0þzw1, ð11Þ

where Fki and Ck

i are functions of material constants and thethickness of the laminated plate, respectively, which are alsoshown in Appendix A. Based on known traction conditions at theupper surface, the derivatives of transverse displacement @wi=@x

and @wi=@y (i¼0, 1) are eliminated. This is a major breakthroughin the finite element formulation of composite plates, in whichsimple 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 excludedfrom the in-plane displacement fields, the classical six-node C0

quadratic triangular element can be used in the finite elementanalysis. The interpolations for the displacement parameters used


in the generalized displacement field are given by

u0 ¼X6

r ¼ 1

Nru0r , v0 ¼X6

r ¼ 1

Nrv0r , w0 ¼X6

r ¼ 1

Nrw0r , w1 ¼X6

r ¼ 1

Nrw1r ,

u11 ¼

X6

r ¼ 1

Nru11r , u1

2 ¼X6

r ¼ 1

Nru12r , uj ¼

X6

r ¼ 1

Nrujr , ðj¼ 123Þ,

v11 ¼

X6

r ¼ 1

Nrv11r , v1

2 ¼X6

r ¼ 1

Nrv12r , vj ¼

X6

r ¼ 1

Nrvjr , ðj¼ 123Þ, ð12Þ

where Ni¼(2Li�1)Li, N4¼4L1L2, N5¼4L2L3, N6¼4L3L1; Li are areacoordinates, (i¼1,2,3).

3.1. The strain and the stiffness matrix

By virtue of linear strain–displacement relationships, thestrain for the kth layer can be written as

ek ¼ @uk ¼ B1 B2 B3 B4 B5 B6� �

de, ð13Þ

where de¼ ½ de

1 de2 de

3 de4 de

5 de6 �

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

dei ¼ u0i v0i w0i w1i u1

1i u12i u1

3i u1i u2i u3i v11i v1

2i v13i v1i v2i v3i

h iT:

@¼

@@x 0 0 @

@z 0 @@y

0 @@y 0 0 @

@z@@x

0 0 @@z

@@x

@@y 0

2664

3775

T

,

Bi ¼

@Ni@x 0 0 0 0 @Ni

@y

0 @Ni@y 0 0 0 @Ni

@x

0 0 0 @Ni

@x@Ni

@y 0

0 0 Ni z @Ni@x z @Ni

@y 0

Fk1@Ni

@x Ck1@Ni

@y 0 @Fk1

@z Ni@Ck

1@z Ni Fk

1@Ni

@y þCk1@Ni

@x

^ ^ ^ ^ ^ ^

Fk6@Ni@x Ck

6@Ni@y 0 @Fk

6@z Ni

@Ck6

@z Ni Fk6@Ni@y þC

k6@Ni@x

Fk7@Ni@x Ck

7@Ni@y 0 @Fk

7@z Ni

@Ck7

@z Ni Fk7@Ni@y þC

k7@Ni@x

^ ^ ^ ^ ^ ^

Fk12

@Ni

@x Ck12

@Ni

@y 0 @Fk12@z Ni

@Ck12

@z Ni Fk12

@Ni

@y þCk12

@Ni

@x

266666666666666666666666664

377777777777777777777777775

T

, ði¼ 126Þ:

By means of the strain matrix B, the element stiffness matrixKe can be readily evaluated using the following equation:

Ke¼Xn

k ¼ 1

Zhk

ZZBT Q kBdxdy

� �dz, ð14Þ

where Qk is the transformed material constant matrix of thekth 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.

txzðzÞ ¼�

Z z

�h2

@sx

@xþ@txy

@ydz,

tyzðzÞ ¼�

Z z

�h2

@txy

@xþ@sy

@ydz, ð15Þ

where sx, sy and txy 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. Theeffects of transverse normal strain and stacking sequence are alsostudied. The finite element meshes for the static analysis of thecomposite plates are shown in Fig. 2. The material constants usedin the examples are given as follows:

Material (1) for laminated composite plates [28]

