a unified chebyshev–ritz formulation for vibration analysis of composite laminated deep open...

Arch Appl Mech (2014) 84:441–471DOI 10.1007/s00419-013-0810-1

ORIGINAL

Tiangui Ye · Guoyong Jin · Zhu Su · Xingzhao Jia

A unified Chebyshev–Ritz formulation for vibration analysisof composite laminated deep open shells with arbitraryboundary conditions

Received: 8 October 2013 / Accepted: 27 November 2013 / Published online: 21 December 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract In this paper, a unified Chebyshev–Ritz formulation is presented to investigate the vibrations ofcomposite laminated deep open shells with various shell curvatures and arbitrary restraints, including cylindri-cal, conical and spherical ones. The general first-order shear deformation shell theory is employed to includethe effects of rotary inertias and shear deformation. Under the current framework, regardless of boundaryconditions, each of displacements and rotations of the open shells is invariantly expressed as Chebyshevorthogonal polynomials of first kind in both directions. Then, the accurate solutions are obtained by using theRayleigh–Ritz procedure based on the energy functional of the open shells. The convergence and accuracy ofthe present formulation are verified by a considerable number of convergence tests and comparisons. A varietyof numerical examples are presented for the vibrations of the composite laminated deep shells with variousgeometric dimensions and lamination schemes. Different sets of classical constraints, elastic supports as wellas their combinations are considered. These results may serve as reference data for future researches. Paramet-ric studies are also undertaken, giving insight into the effects of elastic restraint parameters, fiber orientation,layer number, subtended angle as well as conical angle on the vibration frequencies of the composite openshells.

Keywords Vibration analysis · Chebyshev–Ritz formulation · Laminated open cylindrical shells · Laminatedopen conical shells · Laminated open spherical shells · Arbitrary boundary conditions

1 Introduction

Composite laminated deep open shells, consisting of segments of shells of revolution (e.g., open cylindrical,conical and spherical shells), have a wide range of engineering applications, particularly in aerospace crafts,military hardware and civil constructions. The composite laminated open shells in these applications can besubjected to various boundary conditions, such as classical restraints, elastic supports and their combinations.Understanding the vibration characteristics of these shell components is particularly important for engineersto design suitable structures with low vibration and noise radiation characteristics. Compared with many otherkinds of open shells, the open cylindrical, conical and spherical shells are most frequently encountered in theengineering practices. Therefore, the present work is focused on the vibration analysis of these three types ofopen shells with arbitrary boundary conditions.

In the past few decades, extensive researches regarding the vibration analysis of composite laminatedshells have been carried out. Recently, a modified Fourier method had been used to predict the vibration

T. Ye · G. Jin (B) · Z. Su · X. JiaCollege of Power and Energy Engineering, Harbin Engineering University,Harbin 150001, People’s Republic of ChinaE-mail: [email protected]; [email protected]

T. YeE-mail: [email protected]

442 T. Ye et al.

characteristics of thin and moderately thick composite cylindrical shells with general elastic restraints andinternal line supports by Jin et al. [1,2]. Qu et al. [3] analyzed the free and forced vibrations of compositelaminated shells of revolution, in which a modified variational principle in conjunction with a multi-segmentpartitioning technique is employed to derive the formulation based on the first-order shear deformation theory.Exhaustive descriptions of various theoretical formulations and solution methods on this subject are availablein the review articles [4–7] and monographs [8–10]. However, most of these researches are restricted to theclosed shells (shells of revolution) and the information available for the vibration characteristics of compositelaminated deep open shells; especially, the open conical and spherical ones is very limited despite their practicalimportance. The most likely reason for this lacuna lies in the analytical difficulties involved: for a completelyclosed shell such as circular cylindrical, circular conical and spherical shells, the assumed 2D displacementfield can be reduced to a quasi 1D problem through Fourier decomposition of the circumferential wave motion.However, for an open shell, the assumption of whole periodic wave numbers in the circumferential directionis inappropriate, and thus, a set of complete three-dimensional analysis is required and resort must be made toa full two-dimensional solution scheme. Such a scheme will inevitably be complicated further by the depen-dence of the circumferential arc length on its meridional location [11]. This forms a major deterrent so thatthe analyses of open shells have not been widely available.

To overcome the difficulties, the shallow shell theories that are based on the general shell equations andcertain additional assumptions have been developed. A few of notable works on the free vibration analyses ofcomposite shallow shells can be found in Refs. [12–27]. Recently, the vibration behaviors of thin compositelaminated shallow shells with general elastic boundary conditions were reported by Ye et al. [28] using amodified Fourier series method. However, the shallow shell theories are limited to the open shells which havesmall curvatures (i.e., large radius of curvature). It has been proven that the shallow shell theories will giveinaccurate results for the lower frequencies when applied to deep open shells [8]. Researches on the vibrationcharacteristics of deep open shells are still limited in the open literature. Among those available, Bardell etal. [29] studied the vibration of a general three-layer conical sandwich panel based on the h-p version of thefinite element method. The static and free vibration analysis of laminated shells is performed by radial basisfunctions collocation, according to a layerwise deformation theory by Ferreira et al. [14]. A general surveyand comparison for variety of simply supported shallow spherical, cylindrical, plate and saddle panels in rect-angular planform was made by Chern and Chao [30]. By using the finite element method and classical shelltheories, Selmane and Lakis [31] investigated the dynamic and static behaviors of thin, elastic, anisotropicand non-uniform open cylindrical shells. Free vibration of simply supported laminated spherical panels withrandom material properties is reported by Singh et al. [32] based on the high-order shear deformation shallowtheory. The influence of classical boundary conditions and transverse shear on the vibration of angle-ply lam-inated plates and cylindrical panels was investigated by Soldatos and Messina [33]. This analysis was basedon the unified shear deformable Love-type theory and Ritz method. The vibration characteristics of twistedcantilevered conical composite shells were reported by Lee et al. [34] and Hu et al. [35]. Also, vibration of can-tilevered laminated composite shallow conical shells was presented by Lim et al. [36]. In contrast to the openconical and spherical shells, the researches on the open cylindrical ones are considerable. The free vibrationof two-side simply supported laminated cylindrical panels is analyzed by Zhao et al. [37,38] via the mesh-free kp-Ritz method. Comprehensive studies of thin, laminated cylindrical panels are conducted by Bardellet al. [39] by using the h − p version of the FEM. The natural frequencies of thin two-side simply supportedlaminated cylindrical panels were calculated by Bercin [40]. Some other contributors on this subject are thefollowing: Messina and Soldatos [41,42], Selmane and Lakis [43], Lee and Reddy [44], Messina [45], etc.

In view of the aforementioned issues and concerns, it should be emphasized that most of the existingcontributions were restricted to composite laminated open shells with large radius of curvature or subjectedto a limited set of classical supports. The vibration behaviors of composite laminated deep open shells withcircumferentially varying geometry and different combinations of classical and non-classical boundary con-straints have until now remained unsolved. Moreover, most of the available solution procedures in the openliterature are often only customized for a specific set of restraint conditions, which may not be appropriate forpractical applications because there are hundreds of different combinations of boundary conditions for an openshell. Developing an accurate, robust and efficient method which is capable of simplifying solution algorithms,reducing model input data and universally dealing with various boundary conditions is still of great interest toboth researchers and engineers.

The main purpose of this study is to complement the vibration investigations of composite laminated deepopen shells with arbitrary boundary conditions and develop a unified and accurate formulation to provide someuseful results for the titled problem, which may be used as reference data for future researchers. Under the cur-

Arbitrary boundary conditions 443

rent formulation, the first-order shear deformation shell theory based on the general shell equations is employedto include the effects of rotary inertias and shear deformation. Each displacement and rotation of the open shells,regardless of boundary conditions, is invariantly expressed as Chebyshev polynomials of first kind in both direc-tions. Thereby, all the Chebyshev expanded coefficients are treated equally and independently as the generalizedcoordinates and solved directly by using the Rayleigh–Ritz procedure. The convergence and accuracy of thepresent formulation are checked by a considerable number of convergence tests and comparisons. A variety ofnumerical examples are presented for the free vibration of shallow and deep composite laminated cylindrical,conical and spherical panels with various geometric dimensions and lamination schemes. Different combina-tions of classical boundary conditions (e.g., completely free, shear-diaphragm restrained, simply supported andclamped) and uniform elastic restraints as well as their combinations are considered in the investigation. Para-metric studies are also undertaken, giving insight into the effects of elastic restraint parameter, fiber orientation,layer number, subtended angle as well as conical angle on the vibration frequencies of the composite open shells.

2 Theoretical formulations

2.1 Preliminaries

A doubly curved composite laminated deep open shell with length Lα , width Lβ and uniform total thickness hshown in Fig. 1 is selected as the analysis model. Let (α, β and z) denote the orthogonal curvilinear coordinatesystem such that α and β curves are lines of curvature on the middle surface (z = 0). Rα and Rβ represent thevalues of the principal radii of curvature of the middle surface. The considered open shell is composed of NLorthotropic layers. Unless otherwise stated, all the layers are assumed to be of equal thickness (h/NL). Thecharacters Zk and Zk+1 are used to indicate the distances from the undersurface and the top surface of the k’thlayer to the referenced plane. According to the FSDT assumptions, the displacement field in the shell space isgiven as follows:

u(α, β, z, t) = u0(α, β, t)+ zφα(α, β, t)

v(α, β, z, t) = v0(α, β, t)+ zφβ(α, β, t)

w(α, β, z, t) = w0(α, β, t) (1)

where t is time variable; u, v andw are the generalized displacements along the α, β and z coordinates, respec-tively. u0, v0 andw0 are the displacements of a point on the middle surface, and φα and φβ separately representthe rotations at z = 0 of transverse normal to the mid-surface with respect to theβ, α coordinates. The strain dis-placement relations at any point in the shell space with reference to curvilinear coordinate system are as follows:

εα = ε0α + zχα, εβ = ε0

β + zχβ, γαβ = γ 0αβ + zχαβ

γαz = ∂w0

∂α+ φα, γβz = ∂w0

∂β+ φβ (2)

z

w

vu

Rα

Rβ

β

α

(b)(a)

Lα

Lβ

Middle surface

h

k+1

k

Fig. 1 Geometry and notations of a laminated open shell: a coordinates; b lamination scheme

444 T. Ye et al.

x

zz

θ θ

θ0 θ0

θ0

ϕ0

ϕ1

ϕ

θ

ϕc z

o

o

oL

LR R0

R1

z

x

R

(a) (b) (c)

Fig. 2 Coordinate systems and geometric parameters of three types of open shells: a open cylindrical shell; b open conical shell;c open spherical shell

where ε0α, ε

0β and γ 0

αβ denote the normal and shear strains in the (α, β, z) coordinate system. χα, χβ and χαβ arethe curvature and twist changes; the transverse shear strains γαz and γβz are constant through the thickness. Theparameters (ε0

α, ε0β, γ

0αβ, χα, χβ, χαβ) are defined in terms of the middle surface displacements and rotation

components as [8]:

ε0α = 1

A∂u0∂α

+ v0AB

∂A∂β

+ w0Rα, χα = 1

A∂φα∂α

+ φβAB

∂A∂β

ε0β = 1

B∂v0∂β

+ u0AB

∂B∂α

+ w0Rβ, χβ = 1

B∂φβ∂β

+ φαAB

∂B∂α

ε0αβ = 1

A∂v0∂α

− u0AB

∂A∂β

+ 1B∂u∂β

− v0AB

∂B∂α, χαβ = 1

A∂φβ∂α

− φαAB

∂A∂β

+ 1B∂φα∂β

− φβAB

∂B∂α

(3)

where the quantities A and B are the Lamé parameters. According to Fig. 2, the coordinate systems andgeometric parameters (Rα, Rβ, A and B) of the deep open cylindrical, conical and spherical shells underconsideration are the following:

(a) Cylindrical shells: α = x, β = θ, Rα = ∞, Rβ = R, A = 1, B = R;(b) Conical shells: α = x, β = θ, Rα = ∞, Rβ = x tan α0, A = 1, B = x sin α0;(c) Spherical shells: α = ϕ, β = θ, Rα = R, Rβ = R, A = R, B = R sin ϕ.

