Vibration analysis of ring-stiffened conical–cylindrical–spherical shells based on a modified variational approach

International Journal of Mechanical Sciences 69 (2013) 72–84

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Vibration analysis of ring-stiffened conical–cylindrical–spherical shells based on a modified variational approach Yegao Qu n, Shihao Wu, Yong Chen, Hongxing Hua State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 23 December 2011 Received in revised form 24 December 2012 Accepted 14 January 2013 Available online 29 January 2013

In this paper, free vibration characteristics of conical–cylindrical–spherical shell combinations with ring stiffeners are investigated by using a modified variational method. Reissner–Naghdi’s thin shell theory in conjunction with a multilevel partition technique, viz., stiffened shell combination, shell component and shell segment, is employed to formulate the theoretical model. The displacement fields of each shell segment are expressed as a product of orthogonal polynomials along the meridional direction and Fourier series along the circumferential direction. The ring stiffeners in shell combinations are treated as discrete elements. Convergence and comparison studies for both non-stiffened and stiffened conical–cylindrical–spherical shells with different boundary conditions (e.g., free, clamped and elastic supported boundary conditions) are carried out to verify the reliability and accuracy of the present solutions. Some selected mode shapes are illustrated to enhance the understanding of the research topic. It is found the present method exhibits stable and rapid convergence characteristics, and the present results, including the natural frequencies and the mode shapes, agree closely with those solutions obtained from the finite element analyses. The effects of the number and geometric dimensions of ring stiffeners on the natural frequencies of a submarine pressure hull are also investigated. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Conical–cylindrical–spherical shell Modified variational method Ring stiffener Discrete element Free vibration

1. Introduction The ring-stiffened conical–cylindrical–spherical shell is a structure combination of considerable technical significance. Some important applications of this type of shell combination are found in missiles, rockets, naval hulls of submarines, etc. It is well known that these structures often operate in complex environmental conditions and are exposed to a variety of dynamic excitations which may result in excessive noise levels, vibration and fatigue damage. Therefore, the vibration characteristics of ring-stiffened shell combinations must be known in order to avoid the destructive effect of resonances. However, these shell combinations offer challenging analysis and design problems not only due to the degree of complexity of the governing shell equations, but also due to the difficulties associated with matching the interface conditions between the substructures, such as the continuity conditions on the shell–shell and shell– stiffener junctions. Thus, it is necessary to develop an analytical method for evaluating the vibration characteristics of the stiffened shell combinations. During the past few decades, many researchers have indulged themselves in the vibration analysis of various shell structures, and a large number of analytical and numerical methods have

n

Corresponding author. Tel.: þ86 21 34206332; fax: þ86 21 34206814. E-mail addresses: [email protected], [email protected] (Y. Qu).

0020-7403/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2013.01.026

been proposed. Survey articles and monographs oriented to such contributions may be found in Liew et al. [1], Qatu [2], and Leissa [3]. It should be noted, however, that a great majority of the works reported in the literature have been concerned the vibration problems of elementary shell configurations, such as cylindrical, conical and spherical shells, rather than shell combinations. In order to properly focus on the features and emphasis of the present paper, a brief review of the works pertaining to the vibration analyses of shell combinations, including the conical– cylindrical shell, the cylindrical–spherical shell and shell–plate combinations is reported below. Hu and Raney [4] presented a multi-segment integration technique for free vibration analysis of joined conical–cylindrical shells. They compared their analytical solutions with experimental results, and good agreement was obtained. Irie et al. [5] studied the dynamic behaviors of joined conical–cylindrical and annular plate–cylindrical shell structures using the transfer matrix method. Efraim and Eisenberger [6] proposed a dynamic stiffness matrix method for predicting the vibration characteristics of a coupled conical–cylindrical shell, and obtained the exact vibration frequencies. Caresta and Kessissoglou [7] presented a wave solution in conjunction with a power-series expansion method to describe the vibration characteristics of joined conical–cylindrical shells. Kang [8] investigated the free vibration of joined thick conical–cylindrical shells with variable thickness using a three-dimensional Ritz method. Qu et al. [9] proposed a variational method to deal with the free

Y. Qu et al. / International Journal of Mechanical Sciences 69 (2013) 72–84

vibration problems of cylindrical–conical shell combinations with classical and non-classical boundary conditions. Patel et al. [10] performed a finite element analysis for vibration analysis of laminated composite conical–cylindrical shells based on a twonoded shear flexible axisymmetric shell element. Benjeddou [11] developed a local–global B-spline finite element method to analyze the modal properties of joined shells of revolution. The vibration results of a hermetic capsule, a truncated cone–cylinder combination and a hermetic can were obtained and discussed. Tavakoli and Singh [12] investigated the free vibrations of joined/ hermetic shell structures based on a state space method, in which the differential equations are solved for each shell substructure by using the pade´ approximation for matrix exponentiation. The natural frequencies of cylindrical shells clamped at one end and closed at the other by different types of shells of revolution were determined by Galletly and Mistry [13] using the finite difference method. Cheng and Nicolas [14] performed a free vibration analysis of a structure consisting of a finite circular cylindrical shell closed at one end by a circular plate on the basis of a variational method. The continuity constraints on the shell–plate interface were imposed by means of continuous distributed artificial springs. Lee and Choi [15] analyzed the free vibrations of a simply supported cylindrical shell with an interior rectangular plate by using the receptance method. Based on the transfer matrix method, the natural frequencies and mode shapes for a conical shell with an annular end plate were investigated by Liang and Chen [16]. Redekop [17] carried out a free vibration analysis for a thin torus-cylinder shell assembly using the differential quadrature method. Shang [18] presented an analytical solution to obtain the vibration frequencies of hermetic capsule structures. The junction conditions between the spherical and cylindrical shells were given by the continuity of deformation and the equilibrium relations, and exact vibration frequencies were obtained by using the Legendre and trigonometric functions. Lee et al. [19] investigated the vibration characteristics of joined spherical–cylindrical shells with different boundary conditions. The theoretical results were calculated by the Rayleigh–Ritz method and compared with those solutions obtained from finite element analyses and experimental results. It should be mentioned that all the references cited above have hitherto focused on the free vibrations of joined shells or shell–plate combinations without ring stiffeners. Stiffeners are often used to strengthen a shell combination, and the effect of ring stiffeners upon the vibration behaviors of the shell is known to be significant. Two approaches, i.e., the smearing method and the discrete element method, are commonly used to deal with the ring stiffeners in a shell, depending upon whether the ring stiffeners are treated by averaging their properties over the surface of the shell or by considering them as discrete elements. In the smearing method, stiffeners in a shell are assumed to be smeared over the shell and the stiffened shell is modeled as an equivalent orthotropic shell [20]. This approach provides

wc

wl

uc

ul

reasonably accurate vibration solutions for shells having a large number of closely spaced stiffeners with equal strength. However, as the stiffener spacing increases or the wavelength of vibration becomes smaller than the stiffener spacing, one should not expect the smearing method to produce accurate predictions for dynamic characteristics of stiffened shells. Caresta and Kessissoglou [21] analyzed the vibroacoustic responses of a complex stiffened vessel based on the smearing stiffener method. The vessel is modeled as a cylindrical shell with internal bulkheads and ring stiffeners, and the cylindrical shell is closed by truncated conical shells, which in turn are closed at each end using circular plates. A more advanced treatment of the stiffeners for stiffened shells is the discrete element method [22–24], which may be regarded as more general in the sense that the stiffeners may be few or many in number, non-uniform or uniform in size, and non-uniformly or uniformly spaced. Qu et al. [25] presented a modified variational method for free and forced vibration analysis of ring-stiffened conical–cylindrical shells subjected to different boundary conditions, and used the discrete element stiffener theory to consider the ringstiffening effects. In views of the accuracy and efficiency of the modified variational method, it offers an appropriate analytical tool for linear vibration problems of ring-stiffened conical–cylindrical shell combinations having arbitrary boundary conditions and dynamic loads. In the present study, the modified variational method proposed by Qu et al. [25,26] is further extended to the free vibration analysis of joined conical–cylindrical–spherical shells with ring stiffeners. Reissner–Naghdi’s thin shell theory in conjunction with a multilevel partition technique, viz., stiffened shell combination, shell component and shell segment, is employed to formulate the theoretical model. The displacement components of each shell segment are expanded in the form of a double mixed series, i.e., Fourier series and orthogonal polynomials. The rings in the shell combination are treated as discrete elements, and the geometric and material characteristics of the rings may be different from one another. To test the convergence, efficiency and accuracy of the present method, free vibrations of non-stiffened and stiffened conical–cylindrical–spherical shell combinations are examined under different support conditions, namely the free, clamped and elastic supported boundary conditions. The theoretical results are verified by comparing the present numerical results with those obtained from finite element analyses. Some selected mode shapes are illustrated to enhance the understanding of the research topic. The effects of number and geometric dimension of the ring stiffeners on the vibration behaviors of a submarine pressure hull are also investigated.

