Dynamic stiffness formulation and response analysis of stiffened shells

Computers and Structures 132 (2014) 75–83

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Dynamic stiffness formulation and response analysis of stiffened shells D. Tounsi a, J.B. Casimir b,⇑, S. Abid a, I. Tawfiq b, M. Haddar a a b

Ecole Nationale d’Ingénieurs de Sfax, Université de Sfax, Unité de Mécanique, Modélisation et Production, Route Soukra Km 3.5, BP 1173, 3038 Sfax, Tunisia Institut Supérieur de Mécanique de Paris, LISMMA, 3 rue Fernand Hainaut, 93407 Saint-Ouen, France

a r t i c l e

i n f o

Article history: Received 15 June 2013 Accepted 6 November 2013 Available online 8 December 2013 Keywords: Dynamic Stiffness Method Continuous Element Method Axisymmetric stiffened shell Harmonic analysis

a b s t r a c t This work presents a dynamic analysis of a stiffened cylindrical shell using the Dynamic Stiffness Method, also known as the Continuous Element Method. This approach is based on the determination of the dynamic stiffness matrix of an unmeshed structure. A method for calculating the dynamic stiffness matrix of an axisymmetric shell stiffened with multiple stiffeners at arbitrary locations is given. Thus a stiffened cylindrical shell is subjected to free-free boundary conditions and three types of loads. A finite element model is used in order to validate the numerical results obtained from the method. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Ring stiffened cylindrical shells occupy a prominent position in engineering, such in as aerospace structures (missiles, spacecraft), mechanical structures (vessels, launch vehicles) and marine structures (submarine hulls). Most of these structures are subjected to different dynamic loads which can affect the dynamic characteristics of the shell. Consequently, a large number of computational methods have been developed to study the dynamic behavior of such stiffened shells. The free vibration of stiffened cylindrical shells has been studied since 1950 by different researchers. The most commonly used method for studying the vibratory behavior of stiffened shells is the energy method based on the Rayleigh–Ritz procedure. Mustafa and Ali [1] analyzed a stiffened cylindrical shell to determine natural frequencies using the Rayleigh–Ritz procedure, the stiffener and the shell are assumed to be in the same structure. Mikulas and Mc Elman [2] studied the natural frequencies of eccentricaly stiffened cylindrical shells and Rosen and Singer [3] treated the case of axially loaded stiffened cylindrical shells using Donell and Flugge theories. Furthermore, Jafari and Bagheri [4] presented a comparison between analytical and experimental results for a non-uniformly ring stiffened cylindrical shell. The analytical results were obtained using the Rayleigh–Ritz method with a displacement function proposed by the authors. Recent research includes that performed by Yegao et al. [5] who presented a modified variational approach for the vibration of a ring-stiffened conical cylindrical shell combination. ⇑ Corresponding author. Tel.: +33 1 49452963. E-mail addresses: [email protected] (D. Tounsi), jean-baptiste.casimir@ supmeca.fr (J.B. Casimir), Imad.tawfi[email protected] (I. Tawfiq), mohamed.haddar@ enis.rnu.tn (M. Haddar). 0045-7949/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2013.11.003

All the references cited above focused on the mathematical equation used in the Rayleigh–Ritz method. In this paper, we use another method called the Dynamic Stiffness Method (DSM), a highly efficient method for studying forced vibration and which gives precise and rapid results without any meshing of the structure. It is based on the determination of the dynamic stiffness matrix denoted KðxÞ, which specifies the displacement of the structure and the external harmonic forces applied on it. All the displacements and forces are developed using the Fourier series in terms of the circumferential coordinate. Considering this expansion, the method is very similar to that used in a recent study carried out by Santos et al. [6]. In this latter work, the authors have used a semi-analytical finite element model to analyse laminated 3D axisymmetric shells. Here, the difference lies in the longitudinal coordinate for which no finite element discretization is used. The longitudinal solution is an analytical one. The DSM is used to cover the different limitations of the Finite Element Method (FEM). Indeed, although FEM has provided good results for all vibratory structures, its use is less efficient for the high frequency range due to the discretization of the domain and its boundary. Therefore the DSM is used as it eliminates discretization errors and is able to predict an infinite number of eigensolutions [7–9]. Since the end of the 70’s, different codes have been developed using the DSM. This method was developed for straight beam assemblies [10–12], and for circular and helical beams [13]. Recently, researchers have determined the DSM of sandwich and composite beams [14–16], plates [17–19], axisymmetric shells [20,21] and circular rings [22]. This paper carries on from recent research, and carries on directly from previous works on axisymmetric shells. In this work, circumferential stiffeners are taken into account as local discontinuities of the shell thickness. The number

