Dynamic analysis of composite laminated doubly-curved revolution shell based on higher order shear deformation theory

Composite Structures 225 (2019) 111155

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Dynamic analysis of composite laminated doubly-curved revolution shell based on higher order shear deformation theory Kwangnam Choea, Kwanghun Kimb, Qingshan Wangc,

T

⁎

a

Department of Light Industry Machinery Engineering, Pyongyang University of Mechanical Engineering, Pyongyang 999093, Democratic People’s Republic of Korea Department of Engineering Machine, Pyongyang University of Mechanical Engineering, Pyongyang 999093, Democratic People’s Republic of Korea c State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410083, PR China b

ARTICLE INFO

ABSTRACT

Keywords: Dynamic analysis Composite laminated doubly-curved revolution shell Multi-segment partitioning technique Rayleigh-Ritz method Jacobi orthogonal polynomials

The main content of this paper is to establish an analysis model for dynamic analysis of composite laminated doubly-curved revolution shell based on the Higher order Shear Deformation Theory (HSDT). Firstly, the energy functional of shell is established based on higher order shear theory. Then, the multi-segment partitioning technique is introduced to segment the shell along the generatrix direction, which is mainly to relax the boundary conditions of shell, and then reduce requirements for the displacement function. At the boundary position, the boundary spring parameters are used to obtain the corresponding boundary conditions. Similarly, the connecting spring parameters are introduced to simulate the continuity conditions between the segmented shells. The parameters of boundary spring and connecting spring can be regarded as the weight parameter. Thirdly, all displacement components of the composite laminated doubly-curved revolution shell are expressed by Jacobi orthogonal polynomials. Finally, the whole dynamic characteristics are obtained by Rayleigh-Ritz method with respect to unknown Jacobian expansion coefficient. The convergence, validity and dynamic characteristics of the analytical model established in this paper are given by a series of numerical examples.

1. Introduction As a kind of high strength material, with the development of processing technology, fiber material has been widely concerned in aerospace, ship and other engineering fields. The doubly-curved revolution shell structure, because of its good structural and mechanical properties, has also been widely used in engineering. Therefore, more and more researchers have paid attention to the study of composite laminated doubly-curved revolution shell which combines the material properties of fiber material and geometric properties of doubly-curved structure. Next, the current research on the dynamic characteristics of composite doubly-curved revolution shell structure will be reviewed. Chandrashekhara and Bhimaraddi [1] studied the thermal stress of laminated doubly curved shells using a shear flexible finite element. Guo et al. [2] used the domain decomposition method to study the dynamic analysis of composite laminated doubly-curved shells with various boundary conditions on the basis of the first order shear deformation theory. Jin et al. [3] extended the modified Fourier series solution for the vibration analysis of functionally graded material (FGM) doubly-curved shells of revolution with arbitrary boundary

⁎

conditions. Pang et al. [4–6] applied the Rayleigh–Ritz method to investigate free vibration of isotropic, functionally graded and composite laminated doubly-curved shells of revolution. Reddy and Chandrashekhara [7] applied the finite element method to study the geometrically non-linear transient analysis of laminated, doubly curved shells. Ye et al. [8] presented unified solution for the vibrations analysis of composite laminated doubly-curved shells of revolution with elastic restraints including shear deformation, rotary inertia and initial curvature. Based on the three-dimensional elastic theory, Zhou and Lo [9] presented a three-dimensional solution for free vibration analysis of doubly-curved shells with various boundary conditions. Wang et al. [10–13] used the Rayleigh–Ritz method and modified Fourier series solution to show the free vibration behavior of functionally graded and composite laminated doubly-curved shells and panels of revolution with general elastic restraints. Choe et al. [14,15] studied free vibration characteristics of the coupled functionally graded and composite laminated doubly-curved revolution shell structures with general boundary conditions by using the first order shear deformation theory and unified Jacobi-Ritz method. For the doubly-curved structure, Tornabene and his team [16–27] have made outstanding contributions.

Corresponding author. E-mail addresses: [email protected] (K. Choe), [email protected] (Q. Wang).

https://doi.org/10.1016/j.compstruct.2019.111155 Received 24 February 2019; Received in revised form 25 May 2019; Accepted 18 June 2019 Available online 21 June 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.

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2. Theoretical formulations

The main contributions include a series of Differential Quadrature (DQ)-based structural dynamics solutions and a series of mechanical analysis of composite doubly-curved revolution shell structures. As described above, most of the current studies on the dynamic characteristics of doubly-curved revolution shell structures are concerned with free vibration. However, there are few studies on forced response and transient response. Although free vibration is the basis of study of forced response and transient response, it is not enough for designers to study only free vibration. In addition, from the point of view of elasticity theory, the main elastic theories at present contain three-dimensional theory, higher order shear deformation theory, firstorder shear deformation theory and classical theory. Among them, the calculation accuracy of the three-dimensional theory is the highest but its calculation time is long. The classical theory has the fast calculation time, but because of the existence of premise assumptions, it has low computational accuracy and great limitations. Therefore, the first-order shear deformation theory (FSDT) and the higher-order shear deformation theory (HSDT) are widely used on the basis of the three-dimensional theory and classical theory. Most of the current research on doubly-curved revolution shell structures are based on the first-order shear deformation theory. However, the first-order shear deformation theory has low accuracy due to the introduction of the shear factor. The higher-order shear deformation theory does not need to introduce shear factor, so it has higher accuracy than the first-order shear deformation theory. In addition, through the investigation of existing literature, it can be found that most of the current studies on double curvature are based on the FSDT, and most of them are on the free vibration characteristics of double curvature shells. For example, based on the firstorder shear deformation theory, the author’s team [12] has studied the free vibration behavior of doubly-curved shells, while the research on forced response of doubly-curved shells has not been carried out. At present, in the application of higher-order theory, most of them are free vibration analysis of rotating shells with special curvature radius, such as cylindrical shells or spherical shells. However, for the double-curved shell of revolution with complex radius of curvature, studies on their free vibration and forced vibration analysis have not been presented. Based on the above research background, this paper is mainly to establish the dynamic analysis model for composite laminated doublycurved revolution shells with general boundary conditions based on high-order shear deformation theory, and to study their dynamic characteristics. The proposed method has the higher computational complexity than the first-order shear deformation theory and classical theory, and has the advantage of being able to guarantee higher accuracy than the classical theory while reducing the computation time compared with the three-dimensional theory. In this paper, the multisegment partitioning technique is introduced to relax the boundary conditions of structure, so as to reduce the selection requirements of the displacement function. By introducing penalty parameters, boundary conditions and continuity conditions between segments are obtained. The displacement function components of the composite laminated doubly-curved revolution shells are expressed by the mixed series where the Jacobi orthogonal polynomials are in the generatrix direction and the trigonometric series are in the circumferential direction. The unknown coefficients of Jacobi orthogonal polynomials are taken as generalized variables, and Rayleigh-Ritz method is used to perform variational operations to obtain the dynamic characteristics of composite laminated doubly-curved revolution shells. The convergence, validity and dynamic characteristics of the composite laminated doubly-curved revolution shells established in this paper are given by a series of numerical cases.