E1 ¼ 172:5GPa, E2 ¼ E3 ¼ 6:9GPa, G12 ¼ G13 ¼ 3:45GPa,

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


C11 ¼ 1:0025E2, C12 ¼ 0:25E2, C22 ¼ 25:0625E2, C33 ¼ C11,

C44 ¼ 0:5E2, C55 ¼ 0:2E2, C66 ¼ C44:


C11 ¼ 32:0625E2, C12 ¼ 0:2495E2, C22 ¼ 1:00195E2, C33 ¼ C22,

C44 ¼ 0:2E2, C55 ¼ 0:8E2, C66 ¼ C55:


C11 ¼ 25:0625E2, C12 ¼ 0:25E2, C22 ¼ 1:0025E2, C33 ¼ C22,

C44 ¼ 0:2E2, C55 ¼ 0:5E2, C66 ¼ C55:

Material (5) for sandwich plates [30]Face sheets (h/5�2):

E1 ¼ 200GPa, E2 ¼ E3 ¼ 8GPa, G12 ¼ G13 ¼ 5GPa,

G23 ¼ 2:2GPa,v12 ¼ v13 ¼ 0:25, v23 ¼ 0:35, a1 ¼�2� 10�6=K ,

a2 ¼ a3 ¼ 50� 10�6=K:

Core material (3h/5):

Ec1 ¼ Ec

2 ¼ 1GPa, Ec3 ¼ 2GPa, Gc

12 ¼ 3:7GPa, Gc13 ¼ Gc

23 ¼ 0:8GPa,

vc12 ¼ 0:35, vc

13 ¼ vc23 ¼ 0:25, ac

1 ¼ ac2 ¼ a

c3 ¼ 30� 10�6=K :

where 1 and 2 denote the in-plane directions and 3 denotestransverse direction of laminates.

Example 1. Cylindrical bending of laminated composite platestrips subjected to a sinusoidal loading q¼ q0 sinðpx=aÞ.

The normalized displacements and stresses are given by

sx ¼sxða=2,b=2,zÞh2

q0a2, txz ¼

txzð0,b=2,zÞh

q0a,

~u ¼E2uð0,b=2,zÞ

q0h, ~sx ¼

sxða=2,b=2,zÞ

q0,

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 sxða=2,b=2,zÞ for [151/�151] plate (Mesh 1,

a/h¼4).

z/h Present 3-D [31]

4�4 8�8 12�12 14�14

�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 sxða=2,b=2,zÞ 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 3Comparison of nondimensional stresses at strategic points of laminated composite

plate [01/901/01].

a/h sxa2 ,b

2 ,h2

� �txz 0,b

2,0� �

4

Present (12�12) 1.1763 0.3531

Exact [28] 1.1678 0.3576

10

Present (12�12) 0.7343 0.4196

Exact [28] 0.7369 0.4238

20

Present (12�12) 0.6540 0.4375

Exact [28] 0.6580 0.4374

50

Present (12�12) 0.6299 0.4854

Exact [28] 0.6348 0.4415


~txy ¼txyða=2,b=2,zÞ

q0, ~txz ¼

txzð0,b=2,zÞ

q0:

The following boundary conditions proposed by Bogdanovichand Yushanov [31] are used.

u0 ¼ v0 ¼w0 ¼w1 ¼ 0, at x¼ 0; u0 ¼w0 ¼w1 ¼ 0, at x¼ a:

Material (1) is used in this example for various modelsconsidered, and the entire plate is modeled in the analysis ofthe static response of angle-ply laminated plates. In the Figures,acronyms have been used. GLHTI represents the results obtained


from the proposed six-node triangular element based on theC0-type Global Local Higher-order Theory, and 16 variables aredefined at each node. GLHTN denotes the results obtained fromthe six-node triangular element based on the C0-type Global LocalHigher-order Theory neglecting transverse normal strain [25], and13 variables are defined at each node. FSDT represents the resultsobtained from the six-node triangular element based on the Firstorder Shear Deformation Theory, and there are 5 variables in thedisplacement field of this model. A suffix has been introduced tothe acronyms: �C and –E denote, respectively, transverse shearstresses obtained directly from constitutive equations and byintegrating 3-D equilibrium equation; –M1 and M2 denote,respectively, results obtained from Mesh 1 and Mesh 2. It isnoted that results of GLHTN and FSDT are produced by theauthors based on their own computer codes.