Considering Eq. (2), the stresses for the k’th layer are determined by the generalized Hooke’s law, namely:

⎧⎨

⎩

σασβταβ

⎫⎬

⎭=⎡

⎢⎣

Qk11 Qk

12 Qk16

Qk12 Qk

22 Qk26

Qk16 Qk

26 Qk66

⎤

⎥⎦

⎧⎨

⎩

εαεβγαβ

⎫⎬

⎭,

{τβzταz

}

=[

Qk44 Qk

45

Qk45 Qk

55

]{γβzγαz

}

(4)

in which σα and σβ are the normal stresses in the α, β directions, respectively. τβz, ταz and ταβ are the corre-

sponding shear stresses in the curvilinear coordinate system. Qki j (i, j = 1, 2, 4 ∼ 6) are the lamina stiffness

coefficients, they are defined as follows:

⎡

⎢⎢⎢⎢⎢⎢⎣

Qk11 Qk

12 0 0 Qk16

Qk12 Qk

22 0 0 Qk26

0 0 Qk44 Qk

45 0

0 0 Qk45 Qk

55 0

Qk16 Qk

26 0 0 Qk66

⎤

⎥⎥⎥⎥⎥⎥⎦

= T

⎡

⎢⎢⎢⎢⎣

Qk11 Qk

12 0 0 0Qk

21 Qk22 0 0 0

0 0 Qk44 0 0

0 0 0 Qk55 0

0 0 0 0 Qk66

⎤

⎥⎥⎥⎥⎦

TT (5)


where Qki j (i, j = 1, 2, 4 ∼ 6) are the material constant of the k’th layer in the laminate coordinate system.

For the orthotropic material, they are known in terms of the engineering constants as:

Qk11 = E1

1 − μ12μ21, Qk

12 = μ12 E2

1 − μ12μ21= Qk

21, Qk22 = E2

1 − μ12μ21

Qk44 = G23, Qk

55 = G13, Qk66 = G12 (6)

where E1 is the longitudinal modulus, E2 is the transverse modulus, and μ12 is the major Poisson’s ratio.The Poisson’s ratio μ21 are determined by equation μ12 E2 = μ21 E1. G12,G13 and G23 are shear moduli. Itshould be noted that by letting E1 = E2,G12 = G13 = G23 = E1/(2+2μ12), the present formulation can bereadily used to analyze the isotropic deep open shells with general boundary restraints. T is the transformationmatrix, which is defined as:

T =

⎡

⎢⎢⎢⎣

cos2 ϑ sin2 ϑ 0 0 −2 sin ϑ cosϑsin2 ϑ cos2 ϑ 0 0 2 sin ϑ cosϑ

0 0 cosϑ sin ϑ 00 0 − sin ϑ cosϑ 0

sin ϑ cosϑ − sin ϑ cosϑ 0 0 cos2 ϑ − sin2 ϑ

⎤

⎥⎥⎥⎦

(7)

where ϑ denotes the included angle between the material coordinate of the k’th layer and the α-axis of theshell. The force and moment resultants are obtained by integrating the stresses over the shell thickness:

⎡

⎣NαNβNαβ

⎤

⎦ =h/2∫

−h/2

⎡

⎣σασβταβ

⎤

⎦dz,

⎡

⎣Mα

Mβ

Mαβ

⎤

⎦ =h/2∫

−h/2

⎡

⎣σασβταβ

⎤

⎦zdz,

[Qα

Qβ

]

=h/2∫

−h/2

[ταzτβz

]

dz (8)

The constitutive equations relating the force and moment resultants to the strains and curvatures of the referencesurface are given in the matrix form:

⎡

⎢⎢⎢⎢⎢⎣

NαNβNαβMα

Mβ

Mαβ

⎤

⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎣

A11 A12 A16 B11 B12 B16A12 A22 A26 B12 B22 B26A16 A26 A66 B16 B26 B66B11 B12 B16 D11 D12 D16B12 B22 B26 D12 D22 D26B16 B26 B66 D16 D26 D66

⎤

⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎣

ε0α

ε0β

γ 0αβ

χαχβχαβ

⎤

⎥⎥⎥⎥⎥⎥⎦

,

[Qβ

Qα

]

= kc

[A44 A45A45 A55

] [γβzγαz

]

(9)

where quantities (Nα, Nβ, Nαβ) are the normal and shear force resultants, and (Mα,Mβ,Mαβ) denote thebending and twisting moment resultants. Qβ and Qα are the transverse shear force resultants. kc is the shearcorrection factor. The existence of the shear correction factor is to make sure that the strain energy due to thetransverse shear stresses in Eq. (8) equals the strain energy due to the true transverse shear stresses predictedby the three-dimensional elasticity theory [10]. In the present analysis, the shear correction factor is typicallytaken at 5/6. Ai j , Bi j and Di j are the stretching, stretching-bending coupling and bending stiffness. They aredefined as follows:

Ai j =NL∑

k=1

Qki j (Zk+1 − Zk), (i, j = 1, 2, 6); Ai j = kc

NL∑

k=1

Qki j (Zk+1 − Zk), (i, j = 4, 5)

Bi j = 1

2

NL∑

k=1

Qki j

[Z2

k+1 − Z2k

], Di j = 1

3

NL∑

k=1

Qki j

[Z3

k+1 − Z3k

], (i, j = 1, 2, 4, 5, 6) (10)

It should be noted that for open shells, which are symmetrical with respect to their middle surface, Bi j = 0. Inthis part, the kinematic relations and stress resultants of the composite laminated open shell based on generalshell equations are presented, and in the next subsection, the energy expressions of the open shell are obtainedby using these relations and resultants.

446 T. Ye et al.

2.2 Energy expressions

It is well known that the exact solutions are generally available only for the open shells that are simply sup-ported along at least one pair of opposite edges. For other boundary conditions, however, one may have to useapproximate methods such as the Rayleigh–Ritz method [47]. Since the main purpose of the present work is todevelop an efficient formulation, which can universally deal with various boundary conditions, the Rayleigh–Ritz method is employed due to its simplicity, stability and efficiency in numerical implementation. Therefore,the energy functional of the open shell is established firstly. The strain energy (UV ) of the considered openshell is defined as follows:

UV = 1

2

∫

S

{Nαε

0α + Nβε

0β + Nαβγ

0αβ + Mαχα + Mβχβ + Mαβχαβ + Qαγαz + Qβχβz

}dS (11)

where S represents the middle surface area of the shell. Substituting Eqs. (3) and (9) into Eq. (11), the strainenergy expression of the open shell is divided into three parts (i.e., UV = Us+Ub+Ubs) and written in terms ofshell displacements and rotations. Considering the Lamé parameter A for the open shells under considerationis constant, i.e., ∂A/∂α = 0 and ∂A/∂β = 0, these energy expressions are written as follows:

Us = 1

2

∫ ∫

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A11

(1A∂u0∂α

+ w0Rα

)2 + A22

(1B∂v0∂β

+ u0AB

∂B∂α

+ w0Rβ

)2

+2A12

(1B∂v0∂β

+ u0AB

∂B∂α

+ w0Rβ

) (1A∂u0∂α

+ w0Rα

)

+2A16

(1A∂u0∂α

+ w0Rα

) (1A∂v0∂α

+ 1B∂u0∂β

− v0AB

∂B∂α

)

+2A26

(1B∂v0∂β

+ u0AB

∂B∂α

+ w0Rβ

) (1A∂v0∂α

+ 1B∂u0∂β

− v0AB

∂B∂α

)

+A66

(1A∂v0∂α

+ 1B∂u0∂β

− v0AB

∂B∂α

)2 + A44

(1B∂w0∂β

+ φβ

)2

+2A45

(1B∂w0∂β

+ φβ

) (1A∂w0∂α

+ φα

)+ A55

(1A∂w0∂α

+ φα

)2

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ABdαdβ (12)

Ub = 1

2

∫ ∫

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

D11

(1A∂φα∂α

)2 + 2D12

(1B∂φβ∂β

+ φαAB

∂B∂α

) (1A∂φα∂α

)

+D22

(1B∂φβ∂β

+ φαAB

∂B∂α

)2 + D66

(1A∂φβ∂α

+ 1B∂φα∂β

− φβAB

∂B∂α

)2

+2D16

(1A∂φα∂α

) (1A∂φβ∂α

+ 1B∂φα∂β

− φβAB

∂B∂α

)

+2D26

(1B∂φβ∂β

+ φαAB

∂B∂α

) (1A∂φβ∂α

+ 1B∂φα∂β

− φβAB

∂B∂α

)

⎫⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎭

ABdαdβ (13)

Ubs =∫ ∫

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[B11

(1A∂φα∂α

)+ B12

(1B∂φβ∂β

+ φαAB

∂B∂α

)] (1A∂u0∂α

+ w0Rα

)

+B16

(1A∂φβ∂α

+ 1B∂φα∂β

− φβAB

∂B∂α

) (1A∂u0∂α

+ w0Rα

)

+[

B12

(1A∂φα∂α

)+ B22

(1B∂φβ∂β

+ φαAB

∂B∂α

)] (1B∂v0∂β

+ u0AB

∂B∂α

+ w0Rβ

)

+B26

(1A∂φβ∂α

+ 1B∂φα∂β

− φβAB

∂B∂α

) (1B∂v0∂β

+ u0AB

∂B∂α

+ w0Rβ

)

+[

B16

(1A∂φα∂α

)+ B26

(1B∂φβ∂β

+ φαAB

∂B∂α

)] (1A∂v0∂α

+ 1B∂u0∂β

− v0AB

∂B∂α

)

+B66

(1A∂φβ∂α

+ 1B∂φα∂β

− φβAB

∂B∂α

) (1A∂v0∂α

+ 1B∂u0∂β

− v0AB

∂B∂α

)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ABdαdβ (14)

where Us,Ub and Ubs indicate the stretching, bending and bending–stretching coupling energy expressions,respectively. The kinetic energy (T ) of the open shell is written as follows:

T = 1

2

∫ ∫⎧⎪⎨

⎪⎩

I0

(∂u0∂t

)2 + 2I1

(∂u0∂t

) (∂φα∂t

)+ I2

(∂φα∂t

)2 + I0

(∂v0∂t

)2

+2I1

(∂v0∂t

) (∂φβ∂t

)+ I2

(∂φβ∂t

)2 + I0

(∂w0∂t

)2

⎫⎪⎬

⎪⎭ABdαdβ (15)

and the inertia terms are the following:

[I0, I1, I2] =⎛

⎜⎝

NL∑

k=1

Zk+1∫

Zk

ρk[1, z1, z2]dz

⎞

⎟⎠ (16)


in which ρk is the mass of the k’ th layer per unit middle surface area. The arbitrary boundary conditions ofthe open shell are implemented by introducing three groups of linear springs (ku, kv, kw) and two groups ofrotational springs (Kα, Kβ), which are distributed uniformly along the boundary at each of the shell ends.The stiffness of the boundary springs can take any value from zero to infinity to better model many real-worldrestraint conditions. For instance, the clamped restraint condition is essentially obtained by setting the springstiffness substantially larger than the bending rigidity of the involved open shell. For the sake of simplicity,kuψ, kvψ , kwψ , K α

ψ and K βψ(ψ = α0, β0, αL and βL) are used to indicate the stiffness (per unit length) of the

boundary springs and the subscripts α0, β0, αL and βL separately denote the springs distributed at the edgesα = 0. β = 0, α = Lα and α = Lβ . Hence, the potential energy (Psp) stored in the boundary springs is

Psp = 1

2

∫⎧⎨

⎩

[kuα0u2

0 + kvα0v20 + kwα0w

20 + K α

α0φ2α + K β

α0φ2β

] ∣∣∣α=0

+[kuαLu2

0 + kvαLv20 + kwαLw

20 + K α

αLφ2α + K β

αLφ2β

] ∣∣∣α=Lα

⎫⎬

⎭Bdβ

+1

2

∫ { [kuβ0u2

0 + kvβ0v20 + kwβ0w

20 + K α

β0φ2α + K β

β0φ2β ]β=0

+[kuβLu2

0 + kvβLv20 + kwβLw

20 + K α

βLφ2α + K β

βLφ2β ]β=Lβ

}

Adα (17)

Thus, the Lagrangian energy functional (L) of the shell can be defined in terms of the aforementioned energyexpressions as follows:

L = T − Us − Ub − Psp (18)

Once the Lagrangian energy functional of the open shell is established, the following task is to construct a setof appropriate admissible displacement functions and determine these functions.