2.1. Preliminaries Fig. 1 shows the composite laminated doubly-curved revolution shells with respect to an orthogonal co-ordinate system (φ, θ, z). The geometry of shell is obtained by rotating the generatrix around the geometric central axis o z which is paralleled to the spatial coordinate axis oz , as shown in Fig. 1. Correspondingly, the deformation in the generatrix (φ-), circumferential (θ-) and normal directions (z-) are represented by symbols u, v and w. h denotes the thickness. Rφ and Rθ are principal radius of curvature of the doubly-curved shell. Oφ and Oθ indicate the centers of the two principal radiuses Rφ and Rθ, respectively. Rs is the offset distance of the geometric central axis z with respect to the spatial coordinate axis z. R0 represents the offset distance and horizontal radius. Its calculation formula is as follows: R0 = Rθsin φ. From the detailed geometric model description above, it can be seen that the geometric characteristics of doubly-curved revolution shell structures depend on the geometric characteristics of generatrix. Therefore, it may be diversified. To simplify the study, this paper chooses the most common and typical three generatrix for research. The corresponding geometric model is shown in Fig. 2 and the corresponding generatrix equation is as follows: (1) Paraboloidal shell, seen in Fig. 2(a)

R ( )=

k k R ( )= 2cos3 2 cos

+

Rs sin

(1.a)

where k is the characteristic parameter of the parabolic meridian. Specially,

k=

0

R12

R 02

(1.b)

L

= arctan

2R 0 k

1

2R1 k

= arctan

(1.c)

(2) Elliptical shell, seen in Fig. 2(b)

R ( )=

a2b 2 (a2sin2

+

b2 cos2

)3

R ( )=

a2 a2sin2

+ b2 cos2

+

Rs sin (2.a)

where a and b are the lengths of semimajor and semiminor axes of the elliptic meridian. Specially,

R02

b = aL ( a2

0

= arctan

bR0 a

R12 )

a2

R02

a2

1

(2.b)

= arctan

bR1 a a2

R12

(2.c)

(3) Hyperbolical shell, seen in Fig. 2(c)

R ( )=

a2b2 (a2sin2

b2 cos2 )3

R ( )=

a2 a2sin2

b2 cos2

+

Rs sin (3.a)

where a and b are the lengths of semitransverse and semiconjugate axes of the hyperbolic meridian. Specially,

b = aC R02

0

2

= arctan

a2 = aD

bR0 a R 02

a2

R12

1

(3.b)

a2

=

arctan

bR1 a R12

a2

(3.c)

Composite Structures 225 (2019) 111155

K. Choe, et al.

(a) o

(b)

o' Rs

oφ

φ0 c0

z R0

n

Rθ

φ

x

Rφ

R0

(c)

W

θ φ

oθ φ1

z

c1

Rθ

U

V

u

v

θ

h

Rφ

z' Fig. 1. Geometry and reference system of a doubly-curved revolution shell: (a) the geometric relationship; (b) the partial element.

o Rs o' R0 R0(φ) φ Rφ

L

Rθ

oφ

oθ

R1

Rs

x

R0(φ)

n

Rs

R0 Rφ

φ

R0(φ)

C

n b

Rθ

o

o o' oθ oφ a

x

R0

o' a

D

φ

Rθ

Rφ

x oφ

R1

oθ

R1

n

ς

z

z'

(a)

z

z

z'

(b)

z'

(c)

Fig. 2. The geometric parameters and shell structures of doubly-curved revolution shells; (a) paraboloidal shell, (b) elliptical shell, (c) hyperbolical shell.

3

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K. Choe, et al.

2.2. Energy functional of shell

=

z

In this paper, a higher order shear deformation theory is used for modeling [28,29]. The displacement fields of composite laminated doubly-curved revolution shells can be written as follows:

U ( , , z, t ) = u ( , , t ) + z

( , , t ) + z2

( , , t) + z3

( , , t)

V ( , , z, t ) = v ( , , t ) + z

( , , t ) + z2

( , , t ) + z3

( , , t)

=

1 ( (1 + z R )

=

1 ( (1 + z R )

(4.b) (4.c)

where u, v and w are the displacements on the middle surface in the φ, θ and z directions, respectively. and are the rotations of the normal to the reference surface about the θ and φ direction, respectively. , and , are higher-order terms of Taylor series. The higher-order terms of Taylor series are to maintain the actual profile of shear stress (parabolic variation) according to the thickness of the laminated composite double-curved shells of revolution [30–32]. t is the time. The linear strain–displacement relationships considering z R and z R terms can be expressed as follows:

=

1 1 U V LA W + + (1 + z R ) LA LA LB R

(5.a)

=

1 1 V U LB W + + (1 + z R ) LB LA LB R

(5.b)

i

0 i

1 R 1 R0

(

(

1 R

(

z

1 R

(

z

1 R0

=

z

=

1 1 U (1 + z R ) B

+w

v

+ u cos

+ w sin

)

v

1 R0

1 1 V = (1 + z R ) LA

)

u

(

1 ( (1 + z R )

=

1 ( (1 + z R )

z

=

1 (1 + z R )

z

=

1 ( (1 + z R )

+ zk 0 + z 2k 1 + z 3k 2 )

0

+ zk 0 + z 2k 1 + z 3k 2)

v cos

u+

v sin

w

)

)+

+

w

U LA W + LA LB R

V LB W + LA LB R

(6.b)

0

+ zk 0 + z 2k 1 + z 3k 2 )

0

+ zk 0 + z 2k 1 + z 3k 2 )

0 z

+ zk 0z + z 2k 1z + z 3k 2z

0 z

(6.c) (6.d) (6.e)

+ zk 0z + z 2k 1z + z 3k 2z )

(6.f)

ki0

ki1

ki2

1 R

1 R

1 R

1 R0

(

+

cos

)

1 R0

(

+

cos

)

1 R0

1 R0

cos

(

+

cos

)

cos

)

1 R

1 R

1 R0

)+

(6.a)

in which

1 R0

cos

2

3

+

1 R

2

3

+

1 R0

(

1 2 R

sin

2

1 R0

sin

The force and moment resultants relations to the strains and change of curvatures are given as: (5.c)

Qx Q Px P Qx1 =

(5.d)

Q1

1 W U + LA (1 + z R ) LA (1 + z R ) z LA (1 + z R ) V R (1 + z R )

=

0

1 R u

(5.f)

generally LA = R , LB = R0 and R = 0 , for an doubly curved shell. Substituting U, V, W, LA, LB and R in Eq. (5), it can be rewritten as follows:

(4.a)

W ( , , z, t ) = w ( , , t )

1 W V + LB (1 + z R ) LB (1 + z R ) z LB (1 + z R ) U R (1 + z R )

Px2 P2 (5.e)

4

B45

C¯55 C45

D¯55

A 45 A 44 B45 B44 B¯55 B45 C¯55 C45

C45 C44 D¯55 D45

D45 D44 E¯55 E45

B45 B44 C¯55 C45

C45 D¯55

D45 D44 E¯55 E45

E45 E44 F¯55 F45

C45 D¯55

D45

D45 D44 E¯55 E45

E45 E44 F45 F44 F¯55 F45 G¯55 G45

D45

D44

E45

F45

A¯55

A 45

C44

B¯55

C44 D45

E44

F44

G45

D45

G44

0 xz 0 z 0 k xz k 0z 1 k xz k 1z 2 k xz k 2z

(7.a)

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K. Choe, et al.