2. Theoretical formulation A typical ring-stiffened conical–cylindrical–spherical shell combination under consideration is illustrated in Fig. 1. It is assumed that all the shell components and ring stiffeners are

hl

us

si

o

α

R

R0

o′ i r

b eri

hri

o′′

xri

hc

vl

ws

ϕ

x

73

vc

ϕ0

hs L0

Fig. 1. Geometry and coordinate systems of a stiffened conical–cylindrical–spherical shell.

θ

74

Y. Qu et al. / International Journal of Mechanical Sciences 69 (2013) 72–84

made of homogeneous and isotropic materials. The reference surface of the shell combination is taken to be at its middle surface where orthogonal coordinate systems are fixed. The conical shell component is described with a o  s,y,z coordinate system, in which s is measured along the generator of the cone starting at the cone vertex o, and y is the circumferential coordinate. a is the semi-vertex angle of the cone; R0 is the radius of the cone at its small edge and hc is the shell thickness. A cylindrical coordinate system (o0 ,x,y,z) is considered for the cylindrical shell component, which has radius R, length L0 and thickness hl. The cylindrical shell is circumferentially stiffened by NR rings, which may be placed internally or externally. The ith ring stiffener is located at xir measured from the left end of the cylindrical shell. For the spherical shell component, the coordinates along the meridional and circumferential directions are denoted as j and y, respectively. The angle formed by the extended normal to the surface and the axis of rotation is defined as the meridional angle j. Specially, the spherical shell may have a central cutout with an open angle of j0. The displacement components of each shell component with reference to its coordinate system are presented as ud, vd and wd(d ¼c, l, s), where the subscripts c, l and s represent the conical, cylindrical and spherical shell, respectively. Accordingly, the material parameters for the shell components are denoted by Young’s modulus Ed, mass density rd and Poisson’s ratio md. The material properties of the ith ring stiffener are defined as Young’s modulus Eir , mass density rir and Poisson’s ratio mir . In what follows, the modified variational method proposed by Qu et al. [25,26] for dynamic analysis of elementary shells and shell combinations is extended to deal with the free vibration problems of conical–cylindrical–spherical shells with ring stiffeners. In doing so, a stiffened shell combination is preliminary divided into its components, i.e., conical, stiffened-cylindrical and spherical shells, along the locations of junctions; then these shell components are further decomposed into Nc conical, Nl cylindrical and Nq spherical shell segments along the meridional directions of the these shell components to accommodate the computing requirements of high-order vibration modes. The problem under study will now be formulated according to a modified variational functional of the shell combination. 2.1. Energy functional of ring-stiffened cylindrical shell With regard to the stiffened cylindrical shell component, the stiffening and inertial effects of each ring stiffener are treated in a discrete manner. The energy functional Pl for the stiffened cylindrical shell component includes the energy contribution from both the cylindrical shell and the ring stiffeners, expressed as

Pl ¼ Pl þ

Z

t0 t0

Nl X Nr X

m ðT m r U r Þdt

ð1Þ

r ¼1m¼1

where Pl denotes the energy functional of the non-stiffened cylindrical shell; the integral expression in the right-hand side of Pl represents the total kinetic energy and strain energy of the ring stiffeners. Nl is the number of segments decomposed in the cylindrical shell component, whereas Nr indicates the number of rings in the rth shell segment. For more details about the expression of Pl , we refer the reader to [25]. The kinetic energy Tm and strain energy U m for the mth ring stiffener in rth r r cylindrical shell segment are given by [25] (  m 2 ) Z h i   @w _r 1 m 2p m m 2 m 2 m 2 m m m _ _ _ T r ¼ rr Ar ðu r Þ þ ðv r Þ þ ðw r Þ þ Ix,r þ Iz,r R dy 2 @x 0 ð2Þ

and Um r

¼

1

Z 2p

2R

0

m þ Em r Ar

2

2 m m 4Em Im @wr þ 1 @ ur r z,r @x R @y2



@vm r wm r @y

2

m þ Gm r Jr

!2 þ

m Em r I x,r 2

wm r þ

1 @2 wm r

R @y2 !2 3 @2 wm 1 @um r r 5d y  þ @x@y R @y R

!2

ð3Þ

where the dot above a variable represents differentiation with is the respect to time. R is defined by R ¼ R þ er,m , and em r eccentricity (distance from stiffener centroid to the shell middle m surface, Fig. 1) of the ring stiffener. Im x,r and Iz,r are the second moments of areas about the x and z axes of the rings, respectively; m m Am r is the cross-sectional area; Gr is shear modulus and J r is polar moment of inertia. The relation between the displacement comm m ponents (um r , vr , wr ) of the mth ring centroid and the displacement components (ul, vl, wl) of the middle surface of the rth cylindrical shell segment at the cross-section where the ring is attached is expressed as follows:   em em @wl m @wl r , vm vl þ r um wm ð4Þ r ¼ ul þer r ¼ 1þ r ¼ wl @x R R @y Substituting Eq. (4) into Eqs. (2), (3), one obtains the kinetic m energy T m r and strain energy U r in Eq. (1) written in the form of the displacements of cylindrical shell segments. 2.2. Energy functional of conical shell Similarly, the conical shell component is divided into Nc shell segments along the meridional direction, and the energy functional for the conical shell component is defined by Z t1 X Z t1 X Nc Pc ¼ ðT c,i U c,i Þdt þ Pc, lk dt ð5Þ t0

i¼1

t0

i,i þ 1

where Tc,i and Uc,i are the kinetic energy and strain energy of the ith conical shell segment (see Ref. [25]), respectively. Pc,lk represents the interface potential between adjacent conical shell segments (i) and (iþ1), expressed as [9] Z Pc, lk ¼ fBu Ns Yu þ Bv Nsy Yv þ Bw Q s Yw Br Ms Yr gdl l Z 1 ðB ku Y2u þ Bv kv Y2v þ Bw kw Y2w þ Br kr Y2r Þdl ð6Þ  2 l u where Ns,N sy ,Q s and Ms are internal force and moment resultants (for more details, see [25]). Yu, Yv, Yw and Yr are the essential continuity equations on adjacent conical shell segments or prescribeddisplacement boundaries, defined as Yu ¼ uic uicþ 1 ¼ 0, Yv ¼ vic vicþ 1 ¼ 0, Yw ¼ wic wicþ 1 ¼ 0 and Yr ¼ @wic =@s@wicþ 1 =@s ¼ 0. ku, kv,kw and kr are pre-assigned weighted parameters associated with continuity equations. Bt(t¼ u,v,w,r) are the parameters which define the continuity equations on interconnecting boundaries. For the case of two adjacent shell segments, Bt ¼1; while for the case of boundary conditions, values of Bt are defined in Table 1. An arbitrary set of classical boundary conditions at the end of the conical shell component can be obtained by an appropriate choice of the values of Bt. In Table 1, the shear-diaphragm boundary condition is defined as: uc a0, vc ¼wc ¼0 and qwc/qxa0, whereas the simply-supported boundary condition is designated as: uc ¼vc ¼wc ¼ 0 and qwc/qxa0. In practical engineering applications, the accepted fact is that the boundary conditions of a stiffened shell combination may be not classical in nature. It is of interest to note that the weighted parameters in Eq. (6) may be viewed as physical stiffnesses of per unit length of elastic foundation along line translational and/or line rotational directions. Hence, elastic support conditions can be easily incorporated in the present model by removing the first integral expression of Pc,lk.

Y. Qu et al. / International Journal of Mechanical Sciences 69 (2013) 72–84

Table 1 Values of Bt (t ¼u,v,w,r) for classical boundary conditions. Boundary condition

Bu

Bv

Bw

Br

Free Shear-diaphragm Simply-supported Clamped

0 0 1 1

0 1 1 1

0 1 1 1

0 0 0 1

2.3. Energy functional of spherical shell The spherical shell component is divided into Nq shell segments in the j direction, and the energy functional for the spherical shell is defined by

Ps ¼

Z

Nq t1 X t0

Z ðT s,i U s,i Þdt þ

i¼1

t1

t0

X

Ps, lk dt

ð7Þ

i,i þ 1

where Ts,i and Us,I are the maximum kinetic energy and strain energy of the ith spherical shell segment, respectively. Ps,lk is the interface potential between adjacent spherical shell segments (i) and (iþ1). Since the shell combination under consideration is thin, the maximum kinetic energy of the ith spherical shell segment is given as ZZ 1 T s,i ¼ r hs ½ðu_ is Þ2 þðv_ is Þ2 þ ðw_ is Þ2 R2 Sj dy dj ð8Þ 2 Si s where the dot above a variable represents differentiation with respect to time. Si is the middle surface area of the spherical shell segment, and Sj is defined as Sj ¼ sin j. According to Reissner–Naghdi’s linear shell theory, the maximum strain energy of the ith spherical shell segment can be expressed as  ZZ  Ki 1ms U s,i ¼ s ðejy Þ2 R2 Sj dy dj ðej Þ2 þ ðey Þ2 þ 2ms ej ey þ 2 2 Si  i ZZ  D 1ms ðwjy Þ2 R2 Sj dy dj þ s ðwj Þ2 þðwy Þ2 þ2ms wj wy þ 2 2 Si ð9Þ K is and

