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Postbuckling behavior of shear deformable anisotropic laminated cylindrical

shell under combined external pressure and axial compression

Article  in  Composite Structures · May 2018

DOI: 10.1016/j.compstruct.2018.05.064

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Composite Structures

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

Postbuckling behavior of shear deformable anisotropic laminated cylindricalshell under combined external pressure and axial compression

Zhi-Min Lia,⁎, Tao Liua, De-Qing Yangb

a State Key Laboratory of Mechanical System and Vibration, Shanghai Key Lab of Digital Manufacture for Thin-Walled Structures, School of Mechanical Engineering,Shanghai Jiao Tong University, Shanghai 200240, Chinab School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

A R T I C L E I N F O

Keywords:PostbucklingAnisotropic laminated cylindrical shellHigher order shear deformable shell theoryBoundary layer theory of shell bucklingSingular perturbation technique

A B S T R A C T

Structural design of composite shells are more challenging than conventional metals due to the complex me-chanical behavior and damage mechanisms which composite materials exhibit. Postbuckling analysis for amoderately thick anisotropic laminated cylindrical shell subjected to combined loadings of external pressure andaxial compression is presented which extends the boundary layer theory of shell buckling. The governingequations are based on Reddy’s higher order shear deformation shell theory with von Kármán-Donnell-type ofkinematic nonlinearity. Both nonlinear prebuckling deformations and initial geometric imperfections of the shellare taken into account. A two-step singular perturbation method is used to determine interactive buckling loadsand postbuckling equilibrium paths. A verification study is conducted, and the validity of the formulation isestablished through comparison with results of nonlinear finite element software such as ABAQUS®. The internalphysical mechanism of the shell geometric parameters on the buckling load and the postbuckling equilibriumpath is obtained. The numerical illustrations concern the postbuckling response of perfect and imperfect,moderately thick, anisotropic laminated cylindrical shells with different load-proportional parameters. Theanalytical model can provide an effective tool to investigate postbuckling of composite shell structures.

1. Introduction

Thin-walled shell structures of various types are very importantstructural elements. From the perspective of engineering application, itis necessary to predict different modes of behavior of plates and shellsunder applied loadings. Interest in the structural instability analysis ofrelatively thick composite-material shells has been led by a need formore accurate analysis especially in the case of thick-walled structures.As a kind of typical structure, shells are used in civil and mechanicalengineering include slabs, vaults, chimneys, cooling towers, pipes,tanks, containers and pressure vessels; in shipbuilding-ship and sub-marine hulls (Mouritz et al. [1]). On the one hand, the challenge to theengineer and researcher is the discrepancy between the highly efficientload-carrying capacity of the perfect shell and the real, manufacturedshell/tube. For the complex buckling behavior of composite cylindricalshell structures, it is very important to investigate how different initialimperfections influence the load-carrying capacity. Due to the inherentanisotropy, the buckling behavior of composite structures is morecomplicated than those of their metallic components. High performancecomposite materials, for example, graphite/epoxy, boron/epoxy, glass/

epoxy etc. are currently being used in many engineering applications.Such beneficial properties as high stiffness-to-weight and strength-to-weight ratios, etc., make laminated panels/shells attractive for struc-tural components in aerospace, marine, automobile and other applica-tion. On the other hand, many studies have observed buckling andmany attempts have been made to predict buckling behavior for com-posite cylindrical shells. Leissa [2] summarized technical literaturesdealing with buckling and postbuckling behavior of laminated com-posite plates and shell panels. In these analyses, only perfect initialconfigurations were assumed. Teng [3] provided a review of recentresearch advances and trends in the area of thin shell buckling. Heemphasized and discussed imperfections in real structures and theirinfluence, buckling of shells under local/non-uniform loads and loca-lized compressive stresses, and the use of computerized buckling ana-lysis in the stability design of complex thin shell structures. Dinkler andPontow [4] introduced the perturbation energy concept and its appli-cation to stability of imperfection sensitive structures under time-de-pendent loads. In the first order shear deformation theory, the dis-placement field is assumed to vary linearly with respect to thickness(measured from the midsurface) and the rotations of the normal to the

https://doi.org/10.1016/j.compstruct.2018.05.064Received 4 January 2018; Received in revised form 7 May 2018; Accepted 10 May 2018

⁎ Corresponding author.E-mail address: zmli@sjtu.edu.cn (Z.-M. Li).

Composite Structures 198 (2018) 84–108

Available online 14 May 20180263-8223/ © 2018 Elsevier Ltd. All rights reserved.

T

midsurface are independent variables. Iu and Chia [5] developed a firstorder shear deformation theory to study nonlinear vibration and post-buckling of imperfect, moderately thick, cross-ply laminated cylindricalshells. Fu and Waas [6] applied first order shear deformation theory tostudy the initial postbuckling behavior of thick rings under uniform,external hydrostatic pressure. Han and Simitses [7] investigated buck-ling behavior of symmetric laminates composite cylindrical shell sub-jected to lateral or hydrostatic pressure based on Sanders-type [8] offirst order shear deformation theory. These studies show that first ordershear deformation theories significantly improve the prediction accu-racy of the buckling load compared to the thin shell Kirchhoff-Loveassumption. However, the improvement offered by the higher ordertheories over the first order ones is much smaller, and in the first ordershear deformation theory the conditions of zero shear stress on the topand bottom surfaces of the shell are not met, and this requires a shearcorrection to the transverse shear stiffnesses. Simitses and Anastasiadis[9,10] developed a higher order shear deformation theory, and studiedbuckling loads of moderately thick, symmetrically laminated cylind-rical shells. Reddy and Savoia [11] investigated the postbuckling re-sponse of imperfect laminated cylindrical shells, which can producemuch more accurate results but the boundary conditions cannot beimposed accurately in their solutions. Recently, research on compositepipes/shells over the last few decades has covered the buckling andpostbuckling response due to bending, compression or combined axial-external pressure loadings and buckling failure can also be observed tooccur when maximum compressive stress in the structure reaches thecritical stress under pure compression or when the prebuckling loadsignificantly contributes to the bifurcation load through ovalization(Sun et al., [12]). Corona and Rodrigues [13] carried out a study on thebending response of long and thin-walled cross-ply composite cylindersincluding three phases: pre-buckling response, material failure by Tsai-Wu criterion, and shell-type bifurcation buckling. Yang et al. [14]studied the buckling of cylindrical shells under external pressure withgeneral axisymmetric thickness imperfections. Based on a system oflinearized governing partial differential equations of perfect shells withvariable thickness, the effects led by three patterns of thickness im-perfections on the buckling of the laterally pressured cylindrical shells,which are uniform, axisymmetric modal and parabolic, are respectivelyanalyzed. Papadakis [15] studied a set of stability equations for thickcylindrical shells under external pressure, analyzed and discussed dif-ferences between the benchmark solutions and the analytic expressionsbased on the refined high order theory and the classical shell theory,and estimated the stress and moment resultants of thick shell based on ahigher order shell theory. Schillo et al. [16] carried out experimentaland numerical study for geometrical imperfection measurements of aset of 12 CFRP cylinders with the specified manufacturing method.Loading imperfections are measured and implemented in the finiteelement analysis. Model uncertainties are quantified with respect toloading and geometric imperfections as well as level of detail of the as-built layup. They point out a general assessment of the sensitivity ofunstiffened cylinders towards geometric and load imperfections shouldinclude a wider range of values and different manufacturing methods.In order to compare the accuracy of the predictions from the classicaland the improved shell theories, Kardomateas [17,18] and Kardoma-teas and Philobos [19] studied the buckling of orthotropic cylindricalshells subjected to axial compression, external pressure and combinedloadings by using the three-dimensional (3D) elasticity theory. Torna-bene et al. [20] and Brischetto et al. [21] investigated the cylindricalbending conditions in the free frequency analysis of functionally gradedmaterial (FGM) plates and cylindrical shells for different geometries(plates, cylinders, and cylindrical shells), types of FGM law, laminationsequences, and thickness ratios. 2D numerical approaches (the Gen-eralized Differential Quadrature (GDQ) and the finite element (FE)methods) are compared with an exact 3D shell solution in the case offree vibrations of FGM plates and shells. Based on the Koiter’ theory(Koiter, [22]), Arciniega et al. [23] investigated the buckling and

postbuckling behavior of laminated cylindrical shells subjected to axialcompression and lateral pressure loading using Rayleigh-Ritz method.By the same method, Salahshour and Fallah [24] investigated localelastic buckling of thin long cylindrical shells under external pressure.Based on Donnell’s and Sanders’ theories of thin shells and von Kármánnonlinearity assumptions, the potential energy is derived. The bucklingload and curves of the static equilibrium path are obtained. Wang et al.[25] studied the effect material of nonlinearity on buckling and post-buckling of fiber composite laminated plates and cylindrical shells andobtained a modified Riks solution scheme with updated Lagrangianformulation. Based on the Donnell-Mushtari-Vlasov theory of shells,Semenyuk and Trach [26,27] obtained solutions of buckling and post-buckling behavior of composite cylindrical shells subjected to funda-mental loads. By using finite element method and experimental mea-sure, Priyadarsini et al. [28] investigated buckling characteristics offiber reinforced composite cylinders subjected to axial compressiveloads. The effects of different types of loadings, geometric propertiesand lamina lay-up were involved. Mistry et al. [29] presented experi-mental and numerical investigationfor the±55° filament-wound glass/epoxy pipes subjected to combined external pressure and axial com-pression. Based on a layer-wise and higher order shear deformationtheory, Eslami et al. [30] and Eslami and Shariyat [31] investigatedpostbuckling of laminated cylindrical shells. Imperfections from man-ufacturing process can cause a scattered reduction of the load-carryingcapacity cylindrical shell structures. Wang et al. [32] predicted theload-carrying capacity or buckling load of axially compressed cylind-rical shell structures. The influence of pure geometric imperfectionsincluding imperfection component and amplitude on the buckling be-havior is discussed based on Fourier series method. Some guidance forthe dimensional tolerance in manufacturing process relating to theload-carrying capacity of thin-walled structures is provided. Lindgaardand Lund [33] presented an approach to nonlinear buckling fiber angleoptimization of laminated composite shell structures. The approachaccounts for the geometrically nonlinear behavior of the structure byutilizing response analysis up until the critical point. Sensitivity in-formation is obtained efficiently by an estimated critical load factor at aprecritical state. Liguori et al. [34] presented a strategy completelybased on stochastic simulations for optimizing the stacking sequence ofslender composite shells undergoing buckling. They predicted andevaluated postbuckling behavior of composite shells by using randomnumerical experiments for detecting both the best layup and the worstshape of the geometrical imperfection. However, in the foregoing stu-dies, the shell theories used in these analyses are mostly extensions ofthe various isotropic or orthotropic and symmetric laminates shellmodels in buckling analysis involving seldom anisotropic coupling ef-fects.

It has been shown in Weaver et al. [35] that all anisotropic couplingeffects reduce the buckling loads. Shen [36–38] developed a boundary-layer theory for the buckling and postbuckling of anisotropic laminatedthin shells under mechanical loading of axial compression, externalpressure and torsion. Based on the above studies, Li and Lin [39] ob-tained analytical results of the buckling and postbuckling behavior forshear-deformable anisotropic laminated cylindrical shells subjected tovarying external pressure loads. Li and Shen [40,41] investigated shear-deformable anisotropic laminated cylindrical shells subjected to axialcompression or torsion. They found that there exists a compressivestress along with an associate shear stress and twisting when the shear-deformable anisotropic laminated cylindrical shell is subjected to axialcompression. In contrast, there exists a shear stress along with an as-sociate compressive stress when the shear-deformable anisotropic shellis subjected to torsion (Shen and Xiang [42]). Accordingly, we believethat there exists a circumferential stress due to boundary constraintsalong with an associate shear stress when an anisotropic thin shell issubjected to external pressure loads combined with axial compression(Li and Qiao [43]). Due to accounting for transverse shear strains, theshear deformation theory yield improved global response over the

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

85

classical laminate theory. The distortion of the deformed normal of alaminated shell is dependent not only the shell thickness, but also onthe fiber orientation of individual layers. This interaction results inenhanced midplane stretching, which leads to nonlinear terms in thegoverning equations of equilibrium. Thus, a more accurate prediction ofdisplacements and stresses requires the solution of the laminated shellequations that accounts for large deflections and transverse shear/normal deformation. The main point to take away here is that thebuckling behavior for a moderately thick anisotropic laminated cy-lindrical shell usually occurs well before the allowable normal stress ofthe material is reached. Especially under the action of compositeloading, postbuckling and boundary layer characteristics of anisotropiccylindrical shells will become more complex. In this paper, based onReddy's higher order shear deformation shell theory with von Kármán-Donnell-type of kinematic nonlinearity and including the extension-twist, extension-flexural and flexural-twist couplings, the governingequations of equilibrium are obtained by using the Hamilton’s principletype. The nonlinear prebuckling deformations and initial geometricimperfections of the shell are both taken into account. For simplicity,we take as the similar form of initial geometric imperfection with theinitial buckling mode of the shell. A two-step singular perturbationtechnique is utilized to investigate the postbuckling behavior of mod-erately thick anisotropic laminated composite cylindrical shells. Themathematical and physical descriptions of bending moment and de-flection in boundary layer for shell buckling are given in detail. Thenumerical illustrations show the full nonlinear postbuckling response ofmoderately thick anisotropic laminated composite cylindrical shellsunder lateral pressure, hydrostatic pressure combined with axial com-pression.

2. Theoretical development

Assume that a circular cylindrical shell with mean radius R, length Land thickness h, which consists of N plies of any kind. Introducing acoordinate system (X, Y, Z), X and Y axis are in the axial and cir-cumferential directions of the shell and Z is in the direction of the in-ward normal to the middle surface, as shown in Fig. 1. The corre-sponding displacement are designated U , V and W . ΨX and ΨY are therotations of normal to the mid surface with respect to the Y- and X-axes,respectively. The origin of the coordinate system is located at the end ofthe shell. The displacement components are assumed to be of the fol-lowing form

= + + +

= + + + +

=( )

U U X Y Z X Y Z ξ X Y Z ζ X Y

U V X Y Z X Y Z ξ X Y Z ζ X Y

U W X Y

( , ) Ψ ( , ) ( , ) ( , )

1 ( , ) Ψ ( , ) ( , ) ( , )

( , )

X X XZR Y Y Y

12 3

22 3

3 (1)

where U1, U2 and U3 are the displacements along the (X, Y, Z) co-ordinates, U , V and W are the displacements of a point on the midsurface and are the rotations at Z=0 of normals to the midsurface withrespect to (X, Y) and (U , V ), (ζX , ζY ) can be obtained by applying thestress-free conditions on the top and bottom of the shell. Finally, thereduced form of the displacement field can be demonstrated as:

= + − +

= + + − +

=

∂∂

∂∂

( )( )( )

U U X Y Z X Y c Z

U V X Y Z X Y c Z

U W X Y

( , ) Ψ ( , ) Ψ

1 ( , ) Ψ ( , ) Ψ

( , )

X XWX

ZR Y Y

WY

1 13

2 13

3 (2)

where =c h4/312. For a cylindrical shell, we can obtain the strain of the

shell associated with the displacement field given in Eq. (2) are

⎧⎨⎩

⎫⎬⎭

=⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪+

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪+

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

εεγ

εεγ

Zεεγ

Zεεγ

XY

XY

X

Y

XY

X

Y

XY

X

Y

XY

(0)

(0)

(0)

(1)

(1)

(1)

3

(3)

(3)

(3)(3a)

=⎧⎨⎩

⎫⎬⎭

+⎧⎨⎩

⎫⎬⎭

{ }γγ

γ

γZ

γ

γYZ

ZX

YZ

ZX

YZ

ZX

(0)

(0)2

(2)

(2)(3b)

where

=⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

⎧

⎨

⎪⎪

⎩⎪⎪

+

+ +

+ +

⎫

⎬

⎪⎪

⎭⎪⎪

=⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪=

⎧

⎨

⎪

⎩⎪ +

⎫

⎬

⎪

⎭⎪

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

( )( )ε ε

εεγ

εεγ

{ } ,{ }X

Y

XY

UX

WX

VY

WR

WY

UY

VX

WX

WY

X

Y

XY

X

Y

Y X

(0)

(0)

(0)

(0)

12

2

12

2 (1)

(1)

(1)

(1)

Ψ

Ψ

Ψ Ψ

X

Y

X Y

(4a)

=⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪= −

⎧

⎨

⎪⎪

⎩

⎪⎪

+

+

+ +

⎫

⎬

⎪⎪

⎭

⎪⎪

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂ ∂

( )( )( )

εεεγ

c{ }

2

X

Y

XY

XW

X

YW

Y

Y XW

X Y

(3)

(3)

(3)

(3)1

Ψ

Ψ

Ψ Ψ

X

Y

X Y

22

22

2

(4b)

=⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

+

+

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

= −⎧

⎨⎪

⎩⎪

+

+

⎫

⎬⎪

⎭⎪

∂∂∂∂

∂∂

∂∂

( )( )

γ γγ

γ

γ

γc{ }

Ψ

Ψ,{ } 3

Ψ

ΨYZ

ZX

YWY

XWX

YZ

ZX

YWY

XWX

(0)(0)

(0)(2)

(2)

(2) 1

(4c)

Based on a moderately thick shell theory, we introduce the certainassumptions (Reddy [44] and Qatu [45])

(1) The transverse normal is inextensible (i.e. ɛZ=0).(2) There is no reason for straightness and normality of a transverse

normal during deformation.(3) The transverse normal stress is negligible so that the plane stress

assumption can be invoked.

The plane stress constitutive equation may be written in the form

⎧⎨⎩

⎫⎬⎭

=⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

⎧⎨⎩

⎫⎬⎭

σστ

Q Q QQ Q QQ Q Q

εεγ

XYXY

XY

XY

11 12 16

21 22 26

61 62 66 (5a)

= ⎡⎣⎢

⎤⎦⎥{ }{ }τ

τQ QQ Q

γγ

YZZX

YZ

ZX

44 45

54 55 (5b)

Now, applying Hamilton principle and collecting the coefficient ofδU , δV , δW , δΨX and δΨY , the displacement field the governingFig. 1. Configuration of an anisotropic laminated cylindrical shell.

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

86

equations of the higher order shear deformation theory of shells can beobtained as

∂∂

+ ∂∂

=NX

NY

0X XY(6)

∂∂

+ ∂∂

=NX

NY

0XY Y(7)

⎜ ⎟

∂∂

+ ∂∂

− ⎛⎝

∂∂

+ ∂∂

⎞⎠

+ ⎡⎣⎢

∂∂

+ ∂∂ ∂

+ ∂∂

⎤⎦⎥

+ ∂∂

⎡⎣⎢

∂∂

+ ∂∂

⎤⎦⎥

+ ∂∂

⎡⎣⎢

∂∂

+ ∂∂

⎤⎦⎥

− +

=

QX

QY

c RX

RY

PX

PX Y

PY

XN W

XN W

Y YN W

XN W

YNR

q

3 2

0

X Y X Y X XY Y

X XY XY YY

12

2

2 2

2

(8)

∂∂

+ ∂∂

− + − ⎡⎣⎢

∂∂

+ ∂∂

⎤⎦⎥

=MX

MY

Q c R c PX

PY

3 0X XYX X

X XY1 1

(9)

∂∂

+ ∂∂

− + − ⎡⎣⎢

∂∂

+ ∂∂

⎤⎦⎥

=MX

MY

Q c R c PX

PY

3 0XY YY Y

XY Y1 1

(10)

where Ni and Qi are membrane and transverse shear forces, Mi isbending moment per unit length, and Pi and Ri are higher order bendingmoment and shear force, respectively (Reddy [44]).