Nx N Nx Nx Mx M Mx Mx Nx1

A11

A12

A16

A16

B11

A12 A16

A22 A26 A26 B12 A26 A66 A66 B16

B12

B16

B22 B26 B26 B66

B16

C11

B26 C12 B66 C16

C12

C16

C16

D11 D12

C22 C26 C26 D12 D22 D26 D26 C26 C66 C66 D16 D26 D66 D26

B12 B16 =

Nx1 N 1x Mx2 M2 Mx2 M 2x

B22 B26 B26 B66

B26 C12 B66 C16

E22 E26 E26 E66

E26 E66

B16 B26 B66 C11 C12 C16

B66 C16 C26 C66 C66 D16 D26 D66 D66 E16 C16 D11 D12 D16 D16 E11 E12 E16 E16 F11

E26 E66 F12 F16

E66 F16

k 0x

C12 C16

C26 D12 D22 D26 D26 E12 C16 D16 D26 D66 D66 E16

F22 F26

F26 F66

k1

C22 C26 C26 C66

E22 E26 E26 F12 E26 E66 E66 F16

k 1x

D12 D22 D26 D26 E12 D16 D26 D66 D66 E16

E22 E26 E26 F12 E26 E66 E66 F16

F22 F26

F26 F66

F26 G12 G22 G26 G26 F66 G16 G26 G66 G66

k2

D16

E26 E66

F26

F66

F66 G16

k 2x

D26 D66

N A M B = N1 C M2 D

B C D E

C D E F

D66 E16

E66 F16

A¯16

Ks Ks1 Ks 2

(8.a)

(8.b)

A22 A26 A26 A¯66

A26 B12 B22 B26 B26 ,B = ,C A66 B¯16 B26 B¯66 B66

A16

A26 A66

A66

A16

B¯11

B16

B12 B¯16

B26 B66

B16

(9.a)

C66

D16

D66

E¯11

E12 E¯16

C44

G¯55 G45 G45

G44

E16

E26 E66

k

c0 Bij , Aij = Aij + c0 Bij , B¯ij = Bij

G¯ij = Gij

c0 Hij , Gij = Gij + c0 Hij , c0 =

k Q26

E1k

1

k µk µ12 21

k , Q12 =

k Ek µ12 2

1

k µk µ12 21

k , Q22 =

(

c0 Cij, Bij = Bij + c0 Cij c0 Eij , Dij = Dij + c0 Eij c0 Gij , Fij = Fij + c0 Gij

1 R

1 R

E2k

1

k µk µ12 21

)

k k k k , Q44 = G23 , Q55 = G13 ,

k

=

k (Q11

=

k 4 Q11 n

=

k (Q55

+

k Q22

+

k 4Q66 ) m2n2

k 2(Q12

+

+

k ( m4 Q12

k 2Q66 ) m2n2

+

+ n4 )

k Q22 m4 ,

k

k

k k 2 Q44 = Q44 m2 + Q55 n , Q45

k ) mn Q44

k = (Q11

k Q12

k k 2Q66 ) m3n + (Q12

k = (Q55

k Q44 ) mn

k = (Q11

k Q12

k k 2Q66 ) mn3 + (Q12

k

k k + 2Q66 ) mn3, Q45 Q22

k k k + 2Q66 ) m3n, Q55 Q22

k k 2 = Q55 m2 + Q44 n k Q66

E66

k k = (Q11 + Q22

= sin

k 2Q12

k k 2Q66 ) m2n2 + Q66 (m4 + n4 ), m = cos

k fiber ,

n

k fiber

(10) In this paper, the multi-segment partitioning technique is introduced to relax the boundary conditions of the composite laminated doubly-curved revolution shells, so as to reduce the selection requirements of the displacement function. Based on this, the axis direction is divided into Nφ segments. By introducing penalty parameters, boundary conditions and continuity conditions between segments are obtained. Based on the HSDT, the strain energy and kinetic energy of ith segments are defined as:

(9.b)

A¯55

A 45

A 45

A 44

, Bs =

B¯55

B45

B45

B44

, Cs

G66

C¯55 C45 C45

, Gs =

Qij {1, z, z 2, z 3, z 4 , z 5 , z 6, z7,} dz

c0 Fij , Eij = Eij + c0 Fij , F¯ij = Fij

k Q16

E16

F66

G12 G22 G26 G26 , As = G¯16 G26 G¯66 G66

zk + 1 zk

E¯ij = Eij

k Q22

G¯11 G12 G¯16 G16

=

F44

k

F12 F¯16 F16

G26 G66

F45

k k k k 4 + 2Q66 ) m2n2 + Q22 Q11 = Q11 m4 + 2(Q12 n , Q12

F12 F22 F26 F26 F¯16 F26 F¯66 F66

G16

F¯55 F45

, Fs =

k k = G12 Q66

B66

D12 D22 D26 D26 E12 E22 E26 E26 D= ,E= ,F E¯16 E26 E¯66 E66 D¯16 D26 D¯66 D66

F26 F66

E44

c0 Dij , Cij = Cij + c0 Dij , D¯ij = Dij

k Q11 =

C12 C22 C26 C26 = C¯16 C26 C¯66 C66

D26 D66

E45

E45

C¯ij = Cij

C¯11 C12 C¯16 C16

D12 D¯16

Nk k=1

A¯ij = Aij

A12 A¯16

D¯11

E¯55

{Aij , Bij , Cij, Dij , Eij, Fij , Gij, Hij ,} =

E K K1 K2

A12

C26 C66

, Es =

D44

(7.b)

where the stiffness coefficient are obtained as follows:

A¯11

C16

k x2

G66

D¯55 D45 D45

k x2

(9.d)

Ds Es Fs Gs

D E F G

G26 G66

Ds =

where

G=

k x1

E66 F16 F26 F66 F66 F16 G11 G12 G16 G16

Cs Ds Es Fs

F16

k x1

E26 E66 F12 F16

Bs Cs Ds Es

F¯11

F26 F66

k x0

C16 C26 C66 C66 D16 D26 D66 D66 E16 D11 D12 D16 D61 E11 E12 E16 E16 F11

As Q Bs P = Q1 Cs P2 Ds

D16

k

C22 C26 C26 D12 D22 D26 D26 E12 C26 C66 C66 D16 D26 D66 D66 E16

The above two formulas can be further written in the matrix form:

=

0 x 0 x k x0 0

A16 A26 A66 A66 B16 B26 B66 B66 C16 C26 C66 C66 D16 D26 D66 D16 B11 B12 B16 B16 C11 C12 C16 C16 D11 D12 D16 D16 E11 E12 E16 E16

N1

A=

0 x 0

D16 D16

(9.c) 5

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K. Choe, et al.

f1

f2

f3

Fig. 3. Frequencies of the laminated elliptical shell with to different boundary parameters.

6

Composite Structures 225 (2019) 111155

K. Choe, et al.

Table 1 The spring stiffness values of the arbitrary boundary conditions. B.C

ku,0ku,1

kv,0 kv,1

kw,0 kw,1

k

F SD SS C E1 E2 E3

0 0 1014 1014 108 1014 108

0 1014 1014 1014 108 1014 108

0 1014 1014 1014 108 1014 108

0 0 0 1014 1014 108 108

,0 k

,1

k

,0 k

0 0 1014 1014 1014 108 108

,1

k

,0 k

0 0 0 1014 1014 108 108

,1

k

,0 k

,1

0 0 1014 1014 1014 108 108

f1

f2

f3

Fig. 4. Frequencies of the laminated elliptical shell with to different connective parameters.

7

k

,0 k

0 0 0 1014 1014 108 108

,1

k

,0 k

0 0 1014 1014 1014 108 108

,1

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K. Choe, et al.

Fig. 5. Frequencies of the clamped laminated elliptical shell with to different truncation terms. 0

N 1 2

Ui =

k 0z

P 1 d d = 2 Ti =

+P

+

Q 1 k 1z

2

2

z R

1 2

2 0

i+1 i

+

2

2

I2 [(

) +(

) + 2u

2I3 (u

+

+v

z R

1+

I0 [(u)2 + (v )2 + (w)2] + 2I1 (u

=

Q 1 k 1z

+P

2

0 z

k 2z

+ KTs ·P + KTs1·Q1 + KTs2·P2

T ·Q

0

i

k 0z

(z )[(U ) + (V ) + (W ) ] 1 +

V

+ M k 0 + M k 0+

ET ·N + KT ·M + KT1 ·N1 + KT2 ·M2+

2

i+1

2

1 2

0

+N

+ N2 k 2 + N2 k 2 + N2 k 2 + N 2 k 2 + Q

0

i

0

+N

M k 0 + M k 0 + N1 k 1 + N 1 k 1 + N 1 k 1 + N 1 k 1

2

i+1

0

+N

Fig. 6. Percentage error of the frequencies for the Jacobi parameters α and β.

