A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions

Composites Part B 89 (2016) 230e252

Contents lists available at ScienceDirect

Composites Part B journal homepage: www.elsevier.com/locate/compositesb

A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions Guoyong Jin a, **, Tiangui Ye a, *, Xueren Wang b, Xuhong Miao b a b

College of Power and Energy Engineering, Harbin Engineering University, Harbin, 150001, PR China Naval Academy of Armament, Beijing, 100161, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 March 2015 Received in revised form 30 August 2015 Accepted 20 November 2015 Available online 30 November 2015

This paper describes a unified solution for the vibration analysis of functionally graded material (FGM) doubly-curved shells of revolution with arbitrary boundary conditions. The solution is derived by means of the modified Fourier series method on the basis of the first order shear deformation shell theory considering the effects of the deepness terms. The material properties of the shells are assumed to vary continuously and smoothly along the normal direction according to general three-parameter power-law volume fraction functions. In summary, the energy functional of the shells is expressed as a function of five displacement components firstly. Then, each of the displacement components is expanded as a modified Fourier series. Finally, the solutions are obtained by using the variational operation. The convergence and accuracy of the solution are validated by comparing its results with those available in the literature. A variety of new vibration results for the circular toroidal, paraboloidal, hyperbolical, catenary, cycloidal and elliptical shells with classical and elastic boundary conditions as well as different geometric and material parameters are presented, which may serve as benchmark solution for future researches. Furthermore, the effects of the boundary conditions, shell geometric and material parameters on the frequencies are carried out. © 2015 Elsevier Ltd. All rights reserved.

Keywords: B. Vibration C. Analytical modelling Functionally graded doubly-curved shell

1. Introduction A shell is a three-dimensional (3D) body enclosed by two, relatively close, curved surfaces [1]. Shells are an important type of structural component commonly used in the practical engineering structures that range from the outer space to the deep-sea, e.g., underwater vehicle hulls, aircraft fuselages, cooling towers, nuclear power plant domes, large-span roofs since they not only are aesthetically pleasing but also have comparatively light weight, high load capacity and are able to provide a big span of space. As is well-known, the practical engineering structures may fail and collapse because of material fatigue resulting from vibrations. Therefore, it is of particular importance to understand the structural vibration and reduce it through proper design. The vibration modal information of shells, such as their natural frequencies and mode shapes, play a vital role in the safety evaluation, dynamic analysis and reliable design of the whole engineering structures. * Corresponding author. Tel.: þ86 451 82569458; fax: þ86 451 82518264. ** Corresponding author. E-mail addresses: [email protected] (G. Jin), [email protected] (T. Ye). http://dx.doi.org/10.1016/j.compositesb.2015.11.015 1359-8368/© 2015 Elsevier Ltd. All rights reserved.

Hence, an accurate vibration characteristic determination of the shells is essential. The wide applications of shells have attracted considerable attention in developing accurate shell theories and analysis methods to predict their static and dynamic behaviors [1e25], [26e50], [50e63]. It is well known that the exact 3D equations of elasticity are complicated when written in curvilinear coordinates. The calculation based on these equations demands large, fast computational facilities. Therefore, in the literature, the 3D shell problems are typically reduced as the 2D representations by making simple kinematic assumptions about the variation of displacements through the thickness and the solutions are carried out by analyzing its middle surface only [2]. The history of the study on the 2D shell theories can be dated back to about 100 years ago by Love [3]. A large variety of classical and modern theories have been made and developed based on different approximations and assumptions since then. Among them, there are mainly two major theories: the classical thin shell theories (CSTs) and the shear deformation shell theory (SDST). The CSTs are based on the first approximation of Love-Kirchhoff hypothesis, which do not include the effects of the transverse shear deformation. Many subcategory CSTs, such as the Reissner-Naghdi's linear shell theory, Donner-Mushtari's theory, Flügge's theory and Sanders'theory, to

G. Jin et al. / Composites Part B 89 (2016) 230e252

Nomenclature

U, V, W

4, q, z h z z0 Rs

u, v, w

R4, Rq, R ah, bh dc rc ae, be

curvilinear coordinates shell thickness revolution axis of meridian curve geometric axis of meridian curve offset of revolution axis z with respect to geometric axis z0 principal radii of curvature of the shell radius of circular toroidal shell length of the semitransverse and semiconjugate axes of hyperbolical shell radius of curvature at the apex of the catenary shell radius of the generator circle of the cycloidal shell length of the semimajor and semiminor axes of the elliptical shell

name a few, have been developed. A comprehensive monograph of Leissa [1] which summarized approximately 1000 articles and reports shows that most thin shell theories yield similar results. Many publications studied the vibration characteristics of shells on the basis of CSTs [4e11]. However, the CSTs can be grossly in error in prediction the transverse deflections, bucking loads and natural frequencies for shells which are rather thick or when they are made from materials with a high degree of anisotropic [12]. In order to have an accurate prediction of the dynamic behaviors of shells with larger thickness ratios and to eliminate the deficiencies of the CSTs, the Love's fourth assumption in the CSTs is relaxed by including the effects of the shear deformation and rotary inertia, which resulted in the so-called shear deformation shell theory (SDST). In the SDST, the displacements of a shell can be expanded in terms of shell thickness of a first or a higher order. In the case of first order expansion, the theory is named as the first-order shear deformation shell theory (FSDT) [13]. Since the aim of this paper is to study the vibration characteristics of thin and moderately thick doubly-curved shells of revolution, the FSDT is just adopted to formulate the theoretical formulation. Apart from the aforementioned shell theories, a large variety of analytical and numerical methods, such as the RayleighRitz method, the boundary element method (BEM), the generalized differential quadrature method (GDQM), the finite element method (FEM), and the discrete singular convolution (DSC) approach, have been proposed and developed as well. In the practical engineering applications, shells can have different geometry shapes based on their curvature characteristics. Among them, shells of revolution, such as the cylindrical, conical, spherical, elliptical paraboloidal, toroidal, hyperbolic paraboloidal, cycloidal and catenary shells, are most frequently encountered. In recent years, there are a number of publications concerning the vibration of such revolution shells [3e30]. The development of investigations on this subject has been well reviewed in several monographs by Leissa [1], Qatu [13], Reddy [31], and review or survey articles [12,32e36]. However, the extensive literature review reveals that most of the previous studies regarding vibrations of shells of revolution are confined to conventional shells of revolution, i.e., the circular cylindrical, conical and spherical shells. For example, more than half references documented by Leissa [1] deal with circular cylindrical shells. Since the radii of curvature of a general doubly-curved shell of revolution are functions of the meridional coordinate, the resulting governing equations of such shell consist of a system of partial differential equations with variable coefficients. Inevitably, this introduces inherent complexity for finding the

f4, fq t b, c p Us T Usp

P

K, M G

231

displacement variations of an arbitrary point (4, q, z) lying on the shell space middle surface displacements in the 4, q and z directions rotations of normals to the middle surface with respect to the 4 and q -axes time variable material distribution profile along the normal direction of the shell power-law index of the volume fraction strain energy expression kinetic energy expression deformation strain energy about the boundary springs Lagrangian functional stiffness and mass matrices expansion coefficient vector

solutions. There is a considerable lack of information available for the vibration analysis of general doubly-curved shells of revolution. Among those available, Tornabene [3], Tornabene and his coauthors [37e40,54] studied the static and dynamic behaviors of doublycurved shells of revolution with different classical boundary conditions by means of the GDQM, in which the derivatives in the governing equations and boundary conditions are discretized and transformed into a weighted sum of node values of dependent variables. Edalat et al. [41] investigated free vibrations of thin deep shells with variable radii of curvature using semi-analytical methods including power series method, Galerkin method and beam function method. Tan [42] presented an efficient substructuring analysis method for predicting the natural frequencies of shells of revolution having arbitrary shape of meridian and different classical boundary restraints. Nasir et al. [43] analyzed the free vibration and seismic response of hyperbolic shells and examined the influence of thickness, height and curvature on this response. Kang [2], Kang and Leissa [44e46] studied the vibration behaviors of isotropic shells of revolution having different classical boundary conditions by means of the Ritz method. Furthermore, after the review of the literature, it has been shown that most of the previous efforts were restricted to shells of revolution with limited sets of classical boundary conditions, i.e., the free, simply-supported and clamped boundary conditions as well as their combinations. However, in the practical engineering applications, the boundary conditions of a shell may not always be classical in nature. A variety of possible boundary cases can be encountered. Moreover, the existing analysis techniques are often only customized for a specific set of boundary conditions, and thus typically require constant modifications of the solution procedure to adapt to different boundary cases. This will result in very tedious calculations due to the fact that each boundary of a general doublycurved shell can exist 24 forms of classical boundary conditions [13]. Thus, it is of great importance to develop a unified method which is capable of universally dealing with general doubly-curved shells of revolution with arbitrary boundary conditions. In addition, with the development of new industries and modern processes, FGMs have been increasingly utilized to build various doublycurved shell components in many fields of modern engineering practices to satisfy special functional requirements due to their outstanding material properties. The vibration results of such shells are far from complete. In view of these apparent voids, the present work presents an endeavor to complement the vibration analysis of FGM doubly-curved shells of revolution.

232

G. Jin et al. / Composites Part B 89 (2016) 230e252

In this paper, a unified modified Fourier series solution for the vibration analysis of FGM doubly-curved shells of revolution (e.g., the circular toroidal, paraboloidal, hyperbolical, catenary, cycloidal and elliptical shells) with arbitrary boundary conditions is presented. The material properties of the shells under consideration are assumed to vary continuously and smoothly along the normal direction according to general three-parameter power-law volume fraction functions. In summary, the solution is derived by means of the modified Fourier series method on the basis of the FSDT. Effects of the deepness terms are considered in the solution as well. Specifically, the energy functional of the shells is expressed as a function of five displacement components first, by using the constitutive and kinematic relationships. Regardless of boundary conditions, each of the displacement components is then expanded as the superposition of a standard cosine Fourier series and two supplementary functions introduced to remove any potential discontinuous at the ends. Finally, the solutions are obtained by using the variational operation. Comparisons of the present results are made with those presented in the literature to validate the convergence and accuracy of the proposed solution. A considerable number of numerical examples are presented for FGM doubly-curved shells of revolution with classical and elastic boundary conditions as well as different geometric and material parameters. Furthermore, the effects of the boundary conditions, material distribution as well as shell geometric and material parameters are also investigated. 2. Theoretical formulations 2.1. The model Fig. 1(a) shows the cross section of a doubly-curved shell of revolution with uniform thickness h. The shell is characterized by its middle surface, which is generated by the rotation of the meridian curve c1c2 about the revolution axis z. It should be stressed that axis z0 denotes the geometric axis of the meridian curve c1c2 and Rs is the offset of the revolution axis z with respect to the geometric axis z0 . To describe the shell clearly, the orthogonal

curvilinear coordinate system along the meridional, circumferential and normal directions of the shell is taken as 4, q and z, respectively. The two principal radii of curvature of the shell in its middle surface are represented by R4 and Rq, where R4 describes the curvature in the meridional plane x-z and Rq is in a plane normal to the meridian. C4 and Cq indicate the centers of the two principal radii R4 and Rq. The horizontal radius R0 represents the distance of each point of the middle surface from the revolution axis z. Specially, it can be defined as R0 ¼ Rq sin4. Fig. 1(b) shows a differential element of the shell and u, v, w separately indicate the middle surface displacements of the shell in the 4, q and z directions [3]. In practical engineering applications, doubly-curved shells of revolution can have different geometry shapes based on their curvature characteristics. As for many other kinds of doubly-curved revolution shells, the circular toroidal, paraboloidal, hyperbolical, catenary, cycloidal and elliptical shells are most frequently encountered. Hence, for the sake of brevity, the current work will be confined to the six types of classical doubly-curved shells of revolution. The two principal radii of curvature of each type shells are given as follows [2,3]: 1. Circular toroidal shell, see Fig. 2(a)

R4 ð4Þ ¼ R;

Rq ð4Þ ¼ R þ

Rs sin 4

(1.a)

where R is the radius of the circular cross meridian. 2. Paraboloidal shell, see Fig. 2(b)

R4 ð4Þ ¼

k ; 2 cos3 4

Rq ð4Þ ¼

k Rs þ 2 cos 4 sin 4

(1.b)

where k indicates the characteristic parameter of the paraboloidal meridian. 3. Hyperbolical shell, see Fig. 2(c)

a2h b2h R4 ð4Þ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ; a2h sin2 4  b2h cos2 4

3

(1.c)

a2h Rs Rq ð4Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ sin 4 2 2 2 2 ah sin 4  bh cos 4 where ah and bh are the length of the semitransverse and semiconjugate q axes of the meridian, respectively. Specially, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi hyperbolical bh ¼ ahCh/ c2h  a2h ¼ ahDh/ d2h  a2h . 4. Catenary shell, see Fig. 2(d)

R4 ð4Þ ¼

dc ; cos2 4

Rq ð4Þ ¼

dc arc sinhðtan4Þ Rs þ sin 4 sin 4

(1.d)

where dc is the radius of curvature at the apex of the catenary meridian. 5. Cycloidal shell, see Fig. 2(e)

R4 ð4Þ ¼ 4rc cos 4; Fig. 1. Geometry and coordinate system of a doubly-curved shell of revolution: (a) cross-section; (b) differential element [3].