The convergence characteristics of the GLHTI for an angle-plymoderately thick [151/�15] plate (a/h¼4) are shown inTables 1 and 2. It is found that stresses can rapidly converge towithin 1% of the 3-D elasticity solutions [31] by refining regular orirregular meshes. In Table 3, comparison of nondimensionalstresses at strategic points of laminated composite plate [01/901/01]with different length-to-thickness ratios has been presented. It is

-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 (4x4)GLHTI (8x8)GLHTI (12x12)

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

0.53-D [31]GLHTI (24 elements)GLHTI (96 elements)GLHTI (384 elements)GLHTI (600 elements)

( )zbaσ x ,2/,2/

( )zbaσ x ,2/,2/

hz

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

found that the proposed model is applicable not only to thickplates but also to thin composite plates. In Fig. 3, convergence rateof in-plane stress by different meshes for three-layer [301/�301/301]plate (a/h¼4) is also presented. It is found that converged resultsclose to the 3-D elasticity solution [31] could be obtained fromregular 12�12 meshes and irregular meshes of 384 elements, forwhich mesh configurations had little influence on the accuracy ofresults. Effect of transverse normal strain on transverse shearstresses has been shown in Fig. 4. Numerical results show thatGLHTI including transverse normal strain agree well with the 3-Delasticity solution [31]. However, GLHTN neglecting transversenormal 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 ofGLHTI are in good agreement with the 3-D elasticity solutions[32]. However, the results (FSDT) obtained from the first ordertheory seem to be less accurate in comparison with the 3-Delasticity solutions.

Example 2. Simply-supported laminated composite plate sub-jected to a doubly sinusoidal transverse loading q¼ q0 sinðpx=aÞ

sinðpy=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

0.53-D [31]GLHTI (96 elements)GLHTI (384 elements)GLHTI (600 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)GLHTI (384 elements)GLHTI (600 elements)

( )zbaσ x ,2/,2/

( )zbaσ x ,2/,2/

hz

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

-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)

( )zbyz ,2/,0τ

( )zbyz ,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

0.53-D [32]GLHTI-M1 (12x12)GLHTI-M2 (384 elements)FSDT-M1 (12x12)

( )zbu ,2/,0~

hz

Fig. 5. In-plane displacement for five-layer [�151/�451/01/301/151] 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)GLHTI-M2 (384 elements)FSDT-M1 (12x12)

( )zbax ,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)GLHTI-M2 (384 elements)FSDT-M1 (12x12)

( )zbaxy ,2/,2/~τ

hz

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)GLHTI-M2 (384 elements)

( )zbxz ,2/,0~τ

hz

Fig. 8. Transverse shear five-layer [�151/�451/01/301/151] plate (a/h¼4).


Table 4Boundary conditions for examples 2 and 3.

Boundary

conditions

x¼constant y¼constant

Simply-

supported

boundary

w0 ¼w1 ¼ v11 ¼ v1

2 ¼ v13 ¼ v1 ¼ v2 ¼ v3 ¼ 0 w0 ¼w1 ¼ u1

1 ¼ u12 ¼ u1

3 ¼ u1 ¼ u2 ¼ u3 ¼ 0

Symmetric

axisu0 ¼ u1

1 ¼ u12 ¼ u1

3 ¼ u1 ¼ u2 ¼ u3 ¼ 0 v0 ¼ v11 ¼ v1

2 ¼ v13 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]

( )zau ,0,2/

hz

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

0.53-D [29]GLHTI (12x12)ZZT [29]

( )zy ,0,0σ

hz

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

0.53-D [29]GLHTI (12x12)ZZT [29]

( )zbaxy ,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]

( )zaxz ,0,2/τ

hz

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

different thickness at each ply (a/h¼4).



u ¼ E2h2u=q0a3, ðsy,txyÞ ¼ ðsy,txyÞh2=q0a2, txz ¼ txzh=q0a:

Owing to symmetry in geometry, material properties andloading conditions, one-quarter of the laminated plate is consid-ered in this example. The boundary conditions used are given inTable 4.