0 z+

+Q

+P

2

Table 2 Frequencies of F-C composite laminated hyperbolic shell with to different number of segments.

R R0

k 2z

Rs

R R0 d d

(11)

f (Hz)

+

R R0 d d

2

+ ( ) ]+

2

2I5 (

+

) + I6 [(

) +( )]

2

2

zk+ 1 zk

(z ) 1 +

z R

z R

(1, z , z 2, z 3, z 4 , z 5, z 6) dz

As mentioned earlier, penalty parameters are used to simulate boundary conditions and continuity conditions between segments. In this paper, the penalty parameters are expressed as the artificial spring stiffness, and the appropriate value of the artificial spring stiffness ensures fast convergence of the accurate solution [33–36]. The resulting boundary potential energy and internal coupling potential energy can be expressed as follows:

+

2

1 2

1 2

ku,0 u2 + k v,0 v 2 + k w,0 w 2 + k

0

2

k

2

2

,0

ku,1 u2 + k v,1 v 2 + k w,1 w 2 + k

,1

0

k

2

,1

+k

,0

,1

2

+ +k

,1

2

,0

+ +k

,0

+k

2

2

2

+k

,0

36.304 39.300 52.100 61.051 66.331 71.878

36.303 39.299 52.099 61.051 66.331 71.871

36.388 39.563 52.264 61.059 66.332 72.716

−3 m

1 2 3 4 5 6

27.868 42.897 58.113 118.28 168.32 172.79

27.147 42.861 58.047 118.11 164.13 168.16

26.945 42.817 58.045 117.99 163.15 167.33

26.883 42.794 58.045 117.92 162.89 167.13

26.863 42.781 58.045 117.88 162.81 167.07

27.510 43.010 58.108 118.56 166.38 170.76

+k

,0

+k

,1

+k

,1

Wi =

+

R0 d

+

N

L=

R0 d

2

ui + 1)2 + kv (vi

vi + 1)2 + kw (wi

2

1 Uc, i = 2

2

+k

i

i+1

i

i+1

0

i

i+1

i

i+1

2

k

i

+m

i

i

R R0 d d

1

Ui + Wi )

Uc, i i=1

Ub

(17)

2.3. Solution methodology

wi + 1 ) 2

+k

(

+k

( i

i

i + 1)

2+

In Section 2.2, total energy functions (L ) of the composite laminated doubly-curved revolution shells has been established. The purpose of this section is to establish the whole solution equation based on the energy functional. The application of the multi-segment partitioning technique makes it easy to select the permissible displacement function and ensure convergence of the solution in low degree of polynomial. In

R0 d

2

+k

i

= 1

2

+k

N

(Ti i

(14) ku (ui

fui ui + fvi vi + fwi wi + m

(16)

= 0 2

1 2

where fui , fvi and fwi are the distributed forces in the φ, θ and z directions, respectively; m i and m i are the distributed couples about the middle surface of the segment. The total energy functions (L ) of the composite laminated doublycurved revolution shells can be expressed as:

2

2

10

In order to study the dynamic characteristics of composite laminated doubly-curved revolution shells, the work of external forces can be expressed as follows:

(13)

Ub =

8

36.306 39.308 52.109 61.051 66.331 71.903

(12)

1+

6

36.320 39.351 52.142 61.051 66.331 72.036

where (I0, I1, I2, I3, I4 , I5, I6) =

4

36.388 39.639 52.251 61.062 66.332 72.928

)+

)+

+( ) +2

2 1 2 3 4 5 6

+ 2v ]+

I4 [2

Ref. [8]

1m

R R0 d d dz =

+v

Number of the segment (Nφ)

i + 1)

2 i, i + 1

(15) 8

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Table 3 Comparison of some frequencies for composite laminated elliptical shell with different boundary conditions. h

n

F-F

S-S

C-C

h

Present

Ref. [8]

Present

Ref. [8]

Present

Ref. [8]

0.01 m

1 2 3 4 5 6

78.068 1.2696 4.0812 8.0282 11.959 17.592

78.073 1.2699 4.1094 8.0296 11.958 17.553

74.885 40.523 38.926 35.162 30.790 31.442

74.881 40.538 38.929 35.159 30.786 31.434

81.416 47.070 44.760 35.906 32.688 33.636

81.435 47.081 44.775 35.905 32.689 33.634

0.05 m

1 2 3 4 5 6

83.703 6.2156 16.692 29.531 43.755 59.716

83.705 6.2178 16.693 29.533 43.757 59.726

89.164 54.383 53.603 58.521 70.798 89.722

89.179 54.391 53.606 58.521 70.802 89.733

100.57 66.469 57.364 65.227 73.606 90.336

100.56 66.459 57.364 65.221 73.604 90.333

1 2 3 4 5 6

86.259 11.738 27.597 48.927 69.235 87.351

86.259 11.738 27.598 48.929 69.240 87.359

100.05 68.633 68.953 90.895 116.69 144.81

100.10 68.655 68.953 90.924 116.69 144.81

113.22 84.023 78.536 94.416 120.75 150.11

113.16 83.966 78.517 94.414 120.75 150.08

1 2 3 4 5 6

88.176 15.154 37.945 60.752 85.841 110.81

– – – – – –

109.10 83.052 87.404 114.48 152.59 197.63

– – – – – –

122.02 99.796 100.21 122.04 156.87 200.23

– – – – – –

0.1 m

0.15 m

Table 5 Comparison of some frequencies for composite laminated hyperbolic shell with different boundary conditions.

0.01 m

0.05 m

0.1 m

0.15 m

n

F-C

S-S Present

Ref. [8]

Present

Ref. [8]

1 2 3 4 5 6

112.59 61.384 34.802 20.821 16.682 20.243

112.62 61.475 34.916 20.928 16.746 20.267

168.47 125.69 106.19 94.448 86.528 81.180

168.57 125.78 106.26 94.501 86.574 81.223

169.27 126.71 107.16 95.436 87.665 82.518

169.24 126.72 107.26 95.638 87.935 82.824

1 2 3 4 5 6

113.01 62.753 40.270 40.726 57.597 79.838

113.02 62.787 40.306 40.759 57.637 79.895

165.95 122.62 104.11 94.562 89.775 88.833

166.06 122.78 104.26 94.680 89.855 88.885

170.18 128.45 110.25 101.01 96.810 96.558

170.20 128.48 110.31 101.08 96.894 96.652

1 2 3 4 5 6

113.75 65.261 51.060 66.857 94.162 117.26

113.76 65.282 51.089 66.918 94.274 117.41

163.53 119.76 102.96 98.122 101.25 111.11

163.69 119.99 103.21 98.337 101.42 111.26

172.53 132.56 116.84 112.13 115.19 125.04

172.56 132.61 116.89 112.20 115.28 125.18

1 2 3 4 5 6

114.66 68.424 63.120 89.205 121.76 151.71

114.68 68.452 63.179 89.365 122.03 152.11

161.94 118.32 104.11 105.55 118.37 139.96

162.11 118.57 104.37 105.79 118.63 140.27

175.83 138.33 125.74 126.40 137.67 158.12

C-C

Present

Ref. [8]

Present

Ref. [8]

Present

Ref. [8]