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and the locations of these discontinuities are arbitrary. This paper is organized as follows: in Section 2, the formulation of the dynamic stiffness matrix of the structure is developed, after which we present the results of the forced vibration of a free-free stiffened cylindrical shell. These results are then validated by FEM results and discussed in Section 3. Finally, we present our conclusions in Section 4. 2. Stiffened axisymmetric shell continuous element 2.1. Geometry The geometry of the stiffened structure is composed of an axisymmetric shell with n thickness discontinuities, see Fig. 1. The middle surface of the shell is obtained from the rotation of the planar curve l about axis D, see Fig. 2. The radius of curvature Rs of the planar curve is dependent on the curvilinear abscissa s along this curve. The circumferential radius of the middle surface is denoted Rh , and is dependent on the curvilinear abscissa s. The thickness of the shell is uniform between two discontinuities. A local basis ðer ; eh ; nÞ is defined on each point of the middle surface. 2.2. Dynamic stiffness matrix of the axisymmetric shell with constant thickness In this section, the main principles of the procedure for calculating the dynamic stiffness matrix of an axisymmetric shell with constant thickness are given. This procedure has been fully described and validated with the use of finite element models by Casimir et al. [20]. In the paper cited, behavior relationships and equilibrium equations were combined in order to obtain a system of second order partial differential equations satisfied by the components of a state vector Eðs; h; tÞ. In the case of harmonic regimes, the elimination of the time variable t leads to a system of second order partial

Fig. 2. Middle surface.

differential equations parametrized with the circular frequency x of the regime. This system has the following form (see Eq. 1):

@E @E @2E ¼ ½As ðsÞ  x2 Ms :E þ Bs ðsÞ: þ Cs ðsÞ: 2 @s @h @h

ð1Þ

The components of the state vector E are: ðu; v ; w; b; bh ; N s ; N hs ; T s ; Ms ; M hs Þ where u; v ; w are the components of the displacement of any point of the shell’s middle surface in the local basis ðes ; eh ; nÞ and b; bh are rotations of the middle surface. N s and Nhs are internal tension forces, T s is the internal shear force, Ms and Mhs are internal bending moments. The other internal forces have been eliminated from the system. As ðsÞ; Bs ðsÞ; Ms ðsÞ and Cs ðsÞ are 10  10 matrices. The angular h variable is eliminated by taking into account the Fourier expansions of displacements and internal forces. This leads to a system of first order partial differential equations satisfied by the coefficients of the Fourier expansion of the state vector. This differential system of equations can be partitioned into two uncoupled systems considering symmetry properties of the unknowns. We obtain:

dEim ¼ Dim ðs; xÞ:Eim ds

with i ¼ 1; 2

ð2Þ

Dim ðs; xÞ are 10  10 matrices given by Eq. (3). Dim ðs; xÞ ¼ Fig. 1. Axisymmetric stiffened shell.

As11 ðsÞ  x2 Ms11  m2 Cs11 ðsÞ

ð1Þiþ1 mBs12 ðsÞ

ð1Þi mBs21 ðsÞ

As22 ðsÞ  x2 Ms22  m2 Cs22 ðsÞ

!

