A global-local higher order theory for multilayered shells and the analysis of laminated cylindrical shell panels

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Composite Structures 84 (2008) 350–361

A global-local higher order theory for multilayered shellsand the analysis of laminated cylindrical shell panels

Wu Zhen a,b,*, Chen Wanji b,a

a State Key Laboratory for Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116023, Chinab Department of Aeronautics and Astronautics, Shenyang Institute of Aeronautical Engineering, Shenyang 110034, China

Available online 10 October 2007

Abstract

Based on the global-local superposition technique proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based ondouble superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212.], a global-local higher order laminated shell model is pro-posed for predicting both displacement and stress distributions through the thickness of laminated shells. This shell model satisfies trans-verse shear stress continuity conditions at interfaces as well as free surface conditions of transverse shear stresses. The merit of this modelis that transverse shear stresses can be accurately predicted directly from constitutive equations without smoothing techniques. Cylindri-cal bending of laminated and sandwich shell panels is chosen to assess the present model wherein the results from several 2D laminatedshell models and three-dimensional elasticity solution are available for comparison. In addition, thermal bending and thermal expansionof laminated cylindrical shell panels are also considered in this paper.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Global-local superposition technique; Global-local higher order laminated shell model; Transverse shear stress continuity; Thermal stress

1. Introduction

Since laminated composite and sandwich shells due totheir high specific strength and low specific density werewidely used in the aeronautical and aerospace industries,numerous investigators have used a variety of models forthe analysis of laminated structures. Based on a seriesexpansion in term of the thickness coordinate, the first-order shear deformation shell theory [1] and the globalhigher order theories [2–5] are developed to predict the glo-bal response of laminated shells. However, these modelsare inadequate for accurately predicting the interlaminarstresses from the constitutive equations, as the continuityconditions of transverse stress at interfaces cannot be a pri-

0263-8223/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2007.10.006

* Corresponding author. Address: State Key Laboratory for StructuralAnalysis of Industrial Equipment, Dalian University of Technology,Dalian 116023, China.

E-mail address: [email protected] (W. Zhen).

ori satisfied. Further layerwise displacement model [6], lay-erwise mixed model [7] and three-dimensional models[8–11] are proposed to accurately calculate local stressesof laminated shells. However, these models require hugecomputational cost for multilayered structures with com-plicated geometry.

Di Sciuva [12] developed a linear zig-zag model whichcan guarantee the continuity of transverse shear stresses.Moreover, the number of variables in this model is inde-pendent of the number of layers. Further the improvedzig-zag model [13,14] are presented. In addition, Choet al. [15] proposed a third-order zig-zag model that canprovide parabolic variation through thickness of trans-verse shear stresses. Moreover, this model is further devel-oped to predict the deformation and stresses of thicksmart composite shell under mechanical, thermal andelectric loads by Oh and Cho [16]. Cho and Kim [17] alsopresented a postprocessing method to predict the throughthe thickness stresses of laminated shells. Subsequently a

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W. Zhen, C. Wanji / Composite Structures 84 (2008) 350–361 351

nine-noded doubly curved shell element based on higherorder zig-zag model is presented by Cho and Kim [18],in conjunction with a processing method. Recently Icardi[19] proposed a third-order zig-zag shell model for theanalysis of laminated shells with general lay-up. Compar-ison of numerical results shows that the zig-zag modelsare more accurate in comparison with other global higherorder models. On the other hand, the effects of higherorder expansion of Lame’s coefficients across the thick-ness on transverse stresses of laminated shells have beenstudied by Icardi and Ruotolo [20]. Although these men-tioned zig-zag models can satisfy transverse shear stressescontinuity conditions at interfaces, the three-dimensionalequilibrium equations are usually adopted to reasonablypredict interlaminar stresses. Cho and Kim [21] have stud-ied in detailed the equilibrium equation approach. More-over, they have concluded that the constitutive equationapproach is more attractive because the equilibrium equa-tion approach requires the higher derivatives of transversedisplacement that results in a numerical problem in prac-tice. Moreover, numerical results [15,19,20] show thattransverse shear stresses obtained from postprocessingmethod are still not accurate enough in comparison withthree-dimensional elasticity solutions.

To the best knowledge of the authors, it can not befound that displacement-based laminated shell modelspublished are able to predict accurately transverse shearstresses directly from the constitutive equations withoutany postprocessing methods. To fill in the existing gap,an attempt is made in this paper to propose a laminatedshell model based on the global-local superposition tech-nique proposed by Li and Liu [22]. Initial displacementfields of the present laminated shell model are composedof both global displacements and local displacements. Byenforcing free conditions of the transverse shear stresseson the upper and lower surfaces, and displacementsand transverse shear stresses continuity conditions atinterfaces, the number of unknowns is independent ofnumber of layer of laminates. Significant characteristicof the present model is that transverse shear stressescan be accurately computed directly from the constitutiveequations without smoothing.

2. Mathematical formulation

2.1. Preliminaries [23]

Consider a laminated shell composed of n orthotropiclaminates and the thickness of the shell is h. An orthogonalcurvilinear coordinate system composed of coordinates n1,n2, z is located on the middle surface. R1, R2 are the radii ofcurvature to the reference surface and A1, A2 are mid-sur-face metrics.