To further assess the performance of the finite element basedon the proposed model, a five-layer plate with different thickness

and material properties at each ply is studied. The plies of thecomposite plate are of thickness 0.3h/0.2h/0.15h/0.25h/0.1h andof materials 4/2/4/3/2. Distributions of displacement and stressesthrough the thickness are plotted in Figs. 9–12. Due to the rapidchanges of material properties through the thickness direction,the postprocessing method from the zig-zag theory proposed byCho and Choi [29] did not produce accurate results of in-planedisplacement and transverse shear stresses. However, the presentfinite element based on the proposed model could produce moreaccurate displacement and stress distributions.

-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

0.53-D [30]GLHTI (12x12)GLHTN (12x12)

( )zw ,0,0

hz

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)

( )zx ,0,0σ

hz

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)

( )zbaxy ,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)

( )zaxz ,0,2/τ

hz

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


Example 3. A square sandwich plate (01/core/01) subjected tothermal load DTðx,y,zÞ ¼ T0 sinðpx=aÞsinðpy=bÞ has been analyzed.


w ¼w

a0T0, ðsx,txy,txz,Þ ¼

ðsx,txy,txzÞ

ða0T0ET Þ, a0 ¼ 10�6=K:

Taking into account the symmetry in geometry, materialproperties and loading conditions, one-quarter of the sandwichplate is considered in this example. The boundary conditions usedare 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 thethickness are shown in Figs. 13–16. It is found that the proposedmodel GLHTI can produce much better lateral displacementdistribution through the thickness. However, lateral displacementobtained from GLHTN is zero, which is in contradiction with theexact solution [30]. Due to neglect of transverse normal strain, themodel GLHTN fails to produce accurate in-plane stresses.The distributions of transverse shear stresses along the thicknessare shown in Fig. 16. Numerical results show that the transverseshear stresses computed by the proposed model (GLHTI) are inexcellent 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 tocross-ply but also to angle-ply laminated composite plates.By employing transverse shear free conditions at upper and lowersurfaces and interlaminar continuity conditions of transverse shearstresses, the layer-dependent displacement variables can all beeliminated. As a result, the number of variables involved in theproposed model is independent of the number of layers of laminates.One major advantage of the proposed model is that C0 interpolationfunctions are only required for the finite element implementation, asthe derivatives of transverse displacement have been eliminated fromthe general displacement field. The six-node quadratic C0 triangularelement can be conveniently applied to laminated composite and


sandwich plates of various material characteristics under differentload conditions. It is remarked that the finite element results are notaffected by the mesh configuration, which could rapidly converge tothe 3-D elasticity solution using regular or irregular meshes. Further-more, effects of transverse normal strain on thermal/mechanicalbehaviors of thick multilayer plates have been studied. It is foundthat transverse normal strain has a crucial effect on the lateraldisplacement and in-plane stresses of thick multilayer compositeand sandwich plates under thermal loads.

Acknowledgement

The work described in this paper was supported by theUniversity Development Fund (2009) on Computational Scienceand Engineering at the University of Hong Kong, the NationalNatural Sciences Foundation of China (Nos. 10802052, 11072156),the Program for Liaoning Excellent Talents in University(LR201033), and the Program for Science and Technology ofShenyang (F10-205-1-16).