ð3Þ

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D. Tounsi et al. / Computers and Structures 132 (2014) 75–83

Asij ðsÞ; Bsij ðsÞ; Msij ðsÞ and Csij ðsÞ are submatrices of the matrices As ðsÞ; Bs ðsÞ; Ms ðsÞ and Cs ðsÞ. The components of the state vector Eim are as follows:



s

s

s

s

s

s

E1m ðsÞ ¼ ð um ; wm ; bm ; Nsm ; T sm ; M sm ;

a

vm;

a

a

a

bhm ; Nhsm ; M hsm Þ

E2m ðsÞ ¼ ða um ; a wm ; a bm ; a Nsm ; a T sm ; a M sm ; s v m ; s bhm ; s Nhsm ; s M hsm Þ ð4Þ These components are coefficients of each Fourier expansion. For example, the displacement uðs; hÞ is given by Eq. (5)

us ðs; hÞ ¼ us0 ðsÞ þ

þ1 X

s

usm ðsÞ cos mh þ

m¼1

þ1 X

a

usm ðsÞ sin mh

ð5Þ

m¼1

For each circular frequency x, dynamic transfer matrices Tim ðsÞ are defined according to Eq. (6):

Eim ðsÞ ¼ Tim ðsÞEim ð0Þ for i ¼ 1; 2

ð6Þ

These transfer matrices satisfy the differential system of equations defined by the matrices Dim ðxÞ and can be computed numerically according to Eq. (7):

Tim ðs; xÞ ¼ e

Rs 0

Dim ðl;xÞdl

ð7Þ

Therefore the dynamic stiffness matrices are obtained from the dynamic transfer matrices computed for s ¼ L. 2.3. Dynamic stiffness matrix of the stiffened axisymmetric shell In this section, the previous method is generalized to take into account an arbitrary number of discontinuities in terms of thickness and of the position of the middle surface of the shell. This discontinuous model can be used to simulate the influence of circumferential stiffeners. Fig. 3 gives the longitudinal section of the shell and the parameters of the discontinuous geometry. Lk and hk are respectively the length and the thickness of the kth discontinuity. As for the case of shells with constant thickness, the dynamic stiffness matrices of the stiffened shell are obtained using the previous calculation of the symmetric and antisymmetric dynamic transfer matrices. These matrices are denoted Tim ; i ¼ 1; 2. m is relative to the m-th term of the Fourier series expansion. These matrices relate symmetric and antisymmetric contributions of displacements and internal forces acting on both edges of the shell. The matrix relations involve the projections of the unknowns on the Fourier functional basis and are uncoupled according to the symmetry partition given by Eq. (4). In the case of a shell that can be divided into n different shells as described in Fig. 3, the matrices Tim of the whole shell are given by the following matrix products (see Eq. 8): ðkÞ

Tim ¼ Pnk¼1 Tim

ð8Þ

ðkÞ

Tim ¼ e

R Li 0

ðkÞ

Di ds

ð9Þ

m

ðkÞ

and where Dim is given by Eq. (3). The final step consists in the determination of the dynamic stiffness matrix which is expressed according to [13] as follows: 12

Kim ¼

T

1 im

 11 T

12 T

12 T

im

T im

22

where submatrices kl T

PT :Tim :P ¼

11

T

im

21

T

im

1 im

T

!

1 im

ð10Þ

 11 T

im

im

are defined according to Eq. (11):

12

T

im

22

T

im

! ð11Þ

and P is a permutation matrice that permutes the components of the state vectors s Em and a Em in such a way that these vectors have the following sequences: s

Em ¼ ð s u

a

Em ¼ ð a u

a

v

s

v

a

s

w w

s

bh

a

bh

a

b

s

s

a

b

a

Ns Ns

Nhs

s

s

a

N hs

Ts Ts

s

Ms

a

Ms

a

M hs Þ

ð12Þ

s

ð13Þ

M hs Þ

The matrix P is given in Appendix A. Matrices Kim are the symmetric and antisymmetric dynamic stiffness matrices of the whole shell that take into account all the geometric discontinuities. These matrices depend only on the circular frequency x and the harmonic m of the Fourier expansion. 3. Numerical examples To illustrate the application of the method, numerical examples of the forced vibration analysis of a non-stiffened and stiffened cylindrical shell are given. For this kind of shell, the matrices As ; Bs ; Ms and Cs are not depending on the curvilinear s and are given in Appendix B. The numerical results determined using the procedure presented in the previous section are validated with results obtained from a commercial finite element software. The structure studied is a cylindrical shell stiffened with two stiffeners. The middle surface of the shell is a cylinder whose length and radius are respectively 0:53 m and 0:35 m. The positions and the dimensions of the stiffeners are summarized in Fig. 4. The shell and the stiffeners are made of soft steel whose properties are: Young modulus E ¼ 2:1 GPa, Poisson ratio m ¼ 0:3 and mass density q ¼ 7800 kg/m3. Three types of load are applied on the stiffened edge of the shell and free-free boundary conditions are used:  An axisymmetric radial load.  An antisymmetric radial load.  A concentrated radial load.