For elastic and orthotropic materials of each layer oflaminated composite shells, stress–strain relations for gen-eric kth layer can be written as

r1

r2

r3

s13

s23

s12

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

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

k

¼

Q11 Q12 Q13 0 0 0

Q21 Q22 Q23 0 0 0

Q31 Q32 Q33 0 0 0

0 0 0 Q44 0 0

0 0 0 0 Q55 0

0 0 0 0 0 Q66

2666666664

3777777775

k e1 � a1DT

e2 � a2DT

e3 � a3DT

c13

c23

c12

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

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

k

ð1Þ

where Qkij are the transformed material constants for

the kth layer. DT is the temperature rise and the a1,a2 and a3 are the corresponding linear thermalexpansion coefficients in the direction of principal mate-rial axes.

According to the infinitesimal deformation theory, thestrain–displacement relations are given by

ek1 ¼

1

A1k1

ouk

on1

þ vk

A2

oA1

on2

þ A1wk

R1

� �

ek2 ¼

1

A2k2

ovk

on2

þ uk

A1

oA2

on1

þ A2wk

R2

� �

ek3 ¼

owk

oz

ck13 ¼

1

A1k1

owk

on1

þ A1k1

o

ozuk

A1k1

� �

ck23 ¼

1

A2k2

owk

on2

þ A2k2

o

ozvk

A2k2

� �

ck12 ¼

A2k2

A1k1

o

on1

vk

A2k2

� �þ A1k1

A2k2

o

on2

uk

A1k1

� �

ð2Þ

where, k1 ¼ 11þz=R1

and k2 ¼ 11þz=R2

.

2.2. Global-local higher order laminated shell model

Based on both power series expansion of displacementcomponents along the thickness coordinates of laminatedshell and double superposition hypothesis proposed by Liand Liu [22], the initial displacement fields for cross-plylaminated shells are written as follows:

ukðn1; n2; zÞ ¼ uGðn1; n2; zÞ þ �ukLðn1; n2; fkÞ þ uk

Lðn1; n2; fkÞvkðn1; n2; zÞ ¼ vGðn1; n2; zÞ þ �vk

Lðn1; n2; fkÞ þ vkLðn1; n2; fkÞ

wkðn1; n2; zÞ ¼ wGðn1; n2; zÞ ð3Þ

where uG, vG and wG are global components of displace-ment expansion; �uk

L, �vkL, uk

L and vkL are local in-plane

displacements as given in Eq. (5); the superscript k repre-sents the layer order of laminated plates. The local coordi-nates for a layer are denoted by n1, n2, fk in which fk 2 [�1,1]. Relations between global coordinates and local coordi-nates can be seen in Fig. 1.

Using power series expansion of displacement compo-nents along the thickness coordinate z, the global displace-ment components can be given by

R

z

n

R

zL

h

φ

−nz

R

z

z

z

12

1

k

z

z

zθkζ

Fig. 1. Cylindrical shell panel geometry.

352 W. Zhen, C. Wanji / Composite Structures 84 (2008) 350–361

uGðn1; n2; zÞ ¼X5

i¼0

uiðn1; n2Þzi

vGðn1; n2; zÞ ¼X5

i¼0

viðn1; n2Þzi

wGðn1; n2; zÞ ¼X2

i¼0

wiðn1; n2Þzi

ð4Þ

The local displacement components can be written asfollows:

�ukLðn1; n2; fkÞ ¼ fkuk

1ðn1; n2Þ þ f2kuk

2ðn1; n2Þ�vk

Lðn1; n2; fkÞ ¼ fkvk1ðn1; n2Þ þ f2

kvk2ðn1; n2Þ

ukLðn1; n2; fkÞ ¼ f3

kuk3ðn1; n2Þ

vkLðn1; n2; fkÞ ¼ f3

kvk3ðn1; n2Þ

ð5Þ

where fk = akz � bk; ak ¼ 2zkþ1�zk

; bk ¼ zkþ1þzkzkþ1�zk

.

2.3. Displacement continuity conditions at interfaces

At present, the initial displacement fields include 6n + 15unknown variables, in which n denotes the total number oflayers in the laminated shells. It shows that the number ofvariables depends on the number of layers. Using displace-ment continuity conditions proposed by Li and Liu [22],4(n � 1) variables can be eliminated. This continuity condi-tions can be expressed by

�ukLðn1; n2; zkÞ ¼ �uk�1

L ðn1; n2; zkÞuk

Lðn1; n2; zkÞ ¼ uk�1L ðn1; n2; zkÞ

�vkLðn1; n2; zkÞ ¼ �vk�1

L ðn1; n2; zkÞvk

Lðn1; n2; zkÞ ¼ vk�1L ðn1; n2; zkÞ

ð6Þ

in which k = 2, 3, 4, . . ., n.

Nh ¼1

Rk2Wk

1ooh Wk

2ooh � � � Wk

6ooh Wk

7ooh Wk

8o2

oh2 þ 1 Wk9

o2

oh2

hNz ¼ 0 0 0 0 0 0 0 0 1 2z½ �Nw ¼ 0 0 0 0 0 0 0 1 z z2

� �Nhz ¼ oWk

1

oz �Wk

1

Rk2� � � oWk

7

oz �Wk

7

Rk2

oWk8

oz �Wk

8

Rk2þ 1

Rk2

� ooh � � � oW

oz

�hU ¼ v0 v1

1 v1 v2 v3 v4 v5 w0 w1 w2

�T:

2.4. Transverse shear stress continuity conditions at

interfaces

By enforcing the continuity conditions of transverseshear stresses at interfaces, 2(n � 1) degrees of freedomcan be eliminated. The transverse shear stress continuityconditions are given by

sk13ðn1; n2; zkÞ ¼ sk�1

13 ðn1; n2; zkÞsk

23ðn1; n2; zkÞ ¼ sk�123 ðn1; n2; zkÞ

ð7Þ

Using the free conditions of the transverse shear stresses onthe upper and lower surfaces, the number of the indepen-dent unknowns is reduced to 17. The final displacementfields for cross-ply laminated plates are

uk ¼ Uk1u0 þ Uk

2u11 þ Uk

3u1 þ Uk4u2 þ Uk

5u3

þ Uk6u4 þ Uk

7u5 þ Uk8

ow0

on1

þ Uk9

ow1

on1

þ Uk10

ow2

on1

vk ¼ Wk1v0 þWk

2v11 þWk

3v1 þWk4v2 þWk

5v3

þWk6v4 þWk

7v5 þWk8

ow0

on2

þWk9

ow1

on2

þWk10

ow2

on2

wk ¼ w0 þ zw1 þ z2w2

ð8Þ

in which, the expression of Uki and Wk

i can be found inAppendix.

3. Analytical solutions for laminated cylindrical shell panels

To conveniently compare the proposed model and otheravailable two-dimensional models, the cylindrical bendingof cross-ply laminated cylindrical shell panels subject tosinusoidal loading distribution q = q0 sin(ph//) on the topsurface is mainly considered herein. For cylindrical shellpanel, coordinates and radii are defined as follows:

ðn1; n2; zÞ ¼ ðx; h; zÞ; R1 ¼ 1; R2 ¼ R; A1 ¼ 1; A2 ¼ R

According to strain–displacement relations in Eq. (2) anddisplacement fields of present model in (8), the detailedexpression of strains can be written by

e ¼eh

ez

chz

8><>:

9>=>; ¼

Nh

Nz

Nhz

264

375U ð9Þ

Transverse displacement can be written by

w ¼ NwU ð10Þin which

þ z Wk10

o2

oh2 þ z2i

k10 � Wk

10

Rk2þ z2

Rk2

ooh

i


For a simply supported, infinitely long laminated shellpanel under cylindrical bending, the loads can be expressedas q = q0 sin(ph//) where, p = p//. By assuming

v0 ¼ V 0 cos ph; v11 ¼ V 1

1 cos ph; v1 ¼ V 1 cos ph;

v2 ¼ V 2 cos ph; v3 ¼ V 3 cos ph;

v4 ¼ V 4 cos ph; v5 ¼ V 5 cos ph; w0 ¼ W 0 sin ph;

w1 ¼ W 1 sin ph; w2 ¼ W 2 sin ph ð11Þ

the simply supported boundary conditions are automati-cally satisfied.

By substituting Eq. (11) into Eqs. (9) and (10), strainsand transverse displacement can be respectively written asfollows:

e ¼eh

ez

chz

8><>:

9>=>; ¼

Nh sin ph

Nz sin ph

Nhz cos ph

264

375U ð12Þ

w ¼ Nw sin phU ð13Þ

in which

Nh ¼�1

Rk2Wk

1p Wk2p � � � Wk

6p Wk7p Wk

8p2 þ 1 Wk9p2 þ z Wk

10p2 þ z2� �

Nz ¼ 0 0 0 0 0 0 0 0 1 2z½ �

Nhz ¼ oWk1

oz �Wk

1

Rk2� � � oWk

7

oz �Wk

7

Rk2p

oWk8

oz �Wk

8

Rk2þ 1

Rk2

� � � � p

oWk10

oz �Wk

10

Rk2þ z2

Rk2

� h iNw ¼ 0 0 0 0 0 0 0 1 z z2

� �U ¼ V 0 V 1

1 V 1 V 2 V 3 V 4 V 5 W 0 W 1 W 2

�T

The principle of virtual displacement can be written asZA

Xn

k¼1

Z zkþ1

zk

deTr dz

!dA ¼

ZA

dwq dA ð14Þ

Substituting Eqs. (12) and (13) into Eq. (14), followingequation can be obtained

Xn

k¼1

Z zkþ1

zk

N h N z N hz

� �Qk

N h

N z

N hz

264

375Udz ¼ qþ0 Nþw

� �T ð15Þ

where Qk ¼Q11 Q13 0Q13 Q33 00 0 Q44

24

35

k

. By solving Eq. (15), V0,

V 11, V1, V2, V3, V4, V5, W0, W1 and W2 can be obtained.

Moreover, in-plane and transverse shear stresses may becalculated from Eqs. (1) and (2).

4. Numerical examples

To assess present shell model, three typical problems ofcross-ply composite laminated cylindrical shell panels sub-jected to thermal/mechanical loadings are analyzed. At the

same time, we select several typical analytical results forcomparison with results of present theory.

Matsunaga[5]:

Based on global ninth-order model, in-planestress is predicted from constitutiveequation whereas transverse shear stress iscalculated from equilibrium equation.

Exact [8]:
Three-dimensional elasticity solution. ZZT-C [19]: Based on zig-zag theory, result is predicted
from constitutive equations.
RHSDST-C
[19]:
Based on refined higher order sheardeformation laminated shell theory, result ispredicted from constitutive equations.
RHSDST-E[20]:

Based on refined higher order sheardeformation laminated shell theory, result ispredicted from equilibrium equations.

HSDST-E[20]:

Based on global higher order sheardeformation laminated shell theory, result ispresented from equilibrium equations.

Line missing

FSDST-E[20]:

Based on first-order shear deformationlaminated shell theory, result is predictedfrom equilibrium equations.