2.3 Admissible displacement functions

The selection of appropriate admissible displacement functions is of particular importance in the Rayleigh–Ritzmethod because the accuracy of the solution will usually depend upon how well the actual displacement canbe faithfully represented by them [47]. The main advantage of the Lagrangian energy functional given in Eq.(18) is that the choice of the admissible displacement functions of the open shells is considerably simplified,and any continuous, independent, complete basis functions may be employed to produce accurate results. Thereason lies in the fact that the geometric boundary conditions in an open shell are relaxed and enforced throughboundary springs, which can be seen as penalty parameters [49,50], and there is no need to explicitly satisfy theessential and natural conditions on these boundaries for the admissible displacement functions in advance [51].In this work, the displacements and rotation components of the open shell are generally expanded, regardlessof boundary conditions, as Chebyshev polynomials of first kind, written as:

u0(α, β, t) =∞∑

m=0

∞∑

n=0

AmnTm(α)Pn(β)ejωt ; φα(α, β, t) =

∞∑

m=0

∞∑

n=0

DmnTm(α)P(β)ejωt

v0(α, β, t) =∞∑

m=0

∞∑

n=0

BmnTm(α)P(β)ejωt ; φβ(α, β, t) =

∞∑

m=0

∞∑

n=0

EmnTm(α)P(β)ejωt

w0(α, β, t) =∞∑

m=0

∞∑

n=0

CmnTm(α)P(β)ejωt

(19)

where Amn, Bmn,Cmn, Dmn and Emn are the corresponding Chebyshev expanded coefficients; Tm(α) andPn(β) are the mth and nth order Chebyshev polynomial for the displacement components in the α and βdirections, respectively. The Chebyshev polynomials of first kind are written as follows [52]:

T0(α) = 1, T1(α) = α, Tm(α) = 2αTm−1(α)− Tm−2(α) (m > 2)

P0(β) = 1, P1(β) = β, Pn(β) = 2βPn−1(β)− Pn−2(β) (n > 2) (20)

It should be remarked here that the Chebyshev polynomials of first kind given in Eq. (20) are complete andorthogonal polynomials defined on the interval of [−1, 1]. Thus, a coordinate transformation from α and β (forα ∈ [0, Lα] and β ∈ [0, Lβ ], to α, β ∈ [−1, 1]) needs to be introduced to implement the present analysis, i.e.,

448 T. Ye et al.

α = Lα(α + 1)/2 and β = Lβ(β + 1)/2. It is stressed that only finite terms of the displacement expressionsare used in actual calculations due to the limited speed, the capacity and the numerical accuracy of computers.For the sake of computational simplicity, it is assumed that the numbers of polynomial terms truncated forthe displacements and rotation components are uniformly chose as M and N . The good accuracy and fastconvergence behavior of the Chebyshev series solution will be shown later.

2.4 Solution procedure

With the admissible displacement functions and energy expressions of the open shell given in previous sub-sections, the remained task is to determine the unknown expansion coefficients in the admissible displacementfunctions. In the present work, the Rayleigh–Ritz procedure is adopted to obtain the vibration results of the openshell. Substituting the energy expressions and the displacement functions of the open shell into Eq. (18) and min-imizing the total expression of the Lagrangian energy functional with respect to the undetermined coefficients:

∂L

∂α= 0, α = Amn, Bmn,Cmn, Dmn, Emn, (21)

a total of 5 ∗ (M + 1) ∗ (N + 1) equations are obtained. They can be summed up in a matrix form:⎛

⎜⎜⎜⎜⎝

⎡

⎢⎢⎢⎢⎣

Ku,u Ku,v Ku,w Ku,α Ku,β

KTu,v Kv,v Kv,w Kv,α Kv,β

KTu,w KT

v,w Kw,w Kw,α Kw,β

KTu,α KT

v,α KTw,α Kα,α Kα,β

KTu,β KT

v,β KTw,β KT

α,β Kβ,β

⎤

⎥⎥⎥⎥⎦

− ω2

⎡

⎢⎢⎢⎢⎣

Mu,u 0 0 Mu,α 00 Mv,v 0 0 Mv,β

0 0 Mw,w 0 0MT

u,α 0 0 Mα,α 00 MT

v,β 0 0 Mβ,β

⎤

⎥⎥⎥⎥⎦

⎞

⎟⎟⎟⎟⎠

G = 0 (22)

where superscript T represents the transposition operator. The elements of the sub-matrices Ki, j and Mi, j(i, j = u, v and w) are given in Appendices A and B. G represent the collection of the undetermined coeffi-cients and is given as follows:

G = [Gu0 ,Gv0 ,Gw0 ,Gφα ,Gφβ]T

(23)

where

Gu = [A00, . . . , A0n, . . . , A0N , A10, . . . , A1n, . . . , Amn, . . . , AM N]

Gv = [B00, . . . , B0n, . . . , B0N , B10, . . . , B1n, . . . , Bmn, . . . , BM N]

Gw = [C00, . . . ,C0n, . . . ,C0N ,C10, . . . ,C1n, . . . ,Cmn, . . . ,CM N]

Gφα = [D00, . . . , D0n, . . . , D0N , D10, . . . , D1n, . . . , Dmn, . . . , DM N]

Gφβ = [E00, . . . , E0n, . . . , E0N , E10, . . . , E1n, . . . , Emn, . . . , EM N]

The natural frequencies and mode shapes of the open shell can now be easily obtained by solving a stan-dard matrix eigenproblem. Once the coefficients in the Chebyshev expansions are solved from Eq. (22), thedisplacements and rotation components of the shell can be directly determined by using Eq. (19). It should benoted that Eq. (22) represents free vibration analysis of the open shell. When the forced vibration is involved,by adding the work done by external force in the right-hand side of the Lagrangian energy functional, apply-ing the Rayleigh–Ritz procedure and summing the loading vector F on the right side of Eq. (22), then thecharacteristic equation for the forced vibration of the open shell is readily obtained.

3 Numerical examples and discussion

In this section, free vibration characteristics of the composite laminated deep open cylindrical, conical andspherical shells are investigated. In summary, the investigation is arranged as follows: first, the number ofterms in the displacement expressions is investigated and the comparisons between the present results andthe available benchmarks are made, aiming to verify the convergence and accuracy of the Chebyshev–Ritzformulation. The effects of the elastic restraint parameters are also studied. Then, a variety of new vibration


results including frequencies and modes shapes for the composite laminated open cylindrical and conical shellswith various combinations of classical and uniform elastic restraints as well as different shell parameters (e.g.,geometric parameters, material properties and lamination schemes) are presented. Illustrative figures are givento show the effects of the fiber orientation, layer number, conical angle and subtended angle on the vibrationfrequencies of the shells as well. Finally, illustrative examples are presented for the open spherical shells andsome further numerical results for moderately thick composite laminated open spherical shells with variousboundary conditions and shell parameters are given.

3.1 Convergence study

In this subsection, the convergence of the proposed Chebyshev–Ritz method and effects of the elastic restraintparameters are investigated. For all numerical examples in this subsection, unless otherwise stated, thematerial constants of the open shells are the following: for the cylindrical shells: E1 = 60.7 GPa , E2 =24.8 GPa , μ12 = 0.23,G12 = G13 = G23 = 12 GPa , ρ = 1700 kg/m 3; for the conical shells: E1 =E2 = 70 GPa , μ12 = 0.3,G12 = G13 = G23 = 26.9 GPa , ρ = 2,700 kg/m 3; and for the spherical shells:E1 = 138 GPa , E2 = 8.96 GPa , μ12 = 3,G12 = G13 = 7.1 GPa ,G23 = 3.9 GPa , ρ = 1, 500 kg/m 3.

In Tables 1, 2 and 3, the convergence of the frequencies of the aforementioned open cylindrical, conical andspherical shells with completely free boundary condition is presented, respectively. The geometric dimensionsof these shells are as follows: for the cylindrical shell: R = 2 m , L = 1 m , h = 0.01 m , θ0 = 28.6◦; for theconical shell: R0 = 0.16 m , L = 1.12 m , h = 0.002 m , ϕc = 26.5◦, θ0 = 180◦; and for the spherical shell:R = 2 m , h = 0.01 m , ϕ0 = 75.5◦, ϕ1 = 104.5◦, θ0 = 28.6◦ and 180◦. In all the following computations, thezero frequencies corresponding to the rigid body modes were omitted from the results. Excellent convergenceof frequencies can be observed in the tables. In order to verify the accuracy of the formulation, the numericalresults reported by Zhao et al. [37] by using mesh-free method, Messina and Soldatos [42] based on HSDT aswell as Qatu and Leissa [46] based on shallow shell theory are included in Table 1. These comparisons showedthe present solutions are in good agreement with the reference results, although different theories and methodswere employed in the literature. In Table 2, the h-p FEM and experiment results reported by Bardell et al. [11]are listed as well. A consistent agreement between the present solutions and the reference data is seen from

Table 1 Convergence and comparison of frequency parameters Ω = ωL2α/h

√ρ/E1 of a four-layered, angle-ply

[−60◦/60◦/60◦ −60◦] e-glass/epoxy open cylindrical shell (R = 2 m , L/R = 0.5, h/L = 0.01, θ0 = 28.6◦)

Terms (M × N ) Mode number

1 2 3 4 5 6 7 8 9 10

11 × 11 3.2440 5.5872 8.3682 11.114 12.495 15.270 17.799 21.996 22.052 26.00212 × 12 3.2422 5.5869 8.3672 11.112 12.494 15.269 17.793 21.986 22.044 26.00213 × 13 3.2422 5.5868 8.3639 11.109 12.488 15.269 17.792 21.986 22.039 25.99914 × 14 3.2404 5.5866 8.3629 11.107 12.487 15.269 17.786 21.977 22.032 25.99915 × 15 3.2404 5.5865 8.3598 11.105 12.481 15.268 17.785 21.976 22.027 25.99616 × 16 3.2404 5.5863 8.3590 11.103 12.481 15.268 17.785 21.976 22.021 25.996Zhao et al. [37] 3.3016 5.7328 8.5087 11.133 12.626 15.724 18.231 22.129 22.257 25.828Messina and Soldatos [42] 3.2498 5.5910 8.3873 11.137 12.533 15.328 17.895 22.072 22.142 26.036Qatu and Leissa[46] 3.2920 5.7416 8.5412 11.114 12.591 15.696 18.221 22.058 22.194 25.871

Table 2 Convergence and comparison of frequencies (Hz) for an isotropic open conical shell (R0 = 0.16 m , L = 1.12 m , h =0.002 m , θ0 = 180◦, ϕc = 26.5◦)

Terms (M × N ) Mode number

1 2 3 4 5 6 7 8 9 10

13 × 13 4.273 8.201 11.12 20.68 21.80 32.75 47.16 47.21 64.77 67.8214 × 14 4.273 8.200 11.12 20.68 21.79 32.75 46.67 47.17 64.27 67.1315 × 15 4.273 8.200 11.12 20.67 21.78 32.74 46.66 47.13 63.02 67.0416 × 16 4.273 8.199 11.12 20.67 21.78 32.74 46.65 47.13 62.99 66.9117 × 17 4.273 8.199 11.12 20.67 21.77 32.74 46.65 47.12 62.92 66.9018 × 18 4.273 8.198 11.12 20.67 21.77 32.74 46.65 47.12 62.92 66.89ANSYS [11] 4.31 8.58 11.28 20.63 21.92 32.71 46.68 47.20 63.16 66.46Experiment [11] 4.5 8.9 11.5 20.9 21.7 33.2 46.6 47.4 58.6 63.7

450 T. Ye et al.

Table 3 Convergence and comparison of frequency parameters Ω = ωL2α/h

√ρ/E1 of a four-layered, angle-ply

[30◦/−30◦/−30◦/30◦] graphite/epoxy open spherical shell (R = 2 m , h = 0.01 m , ϕ0 = 75.5◦, ϕ1 = 104.5◦)

Terms (M × N ) Mode number (θ0 = 29◦) Mode number (θ0 = 180◦)

1 2 3 4 5 1 2 3 4 5

12 × 12 2.334 3.391 5.995 6.877 8.507 0.0715 0.2150 0.2960 0.4711 0.651313 × 13 2.334 3.391 5.990 6.871 8.497 0.0715 0.2149 0.2960 0.4709 0.651314 × 14 2.333 3.390 5.989 6.869 8.495 0.0715 0.2149 0.2958 0.4708 0.650915 × 15 2.333 3.389 5.985 6.865 8.488 0.0715 0.2149 0.2958 0.4707 0.650816 × 16 2.332 3.389 5.985 6.863 8.487 0.0715 0.2149 0.2957 0.4707 0.650617 × 17 2.332 3.389 5.985 6.863 8.487 0.0715 0.2148 0.2957 0.4704 0.6506Qatu [8] 2.283 3.323 5.871 6.781 8.394 – – – – –

the table. In Table 3, a comparison is made with the frequency parameters Ω = ωL2/h√ρ/E1 obtained by

Qatu [8] by using the classical shallow theory. The symbols “–” are missing data that were not considered byQatu. The discrepancy between the present and the reference results is acceptable. The small discrepancy inthe results may be attributed to different shell theories used in the literature. It has been proven that the shallowshell theories will give inaccurate results at the lower frequencies when applied to deep open shells.

