Dynamic analysis of laminated doubly-curved shells with general boundary conditions by means of a domain decomposition method

Accepted Manuscript

Dynamic analysis of laminated doubly-curved shells with general boundary conditions by means of a domain decomposition method Jianghua Guo , Dongyan Shi , Qingshan Wang , Jinyuan Tang , Cijun Shuai PII: DOI: Reference:

S0020-7403(17)33335-0 10.1016/j.ijmecsci.2018.02.004 MS 4164

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

22 November 2017 31 January 2018 2 February 2018

Please cite this article as: Jianghua Guo , Dongyan Shi , Qingshan Wang , Jinyuan Tang , Cijun Shuai , Dynamic analysis of laminated doubly-curved shells with general boundary conditions by means of a domain decomposition method, International Journal of Mechanical Sciences (2018), doi: 10.1016/j.ijmecsci.2018.02.004

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ACCEPTED MANUSCRIPT

Highlights 

A domain decomposition method for dynamic analysis of laminated doubly-curved shells is presented.



The proposed method is appropriate for the shells with elastic restraints.



New results including the free and static analysis for the laminated doubly-curved shells are

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CE

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presented.

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Dynamic analysis of laminated doubly-curved shells with general boundary conditions by means of a domain decomposition method Jianghua Guo1, Dongyan Shi1, Qingshan Wang2,3*, Jinyuan Tang3, Cijun Shuai3

1

College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin,

2

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150001, PR China College of Mechanical and Electrical Engineering, Central South University, Changsha, 410083, PR China 3

State Key Laboratory of High Performance Complex Manufacturing, Central South University,

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Changsha 410083, PR China ABSTRACT

The purpose of this paper is to study dynamic analysis of composite laminated doubly-curved

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shells with various boundary conditions by a domain decomposition method. Multi-segment

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partitioning technique is used to establish the formulation based on the first-order shear deformation theory. Meanwhile, the interfacial potential energy is introduced to maintain the continuous

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condition on the contact surface of the adjacent segments. The displacement admissible functions

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for each doubly-curved shell segment are uniformly expanded to the double mixed series which is with the Fourier series along the circumferential direction and the orthogonal polynomials (i.e.

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Chebyshev orthogonal polynomial, Legendre orthogonal polynomials and Ordinary power polynomials) along the meridional direction. A series of numerical examples are given for the free vibration, steady-state vibration and transient vibration of laminated doubly-curved shells subject to different geometric and material constants. By comparing with the literature results and the results conducted by the general finite element program ABAQUS, the numerical results show that the

*

Corresponding Author: Telephone: +86-451-82519797; Email: [email protected]

ACCEPTED MANUSCRIPT present formulation has good computational accuracy and efficiency. Based on the verification, the effect of external forces, geometric and material parameters on dynamic analysis (free, steady-state and transient vibration) of laminated doubly-curved shells are also studied. Keywords: Free Vibration; Forced Vibration; Composite Laminated; Doubly-curved shell; Domain

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Decomposition 1. Introduction

The composite materials have been wildly used in many engineering fields such as aerospace industry, automotive industry, chemical industry, textile and machinery manufacturing field and

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medical domain because of their excellent characteristics, such as high specific strength and stiffness, special vibration damping characteristics, vibration and noise reduction and good fatigue resistance. As a very important part of basic elements of engineering structures, doubly-curved

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shells often suffer complex applied loads and different boundary conditions. It is well known that

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the practical engineering structures may fail and collapse because of material fatigue resulting from vibrations[1]. And in most practical engineering situations, doubly-curved shells endure the

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dynamic loads. Therefore, it is quite necessary to understand the vibration characteristics of

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doubly-curved shells under the dynamic loading, which makes it essential to develop an efficient and accurate vibration analysis approach for doubly-curved shells.

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The main research works about the titled problem at home and abroad are listed below: Reddy

and Chandrashekhara [2] developed a finite element for geometrically non-linear (in the von Karman sense) transient analysis of laminated doubly curved shells based on a dynamic shear deformation theory. Later, Chandrashekhara [3] presented an isoparametric doubly curved quadrilateral shear flexible element to study the free vibration characteristics of laminated doubly curved shells with various classical boundary conditions within the framework of the first-order

ACCEPTED MANUSCRIPT shear deformation theory. Based on the finite element method, Amabili and Reddy [4] presented a consistent higher-order shear deformation non-linear theory to study the large-amplitude vibrations of laminated doubly curved shells. Fazzolari and Carrera [5] presented a hierarchical trigonometric Ritz formulation in the framework of the Carrera unified formulation to investigate the free

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vibration response of doubly-curved anisotropic laminated composite shallow and deep shells with various classical boundary conditions. On the basis of geometrically non-linear theory, Amabili [6] proposed a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells with classical boundary conditions. Messina [7]

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presented a mixed variational approach and global piecewise-smooth functions to deal with free vibrations of multilayered laminated doubly curved shells with various boundary conditions. Oktem et al.[8] presented a hitherto unavailable Levy type analytical solution for the problem of

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deformation of a finite-dimensional general cross-ply thick doubly-curved panel of rectangular

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plan-form, modeled using a higher order shear deformation theory (HSDT). Ghavanloo, E.[9] presented the comprehensive free vibration analysis of doubly-curved shallow shells which were

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made of an orthotropic material. Tornabene and his co-authors [10-23] presented a Generalized

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Differential Quadrature (GDQ) method and a series of works about the bending, vibration, buckling and buckling of the composite laminated doubly-curved shells. For instance, Tornabene et al.

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applied the Radial Basis Function (RBF) method and Generalized Differential Quadrature (GDQ) method to study the free vibrations of doubly-curved laminated composite shells and panels with classical boundary conditions on the basis of the General Higher-order Equivalent Single Layer (GHESL) formulation. Viola et al. [21] investigated the static behavior of doubly-curved laminated composite shells and panels by using the Generalized Differential Quadrature (GDQ) method based on the Carrera Unified Formulation. Ye et al. [24] applied the modified Fourier series method to

ACCEPTED MANUSCRIPT study the vibration behavior of composite laminated doubly-curved shells of revolution in the framework of the first order shear deformation shell theory considering the effects of the rotary inertia and initial curvature. Wang et al.[25, 26] studied the vibration of the four-parameter functionally graded moderately thick doubly-curved shells and panels of revolution with general

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boundary conditions and presented a unified numerical analysis model to solve the free vibration of composite laminated doubly-curved shells and panels of revolution with general elastic restraints by using the Fourier–Ritz method.

Through reviewing the existing research reports, we can know that the importance of

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doubly-curved panels and shells of revolution due to the their special geometric shapes. And the main focus of the exsiting jobs are confined to single mechanical properties such as free vibration analysis, bending analysis, or statics analysis. In addtion, it is observed that the existing method or

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technique is only suitable for a particular type of classical boundary conditions, i.e.,

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simply-supported supports, clamped boundaries and free edges, which leads to constant modifications of the solution procedures and corresponding computation codes to adapt the

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different boundary cases. Then it will lead to very boring calculations and very difficult to carry out

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the application in practical engineering since the composite laminated doubly-curved shells are not always in certain classical cases but subjectd to a variety of possible elastic restraints of hybrid

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boundary conditions in nature[27-32]. Thus, to establish a unified, efficient and accurate formulation which is capable of universally dealing with the dynamic analysis of composite laminated doubly-curved shells with general boundary conditions is necessary and of great significance. Based on the above reasons, this paper aims to developing a domain decomposition method for free vibration and static analysis of composite laminated doubly-curved shells with various

ACCEPTED MANUSCRIPT boundary conditions. Multi-segment technique is adopted to satisfy the accurate requirements of response. The doubly-curved shells are split into a series of free-free segments along the length direction. And the interfacial potential energy derived by means of a modified variational principle and least-squares weighted is introduced to maintain the continue condition of interfaces between

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adjacent segments. The first-order shear deformation theory (FSDT) is employed to formulate the theoretical model. The displacement admissible functions for each doubly-curved shell segment are uniformly expanded to the double mixed series which is with the Fourier series along the circumferential direction and the orthogonal polynomials along the meridional direction. In order to

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identify the applicability and accuracy of the proposed domain decomposition method, a series of numerical examples of the free vibration and static response analysis of laminated doubly-curved shells with various boundary conditions are presented. By comparing with the literature results and

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the results conducted by the general finite element program ABAQUS, the numerical results show

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that the present formulation has a good computational accuracy and efficiency. Based on the verification, the effect of external forces, geometric and material parameters on dynamic analysis

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(free, steady-state and transient vibration) of laminated doubly-curved shells are also studied.

h

Names curvilinear coordinates thickness

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Variable φ, θ, ς

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2 Mathematic formulations

Rs

offset of revolution axis z with respect to geometric axis z’

Rφ, Rθ

principal radial of curvature of the shell

a, b rc

the length of the semi-major and semi-minor axis of the generatrix of the elliptical shell, respectively radius of the generator circle of the cycloidal shell

R

radius of circular toroidal shell

U, V, W

displacement variations of an arbitrary point (φ, θ, ς) lying on the shell space

u, v, w

middle surface displacements in the φ, θ and ς directions

ACCEPTED MANUSCRIPT  , 

rotations of normal to the middle surface with respect to the φ and θ-axes

t

time variable

Π

the total energy functional of laminated doubly-curved shell

2.1. The doubly-curved shell model As the structure of revolution, the geometry of shell is formed by a generatrix(generator)

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rotating around a certain axis which is paralleled to the space coordinate axis. Then, the surface of shell is obtained by sweeping the generatrix defined in the middle surface of shell. Fig.1 (a) shows the geometry and coordinate system of a doubly-curved shell of revolution with uniform thickness h.

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The doubly-curved shell is formed by the generatrix c1c2 in the x-z plane around the rotation axis z. Another axis z’ denotes the geometric central axis of the generatrix c1c2. Rs is the offset distance of the rotation axis z with respect to the geometric central axis z’. The orthogonal curvilinear coordinate system is introduced to simplify the description, as shown in Fig.1 (c), and the symbols

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φ, θ and ς are curvilinear coordinates of the doubly-curved shell along the generatrix,

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circumferential and normal directions of the shell, respectively. u, v and w separately indicate the

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displacement component of the shell in the φ, θ and ς directions, respectively. The two principal radius of curvature of the doubly-curved shell are represented by symbols Rφ and Rθ, where Rφ is

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radius of curvature in the plane φ-ς and Rθ is radius of curvature in the plane θ-ς. Cφ and Cθ indicate

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the centers of the two principal radiuses Rφ and Rθ, respectively. The horizontal radius R0 represents the distance from each point of the middle surface to the revolution axis z and it can be defined as R0 = Rθ sinφ [33, 34]. In actual operation, the geometric surface shapes of doubly-curved shells are determined by the radius of curvature Rφ and Rθ. Through the elliptical, cycloidal, circular toroidal, paraboloid, hyperbolical and catenary shells are frequently encountered in the practical engineering. Nevertheless, for the sake of brevity, this paper will only be confined to the above first three shapes.