ZZT-E [15]:
Based on third-order zig-zag theory, resultis given from equilibrium equations.
Present:
Present results as predicted fromconstitutive equations.
where -C and -E denote constitutive equation and equilib-rium equation, respectively. Other acronyms in figures canbe obtained by using the same method.

Example 1. Cylindrical bending of sandwich cylindricalshell panel.

Sandwich cylindrical shell panels, simply supportedalong coordinate x and subjected to sinusoidalloading distribution q = q0 sin (ph//) on the top surface,are analyzed in this example (see Fig. 1), where / = L/Rand L is the arc length. Based on global ninth-order model, Matsunaga [5] has presented analyticalresults of this example. Material constants are chosen asfollows

Face sheets (0.1h � 2)

-60 -40 -20 0 20 40 60-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Matsunaga [5]Present

θσ

h

z


ET ¼ 0:04EL; GLT ¼ 0:008EL; GTT ¼ 0:02EL;

vLT ¼ 0:25

Core material (0.8h � 2)

EcT ¼ 0:0016EL; Gc

LT ¼ GcTT ¼ 0:0024EL; vc

LT ¼ 0:25

where subscript L is the direction parallel to the fibers andsubscript T indicates the transverse direction. In-plane andtransverse shear stresses are normalized as follows:

�w ¼ wELh3

q0L4; �rh ¼

rhð/=2; zÞq0

; �shz ¼shzð0; zÞ

q0

where, L and h are arc length and thickness of laminatedshell panel, respectively.

In-plane and transverse shear stresses of sandwichcylindrical shell panel are given in Figs. 2–7, which arecompared analytical results presented by Matsunaga [5].

-60 -40 -20 0 20 40 60-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


h

z

θσ

Fig. 2. In-plane stress ð�rhÞ through thickness of sandwich shell (L/h = 5,L/R = 1).

0 0.5 1 1.5 2 2.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


h

z

zθτ

Fig. 3. Transverse shear stress ð�shzÞ through thickness of sandwich shell(L/h = 5, L/R = 1).

Fig. 4. In-plane stress ð�rhÞ through thickness of sandwich shell (L/h = 5,L/R = 0.5).

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


zθτ

h

z

Fig. 5. Transverse shear stress ð�shzÞ through thickness of sandwich shell(L/h = 5, L/R = 0.5).

Numerical results show that the present results agree wellwith the previous published results [5].

Example 2. Cylindrical bending of laminated cylindricalshell panel.

Simply supported laminated shell panel under sinusoidalloading distribution q = q0 sin (ph//) is analyzed, where /= p/3(see Fig. 8). Material constants are chosen as follows:

EL ¼ 25� 106 psi; ET ¼ 106 psi;

GLT ¼ 0:5� 106 psi; GTT ¼ 0:2� 106 psi;

vLT ¼ vTT ¼ 0:25

where subscript L is the direction parallel to the fibers andsubscript T indicates the transverse direction. In-plane andtransverse shear stresses are normalized as follows:

-60 -40 -20 0 20 40 60-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


h

z

θσ

Fig. 6. In-plane stress ð�rhÞ through thickness of sandwich shell (L/h = 5,L/R = 0).

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


zθτ

h

z

Fig. 7. Transverse shear stress ð�shzÞ through thickness of sandwich shell(L/h = 5, L/R = 0).

θ

Rh

φ

z

Fig. 8. Cylindrical shell panel geometry for Example 2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Exact [8]PresentZZT-C [19]RHSDST-C [19]HSDST-C [19]

h

z

zθτ

Fig. 9. Transverse shear stress ð�shzÞ through thickness of three-layer shell(R/h = 4).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Exact [8]PresentZZT-E [15]RHSDST-E [20]HSDST-E [20]FSDST-E [20]

zθτ

h

z



�rh ¼rhð/=2; zÞh2

q0R2; �shz ¼

shzð0; zÞhq0R

For this case, 3D elasticity solution proposed by Ren [8] isavailable. In Fig. 9, the present shear stresses in thick lam-inated cylindrical shell (R/h = 4) are compared with avail-able published results as predicted from constitutiveequations. It is found that present results are in good agree-ment with exact solutions [8]. Subsequently, present resultsare compared with other results computed from equilib-rium equations in Fig. 10. Numerical results show thatpresent results are even more accurate than those frompostprocessing methods. At the same time, in-plane andtransverse shear stresses of moderately thick shells (R/h =10) are compared with available published results in Figs.11–13. Thereinto, other published results in Fig. 12 are pre-dicted directly from constitutive equations whereas otheravailable results in Fig. 13 are computed from equilibriumequations. Again, the present results agree well with exactsolutions [8]. To further assess the performance of present

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Exact [8]PresentZZT-C [19]RHSDST-C [19]HSDST-C [19]

h

z

zθτ


0 0.1 0.2 0.3 0.4 0.5 0.6-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Exact [8]PresentZZT-E [19]RHSDST-E [19]RFSDST-E [19]HSDST-E [19]

θzτ

h

z


-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

0.5Exact [8]PresentZZT [19] RHSDST [19]HSDST [19]

θσ

h

z

Fig. 11. In-plane stress ð�rhÞ through thickness of three-layer shell (R/h =10).


model, five-layer cylindrical shell panels are considered inthis example. Moreover, the corresponding results are pre-sented, in conjunction with exact solution and other analyt-ical in Figs. 14–17.

Example 3. Simply supported laminated cylindrical shellpanel under temperature gradient T ¼ hT 0z

2sinðph=/Þ and

constant temperature T = T0 sin(ph//), where / = L/Rand L is the arc length.