Then, the influence of the elastic restraint stiffness on the open shells is investigated. For simplicity andconvenience in the analysis, five non-dimensional elastic restraint parametersΓu, Γv, Γw, Γα andΓβ , which aredefined as the ratios of the corresponding spring stiffness to the bending stiffness D = E1h3/12(1 −μ12μ21),respectively, are introduced here, i.e., Γu = ku/D, Γv = kv/D, Γw = kw/D, Γα = Kα/D and Γβ = Kβ/D.Also, a frequency parameter�Ω , which is defined as the difference of the dimensionless frequency parametersΩ = ωL2

α/h√ρ/E1 to those of the elastic restraint parameters Γλ(λ = u, v, w, α and β ) equal to 10−1, i.e.,

�Ω = ΩΓλ −ΩΓλ=10−1 are used in the calculations. In Figs. 3 and 4, variation in the 1st, 3rd and 5th mode fre-quency parameters�Ω versus the elastic restraint parametersΓλ for three-layered, cross-ply [0◦/90◦/0◦] opencylindrical, conical and spherical shells with various elastic restraints are presented. The geometric dimensionsof these shells are the following: for the cylindrical shell: R = 1 m, L/R = 2, h/R = 0.1, θ0 = 120◦; forthe conical shell: R0 = 1 m , L/R0 = 2, h/R0 = 0.1, ϕc = 45◦, θ0 = 120◦; and for the spherical shell:R = 1 m , h/R = 0.1, ϕ0 = 30◦, ϕ1 = 90◦, θ0 = 120◦. In Fig. 3, the composite open shells are clampedat edge α = 0, free at edges β = 0 and β = Lβ , while the other edge is elastically restrained by only onekind of spring components with various stiffness (denoted by C–F–Fe–F). It is clearly that in a certain range,the frequency parameters �Ω increase rapidly as the elastic restraint parameters Γλ increase. And beyondthis range, there is little variation in the frequency parameters. For the sake of completeness, in Fig. 4, theshells is assumed to be clamped at edge α = 0 and free at edges β = 0 and β = Lβ as usual, but the otheredge is restrained by all the five groups boundary springs in which four groups of them with infinite stiffness(107 D) and the rest one is assigned at arbitrary stiffness (denoted by C–F–Ce–F). The similar tendency asFig. 3 is seen in the figures. This study shows the active ranges of the elastic restraints stiffness on the vibrationcharacteristic of the composite open shells varied with spring components. In this case, they can be defined asΓu : 101 − 104, Γv : 100 − 103, Γw : 10−1 − 103, Γα : 10−1 − 102 and Γβ : 10−1 − 102, respectively.

3.2 Open shells with arbitrary boundary conditions

The present subsection contains comparisons and new results for the composite laminated shallow and deepopen shells with various boundary conditions and different shell parameters. As pointed out by Qatu [8], thethick shallow and deep open shells can have 24 possible classical boundary conditions at each edge, which willresult in a high number of combinations of boundary conditions. Besides, open shells with elastic constraintsare often encountered and such constraints can be generalized by introducing translational and rotationalsprings, which are distributed uniformly along the boundary at each of the shell ends [8]. And it is impossibleto undertake an all-encompassing survey of every case of the aforementioned boundary conditions. Therefore,only four typical classical boundaries, i.e., completely free (F), shear-diaphragm supported (SD), simply sup-ported (S) and complete clamped (C), and three uniform elastic restraint conditions, i.e., E1, E2 and E3, areconsidered in the current investigation. Taking edge α = 0, for example, the corresponding spring stiffnessfor these seven types of boundaries are the following:


10-1

101

103

105

107

0

0.2

0.4

0.6

0.8

1

Elastic restraint parameter: Log(Γu)

Freq

uenc

y pa

ram

eter

: ΔΩ

1st mode3rd mode5th mode

(a)Cylindrical shell

10-1

101

103

105

107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Fre

quen

cy p

aram

eter

: ΔΩ


(a)Conical shell

10-1

101

103

105

107

0

0.5

1

1.5

2

2.5

3

3.5

4


Fre

quen

cy p

aram

eter

: ΔΩ


(a)Spherical shell

10-1

101

103

105

107

0

1

2

3

4

5

6

7

Elastic restraint parameter: Log(Γv)

Freq

uenc

y pa

ram

eter

: ΔΩ


(b)

10-1

101

103

105

107

0

1

2

3

4

5

6


Fre

quen

cy p

aram

eter

: ΔΩ


(b)

10-1

101

103

105

107

0

0.5

1

1.5

2

2.5

3

3.5


Fre

quen

cy p

aram

eter

: ΔΩ


(b)

10-1

101

103

105

107

0

1

2

3

4

5

6

Elastic restraint parameter: Log(Γw)

Freq

uenc

y pa

ram

eter

: ΔΩ


(c)

10-1

101

103

105

107

0

1

2

3

4

5


Fre

quen

cy p

aram

eter

: ΔΩ


(c)

10-1

101

103

105

107

0

0.5

1

1.5

2

2.5

3

3.5


Fre

quen

cy p

aram

eter

: ΔΩ


(c)

10-1

101

103

105

107

0

0.2

0.4

0.6

0.8

Elastic restraint parameter: Log(Γα)

Fre

quen

cy p

aram

eter

: ΔΩ


(d)

10-1

101

103

105

107

0

0.2

0.4

0.6

0.8

1


Fre

quen

cy p

aram

eter

: ΔΩ


(d)

10-1

101

103

105

107

0

0.1

0.2

0.3

0.4

0.5


Fre

quen

cy p

aram

eter

: ΔΩ


(d)

10-1

101

103

105

107

0

0.5

1

1.5

2

Elastic restraint parameter: Log(Γβ)

Freq

uenc

y pa

ram

eter

: ΔΩ


(e)

10-1

101

103

105

107

0

0.5

1

1.5

2

2.5


Fre

quen

cy p

aram

eter

: ΔΩ


(e)

10-1

101

103

105

107

0

0.2

0.4

0.6

0.8

1


Fre

quen

cy p

aram

eter

: ΔΩ


(e)

Fig. 3 Relationships of frequency parameters �Ω with elastic restraint parameter Γλ for certain three-layered, cross-ply[0◦/90◦/0◦] open shells with C–F–Fe-F boundary condition: a Γu ; b Γv ; c Γw; d Γα ; e Γβ

452 T. Ye et al.

10-1

101

103

105

107

0

0.2

0.4

0.6

0.8

1

1.2


Freq

uenc

y pa

ram

eter

: ΔΩ


(a)Cylindrical shell

10-1

101

103

105

107

0

0.1

0.2

0.3

0.4

0.5


Freq

uenc

y pa

ram

eter

: ΔΩ


(a)Conical shell

10-1

101

103

105

107

0

0.5

1

1.5

2

2.5

3

3.5


Freq

uenc

y pa

ram

eter

: ΔΩ


(a)Spherical shell

10-1

101

103

105

107

0

1

2

3

4

5


Freq

uenc

y pa

ram

eter

: ΔΩ


(b)

10-1

101

103

105

107

0

1

2

3

4


Freq

uenc

y pa

ram

eter

: ΔΩ


(b)

10-1

101

103

105

107

0

1

2

3

4

5


Freq

uenc

y pa

ram

eter

: ΔΩ


(b)

10-1

101

103

105

107

0

0.5

1

1.5

2

2.5

3

3.5


Fre

quen

cy p

aram

eter

: ΔΩ


(c)

10-1

101

103

105

107

0

0.5

1

1.5

2

2.5

3

3.5

4


Fre

quen

cy p

aram

eter

: ΔΩ


(c)

10-1

101

103

105

107

0

0.5

1

1.5


Fre

quen

cy p

aram

eter

: ΔΩ


(c)

10-1

101

103

105

107

0

0.5

1

1.5

2

2.5


Fre

quen

cy p

aram

eter

: ΔΩ


(d)

10-1

101

103

105

107

0

0.5

1

1.5


Fre

quen

cy p

aram

eter

: ΔΩ


(d)

10-1

101

103

105

107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Fre

quen

cy p

aram

eter

: ΔΩ


(d)

10-1

101

103

105

107

0

0.5

1

1.5

2

2.5

3

3.5x 10

-3


Freq

uenc

y pa

ram

eter

: ΔΩ


(e)

10-1

101

103

105

107

0

1

2

3

4

5

6

x 10-3


Fre

quen

cy p

aram

eter

: ΔΩ


(e)

10-1

101

103

105

107

0

2

4

6

8

10

12

14x 10

-4


Fre

quen

cy p

aram

eter

: ΔΩ


(e)

Fig. 4 Relationships of frequency parameters �Ω with elastic restraint parameter Γλ for certain three-layered, cross-ply[0◦/90◦/0◦] open shells with C–F–Fe–F boundary condition: a Γu ; b Γv ; c Γw; d Γα ; e Γβ


(a) F edge: Nα = 0, Nαβ = 0, Qα = 0,Mα = 0,Mαβ = 0 or kuα0 = kvα0 = kwα0 = K α

α0 = K βα0 = 0

(b) SD edge: Nα = 0, v = 0, w = 0,Mα = 0,Mαβ = 0 or kvα0 = kwα0 = 107 D, kuα0 = K α

α0 = K βα0 = 0

(c) S edge: u = 0, v = 0, w = 0,Mα = 0, φβ = 0 or kvα0 = kwα0 = kuα0 = K β

α0 = 107 D, K αα0 = 0

(d) C edge: u = 0, v = 0, w = 0, φα = 0, φβ = 0 or kuα0 = kvα0 = kwα0 = K α

α0 = K βα0 = 107 D

(e) E1 edge: only transverse direction is elastically restrained, i.e.:

kwα0 = 10D, kuα0 = kvα0 = K α

α0 = K βα0 = 107 D

(f) E2 edge: only rotation is elastically restrained, i.e.:

K αα0 = D, ku

α0 = kvα0 = kwα0 = K βα0 = 107 D

(g) E3 edge: transverse direction and rotation are elastically restrained, i.e.:

kwα0 = 10D, K αα0 = D, ku

α0 = kvα0 = K βα0 = 107 D

where D = E1h3/12(1−μ12μ21) is the flexural stiffness of the shell. The appropriateness of defining theaforementioned seven types of boundary conditions in terms of boundary spring components will be shownlater. In order to easily refer to the boundary conditions of an open shell, a clockwise notation starting formα = 0 is employed. For instance, the symbol “F–C–S–E1” represents an open shell free at edge α = 0,clamped at edge β = 0, simply supported at edge α = Lα and elastically constrained at edge β = Lβ .

3.2.1 Open cylindrical and conical shells with arbitrary boundary conditions

As the first case, Table 4 shows the comparison of the non-dimensional frequency parameters Ω =ωL2

α

√ρ/E1 Rh of a two-layered, cross-ply [90◦/0◦]open cylindrical shell subjected to SD–SD–SD–SD bound-

ary conditions, with results provided by Messina and Soldatos [41] based on the conjunction of Ritz methodand the Love-type version of a unified shear deformable shell theory. The shell parameters used in the com-parison are as follows: E1 = 10 GPa , E2 = 25E2, μ12 = 0.25,G12 = G13 = 0.5E2,G23 = 0.2E2, ρ =1700 kg/m 3, R = 5 m , L/R = 5, θ0 = 60◦. The first eight frequencies and four sets of thickness radiusratios, i.e., h/R = 0.1, 0.05, 0.02 and 0.01, are performed in the comparison. It is clearly evident that thepresent solutions are generally in good agreement with the reference results, although a different shell theorywas employed by Messina and Soldatos [41]. The differences between these two results are very small and donot exceed 0.74% for the worst case. In order to further verify the present formulation and validate the accuracyof the proposed Chebyshev–Ritz method, in Table 5, the present solutions for dimensionless frequency param-etersΩ = ωL2

α

√ρh/D of certain four-layered, symmetrically laminated composite open cylindrical shells are

compared with the results reported by Bardell et al. [39] by using the h− p version of the finite element method.The geometric and material constants of the shells are the following: E1 = 10 GPa , E2 = 15.4E2, μ12 =0.3,G12 = G13 = G23 = 0.8E2, ρ = 1,500 kg/m 3, R = 5 m , R/L = 5, h/L = 0.01, θ0 = 11.5◦. Fivelamination schemes, i.e., [0◦], [30◦/− 30◦/− 30◦/30◦], [45◦/− 45◦/− 45◦/45◦], [60◦/− 60◦/− 60◦/60◦]and [90◦/−90◦/−90◦/90◦] are conducted in the comparison. It can be seen that a good agreement is obtainedbetween the current solutions and those given by Bardell et al. [39]. The open cylindrical shells can be regardedas the special cases of the open conical shells. For the general composite open conical shells, there are nosuitable results for comparison in the literature. Therefore, to ensure the accuracy of the proposed Cheby-shev–Ritz formulation, the illustrative examples are solved for isotropic conical panels. The comparison ofthe lowest six frequency parameters Ω = ωL2