1 2 3 4 5

34.255 36.198 36.628 36.884 38.731

34.933 36.578 37.024 37.496 39.063

42.910 43.366 44.128 44.443 46.201

42.923 43.416 44.126 44.553 46.385

43.909 44.433 45.378 45.880 47.928

44.813 45.387 46.366 47.053 49.351

0.05 m

1 2 3 4 5

52.137 57.752 57.872 67.679 70.424

52.796 58.425 58.508 67.966 71.110

67.870 68.531 72.849 78.705 80.945

67.988 68.604 73.000 78.748 81.118

75.951 76.825 81.006 86.602 89.703

77.244 78.08 82.405 87.806 91.302

0.1 m

1 2 3 4 5

70.489 70.549 74.592 82.541 84.577

70.749 71.159 75.292 82.653 84.682

89.470 92.991 96.586 109.31 111.13

89.687 93.172 96.862 109.53 111.45

108.29 110.78 114.81 123.87 128.45

109.27 111.69 115.82 124.64 129.53

0.15 m

1 2 3 4 5

73.675 84.448 85.331 90.653 99.706

– – – – –

109.45 112.33 120.03 126.82 144.27

– – – – –

132.29 135.84 139.32 149.29 158.53

– – – – –

M

Ref. [8]

S-S

0.01 m

C-C

Present

F-C

other words, the displacement function in the whole section can be expressed only by the polynomial of higher degree, however, the displacement function in the case of segmentation can be expressed relatively accurately even in the lower degree than the polynomial in the whole section. In this paper, the Jacobian orthogonal polynomial [14,15,33] proposed by the author will be used to represent it. Relevant polynomial principles can be referred to in detail in Refs. [14,15,33]. The detailed expression of displacement function is as follows:

Table 4 Comparison of some frequencies for composite laminated paraboloidal shell with different boundary conditions. h

n

N

u=

Umn, i Pm( , ) ( )[cos(n ) + sin(n )] ei

t

Vmn, i Pm( , ) ( )[sin(n ) + cos(n )] ei

t

(18.a)

m=0 n=0 M

N

v=

(18.b)

m = 0 n= 0

M

N

Wmn, i Pm( , ) ( )[cos(n ) + sin(n )] ei

w=

t

(18.c)

m = 0 n= 0 M

N

= m=0 n=0

M

m = 0 n= 0

m=0 n=0 M

m=0 n=0

t

mn, i

Pm( , ) ( )[sin(n ) + cos(n )] ei

t

mn, i

Pm( , ) ( )[sin(n ) + cos(n )] e i

t

(18.f)

(18.g)

N

= m = 0 n= 0

9

Pm( , ) ( )[cos(n ) + sin(n )] ei

N

= M

mn, i

(18.e)

N

=

175.90 138.42 125.84 126.54 137.89 158.48

Pm( , ) ( )[cos(n ) + sin(n )] ei

t

mn, i

(18.d)

N

= M

Pm( , ) ( )[cos(n ) + sin(n )] ei

t

mn, i

(18.h)

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Table 6 Comparison of first ten frequency parameters for various composite laminated double-curved shells of revolution with C–C boundary conditions. mode

1 2 3 4 5 6 7 8 9 10

Elliptical shell

Paraboloidal shell

Hyperbolic shell

Present (HSDT)

ABAQUS (S4R)

ABAQUS (C3D20R)

Present (HSDT)

ABAQUS (S4R)

ABAQUS (C3D20R)

Present (HSDT)

ABAQUS (S4R)

ABAQUS (C3D20R)

105.5214 111.3363 137.2656 153.4063 162.7774 167.9680 170.5121 186.3909 196.5262 201.5367

105.76 111.59 137.62 153.69 163.35 168.15 170.98 186.92 196.76 202.08

105.89 111.68 137.67 153.71 163.42 168.34 170.99 186.70 196.86 201.98

155.1646 157.1585 168.9051 177.6597 195.4902 218.4661 232.0048 252.0574 270.1548 272.0968

155.77 157.71 169.70 178.16 196.78 218.85 234.22 252.38 272.41 273.95

155.56 157.47 169.40 177.89 196.17 218.64 232.98 252.35 271.59 273.37

124.9086 143.9221 147.3232 194.2711 198.1803 212.0693 212.6088 231.0638 254.7545 262.1683

125.21 144.11 147.98 195.74 198.99 212.76 213.00 231.83 257.62 263.89

122.53 142.12 143.24 186.12 193.91 204.67 212.33 228.88 243.71 251.19

Table 7 First five frequency parameters for a composite laminated elliptical shell with various elastic boundary conditions. h

f(Hz)

[0°/90°]

[0°/90°/0°]

[0°/90°/0°/90°]

E1-E1

E2-E2

E3-E3

E1-E1

E2-E2

E3-E3

E1-E1

E2-E2

E3-E3

0.01 m

1 2 3 4 5

29.985 75.304 82.055 118.40 129.91

46.525 110.12 124.27 130.10 132.70

29.985 75.304 82.047 118.40 129.91

29.986 76.407 83.692 123.06 136.15

46.525 115.16 129.59 136.30 139.37

29.986 76.406 83.674 123.06 136.15

29.985 75.309 83.631 118.50 130.13

46.525 110.34 124.47 130.33 133.18

29.985 75.309 83.619 118.50 130.13

0.05 m

1 2 3 4 5

16.613 46.228 50.276 79.197 110.46

46.525 111.37 124.79 131.26 138.12

16.602 46.215 50.242 79.182 110.42

16.617 46.464 50.429 79.372 114.88

46.525 117.59 131.53 138.46 144.80

16.606 46.441 50.273 79.358 114.85

16.613 46.266 50.431 79.249 111.67

46.525 111.78 125.79 134.07 145.61

16.602 46.250 50.359 79.235 111.65

0.1 m

1 2 3 4 5

12.624 35.117 37.029 58.502 107.97

46.524 112.03 125.65 136.20 154.88

12.531 35.079 36.996 58.462 107.65

12.629 35.238 37.064 58.726 112.26

46.524 118.69 133.85 144.02 157.94

12.535 35.170 36.886 58.671 111.94

12.624 35.150 37.056 58.611 111.60

46.524 112.59 128.04 144.95 163.16

12.531 35.105 36.978 58.571 111.44

0.15 m

1 2 3 4 5

11.108 29.459 30.577 48.701 109.00

46.520 112.24 126.92 144.02 162.14

10.783 29.404 30.556 48.604 108.27

11.113 29.542 30.593 49.198 113.16

46.520 119.65 136.69 151.54 169.93

10.786 29.456 30.455 48.907 112.44

11.109 29.488 30.587 48.929 114.70

46.520 113.16 131.16 160.11 168.21

10.784 29.429 30.529 48.773 114.45

Table 8 First five frequency parameters for a composite laminated paraboloidal shell with various elastic boundary conditions. h

f(Hz)

[0°/90°]

[0°/90°/0°]

[0°/90°/0°/90°]