Rq ð4Þ ¼

rc ð24 þ sin24Þ Rs þ sin 4 sin 4

(1.e)

where rc represents the radius of the generator circle of the cycloidal meridian.

G. Jin et al. / Composites Part B 89 (2016) 230e252

233

Fig. 2. Meridional section of doubly-curved shells of revolution: (a) circular toroidal; (b) paraboloidal; (c) hyperbolical; (d) catenary; (e) cycloidal; (f) elliptical [3].

6. Elliptical shell, see Fig. 2(f)

effects of the deepness terms, the strains at any point of the shell can be defined as [1,13]:

a2e b2e R4 ð4Þ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ; a2e sin2 4 þ b2e cos2 4

3

(1.f)

a2e Rs þ Rq ð4Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin 4 2 2 a2e sin 4 þ be cos2 4 where ae and be denote the length of the semimajor and semiminor axes of the elliptical meridian.

  1 1   ε04 þ zc4 ; g4z ¼   g0 ε4 ¼  1 þ z R4 1 þ z R4 4z   1 1 ε0q þ zcq ; gqz ¼ g0 εq ¼ ð1 þ z=Rq Þ ð1 þ z=Rq Þ qz     1 1   g04q þ zc4q þ g0q4 þ zcq4 g4q ¼  ð1 þ z=Rq Þ 1 þ z R4

where ε04 , ε0q , g04q and g0q4 represent the normal and shear strains in the middle surface. c4, cq, c4q and cq4 are the analogous curvature and twist changes; g04z and g0qz represent the transverse shear strains. The middle surface strains and curvature changes are as follows [3]:

2.2. Kinematic relations and stress resultants The work is based on the FSDT. According to the FSDT assumptions, the displacement field in the shell space can be written by the following relationships:

Uð4; q; z; tÞ ¼ uð4; q; tÞ þ zf4 ð4; q; tÞ Vð4; q; z; tÞ ¼ vð4; q; tÞ þ zfq ð4; q; tÞ Wð4; q; z; tÞ ¼ wð4; q; tÞ

(3)

(2)

where U, V and W are the displacement variations of an arbitrary point (4, q, z) lying on the shell space. u, v and w represent the middle surface displacements in the meridional, circumferential and normal directions, respectively. f4 and fq denote the rotations of normals to the middle surface with respect to the 4-and q-coordinate, respectively, while t is the time variable. Assuming that the transverse normal is inextensible, the shell deflections are small, the strains are infinitesimal and taking into account the

ε04 ¼ ε0q ¼

 1 vu þw ; R4 v4

c4 ¼

1 vv ; R4 v4

c4q ¼

1 vf4 R4 v4

  1 vv 1 vfq þucos4þwsin4 ; cq ¼ þf4 cos4 R0 vq R0 vq g04q ¼

1 vu vcos4 ; g0q4 ¼ R0 vq  1 vw u þf4 ; g04z ¼ R4 v4 



1 vfq R4 v4

(4)

1 vf4 cq4 ¼ fq cos4 R0 vq  1 vw vsin4 þfq g0qz ¼ R0 vq

According to the generalized Hooke's law, the stressestrain relations can be defined as:

234

G. Jin et al. / Composites Part B 89 (2016) 230e252

8 9 2 s4 > Q > > >s > > 6 11 > < q > = 6 Q12 tqz ¼6 6 0 > > > > 4 0 t > > 4z > > : ; t4q 0

Q12 Q22 0 0 0

0 0 Q44 0 0

9 38 ε4 > 0 > > > > > > 7 0 7< εq > = g 0 7 7> qz > g > 0 5> > > > : 4z > ; g4q Q66

0 0 0 Q55 0

2 (5)

where s4 and sq denote the normal stress components. t4q, t4z andtqz are the analogous shear stresses. Qij (i, j ¼ 1, 2, 4e6) represent the material coefficients of the FGMs. These coefficients are functions of thickness coordinate z and can be written as follows:

Q11 ¼ Q22 ¼

Ez ; 1  m2z

Q12 ¼

mz Ez

; 2

1  mz

Q44 ¼ Q55 ¼ Q66

Ez  ¼  2 1 þ mz

(6)

where Ez, mz and rz denote the locally effective Young's modulus, Poisson's ratio and density of the FGM, respectively. Typically, FGMs are made of a mixture of two constituents by gradually changing the volume fraction of one constituent material in the thickness direction. In this paper, the doubly-curved shells under consideration are assumed to be made form a mixture of ceramic and metal. The material coefficients of the shells are graded continuously and smoothly along the normal direction z and determined by the volume fraction function of the ceramic and the properties of the ceramic and metal materials. The continuous material properties along the shell normal direction z can be expressed as:

Ez ðzÞ ¼ ðEC  EM ÞVC ðzÞ þ EM mz ðzÞ ¼ ðmC  mM ÞVC ðzÞ þ mM rz ðzÞ ¼ ðrC  rM ÞVC ðzÞ þ rM

(7)

where EC, mC, rC and EM, mM, rM are the Young's modulus, Poisson's ratio and density of the ceramic and metal constituents, respectively. VC is the volume fraction of the ceramic constituent. In the current work, it is defined as two simple three-parameter power-law functions:

 p 1 z 1 z c þ þb þ ¼ 1 2 h 2 h  p   1 z 1 z c  þb  ¼ 1 2 h 2 h 

FGM1 : VC ðzÞðb=c=pÞ FGM2 : VC ðzÞðb=c=pÞ



Zh=2 h=2



Zh=2 Aij ¼ h=2

Zh=2 Aij ¼ h=2

Bij ¼

Qij dz;

h=2



3

s4

" #  Zh=2 "s # M4 6 7 4 z 6t4q 7 1þ z dz; zdz ¼ 1þ 4 5 Rq Rq M4q t4q h=2 t4z 2 3 sq  " # Zh=2 " s # Mq 6 7 q z z 6tq4 7 1þ dz; zdz ¼ 1þ 4 5 R4 R4 Mq4 tq4 h=2 tqz

where N4, Nq, N4q and Nq4 are called normal and shear force resultants. M4, Mq, M4q and Mq4 represent the analogous bending and twisting moment resultants. Q4 and Qq denote the transverse shear force resultants. It should be pointed out that although t4q equal to tq4 from the symmetry of the shear stress, it is apparent that the shear force resultants N4q and Nq4 are not equal in consequence of difference of deepness terms in the meridional and circumferential directions. Similarly, the twisting moment resultants M4q and Mq4 are not equal either. Substituting Eqs. (3)e(4) into Eq. (5) and performing the integration operation in Eq. (9), the force and moment resultants of the shell can be written as:

2

3

6 6A 6 11 6 6 Nq 7 6 A12 6 7 6 6 7 6 0 6 N4q 7 6 6 7 6 6N 7 6 0 6 q4 7 6 6 7¼6 6 M4 7 6 B 6 11 6 7 6 M 7 6 6 q 7 6 6 7 6 B12 4 M4q 5 6 6 0 6 6 Mq4 6 0 4 2

N4

3

2

A12 A22

ε04

0

0

0

0

0

A66 A66

0

A66 A66

B12

0

0

B22 0 0 0 B66 B66 0 B66 B66

7 0 7 7 7 B12 B22 0 0 7 7 7 0 0 B66 B66 7 7 7 0 0 B66 B66 7 7 7 D11 D12 0 0 7 7 7 D12 D22 0 0 7 7 0 0 D66 D66 7 7 7 0 0 D66 D66 7 5 B11 B12

0

(10)

3

6 0 7 6 ε 7 6 q 7 6 0 7 6 g4q 7 3" 6 7 " # 2 # 6 0 7 g0qz Q q K A 0 6 gq4 7 s 44 5 4 6 ¼ 7; 6 7 Q4 g04z 0 Ks A55 6 c4 7 6 7 6 cq 7 6 7 6 7 4 c4q 5 cq4 in which Ks is the shear correction factor, typically taken by 5/6. The stiffness coefficients Aij, Bij, Dij, Aij , Bij , Dij , Aij , Bij andDij are determined by the following relations:

Dij ¼

2

(9)

Zh=2 Qij zdz;

Zh=2

h=2

Qq

(8)

Zh=2

3

6 7 6N4q 7 ¼ 4 5 h=2 Q4 2 3 Nq Zh=2 6 7 6Nq4 7 ¼ 4 5



where p is the power-law index of the volume fraction. Parameters b and c indicate the material distribution profile along the normal direction of the shell. For example, Fig. 3 shows the through thickness variation of the volume fraction with various distribution profiles. Carrying the integration of stresses over the cross-section, the force and moment resultants of the shell can be obtained:

Aij ¼

N4

Qij z2 dz

h=2

 z z z Zh=2 Zh=2 1þ 1þ Rq Rq Rq dz; Bij ¼ dz; Dij ¼ dz Qij  Qij z  Qij z2  z z z 1þ 1 þ 1 þ h=2 h=2 R4 R4 R4    z z z 1þ 1þ 1þ Zh=2 Zh=2 R4 R4 R4 dz; Bij ¼ dz; Dij ¼ dz Qij  Qij z  Qij z2  z z z 1þ 1 þ 1 þ h=2 h=2 Rq Rq Rq 1þ

(11)

G. Jin et al. / Composites Part B 89 (2016) 230e252

235

Fig. 3. The through thickness variation of volume fraction Vc with various distribution profiles.

The above equations show the stiffness coefficients in Eq. (11) can be integrated exactly since the two principal radii of curvature, R4 and Rq, are only functions of the meridional coordinate-4 and independent of the normal coordinate-z. However, for better numerical stability, the deepness term 1/(1 þ z/Rb) (where b is 4 or q in the denominator) is expanded in a geometric series as:

1



    2   3  1 þ z Rb ¼ 1  z Rb þ z Rb  z Rb ±/

Substituting Eq. (12) into Eq. (11) results in:

(12)

a1 ¼

sin 4 1  ; R0 R4

b1 ¼

1 sin 4  ; R4 R0

a2 ¼ b2 ¼

1 R24



sin2 4

Zh=2

R20

h=2



a3 ¼

sin 4 R24 R0

1 R34 (14)

Zh=2 Qij z4 dz; Hij ¼

h=2



sin4 sin2 4 sin3 4 ; b3 ¼  3 R0 R4 R4 R20 R0

Zh=2 Qij z3 dz; Fij ¼

Eij ¼

sin 4 ; R4 R0

Qij z5 dz h=2

where

A11 ¼ A11 þ a1 B11 þ a2 D11 þ a3 E11 ; B11 ¼ B11 þ a1 D11 þ a2 E11 þ a3 F11 ; D11 ¼ D11 þ a1 E11 þ a2 F11 þ a3 H11 ; A55 ¼ A55 þ a1 B55 þ a2 D55 þ a3 E55 ; A66 ¼ A66 þ a1 B66 þ a2 D66 þ a3 E66 ; B66 ¼ B66 þ a1 D66 þ a2 E66 þ a3 F66 ; D66 ¼ D66 þ a1 E66 þ a2 F66 þ a3 H66 ;