Dis are

the extensional and flexural rigidities of the where ith spherical shell segment, expressed as K is ¼ Es hs =ð1m2s Þand 3 Dis ¼ Es hs =½12ð1m2s Þ, respectively. For the ith spherical shell segment, the membrane strains (ej,ey,ejy) and curvatures (wj,wy,wjy) of middle surface are given as     1 @uis 1 @vi C j uis þ s þSj wis , ej ¼ þ wis , ey ¼ ~ R @j @y R   1 @uis @ vis 1 @yj ejy ¼ ~ þ Sj , , wj ¼ @j R~ R @j R @y     1 @yy 1 @yj @ yy wy ¼ ~ þC j yj , wjy ¼ þ Sj ð10Þ @j R~ R @y R~ @y in which   1 @wis uis  yj ¼ , R @j

1



@wi



yy ¼ ~ Sj vis  s , R~ ¼ RSj , C j ¼ cos j @y R ð11Þ

A combination of a modified variational principle and the least-squares weighted residual method is employed to obtain the interface potentials. In doing so, Ps,lk is written as Z Ps, lk ¼ ðNj Yu þN jy Yv þ Q j Yw Mj Yr Þdl l Z 1 2 2 2 2  ðk~ u Yu þ k~ v Yv þ k~ w Yw þ k~ r Yr Þdl ð12Þ 2 l

75

where Yu , Yv , Yw and Yr are the essential continuity equations on the interface of adjacent spherical shell segments (i) and (iþ1), written by: Yu ¼ uis uisþ 1 , Yv ¼ vis visþ 1 , Yw ¼ wis wisþ 1 and Yr ¼ ½ð@wis =@juis Þð@wisþ 1 =@juisþ 1 Þ=R. Nj, Njy , Q j and Mj are the force and moment resultants of the ith spherical segment. These force and moment resultants are obtained by integrating the stresses over the shell thickness, defined as !   1ms Di Nj ¼ K is ej þ ms ey , Njy ¼ K is ejy þ s wjy , 2 R     @Mjy 1 @ Qj ¼ Sj M j þ 2 C j My @y R~ @j ð13Þ Mj ¼ Dis ðwj þ ms wy Þ in which M jy ¼

1ms i Ds wjy , 2

 M y ¼ Dis wy þ ms wj

ð14Þ

where the strains and curvatures in the above force and moment resultants are defined by Eq. (10). 2.4. Energy functional of ring stiffened shell combination The total energy functional of the ring-stiffened conical– cylindrical–spherical shell combination is taken as the sum of the energy contributions from all of the shell components and interfaces, written as Z t1 ~ lk Þdt PTol ¼ Pl þ Pc þ Ps þ ðPlk þ P ð15Þ t0

~ lk are the interface potentials on the conical– where Plk and P cylindrical junction and the cylindrical–spherical junction, respectively. According to the Newton’s third law that action equals reaction, the force and moment resultants belonging to the each side of adjacent shell components may be used to calculate the interface potentials. Since the expressions of force and moment resultants of the cylindrical shell are simple, Nx,Nxy ,Q x and Mx (see [25] for more details) are employed for the calcula~ lk . In doing so, the interface potentials are tion of Plk and P written as Z Plk ¼ ðNx Yu,cl þN xy Yv,cl þ Q x Yw,cl Mx Yr,cl Þdl l Z 1 ðk Y2 þ kv,cl Y2v,cl þ kw,cl Y2w,cl þ kr,cl Y2r,cl Þdl ð16Þ  2 l u,cl u,cl and ~ lk ¼ P

Z

ðN x Yu,ls þ Nxy Yv,ls þQ x Yw,ls M x Yr,ls Þdl Z 1 ðk Y2 þ kv,ls Y2v,ls þ kw,ls Y2w,ls þ kr,ls Y2r,ls Þdl  2 l u,ls u,ls l

ð17Þ

where Yu,cl, Yv,cl, Yw,cl and Yr,cl are the essential continuity constraints on the conical–cylindrical interface, given by:Yu,cl ¼ ul cos auc þ sin awc , Yv,cc ¼vl vc, Yw,cl ¼ wl sin auc cos awc and Yr,cl ¼ @wl =@x@wc =@s, and the continuity equations on the cylindrical–spherical interface are expressed as: Yu,ls ¼ul  us, Yv,ls ¼vl  vs, Yw,ls ¼wl  ws and Yr,ls ¼[qwl/qx (qws/qj  us)/R]. ku,cl, kv,cl, kw,cl and kr,cl are pre-assigned weighted parameters defined on the conical–cylindrical interface, and ku,ls, kv,ls, kw,ls and kr,ls are weighted parameters defined on the cylindrical–spherical interface. 2.5. Admissible displacement functions and equations of motion The main advantage of the modified variational functional PTol in Eq. (15) is that the choice of the displacement functions for each shell segment is greatly simplified, and any linearly independent, complete basis functions may be employed. The reason

76

Y. Qu et al. / International Journal of Mechanical Sciences 69 (2013) 72–84

lies in the fact that both the interface continuity and geometric boundary conditions in a shell combination are relaxed and enforced through the modified variational method, and there is no need to explicitly satisfy the natural conditions on these interfaces and boundaries for the displacement functions. The modified variational functional PTol permits the use of the same admissible displacement functions for each shell segment. The displacement components of a shell segment can be written in the forms: ud ðg, y,tÞ ¼

P N X X

T p ðgÞ½cosðnyÞu~ pn, d ðtÞ þ sinðnyÞupn, d ðtÞ

p¼0n¼0

¼ U d ðg, yÞud ðtÞ vd ðg, y,tÞ ¼

P N X X

ð18aÞ

T p ðgÞ½sinðnyÞv~ pn, d ðtÞ þ cosðnyÞvpn, d ðtÞ

where q is global generalized coordinate vector for the shell combination. M and K are, respectively, the disjoint generalized mass and stiffness matrices of all shell segments. K l and K k are generalized interface stiffness matrices introduced by the modified variational method. The elements in the above matrices are listed in Appendix A. By assuming harmonic motion, q ¼ qeiot , the governing equations of motion for a stiffened conical–cylindrical– spherical shell combination can be written in the form ½o2 M þ ðKK l þ K k Þq ¼ 0

ð21Þ

A non-trivial solution is obtained by setting the determinant of the coefficient matrix of Eq. (21) equal to zero. Roots of the determinant are the squares of the eigenvalues for the shell combination. The eigenfunctions, i.e. mode shapes, corresponding to these eigenvalues are determined by back-substitution of the eigenvalues.

p¼0n¼0

¼ V d ðg, yÞvd ðtÞ

ð18bÞ 3. Results and discussions

wd ðg, y,tÞ ¼

P N X X

~ pn, d ðtÞ þsinðnyÞwpn, d ðtÞ T p ðgÞ½cosðnyÞw

3.1. Convergence and comparison studies

p¼0n¼0

¼ W d ðg, yÞwd ðtÞ

ð18cÞ

where ud, vd and wd(d ¼c, l, s) are the generalized coordinate vectors of a shell segment, and Ud(g,y), Vd(g,y), Wd(g,y) are admissible displacement function vectors. Tp(g) is the pth order polynomials. The variable g should be respectively replaced with s, x and j for the conical, cylindrical and spherical shells. The nonnegative integer n represents the circumferential wave number of the corresponding mode shape. P and N are the highest degrees taken in admissible functions. In order to demonstrate the versatility of the proposed formulation, three sets of orthogonal polynomials are used to expand the displacement components of each shell segment in the meridional direction. They are: (a) Chebyshev orthogonal polynomials of first kind (COPFK) [27] T 0 ðgÞ ¼ 1,

T 1 ðgÞ ¼ g,

T p þ 1 ðgÞ ¼ 2gT p ðgÞT p1 ðgÞ

for p Z 2: ð19aÞ

(b) Chebyshev orthogonal polynomials of second kind (COPSK) [27] T 0 ðgÞ ¼ 1,

T 1 ðgÞ ¼ 2g,

T p þ 1 ðgÞ ¼ 2gT p ðgÞT p1 ðgÞ

for p Z 2 ð19bÞ

(c) Legendre orthogonal polynomials of first kind (LOPFK) [27] T 0 ðgÞ ¼ 1, T 1 ðgÞ ¼ g, ðp þ 1ÞT p þ 1 ðgÞ ¼ ð2p þ1ÞgT p ðgÞpT p1 ðgÞ for p Z 2 :

ð19cÞ

It is noted that the COPFK, COPSK and LOPFK are complete and orthogonal series defined on the interval of gA[  1, 1]. Thus, a linear transformation rule for coordinate from g (for the ith shell segment, gA[gi,gi þ 1]) to gðg A ½1,1Þ need to be introduced for each shell segment, i.e.,g ¼ Za g þ Zb , Za ¼ ðgi þ 1 gi Þ=2 and Zb ¼ ðgi þ 1 þ gi Þ=2. By substituting Eq. (18) into Eq. (15) and setting the variation of the modified variation functional PTol to zero (i.e., dPTol ¼0) with respect to the generalized coordinate vectors ud, vd and wd, one obtains the equations of motion for a stiffened conical– cylindrical–spherical shell combination as M q€ þ ½KK l þK k q ¼ 0