E1-E1

E2-E2

E3-E3

E1-E1

E2-E2

E3-E3

E1-E1

E2-E2

E3-E3

0.01 m

1 2 3 4 5

55.721 55.919 56.043 56.628 56.872

57.561 57.655 58.037 58.298 59.155

55.720 55.918 56.035 56.628 56.871

57.940 58.064 58.181 58.524 59.350

60.706 60.746 61.044 61.796 63.059

57.949 58.070 58.180 58.233 59.350

58.498 58.552 59.163 59.351 60.531

60.528 60.870 61.051 62.172 62.392

58.496 58.551 59.160 59.350 60.529

0.05 m

1 2 3 4 5

66.832 67.819 68.613 71.308 74.293

74.818 75.749 77.836 80.238 85.847

66.738 67.738 68.499 71.230 74.165

70.332 70.863 71.972 73.153 77.016

81.882 82.397 83.715 85.731 87.700

70.010 70.574 71.603 72.877 76.622

70.606 70.928 74.342 76.064 81.829

80.963 81.172 86.017 87.623 95.717

70.452 70.739 74.203 75.851 81.686

0.1 m

1 2 3 4 5

68.120 68.858 73.616 73.658 83.118

85.924 89.498 90.618 102.28 104.30

67.525 68.179 73.018 73.323 82.486

71.144 72.792 74.996 76.954 83.744

97.350 98.737 101.38 104.68 114.00

69.531 70.793 74.218 74.816 81.572

70.090 73.462 74.426 83.135 84.538

94.503 95.562 107.11 107.45 127.88

69.169 72.407 73.978 82.098 84.342

0.15 m

1 2 3 4 5

63.474 63.597 69.673 70.601 72.327

93.645 96.113 105.58 109.88 132.31

62.294 63.062 68.825 69.448 72.108

64.541 66.022 69.824 72.337 73.149

108.05 108.55 114.63 118.68 127.57

63.324 63.571 69.224 69.497 72.113

64.009 65.867 69.792 72.327 77.567

102.17 110.11 112.33 136.40 144.60

63.411 64.088 69.551 72.108 75.215

10

Composite Structures 225 (2019) 111155

K. Choe, et al. M

N ( , )

=

mn, i Pm

( )[sin(n ) + cos(n )] ei

t

(18.i)

m=0 n=0

where Umn, i , Vmn, i , Wmn, i ,

mn, i

,

mn, i

,

mn, i

,

mn, i

,

mn, i

and

mn, i

are the

( ) is the mth order Jacobi polyJacobi expanded coefficients; nomial for the displacement components in the generatrix direction. In the course of practical application, Jacobi polynomials are the generalization of some orthogonal polynomials like Legndre, Chebyshev and Gegenbauer polynomials. That is, by setting α = β = −1/2, the first kind of Chebyshev polynomials is obtained, andα = β = 1/2 corresponds to the second kind of Chebyshev polynomials. In others cases, α = β = 0 yields the Legendre polynomials. Therefore, compared with other admissible displacement functions, Jacobi polynomial is more general and applicable. The discretized equation of motion for the composite laminated doubly-curved revolution shells can be obtained by substituting Eq. (18) into the energy functional Eq. (17) and then carrying out variation operation with respect Jacobi expanded coefficients: Fig. 7. Some mode shapes of the composite laminated elliptical shell.

L =0 q xmn, i ,

Pm( , )

q = Umn, i, Vmn, i, Wmn, i,

xmn, i ,

mn, i

,

mn, i

,

xmn, i ,

mn, i

(19)

Then the dynamic characteristic equation of the composite laminated doubly-curved revolution shells can be obtained:

Mq¨ + [K + K C + KB ] q = F

(20)

where q is the Jacobi expanded coefficients. M and K are the disjoint generalized mass matrix and stiffness matrix. KC and KB are the stiffness matrices introduced by the penalty parameters. By assuming harmonic motions, q = q ei t , the governing equations of composite laminated doubly-curved revolution shells can be obtained from Eq. (20):

det[

2M

+ (K + KC + KB )] = 0

(21)

The frequencies and the corresponding mode shapes of composite laminated doubly-curved revolution shells can be solved from Eq. (21). 3. Numerical results and discussion The main purpose of this section is to carry out numerical examples to verify the convergence, validity and reliability of the previous analysis model. This section can be divided into three parts: (1) The convergence of the current method is studied, and the convergence properties of penalty parameters and Jacobi orthogonal polynomial are studied respectively. (2) To carry out validity research and verify the correctness of the current method by comparing with the existing literature. (3) Based on the above research, the dynamic characteristics of composite laminated doubly-curved revolution shells are further studied.

Fig. 8. Some mode shapes of the composite laminated paraboloidal shell.

3.1. Convergence of current method Fig. 3 shows the influence of boundary parameters on the first three order frequency parameters of composite laminated elliptic shells with elastic constraints. The elastic boundary conditions are assumed to be clamped constraints at one end and elastic constraints at the other end. This elastic constraint is defined as changing only one type of boundary parameter at a time, increasing in turn from 102 to 1016 and the other boundary parameters are set to 1016. The geometric and material parameters are set as follows: E1 = 150 GPa, E2 = 10 GPa, ρ = 1500 kg/m3, μ12 = 0.25, G12 = G13 = 5 GPa, G23 = 6 GPa, a = 1 m, b = 2 m, k R0 = 0.4 m, R1 = 1 m, h = 0.06 m, fiber = [0° 90° 0°]. The Jacobian parameters and wave numbers are set as follows: α = 0, β = 0, M = 9,

Fig. 9. Some mode shapes of the composite laminated hyperbolic shell.

11

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K. Choe, et al.

z

z

z

A φ

x

A B fw

A B

φ

C

φ0

fw

φ1

C

y

y

B

fw C

y

fw θ

o

2θ x

o

x

o

fw

x

Fig. 10. The diagrammatic sketch of three applied load types for the composite laminated doubly-curved shells. (a) Point force; (b) Line force; (c) Surface force.

stabilized in the region. Therefore, through the above analysis, in order to obtain accurate and stable calculation results, the connective parameters of all subsequent examples are set to 1014. Through Eq. (18), it can be found that the calculation accuracy of the present method depends on the number of truncation terms of Jacobi polynomial. Thus, it is of necessity to study the convergence of truncation terms of Jacobi polynomial for the calculation results. Fig. 5 gives the influence of truncation terms of Jacobi polynomial on the first five order frequency parameters of composite laminated elliptic shells with clamped boundary condition. From Fig. 5, it is obvious that when the truncation terms are very small, the frequency result is inaccurate. When the truncation terms increase gradually, the calculation results converge rapidly. Then, when the value of truncation terms exceeds a certain threshold value, the calculation results gradually diverge. From Fig. 5, it can be found that when the truncation terms exceed 6, the calculation results almost remain unchanged. Therefore, there is a stable convergence region for the convergence of the whole calculation method. In this paper, the truncation terms of Jacobi polynomial are set to M = 8. Fig. 6 presents the percentage error of the frequencies for composite laminated elliptic shells with different Jacobi parameters α and β. The definition of the relative error is given as: ( , = 0, = 0) = 0, = 0 × 100% . In Fig. 6, the percentage error for various values of Jacoby parameters α and β is clearly different. In case of α = β = 1, the percentage error is bigger than the other cases, especially, and in the 5th mode, the maximum percentage error is 8 × 10−3%. However, this error does not significantly affect the accuracy of the solution, thus in all subsequent examples the Jacobi parameter is uniformly chosen as α = β = 0. Lastly, the effects of the number of segments on the vibration characteristics of composite laminated hyperbolic shell with clampedFree boundary condition are presented in Table 2. The geometric and material parameters are set as follows: E1 = 150 GPa, E2 = 10 GPa, μ12 = 0.25, G12 = G13 = 5 GPa, G23 = 6 GPa, ρ = 1450 kg/m3, a = 1 m, C = 3 m, D = 3 m, R1 = 2 m, h = 0.1 m, Rs = 1 m and −3 m, k fiber = [0° 90°]. For comparison, the literature solutions [8] are also listed in this table. From Table 2, it’s not hard to see that the frequencies of composite laminated hyperbolic shell are stable rapidly with the increase of the number of segments, and when the number is more than 6, the calculation results are stable. Of course, increasing the polynomial degree will improve the accuracy of the results, but it is adequate that the polynomial degree be a certain value to ensure a good

Fig. 11. The comparison of displacement of spherical shell with F-C boundary condition.