α

√ρh/D for an isotropic open conical shell with S–S–S–S and

C–C–C–C boundary conditions is given in Table 6. The shell parameters used in the comparison are as follows:E = 70 GPa , μ12 = 0.3, ρ = 1,500 kg/m 3, R0 = 1 m , L/R0 = 2, h/R0 = 0.006, θ0 = 60◦, ϕc = 90◦.The solutions are compared with the results reported by Bardell et al. [11] by using the hp-FEM as well as those

454 T. Ye et al.

Table 4 Comparison of frequency parameters Ω = ωL2α

√ρ/E1 Rh of an unsymmetrically laminated, two-layered open cylin-

drical shell with SD–SD–SD–SD boundary conditions (L/R = 5, θ0 = 60◦)

Lamination schemes h/R Method Mode number

1 2 3 4 5 6 7 8

[90◦/0◦] 0.10 Present 7.025 7.645 8.882 10.89 13.52 14.05 16.66 20.27Ref. [41] 7.025 7.702 8.932 10.94 13.57 14.05 16.74 20.40Difference (%) 0.00 0.74 0.56 0.46 0.37 0.00 0.48 0.64

0.05 Present 5.922 8.145 9.934 11.12 14.44 18.02 19.87 21.83Ref. [41] 5.932 8.153 9.935 11.12 14.45 18.03 19.87 21.87Difference (%) 0.17 0.10 0.01 0.00 0.07 0.06 0.00 0.18



obtained from the finite element analyses (ANSYS). It is observed that the present analysis agree well withthe reference. The difference between these two results is very small and is less than 1.12 % for the worst case.

The excellent agreement between the present results and the referential ones obtained in Tables 1–2 andTables 4–6 indicates that the proposed Chebyshev–Ritz method is able to calculate the vibration frequenciesof the composite laminated open cylindrical and conical shells with sufficient accuracy. It also verified thatthe defining of the four types of classical boundaries at the beginning of this subsection is appropriate. Havinggained confidence in the present method, some further numerical results for the composite laminated opencylindrical and conical shells with different sets of classical, elastic boundary conditions and their combina-tions as well as different shell parameters, such as geometric properties, lamination schemes, are given in thefollowing discussions.

Tables 7 and 8 give the first five non-dimensional frequenciesΩ = ωL2α

√ρh/D of three-layered, cross-ply

[0◦/90◦/0◦] open cylindrical and conical shells with various boundary conditions and subtended angles. Thematerial properties and geometric dimensions of the shells are as follows: E1 = 10 GPa , E2 = 15E2, μ12 =0.25,G12 = G13 = G23 = 0.5E2,ρ = 1,500 kg/m 3; for the cylindrical panel: R = 1 m , L/R = 2, h/R =0.1; for the conical panel: R0 = 1 m , L/R0 = 2, h/R0 = 0.1, ϕc = 45◦. These results may serve as bench-mark values for future researches. It is obvious from the tables that the increase in the subtended angle willresult in an increase in the natural frequencies of the shells. Meanwhile, we can see that an open shell withgreater restraining rigidity will have higher natural frequencies. As aforementioned, once the coefficients inthe admissible displacement expressions of an open shell are solved from Eq. (22), its mode shapes can bedirectly determined. To enhance the understanding of the vibration behaviors of the open cylindrical and con-ical shells, the first five mode shapes for the shells with C–C–C–C restraint conditions are present in Figs. 5and 6, which is constructed in three-dimension views. Next, Table 9 compares the lowest five dimensionlessfrequenciesΩ = ωL2

α

√ρh/D of a two-layered, cross-ply [0◦/90◦] open conical shell with various subtended

angles and cone angles. The shell subjected to SD–SD–SD–SD, S–S–S–S and C–C–C–C boundary conditionsare considered in the investigation. The material and geometric constants of the shell are the same as theaforementioned three-layered conical panel (Table 8). The results show that the C–C–C–C conical panel hasthe higher, frequency values than the SD–SD–SD–SD one. In addition, the increase in the subtended angleand cone angle will result in an increase in the natural frequencies.

One of the main advantages of the composite laminated structures is that desired mechanical parameters canbe obtained by appropriately selecting their lamination schemes, such as fiber orientations, number of layersand stacking sequences. Therefore, it is of particular importance to understand the effects of lamination schemeon the vibration characteristics of a composite laminated open shell. Figure 7 depicts the variation in the firstthree dimensionless frequenciesΩ = ωL2

α

√ρh/D of a full clamped, three-layered composite open cylindrical

shell with varying fiber orientations in the middle layer, while the top and bottom layers of the shell are ofuniform principal direction which is paralleled to α-axis (denoted by [0◦/ϑ/0◦]). The shell parameters are thesame as the aforementioned three-layered, cross-ply [0◦/90◦/0◦] open cylindrical shell employed in Table 7.Four subtended angle configurations, i.e., θ0 = 60◦, 90◦, 120◦, 150◦ are considered in the study. As it can beseen, the effects of the fiber orientations on the frequency parameters varied with the subtended angles. Whenθ0 = 60◦ and θ0 = 90◦, the frequency parameters Ω increase steadily as ϑ becomes larger. However, when


Tabl

e5

Com

pari

son

offr

eque

ncy

para

met

ersΩ

=ω

L2 α√ ρ

h/

Dof

four

-lay

ered

open

cylin

dric

alsh

ells

with

C–C

–C–C

boun

dary

cond

ition

s(R

=5

m,

R/

L=

5,h/

L=

0.01,

θ 0=

11.5

◦ )

Mod

e[0◦

][30

◦ /−

30◦ /

−30◦/30

◦ ][45

◦ /−4

5◦/

−45◦/45

◦ ][60

◦ /−

60◦ /

−60

◦ /60

◦ ][90

◦ /−

90◦ /

−90

◦ /90

◦ ]Pr

esen

tR

ef.[

39]

Pres

ent

Ref

.[39

]Pr

esen

tR

ef.[

39]

Pres

ent

Ref

.[39

]Pr

esen

tR

ef.[

39]

128

.13

28.2

731

.43

31.4

938

.84

38.8

948

.81

48.8

760

.89

60.9

82

31.0

831

.20

38.7

938

.83

46.1

646

.26

52.6

852

.89

63.2

463

.88

342

.30

42.6

156

.00

56.3

755

.55

55.7

059

.88

60.1

764

.28

64.2

04

60.0

260

.55

57.2

157

.61

65.9

766

.38

67.5

467

.86

70.1

170

.77

564

.74

65.0

470

.44

71.0

977

.87

78.6

677

.18

77.7

471

.36

71.5

06

68.0

768

.76

78.4

579

.57

88.0

089

.02

87.1

487

.90

81.5

182

.21

456 T. Ye et al.

Tabl

e6

Com

pari

son

offr

eque

ncy

para

met

ersΩ

=ω

L2 α√ ρ

h/

Dof

anis

otro

pic

open

coni

cals

hell

with

S–S–

S–S

and

C–C

–C–C

boun

dary

cond

ition

s(R

0=

1m,

L/

R0

=2,

h/

R0

=0.

006,θ 0

=60

◦ ,ϕ

c=

90◦ )

Mod

eS–

S–S–

SC

–C–C

–C

Pres

ent

hp-F

EM

[11]

Err

or(%

)A

NSY

S[1

1]E

rror

(%)

Pres

ent

hp-F

EM

[11]

Err

or(%

)A

NSY

S[1

1]E

rror

(%)

119.1

019.2

0.52

19.1

0.00

35.0

935.2

0.32

35.0

0.25

243.6

343.6

0.06

43.5

0.30

66.1

966.3

0.16

66.0

0.29

350.8

150.9

0.18

50.7

0.21

74.5

974.9

0.41

74.4

0.26

478.9

378.9

0.04

78.6

0.42

108.

1810

8.4

0.20

107.

70.

455

83.1

683.3

0.16

82.9

0.32

112.

8511

3.2

0.31

112.

20.

586

100.

7810

1.1

0.32

100.

60.

1813

3.39

134.

91.

1213

3.0

0.29


Tabl

e7

The

low

estfi

vefr

eque

ncy

para

met

ersΩ

=ω

L2 α√ ρ

h/

Dof

ath

ree-

laye

red,

cros

s-pl

y[0◦/90

◦ /0◦

]ope

ncy

lindr

ical

shel

lwith

vari

ous

boun

dary

cond

ition

san

dsu

bten

ded

angl

es(R

=1

m,

L/

R=

2,h/

R=

0.1)

Subt

ende

dan

gle

(θ0)

Mod

eB

ound

ary

cond

ition

s

S–S–

S–S

C–C

–C–C

E1–

E1–

E1–

E1

E2–

E2–

E2–

E2

E3–

E3–

E3–

E3

F–E

1–F

–E

1F–

E2–F

–E

2F-

E3–F

–E

3SD

-E1–S

D–

E1

SD-E

2–S

D–

E2

45◦

173.2

910

83.2

629

38.2

183

80.2

406

36.9

915

32.2

969

78.7

490

31.8

954

37.8

709

79.8

828

278.0

822

93.8

046

51.6

517

85.9

480

50.9

290

37.1

930

79.2

152

36.8

846

56.4

151

85.5

810

384.8

691

101.

206

70.2

437

88.4

240

69.9

175

52.4

520

83.3

029

52.2

645

78.1

625

87.5

019

485.8

746

111.

677

78.4

578

96.1

813

71.0

621

75.5

263

83.6

009

68.1

201

85.3

006

95.3

444

510

4.99

611

3.75

085.4

237

107.

406

80.0

494

77.4

524

85.0

636

70.6

430

88.0

511

100.

979

90◦

125.9

226

36.2

073

27.8

660

30.5

844

25.8

649

15.9

876

22.8

582

15.3

691

25.8

452

29.5

411

241.7

629

51.4

391

32.2

137

46.0

887

30.7

796

23.6

851

27.2

626

23.0

811

30.7

598

45.3

682

348.8

777

59.0

643

46.4

152

52.0

340

45.6

829

25.5

137

42.2

955

25.3

875

49.6

361

50.4

902

457.3

390

68.3

664

47.1

679

61.0

707

46.1

159

27.8

199

42.3

096

27.7

377

50.0

194

50.6

410

573.3

012

83.4

734

50.4

757

79.6

108

48.4

787

43.7

127

44.4

612

43.1

826

50.0

733

59.7

677

135◦

123.5

462

28.4

288

25.3

762

24.8

346

23.3

236

8.90

282

9.59

418

8.10

483

22.7

972

23.6

472

223.9

800

31.7

386

26.3

539

26.9

701

24.3

611

14.7

709

18.9

623

14.5

617

23.8

653

25.7

078

336.7

580

46.0

048

32.8

598

40.9

773

31.6

645

15.6

516

19.7

151

15.4

631

31.6

216

33.6

603

446.0

628

55.3

353

41.5

853

48.4

516

40.1

247

19.1

403

20.0

410

18.8

048

33.6

603

40.2

035

548.5

158

55.6

624

43.1

197

49.8

557

42.1

916

24.4

959

33.4

695

24.4

958

40.6

088

46.8

978

180◦

120.7

458

26.7

045

23.9

731

22.4

267

21.7

196

4.95

173

4.81

124

4.43

420

20.9

804

20.9

466

222.7

066

27.8

521

24.2

230

24.1

403

22.1

716

9.43

695

10.7

630

9.10

214

21.3

473

22.8

068

328.2

867

33.3

990

30.4

373

30.0

170

28.9

710

9.54

649

10.7

902

9.11

705

25.2

452

25.2

452

432.5

518

39.8

891

30.8

290

35.5

486

29.3

558

14.1

193

15.9

960

14.1

171

28.6

804

29.1

538

544.4

454

52.1

301

38.1

696

46.8

712

37.0

584

16.0

470

19.9

382

15.7

937

29.2

386

34.6

057

215◦

120.1

659

25.9

464

23.1

276

21.7

914

20.8

369

2.85

403

2.66

187

2.55

135

19.9

462

20.1

962

220.9

320

26.1

552

23.3

816

21.9

780

21.2

288

5.78

266

6.22

486

5.50

721

20.1

962

20.2

222

324.8

366

31.2

471

27.1

244

27.1

241

25.2

318

6.27

179

6.46

487

5.86

619

20.2

662

20.5

008

428.4

455

31.6

396

28.5

512

29.1

824

26.9

505

10.1

379

11.4

349

10.0

328

24.8

487

25.9

197

532.3

505

38.3

294

33.3

688

34.6

933

32.3

519

11.0

966

12.3

305

10.6

821

26.4

381

28.3

208

458 T. Ye et al.

Tabl

e8

The

low

est

five

freq

uenc

ypa

ram

eter

sΩ=ω

L2 α√ ρ

h/

Dof

ath

ree-

laye

red,

cros

s-pl

y[0◦/90

◦ /0◦

]ope

nco

nica

lsh

ell

with

vari

ous

boun

dary

cond

ition

san

dsu

bten

ded

angl

es(R

0=

1m,

L/

R0

=2,

h/

R0

=0.