ð20Þ

To check the convergence and accuracy of the present method, some results and considerations about the free vibrations of nonstiffened and stiffened conical–cylindrical–spherical shell combinations are presented in this section. A numerical code written in MATLAB scripts has been used to implement the present study. The first case concerns the free vibration analysis of nonstiffened conical–cylindrical–spherical shells with and without central cutout in the spherical shell component. Free boundary conditions are examined herein. The dimensions of the shell combinations are given as follows: R¼1 m, L0 ¼2.5 m, R0 ¼0.4 m, a ¼301,hc ¼hl ¼hs ¼0.01 m, j0 ¼01, 301, 451. All the shell components are made of aluminum with Young’s modulus E¼69.58 GPa, Poisson’s ratio m ¼0.31 and density r ¼2700 kg m  3. It is noted that, for a given number of admissible functions in each shell segment, the accuracy of the present solutions depends upon both the weighted parameters introduced in Eq. (15) and the number of segments divided in the shell components. It has been verified by Qu et al. [25,26] that the weighted parameters taken to be 102E–107E will lead to reasonably converged solutions. Therefore, k ¼103E has been conservatively employed to present all the results in the following analysis. For simplicity, equal number, N, of shell segments are chosen for the conical, cylindrical and spherical shell components, namely Nc ¼Nl ¼Nq ¼N0, although in most cases, using unequal numbers of segments for these shell components may be more efficient in computation. The frequency convergence criterion of the present method for the non-stiffened shell combinations with respect to the number of shell segments decomposed in the shell components is examined in Table 2, where m denotes the mode number of the shell combination for a given circumferential wave number n. Eight terms (i.e., P¼7) of the first kind Chebyshev orthogonal polynomials for each shell segment are considered. To the best of the authors’ knowledge, no analytical solutions for such complex structures have ever been reported in the literature. Thus, the theoretical results are verified by comparing the present solutions with those obtained from the finite element program ANSYS. A160  40 finite element mesh of SHELL63 elements (circumferential and meridional directions, respectively) is used for the conical and spherical shell components, whereas a160  60 SHELL63 mesh is employed for the cylindrical shell component. It could be observed from Table 2 that the convergence of the present method is excellent, and using two segments for each shell component is adequate for converged results. It should be noted that for higher-order vibration mode frequencies, good accuracy can be achieved if

Y. Qu et al. / International Journal of Mechanical Sciences 69 (2013) 72–84

77

Table 2 Convergence of natural frequencies (Hz) of non-stiffened conical–cylindrical–spherical shells versus the number of shell segments divided in shell components (COPFK: P ¼7, Nc ¼Nl ¼Nq ¼N0).

j0 ¼01

Mode

j0 ¼301

j0 ¼ 451

n

m

N0 ¼ 2

N0 ¼ 3

N0 ¼ 4

ANSYS

N0 ¼2

N0 ¼3

N0 ¼4

ANSYS

N0 ¼2

N0 ¼3

N0 ¼4

ANSYS

0

1 2 3 4 5 6

423.92 430.43 602.77 731.55 766.87 787.35

423.92 429.54 601.67 731.50 766.87 787.34

423.92 429.13 601.59 731.50 766.86 787.34

423.95 428.44 600.67 730.76 765.86 785.33

426.25 440.23 635.20 753.90 782.41 798.46

426.25 438.98 634.43 753.88 782.40 798.45

426.25 438.53 634.38 753.88 782.40 798.45

426.26 436.65 633.69 753.44 781.57 797.21

434.04 451.93 670.30 766.23 794.73 801.14

434.04 447.35 669.83 766.22 794.72 801.13

434.04 446.88 669.80 766.22 794.72 801.13

434.04 444.82 669.28 765.90 793.85 799.60

1

1 2 3 4 5 6

355.07 491.03 588.05 629.49 701.08 723.27

353.64 490.21 585.77 629.22 700.94 723.24

353.46 490.13 585.60 629.20 700.93 723.24

353.00 489.72 584.05 628.78 700.52 722.43

367.20 492.85 596.51 636.99 714.95 728.72

365.58 492.14 594.10 636.92 714.88 728.67

365.36 492.07 593.93 636.91 714.86 728.66

364.79 491.71 592.13 636.63 714.28 728.15

381.83 496.56 605.15 646.58 724.41 746.73

379.96 495.94 602.77 646.57 724.38 746.69

379.71 495.88 602.61 646.57 724.38 746.69

379.02 495.56 600.73 646.34 723.57 746.50

2

1 2 3 4 5 6

24.43 143.22 347.08 491.58 570.98 646.10

24.41 142.96 345.48 489.38 570.71 646.04

24.41 142.92 345.27 489.16 570.69 646.03

24.15 142.81 344.67 488.02 570.26 645.19

23.98 33.07 161.18 361.64 500.16 578.52

23.97 33.06 160.84 359.93 498.06 578.36

23.97 33.06 160.79 359.71 497.86 578.34

23.73 32.88 160.66 359.06 496.72 577.93

14.05 25.36 176.55 375.69 507.59 584.79

14.05 25.35 176.12 373.88 505.60 584.70

14.05 25.35 176.05 373.65 505.42 584.69

13.97 25.06 175.90 372.94 504.32 584.27

3

1 2 3 4 5 6

67.76 92.08 251.00 368.06 436.89 543.77

67.69 92.04 250.19 366.36 436.87 543.37

67.69 92.03 250.08 366.15 436.86 543.33

67.55 91.94 249.78 365.37 436.76 542.59

67.59 89.58 109.63 252.46 368.53 437.77

67.52 89.54 109.59 251.65 366.84 437.74

67.51 89.54 109.58 251.53 366.63 437.74

67.38 89.45 109.58 251.22 365.85 437.61

47.78 69.57 99.64 256.46 368.51 440.78

47.78 69.57 99.64 256.46 368.51 440.78

47.78 69.57 99.63 256.34 368.30 440.77

47.67 69.42 99.55 256.00 367.53 440.63

more segments are divided in each shell component. In the interest of brevity, these converged high-order vibration frequencies are not given in Table 2. In all numerical cases, there is an excellent agreement between the converged values of the present method and the finite element results, thus validating the high accuracy of the present computations. For illustrative purposes, some representative mode shapes for the non-stiffened shell combination with an open angle j0 ¼301 in the spherical shell component are given in Fig. 2, as it is believed that they help in the understanding of certain features for the vibration behaviors of conical–cylindrical–spherical shell combinations. These mode shapes have been reconstructed in three-dimensional view by means of considering the displacement field in Eq. (18) after solving the eigenvalue problem. The present solutions are compared with those obtained from the finite element software ANSYS. It can be seen that the mode shapes obtained by the two methods are generally in good agreement. In order to further validate the present method, the natural frequencies of stiffened conical–cylindrical–spherical shell combinations with different number of shell segments are given in Table 3. Three types of shell designs, designated as Shell Combination I–III, with different distributions of internal ring stiffeners in the cylindrical shell component are examined. As shown in Fig. 3, the Shell Combination I has a uniform stiffener distribution in the cylindrical shell component, whereas non-uniform distributions of the rings are considered for the Shell Combination II and Shell Combination III. The dimensions of the shell combinations are: R¼1 m, L0 ¼5 m, R0 ¼ 0.5 m, a ¼151, hc ¼hl ¼hs ¼ 0.01 m, j0 ¼01. The ring stiffener has a rectangular cross-section with constant width of b¼0.02 m and depth of d¼ 0.02 m, and the total number of rings in the cylindrical shell component is taken as NR ¼9. All the shell components and the ring stiffeners are made of steel with Young’s modulus E¼210 GPa, Poisson’s ratio m ¼0.3 and density r ¼7800 kg m  3. Free boundary conditions

are considered for these shell combinations. Again, the present solutions are compared with those obtained from the finite element software ANSYS, in which SHELL 63 and BEAM 188 elements are used for the finite element discretization of the shell components and ring stiffeners, respectively. Eccentricity of the internal ring stiffeners is modeled through the offset modeling feature of the BEAM 188 element. It is clearly observed from Table 3 that the present method exhibits stable and rapid convergence characteristics as the number of shell segments is increased. In all cases, the present results agree closely with those solutions obtained from the finite element analyses. Review of the convergence rate of the natural frequencies of the stiffened shell combinations indicates that a decomposition of two segments in the each shell component provides reasonably accurate solutions. It should be remarked here that the lower circumferential mode frequencies of the shell combinations are insignificantly affected by the distributions of the ring stiffeners. However, this is not the case for vibration modes with large circumferential wave numbers since local vibration modes of shell combinations may appear by arranging of the locations the rings. Some selected mode shapes of the Shell Combination III are shown in Fig. 4. It may be seen from Fig. 4 that the detailed plots of the deformation shapes are in qualitative agreement with these results obtained from ANSYS. Interestingly, the mode shape corresponding to n ¼5 and m¼1 is a ‘‘local’’ mode in which vibration occurs predominantly in a small region of the shell combination. This is due to the fact that only one ring stiffener is located in this region. The arrangement of ring stiffeners in a shell combination requires special attention of the designers, since high and localized stresses may generally develop within a small region during the modal vibrations of the shell combination. The last example is a submarine pressure hull. To accommodate the design requirements, the cylindrical shell component in the pressure hull is equally divided into eight shell segments

78

Y. Qu et al. / International Journal of Mechanical Sciences 69 (2013) 72–84

Present: n=2, m=1, 23.97Hz

ANSYS:n=2, m=1, 23.73Hz

Present: n=2, m=4, 359.71Hz

ANSYS:n=2, m=4, 359.06Hz

Present: n=3, m=2, 89.54Hz

ANSYS:n=3, m=2, 89.45Hz

Present: n=5, m=4, 267.99Hz

ANSYS:n=5, m=4, 267.81Hz

Fig. 2. Comparison of some mode shapes between present method and FEM (ANSYS) for the non-stiffened conical–cylindrical–spherical shell combination with a cut out (open angle: j0 ¼ 301).