Nφ = 4, n = 2. Through this picture, it shows intuitively that the boundary parameters have a very important influence on the vibration characteristics of the composite laminated doubly-curved revolution shells. In addition, the boundary parameters introduced by Taylor series expansion of higher order terms have little effect on the structural vibration characteristics. According to the specific analysis of this picture, the definition of the boundary parameters of the classical boundary conditions and the elastic boundary conditions studied is shown in Table 1. Fig. 4 shows the influence of connective parameters on the first three order frequency parameters of composite laminated elliptic shells with clamped-clamped boundary condition. The geometric and material parameters are the same as Fig. 3. Also, The Jacobi parameters and wave numbers are consistent with Fig. 3. Similar to the definition of elastic boundary conditions, only one connective parameter is changed at a time, ranging from 102 to 1016, while the remaining other types of connective parameters are set to 1016. From Fig. 4, the frequency parameters of the composite laminated doubly-curved revolution shells increase rapidly with the increase of the connective parameters. When the connective parameters exceed 1014, the frequency parameters are 12

Composite Structures 225 (2019) 111155

K. Choe, et al.

a) B(π/6,0)

b) C(π/3,0)

Fig. 12. The displacement of composite laminated elliptical shell with different applied load types.

a) B(0.3π,0)

b) C(0.38π,0)

Fig. 13. The displacement of composite laminated paraboloidal shell with different applied load types.

a) B(π/2.2,0)

b) C(2π/3.5,0)

Fig. 14. The displacement of composite laminated hyperbolic shell with different applied load types.

13

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Table 9 First five frequency parameters for a composite laminated hyperbolic shell with various elastic boundary conditions. h

f(Hz)

[0°/90°]

[0°/90°/0°]

[0°/90°/0°/90°]

E1-E1

E2-E2

E3-E3

E1-E1

E2-E2

E3-E3

E1-E1

E2-E2

E3-E3

0.01 m

1 2 3 4 5

4.7452 6.3086 11.347 14.309 24.160

44.761 45.058 47.013 48.111 50.551

3.636 4.727 11.366 14.314 24.163

9.253 16.591 21.395 22.790 25.527

44.148 44.774 45.812 46.636 49.258

9.140 16.211 21.394 22.814 25.529

16.315 18.937 21.711 22.019 29.622

49.203 50.246 51.246 53.006 55.867

16.323 18.940 21.748 21.967 29.616

0.05 m

1 2 3 4 5

17.871 34.865 43.486 57.662 72.314

71.890 74.619 84.187 86.582 102.90

17.761 32.510 35.322 40.791 43.535

10.710 41.199 44.869 58.963 64.247

77.236 78.895 83.174 91.968 93.895

9.515 21.621 24.658 41.356 46.017

11.763 39.424 54.838 76.594 86.488

80.275 88.041 89.739 109.62 113.91

12.138 37.430 40.064 54.928 76.991

0.1 m

1 2 3 4 5

38.623 63.106 67.028 75.750 76.229

95.327 96.204 115.78 115.82 141.02

37.996 39.690 63.916 66.412 80.387

29.918 50.096 62.690 70.328 71.228

113.19 115.70 121.89 131.27 139.73

20.203 59.970 62.107 65.569 86.532

53.991 56.203 64.575 72.534 87.932

106.42 117.96 119.79 150.25 154.75

58.686 67.142 72.553 77.874 87.025

0.15 m

1 2 3 4 5

43.587 67.260 68.259 69.354 70.132

110.60 121.25 124.48 154.46 158.84

54.560 56.709 61.550 70.853 75.419

35.005 57.102 62.290 63.825 71.149

57.739 101.07 124.27 127.90 137.31

49.404 51.194 61.074 70.478 71.343

56.230 63.301 71.198 72.445 80.228

128.85 130.38 158.40 161.29 170.77

49.056 70.773 74.853 75.430 76.601

Fig. 16. The comparison of normal displacement of spherical shell. Fig. 15. The sketch of load functions (a) Rectangular pulse; (b) Triangular pulse; (c) Half-sine pulse; (d) Exponential pulse.

expressed as follows: R0 = 2 m, R1 = 4 m, k = 4, Rs = 0 m, h = 0.1 m, k fiber = [0° 90°]. Table 5 depicts the comparison of some frequencies for composite laminated hyperbolic shell with different boundary conditions. The geometric parameters of the hyperbolic shell are expressed as follows: a = 1 m, R1 = 2 m, C = 1 m, D = 4 m, Rs = 1 m, h = 0.1 m, k fiber = [0° 90°]. The material parameters of the above tables are consistent with those of Table 2. To verify the correctness of the above tables, the results [8] by the Ritz method based on the first-order shear deformation theory are also given in the tables, from which, it is obvious that the predicted results by the method in this paper based on the high-order shear deformation theory are in good agreement with the literature results. Next, the free vibration analysis of several composite laminated double-curved shells of revolution including elliptical shell, paraboloidal shell and hyperbolic shell, is performed using the proposed method, and the results are compared with those by the finite element analysis software ABAQUS, because of the lack of literature on the vibration analysis of composite laminated double-curved shell of revolution based on the HSDT. Geometric dimensions of composite

satisfaction. Thus, in the follow-up study, the number of segments is taken as Nφ = 8 unless otherwise specified. 3.2. Free vibration analysis of composite laminated doubly-curved revolution shell The free vibration behavior of composite laminated doubly-curved revolution shell with general boundary condition by the present solutions will be studied. Firstly, the correctness of this method will be studied. Tables 3–5 show some frequencies of composite laminated doubly-curved revolution shell with different boundary conditions. Table 3 gives the comparison of some frequencies for composite laminated elliptical shell with different boundary conditions. The geometric parameters of the elliptical shell are expressed: a = 2 m, b = 4 m, k Rs = −4 m, φ0 = π/6, φ1 = 5π/6, fiber = [0°]. The comparison of some frequencies for the composite laminated paraboloidal shell is presented in Table 4. The geometric parameters of the paraboloidal shell are 14

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a) B(π/6,0)

b) C(π/3,0)

Fig. 17. The displacement of composite laminated elliptical shell with different transient loads.

a) B(0.3π,0)

b) C(0.38π,0)

Fig. 18. The displacement of composite laminated paraboloidal shell with different transient loads.

a) B(π/2.2,0)

b) C(2π/3.5,0)

Fig. 19. The displacement of composite laminated hyperbolic shell with different transient loads.

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laminated double-curved shells of revolution used in this study are the same as Table 3–5, except for the thickness h = 0.15 m. The material properties are: E = 210 GPa, ρ = 7800 kg/m3, μ = 0.3, and the boundary conditions are set to clamped boundary conditions in all cases. The element types used in the finite element model are solid element type (C3D20R) and shell element type (S4R). The number of elements is defined as follows: elliptical shell: 8208 (S4R) and 9360 (C3D20R); paraboloidal shell: 3024 (S4R) and 3000 (C3D20R); hyperbolic shell: 3638 (S4R) and 104,621 (C3D20R). As can be seen from Table 6, the results taken by the present method agree well with those by FEM. In addition, it also can be that for numerical solutions such as the finite element method, the choice of element type and mesh number is crucial to the calculation results and it is not conducive to parametric research. For the semi-analytical method similar to this paper, there is no such problem. After studying the convergence characteristics of parameters, the appropriate parameters can be selected to carry out efficient parameterization research. On the basis of the above comparative study, to enrich the existing numerical results, the vibration characteristics under elastic boundary conditions obtained by the higher-order shear deformation theory and the method in this paper will be further given. Table 7 shows the frequency parameters for a composite laminated elliptical shell with various elastic boundary conditions. Three groups of fiber angles, k fiber = [0° 90°], [0° 90° 0°], [0° 90° 0° 90°], are given in this example. Table 8 gives the first five frequency parameters for a composite laminated paraboloidal shell, in which the elastic boundary conditions including E1-E1, E2-E2 and E3-E3 are considered. Table 8 shows fist five frequency parameters for a composite laminated hyperbolic shell. In the above tables there are four groups of thickness coefficients: 0.01 m, 0.05 m, 0.1 m and 0.15 m, respectively. From the above table, it can be found that for different geometric forms, the influence of thickness coefficient on vibration characteristics is different. For composite laminated elliptical shell, the frequency parameter decreases with the increase of thickness. For composite laminated paraboloidal shell, the frequency parameter increases first and then decreases with the increase of thickness. For composite laminated hyperbolic shell, the frequency parameters increase with the increase of thickness. Figs. 7–9 present some mode shapes of the of composite laminated elliptical shell, composite laminated paraboloidal shell and composite laminated hyperbolic shell with E3-E3 case, respectively. The drawing of mode shapes is based on SURF command in MATLAB software. Through the above mode shapes, the inherent vibration characteristics of the structure can be more intuitively reflected.