A22 B22 D22 A44 A66 B66 D66

¼ A22 þ b1 B22 þ b2 D22 þ b3 E22 ¼ B22 þ b1 D22 þ b2 E22 þ b3 F22 ¼ D22 þ b1 E22 þ b2 F22 þ b3 H22 ¼ A44 þ b1 B44 þ b2 D44 þ b3 E44 ¼ A66 þ b1 B66 þ b2 D66 þ b3 E66 ¼ B66 þ b1 D66 þ b2 E66 þ b3 F66 ¼ D66 þ b1 E66 þ b2 F66 þ b3 H66

(13)

236

G. Jin et al. / Composites Part B 89 (2016) 230e252

2.3. Energy expressions Since the two radii of curvature of a general doubly-curved shell of revolution are functions of the meridional coordinate 4, the resulting governing equations of such shell consist of a system of partial differential equations with variable coefficients. This introduces inherent complexity for solving these equations in closed form. In this work, the Ritz technique is taken up, and used to develop a unified solution code to find vibration results of FGM doubly-curved shells of revolution with arbitrary boundary conditions. Thus, the energy expressions of the shells are developed firstly. The strain energy of deformation can be expressed in terms of force and moment resultants, middle surface strains and curvature changes as:

1 Us ¼ 2

Z Z ( 4 q

) N4 ε04 þNq ε0q þN4q g04q þNq4 g0q4 þM4 c4 þ R0 R4dqd4 Mq cq þM4q c4q þMq4 cq4 þQ4 g04z þQq g0qz (15)

The kinetic energy expression is simply:

T¼

1 2

Z Z

(

Zh=2 rz

4 q

h=2

vf4 vu þz vt vt

2

 þ

vv vf þz q vt vt

2

 þ

1 Usp ¼ 2

i 8h 9 Z < ku u2 þkv v2 þkw w2 þK 4 f24 þK q f2q = 40 40 40 40 40 4¼4 i 0 h R dq : þ ku u2 þkv v2 þkw w2 þK 4 f2 þK q f2 ; 0 41 41 41 q 41 41 4 q

4¼41

(18) 4 q where kuj , kvj , kw j , Kj and Kj (j ¼ 40 and 41) represent the stiffness (per unit length) of the artificial boundary springs along boundaries 4 ¼ 40 and 4 ¼ 41, respectively. Unless otherwise stated, N/m and N/ rad are utilized as the units of the stiffness of stretching and rotational springs, respectively.

2.4. Governing equations and boundary conditions Hamilton's principle is a generalized principle of virtual displacement to dynamics of systems. It can be used to determine the governing and boundary equations of shells under vibration directly. Applying the Hamilton's principle, i.e., letting Rt d 0 ðT  Us  Usp Þdt ¼ 0, the governing equations of the shells under free vibration are:

)  vw 2 z z 1þ R R4 dzdqd4 1þ vt R4 Rq 0

" #)   Z Z ( " 2  2  2 # vf4 2 1 vu vv vw vu vf4 vv vfq vfq 2 þ þ I þ þ I þ þ 2I R0 R4 dqd4 0 1 2 ¼ 2 vt vt vt vt vt vt vt vt vt

(16)

4 q

where

I0 ¼ I0 þ

I1 I I þ 1 þ 2 ; R4 Rq R4 Rq Zh=2

½I0 ; I1 ; I2 ; I3 ; I4  ¼

I1 ¼ I1 þ

I2 I I þ 2 þ 3 ; R4 Rq R4 Rq

I2 ¼ I2 þ

I3 I I þ 3 þ 4 R4 Rq R4 Rq (17)

h i rz 1; z; z2 ; z3 ; z4 dz

h=2

 cos 4 Q4 v2 f4 vN4 vNq4  v2 u þ þ N4  Nq þ ¼ I0 2 þ I1 2 R0 R4 v4 R0 vq R4 vt vt Since the main focus of this work is to complement the vibration studies of doubly-curved shells of revolution with arbitrary boundary conditions, in order to satisfy the request, the artificial spring boundary technique is adopted. More information about the artificial spring boundary technique can be seen at Refs. [47,48]. Specially, any classical boundary conditions can be well modeled by assigning the stiffness of the artificial boundary springs to either zero or infinity. Consistent with the artificial spring boundary technique, the deformation strain energy about the boundary springs (Usp) can be expressed as:

 cos 4 Qq sin 4 vN4q vNq  v2 v v2 f þ þ N4q þ Nq4 þ ¼ I0 2 þ I1 2q R0 R0 R4 v4 R0 vq vt vt N4 Nq sin 4 vQ4 vQq cos 4 v2 w þ þ Q4   þ ¼ I0 2 R0 R0 R4 R4 v4 R0 vq vt  cos 4 v2 f4 vM4 vMq4  v2 u þ þ M4  Mq  Q4 ¼ I1 2 þ I2 2 R0 R4 v4 R0 vq vt vt  cos 4 vM4q vMq  v2 v v2 f þ þ M4q þ Mq4  Qq ¼ I1 2 þ I2 2q R0 R4 v4 R0 vq vt vt

(19)

G. Jin et al. / Composites Part B 89 (2016) 230e252

237

On each boundary of 4 ¼ constant, five boundary equations are obtained:

4 ¼ 40 : N4  ku40 u ¼ 0;

N4q  kv40 v ¼ 0;

Q4  kw 40 w ¼ 0;

4 ¼ 41 : N4 

N4q 

Q4 

ku41 u

¼ 0;

kv41 v

¼ 0;

kw 41 w

¼ 0;

4 M4  K40 f4 ¼ 0;

M4 

¼ 0;

q M4q  K40 fq ¼ 0

(20)

q M4q  K41 fq ¼ 0

symmetry of revolution shells, each displacement/rotation component of the shells is expanded as a set of modified Fourier series as:

2.5. Admissible displacement functions Vibrations are naturally expressible as waves, which are usually described by Fourier series mathematically. For vibration problems of shells, the admissible functions are often expressed in the form of Fourier series because of their orthogonality and excellent stability in numerical calculations. However, the conventional Fourier series expression only applicable for a few simple boundary conditions and can lead unavoidable convergence problem for other boundary conditions [49]. Taking a continuous function defined on [0, L], w(x), for example. The Fourier cosine series expression wðxÞ ¼ P∞ m¼0 Am cosðmpx=LÞ is able to correctly converge to w(x) at any point over [0, L]. However, its first-order derivative w'(x) is an odd function over [-L, L] leading to a jump at end locations, see Fig. 4 [49]. The corresponding Fourier expansion of w'(x) continue on [0, L] and can be differentiated term-by-term only if w(L) ¼ w(0) ¼ 0. Thus, the Fourier series expansion will accordingly have a convergence problem due to the discontinuity at end points when w(x) is required to have up to the first-derivative continuity. Recently, Li [49,50] proposed a modified Fourier series technique for beams with arbitrary boundary conditions. In this technique, each displacement of a beam is expressed as a standard cosine Fourier series with the addition of several supplementary terms. In that case, the standard cosine Fourier series represent a residual beam displacement that is continuous everywhere. In this work, the modified Fourier series technique is further developed and extended to the vibration analysis of general FGM doublycurved shells of revolution with arbitrary boundary conditions, aiming to provide a unified and reasonable accurate alternative to other analytical and numerical techniques. Under the consideration mentioned above as well as considering the circumferential

uð4; q; tÞ ¼

M P

vð4; q; tÞ ¼

M P

Bm cos lm 4 þ

m¼0

wð4; q; tÞ ¼ f4 ð4; q; tÞ ¼ fq ð4; q; tÞ ¼

M P

x L

w '( x)

Fig. 4. An illustration of the possible discontinuities of w'(x) at the ends.

2 X

Am cos lm 4 þ

m¼0

l¼1 2 X

! aln Pl ð4Þ cos nqejut !

bln Pl ð4Þ sin nqejut !

l¼1 2 X

cln Pl ð4Þ cos nqejut l¼1 ! 2 X Dm cos lm 4 þ dln Pl ð4Þ cos nqejut m¼0 l¼1 ! M 2 X X Em cos lm 4 þ eln Pl ð4Þ sin nqejut Cm cos lm 4 þ

(21)

m¼0 M X

m¼0

l¼1

where u is the frequency and j2 ¼ 1, lm ¼ mp/D4 (D4 denotes the included angle in the meridional direction, D4 ¼ 41-40). The circumferential wave number, n, is taken to be an integer to ensure periodicity in the circumferential direction and n ¼ 0 means the axisymmetric vibration modes. Interchanging sinnq and cosnq in Eq. (21), another set of admissible functions of the shell can be obtained. In this case, n ¼ 0 corresponds to torsional vibration modes. It should be stressed that for n > 0, the two set admissible displacement functions result in the same solutions. Am, Bm, Cm, Dm and Em are the expansion coefficients of the standard Fourier series. al, bl, cl, dl and el denote the corresponding coefficients of the auxiliary functions Pl (4). According to Eq. (19), it is obvious that each displacement/rotation component of a doubly-curved shell is required to have up to the second-order derivative. Therefore, two auxiliary functions P1 (4) and P2 (4) are selected to remove all the discontinuities potentially associated with the first-order derivatives at the boundaries so that the function sets are capable of representing any free vibration motion of the doubly-curved shell of revolution. More detail information about the auxiliary functions can be seen in Refs. [47,48]. The two auxiliary functions are given as:

2  4 1 ; P1 ð4Þ ¼ 4 D4

w( x)

o L

4 K41 f4

P2 ð4Þ ¼

 42 4 1 D4 D4

(22)

It is easy to verify that 0

0

P1 ð0Þ ¼ P1 ðD4Þ ¼ P1 ðD4Þ ¼ 0; P1 ð0Þ ¼ 1 0 0 P2 ð0Þ ¼ P2 ðD4Þ ¼ P2 ð0Þ ¼ 0; P2 ðD4Þ ¼ 1

(23)

It should be noted that the modified Fourier series presented in Eq. (21) are complete series defined over the range [0, Df]. Therefore, linear transformation for coordinate from 42[40, 41] to 42½0; D4 needs to be introduced for the practical programming and computing, i.e., 4 ¼ 4 þ 40 . Furthermore, all the five infinite series expressions need to be truncated as finite series to obtain the results with acceptable accuracy due to the limited performance of

238

G. Jin et al. / Composites Part B 89 (2016) 230e252

the computers. In this work, the highest degree taken in the displacement expansions is uniformly taken as M. 2.6. Solution procedure The Lagrangian functional (P) of a general doubly-curved shell of revolution under free vibration can be expressed as:

P ¼ T  Us  Usp

(24)

Substituting Eqs. 15e18 into Eq. (24) together with the modified series expansions of the shell displacements and performing the Ritz operation with respect to each of the expansion coefficients Am, Bm, Cm, Dm, Em, al, bl, cl, dl and el, respectively:

vP ¼0 vq

q ¼ Am ; Bm ; Cm ; Dm ; Em ; al ; bl ; cl ; dl ; el

(25)

the free vibration characteristic equation for a doubly-curved shell of revolution are finally obtained, and can be summed up in a matrix form as:



 K  u2 M G ¼ 0

(26)

verify the accuracy and flexibility of the proposed solution. Typically, FGM is made from a mixture of ceramic and metal. Unless mentioned otherwise, the shells under consideration are assumed to be made of a mixture of Al2O3 (EC ¼ 380 GPa, mC ¼ 0.3, rC ¼ 3800 kg/m3) and aluminum (EM ¼ 70 GPa, mM ¼ 0.3, rM ¼ 2707 kg/m3). For general thick shells, each of the classical boundary conditions can exist in eight possible forms [13]. However, in the engineering applications, the completely free (F), simply supported (S) and completely clamped (C) boundary conditions are widely encountered and are of particular interest. Thus, the three types of boundary conditions are considered in this work. Boundaries for a doubly-curved shell may also be elastically constrained. Two elastically constrained conditions, designated by EI and EII are studied as well. The EI type boundary condition is assumed to be elastically constrained in the meridional direction and is characterized by a stiffness parameter ku (per unit length), whereas the EII one is considered to be circumferentially elastically supported and is prescribed by the stiffness parameter kv. Using the artificial spring boundary technique as described earlier, each classical and elastic boundary condition can be well modeled by assigning the boundary springs at proper stiffness. Taking edge 4 ¼ 40 for example, the corresponding spring stiffness for the mentioned boundary conditions can be defined in terms of spring rigidities as:

where G is the column matrix which contains, in an appropriate order, the unknown expansion coefficients that appear in the series