ACCEPTED MANUSCRIPT The two principal radius of curvature of the above shells are given as follows [1, 24]: 1. Elliptical shell, see Fig.2 (a)

a 2b 2

a

2

3

a2

R   

(1.a)

sin 2   b2 cos 2  

a sin   b cos  2

2

2

2



Rs sin 

(1.b)

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R   

where a and b are the length of the semi-major and semi-minor axis of the generatrix of the elliptical shell, respectively.

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2. Cycloidal shell, see Fig.2 (b) R    4rc cos 

R   

rc  2  sin 2  sin 



Rs sin 

(2.a) (2.b)

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where rc represents the radius of the generator circle of the cycloidal meridian. 3. Circular toroidal shell, see Fig.2 (c)

Rs sin 

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R    R 

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R    R

(3.a) (3.b)

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where R is the radius of curvature of the cross section of middle surface along axis of rotation.

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2.2. Kinematic relations and stress resultants Before the theoretical modeling, the reasonable elastic theory according to the specific needs

of this paper should be selected firstly. In this work, the first-order shear deformation shell theory (FSDT) is adopted to establish the theoretical model. Accordingly, following the FSDT assumptions, the displacement field of the doubly-curved shell problem is expressed as[10]: U  , ,  , t   u  , , t     , , t 

(4.a)

ACCEPTED MANUSCRIPT V  , ,  , t   v  , , t     , , t 

(4.b)

W  , ,  , t   w  , , t 

(4.c)

where u, v, w, ϕφ and ϕθ are functions of the coordinate φ, θ. Among them, u, v and w represent the displacement variations of the corresponding point at the middle surface in the meridional (φ-), circumferential (θ-) and normal (ς -) directions, respectively; ϕφ and ϕθ represent the rotations of the

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normal to the middle surface with reference to the θ and φ direction. t is the time variable.

With the assumption of the first-order shear deformation shell theory, the strain components at

and rotations of normal as [1, 24]:

1   R 1

  

1   R 1

  

1   R



0





   ,  

0 

   

 0 ,   

1 1   R

1 1   R 1

1   R

 0





0



  

(5.a)

  

(5.b)

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1

(5.c)

0 

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 

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any point of the doubly-curved shell can be defined in terms of the reference surface displacements

in which

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 u  1   w  ,    R    

1 R

0 

1  v 1      u cos   w sin   ,       cos    R0   R0    

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 0 

(6.a)

(6.b)

 0 

1 v 1  ,   R  R 

(6.c)

 0 

1  u 1  w   0   v cos   ,      v sin     R0   R0    

(6.d)

 0 

1 R

 1    w      cos   ,  u        R0      

(6.e)

For composite laminated doubly-curved shells, the corresponding stress-strain relations in the

ACCEPTED MANUSCRIPT kth layer can be determined according to the generalized Hooke’s law [19, 33].     Q11k    k     Q12        0    0      k     k Q16

Q12k

0

0

Q22k

0

0

k 44 k 45

0 0 k 26

Q

Q

Q45k

Q

k 55

Q

0

0

Q16k        Q26k       0      0       Q66k      k

(7)

where   ,    and   ,   represent the normal stress and strain along the direction of φ and θ, T

k

respectively.



k



,  ,  

T

k

and   ,   ,   

T

k

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T

are the analogous shear stresses and strain

components of the cylindrical coordinate system. Qijk is transformed elastic coefficient defined in

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the definition of the lamina elastic coefficients Qijk for the kth layer and the angle ϑk between the principle material directions and the x-axis. The kth orthotropic laminated elastic coefficients Qijk are listed in Appendix A. The Qijk can be obtained by the characteristics of the kth orthotropic

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material[26, 35].

Carrying the integration of the stresses over the cross-section, the force and moment resultants

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of a composite laminated doubly-curved shell can be obtained:

 d 

(10.a)

 N  N    zk 1       N    z    1   Q  k 1 k    R     

 d 

(10.b)

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CE

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 N        N zk 1     N    z    1   Q  k 1 k    R      

 M   N zk 1         z   1   M   k 1 k     R

   d 

(10.c)

 M   N zk 1          z   1   M   k 1 k     R

   d 

(10.d)

Substituting equations (8) and (9) into Eq. (10) and performing the integration operation in Eq. (9) result in:

ACCEPTED MANUSCRIPT A12

A16

A16

B11

B12

B16

A22

A26

A26

B12

B22

B26

A26

A66

A66

B16

B26

B66

A26

A66

A66

B16

B26

B66

B12

B16

B16

D11

D12

D16

B22

B26

B26

D12

D22

D26

B26

B66

B66

D16

D26

D66

B26

B66

B66

D16

D26

D66

 A44 Q     Ks  Q   A45

B16   0    B26   0    0  B66       0  B66        D16       D26     D66        D66    

0 A45        0  A55      

where  k 1

A , B , D    

 k 1

ij

ij

ij

ij

ij

k

ij

k

A , B , D     ij

ij

 k 1

k

(11.b)

(12.a)

1   R  1   R 2 1   R   Qijk  , ,  d ,  i, j  1, 2, 4,5, 6  1   R  1   R 1   R  

(12.b)

1   R 1   R 2 1   R  Qijk  , ,  d ,  i, j  1, 2, 4,5, 6  1   R  1   R 1   R

(12.c)

M

ij

Qijk 1,  ,  2  d ,  i, j  1, 2, 4,5, 6 

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A , B , D    

(11.a)

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A  N   11  N  A     12  N   A16    N   A16   M     B11  M      B12  M    B16  M      B  16

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The above equations show that the stiffness coefficients can be integrated exactly since the two principal radius of curvature are only functions of the variable and independent of the normal

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coordinate. However, for better numerical stability, the initial curvature terms are expanded in series

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as:

  1  1   R R  R



2

3

       R

   (higher order terms) 

2

3

AC

1

     1        (higher order terms) 1   R R  R   R  1



(13.a)

(13.b)

Substituting Eq.(13) into Eq.(12) and neglecting the higher order terms result in the elements of the coefficient matrix of Eq. (11) and they are listed in Appendix A. 2.3. Energy expressions As mentioned earlier, the Multi-segment partitioning technique is used to establish the

ACCEPTED MANUSCRIPT formulation in framework of the domain decomposition method. Thus, the laminated doubly-curved shell is divided into N0 free-free laminated doubly-curved shell segments along generatrix line, as shown in Fig. 3(a). The natural continuity conditions of a shell domain need not be imposed as their eventual satisfaction is implied in a variational statement. Thus, admissible functions of the shell

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segments can be handled in a unified way. Thereby, the problem is reduced to modeling the essential continuity constraints of those shell domains on common boundaries. The geometrical boundaries herein are treated as special interfaces as those between adjacent shell domains. In this work, according to Hamilton principle, the total energy functional of entire laminated



t1 N

t0

 T  U i 1

i

i

 Wi dt  

t1

t0

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doubly-curved shell is given as[36]:

  dt

i ,i 1

(16)

where Ti and Ui are the kinetic energy and strain energy of each segment. Wi is the work done by

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external forces. Πλκ is the interfacial potential of adjacent segment i and i+1. The subscript i

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resprents the number of segment. t0 and t1 are two specified times. Based on the displacement field  u , v , w the kinetic energy ith segment is expressed as[36]:

 h2 1  2 2 2   U  V  W   1    i i i       h 2 2 i i  R



PT

Ti 

    1   R0 Ri d  d d   R 







AC

where

CE

1    0  ui 2  vi 2  wi 2   2 1 uii  vii   2 2i  2i 2 i i

0  0 

 R R 0

i

(17)

d d

   1 1       2 , 1  1  2  2  3 , 2  2  3  3  4 R R R R R R R R R R R R

 0 , 1 , 2 , 3 , 4    h/2  k 1,  ,  2 ,  3 ,  4  d h /2

(18.a) (18.b)

where ρ(ς) is density distribution function of the lamina along z-axis. φi and θi are length and width of each segment, respectively. The strain energy of the ith segment is expressed as[36]:

ACCEPTED MANUSCRIPT 0 0  N 0  N 0  N    N    M    1 Ui     0 2 i i  M    M    M    Q  0  Q  

 R0 Ri d d  

(19)

External loads are assumed to act on the entire middle surface of the doubly-curved shell. The virtual work done on the ith shell domain by the distributed load components in the φ, θ and ς directions, namely fu,i, fv,i and fw,i, is presented by[37]:

 f

 

u  f v,i vi  f w,i wi R0 R d d

(19)

u ,i i

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Wi  

The basic essence of the domain decomposition method is to construct the interface or boundary potentials ∏λκ in Eq. (16). There exists a rich body of literature on the establishment of

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modified variational functionals. In this work, a combination of a modified variational principle and least-squares weighted residual method is employed to obtain the interface and boundary potentials. In doing so, the potentials ∏λκ are written as [36]:

 1      2



    u N u  v N v  wQ w   M     M   R0 d 

u

u



 v v v2  w w2w    2    2 R0 d

M



2 u

(20)

ED

where the first integral expression in above equation is to relax the enforcement of the interface and

PT

boundary constraints by using the modified variational functionals. And the second integral expression is obtained by means of the least squares weighted residual method to ensure the

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numerical stability of the present method. The integrations in Eq. (20) are carried out over the

AC

interfaces and geometric boundaries. κi (i=u, v, w, ϑ and φ) are the pre-assigned weighted parameters. λi (i = u, v, w, ϑ and φ) is the parameter which defines different boundary conditions. For the case of two adjacent shell segments, κi=1; while for the case of the specified displacement boundary, values of κi are defined in Table 1. An arbitrary set of classical boundary conditions at the two ends of a doubly-curved shell can be obtained by an appropriate choice of the values of κi. Θu, Θv, Θw, Θϑ and Θφ represent the continuity equation on the common interfaces and geometrical boundaries which is defined as[36]:

ACCEPTED MANUSCRIPT u  ui  ui 1  0, v  vi  vi 1  0, w  wi  wi 1  0

(21.a)

   i   i 1  0,    i   i 1  0

(21.b)

2.4. Equations of displacement admissible function To derive the discretized equations of motion for the doubly-curved shell, the displacement components (ui, vi, wi, ϕφi, ϕθi,) involved in ∏ should be expanded in terms of generalized

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coordinates and admissible functions. It is obvious that ∏ is not constrained to satisfy any continuity conditions or geometrical boundary conditions. The functional ∏ permits the use of the same admissible functions for each shell domain, and these functions are only required to be

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linearly independent, complete and regular enough to be differentiable. Harmonic functions for the circumferential expansion and polynomials for the meridional expansion of each shell domain are adopted in the present analysis. Subscript i is omitted here for the sake of brevity. The displacement

P

ui  , , t    AipTp   sin n e jt

ED

p 1 P

P

PT

vi  , , t    BipTp   cos n e jt p 1

wi  , , t    C ipTp   sin n e jt

CE

p 1

M

components of each doubly-curved shell domain can be written as [38, 39]: (22.a)

(22.b)

(22.c)

P

 i  , , t    DipTp   sin n e jt

(22.d)

AC

p 1 P

 i  , , t    E ipTp   cos n e jt

(22.e)

p 1

where Aip , B ip , C ip , D ip and E ip are the coefficients of the displacement admission function of pth polynomials on ith segment. Tp  x  is the pth order polynomial function expanded for the displacement component in arbitrary direction. P is the highest order truncated in the polynomial function. n is the number of half wave. In order to illustrate the utility and robustness of the

ACCEPTED MANUSCRIPT proposed formulation, four sets of polynomials are applied to expand the displacements of each shell domain in the meridional direction. They are: (a) Chebyshev orthogonal polynomials of first kind (COPFK): T0 ( )  1

(23.a)

T1 ( )  

(23.b)

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Tp 1 ( )  2Tp ( )  Tp 1 ( ) , for p ≥ 2

(23.c)

(b) Chebyshev orthogonal polynomials of second kind (COPSK): T0 ( )  1

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T1 ( )  2

Tp 1 ( )  2Tp ( )  Tp 1 ( ) , for p ≥ 2

(24.a) (24.b) (24.c)

(c) Legendre orthogonal polynomials of first kind (LOPFK):

M

T0 ( )  1 T1 ( )  2

ED

 p  1 Tp1 ( )   2 p  1Tp ( )  pTp1 ( ) , for p ≥ 2

(25.a) (25.b) (25.c)

PT

(d) Ordinary power polynomials (OPP): (26)

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Tp ( )   P , for p = 0, 1, 2, ...