comparative data are from the finite element analysis software ABAQUS, and the calculation conditions are consistent. The contrast between the two is shown in Fig. 11, from which a good agreement can be seen and it means the present method has the ability to conduct the steady-state vibration analysis of composite laminated doubly-curved shells on the basis of the higher order shear deformation theory. Based on the validation of this method, the steady-state vibration analysis will be carried out next. Figs. 12–14 present the displacement of composite laminated elliptical shell, composite laminated paraboloidal shell, composite laminated hyperbolic shell with various applied load types, respectively. The geometric and material parameters are the same as those in Tables 7–9. In Fig. 12, the positions of point force, line load and surface load are A ( , ) = ( 2, 0) , 6, 6]) , reA ( , ) = ([ 3, 2 3], 0) , A ( , ) = ([ 3, 2 3], [ spectively. The positions of displacement responses measured are defined as B ( 6, 0) and C ( 3, 0) , respectively. For Fig. 13, the positions of point force, line load and surface load are A ( , ) = (0.35 , 0) , A ( , ) = ([0.34 , 0.36 ], 0) , A ( , ) = ([0.34 , 0.36 ], [ 6, 6]) , respectively. The positions of displacement responses measured are defined as B (0.3 , 0) and C (0.38 , 0) , respectively. For composite laminated hyperbolic shell, the positions of point force, line load and surface load are A ( , ) = ( 2, 0) , A ( , ) = ([4 10, 6 10], 0) , A ( , ) = ([4 10, 6 10], [ 6, 6]) , respectively. The positions of displacement responses measured are defined as B ( 2.2, 0) and C (2 3.5, 0) , respectively. For composite laminated elliptical shells, it is obvious that the response amplitude under the action of point force is the smallest, while the surface force is the largest. For composite laminated paraboloidal shells, the difference of displacement response under three load forms is not obvious. For composite laminated hyperbolic shells, the natural frequencies caused by point forces are the least, while those caused by surface forces are the most. Through the above analysis, it is not difficult to find that when there are differences between the geometric parameters of the composite laminated doublycurved shells and the form of external force, the steady-state response can also be greatly different. 3.3.2. Transient vibration analysis Before carrying out the transient vibration analysis, it is necessary to verify whether the proposed method is capable of carrying out the transient vibration analysis of composite laminated doubly-curved shells. Fig. 15 shows four types of transient loads used in this paper [2]. The load functions are expressed as follows: Rectangular pulse

3.3. Force vibration analysis of composite laminated doubly-curved revolution shell

f (t ) =

ft 0 t 0 t>

Triangular pulse:

The composite laminated doubly-curved revolution shell, as the foundation construction, may receive external loads in engineering applications. Therefore, it is necessary to study the forced response of structures. For the study of forced response, it can be divided into the stability response analysis in frequency domain and the transient response analysis in time domain.

2t

f (t ) =

2

ft

(

ft

0

t

2

)

ft

2

0

t

2

t t>

Half-sine pulse:

3.3.1. Steady-state vibration analysis Fig. 10 shows three types of loads containing the point force, line force and surface force, acting on the composite laminated doublycurved shells. Firstly, the feasibility of using the present method to solve the steady-state vibration problem of laminated doubly-curved shells is given. For brevity, the isotropic spherical shell is studied as the object. In this case, the harmonic point force fw is applied at the Load Point A ( , ) = (0, 0) in the thickness direction and is perpendicular to the surface. The point load is: fw = fw ( A) , where A)( fw = 1N . ?? is the Dirac delta function. The sweep range is from 300 to 1500, and the interval is 1 Hz. The measuring point for displacement response is set at Point B ( 4, 0) in the vertical direction. The

f (t ) =

ft sin

( ) t

0

0

t t>

Exponential pulse:

f (t ) =

ft e 0

t

0

t t>

where ft is the load amplitude; ?? is the pulse width; t is the time variable. Fig. 16 shows the comparison curve of transient response data based on the current method and finite element software ABAQUS. The 16

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geometric parameters, material constants and boundary condition are the same as those in Fig. 11. In this case, the transient load f (t ) is set to the rectangular pulse and the amplitude of the rectangular pulse is taken as: ft = 1N . The calculating time step is set to t = 0.05 ms , and the loading time and calculating time t are equal to 10 ms. From Fig. 16, it is easy to see that the present method in this paper has high accuracy in predicting the transient response of composite laminated doubly-curved shells. On this basis, the transient vibration analysis of composite laminated doubly-curved shells are further studied. Figs. 17–19 present the displacement of composite laminated elliptical shell, composite laminated paraboloidal shell, composite laminated hyperbolic shell with various applied transient loads, respectively. The geometrical and material constants are the same as those in Tables 7–9. The calculating time step is set to t = 0.02 ms , and the loading time is equal to = 0.2 ms and calculating time is equal to t = 10 ms . The loading position and response acceptance position are consistent with the point load in Figs. 12–14. From the analysis of the pictures above, it can be seen that when the loading time is equal, the transient response caused by the rectangular pulse is the largest, while the minimum is caused by the triangular pulse. Therefore, the magnitude of the transient response of the composite laminated doubly-curved shells is closely related to the form of load action. Slow loading or unloading can reduce the amplitude of transient response, while sudden loading or unloading force can increase the transient response.

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4. Conclusions In this paper, free vibration analysis and force vibration analysis of composite laminated doubly-curved revolution shell with general boundary conditions are first presented on the basis of higher order shear deformation theory. The multi-segment partitioning technique is adopted to reduce the requirement of the displacement allowable function for boundary conditions. On this basis, the penalty parameters including the boundary parameters and continuity parameters between segments are used to simulate geometric boundary conditions and continuity conditions between segments. All energy functions of segments are based on the higher order shear deformation theory. The displacement admissible functions in all segments are expanded by mixed series, Jacobi polynomial expansion in the generatrix direction and triangular series expansion in the circumferential direction. The Jacobian expansion coefficient is regarded as a generalized variable and the Ritz method is used to perform the variational operation to obtain the dynamic characteristics of the composite laminated doubly-curved revolution shell. The convergence, validity and dynamic characteristics of the model are given by a series of numerical examples, which can be used as reference data for future researchers. 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 Foundation of China (Grant Nos. 51705537), Innovation Driven Program of Central South University (Grant number: 2019CX006) and the Natural Science Foundation of Hunan Province of China (2018JJ3661). The authors also gratefully acknowledge the supports from State Key Laboratory of High Performance Complex Manufacturing, Central South University, China (Grant No. ZZYJKT2018-11). References [1] Chandrashekhara K, Bhimaraddi A. Thermal stress analysis of laminated doubly curved shells using a shear flexible finite element. Comput Struct 1994;52(5):1023–30.

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