(i.e., Nl ¼8) in the axial direction by transverse bulkheads, and four ring stiffeners of constant width b¼0.06 m and depth h¼0.08 m are equally spaced on each cylindrical shell segment (i.e., the total number of ring stiffeners: NR ¼Nl  Nr ¼32). The dimensions of the pressure hull used for the analysis are: R¼3.25 m, L0 ¼30 m, Rc ¼1 m, a ¼181, hc ¼hl ¼hs ¼0.04 m. The following material properties are used for the shells and stiffeners: E¼210 GPa, m ¼0.3 and r ¼7800 kg m  3. Three types of edge conditions, i.e., free, clamped and elastic supported boundary conditions, are chosen for the pressure hull at the cone end. In the case of the elastic supported boundary condition, the cone edge is considered to be circumferentially elastic (i.e., vc a0, uc ¼wc ¼qwc/qs¼ 0), and is characterized by a stiffness constant kv per unit length in the circumferential direction. It has been proved in the aforementioned studies that the free vibration behaviors of stiffened shell combinations are well characterized by the Chebyshev orthogonal polynomials of first kind (COPFK). To illustrate the capability of the present formulation in handling

various polynomials, the COPFK, COPSK, and LOPFK are employed in the numerical computations. For all numerical cases, three shell segments are used for the conical and spherical shell components, and the natural frequencies are presented using qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the dimensionless form: On,m ¼ oR rl ð1m2l Þ=El . The frequencies of the present method are compared with those results obtained from ANSYS. Once again, SHELL 63 and BEAM 188 elements are used for the finite element discretization of the shell components and the ring stiffeners in the pressure hull. From the comparison, it can be generally concluded that present results are correct and they can be used as a benchmark to validate other approximate solutions. Another important observation is that the three types of polynomials, i.e., COPFK, COPSK, and LOPFK, gives the same vibration results. This implies, regardless of the boundary condition being considered, that the accuracy of the present method is not particularly affected by the employed polynomials. From the above studies, it can be concluded that the present method is capable of handling the vibration problems of stiffened

Y. Qu et al. / International Journal of Mechanical Sciences 69 (2013) 72–84

79

Table 3 Convergence of natural frequencies (Hz) of stiffened conical–cylindrical–spherical shells versus the number of shell segments divided in shell components (COPFK: P¼ 7, Nc ¼ Nl ¼ Nq ¼ N0). Mode

Shell Combination I

Shell Combination II

Shell Combination III

n

m

N0 ¼ 2

N0 ¼ 3

N0 ¼ 4

ANSYS

N0 ¼2

N0 ¼3

N0 ¼4

ANSYS

N0 ¼2

N0 ¼3

N0 ¼4

ANSYS

0

1 2 3 4 5

239.18 328.41 463.66 556.25 658.41

239.18 327.80 463.66 553.92 658.41

239.18 327.28 463.65 552.39 658.40

239.13 326.24 463.57 549.16 656.75

239.31 328.67 463.44 555.85 658.46

239.31 328.03 463.44 553.61 658.45

239.31 327.41 463.44 552.00 658.44

239.26 326.64 463.33 549.33 658.30

239.60 329.21 462.84 554.99 658.63

239.60 328.60 462.84 552.50 658.63

239.60 328.02 462.84 550.65 658.62

239.56 327.17 462.73 547.99 656.54

1

1 2 3 4 5

167.36 317.17 458.23 517.61 582.94

167.06 316.24 456.60 517.27 581.92

166.89 315.89 455.46 517.26 581.26

166.87 315.35 455.10 517.00 581.70

166.87 317.29 448.85 517.63 585.36

166.82 315.87 455.74 517.60 583.29

166.82 315.86 455.70 517.58 583.23

166.81 315.56 455.12 517.36 582.74

166.51 317.64 453.93 517.73 586.99

166.61 316.22 455.12 518.25 585.02

166.61 316.10 455.15 518.23 584.76

166.60 315.87 454.54 518.03 584.22

2

1 2 3 4 5

16.07 52.12 151.33 258.18 340.17

16.02 52.10 151.08 257.53 339.84

15.98 52.01 150.92 257.08 339.50

15.83 51.83 150.39 255.97 338.85

16.15 52.14 151.39 258.15 340.57

16.09 52.10 151.12 257.52 340.25

16.08 52.04 150.87 256.93 339.97

15.90 51.85 150.48 256.04 339.36

16.31 52.15 151.58 258.13 341.86

16.26 52.13 151.33 257.46 341.53

16.24 52.09 151.09 256.89 341.26

16.05 51.91 150.67 255.95 340.73

3

1 2 3 4 5

41.01 51.36 97.96 171.35 216.15

40.57 51.30 97.80 171.07 215.92

40.39 51.27 97.56 170.65 215.80

39.88 51.17 97.17 170.19 215.53

41.14 51.34 98.16 171.82 216.44

40.98 51.28 97.94 171.35 216.29

40.78 51.23 97.71 170.90 216.15

39.97 51.09 97.19 170.21 215.71

40.96 51.18 98.07 171.78 216.91

40.91 51.12 97.91 171.27 216.77

40.75 51.09 97.73 170.87 216.65

40.01 50.97 97.21 170.13 216.27

5

1 2 3 4 5

101.77 111.33 121.75 138.41 179.16

101.92 110.84 121.52 138.61 178.57

100.65 110.14 121.34 137.64 178.41

98.03 107.39 120.74 135.61 176.39

99.23 112.79 121.82 138.06 179.23

98.96 112.16 121.58 138.13 178.69

97.96 111.68 121.41 137.29 178.61

97.19 108.83 120.85 135.30 176.46

86.15 116.52 121.71 136.05 178.39

85.98 116.58 121.48 136.06 178.14

84.93 116.03 121.30 135.64 177.92

84.15 113.32 120.81 133.70 175.85

shell combinations with complex stiffener distributions and boundary conditions. Although the finite element model gives high accurate solutions, it is generally computationally inefficient when compared with the present method. For example, the frequency results of the finite element model in Table 4 have been obtained by means of a huge system having about 83,748 degrees of freedom (dofs). Conversely, using the present approach, only 288 dofs are needed for each circumferential wave number. Moreover, the present method provides more physical insight into the vibration behaviors of the shell combinations than the finite element analysis. The reason lies in the fact that the present method is semi-analytical for a general circumferential wave number n, and by assigning different values of n one obtains the natural frequencies and mode shapes associated with these circumferential waves. However, to identify the mode shapes corresponding to certain frequencies of the shell combinations in the finite element analysis, the solutions must be extracted for each mode shape and classified one by one, which can be very tiresome work. This is significant, particularly when optimization studies for shell combinations need to be carried out. Although the present method implemented in MATLAB has been optimized neither in terms of memory storage nor in terms of solution technique, it is at least 20 times faster than ANSYS when computing the natural frequencies of a given circumferential wave. Furthermore, for different boundary conditions, it is unnecessary to reformulate the system matrices of the shell combination in the present approach, and only some simple algebraic operations at the geometrical boundaries need to be re-evaluated. This is quite simple and straightforward to deal with. 3.2. Parametric studies for submarine pressure hull Since confidence has been established in the frequency predictions, we use the present method to investigate the effects of stiffener number and geometry on the vibration frequencies of

L0 = 10 × Lr

R

Lr

6 × Lr1 = L0 / 2 Lr1

4 = Lr 2 × L0 / 2 Lr 2

R

Lr1

8 × Lr1 = L0 / 2

2 × Lr 2 = L0 / 2 Lr 2

R

Fig. 3. Distribution of internal ring stiffeners in the cylindrical shell component of the stiffened conical–cylindrical–spherical shell combinations: (a) Shell Combination I; (b) Shell Combination II; and (c) Shell Combination III.

the submarine pressure hull defined in Section 3.1. A parametric study is first made by keeping the cross-sectional dimensions of ring stiffener constant (i.e., b¼0.06 m and h¼0.08 m) and varying the number of the stiffeners in the pressure hull. It is assumed that the stiffeners are equally placed in each cylindrical shell segment. The geometrical dimensions material properties of the

80

Y. Qu et al. / International Journal of Mechanical Sciences 69 (2013) 72–84

Present: n=2, m=5, 341.26 Hz

ANSYS: n=2, m=5, 340.73 Hz

Present: n=3, m=6, 284.95 Hz

ANSYS: n=3, m=6, 284.67 Hz

Present: n=5, m=1, 84.93 Hz

ANSYS: n=5, m=1, 84.15 Hz

Fig. 4. Comparison of some mode shapes between present method and FEM (ANSYS) for the stiffened conical–cylindrical–spherical shell combination (Shell Combination III).