The polynomial functions are defined on the [-1,1] integral. Therefore, a transformation rule

AC

for coordinates from x, y and z to x , y and z which are defined on the integral [0,1] need to be introduced, i.e., x  a x  b , a  ( xi 1  xi ) 2 and b  ( xi 1  xi ) 2 . Substituting Eqs.(17)-(20) into Eq.(16) as well as the displacement admission function Eq.(23)-(26), and then applying the variational operation:

   

t1

t0

yields:

N0

 T  U i 1

i

i

 Wi dt   

t1

t0

  dt  0

i ,i 1

(27)

ACCEPTED MANUSCRIPT N   0ui ui  1i ui   0 vi vi  1i vi   0 wi wi  t1   0 0        R0 R d d dt t0     i 1 i   1uii   2i i  1vii   2i i   0 0  N0  N  0  N 0  N    N    M          0 0 t0 i    M    M    M    Q    Q    i 1 t1

  R0 R d d dt   

 N0     N     N     Q     M     M   u u  v v  w w               R0 d dt  t0     N    N    Q    M    M   v  v w  w         i 1  u  u   i

(28)

t1

Organizing Eq.(28) into the matrix form as follows:

(29)

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Mq + K - K  + K  q = F

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 N0                  u u u u v v v v w w w w        R0 d dt  t0                  i 1         i t1

where q is global generalized coordinate vector for the doubly-curved shell. M and K are mass and stiffness matrices, which are obtained by the assembly of the corresponding segment matrices Mi

M

and Ki. Kλ and Kκ are the generalized interface stiffness matrices introduced by the identified Lagrange multipliers and the least-squares weighted residual terms, respectively. The way stiffness

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matrix K is constructed as shown in Fig. 4 and Fig.5. The elements in the above matrices are listed

PT

in Appendix B.

3. Numerical examples and discussion

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In order to identify the reliability and accuracy of the proposed domain decomposition

AC

approach, a series of numerical examples for the free and forced vibration analysis of laminated doubly-curved shells with various boundary conditions are presented. And the computational results of the present method are compared with those of published articles and conducted by the general finite element program ABAQUS v6.10, which are run on a Intel(R) Core(TM) i7-600 3.40GHz PC. Based on the verification, the influence of the geometry and material parameters on free and forced vibration response of laminated doubly-curved shell is studied. For the convenience of expression, the boundary conditions of free edges, simply-supported and clamped boundary are simplified by

ACCEPTED MANUSCRIPT uppercase letters F, SS and C. The solution procedure by means of the present method is implemented in MATLAB scripts, which are run on the computer which has the same hardware configuration as the program ABAQUS v6.10. 3.1. Free vibration analysis

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In this section, the free vibration analysis of laminated doubly-curved shells with various boundary conditions is examined by the present method. To obtain the free vibration, we should assume that the structure is simple harmonic motion, q=q0ejωt. Then, by substituting q=q0ejωt into Eq.(29), the governing equations of motion for free vibration analysis can by be written in the

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standard form  2M   K - K  + K  q0  0

(30)

Solving the eigenvalue problem of Eq. (30) yields the frequency parameters ω and the

M

corresponding eigenvector q0. The mode shapes corresponding to the certain frequency parameter

ED

are obtained by substituting the corresponding eigenvector q0 back into the displacement fields defined in Eq. (22). As mentioned earlier, the choice of the displacement admission functions is the

PT

key to obtain the accurate results during structural vibration analysis. Thus, the deviation of modal

CE

frequency for four general types of displacement admission functions under the same situations is studied aiming at deciding which one is suitable for this method. Fig.6 presents the frequency f1,1

AC

discrepancies of orthogonal polynomials for the two-layer [00/900] laminated doubly-curved shell with C-C boundary conditions. The geometric and material parameters are given as: (a) a = 2 m, b = 4m, h = 0.1m, Rs = 1 m, φ0 = 1/6π, φ1 = 5/6π ( Fig.6(a) ); (b) rc = 1 m, h = 0.1m, Rs = 2 m, φ0 = 150, φ1 = 750 ( Fig.6(b) ); (c) R = 1 m, h = 0.05m, Rs =1m, φ0 = 1/6π, φ1 = 1/2π, ( Fig.6(c) ). The material constants are defined as below: E1 = 150 GPa, E2 = 10 GPa, ν12=0.25, G12 = G13 = 5 GPa, G23 = 6 GPa and ρ =1450 kg/m3. The discrepancy is defined as: Relative discrepancy = ( fδ - fCOPFK )/fCOPFK,

ACCEPTED MANUSCRIPT where the subscript δ denotes the COPSK, LOPFK and OPP. The number of segments and weighted parameter are set as N0 = 20 and κ = 1×1014, respectively. It can be observed from Fig.6 that the absolute maximum discrepancy does not exceed 5×10-7 for the worst case. It means that the present method has good compatibility, regardless of the types of the displacement admission functions.

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Thus, in the following examples, the Chebyshev orthogonal polynomial of the first kind is set as the displacement admission function.

Then, the convergence research on the number of segments N0 for the present method is carried out. Fig.7 shows the first six natural frequencies fn,m ( n: circumferential wavenumber; m:

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longitudinal wavenumber ) of two-layer [0°/90°] laminated doubly-curved shell with various number of segments under C-C boundary condition. The geometric and material parameters are the same as Fig.6. The weighted parameter is taken as κ = 1×1014. It can be seen from the Fig.7 that the

M

natural modal frequency of each order has a good convergence behavior with the incensement of

ED

the number of segments. When the number of segments is more than five, the vibration characteristics of the laminated doubly-curved shells almost do not change. Thus, the number of

PT

segments is taken as N0=20 in the following examples.

CE

After the convergence research on the number of segments N0, the convergence research on different weight parameter is carried out. The convergence research on different weight parameter κ

AC

(defined from 0 to 1×1018) of a two-layer [0°/90°] laminated doubly-curved shell with C-C boundary condition is shown in Fig.8. It can be seen that the weight parameter plays an important role in current solutions. According to actual trial, the stable and accurate results will be obtained when the weighted parameter is selected as κ>1×1013. However, the weighted parameter doesn’t have to be large enough to get the reasonable responses. Thus, in following studies the value of the weighted parameter is taken as κ=1×1014.

ACCEPTED MANUSCRIPT It should be mentioned that an appropriate pre-assigned weighted parameter κ and the parameter λ can be used to formulate all kind of boundary conditions including classic and elastic boundary conditions. Fig.9 shows the effect of the elastic stiffness K on the frequencies of laminated doubly-curved shells. The geometric and material parameters are the same as Fig. 6. It can be seen from Fig.9 that the frequencies rise with the increase of displacement stiffness Ku, Kv

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and Kw in the interval Ki =1×104~1×1013, and remain unchanged when Ki <1×104 or Ki >1×1013. It indicates that the elastic stiffness Ki <1×104 and Ki >1×1013 can simulate the free and rigid constraints, respectively. In addition, the torsional stiffness has little effect on frequencies. Thus,

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based on the above analysis, Ki =0 and Ki =1×1014 are employed to simulate the free edge and clamped boundary condition.

In order to illustrate the accuracy of the present method, the natural modal frequencies of

M

laminated doubly-curved shells with various boundary conditions are compared with the published

ED

literature results. Table 2 presents the first six natural frequencies for a C-C elliptical shell with [0°/90°] and [0°/90°/0°] lamination schemes. The geometric and material parameters are the same

PT

as Fig. 9. The current results are compared with those reported by Wang et al. [26] by using the Ritz

CE

method on the basis of the FSDT. Table 3 given the first ten frequencies for a C-C elliptical shell with [30°/60°] lamination scheme. The geometrical and material constants of shell are assumed as:

AC

a = 1 m, b = 1 m, h = 0.1m, Rs = 3 m, φ0 = 0, φ1 =2π, θ=2/3π, E1 = 137.9 GPa, E2 = 8.96 GPa, ν12=0.3, G12 = G13 = 7.1 GPa, G23 = 6.21 GPa and ρ =1450 kg/m3. The results are compared with those obtained by Tornabene [10] using the Generalized Differential Quadrature procedure and FEM commercial program. The first five non-dimensional frequency n,m   rc2 h  E2 (n: circumferential wavenumber; m: longitudinal wavenumber) of elliptical shell with [0°/90°] and [0°/90°/0°] lamination schemes under C-F, SS-SS and C-C boundary conditions are performed in

ACCEPTED MANUSCRIPT Table 4. The geometric and material parameters of Fig.9 are used in this table. In order to contrast, the comparison data taken from Wang et al. [26] by means of the Ritz method are also shown in Table 4. Further, Table 5 shows the first ten frequencies for an F-C asymmetric laminated cycloidal shell with [-45°/-20°/70°/20°] lamination scheme. The geometrical constants are given as: rc = 1 m,

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h = 0.1m, Rs =5 m, φ0 = -700, φ1 = -50 and the material parameters are the same as Table 3. In this table, the reliable contrast data are from the Tornabene [10] by using the 2-D GDQ solution. Subsequently, the non-dimensional frequencies of the first ten circumferential wavenumber of [0°/90°] circular toroidal shells subject to C-F, SS-SS and C-C boundary conditions are given in

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Table 6. The geometric parameters are as follows: R = 1 m, h = 0.05m, Rs =0, φ0 = 1/6π, φ1 = 1/2π. The current results are compared with those reported by Qu et al. [36] using the semi-analytical solutions on the basis of the FSDT. Lastly, Table 7 performs the comparison of the first ten