ðkÞ

where Tim are the dynamic transfer matrices of the k-th shell. The dynamic transfer matrices of the k-th shell are calculated by Eq. (7) where the curvilinear abscissa s is given by Li , that is to say:

h2

h3

h4

h1 L3 L1

These loads are harmonically varying dynamic loads with the intensity of constant amplitudes.

L4

L5 hn-1

L2

Ln-1 Δ

Fig. 3. Discontinuous geometry.

hn

Ln

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0.53 m 0.09 m

0.05 m

0.15 m

0.04 m

0.15 m

0.15 m

0.04 m

Fig. 4. Cylindrical stiffened shell.

3.1. Axisymmetric radial load An axisymmetric radial load per unit of length p is applied on the first edge of the shell (see Fig. 5). The only non-zero force component is fn and is defined as follows:

h 2 ½0; 2p;

f n ðhÞ ¼ p

ð14Þ

The Fourier expansion series of this load gives the only non-zero component:

F10 ¼ ð 0 0 p 0 0 0 0 0 0 0 Þ

ð15Þ

For each circular frequency x, the Fourier coefficients of displacements along the two edges of the stiffened shell are solutions of the following systems of linear equations:

½Kim ðxÞUim ¼ Fim

in Fig. 6. Despite a free-free boundary condition, the response for 0 Hz cannot be obtained from infinite values due to the axisymmetry of the displacement response. Very good convergence between the FE results and the CE results is obtained. It is also noteworthy that more than 2500 8-node finite elements are required to achieve the precision of the continuous element model. For benchmark purposes, Table 1 gives numerical values obtained by the CEM formulation for several frequencies. No damping has been introduced in these models. An undamped response is more convenient to compare results. Nevertheless, structural damping given by complex moduli can be used without any problem. For example, the harmonic response corresponding to different levels of damping are shown in Fig. 7. 3.2. Antisymmetric radial load

ð16Þ

These coefficients are used to determine the harmonic response at the point located at h ¼ 0 along the edge of the shell. The radial displacement of this point is s w0 ð0Þ. The validation of the numerical results is achieved by comparing it with Finite Element results. The axisymmetric shell is discretized using 8-node shell finite elements that take into account rotatory inertia and shear deformation. Three finite element models are used. The meshes of the model are refined to observe the convergence of the results towards the continuous element model. A 960 finite element mesh is given in Fig. 5 (the thicknesses of the shell elements are displayed, this is not a volume mesh). For the continuous element model, one term is used in the Fourier expansion series because of the axisymmetry of the problem. Numerical results are obtained over [0 Hz, 20000 Hz] and are given

The second load applied to the stiffened shell structure is an antisymmetric radial load per unit length p (see Fig. 8). This load is defined mathematically by Eq. (17).

8 > < 8h 2  p; 0½; f n ðhÞ ¼ p 8h 20; p½; f n ðhÞ ¼ p > : fn ð0Þ ¼ fn ðpÞ ¼ 0

ð17Þ

The Fourier expansion of the load defined gives:

F1m ¼ 0

ð18Þ

and

F2m ¼



2p ½1 mp

0 0

 ð1Þm  0 0 0 0 0 0 0



ð19Þ

For each circular frequency x, Fourier coefficients of displacements Uim along the two edges of the stiffened shell are obtained from the solution of the system of Eq. (16) over the frequency range [0 Hz, 6000 Hz]. These coefficients are used to compute the harmonic response at the point located at h ¼ p=2 along the middle line of the loaded edge. The radial displacement of this point is given by Eq. (20):

wðp=2Þ ¼

n X

a

w2mþ1 ð1Þm

ð20Þ

m¼0

n is the number of terms of the Fourier expansion. This number depends on the load applied, for example, for an antisymmetric distributed load ten terms is good enough to obtain the convergence of the results. In Fig. 9, it can be seen that the harmonic response obtained from FEM models converges well with the harmonic response obtained from the CEM, thereby improving the efficiency of this method. For benchmark purposes, Table 2 gives numerical values obtained by the CEM formulation for several frequencies. 3.3. Concentratred radial load

Fig. 5. Axisymmetric radial load.