  4 q ku40 ¼ kv40 ¼ kw 40 ¼ K40 ¼ K40 ¼ 0 N4 ¼ N4q ¼ Q4 ¼ M4 ¼ M4q ¼ 0   4 q 7 S : ku40 ¼ kv40 ¼ kw 40 ¼ K40 ¼ 10 D; K40 ¼ 0 u ¼ v ¼ w ¼ M4 ¼ fq ¼ 0   4 q 7 C : ku40 ¼ kv40 ¼ kw 40 ¼ K40 ¼ K40 ¼ 10 D u ¼ v ¼ w ¼ f4 ¼ fq ¼ 0   4 q 7 EI : ku40 ¼ ku D; kv40 ¼ kw 40 ¼ K40 ¼ K40 ¼ 10 D us0; v ¼ w ¼ f4 ¼ fq ¼ 0   4 q 7 EII : kv40 ¼ kv D; ku40 ¼ kw 40 ¼ K40 ¼ K40 ¼ 10 D vs0; u ¼ w ¼ f4 ¼ fq ¼ 0 F:

expansions (21). K and M are the generalized stiffness and mass matrices of the shell, respectively. Both of them are symmetric matrices and they can be expressed as:

2

Kuu

6 KT 6 uv 6 6 T K ¼ 6 Kuw 6 T 6K 4 u4 2

KTuq

Kuv

Kuw

Ku4

Kvv

Kvw

Kv4

KTvw KTv4 KTvq

Kww

Kw4

KTw4 KTwq

K44 KT4q

Kuq

3

Kvq 7 7 7 Kwq 7 7; 7 K4q 7 5 Kqq 3

Mu4 6 Muu M 7 Mvq 7 6 vv 6 7 7; M¼6 6 7 M ww 6 7 4 T 5 Mu4 T M44 Mvq Mqq

(28)

where D ¼ ECh3/12(1m2C). The appropriateness of defining these boundary conditions in terms of spring rigidities will be proved by several examples in the latter subsections. In order to simplify the presentation, the symbolism FeC denotes that the shell is free and clamped at boundaries 4 ¼ 40 and 4 ¼ 41, respectively. 3.1. Convergence study and formulation validation

2

uT 6 T 6v 6 T G¼6 6w 6 T 44

3

(27)

7 7 7 7 7 7 5

qT

The elements in the above matrices are given in Appendix A. In Eq. (21), it has been assumed that the degree of each displacement expansion is uniformly truncated to M terms. Obviously, 5(M þ 3) eigenvalues and 5(M þ 3) eigenvectors corresponding to every eigenvalue can be easily obtained by using the standard eigenvalue decomposition program. 3. Results and discussions In this section, some results and discussions about the free vibration of FGM doubly-curved shells of revolution are presented to

In this subsection, results are compared with those reported in publications of Tornabene and Viola [38], Brebbia et al. [51] and Su et al. [52] as well as the FEM results obtained with commercial programs. As the first example, the first six natural frequencies (Hz) for an SeS supported FGM circular toroid shell with FGM1(b¼0/c/p¼5) and FGM2 (b¼0/c/p¼5) power-law distributions are considered in Table 1, in which five truncation schemes of the admissible displacement functions are performed. The details regarding the geometry of the shell are: Rs ¼ 1.5 m, R ¼ 3 m, h ¼ 0.2 m, 40 ¼ p/3 and 41 ¼ 2p/3. In this table, results reported by Tornabene and Viola [38] with GDQM based on FSDT are selected as the benchmark solutions. From the table, we can see that the maximum difference between the two results is smaller than 0.39%. The differences may be caused by different shell theories and methods used in the two studies. The effects of the deepness terms 1/(1 þ z/R4) and 1/(1 þ z/ Rq) were neglected in Ref. [38]. The table also shows that the modified Fourier series method has satisfactory convergence and accuracy. Table 2 presents the similar study for a CeF FGM paraboloidal shell (k ¼ 4.5, Rs ¼ 6 m, h ¼ 0.2 m, ap ¼ 0, cp ¼ 3 m). It is

G. Jin et al. / Composites Part B 89 (2016) 230e252

239

Table 1 Convergence and comparison of the first six frequencies (Hz) for an SeS FGM circular toroid shell (Rs ¼ 1.5 m, R ¼ 3 m, h ¼ 0.2 m, 40 ¼ p/3 and 41 ¼ 2p/3). Mode

1 2 3 4 5 6

FGM1

FGM2

(b¼0/c/p¼5)

(b¼0/c/p¼5)

M ¼ 11

M ¼ 12

M ¼ 13

M ¼ 14

M ¼ 15

Ref. [38]

M ¼ 11

M ¼ 12

M ¼ 13

M ¼ 14

M ¼ 15

Ref. [38]

329.59 329.59 358.62 358.62 368.99 368.99

329.59 329.59 358.62 358.62 368.99 368.99

329.59 329.59 358.62 358.62 368.99 368.99

329.59 329.59 358.62 358.62 368.99 368.99

329.59 329.59 358.62 358.62 368.99 368.99

330.03 330.03 359.34 359.34 368.79 368.79

328.59 328.59 358.70 358.70 364.41 364.41

328.59 328.59 358.70 358.70 364.41 364.41

328.59 328.59 358.70 358.70 364.41 364.41

328.59 328.59 358.70 358.70 364.41 364.41

328.59 328.59 358.70 358.70 364.41 364.41

327.45 327.45 357.36 357.36 363.00 363.00

Table 2 Convergence and comparison of the first six frequencies (Hz) for a CeF FGM paraboloidal shell (k ¼ 4.5, Rs ¼ 6 m, h ¼ 0.2 m, ap ¼ 0, cp ¼ 3 m). Mode

1 2 3 4 5 6

FGM1

FGM2

(b¼0/c/p¼1)

(b¼0/c/p¼1)

M ¼ 11

M ¼ 12

M ¼ 13

M ¼ 14

M ¼ 15

Ref. [38]

M ¼ 11

M ¼ 12

M ¼ 13

M ¼ 14

M ¼ 15

Ref. [38]

48.52 48.52 52.78 52.78 54.73 54.73

48.52 48.52 52.77 52.77 54.73 54.73

48.51 48.51 52.77 52.77 54.73 54.73

48.51 48.51 52.77 52.77 54.73 54.73

48.51 48.51 52.77 52.77 54.73 54.73

48.36 48.36 52.60 52.60 54.55 54.55

48.25 48.25 52.58 52.58 54.39 54.39

48.24 48.24 52.58 52.58 54.39 54.39

48.24 48.24 52.57 52.57 54.39 54.39

48.23 48.23 52.57 52.57 54.39 54.39

48.23 48.23 52.57 52.57 54.39 54.39

48.13 48.13 52.34 52.34 54.24 54.24

Table 3 Convergence and comparison of the frequencies (Hz) for an FeC hyperbolical shell (a ¼ 25.6 m, b ¼ 63.91 m, C ¼ 18.59 m, D ¼ 82.19 m, Rs ¼ 0 m and h ¼ 0.127 m). n

m¼1

m¼2

M ¼ 11

M ¼ 12

M ¼ 13

M ¼ 14

M ¼ 15

Ref. [51]

M ¼ 11

M ¼ 12

M ¼ 13

M ¼ 14

M ¼ 15

Ref. [51]

0 1 2 3 4

6.231 3.291 1.766 1.375 1.182

6.231 3.291 1.766 1.375 1.182

6.231 3.291 1.766 1.375 1.182

6.231 3.290 1.766 1.375 1.181

6.231 3.290 1.766 1.375 1.181

6.233 3.290 1.766 1.376 1.181

7.753 6.818 3.715 2.005 1.454

7.752 6.814 3.711 2.004 1.453

7.752 6.811 3.709 2.003 1.452

7.752 6.809 3.706 2.001 1.451

7.752 6.807 3.705 2.001 1.451

7.752 6.793 3.689 1.994 1.448

obvious that the present results match well with the reference ones. Table 3 shows the convergence study and comparison between the current results and those reported by Brebbia et al. [51] for an FeC supported hyperbolical shell. The natural frequencies of the lowest five circumferential wave numbers (i.e., n ¼ 0, 1, 2, 3, 4, 5) and the first two meridional modes are considered in the comparison. The material and geometrical properties of the shell are assumed to be: E ¼ 20.69 GPa, m ¼ 0.15, r ¼ 2405 kg/m3, ah ¼ 25.6 m, bh ¼ 63.91 m, Ch ¼ 18.59 m, Dh ¼ 82.19 m, Rs ¼ 0 m and h ¼ 0.127 m. It can be seen that the current results are in good correlation with the reference ones. The slight differences between the two solutions may be due to the fact that the deepness terms 1/ (1 þ z/R4) and 1/(1 þ z/Rq) were ignored in the reference paper. Table 4 gives the similar comparison for a FGM1 (b¼0/c/p¼0.6) elliptical shell (Rs ¼ 0, ae ¼ be ¼ 1 m, h ¼ 0.05 m, 40 ¼ p/8 and 41 ¼ p/2) with FeF and CeC boundary conditions. The current results are compared with those reported by Su et al. [52] by a simple FSDT theory. The comparison shows that the present results are close to the reference ones. Concerning the results for the free vibration of FGM shell with cycloidal meridian, no literature is available. Thus, in order to confirm the convergence and accuracy of the present method for the cycloidal shells, the last example is devoted to the convergence study and comparison for such shells. Table 5 lists the first six frequencies for an FeC restrained Al cycloidal shell (Rs ¼ 1 m, rc ¼ 1 m, h ¼ 0.05/0.1 m, 40 ¼ 0 and 41 ¼ p/2). Results are compared with those obtained by FEM commercial program ANSYS. In the ANSYS program, a 0.02 m size of mesh scheme with SHELL63 element type is employed. As can be seen from the comparison, a

good agreement between the two results is obtained. Comparing Tables 1e5, it is obvious that the frequencies converge monotonically as the truncated number increases. More numerical examples will be presented in the later subsections. In each case, the convergence study is performed and for brevity purposes, only the converged results are presented here.

3.2. Results for FGM doubly-curved shells of revolution with arbitrary boundary conditions Tables 1e5 indicate that the present solution is accurate and convergent. Having gained confidence in the proposed solution, some further results for FGM doubly-curved shells of revolution with various boundary conditions, material distributions and geometry dimensions are given in the following presentation, which may serve as benchmark values for the further researches. Unless otherwise stated, the details regarding the geometry of the shells under consideration are: 1. Circular toroidal shell: R ¼ 1 m, h ¼ 0.1 m, 40 ¼ p/3, 41 ¼ 5p/6, Rs ¼ 0 m. 2. Paraboloidal shell: ap ¼ 3 m, cp ¼ 1 m, k ¼ 4, h ¼ 0.1 m, Rs ¼ 0 m. 3. Hyperbolical shell: ah ¼ 1 m, dh ¼ 2 m, Ch ¼ 2 m, Dh ¼ 3 m, h ¼ 0.1 m, Rs ¼ 0 m. 4. Catenary shell: dc ¼ 2 m, h ¼ 0.1 m, 40 ¼ p/6, 41 ¼ p/3, Rs ¼ 0 m. 5. Cycloidal shell: rc ¼ 1 m, h ¼ 0.1 m, 40 ¼ p/12, 41 ¼ 5p/12, Rs ¼ 1 m. 6. Elliptical shell: ae ¼ 3 m, be ¼ 2 m, h ¼ 0.2 m, 40 ¼ p/3, 41 ¼ 5p/6, Rs ¼ 0 m.