M

frequencies (Hz) for the [30°/45°] circular toroidal shells with F-C boundary condition. The

ED

following geometric parameters: R = 2 m, h = 0.1 m, Rs =0, φ0 = 1/6π, φ1 = 1/2π are used in practical calculation. And the material constants of this table are the same as the Table 3. Also, the

PT

results from the Tornabene et al. [20] by using the local GDQ method are given in Table 7 as the

CE

reference data. From the comparisons, we can see a consistent agreement of the present results and the referential data. Besides, Tables 2-7 also show that it is appropriate to define the classical

AC

boundary conditions in terms of the boundary spring rigidities. Based on the studies of convergence and verification, the influence of lamination schemes on vibration of laminated doubly-curved shells is discussed. Fig.10 and Fig.11 present the frequency fn,m of laminated doubly-curved shells under C-C boundary condition with different lamination schemes [0/ϑ]2 and [0/ϑ/ϑ/0], respectively. The fiber orientation ϑ is changed from 0 to 90o. The geometric and material parameters are the same as Fig.9. It is observed that the fiber orientation has a significant effect on the vibration

ACCEPTED MANUSCRIPT characteristics of the composite laminated panel. At the end of this section, some wonderful modal shapes will be given in Fig.12-14 to improve the understanding of the vibration behavior of laminated doubly-curved shell of potential readers. 3.2. Forced vibration analysis

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This section concentrates on the forced vibration (steady-state and transient vibration) problems of laminated doubly-curved shells under different external excitation forces. Three common loads: point force, line force and surface force are discussed in this section. The diagrammatic sketch of three applied load types for the doubly-curved shells is shown in Fig.15.

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3.2.1 Steady-state vibration analysis

In this section, the feasibility of verifying this method to calculate steady-state vibration analysis of laminated doubly-curved shells will be firstly displayed. For the sake of brevity, the

M

laminated doubly-curved shells are represented by circular toroidal shells to study. The first case

ED

concerns the circular toroidal shell subjected to the harmonic point force fw applied at the Load Point A (φA = 0.5π, θA = 0) in the thickness direction and vertical acting on the surface. The point

PT

load in expressed as: f w  f w sin t      A      A  , where the amplitude of the harmonic

CE

force is taken as: qw =1N and ω is the frequency of the harmonic point force; δ(φ) is the Dirac delta function. The displacement response measured at Point B (φB = 0.5π, θB =π) in the vertical direction

AC

is illustrated in Fig.15(a). The second case is circular toroidal shell under the axisymmetric line force fu  fu sin t     0  , which is applied at the left end of circular toroidal shell in the φ direction, as shown in Fig.15(b). f u ( f u =1N) and φ0 are the amplitude and location of the harmonic force, respectively. The last case concerns the vibration responses of the circular toroidal shell subjected to the normal distributed unit surface force fw over the area (φ1 = 1/6π, φ2 = 5/6π, θ1 = 0, θ2 =π). The displacement response of the Point B obtained in the normal direction is depicted in

ACCEPTED MANUSCRIPT Fig.15(c). The displacement response of the above three case are present in Fig.16. Due to lack of suitable comparison results in the literature, the accuracy of the present model is validated by making comparisons with the finite element analysis ABAQUS v6.11. There is a good agreement between the present results and those obtained by using ABAQUS, which proves that the present

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method has the ability to deal with the steady-state vibration analysis of laminated doubly-curved shells with various boundary conditions.

Based on the verification, the influence of lamination schemes on steady-state vibration of the laminated doubly-curved shells is studied. Fig.17 presents the effect of the fiber orientation on the

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displacement of laminated doubly-curved shells. And Fig.18 presents the effect of the number of layer on the displacement of laminated doubly-curved shells. The boundary conditions of the two examples are set as C-C. The surface load fw ( f u =1N; over the areaφ1 = 1/6π, φ2 = 5/6π, θ1 = 0, θ2

M

=π) are the same as Fig.16. The geometric and material parameters are given as: (1) elliptical: a =

ED

1m, b = 2m, h = 0.05m, Rs = 1 m, φ0 = 1/6π, φ1 = 5/6π; (2) cycloidal: rc = 1 m, h = 0.05m, Rs = 1 m, φ0 = 150, φ1 = 900; (3) circular toroidal: R = 1 m, h = 0.05m, Rs =1m, φ0 = 1/6π, φ1 = 5/6π. It can be

PT

fingered out that the fiber orientation has a more obvious effect on the steady-state vibration

CE

response than the number of layer of laminated doubly-curved shells. 3.2.2 Transient vibration analysis

AC

In this subsection, the methodology outlined previously is applied to obtain the transient

responses of laminated doubly-curved shells subjected to different loads. In this work, four common shock loads, namely rectangular pulse, triangular pulse, half-sine pulse and exponential pulse, are considered. The sketch of load time domain curve is illustrated in Fig.19. These load curves can be described by the following formulas:

ACCEPTED MANUSCRIPT q fr t    0 0

0  t 

(31.a)

t 

2t  q   0  2    ft  t   q0   t   q0   2   0  

0t 

 2

 2

 t 

(31.b)

t 

0  t  t 

(31.c)

(31.d)

AN US

 q e  t fe  t    0  0

CR IP T

  t  q0 sin   0  t   fs t       0 t  

where f r  t  , ft  t  , f s  t  and f e  t  represent the function of rectangular pulse, triangular pulse, half-sine pulse and exponential pulse, respectively; q0 is the load amplitude;  is the pulse width; t

M

is the time variable. In the following analysis process, three types of loads (i.e., the point force, axisymmetric line force and normal distributed surface force) are considered. The Newmark direct

ED

integration method is used to calculate the transient response of the laminated doubly-curved shells. First, the contrastive study is carried out on the displacement of the observation point B (φB =

PT

0.5π, θB =π) on the circular toroidal shell which is subjected to rectangular pulse load f r  t  ( q0 =1N;

CE

 =3ms) and with C-C boundary condition, as shown in Fig.19 . The calculation time t0 and

AC

calculation step Δt are set 10ms and 0.02ms, respectively. The geometric parameters are given as: R = 1 m, h = 0.05m, Rs =1m, φ0 = 1/6π, φ1 = 5/6π. The material is given as: E = 210 GPa, ν=0.3 and ρ =7800 kg/m3. The results are compared with those obtained by the finite element analysis ABAQUS. It is obvious that the theoretical results show a good agreement with the FEM results. After having tested the accuracy of the present method on the transient vibration problems, the effects of different types of loads on the transient vibration responses of laminated doubly-curved shells are presented. The displacement of the observation point B (φB = 0.5π, θB =π) on elliptic,

ACCEPTED MANUSCRIPT cycloid and circular toroidal shells with C-C boundary condition under the different shock loads is examined herein as shown in Fig.21. The surface load is introduced in this process. The geometric and material parameters are the same Fig.6. Comparing with the rectangular pulse, the sine and triangular pulses lead to much smaller magnitude for the deflection response. Hence, the slow

CR IP T

loading (unloading) has the capacity of decreasing the transient response amplitudes, and on the other hand, a suddenly loaded (unloaded) force will increase the transient response. In addition, the transient responses in the half cycle sinusoidal pulse loading case, which indicates that the transient responses can be reduced by smoothing the load shape.

AN US

4. Conclusions

In this article, the free and forced vibrations analysis of isotropic and composite laminated doubly-curved shells is studied by a domain decomposition approach. In the proposed approach, the

M

doubly-curved shells are divided into a series of free doubly-curved shell segments and geometric

ED

boundaries along the length direction. The potential energy function is introduced to satisfy the interface continuity condition by introducing a modified variational principle and least-squares

PT

weighted residual method. By comparing with the literature results and FEM results, the efficiency

CE

and stability of this method are illustrated for free and forced vibrations of the isotropic and composite doubly-curved shells with various boundary conditions. Some new results of laminated

AC

doubly-curved shells with different geometry dimensions, material constants and boundary conditions are presented, which may serve as benchmark solutions for the future researches in this field. Acknowledgments The authors would like to thank the anonymous reviewers for their very valuable comments. The authors gratefully acknowledge the financial support from the National Natural Science

ACCEPTED MANUSCRIPT Foundation of China (Grant No. 51705537, 51535012, U1604255) and the Key research and development project of Hunan province (No. 2016JC2001). Appendix A The Qijk can be obtained by the characteristics of the kth orthotropic material [26, 35].

k k k k Q12   Q11  Q22  4Q66  sin 2  k cos2  k  Q12k sin 4  k  cos4  k  k k k k Q22  Q11 sin 4  k  2  Q12  2Q66  sin 2  k cos2  k  Q22k cos4  k

CR IP T

k k k k Q11  Q11 cos4  k  2  Q12  2Q66  sin 2  k cos2  k  Q22k sin 4  k

k k k k Q16   Q11  Q12  2Q66  sin  k cos3  k  Q12k  Q22k  2Q66k  cos k sin3  k

AN US

k k k k Q26   Q11  Q12  2Q66  sin3  k cos k  Q12k  Q22k  2Q66k  cos3  k sin  k k k k k k Q66   Q11  Q22  2Q12  2Q66  sin 2  k cos2  k  Q66k  cos4  k  sin 4  k 

k Q44k  Q44 cos2  k  Q55k sin 2  k

ED

k Q55k  Q44 cos2  k  Q55k sin 2  k

E1k

k 1  12k 21

12k E2k Q  1  12k 21k E2k

AC

Q22k 

CE

k 12

PT

where

Q11k 

M

k k k Q45   Q55  Q44  sin  k cos k

k 1  12k 21

k k Q44  G23

Q55k  G13k k Q66  G12k

where E1k and E2k are elastic modulus of the kth layer in the principle material direction, 12k k and 21k are Poisson’s ratios. They have the relationship of E1k 12k  E2k 21 . G12k , G23k and G13k are shear

ACCEPTED MANUSCRIPT k k k modulus. For isotropic material, E1k  E2k  E , 12k  12k   and G12  G23  G13  G 

E , 2 1   

and E, μ and G are the Young's modulus, Poisson’s ratio and shear modulus of isotropic material, respectively. The elements of coefficient matrix of Eq. (11) are represented as follows: A22  A22  b1B22  b2 D22  b3 E22