Denoting the initial deflection by W ∗(X, Y), consider the additionaldeflection W (X, Y) and the stress function F (X, Y), we defined theforce components =N F ,X YY , =N F ,Y XX and = −N F ,XY XY , and thecomma denotes partial differentiation with respect to the correspondingcoordinates.

Reddy and Liu [46] derived the Navier-type exact solution con-sidering higher order shear deformation theory (HSDT) with assump-tion of zero tangential stress on the boundary surfaces. Combining withvon Kármán-Donnell-type kinematic relations, the governing differ-ential equations can be expressed in terms of a stress function F , tworotations ΨX and ΨY , and a transverse displacement W , along with theinitial geometric imperfection . Taking the compatibility equation intoaccount, that is

⎜ ⎟ ⎜ ⎟

∂∂

+ ∂∂

−∂∂ ∂

= ⎛⎝

∂∂ ∂

⎞⎠

− ∂∂

∂∂

+ ⎛⎝

∂∂ ∂

⎞⎠

− ∂∂

∂∂

− ∂∂

∂∂

− ∂∂ ∂

∗ ∗

∗

εY

εX

γX Y

WX Y

WX

WY

WX Y

WX

WY

WY

WX R

WX Y

2

1

X Y XY2

2

2

2

2 2 2 2

2

2

2

2 2 2

2

2

2

2

2

2

2

2

(11)

We have

− − + − = + +∼ ∼ ∼ ∼ ∼ ∗L W L L L FR

F L W W F q( ) (Ψ ) (Ψ ) ( ) 1 , ( , )X Y XX11 12 13 14

(12)

+ + − + = − +∼ ∼ ∼ ∼ ∼ ∗L F L L L WR

W L W W W( ) (Ψ ) (Ψ ) ( ) 1 , 12

( 2 , )X Y XX21 22 23 24

(13)

+ + + =∼ ∼ ∼ ∼L W L L L F( ) (Ψ ) (Ψ ) ( ) 0X Y31 32 33 34 (14)

+ + + =∼ ∼ ∼ ∼L W L L L F( ) (Ψ ) (Ψ ) ( ) 0X Y41 42 43 44 (15)

where terms of ∼L ( )ij (i, j=1, 2, 3,4) in Eqs. (12)–(15) are linear op-erators and are defined by

= ⎡⎣

+ + + + ⎤⎦

= ⎡⎣ − ⎤⎦ + ⎡⎣ + − + ⎤⎦= ⎡⎣ + − + ⎤⎦ + ⎡⎣ − ⎤⎦= + + − +

= + + +

= ⎡⎣ − ⎤⎦ + ⎡⎣ − − − ⎤⎦= ⎡⎣ − − − ⎤⎦ + ⎡⎣ − ⎤⎦

= ⎡⎣

+ + − + ⎤⎦

= ⎡⎣ − + ⎤⎦

+ ⎡⎣

− + + − + ⎤⎦

= ⎡⎣ − + ⎤⎦−⎡⎣ − + ⎤⎦

−⎡⎣ − + ⎤⎦= ⎡⎣ + − + + + + ⎤⎦=

= ⎡⎣ − + ⎤⎦

+ ⎡⎣

+ − +

+ − ⎤⎦

=

= ⎡⎣ − + ⎤⎦−⎡⎣ − + ⎤⎦

−⎡⎣ − + ⎤⎦=

= − +

∼

∼

∼

∼

∼

∼

∼

∼

∼

∼

∼

∼ ∼

∼

∼ ∼

∼

∼ ∼

∼

∗ ∂∂

∗ ∗ ∗ ∂∂ ∂

∗ ∂∂

∗ ∗ ∂∂

∗ ∗ ∗ ∗ ∂∂ ∂

∗ ∗ ∗ ∗ ∂∂ ∂

∗ ∗ ∂∂

∗ ∂∂

∗ ∗ ∗ ∂∂ ∂

∗ ∂∂

∗ ∂∂

∗ ∗ ∂∂ ∂

∗ ∂∂

∗ ∗ ∂∂

∗ ∗ ∗ ∗ ∂∂ ∂

∗ ∗ ∗ ∗ ∂∂ ∂

∗ ∗ ∂∂

∗ ∂∂

∗ ∗ ∗ ∂∂ ∂

∗ ∂∂

∂∂

∗ ∗ ∂∂

∗ ∗ ∗ ∗ ∂∂ ∂

∗ ∗ ∗ ∂∂

∗ ∗ ∗ ∂∂

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∂∂ ∂

∂∂

∗ ∗ ∗ ∗ ∂∂ ∂

∗ ∗ ∂∂

∗ ∗ ∗ ∂∂

∗ ∗ ∗ ∂∂

∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

( )( )

L F F F F F

L D F D D F F

L D D F F D F

L B B B B B

L A A A A

L B E B B E E

L B B E E B E

L E E E E E

L A D F

F H F F H H

L A D F D F H

D F H

L D D F F F H H

L L

L A D F

F F H H

F H

L L

L A D F D F H

D F H

L L

L

( ) ( 4 )

( ) ( 2 ) ( 2 )

( ) ( 2 ) ( 2 )

( ) ( 2 )

( ) (2 )

( ) ( ) ( )

( ) ( ) ( )

( ) ( 2 )

( )

( ) (( 2 ) ( 2 ))

( )

( ) ( ) ( 2 ) ( )

( ) ( )

( )

( 2 ) ( 2 )

( ) ( )

( )

( ) ( )

( ) 2

h X X Y Y

h X h X Y

h X Y h Y

X X Y Y

X X Y Y

h X h X Y

h X Y h Y

h X X Y Y

h h X

h h X h X Y

h h h h X

h h Y

h h X Y

h h Y

h h X Y

h Y

h h h h X

h h Y

X Y X Y X Y Y X

114

3 11 12 21 66 22

12 114

3 11 12 664

3 12 66

13 12 664

3 21 66 224

3 22

14 21 11 22 66 12

21 22 12 66 11

22 214

3 21 11 664

3 11 66

23 22 664

3 22 66 124

3 12

244

3 21 11 22 66 12

31 558

5516

55

43 11

43 11 21 66

43 12 66

32 558

5516

55 118

3 1116

9 11

668

3 6616

9 66

33 12 664

3 12 21 6616

9 12 66

34 22

41 448

4416

44

43 12 66

43 12 66

224

3 22

42 33

43 448

4416

44 668

3 6616

9 66

228

3 2216

9 22

44 23

244

42 2

44

233 2

32

23

2 233

44

42 2

44

44

42 2

44

233 2

32

23

2 233

244

42 2

44

2 4

2 233 2

32

2 4 2 422

2 422

2 42

2 4

2 23

2

233

2 4 2 422

2 422

22

22

2 2 22

22

(16)

Note that the geometric nonlinearity in the von Kármán sense isgiven in terms of ∼L () in Eqs. (12) and (13).

Eqs. (12)–(15) are remarkable not only for the coupling betweentransverse bending and in-plane stretching involved in terms of ∗Bij (i,j=1, 2, 6), but also for the flexural-twist and extension-twist couplingindicated by ∗D16,

∗D26,∗A16 and

∗A26 when the fiber angles exist that do notlie parallel to the cylindrical axis or in a circumferential plane forgeneral anisotropic laying-up of laminated cylindrical shells.

The two end edges of the shell are assumed to be simply supportedor clamped, so that the boundary conditions areX=0, L:

= = = =W M PΨ 0, 0 (simply supported)Y X X (17a)

= = =W Ψ Ψ 0 (clamped)X Y (17b)

∫ + + =N dY πRhσ πR qa2 0πR

X X0

2 21 (17c)

where a1= 0 and a1= 1 for uniform lateral and hydrostatic pressureloading case, respectively, as defined in Reddy and Liu [46]. Also wehave the closed (or periodicity) condition

∫ ∂∂

=VY

dY 0πR

0

2

(18a)

or

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

87

∫ ⎜ ⎟

⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎡⎣⎢

⎛⎝

∂∂

+ ∂∂

− ∂∂ ∂

⎞⎠

+ ⎛⎝

− ⎞⎠

∂∂

+ ⎛⎝

− ⎞⎠

∂∂

+ ⎛⎝

− ⎞⎠

⎛⎝

∂∂

+ ∂∂

⎞⎠

− ⎛⎝

∂∂

+ ∂∂

+ ∂∂ ∂

⎞⎠

+ − ⎛⎝

∂∂

⎞⎠

− ∂∂

∂∂

⎤⎦⎥ =

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗

∗

A FX

A FY

A FX Y

Bh

EX

Bh

EY

Bh

EY X

hE W

XE W

YE W

X YWR

WY

WY

WY

dY

43

Ψ

43

Ψ 43

Ψ Ψ

43

2 12

0

πR X

Y X Y

0

222

2

2 12

2

2 26

2

21 2 21

22 2 22 26 2 26

2 21

2

2 22

2

2 26

2 2

(18b)

Because of Eq. (18), the in-plane boundary condition V =0 (atX=0, L) is not needed in Eq. (17).

The average end-shortening relationship is in Li and Shen [40]

∫ ∫

∫ ∫

= −

= − ⎡⎣

+ − + −

+ − + − +

− + + − − ⎤⎦

∂∂

∗ ∂∂

∗ ∂∂

∗ ∂∂ ∂

∗ ∗ ∂∂

∗ ∗ ∂∂

∗ ∗ ∂∂

∂∂

∗ ∂∂

∗ ∂∂

∗ ∂∂ ∂

∂∂

∂∂

∂∂

∗

( )

( )( )

( )

( )( ) ( )

dXdY

A A A B E

B E B E

E E E dXdY2

L πRLπR L U

X

πRLπR L F

YF

XF

X Y h X

h Y h Y X

hW

XW

YW

X YWX

WX

WX

Δ 12 0

20

12 0

20 11 12 16 11

43 11

Ψ

124

3 12Ψ

164

3 16Ψ Ψ

43 11 12 16

12

2

X

X

Y X Y

22

22

22

2 2

22

22

22

(18c)

3. Analytical method and asymptotic solutions

Having developed the theory, we now try to solve Eqs. (12)–(15)with boundary conditions (17)-(18). Before proceeding, it is convenientfirst to define the following dimensionless quantities [in which coeffi-cients γijk in Eqs. (24), (26) and (27) below are defined in Appendix B]

= = = = == =

== = = −

==

= == − = −

=== −

= −

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗

∗

x πX L y Y R β L πR Z L Rh ε π R L D D A AW W ε W W D D A A F ε F D D

ε L π D D A Aγ D D γ A A γ A Aγ γ A A AM P ε M P h L π D D D A A

λ σ Rh D D A A δ L R D D A Aλ σ R h ν ν E E δ L R h ν ν

λ q LR A A π D Dδ L LR π D D A Aλ q LR ν ν πh E E

δ L LR ν ν πh

/ , / , / , / , ( / )[ ]( , ) ( , )/[ ] , /[ ] ,(Ψ ,Ψ ) (Ψ ,Ψ )( / )/[ ] ,

[ / ] , [ / ] , / ,( , ) ( , )/ ,( , ) ( , 4 /3 ) / [ ] ,

/(2/ )[ / ] , (Δ / )/(2/ )[ ] ,( / )[3(1 )/ ] , (Δ / )( / )[3(1 )] ,

(3) [ ] /4 [ ] ,(Δ / )(3) /4 [ ] ,

(3) (1 ) / [2 ] ,

(Δ / )(3) (1 ) / (2)

x y X Y

x x X X

p x p x

p x p x

q

q x

q

q x

2 2 211 22 11 22

1/4

11 22 11 221/4 2

11 221/2

211 22 11 22

1/4

14 22 111/2

24 11 221/2

5 12 22

211 213 26 16 222 2 2 2

11 11 22 11 221/4

11 22 11 221/4

11 22 11 221/4

12 21 11 221/2

12 211/2

3/4 3/211 22

1/811 22

3/8

3/4 1/211 22 11 22

3/8

3/2 3/212 21

3/4 5/211 22

1/2

3/2 1/212 21

3/4 3/2 1/2

(19)

The nonlinear Eqs. (12)–(15) may then be written in dimensionlessform as

− − + − = +

+

∗ε L W εL εL εγ L F γ F γ β L W W F

γ λ ε

( ) (Ψ ) (Ψ ) ( ) , ( , )43

(3)

x y xx

q

211 12 13 14 14 14 14

2

141/4 3/2

(20)

+ + − +

= − + ∗

L F γ L γ L εγ L W γ W

γ β L W W W

( ) (Ψ ) (Ψ ) ( ) ,12

( 2 , )

x y xx21 24 22 24 23 24 24 24

242

(21)

+ − + =εL W L L γ L F( ) (Ψ ) (Ψ ) ( ) 0x y31 32 33 14 34 (22)

− + + =εL W L L γ L F( ) (Ψ ) (Ψ ) ( ) 0x y41 42 43 14 44 (23)

where

= + + + +

= + + +

= + + +

= + + + +

= − + − +

= + + +

= + + +

= + + + +

= + + + + +

= − − −

= − − −

=

= + + + + +

=

= − − −

=

= − +

∂∂

∂∂ ∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

∂∂ ∂

∂∂

∂∂

∂∂ ∂

∂∂

∂∂

∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

∂∂ ∂

∂∂

∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

L γ γ β γ β γ β γ β

L γ γ β γ β γ β

L γ γ β γ β γ β

L γ γ β γ β γ β γ β

L γ β γ β γ β γ β

L γ γ β γ β γ β

L γ γ β γ β γ β

L γ γ β γ β γ β γ β

L γ γ β γ γ β γ β γ β

L γ γ γ β γ β

L γ γ γ β γ β

L L

L γ γ β γ γ β γ β γ β

L L

L γ γ γ β γ β

L L

L

( ) 2

( )

( )

( )

( ) 2 2 2

( )

( )

( )

( )

( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( ) 2

x x y x y x y y

x x y x y y

x x y x y y

x x y x y x y y

x x y x y x y y

x x y x y y

x x y x y y

x x y x y x y y

x y x x y x y y

x x y y

x x y y

x y x x y x y y

x x y y

x y x y x y y x

11 110 111 1122

1133

1144

12 120 121 1222

1233

13 130 131 1322

1333

14 140 141 1422

1433

1444

21 211 2122

2133

2144

22 220 221 2222

2233

23 230 231 2322

2333

24 240 241 2422

2433

2444

31 31 32 310 311 3122

3133

32 31 320 321 3222

33 32 330 331 3322

34 22

41 32 41 410 411 4122

4133

42 33

43 41 430 431 4322

44 23

44

43

42 2

43

44

33

32

32

33

33

32

32

33

44

43

42 2

43

44

44

43

42 2

43

44

33

32

32

33

33

32

32

33

44

43

42 2

43

44

33

32

32

33

22

2 22

22

2 22

33

32

32

33

22

2 22

22

22

2 2 22

22

(24a)

= + += + += − + − += − + −= − − += + − + −

= + −= − −== += − − − −= − − − −= − − − −= − − − −

= + −= − −= − + − += + − −= − + − +

= − + += − + += + − + + + += − + +

= − + −= + − + −= − + − +

= − + += − −=

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

γ γ γ h F F F F F Dγ γ h F F F F Dγ γ D F h D D F F h Dγ γ D F F h D F h Dγ γ D F h D F F h Dγ γ D D F F h D F h Dγ γ γ B B B B B D D A Aγ γ B B B B D D A Aγ γ A A Aγ γ A A A Aγ γ B E h B B E E h D D A Aγ γ B B E E h B E h D D A Aγ γ B E h B B E E h D D A Aγ γ B B E E h B E h D D A Aγ γ γ h E E E E E D D A Aγ γ h E E E E D D A Aγ γ h F H h F F H H h Dγ γ h F F H h F H h Dγ γ D F h H h D F h H h D

γ D F F h H h Dγ D F F h H h Dγ D D F F F h H H h Dγ D F F h H h Dγ γ h F H h F F H h Dγ γ h F F H H h F H h Dγ γ D F h H h D F h H h D

γ D F F h H h Dγ γ B E h B E h D D A Aγ γ h E E D D A A

( , , ) (4/3 )[ , ( 4 )/2, ]/( , ) (4/3 )[2( ), 2( )]/( , ) [ 4 /3 , ( 2 ) 4( 2 )/3 ]/( , ) [3 4( 2 )/3 , 4 /3 ]/( , ) [ 4 /3 , 3 4( 2 )/3 ]/( , ) [( 2 ) 4( 2 )/3 , 4 /3 ]/( , , ) [ , ( 2 ), ]/[ ]( , ) (2 , 2 )/[ ]( , ) ( , )/( , ) ( /2, )/( , ) [ 4 /3 , ( ) 4( )/3 ]/[ ]( , ) [( ) 4( )/3 , 4 /3 )/[ ]( , ) [ 4 /3 , ( ) 4( )/3 ]/[ ]( , ) [( ) 4( )/3 , 4 /3 ]/[ ]( , , ) (4/3 )[ , ( 2 ), ]/[ ]( , ) (4/3 )(2 , 2 )/[ ]( , ) (4/3 )[ 4 /3 , ( 2 ) 4( 2 )/3 ]/( , ) (4/3 )[( 2 ) 4 / , 4 /3 ]/( , ) ( 8 /3 16 /9 , 8 /3 16 /9 )/

2[ 4( )/3 16 /9 ]/[ 4( )/3 16 /9 ]/[( ) 4( 2 )/3 16( )/9 ]/[ 4( )/3 16 /9 ]/

( , ) (4/3 )[ 4 /3 , ( 2 ) 4 / ]/( , ) (4/3 )[( 2 ) 4( 2 )/3 , 4 /3 ]/( , ) ( 8 /3 16 /9 , 8 /3 16 /9 )/

2[ 4( )/3 16 /9 ]/( , ) ( 4 /3 , 4 /3 )/[ ]( , ) (4/3 )( , )/[ ]

110 112 1142

11 12 21 66 22 11

111 1132

16 61 26 62 11

120 122 11 112

12 66 12 662

11

121 123 16 61 162

26 262

11

130 132 16 162

26 62 262

11

131 133 12 66 21 662

22 222

11

140 142 144 21 11 22 66 12 11 22 11 221/4

141 143 26 61 16 62 11 22 11 221/4

211 213 26 16 22

212 214 12 66 11 22

220 222 21 212

11 66 11 662

11 22 11 221/4

221 223 26 61 26 612

16 162

11 22 11 221/4

230 232 26 262

16 62 16 622

11 22 11 221/4

231 233 22 66 22 662

12 122

11 22 11 221/4

240 242 2442

21 11 22 66 12 11 22 11 221/4

241 2432

26 61 16 62 11 22 11 221/4

310 3122

11 112

21 66 12 662

11

311 3132

16 61 162

26 262

11

320 322 11 112

114

66 662

664

11

321 16 16 612

164

11

330 16 16 612

164

11

331 12 66 12 21 662

12 664

11

332 26 26 622

264

11

410 4122

16 162

26 62 262

11

411 4132

12 66 12 662

22 222

11

430 432 66 662

664

22 222

224

11

431 26 26 622

264

11

511 522 11 112

22 222

11 22 11 221/4

611 6222

11 22 11 22 11 221/4

(24b)

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

88

Because of the definition of ε given in Eq. (19), for most of thecomposite materials 0.22 h < ∗ ∗ ∗ ∗D D A A[ ]11 22 11 22

1/4 < 0.35 h, hence whenZ =(L2/Rh) > 3.46, we have ɛ < 1. We may see more details in Liand Qiao [43]. In fact, the shell structure exists ⩾Z 10 , so that we have

≪ε 1. While for ɛ < 1, Eqs. (20)–(23) are the equations of theboundary layer type, from which nonlinear prebuckling deformations,large deflections in the postbuckling range, and initial geometric im-perfections of the shell can be taken into account simultaneously.