240

G. Jin et al. / Composites Part B 89 (2016) 230e252

Table 4 Convergence and comparison of the frequencies (Hz) for an FGM1 (b¼0/c/p¼0.6) elliptical shell with FeF and CeC boundary conditions (Rs ¼ 0, ae ¼ be ¼ 1 m, h ¼ 0.05 m, 40 ¼ p/8 and 41 ¼ p/2, m ¼ 1). n

0 1 2 3 4

FeF

CeC

M ¼ 11

M ¼ 12

M ¼ 13

M ¼ 14

M ¼ 15

Ref. [52]

M ¼ 11

M ¼ 12

M ¼ 13

M ¼ 14

M ¼ 15

Ref. [52]

779.66 773.88 50.05 134.11 244.44

779.66 773.87 50.03 134.06 244.34

779.66 773.87 50.02 134.02 244.26

779.66 773.87 50.00 133.98 244.19

779.66 773.87 50.00 133.96 244.14

777.61 771.92 49.89 133.53 243.64

878.31 913.32 896.72 891.66 929.77

878.12 913.16 896.72 891.65 929.76

878.05 913.10 896.70 891.65 929.75

877.97 913.03 896.70 891.64 929.75

877.95 913.00 896.69 891.64 929.75

875.72 910.57 894.67 889.45 927.36

Table 5 Convergence and comparison of the first six frequencies (Hz) for an Al cycloidal shell (Rs ¼ 1 m, rc ¼ 1 m, 40 ¼ 0 and 41 ¼ p/2). Mode

1 2 3 4 5 6

h ¼ 0.05 m

h ¼ 0.10 m

M ¼ 11

M ¼ 12

M ¼ 13

M ¼ 14

M ¼ 15

ANSYS

M ¼ 11

M ¼ 12

M ¼ 13

M ¼ 14

M ¼ 15

ANSYS

66.101 66.101 77.007 83.476 83.476 116.06

66.052 66.052 76.954 83.450 83.450 115.95

66.008 66.008 76.953 83.445 83.445 115.86

65.976 65.976 76.927 83.431 83.431 115.79

65.947 65.947 76.927 83.426 83.426 115.73

66.206 66.227 76.924 83.595 83.595 116.18

91.483 94.222 94.222 103.63 103.63 143.92

91.472 94.182 94.182 103.60 103.60 143.92

91.471 94.150 94.150 103.59 103.59 143.92

91.466 94.124 94.124 103.57 103.57 143.91

91.465 94.102 94.102 103.56 103.56 143.91

91.521 94.814 94.830 104.07 104.07 144.05

Table 10 illustrates the results for a hyperbolical shell of revolution with different meridian characteristics and Table 11 studies the effects of the power-law index on the lowest three longitudinal mode frequencies (i.e., m ¼ 1, 2, 3) of an FeC hyperbolical shell made of FGM1 (b¼0/c/p) material (ah ¼ 1 m, dh ¼ 2 m, Ch ¼ 1 m, Dh ¼ 3 m, h ¼ 0.2 m, Rs ¼ 1 m). It is worth noting that the frequencies of the shell decrease as the powerlaw index increases. Another interesting observation is that when the power-law index is higher, the increment of the power-law index does not seem to affect the values of the frequencies. The reason for this lies in the fact that when the power-law index is set equal to infinity, the homogeneous isotropic material is obtained as a special case of FGMs. Tables 12 and 13 show the results for the catenary (Rs ¼ 0 and 4 m) and cycloidal (40 ¼ p/12 and p/4) shells with different geometry parameters, respectively. As expected, in most cases, the frequencies for the shells with FGM1 material distribution are larger than those of FGM2. However, for the catenary shell with axis offset Rs ¼ 4 m and FeS, SeS, SeC boundary conditions, the shell with FGM2 material distribution has higher frequencies.

Tables 6e15 show the fundamental mode frequencies (i.e., m ¼ 1) of the lowest five circumferential waves (i.e., n ¼ 0, 1, 2, 3, 4) for the shells mentioned above. The FGM1 (b¼0/c/p¼1) and FGM2 (b¼0/ c/p¼1) power-law distributions and various boundary conditions (FeS, FeC, SeS, SeC, CeC, CeF) are considered. Table 6 presents the frequencies for the circular toroidal shell having different meridional dimensions (i.e., 40 ¼ p/6, p/3 and p/2), while Table 7 gives results for the FGM1 (b¼0/c/p¼1) circular toroidal shell (i.e., 40 ¼ 2p/3, 41 ¼ p/3) with different thicknesses. From Table 6, it can be seen that the frequencies for the shell with FGM1 material distribution in general are larger than those of FGM2. Table 7 reveals that the frequencies of the shell increase with increasing thickness. It is attributed to the stiffness of the shell increases as the thickness increases. In addition, the two tables also show that shells with stronger restraint rigidity always result in higher frequencies. In Table 8, the frequencies of a paraboloidal shell characterized by the characteristic parameters k ¼ 2, 4 and 16 are considered, while Table 9 reports results for a paraboloidal shell (Rs ¼ 4 m) made of FGM1 (b¼0/c/p) material with various values of the power-law index. Table 9 indicates that as the power-law index increases, the frequencies decrease.

Table 6 Frequencies (Hz) for a circular toroid shell with different meridional dimensions (R ¼ 1 m, h ¼ 0.1 m, 41 ¼ 5p/6, Rs ¼ 0 m). 40

p/6

p/3

p/2

n

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

FGM1

FGM2

(b¼0/c/p¼1)

(b¼0/c/p¼1)

FeS

FeC

SeS

CeF

CeC

FeS

FeC

SeS

CeF

CeC

314.49 96.799 250.95 606.84 873.68 364.56 125.91 112.48 289.95 515.04 469.03 202.84 122.65 249.17 442.14

339.81 112.27 251.37 606.86 874.15 397.60 146.35 116.41 290.11 515.06 520.22 237.58 143.94 252.58 442.90

469.28 609.17 715.34 796.97 897.34 691.85 754.56 810.80 853.64 939.61 1078.3 1089.7 979.87 988.01 1087.0

339.81 112.27 251.37 606.86 874.15 576.92 313.46 269.68 610.28 901.39 834.08 599.20 377.02 640.13 985.35

521.66 609.21 716.07 799.13 899.33 784.46 806.82 820.38 866.38 954.71 1305.0 1138.0 1047.2 1063.6 1163.1

288.08 86.033 245.71 594.94 858.99 332.36 111.91 108.53 283.40 503.54 425.95 180.22 111.69 242.89 431.95

335.16 110.74 246.20 594.95 859.15 391.98 144.32 113.82 283.56 503.55 512.74 234.22 141.00 246.65 432.59

426.01 590.48 706.05 786.90 883.71 641.68 719.09 805.03 848.29 930.92 1010.3 1061.1 996.09 1000.0 1090.8

335.16 110.74 246.20 594.95 859.15 568.64 309.53 264.27 598.14 885.06 820.24 591.55 370.00 626.69 965.16

513.92 601.87 706.21 787.23 884.46 771.23 795.94 808.73 853.01 938.45 1282.4 1121.8 1030.7 1045.3 1141.3

G. Jin et al. / Composites Part B 89 (2016) 230e252 Table 7 Frequencies (Hz) for a circular toroid shell with various thicknesses (R ¼ 1 m, 40 ¼ 2p/3, 41 ¼ -p/3, FGM1 Rs

0m

2m

n

0 1 2 3 4 0 1 2 3 4

h/R ¼ 0.01

h/R ¼ 0.05

241

(b¼0/c/p¼1)).

h/R ¼ 0.10

h/R ¼ 0.15

FeC

SeS

CeC

FeC

SeS

CeC

FeC

SeS

CeC

FeC

SeS

CeC

609.34 374.92 163.29 72.399 74.788 661.27 512.55 382.53 296.30 245.63

737.06 763.93 819.32 831.70 832.81 724.38 696.49 625.72 548.81 495.96

759.04 781.62 826.73 840.12 837.23 758.37 725.50 646.55 562.89 503.74

670.56 404.61 197.14 181.27 299.54 709.03 521.34 408.67 362.32 369.27

847.03 872.41 932.80 922.97 916.44 853.46 826.69 770.19 731.68 743.58

973.75 996.24 975.89 934.69 934.08 1001.8 941.08 853.10 808.70 806.26

740.81 429.23 239.43 308.91 520.54 758.74 527.25 427.88 426.36 505.59

1074.5 1106.9 999.55 977.80 1027.7 988.58 923.16 849.61 841.16 901.92

1340.2 1156.3 1061.8 1050.8 1102.3 1089.5 1026.7 961.13 955.34 1011.2

810.00 449.36 286.30 424.54 707.34 780.97 535.20 449.34 488.11 631.44

1320.2 1126.3 1033.0 1053.8 1172.2 991.36 928.85 872.91 908.45 1038.8

1429.7 1227.8 1158.5 1182.7 1287.5 1186.2 1128.1 1082.7 1111.8 1218.4

Table 8 Frequencies (Hz) for a paraboloidal shell with different characteristic parameters (ap ¼ 3 m, cp ¼ 1 m, h ¼ 0.1 m, Rs ¼ 0 m). k

n

2

FGM1

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

4

16

FGM2

(b¼0/c/p¼1)

(b¼0/c/p¼1)

FeS

FeC

SeS

SeC

CeC

FeS

FeC

SeS

SeC

CeC

290.70 164.18 83.443 118.06 134.34 284.58 225.18 122.83 152.83 196.28 133.02 114.50 90.584 126.41 173.48

300.29 165.47 84.263 120.00 139.58 301.98 230.22 125.55 155.74 206.69 133.52 119.47 103.06 143.30 198.18

308.41 227.40 161.04 134.76 135.10 295.03 276.16 227.89 204.53 203.81 158.96 151.56 145.54 156.24 184.13

324.86 229.60 163.99 138.70 140.57 318.85 289.03 237.17 215.43 217.48 175.07 170.00 168.35 183.18 214.93

325.35 230.03 165.03 139.53 140.84 326.03 289.03 238.18 216.95 218.56 196.62 190.85 186.94 197.27 223.62

284.42 162.05 81.572 116.10 131.99 283.38 223.81 119.71 151.01 195.83 130.52 116.40 89.482 127.62 176.45

299.12 165.12 83.871 119.13 139.00 300.54 229.41 125.00 154.64 205.65 133.28 119.25 102.79 142.88 197.60

306.64 227.28 159.95 132.95 132.73 294.14 279.82 230.27 205.85 204.23 174.46 165.85 156.77 164.24 189.76

322.49 229.22 163.71 138.30 139.98 315.07 287.71 237.10 215.34 216.92 180.58 175.43 172.92 185.98 216.00

323.81 229.57 164.73 139.19 140.29 324.54 288.07 237.43 216.21 217.67 196.20 190.43 186.49 196.75 222.99

Table 9 Frequencies (Hz) for a paraboloidal shell with various thicknesses (ap ¼ 3 m, cp ¼ 1 m, k ¼ 4, Rs ¼ 4 m, FGM1 p

0.1

10

n

0 1 2 3 4 0 1 2 3 4

h ¼ 0.01 m

h ¼ 0.05 m

(b¼0/c/p)).

h ¼ 0.10 m

h ¼ 0.15 m

FeF

SeS

CeC

FeF

SeS

CeC

FeF

SeS

CeC

FeF

SeS

CeC

156.45 158.45 2.4747 7.9607 17.006 149.37 151.37 2.4862 8.0035 17.125

184.96 183.77 166.22 126.37 97.991 176.73 175.52 159.40 121.51 94.261

197.67 195.42 180.14 136.19 104.55 189.25 186.99 172.68 130.67 100.55

182.83 190.02 11.053 34.134 63.598 174.98 182.15 11.056 33.904 62.113

218.76 213.17 199.23 166.09 155.67 209.55 203.41 188.76 160.64 152.71

244.54 231.75 202.13 173.04 162.99 234.69 221.87 192.89 166.14 158.46

200.24 213.13 20.062 54.710 87.434 191.82 204.70 19.964 53.774 85.597

241.26 229.49 206.57 195.85 213.15 231.77 218.89 195.27 186.72 205.56

276.82 254.09 216.84 202.07 219.23 266.10 243.66 208.13 195.81 214.88

212.67 230.97 27.660 69.487 110.28 203.80 222.10 27.419 68.313 108.81

257.86 240.93 215.78 219.86 259.16 248.17 229.62 203.92 210.16 251.23

300.76 271.95 236.18 237.50 278.66 289.44 261.41 228.21 232.25 275.14

Table 10 Frequencies (Hz) for a hyperbolical shell with different meridian characteristics (ah ¼ 1 m, dh ¼ 2 m, h ¼ 0.1 m, Rs ¼ 0 m). C/D

C¼0 D¼5m

C¼2m D¼3m

n

0 1 2 3 4 0 1 2 3 4

FGM1

FGM2

(b¼0/c/p¼1)

(b¼0/c/p¼1)