Table 4 Frequency parameters On,m ¼ oR Mode

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rð1m2 Þ=E of a submarine hull with different boundary conditions (kv ¼ 2  109 N/m).

Free

Clamped

Elastic supported

n

m

COPFK

COPSK

LOPSK

ANSYS

COPFK

COPSK

LOPSK

ANSYS

COPFK

COPSK

LOPSK

ANSYS

0

1 2 3 4 5

0.1663 0.2444 0.3310 0.4599 0.4917

0.1663 0.2444 0.3310 0.4599 0.4917

0.1663 0.2444 0.3310 0.4599 0.4917

0.1661 0.2439 0.3306 0.4575 0.4912

0.0402 0.0955 0.1787 0.3095 0.3401

0.0402 0.0955 0.1787 0.3095 0.3401

0.0402 0.0955 0.1787 0.3095 0.3401

0.0401 0.0950 0.1760 0.3089 0.3397

0.0290 0.0955 0.1724 0.3095 0.3354

0.0290 0.0955 0.1724 0.3095 0.3354

0.0290 0.0955 0.1724 0.3095 0.3354

0.0289 0.0950 0.1721 0.3089 0.3350

1

1 2 3 4 5

0.0902 0.1973 0.3092 0.4128 0.5067

0.0902 0.1973 0.3092 0.4128 0.5067

0.0902 0.1973 0.3092 0.4128 0.5067

0.0904 0.1970 0.3082 0.4113 0.5050

0.0055 0.0584 0.1508 0.2553 0.3630

0.0055 0.0584 0.1508 0.2553 0.3630

0.0055 0.0584 0.1508 0.2553 0.3630

0.0054 0.0587 0.1505 0.2548 0.3625

0.0053 0.0564 0.1452 0.2470 0.3539

0.0053 0.0564 0.1452 0.2470 0.3539

0.0053 0.0564 0.1452 0.2470 0.3539

0.0052 0.0566 0.1446 0.2464 0.3531

2

1 2 3 4 5

0.0262 0.0441 0.0768 0.1419 0.2142

0.0262 0.0441 0.0768 0.1419 0.2142

0.0262 0.0441 0.0768 0.1419 0.2142

0.0228 0.0434 0.0753 0.1406 0.2129

0.0270 0.0690 0.1265 0.1768 0.2371

0.0270 0.0690 0.1265 0.1768 0.2371

0.0270 0.0690 0.1265 0.1768 0.2371

0.0235 0.0674 0.1253 0.1760 0.2363

0.0235 0.0671 0.1240 0.1728 0.2341

0.0235 0.0671 0.1240 0.1728 0.2341

0.0235 0.0671 0.1240 0.1728 0.2341

0.0240 0.0673 0.1241 0.1729 0.2343

shell components in the pressure hull follow the same configurations of those defined in Section 3.1. The variations of the natural frequencies with respect to the number, Nr, of ring stiffeners in each cylindrical shell segment for the pressure hull with free and clamped conditions at the cone end are given in Figs. 5 and 6, respectively. From Figs. 5 and 6, it is of interest to note that the increase of number of ring stiffeners does not necessarily lead to a proportionate increase of all frequencies for the pressure hull. This is because the ring stiffeners impose not only stiffness but also mass

effect on the pressure hull. In the case of lower circumferential modes, the frequencies of the pressure hull decrease by the increase in the number of the ring stiffeners. This phenomenon occurs because the mass effects of the rings activate greater than those effects of stiffness on the frequencies of the pressure hull. However, for higher values of circumferential waves, the effect of increase in the number of rings increases the stiffness more than the mass. Therefore it becomes necessary to choose the reasonable number of ring stiffeners in order to improve the vibration resistance in the design practice of a practical submarine hull.

Y. Qu et al. / International Journal of Mechanical Sciences 69 (2013) 72–84

0.06 0.04 0.02

1

2

3

4

5

Nr=0 Nr=4 Nr=7 Nr=9

0.10 0.08 0.06 0.04 0.02

Nr=0 Nr=4 Nr=7 Nr=9

0.10 0.08 0.06 0.04 0.02 0.00

0.00

6

Dimensionless frequency Ωnm

0.08

0.00

0.12

0.12 Nr=0 Nr=4 Nr=7 Nr=9

Dimensionless frequency Ωnm

Dimensionless frequency Ωnm

0.10

81

1

Circumferential wave number n

2

3

4

5

1

6

2

3

4

5

6

Circumferential wave number n

Circumferential wave number n

Fig. 5. Variations of frequency parameters On,m against the number of ring stiffeners for the submarine pressure hull with free boundary condition: (a) m¼ 1; (b) m¼2; and (c) m¼ 3.

0.06 0.04 0.02

1

2

4

3

5

6

Nr=0 Nr=4 Nr=7 Nr=9

0.10 0.08

Dimensionless frequency Ωnm

Dimensionless frequency Ωnm

Dimensionless frequency Ωnm

Nr=0 Nr=4 Nr=7 Nr=9

0.08

0.00

0.12

0.12

0.10

0.06 0.04 0.02 0.00

1

2

3

4

5

0.08 0.06 0.04 0.02 0.00

6

Circumferential wave number n

Circumferential wave number n

Nr=0 Nr=4 Nr=7 Nr=9

0.10

1

2

3

4

5

6

Circumferential wave number n

Fig. 6. Variations of frequency parameters On,m against the number of ring stiffeners for the submarine pressure hull with clamped boundary condition: (a) m ¼1; (b) m¼ 2; and (c) m¼ 3.

1.55 Frequency (Hz)

12

Frequency (Hz)

11 10 9

1.5 1.45 1.4

8 1.35 0.12

7 6

0.1

5 0.12

0.08

0.1 0.08 0.06 Width (m) 0.04

0.12

0.02 0

0

0.02

0.14

0.16

0.1 0.08 0.06 Depth (m) 0.04

Width (m) 0.06

0.04 0.02 0

0

0.02

0.04

0.06

0.12 0.1 0.08 Depth (m)

0.14

0.16

Fig. 7. Variation of the fundamental frequencies with the depth and width of the ring stiffeners for the pressure hull with different boundary conditions: (a) free and (b) clamped.

The influences of geometric dimensions of ring stiffeners on the fundamental frequencies of the pressure hull are then investigated. Ring stiffeners with different widths and depths are examined. The number of rings in each cylindrical shell segment is: Nr ¼4 (i.e., the total ring stiffener number: NR ¼8  4). It should be noted that for, a given wave number n, the natural

frequencies of the pressure hull increase with the increase of mode number, m. Thus, the minimum natural frequency for the pressure hull always corresponds to m¼1. This assertion is valid for the entire range of hull parameters and for all types of boundary conditions. Hence, the value of m is taken to be unity in the analysis. The variations of the fundamental frequencies

82

Y. Qu et al. / International Journal of Mechanical Sciences 69 (2013) 72–84

with the ring depths and widths for the pressure hull with free and clamped cone end conditions are shown in Fig. 7(a and b), respectively. It is obvious that the fundamental frequencies for pressure hull with free boundary conditions can be raised significantly by increasing the depths and widths of the rings. This is because the stiffness effects overcome the inertia effects contributed by the ring stiffeners. For the hull clamped at the end of the cone, the incorporation of more ring stiffeners will lead to smaller fundamental frequency parameters. It should be noted that the higherorder vibration modes of the pressure hull may benefit from a large number of stiffeners; however, these vibration frequencies do not usually dominate the practical pressure hull design.

2

3

6 K ¼ diag6 c,1 ,K c,2 ,. . .,K c,N , 4K |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}c

In this paper, a modified variational approach is employed to study the free vibration of ring-stiffened conical–cylindrical– spherical shell combinations. Reissner–Naghdi’s thin shell theory in conjunction with a multilevel partition technique, viz., stiffened shell combination, shell component and shell segment, is employed to formulate the theoretical model. The ring stiffeners in the shell combination are treated as discrete elements, and these rings may be few or many in number, non-uniform or uniform in size, and non-uniformly or uniformly spaced. The displacement and rotation fields of each shell segment are expanded by means of a double mixed series, i.e., Fourier series for the circumferential variable and orthogonal polynomials for the meridional variable. The versatility and reliability of the present approach for the application of different basis functions have been demonstrated with the following three types of orthogonal polynomials, i.e., Chebyshev orthogonal polynomials of first and second kind, and Legendre orthogonal polynomials of first kind. To check the convergence and accuracy of the present method, some results and considerations about the free vibrations of non-stiffened and stiffened conical–cylindrical–spherical shell combinations are presented. The theoretical results are verified by comparing finite element analysis results from ANSYS. It is found the present method exhibits stable and rapid convergence characteristics, and the present results, including natural frequencies and mode shapes, agree closely with those solutions obtained from the finite element analyses. Regardless of the boundary condition being considered, it seems that the accuracy of the present method is not particularly affected by the employed polynomials. This makes the choice of admissible functions very flexible, and can be considered as one distinguished feature of the present formulation. In conclusion, the present analysis has shown that the modified method can be considered as a valuable tool for prediction of the free vibrations of stiffened shells and is easily applicable to shell combinations with complex geometric configurations.