Appendix

The coefficients Hi and Ni (i¼1–7) are given by

H1 ¼�a1, H2 ¼ 2a1, H3 ¼�3a1, H4 ¼�1, H5 ¼�2z1,

H6 ¼�3z21, H7 ¼�z1,

N1 ¼�a1, N2 ¼ 2a1, N3 ¼�3a1, N4 ¼�1, N5 ¼�2z1,

N6 ¼�3z21, N7 ¼�z1:

For k¼1, the coefficients are written as

F11 ¼ 1, F1

2 ¼ F13 ¼ F1

4 ¼ . . .F114 ¼ 0, G1

2 ¼ 1, G11 ¼

G13 ¼ G1

4 ¼ . . .G114 ¼ 0,

H13 ¼ 1, H1

1 ¼H12 ¼H1

4 ¼ . . .H114 ¼ 0;

L18 ¼ 1, L1

1 ¼ L12 ¼ L1

3 ¼ . . .L114 ¼ 0,

M19 ¼ 1,M1

1 ¼M12 ¼M1

3 ¼ . . .M114 ¼ 0

N110 ¼ 1, N1

1 ¼N12 ¼N1

3 ¼ . . .N114 ¼ 0:

The coefficients for k41 can be determined from the followingrecursive equations:

Fki ¼�ð2þwkÞF

k�1i �2ð1þwkÞG

k�1i �3ð1þwkÞH

k�1i

þgkðLk�1i þ2Mk�1

i þ3Nk�1i ÞþSi;

Lki ¼�ð2þBkÞL

k�1i �2ð1þBkÞM

k�1i �3ð1þBkÞN

k�1i

þykðFk�1i þ2Gk�1

i þ3Hk�1i ÞþSi;

Gki ¼ Fk

i þFk�1i þGk�1

i , Hki ¼�Hk�1

i ;

Mki ¼ Lk

i þLk�1i þMk�1

i , Nki ¼�Nk�1

i

where

S1 ¼H1Wk, S2 ¼H2Wk, S3 ¼H3Wk, S4 ¼ ðH4þ1ÞWk,

S5 ¼ ðH5þ2zkÞWk,

S6 ¼ ðH6þ3z2k ÞWk, S7 ¼ ðH7þzkÞWk, S8 ¼N1rk,

S9 ¼N2rk, S10 ¼N3rk,

S11 ¼ ðN4þ1Þrk, S12 ¼ ðN5þ2zkÞrk,

S13 ¼ ðN6þ3z2k Þrk, S14 ¼ ðN7þzkÞrk;

S1 ¼H1ck, S2 ¼H2ck, S3 ¼H3ck,

S4 ¼ ðH4þ1Þck, S5 ¼ ðH5þ2zkÞck

S6 ¼ ðH6þ3z2k Þck, S7 ¼ ðH7þzkÞck,

S8 ¼N1Zk, S9 ¼N2Zk, S10 ¼N3Zk,

S11 ¼ ðN4þ1ÞZk, S12 ¼ ðN5þ2zkÞZk,

S13 ¼ ðN6þ3z2k ÞZk, S14 ¼ ðN7þzkÞZk:

k¼ 2,3,:::n:

wk ¼Q44k�1Q55k�Q45k�1Q45k

Q44kQ55k�Q245k

!ak�1

ak;

Wk ¼ 1þQ45k�1Q45k�Q44k�1Q55k

Q44kQ55k�Q245k

!1

ak;

gk ¼Q55k�1Q45k�Q45k�1Q55k

Q44kQ55k�Q245k

!ak�1

ak;

rk ¼Q55k�1Q45k�Q45k�1Q55k

Q44kQ55k�Q245k

!1

ak;

Bk ¼Q55k�1Q44k�Q45k�1Q45k

Q44kQ55k�Q245k

!ak�1

ak;

Zk ¼ 1þQ45k�1Q45k�Q55k�1Q44k

Q44kQ55k�Q245k

!1

ak;

yk ¼Q44k�1Q45k�Q45k�1Q44k

Q44kQ55k�Q245k

!ak�1

ak;

ck ¼Q44k�1Q45k�Q45k�1Q44k

Q44kQ55k�Q245k

!1

ak:

Employing the stress free conditions of transverse shearstresses at the upper surface, @w1=@x and @w1=@y can be respec-tively expressed as