B11  B11  a1D11  a2 E11  a3 F11

B22  B22  b1D22  b2 E22  b3 F22

D11  D11  a1E11  a2 F11  a3 H11

D22  D22  b1E22  b2 F22  b3 H 22

A16  A16  a1B16  a2 D16  a3 E16

A26  A26  b1B26  b2 D26  b3 E26

B16  B16  a1D16  a2 E16  a3 F16

B26  B26  b1D26  b2 E26  b3 F26

D16  D16  a1E16  a2 F16  a3 H16

D26  D26  b1E26  b2 F26  b3 H 26

A55  A55  a1B55  a2 D55  a3 E55

A44  A44  b1B44  b2 D44  b3 E44

A66  A66  a1B66  a2 D66  a3 E66

A66  A66  b1B66  b2 D66  b3 E66

AN US

M

B66  B66  b1D66  b2 E66  b3 F66

ED

B66  B66  a1D66  a2 E66  a3 F66

CR IP T

A11  A11  a1B11  a2 D11  a3 E11

D66  D66  b1E66  b2 F66  b3 H 66

PT

D66  D66  a1E66  a2 F66  a3 H 66

where

sin  1 1 sin  sin  1  , a2  2   , a3  R0 R R R0 R R0 R2 R3

b1 

1 sin  sin 2  sin  sin 2  sin 3   , b2  ,  b   3 R R0 R02 R0 R R02 R R03

AC

CE

a1 

E , F , H     ij

ij

ij

 k 1

k

Qijk  3 ,  4 ,  5  d ,  i, j  1, 2, 4,5, 6 

Appendix B The disjoint generalized mass and stiffness matrices of doubly-curved shell are respectively given by

ACCEPTED MANUSCRIPT M = diag M1 ,M 2 ,..., Mi , ...,M N0  K = diag K 1 ,K 2 ,...,K i , ...,K N0 

where the sub-matrices Mi and Ki are the mass and stiffness matrices of ith doubly-curved shell

Ki  

  K A

    K   i

T B

0

0

M iu

M ivv

0

0

0

M iww

0

0

0

Mi 

M iv

0

0

0   M iv   0 R0 R d d 0   Mi   

 KB    K Q R0 R d d  KD   

where

K K

K iuw

K ivv

K ivw

K ivw

K iww

K iv

K iw

K i 

K iw

K i 

K iv

K iu

K iu   K i   i K v  , K D   i K Ki     K iw 

K iu   K iv   K iw   K i   K i   

ED

K iuv

CE

 K iuu  i  K uv  K Q   K iuw  i K u K i  u

i vv i vw

 K iu K iuw    i  K vw  , K B   K iv  i K iww  K w

M

K iuv

K iv

K iw

PT

 K iuu  K A   K iuv K iuw 

AN US

 M iuu   0  Mi     0 i  M iu    0

The elements of the mass matrices are given as:

AC

Miuu  0uiT ui

Miu  1uiT fφi

Mivv  0 viT vi

Miv  1viT fθi

Miww  0 wiT wi Mi   2fφiT fφi

CR IP T

segment.

K i    K i    

ACCEPTED MANUSCRIPT Mi    2fθiT fθi

The elements of the stiffness matrices are given as: A16  uT u uT u  A11 uT u A12 cos   uT T u   u  u       R 2   R0 R     R0 R      

K iuu =

A26 cos   T u uT  A66 uT u A22 cos 2  T u u   u u R0 2 R0 2     R0 2  



uT 

 A12 v A16 v A16 cos   2   R R   R   R0 R   0 

 A cos  v A26 cos  v A26 cos 2   v   uT  22 2     R0   R R   R0 2 0   

uT 

 A26 v A v A66 cos   66   2  R0  R0 R  R0 2 

 v  



 v  

CR IP T

K iuv =

AN US

 A16  A12 cos  A22 sin  cos   T A A26 sin   uT A sin   uT K iuw =  112  12 w   u w   w      2 2 R  R0 R   R0 R   R R   R R  0  0 0       A26  vT v vT v  A26 cos   vT A22 vT v v    v  vT     2 2 R0   R0 R       R0    

K ivv =

A66 vT v A66 cos   vT A66 cos 2  T T v   v  v  v v   R 2   R0 R     R0 2

M



ED

 A cos  A26 sin  cos   T  A A A sin   vT A sin   vT K ivw =  12  22 2  w   162  26 w   16  v w  2 R    R0 R    R   R R R R R 0 0  0       0  

fφ  B16  uT fφ uT fφ  B11 uT fφ B12 cos   uT T  f  u      φ R 2   R0 R     R0 R      

CE

K iu 

PT

A 2 A sin  A22 sin 2   T K iww =  112  12  w w 2  R  R R R 0  0  

AC

B26 cos   T fφ uT  B66 uT fφ B22 cos 2  T  u fφ   fφ   u R0 2 R0 2     R0 2  

K iu 

uT 

 B12 fθ B16 fθ B16 cos   T  B22 cos  fθ B26 cos  fθ B26 cos 2    2  fθ   u    fθ     R0 2  R0 R  R0 R  R0 2  R0 R  R    



uT 

 B26 fθ B fθ B66 cos    66  fθ   2 2  R   R R   R 0  0  0 

ACCEPTED MANUSCRIPT vT 

K iv 

 B12 fφ B22 cos  B26 fφ  vT  f     φ 2 2 R R   R R   0  0 0   

 B16 fφ B26 cos  B fφ   fφ  66  2  R0 R R0 R    R 

 B cos  fφ B26 cos 2  B66 cos  fφ   vT  16  f   φ  R0 R  R0 2 R0 2   



B66 vT fθ B66 cos   vT B66 cos 2  T T fθ   f  v  v fθ   θ R 2   R0 R     R0 2

CR IP T

B26  vT fθ vT fθ  B26 cos   vT f  B22 vT fθ   fθ  vT θ     2 2 R0   R0 R       R0    

K iv 

 B16 B B26 sin   T fφ B sin   T fφ  B12 cos  B22 sin  cos   T K iw   112  12 w   w f    w   φ R   R0 R R0 R    R0 R R0 2 R0 2      

i

K 

AN US

 B f  B B sin   T fθ  B16 cos  B26 sin  cos   T B sin   K iw   12  22 2  wT θ   162  26    w fθ  w 2 R  R0 R   R0 R R   R R   R 0  0  0       T T f  D D11 fφ fφ D12 cos   fφ  2  fφ  fφT φ   16  R   R0 R     R0 R

 fφT fφ fφT fφ          

i

 D cos  fθ D26 cos  fθ D26 cos 2   fφT  D12 fθ D16 fθ D16 cos     2  fθ   fφT  22 2   fθ   2   R0    R0 R  R  R0 R   R R   R 0  0   

ED

K 

M

fφT  D66 fφT fφ D26 cos    D22 cos 2  T  fφ fφ    fφ   2   R   R0 2 R0 2    0 

K  

f  D22 fθT fθ D26  fθT fθ fθT fθ  D26 cos   fθT  2     fθT θ     2 R0   R0 R       R0    

CE

i

PT

fφT  D26 fθ D fθ D66 cos     66  fθ   2    R0  R0 R  R0 2 

AC

D66 fθT fθ D66 cos   fθT D66 cos 2  T T fθ   2  fθ  fθ fθ fθ   R   R0 R     R0 2

K iuu 

A55 T u u R 2

K iuv 

A45 sin  T u v R0 R

 A w A55 w  K iuw  uT  45   R R  R 2     0  

ACCEPTED MANUSCRIPT K iu  

A55 T u fφ R

K iu  

A45 T u fθ R

K ivv 

A44 sin 2  T v v R0 2

A45 sin  T v fφ R0

K iv  

A44 sin  T v fθ R0

K iww 

A45  wT w wT w  A55 wT w A44 wT w     R0 2   R0 R       R 2  

 A45 wT A55 wT     fφ  R  R    0

M

K

i w

AN US

K iv  

CR IP T

 A sin  w A45 sin  w  K ivw   vT  44 2    R0    R R   0   

ED

 A wT A45 wT  K iw   44   fθ  R0   R     

Ki   A45fφT fθ

CE

Ki    A44fθT fθ

PT

Ki   A55fφT fφ

AC

The generalized interface stiffness matrix Kλ and Kκ derived bymeans of the MVP and LSMRM are obtained through the assembly of all inter matrices. The Kλ and Kκ are given below K i   K  A  K B  K D  K Q  

 i

R0 d

ACCEPTED MANUSCRIPT  K uiui   0   0  0   0 K i     K  uiui1   0  0   0   0

0

0

0

K uiui1

0

0

0

K vi vi

0

0

0

0

K vi vi1

0

0

0

K wi wi

0

0

0

0

K wi wi1

0

0

0

K  i i

0

0

0

0

K  i i1

0

0

0

K  i i

0

0

0

0

0

0

0

0

K ui1ui1

0

0

0

K vi vi1

0

0

0

0

K vi1vi1

0

0

0

K wi wi1

0

0

0

0

K wi1wi1

0

0

0

K  i i1

0

0

0

0

K  i1 i1

0

0

0

K  i i1

0

0

0

0

K ui vi

K ui wi

0 0 K uiui1

K vi vi

K vi wi

0 0 K viui1

K vi wi

0

0 0 K wiui1

0

0

0 0

0

0

0

0 0

0

K viui1

K wiui1

0 0

0

K vi vi1

K wi vi1

0 0

0

0

0 0

0

0

0

0

where

AN US K vi vi1

K wi vi1 0 0

0

0

0

ED

M

0

0 0

0

0

0 0

0

0

0

0

K ui i

K ui i

0

0

0 K ui i1

0

0

K vi i

K vi i

0

0

0 K vi i1

0

0

K wi i

K wi i

0

0

0 K wi i1

K vi i

K wi i

0

0

K  iui1

K  i vi1

0

0

K vi i

K wi i

0

0

K  iui1

K  i vi1

0

0

0

0

K  iui1

K  iui1

0

0

0

0

0

0

K  i vi1

K  i vi1

0

0

0

0

0

0

0

0

0

0

0

0

K vi i1

K wi i1

0

0

0

0

0

0

K vi i1

K wi i1

0

0

0

0

0

0

AC

K B

0 0 0  0 0 0 0 0 0  0 0 0  0 0 0 0 0 0  0 0 0  0 0 0 0 0 0  0 0 0 

K ui vi1

0

PT

 0   0   0 K  ui i K u   i i  0   0   0 K u   i  i1  K ui i1

CE

K A

 K uiui   K ui vi K  ui wi  0  0  K  uiui1  K ui vi1   0  0   0

  0   0  0   K  i i1   R0 d 0   0  0   0   K  i1 i1   i

CR IP T

0

K ui i1   K vi i1   K wi i1  0   0   0   0   0  0   0 

0

ACCEPTED MANUSCRIPT 0

0 0 0

0

0 0

0

0

0 0 0

0

0 0

0

0

0 0 0

0

0 0

K  i i

K  i i

0 0 0 K  i i1

0 0

K  i i

K  i i

0 0 0 K  i i1

0 0

0

0

0 0 0

0

0 0

0

0

0 0 0

0

0 0

0

0

0 0 0

0

0 0 K  i i1

K  i i1

0 0 0

0

0 0 K  i i1

K  i i1

0 0 0

0

0

K ui wi

0

0

0 0

K ui wi1

0

K vi wi

0

0

0 0

K vi wi1

K vi wi

K wi wi

K wi i

K wi i

0

K wi i

0

0

0

K wi i

0

0

0

0

0

0

0

0

0

0

K vi wi1

K wi wi1

K  i wi1

0

0

0

0

0

0

0 0 K wi wi1

0 0  0 0  0 0 0 0  0 0  0 0  0 0 0 0  0 0 0 0 

AN US

 0   0   K ui wi  0    0  0   0 K u w  i i1  0  0 

 0  0   K  i i1  K  i i1   0   0  0   0   0  0

CR IP T

0

0 0 K  i wi1 0 0 K  i wi1

0 0

0

0 0

0

K  i wi1

0 0

0

M

K Q

0 0

0

0 0

0

0

0 0

0

ED

K D

0 0  0  0 0   0  0 0  0  0

where the sub-matrices are expanded as:

 A12 vi A16 cos   A16   A26 cos  T A66 uiT  uiT  T v i  u ui   vi    v vi   v  ui   vi  u ui R0    R0    R0   R   R0 T

CE

K ui vi

A16  T ui uiT  A11  T ui uiT  A12 cos  T T u  u   u u  u u    ui     i  ui i u i i i i u R     R0 R0    

PT

K uiui  u

AC

A A sin   T K ui wi  u  11  12  ui w i R R 0     A u T A cos  T A16 uiT  K uiui1  u  11 i  12 ui   ui 1  R  R R    0 0    A u T A cos  T A66 uiT  K ui vi1  v  16 i  26 ui   vi 1  R  R0 R0    

K vi vi

A26  T vi viT  A66  T vi viT  A cos  T  v  vi   v  vi   v 66 v i v i  v i T vi    vi  vi R0    R     R0 

ACCEPTED MANUSCRIPT A A sin   T K vi wi  v  16  26  vi w i R R0     A v T A v T A cos  T  K viui1  u  12 i  16 i  16 vi  ui 1  R   R   R  0  0 

A A sin   T K wiui1  u  11  12  w i ui 1 R R0    A A sin   T K wi vi1  v  16  26  w i vi 1 R R0   

fφi fφ uiT  B16  uiT  B11  B12 cos  T T T  u   f     u f   u   fφ i   u i  u i  φi  u  i φi  R    R0 R0     





AN US

K ui i 

CR IP T

 A v T A v T A cos  T  K vi vi1  v  26 i  66 i  66 vi  vi 1  R   R   R 0  0  

 B f B cos   B16  uiT  T  K ui i  u uiT  12 θi  16 fθi    u   fθ i   u i  R0     R0   R   B cos  T B66 uiT    26 ui   fθ i R0 R0   

M

ED

K ui i1

 B11 uiT B12 cos  T B16 uiT      ui   fφi 1  R  R0 R0    

 B u T B cos  T B66 uiT  K ui i1    16 i  26 ui   fθi 1  R  R R    0 0  

B26  viT  B66  viT  B cos  T T fθ i T fθ i  v   f   v   fθi   v   66 v i fθi  v i  v i  θi   R0    R     R     0

AC

K vi i 

CE

PT

f  B v T B cos  T   B  v T K vi i    12 i  16 v i  fφi  16  v viT φi   i fφi  R0 R      R0    B cos  B f  v viT  26 fφi  66 φi  R0 R0   

 B v T B v T B cos  T  K vi i1    12 i  16 i  16 vi  fφi 1  R   R   R 0  0    B v T B v T B cos  T  K vi i1    26 i  66 i  66 vi  fθi 1  R   R  R0  0  B B sin   T K wi i    11  12  w i fφ i R R0   





ACCEPTED MANUSCRIPT B B sin   T K wi i    16  26  w i fθ i R R0    B B sin   T K wi i1    11  12  w i fφi 1 R R 0   

 B f B f  B cos  K iui1  u  11 φi  12 fφi  16 φi  ui 1  R  R0 R0      B f B cos  B f  K i vi1  v  16 φi  26 fφi  66 φi  vi 1  R  R0 R0    

AN US

 B f B f B cos   K iui1  u  12 θi  16 θi  16 fθi  ui 1  R  R   R 0  0  

CR IP T

B B sin   T K wi i1    16  26  w i fθi 1 R R  0  

 B f B f B cos   K i vi1  v  26 θi  66 θi  66 fθi  vi 1  R  R   R0   0 

fφiT   D12 fθi D16 cos   D16  T fθ i  fθ i     fθ i    fφi   R0    R0   R  

ED

K  i i   fφi

T

M

K i i

T T   D16  T fφi fφi D11  T  fφi D12 cos  T T    fφi    fφi fφi  fφi fφi    fφi  fφ i   fφi     R    R0 R0     

PT

 D cos  T D66 fφiT    26 fφ i   fθi  R R   0 0  

K i i1

 D16 fφiT D26 cos  T D66 fφiT      fφ i   fθi 1  R  R0 R0    

AC

CE

K i i1

 D11 fφiT D12 cos  T D16 fφiT      fφ i   fφi 1  R  R0 R0    

f   D f T D cos  T  D  f T K  i i    12 θi  16 fθi  fφi  16   θi fφi   fθiT φi  R0 R      R0    D cos  D f   fθiT  26 fφi  66 φi  R0 R0   

K i i

D26  T fθi fθiT  D66  T fθi fθiT  D cos  T    fθi     fθi    66 fθ i fθ i  fθ i T fθ i    fθ i  fθ i R0    R    R0  

ACCEPTED MANUSCRIPT  D f T D f T D cos  T  K i i1    12 θi  16 θi  16 fθi  fφi 1  R   R  R0  0   D f T D f T D cos  T  K i i1    26 θi  66 θi  66 fθi  fθi 1  R   R   R  0  0  K ui wi  

K vi wi  

A45 sin  T vi w i R0

K vi wi1 

A45 sin  T vi w i 1 R0

K wi wi 

CR IP T

A55 T ui w i 1 R

AN US

K ui wi1 

A55 T ui w i R

 A55  T wi w iT  A45  T wi wiT w  w  wi   i  wi i  R0       R  

K wi i  A45 wiT fθi

PT

K i wi1   A55fφiT wi 1

ED

 A w iT A55 w iT  K wi wi1    45   w i 1  R  R   0   

CE

K i wi1   A45fθiT wi 1 K uiui  u u uiT ui

AC

K uiui1  uu uiT ui 1 K vi vi  v v viT vi

K vi vi1  v v viT vi 1 K wi wi  w wwiT wi K wi wi1  w w wiT wi 1 K i i    fφiT fφi

M

K wi i  A55 wiT fφi

ACCEPTED MANUSCRIPT K i i1    fφiT fφi 1

K i i    fθiT fθi K i i1    fθiT fθi 1 K ui1ui1  uu ui 1T ui 1 K vi1vi1  v v vi 1T vi 1

CR IP T

K wi1wi1  w wwi 1T wi 1 K i1 i1    fφi 1T fφi 1

AC

CE

PT

ED

M

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K i1 i1    fθi 1T fθi 1

ACCEPTED MANUSCRIPT Reference [1] G.Y. Jin, T.G. Ye, X.R. Wang, X.H. Miao, A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions, Composites Part B-Engineering, 89 (2016) 230-252. [2] J. Reddy, K. Chandrashekhara, Geometrically non-linear transient analysis of laminated, doubly curved shells, International Journal of Non-Linear Mechanics, 20 (1985) 79-90.

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Non-Linear Mechanics, 45 (2010) 409-418.

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CE

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AC

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[10] F. Tornabene, 2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution, Composite Structures, 93 (2011) 1854-1876. [11] F. Tornabene, A. Liverani, G. Caligiana, FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations, International Journal of Mechanical Sciences, 53 (2011) 446-470. [12] F. Tornabene, A. Liverani, G. Caligiana, General anisotropic doubly-curved shell theory: A differential quadrature solution for free vibrations of shells and panels of revolution with a

ACCEPTED MANUSCRIPT free-form meridian, J. Sound Vibrat., 331 (2012) 4848-4869. [13] F. Tornabene, Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations, Composite Structures, 94 (2011) 186-206. [14] F. Tornabene, Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler-Pasternak elastic foundations, Composite Structures,

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94 (2011) 186-206. [15] F. Tornabene, Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method, Comput. Meth. Appl. Mech. Eng., 200 (2011) 931-952. [16] F. Tornabene, A. Liverani, G. Caligiana, Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using generalized

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differential quadrature method, International Journal of Mechanical Sciences, 61 (2012) 71-87. [17] F. Tornabene, N. Fantuzzi, E. Viola, A.J.M. Ferreira, Radial basis function method applied to doubly-curved laminated composite shells and panels with a General Higher-order Equivalent Single Layer formulation, Compos Part B-Eng, 55 (2013) 642-659.

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[18] F. Tornabene, J.N. Reddy, FGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foundations: A GDQ solution for static analysis with a posteriori stress and

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strain recovery, Journal of the Indian Institute of Science, 93 (2013) 635-688. [19] F. Tornabene, E. Viola, N. Fantuzzi, General higher-order equivalent single layer theory for

PT

free vibrations of doubly-curved laminated composite shells and panels, Composite Structures, 104 (2013) 94-117.

CE

[20] F. Tornabene, N. Fantuzzi, M. Bacciocchi, The local GDQ method applied to general higher-order theories of doubly-curved laminated composite shells and panels: The free

AC

vibration analysis, Composite Structures, 116 (2014) 637-660.

[21] E. Viola, F. Tornabene, N. Fantuzzi, Stress and strain recovery of laminated composite doubly-curved shells and panels using higher-order formulations, in:

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Materials, 2015, pp. 205-213. [22] F. Tornabene, N. Fantuzzi, M. Bacciocchi, On the mechanics of laminated doubly-curved shells subjected to point and line loads, International Journal of Engineering Science, 109 (2016) 115-164. [23] F. Tornabene, N. Fantuzzi, M. Bacciocchi, Higher-order structural theories for the static

ACCEPTED MANUSCRIPT analysis of doubly-curved laminated composite panels reinforced by curvilinear fibers, Thin-Walled Struct., 102 (2016) 222-245. [24] T.G. Ye, G.Y. Jin, Y.T. Zhang, Vibrations of composite laminated doubly-curved shells of revolution with elastic restraints including shear deformation, rotary inertia and initial curvature, Composite Structures, 133 (2015) 202-225. [25] Q.S. Wang, D.Y. Shi, Q. Liang, F.Z. Pang, Free vibration of four-parameter functionally graded

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moderately thick doubly-curved panels and shells of revolution with general boundary conditions, Applied Mathematical Modelling, 42 (2017) 705-734.

[26] Q.S. Wang, D.Y. Shi, Q. Liang, F.Z. Pang, Free vibrations of composite laminated doubly-curved shells and panels of revolution with general elastic restraints, Applied Mathematical Modelling, 46 (2017) 227-262.

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[27] Q. Wang, X. Cui, B. Qin, Q. Liang, Vibration analysis of the functionally graded carbon nanotube reinforced composite shallow shells with arbitrary boundary conditions, Composite Structures, 182 (2017) 364-379.

[28] Q. Wang, X. Cui, B. Qin, Q. Liang, J. Tang, A semi-analytical method for vibration analysis of

M

functionally graded (FG) sandwich doubly-curved panels and shells of revolution, International Journal of Mechanical Sciences, 134 (2017) 479-499.