The boundary conditions of Eq. (17) becomex=0, π :

= = = =W M PΨ 0, 0 (simply supported)y x x (25a)

= = =W Ψ Ψ 0 (clamped)x y (25b)

∫ ∂∂

+ + =π

β Fy

dy λ ε λ ε a12

2 23

(3) 0π

p q0

2 22

21/4 3/2

(25c)

and the closed condition of Eq. (18) becomes

∫ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎡

⎣⎢

⎛⎝

∂∂

− ∂∂

− ∂∂ ∂

⎞⎠

+ ⎛⎝

∂∂

+∂∂

⎞⎠

+ ⎛⎝

∂∂

+∂∂

⎞⎠

− ⎛⎝

∂∂

+ ∂∂

+ ∂∂ ∂

⎞⎠

+ − ⎛⎝

∂∂

⎞⎠

− ∂∂

∂∂

⎤

⎦⎥ =

∗

Fx

γ β Fy

γ β Fx y

γ γx

γ βy

γ γ βy x

εγ γ Wx

γ β Wy

γ β Wx y

γ W γ β Wy

γ β Wy

Wy

dy

Ψ Ψ

Ψ Ψ2

12

0

π x y

x y

0

2 2

2 52

2

2 211

2

24 220 522

24 230 24 240

2

2 6222

2

2 526

2

24 242

2

242

(26)

It has been shown (Li and Shen [40]; Li and Lin [39]; Li and Qiao[43]) that the effect of the boundary layer on the solution of a pres-surized shell is of the order ε3/2 and that of a shell in compression is ofthe order ε1, we may obtain the dimensionless unit end-shortening re-lationship

∫ ∫ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟

= − ⎡⎣⎢

⎛⎝

∂∂

− ∂∂

− ∂∂ ∂

⎞⎠

+ ⎛⎝

∂∂

+∂∂

⎞⎠

+ ⎛⎝

∂∂

+∂∂

⎞⎠

− ⎛⎝

∂∂

+ ∂∂

+ ∂∂ ∂

⎞⎠

− ⎛⎝

∂∂

⎞⎠

− ∂∂

∂∂

⎤⎦⎥

−

∗

δπ γ

ε γ β Fy

γ Fx

γ β Fx y

γ γx

γ βy

γ γ βy x

εγ γ Wx

γ β Wy

γ β Wx y

γ Wx

γ Wx

Wx

dxdy

(3)8

Ψ Ψ Ψ Ψ

2 12

qπ π

x y x y

3/4

224

3/20

2

0 242 2

2

2 5

2

2 213

2

24 511 233 24 223

24 611

2

2 2442

2

2 516

2

24

2

24(27a)

∫ ∫

⎜ ⎟ ⎜ ⎟

⎜ ⎟

= − ⎡⎣⎢

∂∂

− ∂∂

− ∂∂ ∂

+ ⎛⎝

∂∂

+∂∂

⎞⎠

+ ⎛⎝

∂∂

+∂∂

⎞⎠

− ⎛⎝

∂∂

+ ∂∂

+ ∂∂ ∂

⎞⎠

− ⎛⎝

∂∂

⎞⎠

− ∂∂

∂∂

⎤⎦⎥

−

∗

δπ γ

ε γ β Fy

γ Fx

γ β Fx y

γ γx

γ βy

γ γ βy x

εγ γ Wx

γ β Wy

γ β Wx y

γ Wx

γ Wx

Wx

dxdy

14

Ψ Ψ Ψ Ψ

2 12

pπ π

x y x y

224

10

2

0 242 2

2

2 5

2

2 213

2

24 511 233 24 223

24 611

2

2 2442

2

2 516

2

24

2

24(27b)

Employing Eqs. (20)–(27), a two-step perturbation method is uti-lized to investigate the postbuckling behavior of perfect and imperfect,shear deformable anisotropic laminated cylindrical shells subjected tocombined loading. The essence of this procedure, in the present case,the solution is to assume that

= + += + +

= + +

= + +

∼

∼

∼

∼W w x y ε W x ξ y ε W x ς y εF f x y ε F x ξ y ε F x ς y ε

ψ x y ε x ξ y ε x ς y ε

ψ x y ε x ξ y ε x ς y ε

( , , ) ( , , , ) ( , , , )( , , ) ( , , , ) ( , , , )

Ψ ( , , ) Ψ ( , , , ) Ψ ( , , , )

Ψ ( , , ) Ψ ( , , , ) Ψ ( , , , )x x x x

y y y y (28)

where ε is a small perturbation parameter [see beneath Eq. (19)] andw x y ε( , , ), f x y ε( , , ) ψ x y ε( , , )x , ψ x y ε( , , )y are called outer solutions or regular

solutions of the shell, ∼W x ξ y ε( , , , ), ∼F x ξ y ε( , , , ), ∼ x ξ y εΨ ( , , , )x , ∼ x ξ y εΨ ( , , , )y and

W x ς y ε( , , , ), F x ς y ε( , , , ), x ς y εΨ ( , , , )x , x ς y εΨ ( , , , )y are the boundary layer so-lutions near the x=0 and x=edges, respectively, and ξ and ζ are theboundary layer variables, defined as

= = −ξ x ε ς π x ε/ , ( )/ (29)

(We may confirm that the width of the boundary layers is of theorder Rh for isotropic cylindrical shell.) In Eq. (28) the regular andboundary layer solutions are taken in the form of perturbation expan-sions as

∑ ∑

∑ ∑

= =

= == =

= =

w x y ε ε w x y f x y ε ε f x y

ψ x y ε ε ψ x y ψ x y ε ε ψ x y

( , , ) ( , ), ( , , ) ( , )

( , , ) ( ) ( , ), ( , , ) ( ) ( , )j

jj

j

jj

xj

jx j y

j

jy j

1

/2/2

0

/2/2

1

/2/2

1

/2/2

(30a)

∑ ∑

∑

∑

= =

=

=

∼ ∼

∼ ∼

∼ ∼

∼ ∼

=

++

=

++

=

++

=

++

W x ξ y ε ε W x ξ y F x ξ y ε ε F x ξ y

x ξ y ε ε x ξ y

x ξ y ε ε x ξ y

( , , , ) ( , , ), ( , , , ) ( , , )

Ψ ( , , , ) (Ψ ) ( , , ),

Ψ ( , , , ) (Ψ ) ( , , )

j

jj

j

jj

xj

jx j

yj

jy j

0

/2 1/2 1

0

/2 2/2 2

0

( 3)/2( 3)/2

0

/2 2/2 2

(30b)

∑ ∑

∑

∑

= =

=

=

=

++

=

++

=

++

=

++

W x ς y ε ε W x ς y F x ς y ε ε F x ς y

x ς y ε ε x ς y

x ς y ε ε x ς y

( , , , ) ( , , ), ( , , , ) ( , , )

Ψ ( , , , ) (Ψ ) ( , , ),

Ψ ( , , , ) (Ψ ) ( , , )

j

jj

j

jj

xj

jx j

yj

jy j

0

/2 1/2 1

0

/2 2/2 2

0

( 3)/2( 3)/2

0

/2 2/2 2

(30c)

The nondimensional pressure may be expressed by

+ = = + + + +λ ε a b λ λ ελ ε λ ε λ43

(3) ( ) ···q y1/4 3/2

1 0 12

23

3 (31)

Substituting Eqs. (28)–(30) into Eqs. (20)–(23), and by identifica-tion of the same order of ε, three sets of perturbation equations areobtained for the regular and boundary layer solutions, respectively. Inpresent case, the order ε of the boundary layer solution for a pressurizedshell is 3/2, and that for a shell in compression is one. Thus, we con-sider the solutions for two kinds of loading conditions.

Case (1) high values of external pressure combined with relativelylow axial load. Let

=PπR q

b02 1

(32a)

or

=λ ε

λ εb2

(3) 2p

q43

1/4 3/21

(32b)

in this case, the boundary condition of Eq. (25c) becomes

∫ ∂∂

+ + =π

β Fy

dy λ ε a b12

23

(3) ( ) 0π

q0

2 22

21/4 3/2

1(33)

Introducing a new variable a1 to replace (a+ b1) in Eqs. (33)–(38b)in the following, the solutions of perturbation equations of each orderare obtained by employing Eqs. (28) and (30), respectively.

For the regular solutions, 0th order equations can be written byO(ɛ0)

− = +γ f γ β L w f γ λ( ) ( , )xx14 0 , 142

0 0 14 0 (34)

+ + + = −L f γ L ψ γ L ψ γ w γ β L w w( ) ( ) ( ) ( ) 12

( , )x y xx21 0 24 22 0 24 23 0 24 0 , 242

0 0

(35)

+ + =L ψ L ψ γ L f( ) ( ) ( ) 0x y32 0 33 0 14 34 0 (36)

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

89

+ + =L ψ L ψ γ L f( ) ( ) ( ) 0x y42 0 43 0 14 44 0 (37)

The solutions of Eqs. (34)–(37) are

= = =w ψ ψ 0x y0 0 0 (38a)

= − ⎛⎝

+ ⎞⎠

−f B β x a y C xy12

120 00

(0) 2 21

200(0)

(38b)

Substituting Eq. (38) into Eq. (34) yields

=λ β B02

00(0) (39)

Eq. (38b) means there exists a circumferential stress along with anassociate shear stress when the anisotropic laminated cylindrical shell issubjected to lateral pressure.

For the regular solutions, 1st order equations can be written byO(ɛ1):

− = + +γ L f γ f γ β L w f L w f γ λ( ) ( ) [ ( , ) ( , )]xx14 14 0 14 1 , 142

1 0 0 1 14 1 (40)

+ + − +

= −

L f γ L ψ γ L ψ γ L w γ w

γ β L w w

( ) ( ) ( ) ( ) ( )12

( , )

x y xx21 1 24 22 1 24 23 1 24 24 0 24 1 ,

242

0 1 (41)

+ + + =L w L ψ L ψ γ L f( ) ( ) ( ) ( ) 0x y31 0 32 1 33 1 14 34 1 (42)

+ + + =L w L ψ L ψ γ L f( ) ( ) ( ) ( ) 0x y41 0 42 1 43 1 14 44 1 (43)

The solutions of Eqs. (40)–(43) are

= = =w ψ ψ 0x y1 1 1 (44a)

= − + −f x y B β x a y C xy( , ) 12

( 12

)1 00(1) 2 2

12

00(1)

(44b)

Substituting Eq. (44) into Eq. (40) yields

=λ β B12

00(1) (45)

Next, we solved the 1.5th-order equationO(ɛ3/2):

− = +γ f γ β L w f L w f( ) [ ( , ) ( , )]xx14 3/2 , 142

3/2 0 0 3/2 (46)

+ + + =L f γ L ψ γ L ψ γ w( ) ( ) ( ) ( ) 0x y xx21 3/2 24 22 3/2 24 23 3/2 24 3/2 , (47)

+ + =L ψ L ψ γ L f( ) ( ) ( ) 0x y32 3/2 33 3/2 14 34 3/2 (48)

+ + =L ψ L ψ γ L f( ) ( ) ( ) 0x y42 3/2 43 3/2 14 44 3/2 (49)

The solutions of Eqs. (40)–(43) are

= = = =w A ψ ψ f, 0, 0x y3/2 00(3/2)

3/2 3/2 3/2 (50)

Second-order equations can be written byO(ɛ2):

− = + +∗γ L f γ f γ β L w W f γ λ( ) ( ) ( , )xx14 14 1 14 2 , 142

2 0 14 2 (51)

+ + + =L f γ L ψ γ L ψ γ w( ) ( ) ( ) ( ) 0x y xx21 2 24 22 2 24 23 2 24 2 , (52)

+ + =L ψ L ψ γ L f( ) ( ) ( ) 0x y32 2 33 2 14 34 2 (53)

+ + =L ψ L ψ γ L f( ) ( ) ( ) 0x y42 2 43 2 14 44 2 (54)

In the case of external pressure loaded shell, the regular solutions donot need to satisfy boundary conditions. The initial geometric im-perfection is defined by Eq. (56). The solutions of Eqs. (51)–(54) may beexpressed by

= + +w x y A A mx ny a mx ny( , ) sin sin cos cos2 00(2)

11(2)

11(2) (55a)

= − + − +f x y B β x a y C xy B mx ny( , ) 12

( 12

) sin sin2 00(2) 2 2

12

00(1)

11(2)

(55b)

= +ψ x y C mx ny c mx ny( , ) cos cos sin sinx 11(2)

11(2) (55c)

= +ψ x y D mx ny d mx ny( , ) sin sin cos cosy 11(2)

11(2)

(55d)

and the initial geometric imperfection has a similar form

= +∗W x y ε ε μ A mx ny a mx ny( , , ) [ sin sin cos cos ]211(2)

11(2) (56)

where μ is the imperfection parameter. The substitution of Eq. (55) intoEqs. (51)–(54) yields

= = =

= =

= + =

= −

+ −

+ −

a A B γ A C γ γ A

c γ γ A D γ γ A

d γ γ A γ β B n β a m

γ βC mnβ

, , ,

, ,

, ( ) ,

2

gg

mg

mg

mg

mg

mg

γ γ m g

μ g g

γ γ m g

μ g g

11(2)

11(2)

11(2)

24 11(2)

11(2)

14 24ΔΔ 11

(2)

11(2)

14 24ΔΔ 11

(2)11(2)

14 24ΔΔ 11

(2)

11(2)

14 24ΔΔ 11

(2)14

200(2) 2 2 1

2 12

(1 )( )

14 00(2)

(1 )( )

220

210

2

210

0100

2

210

0200

2

210

0300

2

210

0400

2

210

14 244

210

2102

2202

14 244

220

2102

2202 (57)

Similarly, by solving until the sixth-order in order to obtain effectivepostbuckling solutions. Then we focus on the boundary layer solutionsnear the x=0 edge. The 2.5th-order equations can be written by

O(ɛ5/2)

∂∂

− ∂∂

−∂∂

+∂

∂−

∂∂

=∼ ∼∼ ∼∼

γWξ

γξ

γξ

γ γFξ

γFξ

Ψ Ψ0x y

110

43/24 120

32

3 130

32

3 14 140

45/24 14

25/22 (58)

∂∂

+ ∂∂

+∂∂

−∂

∂+

∂∂

=∼ ∼ ∼ ∼ ∼Fξ

γ γξ

γ γξ

γ γWξ

γWξ

Ψ Ψ0x y4

5/24 24 220

32

3 24 230

32

3 24 240

43/24 24

23/22

(59)

∂∂

− ∂∂

−∂∂

+∂

∂=

∼∼ ∼∼γ

Wξ

γξ

γξ

γ γFξ

Ψ Ψ0x y

310

33/23 320

22

2 330

22

2 14 220

35/23 (60)

∂∂

− ∂∂

−∂∂

+∂

∂=

∼∼ ∼∼γ

Wξ

γξ

γξ

γ γFξ

Ψ Ψ0x y

410

33/23 330

22

2 430

22

2 14 230

35/23 (61)

Substituting Eqs. (59)–(61) into Eq. (58) yields

∂∂

+∂

∂+ =

∼ ∼∼W

ξc

Wξ

b W2 04

3/24

23/22

23/2

(62)

The solution of Eq. (62) can be expressed by

= − +∼ −W A a φξ a φξ e( cos sin ) ξ3/2 00

(3/2)01(3/2)

10(3/2) ϑ (63)

where c, b, ϑ and φ are given in detail in Appendix C. The regular so-lution and the boundary layer solution are now matching at x=0. Theclamped boundary conditions require that

+ = + = + =∼ ∼∼w W ψ ψΨ Ψ 0x x y y3/2 3/2 3/2 3/2 3/2 3/2 (64)

from which one has

= =a aφ

1, ϑ01(3/2)

10(3/2)

(65)

Similarly, the boundary layer solutions near the =x π edge can bedetermined by the same manner. We match the regular solutions withthe boundary layer solutions at the each end of the shell, the solutionssatisfying the clamped boundary conditions in the asymptotic sense arewritten as

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

= ⎡⎣⎢ ⎛

⎝− ⎞

⎠− ⎛

⎝⎛⎝

− ⎞⎠

+ ⎛⎝

− ⎞⎠

⎞⎠

⎛⎝

− ⎞⎠

− ⎛⎝

⎛⎝

− ⎞⎠

− + ⎛⎝

− ⎞⎠

− ⎞⎠

⎛⎝

− − ⎞⎠

⎤⎦⎥

+ + + +

+ + + + + +

W ε A a γ A a γ φ xε

a γφ

φ xε

xε

A a γ φ π xε

a γφ

φ π xε

π xε

ε A mx ny a mx ny ε A mx ny a mx ny

ε A A mx ny a mx ny A mx A ny O ε

12

12

cos 12

ϑ sin exp ϑ

12

cos 12

ϑ sin exp ϑ

[ sin sin cos cos ] [ sin sin cos cos ]

[ sin sin cos cos cos2 cos2 ] ( )

3/200(3/2) 1

5 00(3/2) 1

51

5

00(3/2) 1

51

5

211(2)

11(2) 3

11(3)

11(3)

400(4)

11(4)

11(4)

20(4)

02(4) 5

(66)

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

90

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

= − ⎛⎝

+ ⎞⎠

− + ⎡

⎣⎢− ⎛

⎝+ ⎞

⎠− ⎤

⎦⎥

+ ⎡

⎣⎢− ⎛

⎝+ ⎞

⎠− + ⎤

⎦⎥

+ ⎡⎣⎢

⎛⎝

− − − ⎞⎠

⎛⎝

− ⎞⎠

+ ⎛⎝

− − − − − ⎞⎠

⎛⎝

− − ⎞⎠

⎤⎦⎥

+ ⎡

⎣⎢− ⎛

⎝+ ⎞

⎠− ⎤

⎦⎥ + ⎡

⎣⎢− ⎛

⎝+ ⎞

⎠− +

+ ⎤

⎦⎥ +

F B β x a y b xy ε B β x a y b xy

ε B β x a y b xy B mx ny

ε A b a γ φ xε

b a γ φ xε

xε

A b a γ φ π xε

b a γ φ π xε

π xε

ε B β x a y b xy ε B β x a y b xy B mx

B ny O ε

12 2

12 2

12 2

sin sin

(12

)cos (12

)sin exp ϑ

(12

)cos (12

)sin exp ϑ

12 2

12 2

cos2

cos2 ( )

00(0) 2 2 1

200(0)

00(1) 2 2 1

200(1)

200(2) 2 2 1

200(2)

11(2)

5/200(3/2)

01(5/2) 1

5 10(5/2) 1

5

00(3/2)

01(5/2) 1

5 10(5/2) 1

5

300(3) 2 2 1

200(3) 4

00(4) 2 2 1

200(4)

20(4)

02(4) 5

(67)

= ⎡⎣⎢

+

+ ⎛⎝

⎛⎝

− ⎞⎠

− ⎛⎝

− ⎞⎠

⎞⎠

⎛⎝

− ⎞⎠

+ ⎛⎝

⎛⎝

− ⎞⎠

− − ⎛⎝

− ⎞⎠

− ⎞⎠

⎛⎝

− − ⎞⎠

⎤⎦⎥

+ + +

+ + +

ε C mx ny c mx ny

c a γ φ xε

c a γ φ xε

xε

c a γ φ π xε

c a γ φ π xε

π xε

ε C mx ny c mx ny ε C mx ny

c mx ny C mx c ny

Ψ cos sin sin cos

12

cos 12

sin exp ϑ

12

cos 12

sin exp ϑ

[ cos sin sin cos ] [ cos sin

sin cos sin2 sin2 ]

x2

11(2)

11(2)

01(2) 1

5 10(2) 1

5

01(2) 1

5 10(2) 1

5

311(3)

11(3) 4

11(4)

11(4)

20(4)

02(4) (68)

= + +

+ + +

+ + +

ε D mx ny d mx ny ε D mx ny

d mx ny ε D mx ny d mx ny

d mx D ny O ε

Ψ [ sin cos cos sin ] [ sin cos

cos sin ] [ sin cos cos sin

sin2 sin2 ] ( )

y2

11(2)

11(2) 3

11(3)

11(3) 4

11(4)

11(4)

20(4)

02(4) 5 (69)

Note that all of the coefficients in Eqs. (66)–(69) are related and canbe written as functions of A11

(2). Meanwhile, because of Eq. (66), theprebuckling deformation of the shell is nonlinear.