FeS

FeC

SeS

SeC

CeC

FeS

FeC

SeS

SeC

CeC

299.48 123.87 84.722 136.13 192.81 261.49 95.498 87.568 130.67 210.90

299.90 124.18 84.920 138.55 200.88 261.51 95.500 87.876 130.70 210.90

470.54 225.81 133.59 139.14 192.84 480.37 271.15 179.64 188.38 239.87

504.33 227.66 135.37 142.02 200.93 525.33 274.39 182.22 198.53 264.92

504.33 228.10 135.93 142.06 200.93 525.51 274.58 182.24 198.79 264.92

296.92 123.25 83.515 132.57 187.01 261.37 96.368 87.581 129.46 208.81

299.87 124.59 84.371 137.21 198.97 262.10 96.539 88.092 129.57 208.81

464.44 223.61 130.00 135.36 187.04 484.74 268.14 176.86 181.80 230.93

500.93 226.82 133.71 140.67 199.02 523.05 273.36 182.20 197.04 262.66

501.91 228.22 136.07 140.99 199.02 523.88 274.77 182.45 197.64 262.68

242

G. Jin et al. / Composites Part B 89 (2016) 230e252

Table 11 Frequencies (Hz) for a hyperbolical shell with different material distributions (ah ¼ 1 m, dh ¼ 2 m, Ch ¼ 1 m, Dh ¼ 3 m, h ¼ 0.2 m, Rs ¼ 1 m). n

1 2 3

m

(b¼0/c/p),

FGM1

1 2 1 2 1 2

FeC

p¼0

p ¼ 0.01

p ¼ 0.1

p ¼ 0.5

p¼1

p¼2

p ¼ 10

p ¼ 100

p ¼ 1010

171.47 257.22 126.89 182.15 152.14 174.09

171.41 257.13 126.84 182.06 152.04 173.94

170.93 256.37 126.38 181.25 151.24 172.72

169.21 253.73 124.89 178.71 148.86 169.26

167.73 251.58 123.81 177.12 147.74 167.86

165.93 249.11 122.79 176.01 147.69 168.37

162.43 244.24 120.93 174.29 147.82 170.20

160.83 241.39 119.19 171.28 143.56 164.55

160.61 240.93 118.86 170.62 142.50 163.06

Table 12 Frequencies (Hz) for a catenary shell with different geometry parameters (dc ¼ 2 m, h ¼ 0.1 m, 40 ¼ p/6, 41 ¼ p/3). Rs

n

0m

0 1 2 3 4 0 1 2 3 4

4 m

FGM1

FGM2

(b¼0/c/p¼1)

(b¼0/c/p¼1)

FeS

FeC

SeS

SeC

CeC

FeS

FeC

SeS

SeC

CeC

327.39 261.17 143.18 154.20 225.65 193.57 116.37 81.967 81.001 98.943

344.44 267.53 147.58 157.39 232.73 193.70 118.83 86.295 83.580 99.202

354.84 338.87 287.94 261.76 261.50 274.10 260.31 233.94 219.82 228.28

389.27 357.34 301.32 277.07 279.79 274.71 261.88 238.47 226.64 234.23

402.45 357.66 303.94 280.81 283.10 310.06 289.88 257.15 241.13 248.20

324.65 259.27 138.91 151.54 224.29 193.92 117.62 84.814 83.007 99.176

342.46 266.44 146.92 156.12 230.97 193.94 118.20 85.839 83.349 99.181

351.80 343.65 292.66 265.16 263.57 276.24 266.52 246.03 233.35 241.15

383.46 356.13 301.80 277.54 279.50 276.28 266.54 247.03 236.22 244.21

400.48 356.21 302.74 279.60 281.69 310.49 290.05 256.99 240.89 248.11

Table 13 Frequencies (Hz) for a cycloidal shell with different geometry parameters (rc ¼ 1 m, h ¼ 0.1 m, 41 ¼ 5p/12, Rs ¼ 1 m). 40

p/12

p/4

n

0 1 2 3 4 0 1 2 3 4

FGM1

FGM2

(b¼0/c/p¼1)

(b¼0/c/p¼1)

FeS

FeC

SeS

SeC

CeC

FeS

FeC

SeS

SeC

CeC

134.63 139.72 77.262 74.303 114.37 198.90 190.01 167.80 141.80 121.28

138.24 141.12 80.811 75.093 114.37 211.33 204.60 188.11 169.44 155.34

180.90 187.45 205.32 228.11 248.41 478.89 478.43 477.36 476.48 476.75

197.11 203.01 218.83 238.44 254.59 531.02 530.86 530.63 530.98 532.66

224.96 229.94 242.89 256.40 265.19 653.97 653.51 652.44 651.47 651.41

133.52 139.10 75.692 73.777 113.93 200.24 190.66 166.34 137.42 114.60

137.83 140.78 80.458 74.789 113.93 210.73 203.95 187.33 168.51 154.28

178.10 185.68 206.28 231.72 252.07 549.62 548.65 546.13 543.07 540.67

197.50 204.04 221.77 243.64 259.40 570.33 569.80 568.53 567.33 567.15

223.63 228.61 241.58 255.11 263.91 648.77 648.30 647.17 646.12 645.97

Table 14 Frequencies (Hz) for an elliptic shell with different meridional dimensions (ae ¼ 3 m, be ¼ 2 m, h ¼ 0.2 m, 41 ¼ 5p/6, Rs ¼ 0 m). 40

p/6

p/3

p/2

n

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

FGM1

FGM2

(b¼0/c/p¼1)

(b¼0/c/p¼1)

FeS

FeC

SeS

SeC

CeC

FeS

FeC

SeS

SeC

CeC

119.92 66.511 54.449 130.60 216.99 148.10 89.048 44.863 76.757 135.26 177.87 120.04 67.185 72.333 119.16

133.30 76.616 56.246 130.61 217.00 169.89 103.93 54.432 77.922 135.31 204.40 141.59 88.151 83.292 122.79

182.38 211.66 278.93 350.25 409.99 300.45 315.14 354.56 405.74 455.14 492.60 493.76 489.64 486.73 497.24

199.42 226.29 289.97 359.16 417.97 344.58 357.47 393.99 443.49 487.89 577.38 555.74 528.24 518.35 525.76

219.68 244.37 304.79 372.20 429.99 390.95 402.61 433.63 467.90 493.87 581.63 566.56 546.51 540.40 549.68

111.79 60.645 53.079 128.84 214.03 137.02 80.968 40.316 75.325 133.08 165.35 109.42 57.987 68.846 117.18

131.82 75.399 55.434 128.85 214.03 167.58 102.21 53.310 76.650 133.12 202.36 139.61 86.449 81.591 120.58

168.13 201.55 275.59 351.31 412.03 281.17 298.76 345.37 404.25 457.24 475.96 484.62 500.65 504.92 514.01

190.41 219.33 286.59 358.10 417.13 329.75 343.84 383.03 433.78 476.88 565.66 557.84 534.63 524.26 530.77

216.27 240.98 301.14 367.60 424.04 383.00 394.67 425.71 460.00 485.65 573.16 558.06 537.86 531.52 540.42

G. Jin et al. / Composites Part B 89 (2016) 230e252 Table 15 Frequencies (Hz) for an FGM2 40

p/6

p/3

n

0 1 2 3 4 0 1 2 3 4

(b¼0/c/p¼1)

243

elliptic shell with various thicknesses (ae ¼ 3 m, be ¼ 2 m, 41 ¼ 5p/6, Rs ¼ 4 m).

h/ae ¼ 0.01

h/ae ¼ 0.05

h/ae ¼ 0.10

h/ae ¼ 0.15

FeC

SeS

CeC

FeC

SeS

CeC

FeC

SeS

CeC

FeC

SeS

CeC

201.25 98.136 91.178 77.097 89.187 273.62 223.42 166.53 167.25 178.40

258.11 279.73 214.68 207.64 210.73 316.31 300.30 267.56 239.42 241.48

301.54 302.79 236.64 215.66 221.12 358.64 336.23 291.18 258.26 254.10

220.08 103.13 108.82 152.50 221.96 338.04 259.96 223.19 331.15 512.09

316.06 322.80 332.44 420.74 529.54 414.50 392.19 385.26 442.39 533.50

391.26 365.80 361.81 433.74 555.19 501.08 463.11 434.22 474.36 567.78

233.14 108.00 136.87 234.85 370.37 374.91 283.23 297.35 524.85 836.58

354.82 360.55 428.44 585.30 789.14 484.61 470.00 498.63 614.07 797.24

452.52 428.56 467.86 623.28 838.12 602.52 546.48 534.34 653.22 848.43

242.92 112.34 166.62 312.01 499.24 398.85 296.33 360.09 680.45 1044.6

384.04 398.62 509.07 728.16 992.67 539.08 524.83 569.34 750.75 999.61

496.68 484.53 540.34 774.22 1040.8 666.39 586.79 604.78 792.79 1048.5

Fig. 5. Mode shapes for the FeC FGM1 circular toroidal shell of Table 6.

Fig. 6. Mode shapes for the SeS FGM1 paraboloidal shell of Table 8.

244

G. Jin et al. / Composites Part B 89 (2016) 230e252

Fig. 7. Mode shapes for the FeC FGM1 hyperbolical shell of Table 10.

Fig. 8. Mode shapes for the FeC FGM1 catenary shell of Table 12.

Fig. 9. Mode shapes for the CeC FGM1 cycloidal shell of Table 13.

G. Jin et al. / Composites Part B 89 (2016) 230e252

245

Fig. 10. Mode shapes for the CeC FGM1 elliptical shell of Table 14.

Then, Table 14 considers frequencies for an elliptical shell having different meridional dimensions (i.e., 40 ¼ p/6, p/3 and p/2), while Table 15 shows results for the FGM2 (b¼0/c/p¼1) elliptical shell having different thicknesses and meridional dimensions (i.e., 40 ¼ p/6 and p/3, Rs ¼ 4 m). It is noteworthy that the decrease of the meridional dimension always results in increases of shell frequencies. In addition, the frequencies of the shell increase as the thickness of the shell increases. In order to enhance our understanding on the vibrations of FGM doubly-curved shells of revolution, some selected mode shapes of the shells studied in Tables 6, 8, 10, 12e14 are given in Figs. 5e10. The mode shapes of all the shells under consideration are evaluated for the FGM1 (b¼0/c/p¼1) material distribution. Each of these mode shapes is reconstructed in 3D view by substituting the correspond eigenvector back into the admissible displacement functions and using the SURF tool of the MATLAB. The above examples are presented as FGM doubly-curved shells of revolution with classical boundary conditions. It is obvious that the frequencies and mode shapes for the shells with arbitrary classical boundary conditions and their combinations can be determined easily by the proposed solution. For the sake of completeness, FGM doubly-curved shells of revolution with elastic boundary conditions are considered in the following discussion. Fig. 11 shows the first and ninth longitudinal mode frequencies of the first circumferential wave number (i.e., n ¼ 1, m ¼ 1, 9) for the revolution shells with EIC boundary conditions. The shells under consideration are assumed to be made of FGM1 (b¼0/c/p¼1) and FGM2 (b¼0/c/p¼1) material. From the figure, it can be seen that the variation of the normal restraint rigidity ku has a great impact on the frequencies of the shells. The increase of ku in the stiffness rage of 101e105 results in a rapid increase of the frequencies, whereas the frequencies of the shells keep at a relatively constant value beyond this range. Fig. 12 shows the similar study for the shells with EIIC boundary conditions, a similar tendency as Fig. 11 can be seen in this figure. The two figures also certify that any classical boundary conditions can be obtained by setting the spring's stiffness to 107 or zero. The effects of the power-law index p and the material distribution profile parameters b and c are studied as well. Fig. 13 shows the first longitudinal mode frequencies of the first circumferential wave number (i.e., n ¼ 1, m ¼ 1) versus the power-law index p for the shells with FeC boundary conditions. The material distribution using in the study are FGM1 (b¼0/c/p) and FGM2 (b¼0/c/p). It is obvious that the frequencies for a shell made only of aluminum are smaller than those of Al2O3. It is attributed to the Young's modulus-to-