ED

[29] Q. Wang, B. Qin, D. Shi, Q. Liang, A semi-analytical method for vibration analysis of functionally graded carbon nanotube reinforced composite doubly-curved panels and shells of

PT

revolution, Composite Structures, 174 (2017) 87-109. [30] Y. Zhou, Q. Wang, D. Shi, Q. Liang, Z. Zhang, Exact solutions for the free in-plane vibrations

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of rectangular plates with arbitrary boundary conditions, International Journal of Mechanical Sciences, 130 (2017) 1-10.

AC

[31] Q. Wang, D. Shao, B. Qin, A simple first-order shear deformation shell theory for vibration analysis of composite laminated open cylindrical shells with general boundary conditions, Composite Structures, 184 (2018) 211-232.

[32] Q. Wang, K. Choe, D. Shi, K. Sin, Vibration analysis of the coupled doubly-curved revolution shell structures by using Jacobi-Ritz method, International Journal of Mechanical Sciences, 135 (2018) 517-531. [33] F. Tornabene, N. Fantuzzi, M. Bacciocchi, The local GDQ method applied to general higher-order theories of doubly-curved laminated composite shells and panels: The free

ACCEPTED MANUSCRIPT vibration analysis, Composite Structures, 116 (2014) 637-660. [34] F. Tornabene, N. Fantuzzi, M. Bacciocchi, E. Viola, A new approach for treating concentrated loads in doubly-curved composite deep shells with variable radii of curvature, Composite Structures, 131 (2015) 433-452. [35] D. Shao, S.H. Hu, Q.S. Wang, F.Z. Pang, A unified analysis for the transient response of composite laminated curved beam with arbitrary lamination schemes and general boundary

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restraints, Composite Structures, 154 (2016) 507-526. [36] Y.G. Qu, X.H. Long, S.H. Wu, G. Meng, A unified formulation for vibration analysis of composite laminated shells of revolution including shear deformation and rotary inertia, Composite Structures, 98 (2013) 169-191.

[37] Y.G. Qu, Y. Chen, X.H. Long, H.X. Hua, G. Meng, Free and forced vibration analysis of

Acoustics, 74 (2013) 425-439.

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uniform and stepped circular cylindrical shells using a domain decomposition method, Applied

[38] Y.G. Qu, X.H. Long, G.Q. Yuan, G. Meng, A unified formulation for vibration analysis of functionally graded shells of revolution with arbitrary boundary conditions, Composites Part

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B-Engineering, 50 (2013) 381-402.

[39] Y. Qu, S. Wu, Y. Chen, H. Hua, Vibration analysis of ring-stiffened conical–cylindrical–

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spherical shells based on a modified variational approach, International Journal of Mechanical

AC

CE

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Sciences, 69 (2013) 72-84.

ACCEPTED MANUSCRIPT List of Collected Table Captions



Table 1 Values of i i  u, v, w,  , 



for various boundary conditions

Table 2 Comparison of the first six natural frequencies (Hz) for a C-C elliptical shell with [0°/90°] and [0°/90°/0°] lamination schemes (a=2m, b=4m, h=0.1m, ϕ0=π/6, ϕ1=5π/6, Rb=1m). Table 3 Comparison of the first ten natural frequencies (Hz) for a C-C elliptical shell with [30°/60°] lamination schemes (a =1 m, b = 1 m, h = 0.1 m, Rs = 3 m, φ0 = 0, φ1 =2π, θ=2/3π,).

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Table 4 Comparison of the first five non-dimensional frequency Ωn,m of cycloidal shells with [0°/90°], [0°/90°/0°] and [0°/90°/0°/90°] lamination schemes under C-F, S-S and C-C boundary conditions ( rc = 1 m, h = 0.1m, Rs = 2 m, φ0 = 150, φ1 = 750 ).

Table 5 Comparison of the first ten natural frequencies (Hz) for an F-C asymmetric laminated cycloidal shell with

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[45°/0°/45°] lamination scheme (rc = 1 m, h = 0.1m, Rs =5 m, φ0 = -700, φ1 = -50).

Table 6 Comparison of the non-dimensional frequency Ωn,m of circular toroidal shells with [0°/90°] lamination schemes (R = 1 m, h = 0.05m, Rs =1m, φ0 = 1/6π, φ1 = 1/2π, m=1).

Table 7 Comparison of the first ten natural frequencies (Hz) for an F-C circular toroidal shells with [30°/45°]

AC

CE

PT

ED

M

lamination scheme (R = 2 m, h = 0.1 m, Rs =0, φ0 = 1/6π, φ1 = 1/2π).

ACCEPTED MANUSCRIPT



Table 1 Values of i i  u, v, w,  , 



for various boundary conditions

Boundary set

Essential conditions

u

v

w





Free

No constrains

0

0

0

0

0

Simply-supported

u w0

1

1

1

0

1

Clamped

u  v  w      0

1

1

1

1

1

m=2

m=3

m=4

m=5

m=6

Present

72.638

114.14

114.14

116.05

116.05

116.95

Wang[27]

72.649

114.18

114.18

116.10

116.10

117.02

Error

0.02%

0.03%

0.03%

0.04%

0.04%

0.06%

Jin[27]

72.637

114.15

114.15

116.05

116.05

116.95

Error

0.00%

0.01%

0.01%

0.00%

0.00%

0.00%

Present

72.638

119.25

119.25

123.11

123.11

124.22

Wang[27]

72.649

119.25

119.25

123.11

123.11

124.21

Error

0.02%

0.00%

0.00%

0.00%

0.00%

0.01%

Jin[27]

72.637

119.25

119.25

123.11

123.11

124.22

Error

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

ED

[0°/90°/0°]

m=1

AN US

[0°/90°]

Mode

M

Lamination schemes

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Table 2 Comparison of the first six natural frequencies (Hz) for a C-C elliptical shell with [0°/90°] and [0°/90°/0°] lamination schemes (a=2m, b=4m, h=0.1m, ϕ0=π/6, ϕ1=5π/6, Rb=1m).

GDQ-RM

Mode number

1

2

3

4

5

6

7

8

9

10

103.481

150.664

162.837

185.505

194.526

224.148

230.685

235.097

237.502

262.718

103.462

150.841

162.485

184.608

194.456

223.642

231.560

236.832

239.931

264.295

AC

GDQ-TL

CE

Method

PT

Table 3 Comparison of the first ten natural frequencies (Hz) for a C-C elliptical shell with [30°/60°] lamination schemes (a =1 m, b = 1 m, h = 0.1 m, Rs = 3 m, φ0 = 0, φ1 =2π, θ=2/3π,).

Nastran

103.148

149.930

161.687

184.064

193.853

222.550

230.177

235.625

238.245

262.822

Abaqus

103.298

150.494

162.090

185.098

194.040

223.698

231.317

237.013

238.246

264.464

Straus

103.542

150.681

162.665

185.729

194.400

225.352

232.238

238.636

241.084

267.470

GEM

103.154

149.886

161.687

184.059

193.802

222.390

229.852

235.348

237.903

262.383

Present

103.472

150.754

162.647

185.079

194.485

224.247

230.387

236.246

237.785

262.904

ACCEPTED MANUSCRIPT Table 4 Comparison of the first five non-dimensional frequency Ωn,m of cycloidal shells with [0°/90°], [0°/90°/0°] and [0°/90°/0°/90°] lamination schemes under C-F, S-S and C-C boundary conditions ( rc = 1 m, h = 0.1m, Rs = 2 m, φ0 = 150, φ1 = 750 ). Boundary conditions Mode

[0°/90°]

[0°/90°/0°]

SS-SS

C-C

Present

Wang[27]

Error

Present

Wang[27]

Error

Present

Wang[27]

Error

1

0.376

0.370

1.56%

3.966

3.962

0.10%

4.942

4.938

0.09%

2

0.513

0.512

0.29%

3.976

3.973

0.08%

4.963

4.959

0.08%

3

0.551

0.542

1.67%

4.099

4.094

0.13%

5.038

5.033

0.10%

4

0.834

0.823

1.38%

4.104

4.101

0.07%

5.079

5.076

0.06%

5

1.029

1.029

0.04%

4.327

4.325

0.05%

5.233

5.227

0.12%

1

0.527

0.524

0.61%

5.555

5.552

0.05%

7.297

7.296

0.02%

2

0.686

0.685

0.09%

5.556

5.554

0.04%

7.307

7.306

0.01%

3

0.690

0.685

0.74%

5.620

5.617

0.05%

7.337

7.337

0.01%

4

0.972

0.966

0.59%

5.635

5.633

0.03%

7.359

7.358

0.02%

5

1.207

1.207

0.01%

5.777

5.669

1.90%

7.419

7.396

0.32%

1

0.462

0.457

1.00%

4.624

4.619

0.12%

5.965

5.959

0.11%

2

0.590

0.589

0.18%

4.680

4.676

0.10%

6.005

5.999

0.10%

3

0.680

0.673

1.09%

4.709

4.703

0.13%

6.029

6.022

0.11%

4

1.025

1.015

1.02%

4.860

4.856

0.08%

6.136

6.130

0.10%

5

1.117

1.117

0.04%

4.913

4.906

0.14%

6.182

6.175

0.11%

M

[0°/90°/0°/90°]

C-F

CR IP T

schemes

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Lamination

1 43.875

GDQ-TL

43.742

Nastran

43.695

Abaqus Straus

4

5

6

7

8

9

10

45.513

45.513

47.560

47.560

49.004

49.004

51.316

60.100

43.742

45.277

45.277

47.291

47.291

48.980

48.980

51.038

60.093

43.695

45.432

45.432

47.538

47.538

48.849

48.849

51.319

59.992

42.764

42.780

44.591

44.675

46.760

46.847

47.451

47.451

50.456

57.863

43.381

43.381

44.977

44.977

47.309

47.309

48.544

48.544

51.471

59.870

43.709

43.709

45.448

45.448

47.563

47.563

48.877

48.877

51.362

60.044

43.801

45.526

45.526

47.522

47.522

49.003

49.003

51.311

60.079

AC

GEM Present

3

Mode number

43.875

CE

GDQ-RM

2

PT

Method

ED

Table 5 Comparison of the first ten natural frequencies (Hz) for an F-C asymmetric laminated cycloidal shell with [-45°/-20°/70°/20°] lamination scheme (rc = 1 m, h = 0.1m, Rs =5 m, φ0 = -700, φ1 = -50).