Next, substituting Eqs. (66)–(69) into the boundary condition (25c)and into Eq. (27a), the postbuckling equilibrium paths can be written as

= + + ⋯−λ ε λ λ A ε14

(3) [ ( ) ]q q q3/4 3/2 (0) (2)

11(2) 2 2

(70a)

= + + ⋯−λ ε λ λ A ε[ ( ) ]s s s3/2 (0) (2)

11(2) 2 2 (70b)

and

= + + ⋯δ δ δ A ε( )q q q(0) (2)

11(2) 2 2 (71)

in Eqs. (70) and (71), (A ε11(2) 2) is taken as the second perturbation

parameter relating to the dimensionless maximum deflection. If themaximum deflection is assumed to be at the point =x y π m π n( , ) ( /2 , /2 ),then

= − + ⋯A ε W WΘm m11(2) 2

12 (72a)

from here, we obtain the dimensionless maximum deflection of the shell

= ⎡⎣⎢

+ ⎤⎦⎥

∗ ∗ ∗ ∗WC

ε hD D A A

Wh

1[ ]

Θm3 11 22 11 22

1/4 2(72b)

The experimental results showed that there is no such a large dif-ference of buckling pressure between these two kinds of boundaryconditions (simply supported and clamped). On the contrary, the ex-perimental results obtained for the shell with clamped edges match wellwith the theoretical solutions of the shell with (simply supported) SSedges. It is well known that the classical buckling pressure and stress ofa cylindrical shell under lateral pressure, i.e.

=−

⎛⎝

⎞⎠

q πEν

RL

hR

23 3 (1 )cr 2 3/4

5/2

(73a)

=−

⎛⎝

⎞⎠

σ πEν

RL

hR

23 3 (1 )cr

e2 3/4

5/2

(73b)

From Eq. (73b), it can be seen that the buckling stress is aboutproportional to (h/R)3/2, when other things are equal. It is known thatthe assumptions of the classical theory, though initially formulated as

geometrical, have a physical aspect as well. In the case of stronglyanisotropic shells could lead to crucial errors in computations if thetransverse tangential and normal stresses are ignored in Eqs. (66)–(69).The asymptotic analysis reveals the limits of applicability of governingequations for anisotropic shells.

All symbols used in Eqs. (70)–(72) and Eqs. (80)–(82) below are alsodescribed in detail in Appendix C.

Case (2) high values of axial compression combined with relativelylow external pressure. Let

=πR q

Pb

2

02 (74a)

or

=λ ε

λ εb

(3)

22

q

p

43

1/4 3/2

2(74b)

in this case, the boundary condition of Eq. (25c) becomes

∫ ∂∂

+ + =π

β Fy

dy λ ε ab12

2 (1 ) 0π

p0

2 22

2 2(75)

Similarly, by taking = +a b ab2 /(1 )2 2 2 and using a singular pertur-bation procedure, the solutions satisfying the clamped boundary con-ditions in the asymptotic sense are conducted as

= ⎡⎣⎢

− ⎛⎝

+ ⎞⎠

⎛⎝

− ⎞⎠

− ⎛⎝

− + − ⎞⎠

⎛⎝

− − ⎞⎠

⎤⎦⎥

+ + +

− ⎛⎝

+ ⎞⎠

⎛⎝

− ⎞⎠

− ⎛⎝

− + − ⎞⎠

⎛⎝

− − ⎞⎠

+ + + +

+ + + +

W ε A A a φ xε

a φ xε

xε

A a φ π xε

a φ π xε

π xε

ε A mx ny A ny

A ny a φ xε

a φ xε

xε

A ny a φ π xε

a φ π xε

π xε

ε A mx ny A ny ε A A mx

A ny A mx ny A ny O ε

cos sin exp ϑ

cos sin exp ϑ

[ sin sin cos2

( cos2 ) cos sin exp ϑ

( cos2 ) cos sin exp ϑ ]

[ sin sin cos2 ] [ cos2

cos2 sin sin3 cos4 ] ( )

00(1)

00(1)

01(1)

10(1)

00(1)

01(1)

10(1)

211(2)

02(2)

02(2)

01(1)

10(1)

02(2)

01(1)

10(1)

311(3)

02(3) 4

00(4)

20(4)

02(4)

13(4)

04(4) 5 (76)

= − + + ⎡⎣

− + ⎤⎦

+ ⎡⎣

− + +

+ ⎛⎝

+ ⎞⎠

⎛⎝

− ⎞⎠

+ ⎛⎝

− + − ⎞⎠

⎛⎝

− − ⎞⎠

⎤⎦⎥

+ ⎡⎣

− + +

+ ⎛⎝

+ ⎞⎠

⎛⎝

− ⎞⎠

+ ⎛⎝

− + − ⎞⎠

⎛⎝

− − ⎞⎠

⎤⎦⎥

+ ⎡⎣

− + + +

+ + +

F B a β x y ε B a β x y

ε B a β x y B mx ny

A b φ xε

b φ xε

xε

A b φ π xε

b φ π xε

π xε

ε B a β x y B ny

A ny b φ xε

b φ xε

xε

A ny b φ π xε

b φ π xε

π xε

ε B a β x y B mx ny B mx

B ny B mx ny O ε

12

( ) 12

( )

12

( ) sin sin

cos sin exp ϑ

cos sin exp ϑ

12

( ) cos2

( cos2 ) cos sin exp ϑ

( cos2 ) cos sin exp ϑ

12

( ) sin sin cos2

cos2 sin sin3 ] ( )

00(0)

22 2 2

00(1)

22 2 2

200(2)

22 2 2

11(2)

00(1)

01(2)

10(2)

00(1)

01(2)

10(2)

300(3)

22 2 2

02(3)

02(2)

01(3)

10(3)

02(2)

01(3)

10(3)

400(4)

22 2 2

11(4)

20(4)

02(4)

13(4) 5 (77)

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

91

= ⎡⎣

⎛⎝

− ⎞⎠

+ − ⎛⎝

− − ⎞⎠

⎤⎦⎥

+

+ ⎡⎣

⎛⎝

− ⎞⎠

+ − ⎛⎝

− − ⎞⎠

⎤⎦⎥

+

+ + +

+ +

ε A c φ xε

xε

A c φ π xε

π xε

ε C mx ny

ε A ny c φ xε

xε

A ny c φ π xε

π xε

ε C mx ny

ε C mx ny C mx c ny

C mx ny O ε

Ψ sin exp ϑ

sin exp ϑ [ cos sin ]

( cos2 ) sin exp ϑ

( cos2 ) sin exp ϑ [ cos sin ]

[ cos sin sin2 sin2

cos sin3 ] ( )

x3/2

00(1)

10(3/2)

00(1)

10(3/2) 2

11(2)

5/202(2)

10(5/2)

02(2)

10(5/2) 3

11(3)

411(4)

20(4)

02(4)

13(4) 5 (78)

= + +

− ⎛⎝

+ ⎞⎠

⎛⎝

− ⎞⎠

− ⎛⎝

− + − ⎞⎠

⎛⎝

− − ⎞⎠

⎤⎦⎥

+ + +

+ +

ε D mx ny ε D mx ny D ny

A nβ ny d φ xε

d φ xε

xε

A nβ ny d φ π xε

d φ π xε

π xε

ε D mx ny D mx D ny

D mx ny O ε

Ψ [ sin cos ] [ sin cos sin2

( 2 sin2 ) cos sin exp ϑ

( 2 sin2 ) cos sin exp ϑ

[ sin cos sin2 sin2

sin cos3 ] ( )

y2

11(2) 3

11(3)

02(3)

02(2)

01(3)

10(3)

02(2)

01(3)

10(3)

411(4)

20(4)

02(4)

13(4) 5 (79)

It's important to note that here that we have given the followingrelationship for the first time,

= − +

= −

= +

= +

− + − + +

+ − −

− + + ++ − −

−

−

A n β μ A

B γ m n β A

A γ γ m μ R A

a γ γ m μ R A

(1 ) ( ) ,

( ) ,

(1 ) ( ) ,

(1 ) ( )

g g

g

μ g g μ g g

μ g g C g g

g gg

C μ μμ g g C g g

g g

g g

g g

g g

20(4) 1

42 2 ( ) [2(1 2 )( ) (1 ) ]

[4(1 )( ) ] 11(2) 2

20(4) 1

2 242 2 2 ( ) [ (1 2 ) 2(1 ) ]

[4(1 )( ) ] 11(2) 2

13(4)

14 244 2 ( )

8 1 11(2) 3

13(4)

14 244 2 ( )

8 2 11(2) 3

2102

2202

2102

2102

2202

210 200

2102

2202 2 210 200

2102

2202

210

2 2

2102

2202 2 210 200

2102

2202

09 2102

2102

2202

09 2102

(80)

Due to the change of the above relationship, it can be seen that thereare obvious differences between Case (2) high values of axial com-pression combined with relatively low external pressure and pure axialcompression for the influence of the buckling loading and postbucklingequilibrium path of an anisotropic laminated cylindrical shell.

Next, substituting Eqs. (76)–(79) into the boundary condition (75)and into Eq. (27b), the postbuckling equilibrium paths can be written as

=+

− + + ⋯λab

λ λ A ε λ A ε11

[ ( ) ( ) ]p p p p2

(0) (2)11(2) 2 (4)

11(2) 4

(81a)

= − + + ⋯λ λ λ A ε λ A ε( ) ( )s s s s(0) (2)

11(2) 2 (4)

11(2) 4 (81b)

and

= + + + ⋯δ δ δ A ε δ A ε( ) ( )p p p p(0) (2)

11(2) 2 (4)

11(2) 4 (82)

in Eqs. (81) and (82), similarly, (A ε11(2) ) is taken as the second step

perturbation parameter in this case, and we have

= − + ⋯A ε W WΘm m11(2)

32 (83a)

and the dimensionless maximum deflection of the shell is written as

= ⎡⎣⎢

+ ⎤⎦⎥

∗ ∗ ∗ ∗WC

hD D A A

Wh

1[ ]

Θm3 11 22 11 22

1/4 4(83b)

Eqs. (68)–(70) and (81)–(83) can be employed to obtain numericalresults for the postbuckling load-shortening or load-deflection curves ofmoderately thick anisotropic laminated cylindrical shells subjected tocombined loading of external pressure and axial compression. By in-creasing b1 and b2, respectively, the interaction curve of a moderatelythick laminated cylindrical shell under combined loading can be

constructed with these two lines. Eq. (66) is the large deflection of re-latively higher external pressure-loaded shell, while Eq. (76) is the largedeflection of relatively higher axially loaded shell, and they have agreat difference. Hence, the postbuckling equilibrium path of an axiallyloaded shell is unstable; on the contrary, the postbuckling equilibriumpath of a pressure loaded shell is weakly stable. Note that since

=b b1/2 1, only one load-proportional parameter should be determinedin advance. The buckling load of a perfect shell can readily be obtainednumerically, by setting =∗W h/ 0 (or =μ 0), while taking =W h/ 0(note that ≠W 0m ). In this case, the minimum buckling load is acquiredby considering Eq. (70) or (81) by changing values of the bucklingmode (m, n), which determine the number of half-waves in the X-di-rection and of full waves in the Y-direction. From Eqs. (66) and (76),the nonlinear prebuckling deformation of the shell can be shown. InEqs. (67) and (77), we can confirm that there exists a circumferentialstress or a compressive stress along with an associate shear stress whenthe shell is subjected to combined loading of external pressure and axialcompression. Such a shear stress, no matter how small it is, will affectthe buckling and postbuckling behavior of the moderately thick ani-sotropic shell as shown in Eqs. (70) and (71) and Eqs. (77)-(82).

It is well known the classical buckling theory of axially loaded thincylindrical shells predicts that the buckling stress −σcr axial(= −E h R υ( / )/[3(1 )]2 1/2) is directly proportional to (h/R)1, with otherthings being equal. However, the experimental data showed that thebuckling stress is about proportional to (h/R)3/2, with other thingsbeing equal. For a homogeneous isotropic cylindrical shell

= −ε π R L h R υ( / ) ( / )/[12(1 )]2 2 2 ; this means e is also proportional to h/R,with other things being equal. From Eq. (C-3) in Appendix C, it can beseen that the critical value of λp is also a function of h/R, when ≠W 0m .Usually the critical value of λp is less than 1, hence the buckling stressσcr (= −λ σp cr axial) may be proportional to h R( / ) j and j must be greaterthan 1. This provides an explanation for the experimental results.

4. Numerical results and comments

In this section, the performance of postbuckling behavior for mod-erately thick laminated cylindrical shells subjected to combinedloading. We made an illustrative numerous examples of perfect andimperfect, moderately thick, anisotropic laminated cylindrical shells,where the outmost layer is the first mentioned orientation.

As part of the validation of the present method, the buckling pres-sures (in kPa) for simply supported symmetric laminated thick cylind-rical shells with different geometric parameters and stacking sequencessubjected to lateral pressure are calculated and compared in Table 1with the analytical results of Arciniega et al. [23]. In calculation, ma-terial properties adopted are: E11= 127.8 GPa, E22= 9.40 GPa,G12=G13= 4.20 GPa, G23= 3.10 GPa, v12 =0.28, and R=0.1905m,L/R=1. The results show that the present results agree well with, andthe relative error is less than 5.2%, but slightly lower than those ofArciniega et al. [23]. Meanwhile, the buckling loads (in N/m) forsymmetric laminated thick cylindrical shells with clamped boundaryconditions and with different geometric parameters (Z =375) anddifferent stacking sequences subjected to axial compression are calcu-lated and compared in Table 2 with the analytical results of Simitsesand Anastasiadis [10], Eslami et al. [30] and Eslami and Shariyat [31].The material properties adopted are (Simitses and Anastasiadis [10];Eslami et al. [30] and Eslami and Shariyat [31]): E11= 206.844 GPa,E22= 18.6159 GPa, G12=G13= 4.48162 GPa, G23= 2.55107 GPa,and v12 =0.21. It can be seen that the present results agree well with,but lower than those of Simitses and Anastasiadis [10], Eslami et al.[30] and Eslami and Shariyat [31].

The buckling pressure for cross-ply multi-layers laminated cylindersunder hydrostatic pressure with clamped boundary conditions are listed

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92

in Table 3 and compared with experimental results of Hur et al. [47]. Incalculation, material properties adopted are: E11= 162.0 GPa,E22= 9.6 GPa, G12=G13= 6.1 GPa, G23= 3.5 GPa, v12 =0.298, andall plies are of equal thickness and the inner radius is 158mm. Theresults show that the present results are lower than nonlinear FEM(MSC.Marc®) results, but higher than experimental results of Hur et al.[47] except for cylinder of CTM1.

In order to obtain the accurate prediction of bearing capacity for anarbitrary reinforced shell structure, Smerdov [48] carried out a struc-tural optimization study for buckling of shell structures. The bucklingloads for multi-layered laminated cylindrical shells subjected to lateralpressure are calculated and compared with the numerical results ofSmerdov [48] in Table 4. In calculation, the material propertiesadopted are: E11= 146.0 GPa, E22= 10.8 GPa, G12= 5.78 GPa,v12 =0.29, R=82.5mm, L/R=1.74, R/h=165. The results showthat the present results are lower than those of Smerdov [48], espe-cially, for (87.3/45)T, (77/52)T, (90/51.8)T, (90/51.8)T, (902/60.3)Tshells. Moreover, we examine the buckling pressure qcr (in MPa) forfilament-wound shells with clamped boundary conditions subjected tocombined lateral pressure and axial compression. The results are listedin Table 5 and compared with experimental results of Mistry et al. [29],using their material properties, i.e. Ef=72GPa, vf=0.25,Em=3.8 GPa, vm=0.39 and Vf =0.55. Note that in these two tables

=σ σ/ 1: 0y x means pure lateral pressure and =σ σ/ 0: 1y x means pure

Table 1Comparisons of buckling loads qcr (kPa) for perfect laminated cylindrical shellsunder lateral pressure (E11= 127.8 GPa, E22= 9.40 GPa,G12=G13= 4.20 GPa, G23= 3.10 GPa, v12 =0.28, L/R=1, R=0.1905m).

R/h Lay-up Arciniega et al. [23] TSDT Present Relative Errorb

10 (0/90)S 90720.8835 87485.27(4)a 3.567%(90/0)S 140920.8329 136678.4(4) 3.011%(45/−45)S 171119.6585 170238.4(4) 0.515%(30/−60)S 140567.8288 134418.6(8) 4.375%(60/−30)S 190074.4261 181367.0(8) 4.581%

50 (0/90)S 1356.8921 1336.7460(7) 1.485%(90/0)S 3273.0872 3167.7940(5) 3.217%(45/−45)S 2904.3274 2814.8620(7) 3.080%(30/−60)S 2330.1198 2235.6310(8) 4.055%(60/−30)S 3732.8465 3547.3310(7) 4.970%

100 (0/90)S 236.2278 233.8530(8) 1.005%(90/0)S 616.5122 610.4680(7) 0.980%(45/−45)S 446.4328 439.1308(8) 1.636%(30/−60)S 361.3969 359.3473(8) 0.567%(60/−30)S 330.1454 313.1208(7) 5.157%

a The number in parentheses indicate the full wave number in the cir-cumferential direction.

b Error= × −q q q100 | |/crA

cr crA.