Material constants are chosen as follows:

EL ¼ 144:8 GPa; ET ¼ 9:65 GPa;

GLT ¼ 4:14 GPa; GLT ¼ 3:45 GPa; vLT ¼ 0:3;

aL ¼ 0:139� 10�6=K; aT ¼ 9� 10�6=K

The in-plane and transverse shear stresses are normalizedas follows:

-2 -1.5 -1 -0.5 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

Exact [8]PresentZZT [17]FSDST [17]

θσ

h

z

Fig. 14. In-plane stress ð�rhÞ through thickness of five-layer shell (R/h = 5).

0 0.1 0.2 0.3 0.4 0.5 0.6-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Exact [8]PresentZZT [17]

h

z

zθτ

Fig. 15. Transverse shear stress ð�shzÞ through thickness of five-layer shell(R/h = 5).

-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

Exact [8]Present

θσ

h

z

Fig. 16. In-plane stress ð�rhÞ through thickness of five-layer shell (R/h = 10).

0 0.1 0.2 0.3 0.4 0.5 0.6-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Exact [8]PresentZZT [15]

zθτ

h

z

Fig. 17. Transverse shear stress ð�shzÞ through thickness of five-layer shell(R/h = 10).

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


h

z

θσ

Fig. 18. In-plane stress ð�rhÞ through thickness of three-layer shell undertemperature gradient (L/h = 5, L/R = 1).

-8 -6 -4 -2 0 2 4 6 8

x 10-3

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


θzτ

h

z

Fig. 19. Transverse shear stress through thickness of three-layer shellunder temperature gradient (L/h = 5, L/R = 1).

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


θσ

h

z

Fig. 20. In-plane stress ð�rhÞ through thickness of three-layer shell undertemperature gradient (L/h = 5, L/R = 0.5).


�rh ¼rh

a0T 0EL

; �shz ¼shz

a0T 0EL

; a0 ¼ 10�6=K

Firstly laminated cylindrical shell panel subjected to tem-perature gradient is analyzed and corresponding resultsare given in Figs. 18–21. Numerical results show that pres-ent results are in good agreement with analytical results [5].On the other hand, both in-plane and transverse shearstresses of laminated cylindrical shell panel under constanttemperature are presented in Figs. 22–25.

5. Conclusions

A global-local higher order laminated shell model hasbeen derived for analyzing the bending and the thermalexpansion problems of multilayered shells. Present modelis able to a priori satisfy free surface conditions and

h

z

-6 -4 -2 0 2 4

x 10-3

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


θzτ

Fig. 21. Transverse shear stress through thickness of three-layer shellunder temperature gradient (L/h = 5, L/R = 0.5).

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

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


θσ

h

z

Fig. 22. In-plane stress ð�rhÞ through thickness of three-layer shell underconstant temperature (L/h = 5, L/R = 1).

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


zθτ

h

z

Fig. 23. Transverse shear stress through thickness of three-layer shellunder constant temperature (L/h = 5, L/R = 1).

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

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


h

z

θσ

Fig. 24. In-plane stress ð�rhÞ through thickness of three-layer shell underconstant temperature (L/h = 5, L/R = 0.5).

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


h

z

zθτ

Fig. 25. Transverse shear stress ð�shzÞ through thickness of three-layer shellunder constant temperature (L/h = 5, L/R = 0.5).


transverse shear stress continuity at interfaces. Moreover,number of unknown variables in this model does notdepend on number of layers of shells.

Comparison of present results with available publishedresults shows that for moderately thick laminated shells,present model is able to predict more accurate in-planeand transverse shear stress distributions through the thick-ness of shells directly from constitutive equations. Draw-back of the present model is that complicatedmathematical formulation is involved whereas completeexpressions of the present model have been given. In addi-tion, this model will be extended to more general laminatedshells in our future work.

Acknowledgement

This work is supported by National Natural SciencesFoundation of China (No. 50479058).


Appendix

Using continuity conditions of transverse shear stressesat interfaces, the following expressions can be written as