1,ϕ

c=

45◦ )

Subt

ende

dan

gle

(θ0)

Mod

eB

ound

ary

cond

ition

s

S–S–

S–S

C–C

–C–C

E1−

E1–

E1–

E1

E2–

E2–

E2–

E2

E3–

E3–

E3−

E3

F-E

1–F

–E

1F-

E2–F

–E

2F-

E3–F

–E

3SD

-E1–S

D–

E1

SD-E

2–S

D–

E2

45◦

129.9

531

37.8

306

28.8

198

34.1

433

27.4

694

20.8

519

22.8

218

19.9

823

28.7

481

30.9

805

232.5

418

44.1

757

31.4

313

37.6

354

30.8

347

22.2

939

23.6

698

22.2

018

31.2

546

36.4

947

350.0

048

58.3

630

39.4

802

52.2

366

38.8

227

29.9

684

35.2

348

29.7

024

44.4

719

50.1

424

452.4

097

67.4

064

42.8

628

59.4

472

41.4

628

35.5

898

42.1

113

35.0

797

47.3

355

58.3

983

554.1

443

70.3

470

45.0

697

62.7

026

44.0

163

36.7

203

44.7

549

36.2

172

50.9

200

60.7

600

90◦

116.7

023

24.1

034

19.5

931

19.1

932

17.7

335

7.24

791

7.25

426

6.91

093

17.8

157

17.7

056

219.8

104

27.6

709

21.2

025

22.8

720

19.5

758

10.4

828

11.0

037

10.2

467

19.3

284

20.4

278

329.9

098

37.8

984

28.9

555

34.3

015

27.8

802

17.4

350

18.3

831

16.9

269

25.2

241

25.4

229

433.3

470

39.2

989

31.3

100

34.4

347

30.8

400

18.5

249

18.7

199

18.4

149

29.3

059

33.1

349

540.4

619

49.8

597

33.0

667

42.6

416

32.4

185

19.7

646

20.6

545

19.2

693

36.4

992

40.8

596

135◦

115.2

810

22.5

655

17.9

102

17.4

019

15.9

187

3.48

686

3.45

428

3.30

912

15.6

272

15.6

413

216.5

357

22.8

877

18.3

448

18.0

846

16.5

362

5.17

746

5.21

019

5.03

296

16.2

094

16.4

474

319.9

434

26.7

792

22.2

250

22.3

346

20.7

278

9.59

960

9.65

161

9.31

237

18.8

795

18.8

827

422.9

775

30.1

860

23.5

819

25.7

042

22.2

468

10.5

397

11.3

673

10.3

621

20.9

632

20.9

373

529.8

987

37.7

534

29.1

453

33.2

195

28.1

523

13.5

280

14.3

986

13.1

983

23.8

348

25.1

830

180◦

114.8

195

21.8

554

17.3

066

16.7

122

15.2

739

1.88

869

1.83

388

1.78

656

14.9

136

14.9

495

215.1

287

21.9

830

17.3

403

16.7

362

15.3

764

2.86

904

2.85

679

2.79

299

14.9

552

14.9

825

317.4

443

24.5

192

19.7

835

19.7

118

18.0

272

5.61

851

5.56

669

5.46

221

16.3

091

16.3

458

419.3

675

25.0

376

20.5

301

20.8

068

19.0

260

6.04

017

6.42

453

5.85

254

18.5

472

18.5

567

522.3

378

28.5

277

24.0

125

24.3

908

22.7

765

8.40

367

8.50

183

8.16

553

18.6

167

18.8

810

215◦

114.5

709

21.5

261

16.9

448

16.2

410

14.9

048

1.09

833

1.05

464

1.03

971

14.4

567

14.4

500

214.5

881

21.6

674

17.0

018

16.3

805

14.9

487

1.71

266

1.70

278

1.67

007

14.5

267

14.5

651

316.4

698

23.0

782

18.7

185

18.3

505

16.8

709

3.55

957

3.52

774

3.41

199

15.1

623

15.1

651

416.9

153

23.4

311

18.7

862

18.4

918

17.0

601

3.57

925

3.68

648

3.48

758

16.6

180

16.7

398

519.3

833

26.1

952

21.3

274

21.6

336

19.7

653

5.53

400

5.50

461

5.38

513

16.8

430

16.8

847


Fig. 5 Mode shapes for the open composite cylindrical shell with C–C–C–C boundary conditions

Fig. 6 Mode shapes for the composite open conical shell with C–C–C–C boundary conditions

θ0 = 150◦, the frequency parameter traces climb up and then decline, and reach their crests around ϑ = 45◦.Figure 8 gives the variations in the lowest three dimensionless frequencies Ω = ωL2

α

√ρh/D of a [0◦/ϑ]n

layered conical panel with C–C–C–C boundary conditions against the number of layers n (where n=1 meansthe single-layered scheme [0◦]; sn = 2 means the two-layered scheme [0◦/ϑ], and so forth). Four types of fiberorientations, i.e., ϑ = 30◦, 45◦, 60◦, 90◦, are considered. The layers of the conical shells are the same as the

460 T. Ye et al.

0° 15° 30° 45° 60° 75° 90°

40

50

60

70

80

Fiber orientation: ϑ

Fre

quen

cy p

aram

eter

: Ω

1st 2nd 3rd0° 15° 30° 45° 60° 75° 90°

40

50

60

70

80

90

100

110


Fre

quen

cy p

aram

eter

: Ω

1st 2nd 3rd

(b) θ0=90°(a) θ0=60°

0° 15° 30° 45° 60° 75° 90°25

30

35

40

45

50

55


Fre

quen

cy p

aram

eter

: Ω

1st 2nd 3rd0° 15° 30° 45° 60° 75° 90°

25

30

35

40


Fre

quen

cy p

aram

eter

: Ω

1st 2nd 3rd

(c) θ0=120° (d) θ0=150°

Fig. 7 Relationships of frequency parameters Ω = ωL2α

√ρh/D with fiber orientations for a three-layered [0◦/ϑ/0◦] open

cylindrical shell with C–C–C–C boundary conditions

5 10 15 20 25 30

22

24

26

28

30

32

Layer number: n

Fre

quen

cy p

aram

eter

: Ω

1st 2nd 3rd5 10 15 20 25 30

25

30

35

Layer number: n

Fre

quen

cy p

aram

eter

: Ω

1st 2nd 3rd

(a) ϑ=30° (b) ϑ=45°

5 10 15 20 25 30

25

30

35

40

Layer number: n

Fre

quen

cy p

aram

eter

: Ω

1st 2nd 3rd

5 10 15 20 25 30

25

30

35

40

45

Layer number: n

Fre

quen

cy p

aram

eter

: Ω

1st 2nd 3rd

(c) ϑ=60° (d) ϑ=90°

Fig. 8 Relationships of frequency parametersΩ = ωL2α

√ρh/D with number of layers n for a [0/ϑ]n layered conical shell with

C–C–C–C restraints: a ϑ = 30◦; b ϑ = 45◦; c ϑ = 60◦; d ϑ = 90◦

aforementioned open conical shell used for Tables 8 and 9. The results demonstrate that the frequency parame-ters increase rapidly and may reach their crests around n = 8, and beyond this range, the frequency parametersremain unchanged. The fluctuations on the curves are due to the fact that the shells are symmetrically laminatedwhen n is an odd number, and the shells are unsymmetrically laminated when n equal to an even number.


Tabl

e9

The

low

est

five

freq

uenc

ypa

ram

eter

sΩ

=ω

L2 α√ ρ

h/

Dof

atw

o-la

yere

d,cr

oss-

ply

[0◦/90

◦ ]op

enco

nica

lsh

ell

with

vari

ous

subt

ende

dan

gles

and

cone

angl

es(R

0=

1m,

L/

R0

=2,

h/

R0

=0.

1)

Con

ean

gle

(ϕc)

Mod

eθ 0

=90

◦θ 0

=13

5◦θ 0

=18

0◦

SD-S

D-

SD-S

DS–

S–S–

SC

–C–C

–CSD

-SD

-SD

-SD

S–S–

S–S

C–C

–C–C

SD-S

D-

SD-S

DS–

S–S–

SC

–C–C

–C

15◦

117

.167

624

.767

830

.438

815

.219

521

.887

823

.640

915

.413

519

.496

021

.220

72

19.6

945

34.4

967

42.9

727

19.7

142

22.4

416

25.8

260

17.1

750

20.5

506

22.6

686

332

.313

840

.331

546

.056

522

.388

432

.857

238

.692

319

.725

627

.435

728

.830

34

35.7

906

45.4

140

54.7

992

27.1

456

35.5

398

41.3

657

20.5

974

28.7

384

33.2

352

536

.909

360

.425

467

.077

730

.458

137

.922

442

.963

826

.666

834

.018

039

.516

530

◦1

15.9

104

21.1

735

25.0

914

13.8

939

18.5

233

21.1

854

13.6

790

17.3

619

18.9

734

216

.112

926

.509

733

.130

115

.925

819

.750

021

.571

315

.934

417

.555

519

.810

93

27.1

790

35.5

128

41.4

120

20.1

581

26.6

422

30.7

187

16.1

190

22.9

041

25.4

230

429

.331

738

.831

246

.995

222

.890

331

.318

737

.480

118

.203

024

.130

726

.459

65

32.0

561

44.9

850

52.4

736

24.9

739

32.8

993

38.4

122

20.9

119

30.1

536

34.0

252

45◦

113

.596

518

.185

421

.659

612

.248

315

.931

618

.666

212

.022

515

.055

516

.924

22

14.0

524

21.6

759

27.3

025

13.6

097

16.8

527

18.8

354

13.6

174

15.1

369

17.5

040

322

.146

931

.323

737

.806

716

.481

122

.590

026

.098

814

.057

819

.289

022

.121

44

26.9

242

34.1

011

39.3

996

18.8

779

25.8

526

31.0

656

15.8

363

20.3

512

22.5

772

527

.735

335

.543

441

.803

225

.141

328

.932

535

.211

917

.372

225

.403

428

.695

460

◦1

11.0

772

15.3

239

19.0

842

10.1

543

13.5

156

15.9

926

10.1

683

12.5

006

14.8

612

211

.871

717

.772

222

.429

411

.883

813

.787

616

.718

911

.082

912

.804

515

.404

33

19.3

125

26.1

572

30.1

185

12.3

946

19.2

306

23.0

050

11.8

912

16.4

014

18.5

954

423

.835

827

.845

934

.900

316

.516

120

.868

325

.094

412

.455

916

.466

819

.951

05

24.8

777

30.0

418

36.3

806

22.3

716

25.6

350

30.4

713

15.2

166

21.6

778

25.2

826

75◦

17.

8891

512

.509

415

.821

78.

0182

610

.424

413

.460

47.

8966

59.

8272

012

.871

82

10.6

232

12.7

665

17.2

753

8.27

453

10.8

437

14.4

250

8.40

074

9.94

901

12.9

913

317

.793

320

.495

025

.713

910

.635

114

.138

917

.428

88.