Table 2Comparison of buckling loads (N/m×106) for moderately thick laminated cylindrical shell under axial compression (R/h=15, Z =375, h=12.7mm).

Lay-up Simitses and Anastasiadis [10] Eslami et al. [30] Eslami and Shariyat [31] Present

CL FOSD HOSD

(03)S 12.52 12.71 12.45 12.37 12.41 12.36(3)b

(0/90/0)S 15.77 14.98 14.76 14.52 14.64 14.71(3)(90/0/90)S 15.11 14.01 13.72 13.48 13.69 14.25(3)(9 0 3)S 11.38 10.67 10.54 10.49 10.51 11.02(3)(452/−45)S 14.50 12.76 12.41 12.35 12.38 12.51(3)(45/−452)S 17.45 14.93 14.38 14.32 14.34 11.67(2)(−45/45/−45)S 20.66 17.06 16.12 15.86 15.97 15.54(2)(−452/45)S 14.50 12.76 12.41 12.35 12.39 12.51(3)(302/−60)S 12.03 10.66 10.12 9.89 9.92 10.59(3)

b The number in parentheses indicate the full wave number in the circumferential direction.

Table 3Comparisons of buckling loads qcr (MPa) for (0/90)12T shells under hydrostaticpressure (E11= 162.0 GPa, E22= 9.6 GPa, G12=G13= 6.1 GPa,G23= 3.5 GPa, v12 =0.298).

ID h (mm) L (mm) Exp. Hur et al.[47]

FEM Hur et al. [47] Present

CTM1 2.52 564 0.60(4) 0.691(4) 0.5664(4)c

CTM2 2.52 569 0.51(4) 0.684(4) 0.5631(4)CTM3 2.69 600 0.55(4) 0.653(4) 0.6477(4)CTM4 2.68 600 0.55(4) 0.653(4) 0.6476(4)CTM5 2.65 600 0.55(4) 0.653(4) 0.6474(4)

c The number in parentheses indicate the full wave number in the cir-cumferential direction.

Table 4Comparisons of buckling loads qcr (kPa) for laminated shells under lateralpressure (E11= 146.0 GPa, E22= 10.8 GPa, G12= 5.78 GPa, ν12 =0.29,R=82.5mm, R/h=165).

L/R Lay-up Smerdov [48] Present

0.5 (68/54)T 439.56 400.4386(13)b

(67.5/2.6/90)T 478.63 427.9058(12)(75/41.3/0/90)T 488.4 441.1397(12)

1 (77/52)T 203.52 146.9375(8)(76.6/15.8/90)T 235.39 224.7008(8)(83.9/34.6/0/90)T 245.2 241.3022(8)

1.74 (90/51.8)T 118.42 72.89089(6)(90/17.1/70.1)T 143.28 113.0501(7)(90/14.2/26/90)T 146.2 142.1338(7)

3 (90/65)T 70.143 49.57425(5)(90/12/90)T 83.665 83.89699(5)(90/0/25.3/90)T 84.51 84.11853(5)

5 (90/68)T 42.587 32.32274(4)(90/5/90)T 50.797 52.54230(4)(90/0/20.2/90)T 51.31 51.72057(4)

10 (90/77)T 22.695 21.37355(3)(90/0/90)T 25.79 27.83218(3)(90/0/13.1/90)T 25.79 26.96911(3)

b The number in brackets indicates the number of waves in the cir-cumference direction (n).

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93

axial compression. The results show that the present solutions agreereasonably well with experimental results of Mistry et al. [29].

In addition, we first examine the interactive buckling loads qcr (inMPa) and Pcr (in kN) for± 55° filament-wound E-glass/epoxy tubessubjected to combined lateral pressure and axial compression withclamped boundary conditions. The results are listed in Table 6 andcompared with experimental results of Kaddour et al. [49]. The com-puting data adopted here are Ef1 = 72.4 GPa, Ef2 = 71.3 GPa,Gf12= 28.6 GPa, νf12= 0.25 GPa, Em=4.0 GPa and νm=0.35 andfiber volume fraction Vf =0.68. As a second example, we examine thebuckling pressure qcr (in MPa) for filament-wound shells with clampedboundary conditions subjected to combined lateral pressure and axialcompression. The results show that the present solutions agree rea-sonably well with experimental results of Kaddour et al. [49]. It waswell known that computationally generated design curves can obtainthe buckling loads of anisotropic composite shell structures subjected tocombined loading. These curves will provide an effective guidance forthe future capacity design of shell structures under complex loadingconditions.

Next, we discussed and analyzed the effect of the stacking sequenceof the laminate on the shape of the interaction buckling for the twodifferent stacking sequences of (0/90/0)10T and (02/90/45/-45/-45/45/90/02)2T with the graphite-epoxy material properties. Fig. 2 showsthe comparison on the interaction buckling curve of above cylindricalshells subject to combined axial compression and pressure with thetheoretical and experimental results by Tafreshi and Bailey [50], inwhich =R q q/q cr and =R σ σ/p x cr r, where qcr and σcr are the criticalbuckling loads under lateral pressure alone or axial compression alone.

The material and geometric properties of graphite-epoxy are used in thenumerical studies (Tafreshi and Bailey [50]): E11/E22= 40,E22= 5.17 GPa, G12/E22=G13/E22= 0.5, and v12 =0.25, and L/R=5, R/h=30 and h=1.0mm. It can be observed that the shape of

Table 5Comparison of under combined external pressure and axial loading.

No. Exp. Mistry et al. [29]1 Present Exp. Mistry et al. [29]2 Present Exp. Mistry et al. [29]3 Present Exp. Mistry et al. [29]4 Present

1 2.5 2.52(1,2)c 2.4 2.47(1,3) 2.0 2.27(1,3) 2.0 2.28(1,3)1.2 1.22(1,2) 1.1 1.46(1,2) 1.0 1.06(1,2) 1.0 1.03(1,2)0.7 0.78(1,2) 0.7 0.82(1,2) 0.7 0.75(1,2) 0.7 0.78(1,2)

2 5.4 5.48(1,2) 6.2 6.27(1,2) 5.5 5.73(1,2) 5.2 5.43(2,2)5.0 5.11(1,2) 5.2 5.48(1,2) 4.9 5.28(1,2) 4.1 4.34(2,2)4.4 4.44(1,2) 4.9 4.97(1,2) 5.1 5.15(1,2) 3.1 3.29(2,2)

3 4.5 4.50(1,3) 4.4 4.43(1,2) 4.1 4.18(1,2) 3.6 3.86(2,3)1.8 2.03(1,2) 1.9 1.96(1,2) 2.0 2.15(1,2) 1.5 1.77(1,2)1.3 1.45(1,2) 1.3 1.35(1,2) 1.3 1.30(1,2) 1.2 1.44(1,2)

1–4Indicate the ratio of stresses σy/σx (=1:0, 2:1, 1:1, 1:2); cThe number in brackets indicates the buckling mode (m, n).

Table 6Comparison of critical external pressure (qcr) (MPa) and axial compression Pcr (kN) for± 55° GRP tubes subjected to combined external pressure and axial com-pression.

Test No. h R/h L/R Z σy/σx (qcr, Pcr) Kaddour et al. [49] Present

5 9.43 5.80 6.76 265.33 0:1 (0, 495) (0, 569.77)1 9.57 5.725 6.75 261.11 0:1 (0, 494) (0, 567.68)16 4.78 5.835 6.63 256.72 1:2 (14.14, 122.3) (15.79, 131.06)36 4.94 5.662 6.61 247.70 1:1 (40.4, 117) (42.69, 122.37)11 4.65 5.984 6.65 264.52 1:1 (37.7, 109) (44.32, 112.25)43 4.76 5.857 6.64 257.90 2.5:2 (59.12, 103) (61.45, 108.42)65 4.66 5.972 6.65 263.90 1.5:1 (77.12, 37.74) (80.05, 44.39)38 4.67 5.96 6.65 263.29 3:2 (74.99, 71.1) (79.86, 78.52)17 4.78 5.835 6.63 256.72 3:2 (66.9, 63.4) (71.56, 69.81)61 7.13 4.076 6.36 165.15 1.7:1 (135.7, 67.2) (136.17, 75.19)48 9.595 5.711 6.75 260.36 2.4:1 (82.69, 159.8) (86.48, 169.47)44 9.9 5.551 6.73 251.65 2.4:1 (89.45, 172.4) (89.98, 180.24)62 5.87 4.844 6.51 205.05 2.5:1 (84.57, 130.4) (85.43, 137.67)64 5.2 5.404 6.58 234.23 2.5:1 (71.29, 109.2) (72.22, 115.26)63 4.24 6.514 6.70 292.25 2.7:1 (74.0, 63.47) (80.34, 64.02)18 4.78 5.835 6.63 256.72 3:1 (79.9, 76.8) (81.13, 94.51)7 4.73 5.89 6.64 259.67 4:1 (55.68, 81.8) (57.66, 89.20)31 9.51 5.76 6.76 262.91 5:1 (42.6, 293) (47.38, 298.08)8 4.64 6.00 6.65 265.14 6:1 (39.39, 79.2) (43.28, 83.91)

Fig. 2. Comparison on the interaction buckling curves for laminated cylindricalshells under combined loading.

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94

the interaction buckling curve is mainly influenced by the stackingsequence. The difference between the interaction curves is more ap-parent when the cylinder is subject to high level of external pressure.

A parametric study is performed to predict different influencingfactors on the buckling and postbuckling behavior of moderately thickanisotropic laminated cylindrical shells under combined loadings.Typical results are shown in Table 7 and Figs. 4–8. Graphite/epoxycomposite material was selected for the shells in these examples. Forthese examples R/h=40, all plies are of equal thickness and the totalthickness of the shell is h=4.0mm except for Fig. 7 and the materialproperties, adopted as in Yuan and Hsieh [51], are: E11= 138.0 GPa,E22= 8.9 GPa, G12=G13= 5.17 GPa, G23= 2.89 GPa, and v12 =0.30.It should be appreciated that in all of the figures ∗W h/ denotes thedimensionless maximum initial geometric imperfection of the shell.Note that now the coupling stiffnesses ∗Bij are present, even though theskin of the shell is symmetric.

In order to prove the validity and appropriateness of the presentmethods, the (± 45)2S laminated cylindrical shell with simply sup-ported boundary conditions and the same geometric (Z =100) andmaterial parameters subjected to lateral pressure are conducted usingthe commercial nonlinear finite element analysis program ABAQUS®and a homogeneous mesh of finite elements (total number of nodes is4416, total number of elements is 1440) and compared to the presentresults by a two-step methods. Shell behavior that can be properlydescribed with shear flexible shell theory and results in smooth dis-placement fields can be analyzed accurately with the second-orderABAQUS®/Standard thick shell element S8R by considering the effect oftransverse shear. S8R is an eight-node shear deformable shell elementwith reduced integration, which allows large rotations and smallstrains. Therefore, this element is recommended for use in regular meshgeometries for thick shell applications. A comparison between thepresent results by a two-step method and those obtained by the mod-ified Riks method available in ABAQUS® is present in Fig. 3. It shouldbe mentioned that the presented method is in good agreement with thenonlinear FEM results that are slightly higher, so in the following, onlythe analytical results are presented.

Table 7 gives the extension/twist and flexural/twist couplings de-noted by ∗Ai6 and

∗Di6 (i=1,2), on the buckling loads (Pcr/qcr) (kN/MPa)and associated shear stress τs (in MPa) for perfect (0/90)2S, (± 45)2S,

Fig. 3. Comparison of postbuckling behavior of anisotropic laminated cylind-rical shells under lateral pressure.

Table 7Comparison of buckling loads (Pcr/qcr) (kN/MPa) and stresses (τs) (MPa) forperfect (0/90)2S, (± 45)2S, (30/60)S, (30/60)T, (−45/0/45/90)S, (45/0/−45/90)S, (−452/−302/602/152)T, (152/602/−302/−452)T, (45/−30/30/90/0/60/−45/-60)T and (73/30/−50/46/58/−16/−45/20)T laminated cylindricalshells under combined loading of axial compression and lateral pressure (R/h=40, Z =500).

Lay-up ∗Ai6∗Di6 (Pcr/qcr) τs

(0/90)2S 0 0 (1687.56/0) 00 0 (1607.16/

0.999)0

0 0 (425.48./1.058)

0

0 0 (0/1.185) 0

(± 45)2S ∼e−16 ∼e+1 (2339.41/0) –23.373∼e−16 ∼e+1 (2253.37/

0.701)–23.373

∼e−16 ∼e+1 (364.39/0.939) −0.1072∼e−16 ∼e+1 (0/1.144) −0.1072

(30/60)S ∼e−8 ∼e+2 (1872.74/0) +0.3082∼e−8 ∼e+2 (1781.47/

0.361)+0.2393

∼e−8 ∼e+2 (290.31/0.722) +0.2632∼e−8 ∼e+2 (0/0.811) +0.2631

(30/60)T ∼e−8 ∼e+1 (1516.43/0) +5.2830∼e−8 ∼e+1 (1443.65/

0.337)+5.2674

∼e−8 ∼e+1 (270.35/0.672) −0.1930∼e−8 ∼e+1 (0/0.755) −0.1929

(−45/0/45/90)S ∼e−16 ∼e+1 (2793.44/0) +15.302∼e−16 ∼e+1 (2662.58/

0.470)+15.302

∼e−16 ∼e+1 (377.81/0.940) +0.1432∼e−16 ∼e+1 (0/1.055) +0.1432

(45/0/−45/90)S ∼e−16 ∼e+1 (2793.44/0) −15.302∼e−16 ∼e+1 (2662.58/

0.469)−15.302

∼e−16 ∼e+1 (377.81/0.939) −0.1432∼e−16 ∼e+1 (0/1.055) −0.1432

(152/602/−302/−452)T ∼e−9 ∼e+1 (2162.23/0) +14.302∼e−9 ∼e+1 (2079.85/

0.225)+13.969

∼e−9 ∼e+1 (180.53/0.555) +4.680e−2∼e−9 ∼e+1 (0/0.637) +4.674e−2

(−452/−302/602/152)T ∼e−9 ∼e+1 (1732.40/0) +4.164e−2∼e−9 ∼e+1 (1658.60/

0.203)+3.495e−2

∼e−9 ∼e+1 (20.066/0.508) +2.387e−2∼e−9 ∼e+1 (0/0.583) +2.381e−2

(45/−30/30/90/0/60/−45/−60)T

∼e−16 ∼e+1 (3076.88/0) +15.313∼e−16 ∼e+1 (2935.22/

0.564)+15.305

∼e−16 ∼e+1 (453.228/1.127)

+7.534e−2

∼e−16 ∼e+1 (0/1.266) +7.531e−2

(73/30/−50/46/58/−16/−45/20)T

∼e−9 ∼e+1 (2275.28/0) +2.449∼e−9 ∼e+1 (2169.50/

0.437)+2.357

∼e−9 ∼e+1 (351.475/0.874)

−3.209e−2

∼e−9 ∼e+1 (0/0.982) −3.207e−2

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95

(30/60)S, (30/60)T, (−45/0/45/90)S, (45/0/−45/90)S, (−452/−302/602/152)T, (152/602/−302/−452)T, (45/−30/30/90/0/60/−45/−60)T and (73/30/−50/46/58/−16/−45/20)T laminated cylindricalshells with shell geometric parameter Z =500, subjected to combinedloading case, i.e. lateral pressure alone (b1= 0), combined loading case(1) (b1= 5), combined loading case (2) (b2= 0.05) and axial com-pression alone (b2= 0.0). As expected the shear stresses τs are zero-valued for the (0/90)2S shell because of no couplings. Due to the geo-metrical symmetry the extension/twist couplings ∗Ai6 (i=1, 2) are ap-proximately zero-valued for the (± 45)2S shell, and the shear stress τsare negative. More generally, in accordance with the explanation pre-sented before, the presence of ply packages with high Poisson’s ratioand their positions strongly influence the buckling loads and modes. Wenoticed well that the (152/602/−302/−452)T shell has the reversestacking sequence to the (−452/−302/602/152)T shell, the shear stressτs are both positive for the (152/602/−302/−452)T and (−452/−302/602/152)T shells under three sets of combined loading conditions, i.e.lateral pressure alone (b1= 0), combined loading case (1) (b1= 5),combined loading case (2) (b2= 0.05). However, the shear stress τs forboth shells are opposite under axial compression alone; whereas the(−45/0/45/90)S and (45/0/−45/90)S shells are both the relativelyhigher shear stress τs among these laminated shells except that of(± 45)2S shell, implying that the shear stiffness of laminates can be

increased by orienting the layers at angle to the laminate coordinates.Note that the coupling between normal forces and shearing strain,shearing force and normal strains, normal moments and twist, andtwisting moment and normal curvatures is not zero for these laminates(i.e., ∗A16,

∗A26,∗D16 and ∗D26 are not zero). Moreover, we may know that

laminate stiffnesses Aij depend only on the thicknesses and stiffnesses ofthe layers, but not on their placement in the laminate. On the otherhand, laminate stiffnesses Dij depend not only on the layer thicknessand stiffnesses but also on their location relative to the midplane. It isseen that both (−45/0/45/90)S and (45/0/−45/90)S shells will havethe same in-plane stiffnesses Aij, and exist the relationship,A11= A12= (A11− A12)/2 and = =A A 016 26 , which are oriented atthe same angle relative to adjacent laminae exhibit “in-plane isotropy”,when the bending-stretching coupling coefficients are zero, the rela-tions between force resultants and membrane strains are the same asthose for isotropic shells.

Variations of the fiber orientations and stacking sequences on thepostbuckling load-shortening and load-deflection curves for (0/90)2S,(± 45)2S, (−452/−302/602/152)T and (152/602/−302/−452)T lami-nated cylindrical shells are given in Figs. 4 and 5, respectively, underfour sets of combined loading conditions, i.e. hydrostatic pressure alone(b1= 0), combined loading case (1) (b1= 5), combined loading case(2) (b2= 0.05) and axial compression alone (b2= 0.0). From Fig. 4, it

Fig. 4. Comparisons of postbuckling behavior of (0/90)2S, (45/−45)2S, (−452/−302/602/152)T and (152/602/−302/−452)T cylindrical shells subjected to axialcompression combined with lateral pressure: (a, c) load-shortening and (b, d) load-deflection.