uk1 ¼ akuk�1

1 þ bkuk�12 þ ckuk�1

3 þ hk

A1k1

ow0

on1

þ zkow1

on1

þ z2k

ow2

on1

� �

þ hk �u0

R1k1

þ 1� zk

R1k1

� �u1 þ 2zk �

z2k

R1k1

� �u2

�

þ 3z2k �

z3k

R1k1

� �u3þ 4z3

k �z4

k

R1k1

� �u4þ 5z4

k �z5

k

R1k1

� �u5

�

vk1 ¼ 1kvk�1

1 þ qkvk�12 þ gkvk�1

3 þ vk

A2k2

ow0

on2

þ zkow1

on2

þ z2k

ow2

on2

� �

þ vk �v0

R2k2

þ 1� zk

R2k2

� �v1 þ 2zk �

z2k

R2k2

� �v2

�

þ 3z2k �

z3k

R2k2

� �v3þ 4z3

k �z4

k

R2k2

� �v4þ 5z4

k �z5

k

R2k2

� �v5

�

in which

ak ¼ �2ak þ 1

R1k1

� Qk

44 þ ak�1 � 1R1k1

� Qk�1

44

akQk44

bk ¼ �2ak þ 1

R1k1

� Qk

44 þ 2ak�1 � 1R1k1

� Qk�1

44

akQk44

ck ¼ �3ak þ 1

R1k1

� Qk

44 þ 3ak�1 � 1R1k1

� Qk�1

44

akQk44

hk ¼Qk

44 � Qk�144

akQk44

1k ¼ �2ak þ 1

R2k2

� Qk

55 þ ak�1 � 1R2k2

� Qk�1

55

akQk55

qk ¼ �2ak þ 1

R2k2

� Qk

55 þ 2ak�1 � 1R2k2

� Qk�1

55

akQk55

gk ¼ �3ak þ 1

R2k2

� Qk

55 þ 3ak�1 � 1R2k2

� Qk�1

55

akQk55

vk ¼Qk

55 � Qk�155

akQk55

The transverse shear strains for kth ply can be written as

ck13 ¼

1

A1k1

ow0

on1

þ zow1

on1

þ z2 ow2

on1

� �� u0

R1k1

þ 1� zR1k1

� �u1

þ 2z� z2

R1k1

� �u2 þ 3z2 � z3

R1k1

� �u3

þ 4z3 � z4

R1k1

� �u4 þ 5z4 � z5

R1k1

� �u5

þ ak �fk

R1k1

� �uk

1 þ 2akfk �f2

k

R1k1

� �uk

2

þ 3akf2k �

f3k

R1k1

� �uk

3ck23 ¼

1

A2k2

ow0

on2

þ zow1

on2

þ z2 ow2

on2

� �

� v0

R2k2

þ 1� zR2k2

� �v1 þ 2z� z2

R2k2

� �v2 þ 3z2 � z3

R2k2

� �

v3 þ 4z3 � z4

R2k2

� �v4 þ 5z4 � z5

R2k2

� �v5 þ ak �

fk

R2k2

� �

vk1 þ 2akfk �

f2k

R2k2

� �vk

2 þ 3akf2k �

f3k

R2k2

� �vk

3

Using transverse shear free condition at lower surface, thefollowing equations can be given by

u13 ¼�

1

ð3a1þ 1R1k1Þ a1 þ

1

R1k1

� �u1

1 � 2a1þ1

R1k1

� ��

u12�

1

R1k1

u0þ 1� z1

R1k1

� �u1þ 2z1�

z21

R1k1

� �

u2þ 3z21�

z31

R1k1

� �u3þ 4z3

1�z4

1

R1k1

� �u4 þ 5z4

1�z5

1

R1k1

� �

u5þ1

A1k1

ow0

on1

þ z1

ow1

on1

þ z21

ow2

on1

� ��v1

3 ¼�1

3a1þ 1R2k2

� a1þ

1

R2k2

� �v1

1� 2a1 þ1

R2k2

� �v1

2�1

R2k2

v0þ 1� z1

R2k2

� �v1

�

þ 2z1�z2

1

R2k2

� �v2þ 3z2

1�z3

1

R2k2

� �v3þ 4z3

1 �z4

1

R2k2

� �v4

þ 5z41�

z51

R2k2

� �v5þ

1

A2k2

ow0

on2

þ z1

ow1

on2

þ z21

ow2

on2

� ��

After eliminating the unknown variables u13and v1

3, theexpressions of uk

1, uk2, uk

3, vk1, vk

2and vk3 can be given by

uk1 ¼ F k

1u11 þ F k

2u12 þ F k

3u0 þ F k4u1 þ F k

5u2 þ F k6u3 þ F k

7u4

þ F k8u5 þ F k

9

ow0

on1

þ F k10

ow1

on1

þ F k11

ow2

on1

uk2 ¼ Gk

1u11 þ Gk

2u12 þ Gk

3u0 þ Gk4u1 þ Gk

5u2 þ Gk6u3 þ Gk

7u4

þ Gk8u5 þ Gk

9

ow0

on1

þ Gk10

ow1

on1

þ Gk11

ow2

on1

uk3 ¼ H k

1u11 þ Hk

2u12 þ H k

3u0 þ H k4u1 þ Hk

5u2 þ H k6u3 þ Hk

7u4

þ Hk8u5 þ H k

9

ow0

on1

þ H k10

ow1

on1

þ Hk11

ow2

on1

vk1 ¼ Lk

1v11 þ Lk

2v12 þ Lk

3v0 þ Lk4v1 þ Lk

5v2 þ Lk6v3 þ Lk

7v4 þ Lk8v5

þ Lk9

ow0

on2

þ Lk10

ow1

on2

þ Lk11

ow2

on2

vk2 ¼ Mk

1v11 þMk

2v12 þMk

3v0 þMk4v1 þMk

5v2 þMk6v3 þMk

7v4

þMk8v5 þMk

9

ow0

on2

þMk10

ow1

on2

þMk11

ow2

on2

vk3 ¼ N k

1v11 þ N k

2v12 þ Nk

3v0 þ Nk4v1 þ N k

5v2 þ N k6v3 þ Nk

7v4

þ Nk8v5 þ N k

9

ow0

on2

þ N k10

ow1

on2

þ N k11

ow2

on2

in which, the coefficients for k = 1 can be easily given by


F 11 ¼ 1; F 1

2 ¼ F 13 ¼ F 1

4 ¼ � � � F 111 ¼ 0;

G12 ¼ 1;G1

1 ¼ G13 ¼ G1

4 ¼ � � � G111 ¼ 0

L11 ¼ 1; L1

2 ¼ L13 ¼ L1

4 ¼ � � � L111 ¼ 0;

M12 ¼ 1;M1

1 ¼ M13 ¼ M1

4 ¼ � � � M111 ¼ 0

H 11 ¼�ða1 þ 1=R1k1Þð3a1 þ 1=R1k1Þ

; H 12 ¼ð2a1 þ 1=R1k1Þð3a1 þ 1=R1k1Þ

;