The last load applied to the stiffened shell structure is a concentrated radial force (see Fig. 10).

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D. Tounsi et al. / Computers and Structures 132 (2014) 75–83

Fig. 6. Harmonic response to an axisymmetric radial load.

Table 1 Numerical values of the radial displacement of the point located at h ¼ 0. Frequency (Hz)

5000

w (dB)

189.7

10000 203.4

15000 207.5

20000

  F10 ¼ 0 0 p1R 0 0 0 0 0 0 0 ;

ð22Þ

 F1m ¼ 0 0

ð23Þ

235.1

1 2pR

0 0 0 0 0 0 0



and

F2m ¼ 0

This load is defined mathematically by Eq. (21).

8h 2  p; p½;

f n ðhÞ ¼ d

ð21Þ

d is the Dirac delta function. The Fourier expansion of the load defined gives:

ð24Þ

The response is evaluated at the excited point (h ¼ 0) according to Eq. (20). Fig. 11 gives a comparison between the CEM response and FEM response at this point.

Fig. 7. Harmonic response for different levels of damping.

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D. Tounsi et al. / Computers and Structures 132 (2014) 75–83

Fig. 8. Antisymmetric radial load.

Fig. 10. Concentrated radial load.

For benchmark purposes, Table 3 gives numerical values obtained by the CEM formulation for several frequencies. 6000 finite elements are necessary to converge towards the CEM results. The convergence of these results allows validating the CEM formulation. Twenty terms are used in CEM series. This number is evaluated from a convergence study of the results. Fig. 12 provides a representation of this study. It can be seen that 10 terms are not enough, especially in the neighborhood of

antiresonance frequencies. The use of more terms is not necessary, since the response curves are identical in this frequency range if more terms are taken into account. 3.4. Modal analysis and post-processing Modal analysis can be achieved with the use of the WilliamsWittrick algorithm [23]. In our work, the harmonic response for a concentrated load is used to evaluate the bending eigenfrequencies

Fig. 9. Harmonic response to an antisymmetric radial load.

Table 2 Numerical values of radial displacement of the point located at h ¼ 0. Frequency (Hz)

1000

2000

3000

4000

5000

6000

w (dB)

189.8

182.3

200.3

196

224

210.5

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D. Tounsi et al. / Computers and Structures 132 (2014) 75–83

Fig. 11. Harmonic response to a concentrated radial load.

Table 3 Numerical values of radial displacement of the point located at h ¼ 0. Frequency (Hz)

1000

2000

3000

4000

5000

6000

w (dB)

173.4

176.9

190.3

188.4

175.7

182.5

Fig. 12. Convergence study for the number of terms in CEM series.

of the structure by considering the frequencies for which the undamped response becomes infinite. Table 4 gives the first 6 eigenfrequencies obtained for CEM and FEM. The mode shapes are obtained using a post-processing procedure. For a given eigenfrequency, Eq. (4) is used to calculate the state vector along the shell and, by employing Eq. (2), the displacements of any point inside the shell are processed.

Table 4 Bending eigenfrequencies. Frequency

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

CEM FEM (325 FE) FEM (960 FE) FEM (2500 FE)

328 327 327 327

442 442 442 442

897 898 897 897

1081 1082 1082 1082

1669 1673 1670 1670

1875 1879 1876 1875

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D. Tounsi et al. / Computers and Structures 132 (2014) 75–83

4. Conclusion

s

B ¼

It has long been known that the Dynamic Stiffness Method is a computationally, efficient and accurate numerical approach for computing the harmonic response of simple structures. This paper demonstrated how this method can be used to deal with more complex structures such as axisymmetric stiffened shell structures. A continuous element of this type was successfully compared with the Finite Element Method in order to validate the formulation. Computational time depends on the number of terms in the CEM series. For example, 1000 frequencies are processed in 5 s, 10 s and 15 s for respectively 10, 20 and 30 terms on a 2.53 GHz Intel 2 core CPU P8700. When comparing the computation time with that of FEM, it was clear that the convergence of finite elements requires considerably more computational resources to achieve comparable accuracy. Optimization procedures, for example, could greatly benefit from this rapidity and accuracy over large frequency ranges. Future work will involve coupling between axisymmetric shells and longitudinal stiffeners.