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

96

can be observed that the slope of the postbuckling load-shorteningcurve for the (0/90)2S shell is the largest among the three, and bothbuckling load and critical end-shortening for the (± 45)2S shell aremuch higher than others, and placing these plies at or near the externalsurface is advantageous yielding high buckling loads. It can also be seenthat the well-known “snap-through” behavior of shells occurs. In Fig. 5,we found that the slope of the postbuckling load-shortening curve forthe (0/90)2S, (± 45)2S, (−452/−302/602/152)T and (152/602/−302/−452)T shells is almost perpendicular to the axis in the initial bucklingstage, whereas the (0/90)2S shell becomes stiffer in the deep post-buckling region when the deflection is sufficiently large. It is noted thatan increase in pressure is usually required to obtain an increase in de-formation, and the postbuckling equilibrium path is stable for moder-ately long shells, and the shell structures is virtually imperfection-in-sensitive.

The effect of shell geometric parameter Z (=100, 200, 500) on thepostbuckling load-shortening and load–deflection curves for (152/602/−302/−452)T laminated cylindrical shells are compared and plotted inFig. 6 under four sets of combined loading conditions, i.e. hydrostaticpressure alone (b1= 0), combined loading case (1) (b1= 5), combinedloading case (2) (b2= 0.05) and axial compression alone (b2= 0.0).The results show that the slope of the postbuckling load-shorteningcurve for the shell with Z =100 is larger than others as well as hint the

shell possesses much higher postbuckling strength. For the same se-quence of stacking, we investigated the influence of radius-to-thicknessratio R/h (=40, 80, 160) in Fig. 7 under four sets of combined loadingconditions. As can be observed, the slope of the postbuckling load-shortening curve for the shell with R/h=40 (radius R=160mm, re-mained constant) are larger than others, and the shell has considerablepostbuckling strength and the postbuckling path obviously goes up dueto the shear stiffnesses increasing for the shells with a relatively thickwall. On the other hand, the results show that the buckling pressure forthe shell with the same radius R=160mm, and radius-to-thicknessratio (R/h=40) has a remarkably higher postbuckling strength thanthe others when the shells are subjected to high values of externalpressure combined with relatively low axial load. It is noted that thelarger the wall thickness when the radius remains constant, the heavierduty equipment is required, and eventually controlling the quality andoperating the equipment getting more difficult, especially for control-ling the wall thickness which is the technical bottleneck for producinglarge diameter cylinders safely for future developments in compositestructural manufacture in the ocean/oil engineering and pressure vesseldesign.

Effects of imperfection sensitivity curves for the same four lami-nated cylindrical shells and under combined loading case (2), withload-proportional parameter b2= 0.0 and 0.05 are calculated and

Fig. 5. Comparisons of postbuckling behavior of (0/90)2S, (45/−45)2S, (−452/−302/602/152)T and (152/602/−302/−452)T cylindrical shells under hydrostaticpressure combined with axial compression: (a, c) load-shortening and (b, d) load-deflection.

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

97

compared in Fig. 8. Here, ∗λ is the maximum value of σx for the im-perfect shell, made dimensionless by dividing by the critical value of σxfor the perfect shell. From Fig. 8, the imperfection sensitivity of the(± 45)2S shell is weaker than others, and that for larger imperfectionamplitudes, e.g. >∗W h/ 0.5, the postbuckling behavior becomes stable,so that no imperfection sensitivity can be predicted for the (0/90)2Sshell. Under the pure axial compression (b2= 0.0), the imperfectionsensitivity for the (−452/−302/602/152)T shell is larger than that ofthe (152/602/−302/−452)T one. In contrast, the imperfection sensi-tivity for the (152/602/−302/−452)T shell is larger than that of the(−452/−302/602/152)T one under combined loading case (2)(b2= 0.05). These results also imply that by increasing the load-pro-portional parameter b2 for these shells, the imperfection sensitivity willdecline. It should be mentioned, when the magnitude of the initialimperfection amplitude becomes larger, the prebuckling stiffness de-creases dramatically and imperfection sensitivity will become weakerfor the case of combined loadings.

In present study, both the loads and imperfection are expressed withtrigonometric series, and the initial geometric imperfection which isassumed to have a similar form with deflection plays a significant rolein the influence of the postbuckling equilibrium path. It is confirmed

that the extreme sensitivity to imperfections of axially compressed cy-lindrical shell rather than the same externally pressurized shell.

As Shen [52] pointed out “all the theoretical and numerical resultsreveal that with the help of boundary layer theory: (1) the nonlinearprebuckling deformations, large deflection in the postbuckling rangeand initial geometric imperfections of the shell could be consideredsimultaneously; (2) it is unnecessary to guess the forms of solutionswhich can be obtained step by step, and such solutions satisfy bothgoverning equations and boundary conditions accurately in theasymptotic sense; (3) both full nonlinear postbuckling analysis andimperfection sensitivity analysis could be performed.” The presentstudy is capable of treating both thick and thin shells. The results of theboundary layer theory of shell buckling also implied that the effect ofshear deformation on the response of the laminated shell type struc-tures, made of fiber reinforced composite materials, has thus become animportant issue, and a reliable prediction of this response necessitatesthe inclusion of shear deformation. When the transverse deflectionsexperienced by shells are not small in comparison to the laminatethickness, the interaction between membrance stresses and the curva-tures of the laminate must be considered.

Fig. 6. Effect of shell geometric parameter on the postbuckling behavior for (152/602/−302/−452)T cylindrical shells under axial compression combined withlateral pressure (b2= 0.0, 0.05) or hydrostatic pressure combined with axial compression (b1= 0, 5): (a, c) load-shortening and (b, d) load-deflection.

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

98

5. Concluding remarks

A fully nonlinear postbuckling analysis is presented for moderatelythick anisotropic laminated cylindrical shells based on Reddy’s higherorder shear deformation theory with von Kármán-Donnell-type of ki-nematic nonlinearity and including the extension/twist, extension/flexural and flexural/twist couplings. In this work, we extended theboundary layer theory of shell buckling to shear deformable laminatedcylindrical shells subjected to combined loading of external pressureand axial compression. A two-step perturbation method is employed todetermine interactive buckling loads and postbuckling equilibriumpaths. We reveal that interesting phenomena for a shell buckling whichexist a circumferential stress or a compressive stress along with an as-sociate shear stress when the shear deformable laminated cylindricalshell is subjected to the combined loading of external pressure and axial

compression. The results show that the shape of interaction bucklingcurves depends significantly on the laminate stacking sequence, shellgeometric parameters, and total number of plies and the postbucklingcharacteristics additionally depend significantly upon the load-pro-portional parameter b2 (or b1). These results provide a new and pow-erful way to solve the buckling problem of the shell structure by theactions of various complex loads.

Acknowledgement

The authors wish to thank Professor Shen H-S of Shanghai Jiao TongUniversity for his considerable support. The work described in thispaper is supported in part by grants from the National Natural ScienceFoundation of China (Nos. 51279222, 51775346 and 51479115). Theauthors are also grateful for these financial supports.

Fig. 7. Effect of radius-to-thickness ratio on the postbuckling behavior for (152/602/−302/−452)T cylindrical shells under axial compression combined with lateralpressure (b2= 0.0, 0.05) or hydrostatic pressure combined with axial compression (b1= 0, 5): (a, c) load-shortening and (b, d) load-deflection.

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

99

Appendix A

In Eq. (16) [ ∗Aij ], [∗B( ij ], [

∗Dij], [∗E( ij], [

∗Fij ] and [ ∗Hij ] (i, j=1, 2, 6) are reduced stiffness matrices, defined as

= = − = − = − = − = −∗ − ∗ − ∗ − ∗ − ∗ − ∗ −A A B A B D D BA BE A E F F EA B H H EA E, , , ,1 1 1 1 1 1 (A-1)

and

=⎡

⎣⎢⎢

⎤

⎦⎥⎥

=⎡

⎣⎢⎢

⎤

⎦⎥⎥

=⎡

⎣⎢⎢

⎤

⎦⎥⎥

=⎡

⎣⎢⎢

⎤

⎦⎥⎥

=⎡

⎣⎢⎢

⎤

⎦⎥⎥

=⎡

⎣⎢⎢

⎤

⎦⎥⎥

A AA A

A

B BB B

B

D DD D

D

E EE E

E

F FF B

F

H HH H

H

A B D

E F H

00

0 0,

00

0 0,

00

0 0

00

0 0,

00

0 0,

00

0 0

11 12

12 22

66

11 12

12 22

66

11 12

12 22

66

11 12

12 22

66

11 12

12 22

66

11 12

12 22

66 (A-2)

where Aij, Bij etc., are the laminate stiffnesses, defined by

∫∑= == −

A B D E F H Q Z Z Z Z Z dZ i j( , , , , , ) ( ) (1, , , , , ) ( , 1,2,6)ij ij ij ij ij ijk

N

t

tij k

1

2 3 4 6k

k

1 (A-3a)

∫∑= == −

A D F Q Z Z dZ i j( , , ) ( ) (1, , ) ( , 4,5)ij ij ijk

N

t

tij k

1

2 4k

k

1 (A-3b)

and Qij are the transformed elastic constants, defined by

Fig. 8. Comparisons of imperfection sensitivities of moderately thick laminated cylindrical shells.

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

100

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

=

⎡

⎣

⎢⎢⎢⎢⎢⎢

+ −

− − − −− − −

− −

⎤

⎦

⎥⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

QQQQQQ

c c s s c sc s c s c s c ss c s c c sc s cs c s cs cs c scs c s cs c s cs c sc s c s c s c s

QQQQ

2 44

2 42 ( )

2 ( )2 ( )

11

12

22

16

26

66

4 2 2 4 2 2

2 2 4 4 2 2 2 2

4 2 2 4 2 2

3 3 3 3 2 2

3 3 3 3 2 2

2 2 2 2 2 2 2 2 2

11

12

22

66

(A-4a)

and

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

=⎡

⎣⎢−

⎤

⎦⎥⎡

⎣⎢⎤⎦⎥

QQQ

c scs cs

s c

QQ

44

45

55

2 2

2 2

44

55(A-4b)

where

= = =

= = =− − −Q Q Q

Q G Q G Q G

, ,

, ,

Eν ν

Eν ν

Eν ν11 (1 ) 22 (1 ) 12 (1 )

44 23 55 13 66 12

1112 21

2212 21

1112 21

(A-4c)

in which E 11, E 22, G 12, G 13, G 23,ν12 andν21 have their usual meanings and

= =c θ s θcos , sin (A-4d)

where θ is lamination angle with respect to the shell X-axis.

Appendix B

In Eqs. (24)–(26)

= + += + += − + − += − + −= − − += + − + −

= + −= − −== += − − − −= − − − −= − − − −= − − − −= − −

= − + − += − + − += + − −= − + − +

= − + += − += − + += + − + + + += − + +

= − + −= + − + −= − + − +

= − + += − −=

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗

∗

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

γ γ γ h F F F F F Dγ γ h F F F F Dγ γ D F h D D F F h Dγ γ D F F h D F h Dγ γ D F h D F F h Dγ γ D D F F h D F h Dγ γ γ B B B B B D D A Aγ γ B B B B D D A Aγ γ A A Aγ γ A A A Aγ γ B E h B B E E h D D A Aγ γ B B E E h B E h D D A Aγ γ B E h B B E E h D D A Aγ γ B B E E h B E h D D A Aγ γ h E E E E D D A Aγ γ L π A D h F h A D h F h Dγ γ h F H h F F H H h Dγ γ h F F H h F H h Dγ γ D F h H h D F h H h D

γ D F F h H h Dγ L π A D h F h Dγ D F F h H h Dγ D D F F F h H H h Dγ D F F h H h Dγ γ h F H h F F H h Dγ γ h F F H H h F H h Dγ γ D F h H h D F h H h D

γ D F F h H h Dγ γ B E h B E h D D A Aγ γ h E E D D A A

( , , ) (4/3 )[ , ( 4 )/2, ]/( , ) (4/3 )[2( ), 2( )]/( , ) [ 4 /3 ,( 2 ) 4( 2 )/3 ]/( , ) [3 4( 2 )/3 , 4 /3 ]/( , ) [ 4 /3 , 3 4( 2 )/3 ]/( , ) [( 2 ) 4( 2 )/3 , 4 /3 ]/( , , ) [ , ( 2 ), ]/[ ]( , ) (2 , 2 )/[ ]( , ) ( , )/( , ) ( /2, )/( , ) [ 4 /3 ,( ) 4( )/3 ]/[ ]( , ) [( ) 4( )/3 , 4 /3 ]/[ ]( , ) [ 4 /3 ,( ) 4( )/3 ]/[ ]( , ) [( ) 4( )/3 , 4 /3 ]/[ ]( , ) (4/3 )(2 , 2 )/[ ]( , ) ( / )( 8 / 16 / , 8 / 16 / )/( , ) (4/3 )[ 4 /3 , ( 2 ) 4( 2 )/3 ]/( , ) (4/3 )[( 2 ) 4 / , 4 /3 ]/( , ) ( 8 /3 16 /9 , 8 /3 16 /9 )/

2[ 4( )/3 16 /9 ]/( / )[ 8 / 16 / ]/ ,[ 4( )/3 16 /9 ]/[( ) 4( 2 )/3 16( )/9 ]/[ 4( )/3 16 /9 ]/

( , ) (4 /3 )[ 4 /3 , ( 2 ) 4 / ]/( , ) (4 /3 )[( 2 ) 4( 2 )/3 , 4 /3 ]/( , ) ( 8 /3 16 /9 , 8 /3 16 /9 )/

2[ 4( )/3 16 /9 ]/( , ) ( 4 /3 , 4 /3 )/[ ]( , ) (4/3 )( , )/[ ]

110 112 1142

11 12 21 66 22 11

111 1132

16 61 26 62 11

120 122 11 112

12 66 12 662

11

121 123 16 61 162

26 262

11

130 132 16 162

26 62 262

11

131 133 12 66 21 662

22 222

11

140 142 144 21 11 22 66 12 11 22 11 221/4

141 143 26 61 16 62 11 22 11 221/4

211 213 26 16 22

212 214 12 66 11 22

220 222 21 212

11 66 11 662

11 22 11 221/4

221 223 26 61 26 612

16 162

11 22 11 221/4

230 232 26 262

16 62 16 622

11 22 11 221/4

231 233 22 66 22 662

12 122

11 22 11 221/4

241 2432

26 61 16 62 11 22 11 221/4

31 412 2

55 552

554

44 442

444

11

310 3122

11 112

21 66 12 662

11

311 3132

16 61 162

26 262

11

320 322 11 112

114

66 662

664

11

321 16 16 612

164

11

322 2

45 452

454

11

330 16 16 612

164

11

331 12 66 12 21 662

12 664

11

332 26 26 622

264

11

410 4122

16 162

26 62 262

11

411 4132

12 66 12 662

22 222

11

430 432 66 662

664

22 222

224

11

431 26 26 622

264

11

511 522 11 112

22 222

11 22 11 221/4

611 6222

11 22 11 22 11 221/4

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

101

Appendix C

In Eqs. (70)–(72)

= + = −

= + + + − +

= − − +

= ⎡⎣

+ − − ⎤⎦

= + + ⎡⎣

+ − ⎤⎦

− ⎡⎣

+

+ − ⎤⎦

+ ⎡⎣

+ + − ⎤⎦

⎫⎬⎭

=

= − + + ⎡⎣

+ − ⎤⎦

− ⎡⎣

+

+ − ⎤⎦

+ ⎡⎣

+ + − ⎤⎦

⎫⎬⎭

= −

= ⎡⎣ − + − − ⎤⎦

− ⎡⎣

+ − ⎤⎦

= + − +

− + −

+ + ⎡⎣

⎤⎦

⎫⎬⎭

−

+ −

+ + + +

+ − +

++ + +

− +

+ +

− ++

++ +

−

++ +

− +

+ + +

− +

+ +

++ + +

− ++ + +

− +

+ + + +

−

+

−

+ −

+ + + +

+ − +

++ + +

− +

+ +

− ++

++ +

−

++ +

− +

+ + +

− +

+ +

++ + +

− ++ + +

− +

+ + + +

−

−

+ − + − − + + −

+ + − + + + + + +

− − + + +

− + −

{

{

{

( )( )

C n β a m C ε

C n β μ n β C μ C μ

a γ λ λ

C a γ λ λ

λ γ ε

ε

ε

ε

λ R

λ γ ε

ε

ε

ε

λ R

δ a γ γ a γ b φb ε λ b φb ε λ

δ m μ ε ε ε

ε ε ε

m n β C μ ε

, 1

(1 2 ) (1 ) (1 )

Θ 1 ,

Θ 1

( ) (1 )(ϑ ) (ϑ )

(1 2 ) 2

2 (1 )

g g g gm g

RR

γ qγγ s

C γ qγγ s

qmC

γ m g

μ g g

g g g g μ g g g g

μ g g m μgγ

γμ

g g g g g g g g g g

g gγ μ

μ

g g g g g g

g gμ

m μg g g g

γγ g

μg g g g g g

g g

γ gμ

g g g g g g

g gγ μ

μ

g g g g g g g g

g gμ

m μ

g g g g g gγ

γ gμ

g g g g g g g g

g g

γ gμ

g g g g g g g g

g gγ μ

μ

g g g g g g g g g g

g g

qγ m n β g g

C g g g

smnβ

γ m g

μ g g

g g g g μ g g g g

μ g g m μgγ

γμ

g g g g g g g g g g

g gγ μ

μ

g g g g g g

g gμ

m μg g g g

γγ g

μg g g g g g

g g

γ gμ

g g g g g g

g gγ μ

μ

g g g g g g g g

g gμ

m μ

g g g g g gγ

γ gμ

g g g g g g g g

g g

γ gμ

g g g g g g g g

g gγ μ

μ

g g g g g g g g g g

g g

sγ m nβg

g g

q γ πγ

γ q γγ γ γ

πγ s

qg g

g

g g g g g g

g m

g g g g g g g g

g

m

g g g g g g g g g g

g m

g g g g g g g g

g m

g g g g g g

g

g g

g

g g μ μ g g

g g C μ a m g g

12 2 1

2 12

3

418

2 2 18

2 21

14 1

2

31 1

2 1 5(0) (0)

41

41 1

2 1 5(2) (2)

(0)(1 )( ) 24

(1 )( )

(1 ) ( )1

(1 )

(1 )( ) ( )

(1 )

2 ( ) 2(1 ) (1 )

2 ( )

(1 )( ) 2

(1 )

( 3 ) (3 ) 3(1 )

( ) 2

(1 )(3 ) ( 3 )

(1 )( 3 ) (3 )

(1 )

4 ( ) ( 6 ) 4

(2) 2 ( )

( ) 6

(0)2 (1 )( ) 24

(1 )( )

(1 ) ( )1

(1 )

(1 )( ) ( )

(1 )

2 ( ) 2(1 ) (1 )

2 ( )

(1 )( ) 2

(1 )

( 3 ) (3 ) 3(1 )

( ) 2

(1 )( 3 ) (3 )

(1 )(3 ) ( 3 )

(1 )

4 ( ) ( 6 ) 4

(2) 2

( ) 6

(0) 1 12 1 24

25

2 12 1 5 01

(5/2)10(5/2) 1/2 3

2 01(5/2)