4314

512

.518

615

.904

04

20.5

913

24.8

100

30.7

926

15.1

784

16.8

320

21.1

273

10.6

423

12.7

082

15.9

064

523

.445

425

.139

632

.863

920

.620

422

.221

827

.182

213

.944

715

.951

819

.437

4

462 T. Ye et al.

3.2.2 Open spherical shells with arbitrary boundary conditions

In the following numerical examples, the present formulation is applied to the composite laminated openspherical shells with general restraints for which very limited amount of results are available in the litera-ture. Since there are no suitable comparison results in the literature, two illustrative examples are presentedfor the vibrations of composite laminated shallow spherical panels. As the first case, Table 10 shows thecomparison of the first nine non-dimensional frequencies Ω = ωL2

α/h√ρ/E1 of certain three-layered shal-

low spherical shells subjected to SD–SD–SD–SD boundary conditions, with results provided by Fazzolariand Carrera [12] by using the Carrera unified formulation. The shell parameters used in the comparisonare the following: E1 = 60.7 GPa , E2 = 24.8 GPa , μ12 = 0.23,G12 = G13 = G23 = 12 GPa , ρ =1700 kg/m 3, R = 2 m , h = 0.05 m , ϕ0 = 75.5◦, ϕ1 = 104.5◦, θ0 = 28.6◦. Four sets of lamination schemes,i.e., [0◦], [15◦/ − 15◦/15◦], [30◦/ − 30◦/30◦] and [45◦/ − 45◦/45◦], are performed in the comparison. It isclearly evident that the present solutions match well with the reference data, although a different shell theoryis used by Fazzolari and Carrera. The differences between these two results are attributed to different shelltheories used in the literature. It has been proven that the shallow shell theories will give inaccurate results forthe lower frequencies when applied to deep open shells [8]. In order to further verify the present formulation,in Table 11, the first six natural frequencies of a four-layered, angle-ply [45◦/− 45◦/45◦/− 45◦] open spher-ical shell with as many as eight combinations of classical restraints are presented, together with the modifiedFourier series solutions obtained by Ye et al.[28]. The shell parameters used in the comparison are as follows:E1 = 10GPa, E2 = 15 GPa , μ12 = 0.23,G12 = G13 = G23 = 0.5E2, ρ = 1700 kg/m 3, R = 10 m , h =0.01 m , ϕ0 = 87.1◦, ϕ1 = 92.9◦, θ0 = 11.5◦. Our results are in closed agreement with the existing ones.

The excellent agreement between the present results and the referential ones shown in Table 3 and Tables 10–11 indicates that the proposed Chebyshev–Ritz formulation is accurate. Having gained confidence in pres-ent formulation, some further vibration results for the composite laminated deep open spherical shells withvarious combinations of boundary conditions and shell parameters are presented. In the following exam-ples, unless otherwise stated, the geometric and material constants of the spherical shells are as follows:E2 = 10 GPa , E1/E2 = 15, μ12 = 0.25,G12 = G13 = 0.5E2,G23 = 0.2E2, ρ = 1,500 kg/m 3, R =1 m , h = 0.1 m , ϕ0 = 30◦, ϕ1 = 90◦. In Table 12, the first five frequency parameters Ω = ωL2

α/h√ρ/E1

of a three-layered, cross-ply [0◦/90◦/0◦] open spherical shell with various boundary conditions and subtendedangles are presented. It is obvious form the table that the increase of the subtended angle will result in anincrease in the natural frequencies of the shell. Meanwhile, we can see that an open spherical shell with greaterrestraining rigidity will have higher vibration frequencies. For any given frequency parameters, the correspond-ing mode shapes of the open spherical shell can be readily determined by Eq. (19) after solving the standardmatrix eigenproblem [see Eq. (22)]. For instance, the first four mode shapes for the spherical shell subjectedto C–C–C–C boundary conditions and with subtended angle θ0 = 45◦, 90◦ and 135◦ are plotted in Fig. 9.

4 Conclusions

The free vibration analysis of generally supported composite laminated deep open shells with various shellcurvatures including cylindrical, conical and spherical ones has been carried out by using the unified Cheby-shev–Ritz formulation. The first-order shear deformation shell theory based on the general shell equations isemployed in the present analysis. Under the current framework, each of displacements and rotations of the openshell, regardless of boundary conditions, is invariantly expressed as Chebyshev polynomials of first kind inboth directions. Thereby, all the Chebyshev expansion coefficients are treated equally and independently as thegeneralized coordinates and solved directly by using the Rayleigh–Ritz procedure. The convergence and accu-racy of the present formulation are checked by a considerable number of convergence tests and comparisons.A variety of numerical examples are presented for the free vibration of shallow and deep composite laminatedcylindrical, conical and spherical panels with various geometric dimensions and lamination schemes. Differ-ent combinations of classical boundary conditions (e.g., completely free, shear-diaphragm supported, simplysupported and clamped) and uniform elastic restraints are considered. These results may serve as benchmarksolutions for future researches. Parametric studies are also undertaken, giving insight into the effects of elasticrestraint parameter, fiber orientation, layer number, subtended angle as well as conical angle on the vibrationfrequencies of the composite open shells.

It should be stressed that the presented formulation is general since various boundary conditions, differentlamination schemes and geometric dimensions (shallow and deep), different numbers of layers (which may beisotropic or orthotropic) and forms of trial functions, such as the Chebyshev orthogonal polynomials of second


Tabl

e10

Com

pari

son

offr

eque

ncy

para

met

ersΩ

=ω

L2 α/

h√ ρ

/E

1of

cert

ain

thre

e-la

yere

dop

ensp

heri

cals

hells

with

SD–S

D–S

D–S

Dbo

unda

ryco

nditi

ons(

R=

2m,

h=

0.05

m,ϕ

0=

75.5

◦ ,ϕ

1=

104.

5◦,θ 0

=28.6

◦ )

Lay

out

The

ory

Mod

enu

mbe

r

12

34

56

78

9

[0◦]

Pres

ent

8.14

234

12.3

699

13.9

511

18.1

169

20.0

452

25.8

037

26.2

185

27.4

026

28.2

361

Shal

low

[12]

8.15

995

12.4

546

14.0

268

18.3

345

20.1

208

26.0

524

15◦ /

−15

◦ /15

◦ ]Pr

esen

t8.

4670

412

.634

613

.906

018

.305

420

.444

125

.677

626

.149

929

.205

729

.999

4Sh

allo

w[1

2]8.

4908

312

.750

314

.008

318

.561

820

.570

325

.824

430

◦ /−

30◦ /

30◦ ]

Pres

ent

9.09

300

12.9

552

13.8

470

18.6

620

21.3

978

24.3

233

26.6

959

30.0

401

32.5

608

Shal

low

[12]

9.14

088

13.1

156

14.0

273

18.9

934

21.6

139

24.5

616

[45◦ /

−45

◦ /45

◦ ]Pr

esen

t9.

3961

412

.994

513

.846

318

.834

422

.425

223

.058

926

.881

830

.110

934

.187

7Sh

allo

w[1

2]9.

4812

513

.163

514

.077

219

.201

122

.669

523

.346

5

464 T. Ye et al.

Tabl

e11

Com

pari

son

offr

eque

ncy

para

met

ersΩ

=ω

L2 α/

h√ ρ

/E

1of

afo

ur-l

ayer

ed,a

ngle

-ply

[45◦ /

−45

◦ ] 2op

ensp

heri

cals

hell

with

vari

ous

rest

rain

ts(R

=10

m,

h=

0.01

m,ϕ

0=

87.1

◦ ,ϕ

1=

92.9

◦ ,θ 0

=11.5

◦ )

The

ory

Mod

eB

ound

ary

cond

ition

s

F–F–

F–F

F–S–

F–S

F–F–

F–C

F–C

–F–C

F–C

–C–C

S–S–

S–S

S–C

–S–C

C–C

–C–C

Shal

low

CST

[28]

12.

3732

7.68

860.

5360

10.1

1912.5

5031.7

9932.3

1035.8

982

5.81

3110.8

953.

0887

11.4

2915.2

2034.1

7834.5

9936.7

003

8.22

6113.3

973.

1118

13.4

2522.3

5134.9

4735.1

1340.6

454

12.1

4116.7

039.

0389

17.7

2926.4

4037.5

9438.4

8742.5

975

14.1

5217.8

309.

4796

20.3

8832.8

5742.4

2844.8

9848.4

696

19.3

6922.3

2816.0

3724.0

3435.3

9443.6

2545.8

5852.5

39Pr

esen

tFSD

T1

2.37

127.

6402

0.53

4710.0

3312.5

2931.7

2732.2

2935.8

272

5.78

5310.8

563.

0688

11.4

0215.1

2534.1

2734.5

4236.6

123

8.16

4613.3

783.

0946

13.4

0322.1

4734.8

8435.0

4340.4

884

12.0

7216.6

298.

9686

17.6

1026.3

3537.5

2138.3

9442.3

615

14.0

6317.7

519.

4149

20.2

2132.5

1942.3

0344.7

4148.1

836

19.2

2422.1

6115.9

1023.9

1135.2

1543.5

0045.6

6852.2

18


Tabl

e12

The

low

estfi

vefr

eque

ncy

para

met

ersΩ

=ω

L2 α/

h√ ρ

/E

1of

ath

ree-

laye

red,

cros

s-pl

y[0◦/90

◦ /0◦

]ope

nsp

heri

cals

hell

with

vari

ous

boun

dary

cond

ition

san

dsu

bten

ded

angl

es(R

=1

m,

h/

R=

0.01,ϕ

0=

30◦ ,ϕ

1=

90◦ ,)

Subt

ende

dan

gle

(θ0)

Mod

eB

ound

ary

cond

ition

s

S–S–

S–S

C–C

–C–C

E1–

E1–

E1–

E1

E2–

E2–

E2–

E2

E3–

E3–

E3–

E3

F-E

1–F

–E

1F-

E2–F

–E

2F-

E3–F

–E

3SD

-E1–S

D–

E1

SD-E

2–S

D–

E2

45◦

19.

7199

110.6

160

7.75

043

9.97

924

7.53

871

4.84

501

6.10

886

4.83

664

6.99

260

6.90

766

210.4

420

11.0

496

8.47

375

10.6

548

8.40

413

6.20

637

7.00

090

5.96

863

7.35

915

9.52

070

310.5

030

11.9

162

9.28

904

11.0

468

9.00

117

7.05

572

7.34

667

6.80

885

9.66

062

9.79

653

412.3

306

13.6

860

10.2

981

12.8

290

9.48

521

7.15

364

9.51

640

7.06

314

9.77

147

11.6

611

514.7

358

15.5

032

11.0

335

14.9

427

10.8

029

8.33

472

10.4

855

8.24

406

10.7

090

12.5

133

90◦

17.

8795

48.

4745

76.

4939

68.

0607

86.

3505

32.

5075

52.

5391

32.

4010

54.

2894

64.

2734

72

8.61

756

9.23

281

6.53

775

8.76

147

6.43

753

2.92

650

3.30

916

2.92

217

5.06

022

5.15

764

38.

8602

39.

5547

07.

9947

19.

0546

87.

7893

14.

2096

64.

3612

04.

1779

66.

1819

06.

7287

74

9.19

082

9.60

484

8.25

598

9.29

629

7.87

940

4.24

592

4.86

417

4.20

653

7.90

007

8.47

822

510.1

869

10.7

409

8.81

178

10.3

5091

8.30

278

5.28

520

6.34

762

5.28

191

8 .54

197

8.69

632

135◦

17.

3429

17.

7322

86.

0025

27.

4461

25.

8907

41.

4068

21.

3856

11.

3435

73.

2342

43.

2322

32

7.87

965

8.41

371

6.03

433

8.01

320

5.93

277

1.64

706

1.78

815

1.62

269

4.34

648

4.32

489

38.

3475

28.

9437

06.

9418

78.

4835

46.

7900

32.

7117

92.

7667

02.

6742

04.

8954

34.

9804

74

8.64

326

9.11

783

7.32

004

8.76

981

7.18

292

3.07

157

3.21

054

3.02

063

5.95

350

6.23

238

58.

6815

59.

1473

17.

7453

18.

7737

37.

4041

03.

7105

73.

7390

73.

6502

46 .

4351

96.

4391

918

0◦1

6.95

809

7.22

125

5.81

021

7.02

677

5.70

471

0.84

237

0.82

639

0.80

409

2.72

744

2.72

822

27.

5868

28.

0518

35.

8127

47.

6944

05.

7111

00.

9718

61.

0290

60.

9460

63.

9994

63.

9897

03

7.88

846

8.41

494

6.48

874

8.01

283

6.35

717

1.76

551

1.84

907

1.73

317

4.35

630

4.35

580

48.

3160

48.

8519

86.

5488

88.

4498

66.

4487

92.

0230

42.

0703

91.

9738

24.

9818

55.

0224

15

8.45

045

8.89

242

7.26

833

8.52

388

6.99

451

2.80

979

2.93

758

2.77

912

5 .23

833

5.23

718

215◦

16.

6430

96.

8222

45.

7109

76.

6892

35.

6057

50.

5279

30.

5168

40.

5042

02.

4516

22.

4527

82

7.40

722

7.80

398

5.71

355

7.49

692

5.61

165

0.58

619

0.61

330

0.56

545

3.58

568

3.58

252

37.

6559

38.

1433

66.

1718

87.

7642

56.