0

Bs12

Bs21

0

Bs12

KRm mD 0 B 0 0 0 B B B K m mKR 0 1 B ¼ 2 B 0 KR KR  D B 0 B @ B57 B58 0 0

Bs21

0

0

0 0

0

0

0

0

0

0 0

0

0

0

0C C C 0C C C 0C C C 0C C 0C C C 0C C C 0C C C 0A

0

0 1

0 0

0

0

0

0

0

0

0 1 0

0

0

0

0

0

0 0

0

1 0

0

0

0

0 0

0

0 1

1 0

0

0 0

0

0

0

0

0

0 1 0

0

0

0

0

0

0

0 0

1 0

0

0

0

0

0 0

0

0 1

0

0

KR

0

0

0

0

KRm 0

0

0

Dm

B58

0

0

Ms22

0

0

As22

ðB:1Þ

0

0

0

0 KR2 D kGh

0 0 0 KR2  D

1 R 0 C C C KR2 C D C C 0 C C K m C A 0 ðB:2Þ

0

As22

0 B B 0 B 1 B ¼ 2 B kGhðKR2 DÞ KR  D B R2 @ kGhðKR2 DÞ R

0

2R2 1m

0

2R 1m

kGhðKR2 DÞ R 2

kGhðKR  DÞ

0 0

2R 1m

1

C C 0 C A 0

ðB:3Þ

Eh shell, D ¼ 12ð1 m2 Þ ; k is the shear correction factor and G the shear

modulus.

0

0

0 0 0C C C 0 0 0C C C 0 0 0C C 0 0 0C A 0 0 0

0

0

0

qh

0

h  q12R

0

qh

0

0

qh3

3

12

0 0

ðB:8Þ

1

0 0C C C 0 0C A 0 0

ðB:9Þ

Cs11

0

0

Cs22

!

Cs11

0

0

B 0 0 B B B 0 0 B 1 B KDð1mÞ ¼ 2 0 2 KR  D B B 2R 2 B DÞ B 0  kGhðKR R2 @ D2 ð1mÞ 0 2R3 0

Cs22

ðB:10Þ 1

0

0 0 0

0

0 0 0C C C 0 0 0C C 0 0 0C C C C 0 0 0C A 0 0 0

ðB:11Þ

1 0 0 0 0C C C 0 0C A 0 0

ðB:12Þ

0 D2 ð1mÞ 2R3

0 D2 ð1mÞ 2R2

0 0 B 0 0 1B 2 2 2 ¼ B KD KD 2 B K ð m  1Þ þ ð m  1Þ þ DR3 a@ R R2 2 2 KD ðm2  1Þ þ DR3 KDðm2  1Þ þ DR2 R

C

2KR2 C ð1mÞD C

with K ¼ 1Ehm2 where h is the thickness of the shell, m the Poisson’s ration and E the Young’s modulus. R is the radius of the cylindrical 3

0

0

!

0 KRm 0 R2 2 B0 0 0 KR þ D B B B0 1 0 R Km B ¼ 2 B 0 0 0 KR  D B B0 B 0 K 2 ð1  m2 Þ  KD 0 K mR @ 2 R 0

B58 ¼ KD ðm2  1Þ R

with q the mass density of the material.

0

As11

0 0 0

0 0 B 0 0 B ¼B h3 B qh  q12R @ h3 qh3  q12R 12

Ms22

and

1

0

0

ðA:1Þ

ðB:6Þ

ðB:7Þ

0

 12R

C C C A

0 KRm

0

qh3

1

!

Ms11

B B B B B ¼B B B B @

Cs ¼

As11

K m

with B57 ¼ K 2 ðm2  1Þ þ KD  kGh ðKR2  DÞ R2 R2 2 D2  kGh ðKR  DÞ þ . 3 R R s

ðB:5Þ

KR 0

Ms11

Appendix B

As ¼

D 0

1

0

0 C C C 0 C C K C C C 0 A

D

0 0

1

0

KR 0 K 0 1B B ¼ B a@ 0 B57

Permutation matrix

0 1

0

0

Appendix A

1 B0 B B B0 B B B0 B B B0 ½P ¼ B B0 B B B0 B B B0 B B @0

ðB:4Þ

0

M ¼

0

!