10(5/2) 1/2

(2) (3)32

2 ( ) 3/2 ( ) 2 1/2 3 ( )( ) 4 1/2

2 ( )( 3 ) 2 (3 ) 3/2 1 ( )( ) 4( ) 5/2 1 8 ( ) 5/2

2 4 412 2 ( ) 2( )(1 2 ) (1 )

16( ) (1 ) 2

23

210 310 220 3202

210

45

24

211

24

3 24

211

24

2

124

2210

2102

2202

220 320 210 310 210 31 220 322

2102

2202 2

110

14

24 31 220 320 210 310 32 220 310 210 320

2102

2202

242

220 310 320 210 3102

3202

2102

2202 4 2

110 310 120 320

14

24 31 220 310 320 210 3102

3202

2102

2202

24 32 220 3102

3202

210 310 320

2102

2202

242

210 310 3102

3202

220 320 3102

3202

2102

2202

2

6 3110 310

23202

120 310 320

14

24 31 220 320 3102

3202

210 310 3102

3202

2102

2202

24 32 220 310 3102

3202

210 320 3102

3202

2102

2202

242

220 310 320 3102

3202

210 3104

3102

3202

3204

2102

2202

244 2 2

2102

2202

1 210 2102

2202 2

242

220

2102

2202

210 320 220 310 220 31 210 322

2102

2202 2

120

14

24 31 220 310 210 320 32 220 320 210 310

2102

2202

242

210 310 320 220 3102

3202

2102

2202 4 2

110 320 120 310

14

24 31 210 310 320 220 3102

3202

2102

2202

24 32 210 3102

3202

220 310 320

2102

2202

242

220 310 3102

3202

210 320 3102

3202

2102

2202

2

6 3120 310

23202

110 310 320

14

24 31 220 310 3102

3202

210 320 3102

3202

2102

2202

24 32 220 320 3102

3202

210 310 3102

3202

2102

2202

242

210 310 320 3102

3202

220 3104

3102

3202

3204

2102

2202

243

220

2102

2202

24

5

24

3/4

24

213 5 211

24

3/4 2102

2202

2102

310 2102

2202

210 220 320

2102 2

2102

2202

3102

3202

210 220 310 320

2102

4310 210

22202

3102

3202

320 210 220 3102

3202

2102 6

2102

2202

3102

3202

2102

2202

3102

3202

2102 6

210 220 310 320 3102

3202

2102

2102

2202

2102

2102

2202

210 200

2102

2202 1 1 2

210 200

(C-1)

in the above equations [with gij and gijk being defined as in Li and Lin [39]

= + + += − + + +

= − + −

= ⎡⎣− ⎤⎦

+ − −

= ⎡⎣

⎤⎦

=

= − = + = =

= =

= + + − +

+ +

R C μ a m μR g g μ g g μ

R C g g μ a m g g

R C n β g g n β

b c γ γ γ

b c φ b c a a g

b γ g b γ g

b a φ b a φc φ φ

4 (1 ) (1 2 )2( )(1 2 ) (1 )

16 ( )(1 ) 2

2 ( )

, ,

ϑ [( )/2] , [( )/2] , 1,

,

[( ) 2ϑ (2ϑ ϑ )]

μ RR

g μg n β

RR

γ γ γg

gg

φ

φ

b

3 12

12

4 2102

2202

210 200

5 1 2102

2202

12

210 200

6 1(1 )

42 2

2102

2202 (1 2 )

322 2

1/2

14 24 320 2

1/2 1/201(3/2)

10(3/2) ϑ

17

01(5/2)

24 19 10(5/2)

24ϑ

20

111

10(1) 2 2

10(1) 4 2 2 4

45

13

2102 2

35

14 24 3202

16

15

16

(C-2)

and in Eqs. (80)–(83)

=+

C mm a n β2

2

22

2 2

= −−

=++

Cg g g g

m gε C

m a n βm a n β

19

3210 310 220 320

2210

9

22

2 2

22

2 2

=−

+γ aγ

λγγ

λΘ2( )

p s15 2

24

(0) 211

24

(0)

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

102

= ⎡⎣⎢

+− +

−+ ⎤

⎦⎥−

Cγ γ

m μn β g g

ε γ γm gn β g

γ aγ

λγγ

λΘ 1 (1 )16 32

2( )p s2

314 24

4

2 209 210

114 24

211

2 209

5 2

24

(2) 211

24

(2)

= ⎧⎨⎩ + −

−λ Cγ m gμ g g

ε12 (1 )( )p

(0)2

242

210

2102

2202

1

+− + + −

+ −γ

g g g g μ g g g gμ g g

(1 )( )(1 ) ( )24

220 320 210 310 210 31 220 322

2102

2202

++

⎡⎣⎢

++

− + −−m μ

gγ

γμ

g g g g g g g g g gg g

1(1 ) (1 )

( ) ( )2

110

14

24 31 220 320 210 310 32 220 310 210 320

2102

2202

++

− +−

⎤

⎦⎥

γ μμ

g g g g g gg g

ε(1 )

2 ( )242

220 310 320 210 3102

3202

2102

2202

++

⎡⎣⎢

++

+− +

−μ

m μg g g g

γγ g

μg g g g g g

g g(1 ) (1 )2 ( )

4 2110 310 120 320

14

24 31 220 310 320 210 3102

3202

2102

2202

++

+ −−

γ gμ

g g g g g gg g(1 )

( ) 224 32 220 3102

3202

210 310 320

2102

2202

−+

+ − +−

⎤

⎦⎥

γ μμ

g g g g g g g gg g

ε(1 )

( 3 ) 2 (3 )242

210 310 3102

3202

220 320 3102

3202

2102

2202

2

++

⎡

⎣⎢

+ +μm μ

g g g g g gγ(1 )

( ) 22

6 3110 310

23202

120 310 320

14

++

+ − +−

γ gμ

g g g g g g g gg g(1 )

( 3 ) (3 )24 31 220 320 3102

3202

210 310 3102

3202

2102

2202

++

+ − +−

γ gμ

g g g g g g g gg g(1 )

( 3 ) (3 )24 32 220 310 3102

3202

210 320 3102

3202

2102

2202

++

+ − + +−

⎤

⎦⎥

⎫⎬⎭

γ μμ

g g g g g g g g g gg g

ε(1 )

4 ( ) ( 6 )242

220 310 320 3102

3202

210 3104

3102

3202

3204

2102

2202

3

= ⎧⎨⎩

+−

−λ Cγ γ m g g

g g gε1

2( )

4 ( )p(2)

214 24

2 62102

2202

09 2102

2202 2

1

−−

+ −γ γ m

g g g gg g g g g

8 ( )[(3 ) ( )14 24

2 4

09 210 2102

2202 2 210

22202

220 320 32

+ + −g g g g g( 3 ) ( )]2102

2202

210 310 31

+− +

+ −γ γ m g

g g g g μg g g g g g

2 ( ) (1 )[(2 ) ]14 24

2 4220

09 210 2102

2202 2 210

22202

320 210 220 310

−+

− ++ − +

γ γ m g gg g g g μ

g μ g μ( )

8 ( )(1 )[ (1 2 ) (1 ) ]14 24

2 42102

2202

09 210 2102

2202 14 302

2

−−

+ +μγ m

g g g gg g g g g g ε

8 ( )[ ( ) 2 ]24

2

09 210 2102

2202 110 210

22202

120 210 220

−+

+γ m g g

gg μ ε

16(1 2 )24

22102

2202

2102 13

++

− +− + + + + +

− + −γ m n β g gg g g μ

g g μ μ g g μg g μ C g g

ε( )

( )(1 )( )[2(1 ) 3(1 2 )] (1 )

4( )(1 )24

2 4 42102

2202

210 2102

2202

2102

2202 2

210 200

2102

2202

2 210 200

−+−

−γ γ m g g

g g g gg g g g g ε

( )16 ( )

[ 4 ]14 242 2

2102

2202

09 210 2102

2202 210 302 11 310 14

+−

γ γ m g g gg g g

ε2 ( )

14 242 2

320 302 220

09 2102

2202

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

103

−+

− ++ +

γ γ m g μg g g g μ

g g g g g g ε(1 2 )

8 ( )(1 )[( ) 2 ]14 24

2 214

09 210 2102

2202 2 210

22202

310 210 220 320

⎜+− +

⎡⎣⎢ + ⎛

⎝

++

−γ γ m

g g g g μg g μ

μμ

g g g g16 ( ) (1 )

( 3 ) 221

( )14 242 2

09 210 2102

2202 2 210

22202

310 31 210 310

⎟ ⎜++

+ − + − ⎞⎠

− + ⎛⎝

− −−+μ

μ g μ g g g g g g g g g g g μ gμ

μg2

1( (2 ) ) ( ) (3 ) 2 (2(4 )

21

)2320

232 210 320 210

2310 31 11 210

22202

220 320 310

2

31

⎟+++

+−+

+ ⎞⎠

⎤⎦⎥μ

μμ

g g gμμ

g g g g g ε221

(1

1)220 32 310 320 32 210 220 11

+− +

+ − − ⎫⎬⎭

γ γ m gg g g g μ

g g g g g g g g ε2 ( ) (1 )

[( ) ( 3 )]14 242 2

220

09 210 2102

2202 2 2 210

22202

310 320 210 220 3102

3202

⎜ ⎟

⎜ ⎟

=+

−⎧⎨⎩

+−

− ++

− −+ + − +

+ ⎛

⎝+

+ −−

+−−

++ ⎞

⎠

+ ⎛

⎝+

+ −−

+−−

− ⎞

⎠

⎫⎬⎭

−

λC γ γ m μ

g g g gg g g

g gμ

g g g g gg g g g g g g g g g

μg g

gg g g g

g gg

g g g gg g

g gg

R

μg g

gg g g g

g gg

g g g gg g

g R ε

12

(1 )64 ( )

4(3 )

( )1

(1 )( )( )

[ ( )(5 ) 2 (3 )]

(1 ) 2 3

(1 ) 2 2

p(4) 2 14

2243 10

092

210 2102

2202

2202

2102

2202

2102

2202 2

2102

2202

232

242

21023 210

22202

2102

2202

24 210 220 2102

2202

2102

2202

210

210 23 220 24

232

242 210

210 23 220 24

232

242

2102

2202

2101

2102

2202

210

220 23 210 24

232

242 210

220 23 210 24

232

242 220 2

1

= − ⎧⎨⎩ + −

−λ mnβ

γ m gμ g g

ε2 (1 )( )s

(0) 242

220

2102

2202

1

+− + + −

+ −γ

g g g g μ g g g gμ g g

(1 )( )(1 ) ( )24

210 320 220 310 220 31 210 322

2102

2202

++

⎡⎣⎢

++

− + −−m μ

gγ

γμ

g g g g g g g g g gg g

1(1 ) (1 )

( ) ( )2

120

14

24 31 220 310 210 320 32 220 320 210 310

2102

2202

++

− +−

⎤

⎦⎥

γ μμ

g g g g g gg g

ε(1 )

2 ( )242

210 310 320 220 3102

3202

2102

2202

−+

⎡⎣⎢

++

+− +

−μ

m μg g g g

γγ g

μg g g g g g

g g(1 ) (1 )2 ( )

4 2110 320 120 310

14

24 31 210 310 320 220 3102

3202

2102

2202

++

+ −−

γ gμ

g g g g g gg g(1 )

( ) 224 32 210 3102

3202

220 310 320

2102

2202

−+

+ − +−

⎤

⎦⎥

γ μμ

g g g g g g g gg g

ε(1 )

( 3 ) 2 (3 )242

220 310 3102

3202

210 320 3102

3202

2102

2202

2

−+

⎡

⎣⎢

+ +μm μ

g g g g g gγ(1 )

( ) 22

6 3120 310

23202

110 310 320

14

++

+ − +−

γ gμ

g g g g g g g gg g(1 )

( 3 ) (3 )24 31 220 310 3102

3202

210 320 3102

3202

2102

2202

++

+ − +−

γ gμ

g g g g g g g gg g(1 )

(3 ) ( 3 )24 32 220 320 3102

3202

210 310 3102

3202

2102

2202

++

+ − + +−

⎤

⎦⎥

⎫⎬⎭

γ μμ

g g g g g g g g g gg g

ε(1 )

4 ( ) ( 6 )242

210 310 320 3102

3202

220 3104

3102

3202

3204

2102

2202

3

= − ⎧⎨⎩ −

−λ mnβ

γ γ m g gg g g

ε2 2 ( )s

(2) 14 242 6

210 220

09 2102

2202 2

1

−−

+ −γ γ m

g g g gg g g g g

8 ( )[(3 ) ( )14 24

2 4

09 210 2102

2202 2 210

22202

220 310 31

+ + −g g g g g( 3 ) ( )]2102

2202

210 320 32

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

104

−− +

−γ γ m g

g g g μg g g g

2 ( ) (1 )( 3 )14 24

2 4220

09 2102

2202 2 210 310 220 320

−− +

+ − +γ γ m g

g g g μg μ g u

4 ( )(1 )[ (1 2 ) (1 ) ]14 24

2 4220

09 2102

2202 14 302

2

−−

+ +μγ m

g g g gg g g g g g ε

8 ( )[ ( ) 2 ]24

2

09 210 2102

2202 120 210

22202

110 210 220

− +γ m g

gg μ ε

8(1 2 )24

2220

21013

+− +

− + + + + +− + −

γ m n β gg g μ

g g μ μ g g μg g μ C g g

ε2

( )(1 )( )[2(1 ) 3(1 2 )] (1 )

4( )(1 )24

2 4 4220

2102

2202

2102

2202 2

210 200

2102

2202

2 210 200

−−

−γ γ m gg g g

g g g g g ε8 ( )

[ 4 ]14 242 2

220

09 2102

2202 210 302 11 310 14

++

−γ γ m g g g g

g g g gε

( )4 ( )

14 242 2

320 302 2102

2202

09 210 2102

2202

−+

− ++ +

γ γ m g μg g g g μ

g g g g g g ε(1 2 )

8 ( )(1 )[( ) 2 ]14 24

2 214

09 210 2102

2202 2 210

22202

320 210 220 310

⎜+− +

⎡⎣⎢ + ⎛

⎝

++

−γ γ m

g g g g μg g μ

μμ

g g g g16 ( ) (1 )

(3 ) 221

( )14 242 2

09 210 2102

2202 2 210

22202

310 31 210 310

⎟ ⎜++

+ − + − ⎞⎠

− + ⎛⎝

− −−+μ

μ g μ g g g g g g g g g g g μ gμ

μg2

1( (2 ) ) ( ) ( 3 ) 2 (2(4 )

21

)2320

232 210 320 210

2310 31 11 210

22202

220 320 310

2

31

⎟+++

+−+

+ ⎞⎠

⎤⎦⎥μ

μμ

g g gμμ

g g g g g ε221

(1

1)220 32 310 320 32 210 220 11

−− +

− − + ⎫⎬⎭

γ γ m gg g g g μ

g g g g g g g g g ε2 ( ) (1 )

[ 2 (2 3 ) ]14 242 2

220

09 210 2102

2202 2 2 210

23102

210 220 310 320 2102

2202

3202

= −+

−⎧⎨⎩

+−

−λ mnβ

γ γ m μg g g

g g gg g

gg2

(1 )64 ( )

4( 3 )

( )s(4) 14

2243 10

092

2102

2202

220 2102

2202

2102

2202 2

220

2102

++

− −− + +

μg g g g g

g g g g g g g g(1 )

( )( )[ ( ) 2 (3 )

2102

2202

232

242

2102 24 210

22202 2

210 220 23 2102

2202

− +g g g g2 ( 3 )]2102

24 2102

2202

⎜ ⎟+ ⎛

⎝

+ − − +−

+−−

+ ⎞

⎠

μg

g g g g g g g gg g

gg

g g g gg g

gg

R(1 ) (3 ) ( )

2 62102

220 23 2102

2202

210 24 2102

2202

232

242

220

210

210 23 220 24

232

242

220

2101

⎜−⎛

⎝

+ − − +−

μg

g g g g g g g gg g

(1 ) (3 ) ( )

2102

220 24 2102

2202

210 23 2102

2202

232

242

⎟−−−

++ ⎞

⎠

⎫⎬⎭

−gg

g g g gg g

g gg

R ε2 220

210

220 23 210 24

232

242

2102

2202

2102 2

1

= ⎡⎣⎢

− −−

− ⎤⎦⎥

+ ⎡⎣⎢

+ − ⎤⎦⎥

δγ

γ a γπ

γ γ aγ

b φb ε λγ

γγ γπγ

b φb ε λ1 ( ) 2 ( )(ϑ ) 1

22

(ϑ )p p s(0)

24242

2 55 5 2

2401(2)

10(2) 1/2

24213

5 211

2401(2)

10(2) 1/2

= ⎡

⎣⎢ +

++δ b

παgg

ε mg g

gμ ε1

16( )

(1 2 )p(2) 11 220

2

2102

12 2 210

22202

2102

++ +g g g g g g

gε2

( ) 2310 2102

2202

210 220 320

2102

2

++ + +

mg g g g g g g g

gε3 ( )( ) 4

22102

2202

3102

3202

210 220 310 320

2102

3

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

105

++ + + + ⎤

⎦⎥m

g g g g g g g g g gg

ε2 ( )( 3 ) 2 (3 )4

310 2102

2202

3102

3202

320 210 220 3102

3202

2102

4

=+

++

+ +

+ + ⎡

⎣⎢

− − + + +− + −

⎤

⎦⎥

⎫⎬⎭

−{δ bπ

γ γm μn β g g

ε mg g

g gμ m g ε ε

m n β μg g

gg g μ μ g g

g g μ g gε

1128 32 ϑ

(1 )2

( )(1 ) ( )

4 (1 )( ) 2( )(1 2 ) (1 )