density ratio of the Al2O3 is higher than the aluminum. The figure also shows that the frequencies decrease as the power-law index p increases. In Fig. 14, the first longitudinal mode frequencies of the third circumferential wave number (i.e., n ¼ 3, m ¼ 1) versus the power-law index p for the FGM1 (b¼0/c/p) and FGM2 (b¼0/c/p) hyperbolical shell with various boundary conditions are reported. From the figure, we can see that for different power-law indexes, the frequencies are higher than those of aluminum and smaller than the Al2O3 ones. Finally, Fig. 15 shows the first longitudinal mode frequencies of the second circumferential wave number (i.e., n ¼ 2, m ¼ 1) versus the power-law index p for the FGM1 (b/c¼4/p) shells with CeC boundary condition and different material distribution profile parameters, i.e., b ¼ 1/4, 3/4 and 1, c ¼ 4. It is interesting to note that frequencies attain a minimum value for a shell made only of metal, due to the fact that aluminum has a much smaller Young's modulus than zirconia. Fig. 15(a)-(d) shows that the frequencies of the shells increase as the parameter b increases. For low values of the parameter p, the most of frequencies exhibits a fast descending behavior by varying the power-law index p from p ¼ 0 to p ¼ 10, while for values of p greater than p ¼ 10, frequencies slowly decrease by increasing the power-law exponent p and tend to the metal limit case. This is expected because the ceramic content decreases by increasing the index p and the FGM shell approaches the case of the fully metal shell. In Fig. 15(e)-(f), in particular for b ¼ 1, the frequencies do not present a variation tendency as previously described, but present a fast rising behavior up to a maximum value by increasing the power-law index p and exceed the ceramic limit case for some values of p. This is because the ceramic content decreases by increasing the index p. Consequently, the Young's modulus and density of the FGM decrease. Thus, the mass of the FGM shell decreases and approaches the case of the fully metal shell. However, the stiffness of the FGM shell not only depends on the Young's modulus of the FGM but also depends on the type of vibration mode [63]. In such case, the stiffness of the FGM shell decreases more slowly than the mass of the shell or it may not decrease. Thus the frequencies of the FGM shell can increases and larger than that of ceramic shell for some values of p. 4. Conclusions This paper introduces a unified modified Fourier series solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions. The circular toroidal,

1330

820 FGM1: m=1 FGM2: m=1

780

3060

760

3040

740

3020

720

3000

700

2980

680

640

FGM2: m=9

620 -2 10

10

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

1310

287

1300 1290

286

1280 285

1270 1260 1250

283

2940 FGM1: m=9

288

284

2960

660

1320

FGM2: m=1

Frequency (Hz)

Frequency (Hz)

800

FGM1: m=1

289

3080

FGM1: m=9

2920

FGM2: m=9

282

7

10

-2

10

-1

10

0

10

(a) Circular toroidal

2

10

3

10

4

10

5

10

6

10

7

1230

(b) Paraboloidal

280

358

FGM1: m=1

FGM2: m=1

356 900

240 870

1732 1730

Frequency (Hz)

880

1734

354

890 250

1736

FGM1: m=1

910

FGM2: m=1

260

Frequency (Hz)

10

Spring Parameter(log(k u ))

Spring Parameter(log(k u ))

270

1

1240

352 1728 350

1726

348

1724

346

1722

230 860 220 FGM1: m=9

210 10

FGM2: m=9 -2

10

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

1720

344

850

FGM1: m=9 FGM2: m=9

342

840

7

10

-2

10

-1

10

0

10

Spring Parameter(log(k u ))

1

1260

420

1250

400

FGM1: m=1

229 1240 228

3

10

4

10

5

10

6

10

7

FGM1: m=1

227

1220

226

1210

225 224 FGM1: m=9

223

1840

380

1820

360

1800

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

7

340

1760 300 280

1190

260

1180

1780

320

1200

FGM2: m=9 10

1716

FGM2: m=1

1230

Frequency (Hz)

Frequency (Hz)

10

(d) Catenary

FGM2: m=1

222 -2 10

2

Spring Parameter(log(k u ))

(c) Hyperbolical 230

10

1718

1740 1720

240

FGM1: m=9

1170

10

1700

FGM2: m=9

220 -2

10

-1

10

0

10

1

10

2

10

3

10

4

Spring Parameter(log(k u ))

Spring Parameter(log(k u ))

(e) Cycloidal

(f) Elliptical

10

5

10

6

10

7

1680

Fig. 11. Variations of the first and ninth frequencies of the FGM1 (b¼0/c/p¼1) and FGM2 (b¼0/c/p¼1) shells versus the elastic restraint parameter ku for the EI-C boundary conditions (n ¼ 1).

3100

FGM1: m=1

800

FGM2: m=1

288

3000

700

2900 2850

1320

286 1310

Frequency (Hz)

2950

1330

FGM1: m=1 FGM2: m=1

3050

750

Frequency (Hz)

290

650 2800

284 1300

282 280

1290

278 1280

2750

600

276

FGM1: m=9 FGM2: m=9 550 -2 10

10

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

FGM1: m=9 2700

FGM2: m=9

274

7

10

-2

10

-1

10

0

10

Spring Parameter(log(k v ))

1

2

10

3

10

4

10

5

10

6

10

7

Spring Parameter(log(k v ))

(a) Circular toroidal

(b) Paraboloidal 360

915

FGM1: m=1 FGM2: m=1

270

10

1270

FGM1: m=1 FGM2: m=1

910

1736

355 1734

905 260 350

895

250

890 240

Frequency (Hz)

Frequency (Hz)

900

885

1732

1730

345

1728 340 1726

880 230 FGM1: m=9 FGM2: m=9 220 -2 10

10

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

7

335

875

FGM1: m=9 FGM2: m=9

870

10

-2

10

-1

10

0

10

Spring Parameter(log(k v ))

1

2

10

3

10

4

10

5

10

6

10

7

Spring Parameter(log(k v ))

(c) Hyperbolical

(d) Catenary 410

230.2

FGM1: m=1

1255 1250

229.8 229.6

1245

229.4

1240

1850

FGM1: m=1

400

FGM2: m=1

230

FGM2: m=1 1800

390 380

1750

229.2

1235

229 1230 228.8

370 360

1700

350 1650

340 1225

228.6

330 FGM1: m=9

228.4

FGM2: m=9

228.2 10

Frequency (Hz)

Frequency (Hz)

10

1724

-2

10

-1

10

0

10

1

10

2

10

3

10

4

10

5

Spring Parameter(log(k v ))

(e) Cycloidal

10

6

10

7

1220 1215

FGM1: m=9

320 310 -2 10

1600

FGM2: m=9 10

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

7

1550

Spring Parameter(log(k v ))

(f) Elliptical

Fig. 12. Variations of the first and ninth frequencies of the FGM1 (b¼0/c/p¼1) and FGM2 (b¼0/c/p¼1) shells versus the elastic restraint parameter kv for the EII-C boundary conditions (n ¼ 1).

248

G. Jin et al. / Composites Part B 89 (2016) 230e252

Fig. 13. Variations of the first frequencies of the FeC supported FGM1

(b¼0/c/p)

and FGM2

(b¼0/c/p)

shells versus the power-law index p (n ¼ 1).

Fig. 14. Variations of the first frequencies of the FGM1 conditions (n ¼ 3).

(b¼0/c/p)

and FGM2

Fig. 15. Variations of the first frequencies of the CeC supported FGM1

(b¼0/c/p)

(b/c¼4/p)

hyperbolical shells versus the power-law index p for the FeS, FeC, SeS and CeC boundary

shells versus the power-law index p for various values of the parameter b (n ¼ 2, b ¼ 1/4, 3/4, 1).

250

G. Jin et al. / Composites Part B 89 (2016) 230e252

paraboloidal, hyperbolical, catenary, cycloidal and elliptical shells are covered. The material properties of the shells are assumed to vary continuously and smoothly along the normal direction according to general three-parameter power-law volume fraction functions. Specifically, the energy functional of the shells is expressed as a function of five displacement components by using the constitutive and kinematic relationships on the basis of the first order shear deformation shell theory. Regardless of boundary conditions, each of the displacement components is then expanded as the superposition of a standard cosine Fourier series and two supplementary functions introduced to remove any potential discontinuous at the ends. The general boundary conditions are accounted for by using the artificial spring technique. Finally, the solutions are obtained using the variational operation. The convergence and accuracy of the solution are validated by comparing the present results with those available in the literature. A good agreement is observed. A variety of new vibration results including frequencies and mode shapes for the shells with classical and elastic boundary conditions as well as different geometric and material parameters are presented, which may serve as benchmark solution for the future researches. The effects of the boundary

(

conditions, geometric and material parameters are also carried out. Even though this study is focused on the free vibration of FGM doubly-curved shells of revolution, the current solution can be readily applied to the static analysis (letting frequency parameter u equal to zero) and forced vibration analysis. When the forced vibration is involved, by adding the work done by the external loads in Eq. (24) and summing the loading vector F on the right side of Eq. (26), thus, the characteristic equation for the forced vibration can be readily obtained. Acknowledgment The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Nos.51175098 and 51279035). Appendix A. Detailed expressions for the matrices K, M and G The detailed expressions for the stiffness matrix K, mass matrix M and expansion coefficient vector G in Eq. (27) are given as follows:

 T

Z n o R4 c24 T R0 vUT vU vU vU R4 vUT vU R þ A12 c4 þ A22 þ A55 0 UT U dS þ U þ UT U U þ A66 ku40 UT U ds0 j4¼40 v4 R4 v4 v4 v4 R0 R0 vq vq R4 Z n o þ ku41 UT U ds1 j4¼41 ( vUT vV R4 c4 T vV vUT vV R4 c4 vUT þ A22 þ A66  A66 VgdS Kuv ¼ ∬ A12 U vq v4 vq R0 vq v4 R0 vq (

 R vUT vUT R4 s4 c4 T R vW dS Kuw ¼ ∬ A11 0 W þ A12 s4 W þ c4 UT W þ A22 U W  A55 0 UT v4 R4 v4 v4 R0 R4 (  o R4 c24 T R vUT vF vUT vF R4 vUT vF þ B12 c4 þ B22  A55 R0 UT F dS Ku4 ¼ ∬ B11 0 F þ c4 UT U F þ B66 v4 R4 v4 v4 v4 R0 R0 vq vq ( vUT vQ R4 c4 T vQ vUT vQ R4 c4 vUT þ B22 þ B66  B66 QgdS U Kuq ¼ ∬ B12 vq v4 vq R0 vq v4 R0 vq ( )  T Z n o R4 c24 T R4 s24 T R4 vVT vV R vVT vV vV vV þ A66 0  A66 c4 þ A66 V þ VT Kvv ¼ ∬ A22 V V þ A44 V V dS þ kv40 VT V ds0 j4¼40 v4 R0 vq vq R4 v4 v4 v4 R0 R0 Z n o þ kv41 VT V ds1 j4¼41 ( (

vVT R4 s4 vVT R4 s4 T vW vVT vF R4 c4 vVT dSKv4 ¼ ∬ B12 þ B22 W þ A22 W  A44 F Kvw ¼ ∬ A12 V vq vq R0 vq R0 vq v4 R0 vq

vVT vF R4 c4 T vF  B66 dS V þ B66 vq v4 vq R0 ( )  T R4 c24 T R4 vVT vQ R0 vVT vQ vV T vQ T Kvq ¼ ∬ B22 þ B66  B66 c4 þ B66 QþV V Q  A44 R4 s4 V Q dS v4 R0 vq vq R4 v4 v4 v4 R0 (

Z n o R4 s24 T R R4 vWT vW R vWT vW T þ A55 0 dS þ kw Kww ¼ ∬ A11 0 WT W þ 2A12 s4 WT W þ A22 W W þ A44 ds0 40 W W R4 R0 R0 vq vq R4 v4 v4 j4¼40 Z n o T þ kw ds1 41 W W Kuu ¼ ∬