@w1

@x¼ r1u1

1þr2u12þr3u1

3þs1u1þs2u2þs3u3þc1v11

þc2v12þc3v1

3þd1v1þd2v2þd3v3,

@w1

@y¼ e1u1

1þe2u12þe3u1

3þ f1u1þ f2u2þ f3u3þg1v11

þg2v12þg3v1

3þh1v1þh2v2þh3v3,

where

r1 ¼b1a14�ða1þH1Þðb14þN7þznþ1Þ

D,


D,


D,

s1 ¼b4a14�ða4þH4þ1Þðb14þN7þznþ1Þ

D,

s2 ¼b5a14�ða5þH5þ2znþ1Þðb14þN7þznþ1Þ

D,

s3 ¼b6a14�ða6þH6þ3z2

nþ1Þðb14þN7þznþ1Þ

D,

c1 ¼ðb8þN1Þa14�a8ðb14þN7þznþ1Þ

D,


D,


D,

d1 ¼ðb11þN4þ1Þa14�a11ðb14þN7þznþ1Þ

D,

d2 ¼ðb12þN5þ2znþ1Þa14�a12ðb14þN7þznþ1Þ

D,

d3 ¼ðb13þN6þ3z2

nþ1Þa14�a13ðb14þN7þznþ1Þ

D

e1 ¼ða1þH1Þb7�b1ða7þH7þznþ1Þ

D,


D,



D,

f1 ¼ða4þH4þ1Þb7�b4ða7þH7þznþ1Þ

D,

f2 ¼ða5þH5þ2znþ1Þb7�b5ða7þH7þznþ1Þ

D,

f3 ¼ða6þH6þ3z2

nþ1Þb7�b6ða7þH7þznþ1Þ

D,

g1 ¼a8b7�ðb8þN1Þða7þH7þznþ1Þ

D,


D,


D,

h1 ¼a11b7�ðb11þN4þ1Þða7þH7þznþ1Þ

D,

h2 ¼a12b7�ðb12þN5þ2znþ1Þða7þH7þznþ1Þ

D,

h3 ¼a13b7�ðb13þN6þ3z2

nþ1Þða7þH7þznþ1Þ

D:

ai ¼ anFni þ2anGn

i þ3anHni ; bi ¼ anLn

i þ2anMni þ3anNn

i

D¼ ðanFn7þ2anGn

7þ3anHn7þH7þznþ1ÞðanLn

14þ2anMn14

þ3anNn14þN7þznþ1Þ�ðanFn

14þ2anGn14þ3anHn

14Þ

�ðanLn7þ2anMn

7þ3anNn7Þ

Finally, coefficients Fki and Ck

i are given by

Fki ¼ Rk

i zkþSki z

2kþTk

i z3kþZi,

Cki ¼ Ok

i zkþPki z

2kþQk

i z3kþZi,

where

Z4 ¼ z, Z5 ¼ z2, Z6 ¼ z3, Zi ¼ 0 ðia4,5,6Þ,

Z10 ¼ z, Z11 ¼ z2, Z12 ¼ z3, Zi ¼ 0 ðia10,11,12Þ,

Rk1 ¼ Fk

1þFk7r1þFk

14e1, Sk1 ¼ Gk

1þGk7r1þGk

14e1,

Tk1 ¼Hk

1þHk7r1þHk

14e1,

Rk2 ¼ Fk

2þFk7r2þFk

14e2, Sk2 ¼ Gk

2þGk7r2þGk

14e2,

Tk2 ¼Hk

2þHk7r2þHk

14e2,

Rk3 ¼ Fk

3þFk7r3þFk

14e3, Sk3 ¼ Gk

3þGk7r3þGk

14e3,

Tk3 ¼Hk

3þHk7r3þHk

14e3,

Rk4 ¼ Fk

4þFk7s1þFk

14f1, Sk4 ¼ Gk

4þGk7s1þGk

14f1,

Tk4 ¼Hk

4þHk7s1þHk

14f 1,

Rk5 ¼ Fk

5þFk7s2þFk

14f2, Sk5 ¼ Gk

5þGk7s2þGk

14f2,

Tk5 ¼Hk

5þHk7s2þHk

14f2,

Rk6 ¼ Fk

6þFk7s3þFk

14f3, Sk6 ¼ Gk

6þGk7s3þGk

14f3,

Tk6 ¼Hk

6þHk7s3þHk

14f3,

Rk7 ¼ Fk

8þFk7c1þFk

14g1, Sk7 ¼ Gk

8þGk7c1þGk

14g1,

Tk7 ¼Hk

8þHk7c1þHk

14g1,

Rk8 ¼ Fk

9þFk7c2þFk

14g2, Sk8 ¼ Gk

9þGk7c2þGk

14g2,

Tk8 ¼Hk

9þHk7c2þHk

14g2,

Rk9 ¼ Fk

10þFk7c3þFk

14g3, Sk9 ¼ Gk

10þGk7c3þGk

14g3,

Tk9 ¼Hk

10þHk7c3þHk

14g3,

Rk10 ¼ Fk

11þFk7d1þFk

14h1, Sk10 ¼ Gk

11þGk7d1þGk

14h1,

Tk10 ¼Hk

11þHk7d1þHk

14h1,

Rk11 ¼ Fk

12þFk7d2þFk

14h2, Sk11 ¼ Gk

12þGk7d2þGk

14h2,

Tk11 ¼Hk

12þHk7d2þHk

14h2,

Rk12 ¼ Fk

13þFk7d3þFk

14h3, Sk12 ¼ Gk

13þGk7d3þGk

14h3,

Tk12 ¼Hk

13þHk7d3þHk

14h3;

Ok1 ¼ Lk

1þLk7r1þLk

14e1, Pk1 ¼Mk

1þMk7r1þMk

14e1,

Qk1 ¼Nk

1þNk7r1þNk

14e1,

Ok2 ¼ Lk

2þLk7r2þLk

14e2, Pk2 ¼Mk

2þMk7r2þMk

14e2,

Qk2 ¼Nk

2þNk7r2þNk

14e2,

Ok3 ¼ Lk

3þLk7r3þLk

14e3, Pk3 ¼Mk

3þMk7r3þMk

14e3,

Qk3 ¼Nk

3þNk7r3þNk

14e3,

Ok4 ¼ Lk

4þLk7s1þLk

14f1, Pk4 ¼Mk

4þMk7s1þMk

14f1,

Qk4 ¼Nk

4þNk7s1þNk

14f 1,

Ok5 ¼ Lk

5þLk7s2þLk

14f2, Pk5 ¼Mk

5þMk7s2þMk

14f2,

Qk5 ¼Nk

5þNk7s2þNk

14f2,

Ok6 ¼ Lk

6þLk7s3þLk

14f3, Pk6 ¼Mk

6þMk7s3þMk

14f3,

Qk6 ¼Nk

6þNk7s3þNk

14f3,

Ok7 ¼ Lk

8þLk7c1þLk

14g1, Pk7 ¼Mk

8þMk7c1þMk

14g1,

Qk7 ¼Nk

8þNk7c1þNk

14g1

Ok8 ¼ Lk

9þLk7c2þLk

14g2, Pk8 ¼Mk

9þMk7c2þMk

14g2,

Qk8 ¼Nk

9þNk7c2þNk

14g2,

Ok9 ¼ Lk

10þLk7c3þLk

14g3, Pk9 ¼Mk

10þMk7c3þMk

14g3,

Qk9 ¼Nk

10þNk7c3þNk

14g3,

Ok10 ¼ Lk

11þLk7d1þLk

14h1, Pk10 ¼Mk

11þMk7d1þMk

14h1,

Qk10 ¼Nk

11þNk7d1þNk

14h1,

Ok11 ¼ Lk

12þLk7d2þLk

14h2, Pk11 ¼Mk

12þMk7d2þMk

14h2,

Qk11 ¼Nk

12þNk7d2þNk

14h2,

Ok12 ¼ Lk

13þLk7d3þLk

14h3, Pk12 ¼Mk

13þMk7d3þMk

14h3,

Qk12 ¼Nk

13þNk7d3þNk

14h3:

References

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Documents

A higher order finite element including transverse normal strain for linear elastic composite plates with general lamination configurations