7 , K s,1 ,K s,2 ,. . .,K s,Ns 7 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 5

N c conical shell segments N l cylindrical shell segments N s segments shell segments

ðA:2Þ where the sub-matrices Mc,i and Kc,i(iA[1,Nc]) are the mass and stiffness matrices of the ith conical shell segment, and Ml,i and Kl,i(iA[1,Nl]) are the mass and stiffness matrices of the ith stiffened-cylindrical shell segment. For more details of these matrices, the readers may refer to the recently published Ref. [25]. The mass and stiffness matrices of the ith spherical shell segment are given as 2 i 3 M uu 0 0 ZZ 6 7 2 6 0 0 7 Mivv M is ¼ 4 5R Sj dj dy, Si

4. Conclusions

K l,1 ,K l,2 ,. . .,K l,Nl |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

K is ¼

ZZ

0

0

2

Kiuu 6 i,T 6 K uv 4 Si K i,T uw

K iuv K ivv K i,T vw

M iww 3 K iuw 7 K ivw 7 5dj dy K iww

ðA:3a; bÞ

The elements of the mass and stiffness matrices are: M iuu

i ¼ rs hs Ui,T s Us ,

i M ivv ¼ rs hs V i,T s V s,

i M iww ¼ rs hs W i,T s Ws

ðA:4a2cÞ According to the Reissner–Naghdi’s linear shell theory, the elements of the stiffness matrices are: K iuu ¼ Sj K is U M1 þ Dis U B1 ,

K iuv ¼ Sj K is V M1 þ Dis V B1 ,

K iuw ¼ Sj K is W M1 þDis W B1 ,

K ivv ¼ Sj K is V M2 þ Dis V B2 ,

K ivw ¼ Sj K is W M2 þDis W B2 ,

K iww ¼ Sj K is W M3 þ Dis W B3

ðA:5a2fÞ

in which @Ui,T s @j

UM1 ¼

þ

UB1 ¼

i 1ms @Ui,T s @Us 2 @ y @ y 2Sj

@Ui,T s @j þ

! ! @Uis @Ui,T s Ui þ ms T j Uis þ T j T j Ui,T s þ ms @j @j ðA:6aÞ

! ! 2 Sj @Uis ms C j i C j i,T @Uis i þ þ U U T U þ m j s s s s @j R2 @j R2 R2

i 1ms @Ui,T s @Us 2 @ y @ y 2R Sj

ðA:6bÞ

i Ui,T @Vis ms @Ui,T 1ms @Ui,T s @V s s þ þ VM1 ¼ s C j @y Sj @j @y 2Sj @y

V B1 ¼

1 R2

T j U i,T s þ

@U i,T ms s @j

!

@Vis T j Vis @j

@Vis 1ms @U i,T s þ @y 2R2 @y

!

@V is T j V is @j

ðA:6cÞ !

ðA:6dÞ Appendix A. Generalized mass and stiffness matrices of stiffened conical–cylindrical–spherical shell

  @Ui,T i s W is þT j U i,T W M1 ¼ 1 þ ms s Ws @j

The disjoint generalized mass and stiffness matrices of the stiffened conical–cylindrical–spherical shell combination are, respectively, given by

W B1 ¼

2

3

6 7 M ¼ diag6 , M s,1 ,M s,2 ,. . .,M s,Ns 7 c,1 ,M c,2 ,. . .,M c,N , M l,1 ,M l,2 ,. . .,M l,N 4M |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}c |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}l |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 5 N c conical shell segments N l cylindrical shell segments N s spherical shell segments

ðA:1Þ

" 1ms @U i,T s 2 Sj @y R

!

! @W is @2 W is  @y @y@j ! i,T 2 @U s @ W is  Sj þ C j ms U i,T s @j @ j2 ! !# @U i,T @2 W is @2 W is s  T j U i,T þ m þ C j s s 2 @j @j2 Sj @y 1

ðA:6eÞ

Tj

ðA:6fÞ

Y. Qu et al. / International Journal of Mechanical Sciences 69 (2013) 72–84 i 1 @V i,T 1ms s @V s V M2 ¼ 2 þ 2 Sj @y @y

@V i,T s T j V i,T s @j

i 1 @V i,T 1ms s @V s þ V B2 ¼ 2 R Sj @y @y 2R2

!

@V i,T s T j V i,T s @j

@V is T j V is @j !

! ðA:6gÞ

@V is C j V is Sj @j

!

Dis

W B2 ¼

R2 

1 @V i,T s R2 @y

R3

ðA:6iÞ 1ms 2R

K vi vi ¼

! @W is @2 W is  Tj @j @y@j ! @2 W is @W is @2 W is þ Tj þ ms 2 @j @ j2 S2j @y

@V i,T s T j V i,T s @j

1ms

!

ðA:6jÞ

W M3 ¼ 2ð1 þ m

K vi wi ¼

! !   2 i,T 2 1ms @ Ws @W i,T @2 W is @W is s T T þ j j @j@y @j @j@y @j R2 Sj

6 KT 6 ui vi 6 Z 2p 6 K Tui wi 6 i 6 Kl ¼ 6 KT 6 ui ui þ 1 0 6 6 T 6 K ui vi þ 1 4 K Tui wi þ 1

K u i wi

K ui ui þ 1

R3 Sj

3

K u i wi þ 1

K v i wi þ 1 7 7 7 K wi wi þ 1 7 7 7 7RSj dy 0 7 7 7 0 7 5 0

K vi vi

K v i wi

K vi ui þ 1

K vi vi þ 1

K wi wi

K wi u i þ 1

K wi vi þ 1

K Tvi ui þ 1

K Twi ui þ 1

0

0

K Tvi vi þ 1

K Twi vi þ 1

0

0

K Tvi wi þ 1

K Twi wi þ 1

0

K vi ui þ 1 ¼ Ai

K vi vi þ 1

i

The elements of the interface stiffness matrix K l are: !" ! !# i 1 Di @U i,T i i,T @U s i s K is þ 2s K ui ui ¼ þ ms T j U i,T þ U þ m T U U s j s s s s R @j @j R

K wi wi

K ui vi ¼

K ui wi ¼

1 RSj

K is þ

R

i,T s @U s

1m 2

@y

i i,T @V s sUs

V is þ m

@y

R3 Sj

" 2ms ¼ 3 R Sj Sj

K ui ui þ 1 ¼ 

1 Di K is þ 2s R R

@V i,T s T j V i,T s 

! V isþ 1

@j

ðA:8jÞ

Tj

@2 W i,T s 2

@y



@3 W i,T s

ðA:8kÞ

!

2

@j@y

!

#

þ

"

Dis

W i,T s

3

R Sj

2

þ Cj

þ

þ

1 @

W is 2

S2j @y

Dis R

"

Tj

@ W is  @ 2 2

@2 W is 2

@3 W is

@j@y # @ W is Sj @j3

!

j



@y

!

þ

2

3

ms @W is Sj @j

3

@2 W i,T m @2 W i,T @W i,T s s s þ 2s þ ms T j 2 2 @j @j Sj @y

Dis @W i,T s R 3 @j

K wi u i þ 1 ¼ 

2ms Sj

!

@W is @j !

@2 W is ms @2 W is @W is þ þ m T j s @j @ j2 S2j @y2

 K is  Dis 1 þ ms W i,T s  3 R R

@2 W i,T m @2 W si,T @W i,T s s þ 2s þ ms T j @j @j2 Sj @y2

ðA:8lÞ !# U is

ðA:8mÞ ðA:8bÞ

!

@U i,T s þ ms T j U i,T U isþ 1 s @j

!

m @W i,T 1 @2 W i,T @2 W i,T @3 W i,T s s s s þ Cj 2  Sj W is þ s 2 2 Sj @j @j @ j3 Sj @y

K wi vi þ 1 ¼ 

Dis 1ms R 3 Sj

Kwi wi þ 1 ¼ 

2ms R3 Sj Sj

Cj

!

ðA:8iÞ

! # i,T iþ1 @V i,T @2 V i,T iþ1 i @V s @W s s s Tj  W s þ ms @y @j@y @y @j

!

! @U i,T @2 U i,T ms i,T 1ms @2 U i,T s s s þ S  U þ W is j @j Sj s Sj @y2 @ j2 R3 Sj ! i  Ki  Dis @U i,T i i,T @W s s þ s 1 þ ms U i,T W  þ m T U j s s s s 3 R @ j @ j R ! i i 2 D @ W s ms @2 W is i  3s U i,T þ þ m T W ðA:8cÞ j s s s @ j2 R S2j @y2 Dis

"

ðA:8hÞ

U isþ 1

Sj @y

Dis

ðA:8aÞ !

ms @V i,T s

Dis

ðA:7Þ

Dis 2

! !# i @V i,T i i,T @V s i s T j V i,T þV T V V j s s s s @j @j

R2

1ms 1 Di K is þ 2s ¼ 2 R R

K vi wi þ 1 ¼

K ui vi þ 1

K Tvi wi

0

!"