A11

j4¼41

G. Jin et al. / Composites Part B 89 (2016) 230e252

( Kw4 ¼ ∬

B11 (

251



 R0 T vF vF R4 s4 c4 T vWT þ B12 c4 WT F þ s4 WT þ B22 F dS W W F þ A55 R0 v4 v4 R4 R0 v4

vQ R4 s4 T vQ vWT þ B22 þ A44 R4 QgdS W vq vq R0 vq ! ( o R4 c24 T R vFT vF vFT vF R4 vFT vF þ D12 c4 þ D22 þ A55 R0 R4 FT F dS F þ FT K44 ¼ ∬ D11 0 F F þ D66 v4 R4 v4 v4 v4 R0 R0 vq vq Z n Z n o o 4 4 FT F ds0 þ K41 FT F ds1 þ K40 j4¼40 j4¼41 ( vFT vQ R4 c4 T vQ vFT vQ R4 c4 vFT þ D22 þ D66  D66 QgdS K4q ¼ ∬ D12 F vq v4 vq R0 vq v4 R0 vq ! ( ) R4 c24 T R4 vQT vQ R0 vQT vQ vQT T vQ T þ D66  D66 c4 þ D66 Kqq ¼ ∬ D22 QþQ Q Q þ A44 R0 R4 Q Q dS v4 R0 vq vq R4 v4 v4 v4 R0 Z n Z n o o q q QT Q ds0 þ K41 QT Q ds1 þ K40 j4¼40 j4¼41 n o n o n o n o Muu ¼ ∬ I0 UT U dS; Mu4 ¼ ∬ I1 UT F dS; Mvv ¼ ∬ I0 VT V dS; Mvq ¼ ∬ I1 VT Q dS n o n o n o Mww ¼ ∬ I0 WT W dS; ; M44 ¼ ∬ I2 FT F dS; Mqq ¼ ∬ I2 QT Q dS Kwq ¼ ∬

B12 WT

where

U ¼ W ¼ F ¼ ½cos l0 4;/;cos lm 4;/;cos lM 4;P1 ð4Þ;P2 ð4Þcos nq V ¼ Q ¼ ½cos l0 4;/;cos lm 4;/;cos lM 4;P1 ð4Þ;P2 ð4Þsin nq u¼ ½A0 ;/;Am ;/;AM ;a1 ;a2 ejut ; 4¼ ½D0 ;/;Dm ;/;DM ;d1 ;d2 ejut v¼ ½B0 ;/;Bm ;/;BM ;b1 ;b2 ejut ; q¼ ½E0 ;/;Em ;/;EM ;e1 ;e2 ejut w¼ ½C0 ;/;Cm ;/;CM ;c1 ;c2 ejut

References [1] Leissa AW. Vibration of shells (NASA SP-288). Washington, DC: US Government Printing Office; 1973. [2] Kang JH. Field equations, equations of motion, and energy functionals for thick shells of revolution with arbitrary curvature and variable thickness form a three-dimensional theory. Acta Mech 2007;188:21e37. [3] Tornabene F. 2-D GDQ solution for free vibrations of anisotropic doublycurved shells and panels of revolution. Compos Struct 2011;93:1854e76. [4] Lam KY, Loy CT. Influence of boundary conditions and fibre orientation on the natural frequencies of thin orthotropic laminated cylindrical shells. Compos Struct 1995;31(1):21e30. [5] Liu B, Xing YF, Qatu MS, Ferreira AJM. Exact characteristic equations for free vibrations of thin orthotropic circular cylindrical shells. Compos Struct 2012;94:484e93. [6] Qu Y, Hua H, Meng G. A domain decomposition approach for vibration analysis of isotropic and composite cylindrical shells with arbitrary boundaries. Compos Struct 2013;95:307e21. [7] Warburton GB. Vibration of thin cylindrical shells. J Mech Eng Sci 1965;7: 399e407. [8] Bert CW, Malik M. Free vibration analysis of thin cylindrical shells by the differential quadrature method. ASME J Press Vess Technol 1996;118:1e12. [9] Shu C. An effective approach for free vibration analysis of conical shells. Int J Mech Sci 1996;38:935e49. [10] Tong L. Free vibration of orthotropic conical shells. Int J Eng Sci 1993;31: 719e33. [11] Kalnins A. On vibrations of shallow spherical shells. J Acoust Soc Am 1961;33: 1102e7. [12] Toorani MH, Lakis AA. General equations of anisotropic plates and shells including transverse shear deformations, rotary inertia and initial curvature effects. J Sound Vib 2000;237:561e615. [13] Qatu MS. Vibration of laminated shells and plates. San Diego: Elsevier; 2004. [14] Reddy JN. Exact solutions of moderately thick laminated shells. J Eng Mech 1984;110(5):794e809. [15] Reddy JN, Chandrashekhara K. Non-linear analysis of laminated shells including transverse shear strains. AIAA J 1985;23:440e1.

[16] Khdeir AA, Reddy JN, Frederick D. A study of bending, vibration and buckling of cross-ply circular cylindrical shells with various shell theories. Int J Eng Sci 1989;27(1):1337e51. [17] Soldatos KP, Messina A. The influence of boundary conditions and transverse shear on the vibration of angle-ply laminated plates, circular cylinders and cylindrical panels. Comput Method Appl Mech Engrg 2001;190:2385e409. [18] Ferreira AJM, Roque CMC, Jorge RMN. Nature frequencies of FSDT cross-ply composite shells by multiquadrics. Compos Struct 2007;77:296e305. [19] Qu Y, Long X, Wu S, Meng G. A unified formulation for vibration analysis of composite laminated shells of revolution including shear deformation and rotary inertia. Compos Struct 2013;98:169e91. [20] Soldatos KP, Messina A. Vibration studies of cross-ply laminated shear deformable circular cylinders on the basis of orthogonal polynomials. J Sound Vib 1998;218(2):219e43. [21] Ich Thinh T, Nguyen MC. Dynamic stiffness matrix of continuous element for vibration of thick cross-ply laminated composite cylindrical shells. Compos Struct 2013;98:93e102. [22] Ferreira AJM, Roque CMC, Jorge RMN. Modelling cross-ply laminated elastic shells by a higher-order theory and multiquadrics. Comput Struct 2006;84(19 and 20):1288e99. [23] Zhao X, Ng TY, Liew KM. Free vibration of two-side simply-supported laminated cylindrical panels via the mesh-free kp-Ritz method. Int J Mech Sci 2004;46(1):123e42. [24] Shu C, Du H. Free vibration analysis of laminated composite cylindrical shells by DQM. Compos Part B-Eng 1997;28(3):267e74. [25] Ng TY, Li H, Lam KY. Generalized differential quadrature for free vibration of rotating composite laminated conical shells with various boundary conditions. Int J Mech Sci 2003;45:567e87. € The determination of frequencies of laminated conical shells via [26] Civalek O. the discrete singular convolution method. J Mech Mater Struct 2006;1(1): 163e82. € Numerical analysis of free vibrations of laminated composite [27] Civalek O. conical and cylindrical shells: discrete singular convolution (DSC) approach. J Comput Appl Math 2007;205:251e71. [28] Ferreira AJM, Carrera E, Cinefra M, Roque CMC, Polit O. Analysis of laminated shells by a sinusoidal shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations. Compos Part BeEng 2011;42(5):1276e84. [29] Jin G, Ye T, Chen Y, Su Z, Yan Y. An exact solution for the free vibration analysis of laminated composite cylindrical shells with general elastic boundary conditions. Compos Struct 2013;106:114e27. [30] Jin G, Ye T, Ma X, Chen Y, Su Z, Xie X. A unified approach for the vibration analysis of moderately thick composite laminated cylindrical shells with arbitrary boundary conditions. Int J Mech Sci 2013;75:357e76. [31] Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis. Florida: CRC Press; 2003. [32] Qatu MS. Recent research advances in the dynamic behavior of shells: 1989e2000. Part 1: laminated composite shells. Appl Mech Rev 2002;55(4): 325e49.

252

G. Jin et al. / Composites Part B 89 (2016) 230e252

[33] Qatu MS. Recent research advances in the dynamic behavior of shells: 1989e2000. Part 2: homogeneous shells. Appl Mech Rev 2002;55(5): 415e34. [34] Qatu MS, Sullivan RW, Wang WC. Recent research advances on the dynamic analysis of composite shells: 2000e2009. Compos Struct 2010;93(1):14e31. [35] Carrera E. Historical review of zigezag theories for multilayered plates and shells. Appl Mech Rev 2003;56(3):287e308. [36] Liew KM, Zhao X, Ferreira AJM. A review of meshless methods for laminated and functionally graded plates and shells. Compos Struct 2011;93(8): 2031e41. [37] Viola E, Tornabene F. Free vibration of three parameter functionally graded parabolic panels of revolution. Mech Res Commun 2009;36:587e94. [38] Tornabene F, Viola E. Free vibration analysis of functionally graded panels and shells of revolution. Meccanica 2009;44:255e81. [39] Tornabene F, Liverani A, Caligiana G. FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: a 2-D GDQ solution for free vibration. Int J Mech Sci 2011;53:446e70. [40] Tornabene F, Viola E. Free vibration of four-parameter functionally graded parabolic panels and shells of revolution. Eur J Mech A-Solid 2009;28: 991e1013. [41] Edalat P, Khedmati MR, Guedes Soares G. Free vibration analysis of open thin deep shells with variable radii of curvature. Meccanica 2014;49:1385e405. [42] Tan DY. Free vibration analysis of shells of revolution. J Sound Vib 1998;213(1):15e33. [43] Nasir AM, Thambiratnam DP, Butler D, Austin P. Dynamics of axisymmetric hyperbolic shells structures. Thin Wall Struct 2002;40:665e90. [44] Kang JH, Leissa AW. Free vibration analysis of complete paraboloidal shells of revolution with variable thickness and solid paraboloids from a threedimensional theory. Comput Struct 2005;83:2594e608. [45] Kang JH, Leissa AW. Three-dimensional vibration analysis of thick hyperboloidal shells of revolution. J Sound Vib 2005;282:277e96. [46] Leissa AW, Kang JH. Three-dimensional vibration analysis of thick shells of revolution. J Eng Mech 1999;125:1365e71. [47] Ye T, Jin G, Su Z. Three-dimensional vibration analysis of laminated functionally graded spherical shells with general boundary conditions. Compos Struct 2014;116:571e88. [48] Su Z, Jin G, Ye T. Three-dimensional vibration analysis of thick functionally graded conical, cylindrical shell and annular plate structures with arbitrary elastic restraints. Compos Struct 2014;118:432e47.

[49] Li WL. Free vibrations of beams with general boundary conditions. J Sound Vib 2000;237(4):709e25. [50] Li WL. Dynamic analysis of beams with arbitrary elastic supports at both ends. J Sound Vib 2001;246(4):751e6. [51] Brebbia CA, Tottenham H, Warburton GB, Wilson J. Vibrations of engineering structures. Berlin Heidelberg: Springer-Verlag; 1985. [52] Su Z, Jin G, Shi S, Ye T. A unified accurate solution for vibration analysis of arbitrary functionally graded spherical shell segments with general end restraints. Compos Struct 2014;111:271e84. [53] Qu Y, Long X, Yuan G, Meng G. A unified formulation for vibration analysis of functionally graded shells of revolution with arbitrary boundary conditions. Compos Part B-Eng 2013;50:381e402. [54] Tornabene F, Fantuzzi N, Bacciocchi M. Free vibrations of free-form doublycurved shells made of functionally graded materials using higher-order equivalent single layer theories. Compos Part B-Eng 2014;67:490e509. [55] Xie X, Zheng H, Jin G. Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions. Compos Part B-Eng 2015;77:59e73. [56] Wu CP, Chang YT. A unified formulation of RMVT-based finite cylindrical layer methods for sandwich circular hollow cylinders with an embedded FGM layer. Compos Part B-Eng 2012;43(8):3318e33. [57] Sofiyev AH. On the vibration and stability of shear deformable FGM truncated conical shells subjected to an axial load. Compos Part B-Eng 2015;80:53e62. [58] Kim YW. Free vibration analysis of FGM cylindrical shell partially resting on Pasternak elastic foundation with an oblique edge. Compos Part B-Eng 2015;70:263e76. [59] Hui D. Effects of geometric imperfections on large-amplitude vibrations of rectangular plates with hysteresis damping. J Appl Mech 1984;51(1): 216e20. [60] Hui D. Large-amplitude vibrations of geometrically imperfect shallow spherical shells with structural damping. AIAA J 1983;21(12):1736e41. [61] Hui D. Asymmetric postbuckling of symmetrically laminated cross ply, short cylindrical panels under compression. Compos Struct 1985;3(1):81e95. [62] Kireitseu M, Hui D, Tomlinson G. Advanced shock-resistant and vibration damping of nanoparticle-reinforced composite material. Compos Part B-Eng 2008;39(1):128e38. [63] Tornabene F. Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution. Comput Method Appl Mech Engrg 2009;198:2911e35.