0666

01.

1787

01.

2186

11.

1480

54.

1075

94.

0984

74

7.91

656

8.42

219

6.21

044

8.03

469

6.10

606

1.36

431

1.37

908

1.32

782

4.44

239

4.44

621

58.

2346

58.

7466

56.

7609

68.

3541

86.

6250

92.

0341

92.

0595

61.

9938

84 .

9207

64.

9181

5

466 T. Ye et al.

Fig. 9 Mode shapes for the composite open spherical shell with C–C–C–C boundary conditions

kind, the ordinary power polynomials and the Legendre orthogonal polynomials, can be easily accommodatedin the formulation. In addition, it can be readily applied to composite laminated deep open shells with more com-plex boundaries such as point supports, non-uniform elastic restraints, partial supports and their combinations.

Acknowledgments The authors would like to thank the reviewers for their Constructive comments. The authors gratefullyacknowledge the financial support from the National Natural Science Foundation of China (Nos. 51175098 and 51279035).

Appendix 1: Stiffness and mass matrices of laminated open cylindrical and conical shells

This appendix contains the detail expressions of the stiffness (K) and mass (M) matrices of a laminated openconical shell. By setting ϕc = 0, these expressions are applicable to the open cylindrical one. For the sake ofsimplicity, two indexes are predefined:

p = n + (N + 1)m + 1, p′ = n′ + (N + 1)m′ + 1 (A.1)

The elements in these two matrices are calculated according to the expressions given as follows:

{Kp,p′

u,u

}= [A11�

111 + A12s(�010 +�100)+ A22s2�00−1]�00 + A16�

010�10 + A16�100�01

+ A26s�00−1�10 + A26s�00−1�01 + A66�00−1�11 +

[R0ku

x0(−1)m+m′ + R1kux L

]�00

+[kuθ0(−1)n+n′ + ku

θL

]�001

{Kp,p′

u,v

}= A12�

010�10+ A16�111�00− A16s�00−1�10+ A22s�100�00+ A26s�100�00 − A26s2�00−1�00

+ A26�00−1�11 + A66�

100�01 − A66s�00−1�01

{Kp,p′

u,w

}= A12c�010�00 + A22sc�00−1�10 + A26c�00−1�01


{Kp,p′

u,x

}= [B11�

111 + B12s(�010 +�100)+ B22s2�00−1]�00 + B16�

010�10 + B16�100�01

+ B26s�00−1�10 + B26s�00−1�01 + B66�00−1�11

{Kp,p′

u,θ

}= B12�

010�10+B16�111�00−B16s�00−1�10+B22s�100�00+B26s�100�00 − B26s2�00−1�00

+ B26�00−1�11 + B66�

100�01 − B66s�00−1�01

{Kp,p′v,v

}= A22�

00−1�11+ A26[(�100 − s�00−1)�01 + (�010 − s�00−1)]�10 + A44c2�00−1�00

+ A66�111�00 + (s�100 + s�010 + s2�00−1) A66�

00

+[

R0kvx0(−1)m+m′ + R1kvx L

]�00 + [kvθ0(−1)n+n′ + kvθL ]�001

{Kp,p′v,w

}= A22c�00−1�01 + A26c�010�00 − A26sc�00−1�00 − A44c�00−1�10 − A45c�100�00

{Kp,p′v,x

}= −A45c�000�00+B21�

100�01+B22s�00−1�01+B26�00−1�11 + B61�

111�00 − B61s�100�00

+ B62s�010�00 − B62s2�00−1�00 + B66�010�10

{Kp,p′v,θ

}= B22�

00−1�11 + B26[(�100 − s�00−1)�01 + (�010 − s�00−1)]�10 − A44c�000�00

+ B66�111�00 − B66s�00−1�10 + (s�100 + s�010 + s2�00−1) B66�

00

{Kp,p′w,w

}= A22c2�00−1�00 + A44�

00−1�11 + A45s�010�10 + A45s�100�01 + A55�111�00

+[

R0kwx0(−1)m+m′ + R1kwx L

]�00 +

[kwθ0(−1)n+n′ + kwθL

]�001

{Kp,p′w,x

}= A45�

000�01 + A55�011�11 + B21c�100�00 + B22sc�00−1�00 + B26c�00−1�10

{Kp,p′w,θ

}= A44�

000�01 + A45�011�00 + B22c�00−1�10 + B26c�100�00 − B26cs�00−1�00

{K p,p′

x,x

}= [D11�

111 + D12s(�010 +�100)+ D22s2�00−1]�00 + D16(�

010�10 +�100�01)

+ A55�001�00+D26s�00−1 (�10+�01)+ D66�

00−1�11 +[

R0 K xx0(−1)m+m′ + R1 K x

x L

]�00

+[Kxθ0(−1)n+n′ + K x

θL

]�001

{Kp,p′

x,θ

}= D12�

010�10 + D16�111�00 − D16s�00−1�10 + D22s�100�00+D26s�100�00

− D26s2�00−1�00 + D26�00−1�11 + D66�

100�01 − D66s�00−1�01 + A45�001�00

{Kp,p′θ,θ

}= D22�

00−1�11+D26[(�100 − s�00−1)�01+(�010−s�00−1)]�10+ A44�001�00+D66�

111�00

+ (s�100 + s�010 + s2�00−1)D66�00

+[

R0 K θx0(−1)m+m′ + R1 K θ

x L

]�00 +

[K θθ0(−1)n+n′ + K θ

θL

]�001

{Mp,p′

u,u

}={

Mp,p′v,v

}={

Mp,p′w,w

}= I0�

001�00;{

Mp,p′x,x

}={

Mp,p′θ,θ

}= I2�

001�00;{

Mp,p′u,x

}={

Mp,p′v,θ

}= 2I1�

001�00

468 T. Ye et al.

where

�abc =1∫

−1

[2a+b−1

La+b−1 λc daTm(x)

dxadbTm′(x)

dxb

]

dx, �e f =1∫

−1

[2e+ f −1

θe+ f −10

de Pn(θ)

dθe

d f Pn′(θ)

dθf

]

dθ

RL = R0 + sL , λ = sL

2(x + 1)+ R0, s = sin(ϕc), c = cos(ϕc)

Appendix 2: Mass and stiffness matrices of a laminated open spherical shell

This appendix contains the detail expressions of the stiffness (K) and mass (M) matrices for a laminated openspherical shell. The elements of these matrices are calculated by the expressions given as follows:

{Kp,p′

u,u

}= [A11�

11ms + A12

(�01

mc+�10mc

)+ A22�00ct

]�00+ A16�

01m0�

10+ A16�10m0�

01+ A26�00dt

(�10+�01)

+ A66�00ds�

11 + A55�00ms�

00 +[

Rskuϕ0(−1)m+m′ + RLku

ϕL

]�00 + [ku

θ0(−1)n+n′ + kuθL ]�00

ms{

Kp,p′u,v

}= A12�

01m0�

10 + A16�11ms�

00 − A16�01mc�

00 + A22�00dt�

10 + A26�10mc�

00 − A26�00ct �

00

+ A26�00ds�

11 + A66�10m0�

01 − A66�00dt�

01 + A45�00ms�

00

{Kp,p′

u,w

}= [A11�

01ms + A12�

01ms + A12�

00mc + A22�

00mc − A55�

10ms]�00 + A16�

00m0�

01 + A26�00m0�

01

− A45�00m0�

10

{Kp,p′

u,ϕ

}= [B11�

11ms + B12

(�01

mc +�10mc

)+ B22�00ct ]�00 + B16�

01m0�

10 + B16�10m0�

01

+ B26�00dt

(�10 +�01)+ B66�

00ds�

11 + B55�00ms�

00 − A55 R�00ms�

00

{Kp,p′

u,θ

}= B12�

01m0�

10 + B16�11ms�

00 − B16�01mc�

00 + B22�00dt�

10 + B26�10mc�

00 − B26�00ct �

00

+ B26�00ds�

11 + B66�10m0�

01 − B66�00dt�

01 + B45�00ms�

00 − A45 R�00ms�

00

{Kp,p′v,v

}= A22�

00ds�

11 + A26�10m0�

01 − A26�00dt�

01 + A26�01m0�

10 − A26�00dt�

10 + A66�11ms�

00

+ A66�10mc�

00 + A66�00 (�01

mc +�00)+ A44�00ms�

00 +[

Rskvϕ0(−1)m+m′ + RLkvϕL

]�00

+[kvθ0(−1)n+n′ + kvθL ]�00ms

{Kp,p′v,w

}= A12�

00m0�

01 + (A16�01ms − A16�

00mc + A26�

01ms − A26�

00mc − A45�

10ms

)�00 + A22�

00m0�

01

−A44�00m0�

10

{Kp,p′v,ϕ

}= B12�

10m0�

01 + B22�00dt�

01 + B26�00ds�

11 + B16�11ms�

00 − B16�10mc�

00 + B26�01mc�

00

− B26�00ct �

00 + B66�01m0�

10 − B66�00dt�

10 − A45 R�00ms�

00

{Kp,p′v,θ

}= B22�

00ds�

11 + B26�10m0�

01 − B26�00dt�

01 + B26�01m0�

10 − B26�00dt�

10 + B66�11ms�

00

+ B66�10mc�

00 + B66�00 (�01

mc +�00)− A44 R�00ms�

00

{Kp,p′w,w

}= A11�

00ms�

00 + A12�00ms�

00 + A12�00ms�

00 + A22�00ms�

00 + A44�00ds�

11 + A45�01ms�

10

+ A45�10m0�

01 + A55�11ms�

00 + [Rskwϕ0(−1)m+m′ + RLkwϕL ]�00 + [kwθ0(−1)n+n′ + kwθL ]�00ms

{Kp,p′w,ϕ

}= (B11�

10ms + B12�

00ms + B21�

10m0 + B22�

00ms)�

00 +�00mc�

10 (B16 + B26)+ A45 R�00m0�

01

+A55 R�01ms�

00


{Kp,p′w,θ

}= (B16�

10ms − B16�

00mc + B26�

10ms − B26�

00mc

)�00 +�00

m0�10 (B12 + B22)+ A44 R�00

m0�01

+A45 R�01ms�

00

{Kp,p′ϕ,ϕ

}= [D11�

11ms + D12

(�01

mc +�10mc

)+ D22�00ct

]�00 + D16�

01m0�

10 + D16�10m0�

01

+ D26�00dt

(�10 +�01)+ D66�

00ds�

11 + A55 R2�00ms�

00 +[

Rs K ϕϕ0(−1)m+m′ + RL K ϕ

ϕL

]�00

+[Kϕθ0(−1)n+n′ + K ϕ

θL ]�00ms

{Kp,p′ϕ,θ

}= D12�

01m0�

10 + D16�11ms�

00 − D16�01mc�

00 + D22�00dt�

10 + D26�10mc�

00 − D26�00ct �

00

+ D26�00ds�

11 + D66�10m0�

01 − D66�00dt�

01 + A45 R2�00ms�

00

{Kp,p′θ,θ

}= D22�

00ds�

11 + D26(�10

m0 −�00dt

)�01 + D26

(�01

m0 −�00dt

)�10 + D66

(�11

ms +�10mc

)�00

+ A44 R2�00ms�

00 + D66�00 (�01

mc +�00)+[

Rs K ϕϕ0(−1)m+m′ + RL K ϕ

ϕL

]�00

+[

K ϕθ0(−1)n+n′ + K ϕ

θL

]�00

ms{

Mp,p′u,u

}={

Mp,p′v,v

}={

Mp,p′w,w

}= I0�

00ms�

00;{

Mp,p′ϕ,ϕ

}={

Mp,p′θ,θ

}= I2�

00ms�

00;{

Mp,p′u,ϕ

}={

Mp,p′v,θ

}= 2I1�

00ms�

00

where

�abη =

1∫

−1

[2a+b−1

�ϕa+b−1 ηdaTm(ϕ)

dϕadbTm′(ϕ)

dϕb

]

dϕ, �cd =1∫

−1

[2c+d−1

θc+d−10

dc Pn(θ)

dθc

dd Pn′(θ)

dθd

]

dθ

η = m0,ms,mc, ds, ct, dt; �ϕ = ϕ1-ϕ0; s = sin

(ϕ1 + ϕ0

2ϕ + �ϕ

2

)

; c = cos

(ϕ1 + ϕ0

2ϕ + �ϕ

2

)

t = tan

(ϕ1 + ϕ0

2ϕ + �ϕ

2

)

; m0 = 1, ms = s, mc = c, ds = 1

s, ct = c

t, dt = 1

t

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a unified chebyshev–ritz formulation for vibration analysis of composite laminated deep open...

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