43.801

ACCEPTED MANUSCRIPT Table 6 Comparison of the non-dimensional frequency Ωn,m of circular toroidal shells with [0°/90°] lamination schemes (R = 1 m, h = 0.05m, Rs =1m, φ0 = 1/6π, φ1 = 1/2π, m=1). Boundary conditions n

F-C

SS-SS

C-C

Qu[31]

Error

Present

Qu[31]

Error

Present

Qu[31]

Error

0

1.57929

1.58011

0.05%

2.10406

2.10516

0.05%

2.10406

2.10516

0.05%

1

0.96576

0.96587

0.01%

2.24353

2.24335

0.01%

2.33857

2.33862

0.00%

2

0.49803

0.49802

0.00%

2.13095

2.13028

0.03%

2.40336

2.40264

0.03%

3

0.66420

0.66416

0.01%

2.12260

2.12180

0.04%

2.42355

2.42262

0.04%

4

1.07537

1.07524

0.01%

2.20473

2.20384

0.04%

2.49114

2.49011

0.04%

5

1.53073

1.53045

0.02%

2.34404

2.34307

0.04%

2.59245

2.59138

0.04%

6

2.00591

2.00543

0.02%

2.51910

2.51806

0.04%

2.71104

2.70995

0.04%

7

2.47222

2.47150

0.03%

2.71052

2.70943

0.04%

2.84846

2.84735

0.04%

8

2.85427

2.85329

0.03%

2.90671

2.90559

0.04%

3.01379

3.01265

0.04%

9

3.13204

3.13092

0.04%

3.11526

3.11411

0.04%

3.21123

3.21006

0.04%

AN US

CR IP T

Present

Table 7 Comparison of the first ten natural frequencies (Hz) for an F-C circular toroidal shells with [30°/45°] lamination scheme (R = 2 m, h = 0.1 m, Rs =0, φ0 = 1/6π, φ1 = 1/2π). Method

Mode number

1

2

3

4

5

6

7

8

9

10

82.581

43.034

43.034

60.481

60.481

82.581

98.945

98.945

143.972

143.972

43.081

43.081

60.483

60.483

82.629

82.629

98.914

98.914

143.906

143.906

ED1

43.252

43.252

61.756

61.756

82.845

82.845

102.015

102.015

149.498

149.498

ED2

42.870

42.870

60.568

60.568

82.527

82.527

99.185

99.185

144.283

144.283

ED3

42.979

42.979

60.610

60.610

82.633

82.633

99.213

99.213

144.321

144.321

ED4

42.860

42.860

60.542

60.542

82.551

82.551

99.159

99.159

144.266

144.266

Abaqus

42.922

42.922

60.564

60.564

82.584

82.584

99.168

99.168

144.280

144.280

Present

43.031

60.406

60.408

82.610

82.610

99.127

99.127

144.281

144.281

AC

ED

PT

CE

43.031

M

FSDT TSDT

ACCEPTED MANUSCRIPT List of Collected Figure Captions Fig.1 Geometry and coordinate system of a doubly-curved shell of revolution Fig.2 Meridional section of doubly-curved shells of revolution: (a) elliptic; (b) cycloid; (c) circular toroidal Fig.3 Expressions and assemble scheme of the interface stiffness matrix Κ i of arbitrary segment Fig.4 The schematic diagram of multi-segment doubly-curved shell Fig.5 Assembling scheme from segment to laminated level related to the global interface stiffness matrix K 

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Fig.6 Frequency discrepancies of orthogonal polynomials for the two-layer (00/900) laminated doubly curved shells with C-C boundary conditions: (a) elliptic; (b) cycloid; (c) circular toroidal

Fig.7 Convergence of the six natural frequencies fn,m of two-layer [0°/90°] laminated doubly curved shells with various number of segments N0 under C-C boundary condition: (a) elliptic; (b) cycloid; (c) circular toroidal

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Fig.8 Convergence of the six natural frequencies fn,m of two-layer [0°/90°] laminated doubly curved shells with various weight parameter κ under C-C boundary condition: (a) elliptic; (b) cycloid; (c) circular toroidal Fig.9 The effect of elastic stiffness on the frequencies on laminated doubly-curved shells Fig.10 The influence of lamination schemes on the frequency parameters fn,m of laminated doubly-curved shells

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with layer [0/ϑ]2

Fig.11 The influence of lamination schemes on the frequency parameters fn,m of laminated doubly-curved shells

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with layer [0/ϑ/ϑ/0]

Fig 12 Mode shapes of a two-layer [0°/90°] laminated elliptical shell for C-C boundary condition

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Fig 13 Mode shapes of a two-layer [0°/90°] laminated cycloidal shell for C-C boundary condition Fig.14 Mode shapes of a two-layer [0°/90°] laminated circular toroidal shell for C-C boundary condition

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Fig. 15 The diagrammatic sketch of three applied load types for the doubly-curved shells. (a) Point force; (b) Line force; (c) Surface force

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Fig. 16 The comparison of displacement of circular toroidal shells under three types of load. (a) Point force; (b) Line force; (c) Surface force Fig. 17 The displacement of doubly-curved shells with different fiber orientation (a) Elliptic; (b) Cycloid; (c) Circular toroidal Fig. 18 The displacement of doubly-curved shells with different number of layer (a) Elliptic; (b) Cycloid; (c) Circular toroidal Fig.19 The sketch of load time domain curve. (a) Rectangular pulse; (b) Triangular pulse; (c) Half-sine pulse; (d) Exponential pulse

ACCEPTED MANUSCRIPT Fig.20 The comparison of displacement response of circular toroidal shell (a) Point force; (b) Line force; (c) Surface force Fig.21 The displacement response of doubly-curved shells under different loads (a) Elliptic; (b) Cycloid; (c)

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Circular toroidal

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0 c1 R

R0

C

R0



Rs



R c2

1 C z

(b)

x

z

CR IP T

o

(a)



(c)

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W



U

v

u



R

V



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R

ED

Fig.1 Geometry and coordinate system of a doubly-curved shell of revolution

(b)

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(a)

O

O1

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b

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Rb

C1

C2

z

z1

R

(c) Rb

n

n

b  2rc

R0

x

Rb

O

O1

x

R0  

rc



a

R

d

C1

a   rc

C2 z

R

C1 C2

z1

n

R z

z1

Fig.2 Meridional section of doubly-curved shells of revolution: (a) elliptic; (b) cycloid; (c) circular toroidal

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N1

N2 N3 Ns

N s

Ms

Qs

Qs

Ns

Ms N s

Ni N i 1

Ns

N s

Ms

Qs

Qs

Ns

Ms

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N s

N N0

Fig.3 The schematic diagram of multi-segment doubly-curved shell K ui wi

Segment i

= Κ i

K uii

K uiui1 K ui wi1 K uii1

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K uiui K ui wi

K wi wi

K wii K wiui1 K wi wi1 K wii1

K uii

K wii

K ii

K iui1 K i wi1 K ii1

K uiui1 K wiui1 K iui1

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K ui wi1 K wi wi1 K i wi1

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K uii1 K wii1 K ii1

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Fig.4. Expressions and assemble scheme of the interface stiffness matrix Κ i of arbitrary segment.

Κ 2

vi

wi

x  xi

Κ N 1

Κ i1

Κ i

ui

Global stiffness matrix K 

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Κ1

x  xi

 ui 1 x  x

x  xi

 vi 1 x  x

Κ N

i

i

 wi 1 x  x

i

Fig.5. Assembling scheme from segment to laminated level related to the global interface stiffness matrix K  .

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Fig.6 Frequency discrepancies of orthogonal polynomials for the two-layer (00/900) laminated doubly curved

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shells with C-C boundary conditions: (a) elliptic; (b) cycloid; (c) circular toroidal

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Fig.7 Convergence of the six natural frequencies fn,m of two-layer [0°/90°] laminated doubly curved shells with

AC

CE

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ED

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various number of segments N0 under C-C boundary condition: (a) elliptic; (b) cycloid; (c) circular toroidal

Fig.8 Convergence of the six natural frequencies fn,m of two-layer [0°/90°] laminated doubly curved shells with various weight parameter κ under C-C boundary condition: (a) elliptic; (b) cycloid; (c) circular toroidal

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f1,2

f1,3 Elliptical

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f1,1

f1,1

f1,2

f1,3

f1,2

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f1,1

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Cycloidal

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f1,3 Circular toroidal Fig.9 The effect of elastic stiffness on the frequencies on laminated doubly-curved shells

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Elliptical

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Cycloidal

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Circular toroidal

Fig.10 The influence of lamination schemes on the frequency parameters fn,m of laminated doubly-curved shells with layer [0/ϑ]2

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AC

CE

PT

ED

Elliptical

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Cycloidal

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Circular toroidal

Fig.11 The influence of lamination schemes on the frequency parameters fn,m of laminated doubly-curved shells with layer [0/ϑ/ϑ/0]

f1,2

f1,3

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f1,1

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f2,2

f2,3

f3,2

f3,3

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ED

f2,1

f3,1

Fig 12 Mode shapes of a two-layer [0°/90°] laminated elliptical shell for C-C boundary condition

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f2,1

f2,2

f1,3

CR IP T

f1,2

f2,3

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f1,1

f3,1

f3,2

f3,3

AC

CE

PT

ED

Fig 13 Mode shapes of a two-layer [0°/90°] laminated cycloidal shell for C-C boundary condition

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f2,2

f2,3

PT

f3,1

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f2,1

f1,3

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f1,2

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f1,1

f3,2

f3,3

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Fig.14 Mode shapes of a two-layer [0°/90°] laminated circular toroidal shell for C-C boundary condition

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(a)

(b)

x A

(c)

x

x

fw

o

y

o

B

o

y

o

B

o

y

1

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fw

2



o

Surface force

Point force

A

B

Line force

1

B

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 B



fw

fw

2

fu

z

z

z

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Fig.15. The diagrammatic sketch of three applied load types for the doubly-curved shells. (a) Point force; (b) Line

AC

CE

PT

ED

force; (c) Surface force

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AN US

CR IP T

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AC

CE

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Fig. 16 The comparison of displacement of circular toroidal shells under three types of load. (a) Point force; (b) Line force; (c) Surface force

CR IP T

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Fig. 17 The displacement of doubly-curved shells with different fiber orientation (a) Elliptic; (b) Cycloid; (c)

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Circular toroidal

Fig. 18 The displacement of doubly-curved shells with different number of layer (a) Elliptic; (b) Cycloid; (c) Circular toroidal

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q0

q0

0

0 0



t0

fs t 

(c)

q0

0



fe  t 

t0 t

(d)

q0

0

0

0



t0

t

o'

0

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o'

o'

t

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o'

(b)

ft  t 

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(a)

fr t 



t0 t

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Fig.19 The sketch of load time domain curve. (a) Rectangular pulse; (b) Triangular pulse; (c) Half-sine pulse; (d)

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Exponential pulse

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Fig.20 The comparison of displacement response of circular toroidal shell (a) Point force; (b) Line force; (c)

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Surface force

Fig.21 The displacement response of doubly-curved shells under different loads (a) Elliptic; (b) Cycloid; (c) Circular toroidal

ACCEPTED MANUSCRIPT Graphical abstract

o

o

0 c1 R

R0

C

R0



Rs



R

1 C z

(b)

x

CR IP T

(a)

c2

z



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(c)



M



ED

R

W U u



V v

R

A domain decomposition method for dynamic analysis of composite laminated doubly-curved shells

AC

CE

PT

with various boundary conditions with general boundary conditions