4( )(1 )

p(4) 11

142

242

8 2

4 4092

2102

3/2 4 2102

2202

09 2103

2 2310

2 4 4 2 2102

2202

2102

2102

2202

210 200

2102

2202

210 200

23

(C-3)

in the above equations

= + + ++ + +

++ + +

g m γ m n β γ n β γ γm γ m γ n β nβ γ m γ n β

γ γm γ m γ n β nβ γ m γ n β

( 2 )( )Δ ( )Δ

Δ( )Δ ( )Δ

Δ

2104

2122 2 2

2144 4

14 24220

2222

2 201 221

2223

2 202

00

14 24230

2232

2 204 231

2233

2 203

00

= + −+ + +

−+ + +

g γ m nβ γ mn β γ γm γ m γ n β nβ γ m γ n β

γ γm γ m γ n β nβ γ m γ n β

2( )( )Δ ( )Δ

Δ( )Δ ( )Δ

Δ220 2113

2133 3

14 24220

2222

2 202 221

2223

2 201

0014 24

2302

2322 2

03 2312

2332 2

04

00

= + + −+ + +

−+ + +

g γ m γ m n β γ n βm γ m γ n β nβ γ m γ n β m γ m γ n β nβ γ m γ n β

( 2 )( )Δ ( )Δ

Δ( )Δ ( )Δ

Δ310 2404

2422 2 2

2444 4 220

2222

2 205 221

2223

2 207

00

2302

2322 2

08 2312

2332 2

06

00

= + −+ + +

−+ + +

g γ m nβ γ mn βm γ m γ n β nβ γ m γ n β m γ m γ n β nβ γ m γ n β

( )( )Δ ( )Δ

Δ( )Δ ( )Δ

Δ320 2413

2433 3 220

2222

2 207 221

2223

2 205

00

2302

2322 2

06 2312

2332 2

08

00

= + + −+ + +

−+ + +

g γ m γ m n β γ n βm γ m γ n β nβ γ m γ n β m γ m γ n β nβ γ m γ n β

( )( )Δ ( )Δ

Δ( )Δ ( )Δ

Δ31 1404

1422 2 2

1444 4 120

2122

2 201 121

2123

2 202

00

1302

1322 2

04 1312

1332 2

03

00

= + −+ + +

−+ + +

g γ m nβ γ mn βm γ m γ n β nβ γ m γ n β m γ m γ n β nβ γ m γ n β

( )( )Δ ( )Δ

Δ( )Δ ( )Δ

Δ32 1413

1433 3 120

2122

2 202 121

2123

2 201

00

1302

1322 2

03 1312

1332 2

04

00

= + + −+ + +

−+ + +

g γ m γ m n β γ n βm γ m γ n β nβ γ m γ n β m γ m γ n β nβ γ m γ n β

( 2 )( )Δ ( )Δ

Δ( )Δ ( )Δ

Δ110 1104

1122 2 2

1144 4 120

2122

2 205 121

2123

2 207

00

1302

1322 2

08 1312

1332 2

06

00

= − + ++ + +

++ + +

g γ m nβ γ mn βm γ m γ n β nβ γ m γ n β m γ m γ n β nβ γ m γ n β

( )( )Δ ( )Δ

Δ( )Δ ( )Δ

Δ120 1113

1133 3 120

2122

2 207 121

2123

2 205

00

1302

1322 2

06 1312

1332 2

08

00

= ++

g m γ γ γ mγ γ m

16 644200

414 24 220

26

31 3202

=+ + −

+ +g

γ γ γ n β γ γ γ n βγ γ γ n β γ γ γ n β

( 4 ) ( 4 )( 4 ) 4302

244 41 4322 2

233 41 4132 2

214 41 4322 2

14 24 2332 2 2

=− −+ +

gγ γ γ n β γ γ γ γ n βγ γ γ n β γ γ γ n β

( 4 ) 4( 4 ) 4303

214 41 4132 2

14 24 233 2442 2

214 41 4322 2

14 24 2332 2 2

= + +g γ γ g γ γ γ g09 114 133 303 14 24 144 302

=+

+ +g

γ γ n βγ γ γ n β γ γ γ n β

4( 4 ) 413

41 4322 2

214 41 4322 2

14 24 2332 2 2

= −+ −

+ +g

γ γ γ n β γ γ n βγ γ γ n β γ γ γ n β

( 4 ) 4( 4 ) 414

144 41 4322 2

133 2332 2

214 41 4322 2

14 24 2332 2 2

=−

+ −+

g gg g

gμ

g g g gg

(1 2 ) 211 142102

2202

2102

210 310 220 320

2102

= + + ++ + +

g m γ m n β γ n β γ γm γ m γ n β nβ γ m γ n β

( 18 81 )( 9 )Δ 3 ( 9 )Δ

Δ234

2122 2 2

2144 4

14 24220

2222

2 2131 221

2223

2 2132

130

++ + +

γ γm γ m γ n β nβ γ m γ n β( )Δ ( )Δ

Δ14 24230

2232

2 2134 231

2233

2 2133

130

= + −+ + +

−+ + +

g γ m nβ γ mn β γ γm γ m γ n β nβ γ m γ n β

γ γm γ m γ n β nβ γ m γ n β

(6 54 )( 9 )Δ 3 ( 9 )Δ

Δ( 9 )Δ 3 ( 9 )Δ

Δ

24 2113

2133 3

14 24220

2222

2 2132 221

2223

2 2131

130

14 24230

2232

2 2133 231

2233

2 2134

130

= − − − +η C g g g g g g μ( ) ( )(1 ),1 9 210 232

242

23 2102

2202

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

106

= − − − +η g g g g g g μ3 ( ) ( )(1 ),2 220 232

242

24 2102

2202

= − + − +η g g g g g g μ( ) ( )(1 ),3 232

242

210 23 220 24

= − +η g g g g μ( )(1 ),4 210 23 220 24

=+−

=+−

Rη η η η

η ηR

η η η ηη η

,13 1 4 2

12

22 2

3 2 4 1

12

22

,

= = =

r r r rr r r rr r r rr a r r

e r r re r r re r r re r r r

r r e rr r e rr r e rr r e r

Δ , Δ , Δ00

11 12 13 1421 22 23 2431 32 33 3441 42 43 44

01

1 12 13 142 22 23 243 32 33 344 42 43 44

02

11 12 1 1421 22 2 2431 32 3 3441 42 4 44

= = =

r e r rr e r rr e r rr e r r

r r r er r r er r r er r r e

f r r rf r r rf r r rf r r r

Δ , Δ , Δ03

11 1 13 1421 2 23 2431 3 33 3441 4 43 44

04

11 12 13 121 22 23 231 32 33 341 42 43 4

05

1 12 13 14

2 22 23 24

3 32 33 34

4 42 43 44

= = =

r r f rr r f rr r f rr r f r

f r r rf r r rf r r rf r r r

r r f rr r f rr r f rr r f r

Δ , Δ , Δ06

11 12 1 14

21 22 2 24

31 32 3 34

41 42 4 44

07

2 12 13 14

1 22 23 24

4 32 33 34

3 42 43 44

08

11 12 2 14

21 22 1 24

31 32 4 34

41 42 3 44

= = =

s s s ss s s ss s s ss s s s

a s s sb s s sc s s sd s s s

b s s sa s s sd s s sc s s s

Δ , Δ , Δ130

11 12 13 1421 22 23 2431 32 33 3441 42 43 44

131

1 12 13 14

1 22 23 241 32 33 34

1 42 43 44

132

1 12 13 141 22 23 24

1 32 33 341 42 43 44

= =

s s a ss s b ss s c ss s d s

s s b ss s a ss s d ss s c s

Δ , Δ133

11 12 1 14

21 22 1 2431 32 1 34

41 42 1 44

134

11 12 1 1421 22 1 24

31 32 1 3441 42 1 44

= = = = = = = = + += = + + = = = == = + + = += + = + = += − − − = − − −= − − − = − − −= = = = = = = = + += = + + = == = = = + += + = += + = +

= − + − + −

= − − − − −

= −

= − + − + − + − − − +

= − − − − − × − + − + −

+ − − − − − × − − − − −

= − = −

= − = +

= = = − − = +

= = = =

= = = =

= + + − +

− −

r r r r γ mnβ r r r r γ γ m γ n βr r γ γ m γ n β r r γ mnβ r r γ mnβr r γ γ m γ n β e γ m γ n β nβe γ m γ n β m e γ m γ n β nβ e γ m γ n β mf γ γ m γ n β nβ f γ γ m γ n β mf γ γ m γ n β nβ f γ γ m γ n β ms s s s γ mnβ s s s s γ γ m γ n βs s γ γ m γ n β s s γ mnβs s γ mnβ s s γ γ m γ n βa γ m γ n β nβ b γ m γ n β mc γ m γ n β nβ d γ m γ n β mC γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ

C γ γ γ γ γ γ γ γ γ γ γ γ γ γ

C γ γ γ

g γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ

g γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ

γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ

b γ γ γ c γ γ γ γ γ

b c φ b c

g g g g g g g g

a a a a

b b γ g b b γ g

b b c φ φ

, ,, , ,, ( ) ,

( ) , ( ) , ( ) ,( ) , ( ) ,( ) , ( ) ,

3 , 9 ,9 , 3 ,

3 , 93( 9 ) , ( 9 ) ,3( 9 ) , ( 9 )

( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( )( )

[ ( ) ( ) ( )] [( ) ( ) ( )]

[ ( ) ( ) ( )] [ ( ) ( ) ( )]

( )[ ] , ( ) ,

ϑ [( )/2] , [( )/2]

, , ,

1,

,

[ϑ ( 2 ) (2ϑ ϑ )]

γ γ γg

gg

CC

CC

φb

φb

φ

φ

b

11 22 34 43 331 12 21 33 44 32 3302

3322 2

13 24 41 4302

4322 2

14 23 431 31 42 321

32 41 31 3202

3222 2

1 2312

2332 2

2 2302

2322 2

3 2212

2232 2

4 2202

2222 2

1 41 4112

4132 2

2 32 4102

4122 2

3 32 3112

3132 2

4 31 3102

3122 2

11 22 34 43 331 12 21 33 44 32 3302

3322 2

13 24 41 4302

4322 2

14 23 431

31 42 321 32 41 31 3202

3222 2

1 2312

2332 2

1 2302

2322 2

1 2212

2232 2

1 2202

2222 2

16 320 430 3302

14 24 220 220 430 230 330 14 24 230 230 320 220 230

17 240 320 430 3302

220 310 430 330 410 230 320 410 310 330

18 320 430 3302

15 120 220 430 230 330 130 230 320 220 330 220 310 430 330 410 230 320 410 310 330 320 430 3302

140 240

16 110 320 430 3302

120 310 430 330 410 130 320 410 310 330 320 430 3302

14 24 220 220 430 230 330 14 24 230 230 320 220 330

14 24 140 320 430 3302

120 220 430 230 330 130 230 320 220 330 240 320 430 3302

220 310 430 330 410 230 320 410 310 330

320 430 3302 1/2

14 24 320 430 3302

2

1/2 1/2

17 18 19 173ϑ

18 20 173 ϑ

18

01(1)

01(3/2)

10(1)

10(3/2) ϑ

01(2)

01(5/2)

24 19 10(2)

10(5/2)

24ϑ

20

111 2 4 2 2 4

14 24 3202

16

15

16

1716

1816

2 2

2

2 2

2

(C-4)

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

107

References

[1] Mouritz AP, Gellert E, Burchill P, Challis K. Review of advanced composite struc-tures for naval ships and submarines. Compos Struct 2001;53:21–41.

[2] Leissa AW. Buckling of laminated composite plates and shell panels. AFWAL-TR-85-3069; 1985.

[3] Teng JG. Buckling of thin shells: recent advances and trends. Appl Mech Rev1996;49(4):263.

[4] Dinkler D, Pontow J. A model to evaluate dynamic stability of imperfection sensi-tive shells. Comput Mech 2006;37(6):523–9.

[5] Iu VP, Chia CY. Effect of transverse shear on nonlinear vibration and postbuckling ofanti-symmetric cross-ply imperfect cylindrical shells. Int J Mech Sci1988;30(10):705–18.

[6] Fu L, Waas AM. Initial post-buckling behavior of thick rings under uniform externalhydrostatic pressure. J Appl Mech 1995;62(2):338–45.

[7] Han B, Simitses GJ. Analysis of anisotropic laminated cylindrical shells subjected todestabilizing loads. Part II: Numerical results. Compos Struct 1991;19(2):183–205.

[8] Sanders JL. Nonlinear theories for thin shells. Q Appl Math 1963;21(1):21–36.[9] Simitses GJ, Anastasiadis JS. Buckling of axially-loaded, moderately-thick, cylind-

rical laminated shells. Compos Eng 1991;1(6):375–91.[10] Simitses GJ, Anastasiadis JS. Shear deformable theories for cylindrical laminates –

Equilibrium and buckling with applications. AIAA J 1992;30(3):826–34.[11] Reddy JN, Savoia M. Layer-wise shell theory for postbuckling of laminated circular

cylindrical shells. AIAA J 1992;30(8):2148–54.[12] Sun XS, Tan VBC, Chen Y, Tan LB, Jaiman RK, Tay TE. Stress analysis of multi-

layered hollow anisotropic composite cylindrical structures using the homo-genization method. Acta Mech 2014;225(6):1649–72.

[13] Corona E, Rodrigues A. Bending of long cross-ply composite circular cylinders.Compos Eng 1995;5(2):163–82.

[14] Yang LC, Chen ZP, Chen FC, Guo WJ, Cao GW. Buckling of cylindrical shells withgeneral axisymmetric thickness imperfections under external pressure. Eur J MechA/Solids 2013;38:90–9.

[15] Papadakis G. Buckling of thick cylindrical shells under external pressure: a newanalytical expression for the critical load and comparison with elasticity solutions.Int J Solids Struct 2008;45(20):5308–21.

[16] Schillo C, Röstermundt D, Krause D. Experimental and numerical study on the in-fluence of imperfections on the buckling load of unstiffened CFRP shells. ComposStruct 2015;131:128–38.

[17] Kardomateas GA. Buckling of thick orthotropic cylindrical shells under externalpressure. J Appl Mech 1993;60(1):195–202.

[18] Kardomateas GA. Bifurcation of equilibrium in thick orthotropic cylindrical shellsunder axial compression. J Appl Mech 1995;62(1):43–52.

[19] Kardomateas GA, Philobos MS. Buckling of thick orthotropic cylindrical shellsunder combined external pressure and axial compression. AIAA J1995;33(10):1946–53.

[20] Tornabene F, Brischetto S, Fantuzzi N, Bacciocchi M. Boundary conditions in 2Dnumerical and 3D exact models for cylindrical bending analysis of functionallygraded structures. Shock Vib 2016:1–17.

[21] Brischetto S, Tornabene F, Fantuzzi N, Bacciocchi M. Interpretation of boundaryconditions in the analytical and numerical shell solutions for mode analysis ofmultilayered structures. Int J Mech Sci 2017;122:18–28.

[22] Koiter WT. On the stability of elastic equilibrium. University of Delft; 1945. [Ph.D.thesis].

[23] Arciniega RA, Gonçalves PB, Reddy JN. Buckling and postbuckling analysis of la-minated cylindrical shells using the third-order shear deformation theory. Int JStruct Stab Dyn 2004;04(03):293–312.

[24] Salahshour S, Fallah F. Elastic collapse of thin long cylindrical shells under externalpressure. Thin-Walled Struct 2018;124:81–7.

[25] Wang SS, Srinivasan S, Hu HT, HajAli R. Effect of material nonlinearity on bucklingand postbuckling of fiber composite laminated plates and cylindrical shells. ComposStruct 1995;33(1):7–15.

[26] Semenyuk NP, Trach VM. Buckling and the initial postbuckling behavior of cy-lindrical shells made of composites with one plane of symmetry. Mech ComposMater 2007;43(2):141–58.

[27] Semenyuk NP, Trach VM. Stability and initial postbuckling behavior of anisotropic

cylindrical shells under external pressure. Int Appl Mech 2007;43(3):314–28.[28] Priyadarsini RS, Kalyanaraman V, Srinivasan SM. Numerical and experimental

study of buckling of advanced fiber composite cylinders under axial compression.Int J Struct Stab Dyn 2012;12(04):1250028.

[29] Mistry J, Gibson AG, Wu YS. Failure of composite cylinders under combined ex-ternal pressure and axial loading. Compos Struct 1992;22(4):193–200.

[30] Eslami MR, Shariyat M, Shakeri M. Layerwise theory for dynamic buckling andpostbuckling of laminated composite cylindrical shells. AIAA J1998;36(10):1874–82.

[31] Eslami MR, Shariyat M. A high-order theory for dynamic buckling and postbucklinganalysis of laminated cylindrical shells. J Pressure Vessel Technol-Trans1999;121(1):94–102.

[32] Wang B, Zhu SY, Hao P, Bi XJ, Du KF, Chen BQ, et al. Buckling of quasi-perfectcylindrical shell under axial compression: a combined experimental and numericalinvestigation. Int J Solids Struct 2018;130–131:232–47.

[33] Lindgaard E, Lund E. Nonlinear buckling optimization of composite structures.Comput Methods Appl Mech Eng 2010;199(37–40):2319–30.

[34] Liguori FS, Madeo A, Magisano D, Leonetti L, Garcea G. Post-buckling optimisationstrategy of imperfection sensitive composite shells using Koiter method and MonteCarlo simulation. Compos Struct 2018;192:654–70.

[35] Weaver PM, Driesen JR, Roberts P. Anisotropic effects in the compression bucklingof laminated composite cylindrical shells. Compos Sci Technol 2002;62(1):91–105.

[36] Shen HS. Boundary layer theory for the buckling and postbuckling of an anisotropiclaminated cylindrical shell, Part I: prediction under axial compression. ComposStruct 2008;82(3):346–61.

[37] Shen HS. Boundary layer theory for the buckling and postbuckling of an anisotropiclaminated cylindrical shell, Part II: prediction under external pressure. ComposStruct 2008;82(3):362–70.

[38] Shen HS. Boundary layer theory for the buckling and postbuckling of an anisotropiclaminated cylindrical shell, Part III: Prediction under torsion. Compos Struct2008;82(3):371–81.

[39] Li ZM, Lin ZQ. Non-linear buckling and postbuckling of shear deformable aniso-tropic laminated cylindrical shell subjected to varying external pressure loads.Compos Struct 2010;92(2):553–67.

[40] Li ZM, Shen HS. Postbuckling of shear-deformable anisotropic laminated cylindricalshells under axial compression. Int J Struct Stab Dyn 2008;8(03):389–414.

[41] Li ZM, Shen HS. Buckling and postbuckling analysis of shear deformable anisotropiclaminated cylindrical shells under torsion. Mech Adv Mater Struct2009;16(1):46–62.

[42] Shen HS, Xiang Y. Buckling and postbuckling of anisotropic laminated cylindricalshells under combined axial compression and torsion. Compos Struct2008;84(4):375–86.

[43] Li ZM, Qiao PZ. Buckling and postbuckling of anisotropic laminated cylindricalshells under combined external pressure and axial compression in thermal en-vironments. Compos Struct 2015;119:709–26.

[44] Reddy JN. Mechanics of laminated plates and shells theory and analysis. CRC Press;1997.

[45] Qatu MS. Vibration of laminated shells and plates. Elsevier Academic Press; 2004.[46] Reddy JN, Liu CF. A higher-order shear deformation theory of laminated elastic

shells. Int J Eng Sci 1985;23(3):319–30.[47] Hur S, Son H, Kweon J, Choi J. Postbuckling of composite cylinders under external

hydrostatic pressure. Compos Struct 2008;86:114–24.[48] Smerdov AA. A computational study in optimum formulations of optimization

problems on laminated cylindrical shells for buckling I. Shells under axial com-pression. Compos Sci Technol 2000;60(11):2057–66.

[49] Kaddour AS, Soden PD, Hinton MJ. Failure of± 55 degree filament wound glass/epoxy composite tubes under biaxial compression. J Compos Mater1998;32(18):1618–45.

[50] Tafreshi A, Bailey C. Instability of imperfect composite cylindrical shells undercombined loading. Compos Struct 2007;80(1):49–64.

[51] Yuan FG, Hsieh CC. Three-dimensional wave propagation in composite cylindricalshells. Compos Struct 1998;42(2):153–67.

[52] Shen HS. A two-step perturbation method in nonlinear analysis of beams, plates andshells. Beijing: Higher Education Press; 2013.

Z.-M. Li et al. Composite Structures 198 (2018) 84–108

108

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