References [1] Mustafa BAJ, Ali R. An energy method for free vibration analysis of stiffened circular cylindrical shells. Comput Struct 1989;32:355–63. [2] M.M. Mikulas Jr., J.A. McElman, On the free vibration of eccentrically stiffened cylindrical shells and plates, NASA, TN-D 3010, 1965. [3] Rosen A, Singer J. Vibrations of axially loaded stiffened cylindrical shells. J Sound Vib 1974;34(3):357–78. [4] Jafari AA, Bagheri M. Free vibration of non-uniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods. Thin Walled Struct 2006;44:82–90.

D. Tounsi et al. / Computers and Structures 132 (2014) 75–83 [5] 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. [6] Santos H, Mota Soares CM, Mota Soares CA, Reddy JN. A semi-analytical finite element model for the analysis of laminated 3D axisymmetric shells: Bending, free vibration and buckling. Compos Struct 2005;71:273–81. [7] Kulla PH. Continuous elements, some practical examples. ESTEC workshop proceedings: modal representation of flexible structures by continuum methods, ESA editor, Noordwijk, Netherlands; 1989. [8] Leung AYT. Dynamic stiffness and substructures. New-York: Springer; 1993. [9] Lee U. Spectral element method in structural dynamics. Wiley; 2009. [10] Akesson BA. PFVIBAT – a computer program for plane frame vibration analysis by an exact method. Int J Numer Methods Eng 1976;10:1221–31. [11] C. Duforêt, Dynamic study of an assembling of rods in medium and higher frequency ranges in computer code ETAPE. In: Proceedings of the third colloquium on new trends in structure calculations, Bastia, Corsica; 1985, p. 229–246. [12] Anderson MS, Williams FW. BUNVIS-RG Exact frame buckling and vibration program, with repetitive geometry and substructuring. J Spacecraft Rockets 1987;24(4):353–61. [13] Casimir JB, Duforet C, Vinh T. Dynamic behavior of structures in large frequency range by continuous element method. J Sound Vib 2003;267:1085–106. [14] Banerjee JR, Sobey AJ. Dynamic stiffness formulation and free vibration analysis of a three-layered sandwich beam. Int J Solids Struct 2005;42:2181–97.

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[15] Banerjee JR, Su H. Dynamic stiffness formulation and free vibration analysis of a three-layered sandwich beam. Int J Solids Struct 2005;42:2181–97. [16] Howson WP, Zare A. Exact dynamic stiffness matrix for flexure vibration of three-layered sandwich beams. J Sound Vib 2005;282:753–67. [17] Casimir JB, Kevorkian S, Vinh T. The dynamic stiffness matrix of two dimensional elements: application to Kirchhoff’s plate continuous elements. J Sound Vib 2005;287:571–89. [18] Boscolo M, Banerjee JR. Dynamic stiffness formulation for composite Mindlin plates for exact modal analysis of structures. Part I: theory. Comput Struct 2012;96-97:61–73. [19] Boscolo M, Banerjee JR. Dynamic stiffness formulation for composite Mindlin plates for exact modal analysis of structures. Part II: results and applications. Comput Struct 2012;96–97:74–83. [20] Casimir JB, Nguyen MC, Tawfiq I. Thick shells of revolution: derivation of the dynamic stiffness matrix of continuous elements and application to a tested cylinder. Comput Struct 2007;85:1845–957. [21] Khadimallah MA, Casimir JB, Chafra M, Smaoui H. Dynamic stiffness matrix of an axisymmetric shell and response to harmonic distributed loads. Comput Struct 2011;89:467–75. [22] Tounsi D, Casimir JB, Haddar M. Dynamic stiffness formulation for circular rings. Comput Struct 2012;112–113:258–65. [23] W Williams F, H Wittrick W. An automatic computational procedure for calculating natural frequencies of skeletal structures. Int J Mech Sci 1970;12:781–91.