H 13 ¼

1=R1k1

ð3a1 þ 1=R1k1Þ; H 1

4 ¼�ð1� z1=R1k1Þð3a1 þ 1=R1k1Þ

;

H 15 ¼�ð2z1 � z2

1=R1k1Þð3a1 þ 1=R1k1Þ

; H 16 ¼�ð3z2

1 � z31=R1k1Þ

ð3a1 þ 1=R1k1Þ;

H 17 ¼�ð4z3

1 � z41=R1k1Þ

ð3a1 þ 1=R1k1Þ; H 1

8 ¼�ð5z4

1 � z51=R1k1Þ

ð3a1 þ 1=R1k1Þ;

H 19 ¼

�1=A1k1

ð3a1 þ 1=R1k1Þ; H 1

10 ¼�z1=A1k1

ð3a1 þ 1=R1k1Þ;

H 111 ¼

�z21=A1k1

ð3a1 þ 1=R1k1Þ;

N 11 ¼�ða1 þ 1=R2k2Þð3a1 þ 1=R2k2Þ

; N 12 ¼ð2a1 þ 1=R2k2Þð3a1 þ 1=R2k2Þ

;

N 13 ¼

1=R2k2

ð3a1 þ 1=R2k2Þ; N 1

4 ¼�ð1� z1=R2k2Þð3a1 þ 1=R2k2Þ

;

N 15 ¼�ð2z1 � z2

1=R2k2Þð3a1 þ 1=R2k2Þ

; N 16 ¼�ð3z2

1 � z31=R2k2Þ

ð3a1 þ 1=R2k2Þ;

N 17 ¼�ð4z3

1 � z41=R2k2Þ

ð3a1 þ 1=R2k2Þ; N 1

8 ¼�ð5z4

1 � z51=R2k2Þ

ð3a1 þ 1=R2k2Þ;

N 19 ¼

�1=A2k2

ð3a1 þ 1=R2k2Þ; N 1

10 ¼�z1=A2k2

ð3a1 þ 1=R2k2Þ;

N 111 ¼

�z21=A2k2

ð3a1 þ 1=R2k2Þ;

where, k1 ¼ 11þz1=R1

and k2 ¼ 11þz1=R2

.The coefficients for k > 1 can be given by

F ki ¼ akF k�1

i þ bkGk�1i þ ckHk�1

i þ S0i; Lki ¼ 1kLk�1

i

þ qkMk�1i þ gkN k�1

i þ SSi

Gki ¼ F k

i þ F k�1i þ Gk�1

i ; Mki ¼ Lk

i þ Lk�1i þMk�1

i ;

H ki ¼ �Hk�1

i ; Nki ¼ �Nk�1

i

i ¼ 1; 2; . . . ; 11; k ¼ 2; 3; . . . ; n

where

S01 ¼ S02 ¼ 0; S03 ¼ �hk

R1k1

; S04 ¼ 1� zk

R1k1

� �hk;

S05 ¼ 2zk �z2

k

R1k1

� �hk;

S06 ¼ 3z2k �

z3k

R1k1

� �hk;

S07 ¼ 4z3k �

z4k

R1k1

� �hk; S08 ¼ 5z4

k �z5

k

R1k1

� �hk;

S09 ¼hk

A1k1

;

S010 ¼zkhk

A1k1

; S011 ¼z2

khk

A1k1

SS1 ¼ SS2 ¼ 0;

SS3 ¼ �vk

R2k2

; SS4 ¼ 1� zk

R2k2

� �vk;

SS5 ¼ 2zk �z2

k

R2k2

� �vk; SS6 ¼ 3z2

k �z3

k

R2k2

� �vk;

SS7 ¼ 4z3k �

z4k

R2k2

� �vk; SS8 ¼ 5z4

k �z5

k

R2k2

� �vk;

SS9 ¼vk

A2k2

; SS10 ¼zkvk

A2k2

; SS11 ¼z2

kvk

A2k2

Using free condition of transverse shear stresses at theupper surface, the following expression can be given by

u12 ¼ A1u1

1 þ B1u0 þ C1u1 þ D1u2 þ E1u3 þ F 1u4 þ G1u5

þ H1ow0

on1

þ I1ow1

on1

þ J1ow2

on1

v12 ¼ A2v1

1 þ B2v0 þ C2v1 þ D2v2 þ E2v3 þ F 2v4 þ G2v5

þ H2ow0

on2

þ I2ow1

on2

þ J2ow2

on2

in which

A1 ¼ D1ð1ÞD11

; B1 ¼ D1ð3Þ � 1=R1k1

D11

;

C1 ¼ D1ð4Þ þ ð1� znþ1=R1k1ÞD11

;

D1 ¼D1ð5Þ þ ð2znþ1 � z2

nþ1=R1k1ÞD11

;

E1 ¼D1ð6Þ þ ð3z2

nþ1 � z3nþ1=R1k1Þ

D11

;

F 1 ¼D1ð7Þ þ ð4z3


D11

;

G1 ¼ D1ð8Þ þ ð5z4nþ1 � z5

nþ1=R1k1ÞD11

H1 ¼ D1ð9Þ þ 1=A1k1

D11

; I1 ¼ D1ð10Þ þ znþ1=A1k1

D11

;

J1 ¼ D1ð11Þ þ z2nþ1=A1k1

D11

A2 ¼ D2ð1ÞD22

; B2 ¼ D2ð3Þ � 1=R2k2

D22

;

C2 ¼ D2ð4Þ þ ð1� znþ1=R2k2ÞD22

;

D2 ¼D2ð5Þ þ ð2znþ1 � z2

nþ1=R2k2ÞD22

;