! @2 V i,T @V i,T s s T j W is @j@y @y

ðA:6lÞ

where Tj is defined as T j ¼ cot j. The generalized interface stiffness matrix K l and K k introduced by the modified variational method are obtained through the assembly of all interface matrices. For more details of the interface matrices of the conical and cylindrical shells, see i reference [25]. The interface matrix K l of the spherical shell at the interface location of j ¼ ji is given below K ui vi

Dis

! # i   @W is @2 W is @Vi,T s @W s T  m  þ 1ms V i,T j s s @y @y@j @y @j

ðA:6kÞ

! @2 W i,T ms @2 W i,T @W i,T @2 W is s s s þ þC m W B3 ¼ 2 Sj j s Sj @y2 @j @j2 @ j2 R ! ! i,T i,T i,T 2 2 1 1 @ Ws @W s @ Ws @2 W is @W is þ 2 þ Tj þ ms þ Cj 2 @j @j @j2 R S2j @y2 Sj @y

"

Dis

1

K ui ui

K is þ

ðA:8gÞ

i,T i s ÞW s W s

2

! @U i,T @2 U i,T ms i,T 1ms @2 U i,T s s s þ  U þ W isþ 1 s @j @ j2 S2j S2j @y2 ! @U i,T @W isþ 1 s þ ms T j U i,T ðA:8fÞ s @j @j

Dis



ðA:8eÞ

Cj

R3

ðA:6hÞ 1 þ ms @V i,T s W M2 ¼ W is Sj @y

1mis 1 @U i,T s V isþ 1 2 Sj @y

K ui vi þ 1 ¼ 

K ui wi þ 1 ¼

83

ðA:8dÞ

Ds

þ Cj



Ds R3



! @2 W i,T @W i,T s s þ Tj V isþ 1 @y@j @y

"

1 @

2

Tj

W i,T s 2

S2j @y @2 W i,T s @ 2

j

þ

@2 W i,T s @y



2

@ W i,T s  @ 2 2

!

j

ms @2 W i,T s S2j @y2

@3 W i,T s @j@y

2

! þ

ðA:8nÞ

ms @W i,T s Sj @j

# @3 W i,T s W isþ 1 @ j3 ! @W i,T @W isþ 1 s þ ms T j @j @j Sj

i

ðA:8oÞ

The generalized interface stiffness matrix K k introduced by the least-squares weighted residual terms at the interface location of

84

Y. Qu et al. / International Journal of Mechanical Sciences 69 (2013) 72–84

j ¼ ji for the spherical shell is given below 2

K ui ui

6 6 0 6 6 T Z 2p 6 6 K ui wi i 6 T Kk ¼ 6 6 K ui ui þ 1 0 6 6 6 0 6 4 T K ui wi þ 1

0

K ui wi

K ui ui þ 1

0

K ui wi þ 1

Kv i v i

0

0

K vi vi þ 1

0

0

K wi wi

K wi ui þ 1

0

Kwi wi þ 1

0

T K wi ui þ 1

K ui þ 1 ui þ 1

0

K ui þ 1 wi þ 1

K vi vi þ 1

0

0

K vi þ 1 vi þ 1

0

0

K Twi wi þ 1

T K ui þ 1 wi þ 1

0

Kw i þ 1 w i þ 1

T

3 7 7 7 7 7 7 7 7RSj dy 7 7 7 7 7 5

ðA:9Þ where the sub-matrices are given as i K ui ui ¼ ðk~ u þ k~ r ÞU i,T s Us,

K ui wi ¼ k~ r U i,T s

iþ1 K ui ui þ 1 ¼ ðk~ u k~ r ÞU i,T , s Us

i K vi vi ¼ k~ v V i,T s V s,

@W is , @j

K ui wi þ 1 ¼ k~ r Ui,T s

@W isþ 1 @j

K vi vi þ 1 ¼ k~ v V isþ 1,T V isþ 1

i ~ K wi wi ¼ k~ w W i,T s W s þ kr

i @W i,T s @W s , @j @j

iþ1 K wi wi þ 1 ¼ k~ w W i,T k~ r s Ws

Kwi ui þ 1 ¼ k~ r

ðA:10e2fÞ @W i,T s U isþ 1 , @j

iþ1 @W i,T s @W s @j @j

K ui þ 1 ui þ 1 ¼ ðk~ u þ k~ r ÞUisþ 1,T Uisþ 1 ,

ðA:10a2dÞ

K ui þ 1 wi þ 1 ¼ kr U isþ 1,T

ðA:10g2iÞ @W isþ 1 @j ðA:10j2kÞ

K vi þ 1 vi þ 1 ¼ k~ v V isþ 1,T V isþ 1 , K wi þ 1 wi þ 1 ¼ k~ w W isþ 1,T W isþ 1 þ k~ r

@W isþ 1,T @W isþ 1 @j @j

ðA:10l2mÞ

References [1] Liew KM, Lim CW, Kitipornchai S. Vibration of shallow shells: a review with bibliography. Appl Mech Rev 1997;50:431–44. [2] Qatu MS. Recent research advances in the dynamic behavior of shells: 1989– 2000, part 2: homogeneous shells. Appl Mech Rev 2002;55:415–34. [3] Leissa AW. Vibration of the shells (NASA SP-288). Washington, DC: US Government Printing Office; 1973. [4] Hu WCL, Raney JP. Experimental and analytical study of vibrations of joined shells. AIAA J 1967;5:976–80.

[5] Irie T, Yamada G, Muramoto Y. Free vibration of joined conical–cylindrical shells. J Sound Vib 1984;95:31–9. [6] Efraim E, Eisenberger M. Exact vibration frequencies of segmented axisymmetric shells. Thin-Walled Struct 2006;44:281–9. [7] Caresta M, Kessissoglou NJ. Free vibrational characteristics of isotropic coupled cylindrical–conical shells. J Sound Vib 2010;329:733–51. [8] Kang JH. Three-dimensional vibration analysis of joined thick conical– cylindrical shells of revolution with variable thickness. J Sound Vib 2012;331: 4187–98. [9] Qu Y, Chen Y, Long X, Hua H, Meng G. A variational method for free vibration analysis of joined cylindrical–conical shells. J Vib Control 2012. http://dx.doi. org/10.1177/1077546312456227. [10] Patel BP, Ganapathi M, Kamat S. Free vibration characteristics of laminated composite joined conical–cylindrical shells. J Sound Vib 2000;237:920–30. [11] Benjeddou A. Vibrations of complex shells of revolution using B-spline finite elements. Comput Struct 2000;74:429–40. [12] Tavakoli MS, Singh R. Eigensolutions of joined/hermetic shell structures using the state space method. J Sound Vib 1989;130:97–123. [13] Galletly GD, Mistry J. The free vibrations of cylindrical shells with various end closures. Nucl Eng Design 1974;30:249–68. [14] Cheng L, Nicolas J. Free vibration analysis of a cylindrical shell–circular plate system with general coupling and various boundary conditions. J Sound Vib 1992;155:231–47. [15] Lee YS, Choi MH. Free vibrations of circular cylindrical shells with an interior plate using the receptance method. J Sound Vib 2001;248:477–97. [16] Liang S, Chen HL. The natural vibration of a conical shell with an annular end plate. J Sound Vib 2006;294:927–43. [17] Redekop D. Vibration analysis of a torus-cylinder shell assembly. J Sound Vib 2004;277:919–30. [18] Shang XC. Exact analysis for free vibration of a composite shell structurehermetic capsule. Appl Math Mech (English Edition) 2001;22:1035–45. [19] Lee YS, Yang MS, Kim HS, Kim JH. A study on the free vibration of the joined cylindrical–spherical shell structures. Comput Struct 2002;80:2405–14. [20] Hoppmann II WH. Some characteristics of the flexural vibrations of orthogonally stiffened cylindrical shells. J Acoust Soc Am 1958;30:77–82. [21] Caresta M, Kessissoglou NJ. Acoustic signature of a submarine hull under harmonic excitation. Appl Acoust 2010;71:17–31. [22] Mustafa BAJ, Ali R. An energy method for free vibration analysis of stiffened circular cylindrical shells. Comput Struct 1989;32:355–63. [23] Wang CM, Swaddiwudhipong S, Tian J. Ritz method for vibration analysis of cylindrical shells with ring stiffeners. J Eng Mech 1997;123:134–42. [24] Jafari AA, Bagheri M. Free vibration of non-uniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods. ThinWalled Struct 2006;44:82–90. [25] Qu Y, Chen Y, Long X, Hua H, Meng G. A modified variational approach for vibration analysis of ring-stiffened conical–cylindrical shell combinations. Eur J Mech, A/Solids 2013;37:200–15. [26] 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;98:169–91. [27] Jeffery A, Dai HH. Handbook of mathematical formulas and integrals. 4th ed. California: Academic Press; 2008.