E2 ¼D2ð6Þ þ ð3z2


D22

;

F 2 ¼D2ð7Þ þ ð4z3


D22

;

G2 ¼D2ð8Þ þ ð5z4


D22


H2 ¼ D2ð9Þ þ 1=A2k2

D22

; I2 ¼ D2ð10Þ þ znþ1=A2k2

D22

;

J2 ¼ D2ð11Þ þ z2nþ1=A2k2

D22

D1ðiÞ ¼ ðan � 1=R1k1ÞF ni þ ð2an � 1=R1k1ÞGn

i

þ ð3an � 1=R1k1ÞH ni ;

D2ðiÞ ¼ ðan � 1=R2k2ÞLni þ 2ðan � 1=R2k2ÞMn

i

þ 3ðan � 1=R2k2ÞNni ;

D11 ¼ ðan � 1=R1k1ÞF n2 þ ð2an � 1=R1k1ÞGn

2

þ ð3an � 1=R1k1ÞH n2;

D22 ¼ ðan � 1=R2k2ÞLn2 þ ð2an � 1=R2k2ÞMn

2

þ ð3an � 1=R2k2ÞNn2:

Finally, the coefficients Uki and Wk

i can be obtained:

Uki ¼ Rk

i fk þ Ski f

2k þ T k

i f3k þ Zi; Wk

i

¼ Oki fk þ P k

i f2k þ Qk

i f3k þ ZZi

where

Z1 ¼ 1; Z3 ¼ z; Z4 ¼ z2; Z5 ¼ z3;

Z6 ¼ z4; Z7 ¼ z5; Zi ¼ 0 ði 6¼ 1; 3; 4; 5; 6; 7ÞZZ1 ¼ 1; ZZ3 ¼ z; ZZ4 ¼ z2; ZZ5 ¼ z3;

ZZ6 ¼ z4; ZZ7 ¼ z5; ZZi ¼ 0 ði 6¼ 1; 3; 4; 5; 6; 7ÞRk

1 ¼ F k3 þ F k

2B1 Sk1 ¼ Gk

3 þ Gk2B1 T k

1 ¼ Hk3 þ Hk

2B1

Rk2 ¼ F k

1 þ F k2A1 Sk

2 ¼ Gk1 þ Gk

2A1 T k2 ¼ Hk

1 þ Hk2A1

Rk3 ¼ F k

4 þ F k2C1 Sk

3 ¼ Gk4 þ Gk

2C1 T k3 ¼ Hk

4 þ Hk2C1

Rk4 ¼ F k

5 þ F k2D1 Sk

4 ¼ Gk5 þ Gk

2D1 T k4 ¼ Hk

5 þ Hk2D1

Rk5 ¼ F k

6 þ F k2E1 Sk

5 ¼ Gk6 þ Gk

2E1 T k5 ¼ Hk

6 þ Hk2E1

Rk6 ¼ F k

7 þ F k2F 1 Sk

6 ¼ Gk7 þ Gk

2F 1 T k6 ¼ H k

7 þ H k2F 1

Rk7 ¼ F k

8 þ F k2G1 Sk

7 ¼ Gk8 þ Gk

2G1 T k7 ¼ H k

8 þ Hk2G1

Rk8 ¼ F k

9 þ F k2H1 Sk

8 ¼ Gk9 þ Gk

2H1 T k8 ¼ Hk

9 þ Hk2H1

Rk9 ¼ F k

10 þ F k2I1 Sk

9 ¼ Gk10 þ Gk

2I1 T k9 ¼ Hk

10 þ Hk2I1

Rk10 ¼ F k

11 þ F k2J1 Sk

10 ¼ Gk11 þ Gk

2J1 T k10 ¼ Hk

11 þ Hk2J1

Ok1 ¼ Lk

3 þ Lk2B2 P k

1 ¼ Mk3 þMk

2B2 Qk1 ¼ N k

3 þ Nk2B2

Ok2 ¼ Lk

1 þ Lk2A2 P k

2 ¼ Mk1 þMk

2A2 Qk2 ¼ N k

1 þ Nk2A2

Ok3 ¼ Lk

4 þ Lk2C2 P k

3 ¼ Mk4 þMk

2C2 Qk3 ¼ Nk

4 þ N k2C2

Ok4 ¼ Lk

5 þ Lk2D2 P k

4 ¼ Mk5 þMk

2D2 Qk4 ¼ N k

5 þ N k2D2

Ok5 ¼ Lk

6 þ Lk2E2 P k

5 ¼ Mk6 þMk

2E2 Qk5 ¼ N k

6 þ N k2E2

Ok6 ¼ Lk

7 þ Lk2F 2 P k

6 ¼ Mk7 þMk

2F 2 Qk6 ¼ N k

7 þ N k2F 2

Ok7 ¼ Lk

8 þ Lk2G2 P k

7 ¼ Mk8 þMk

2G2 Qk7 ¼ N k

8 þ N k2G2

Ok8 ¼ Lk

9 þ Lk2H2 P k

8 ¼ Mk9 þMk

2H2 Qk8 ¼ N k

9 þ N k2H2

Ok9 ¼ Lk

10 þ Lk2I2 P k

9 ¼ Mk10 þMk

2I2 Qk9 ¼ Nk

10 þ N k2I2

Ok10 ¼ Lk

11 þ Lk2J2 P k

10 ¼ Mk11 þMk

2J2 Qk10 ¼ N k

11 þ N k2J2

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A global-local higher order theory for multilayered shells and the analysis of laminated cylindrical shell panels