the effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical...

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The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells Zenon J.G.N. del Prado a,n , Ana Larissa D.P. Argenta a , Frederico M.A. da Silva a , Paulo B. Gonçalves b a Federal University of Goiás, School of Civil Engineering, Avenida Universitária, 1488, Setor Leste Universitário, 74605-200 Goiânia, GO, Brazil b Catholic University of Rio de Janeiro, Department of Civil Engineering, Rua Marquês de São Vicente, 225, Gávea, 22453-900 Rio de Janeiro, RJ, Brazil article info Article history: Received 14 December 2013 Received in revised form 26 March 2014 Accepted 27 March 2014 Keywords: Cylindrical shell Orthotropic material Lateral load Dynamic instability Non-linear vibration abstract The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the inuence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell's non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the RungeKutta method. The obtained results show that the material properties and geometric relations have a signicant inuence on the instability loads and resonance curves of the orthotropic shell. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Circular cylindrical shells subjected to various loading condi- tions are widely used in several engineering areas and industrial applications and their analysis has become an important research area in applied mechanics. These structures present large capacity to withstand both axial loads and lateral pressures; however, they may display complex dynamic behavior due mainly to geometric non-linearity and sensitivity to small imperfections. Although a large number of papers has been published on their non-linear behavior, only a small number of investigations is concerned with the analysis of orthotropic shells. However many natural or articial materials present orthotropic properties and, as shown in this paper, optimal orthotropic materials can be developed to withstand a specic load. Also corrugated and densely stiffened materials can be described as an equivalent orthotropic materials, as shown by Briassoulis [1], Shen [2], Siad [3], Andrianov et al. [4] and Torkamani et al. [5]. One of the rst works on the dynamic behavior of orthotropic cylindrical shells was published by Jain [6] who studied the free vibrations of orthotropic empty cylindrical shells and shells partially or completely lled with an incompressible and non- viscous uid. Warburton and Soni [7] and Bradford and Dong [8] analyzed, respectively, the resonant response and lateral vibra- tions of orthotropic cylindrical shells. Chen et al. [9] and Chen and Ding [10] studied the free vibrations of both uid-lled isotropic and orthotropic cylindrical shells, respectively. Using the SandersKoiter non-linear shell theory, Selmane and Lakis [11] studied the inuence of geometric non-linearities associated with the shell and the uid ow on the dynamics of empty and uid- lled elastic thin orthotropic cylindrical shells. Del Prado et al. [1215] and Argenta et al. [16], using Donnell's non-linear shallow shell theory, without considering the effect of shear deformation, studied the inuence of the ratio of Young's modulus in the circumferential and axial direction as well as geometric relations, on the non-linear vibrations of a simply supported orthotropic cylindrical shell, sub- jected to axial and lateral time-dependent loads. Ip et al. [17], using Love's rst-approximation shell theory, studied the free vibrations of ber-reinforced composite cylindrical shells and the inuence of the shell thickness on the exural and stretching energy. An experimental and numerical analysis of a base-excited thin orthotropic cylindrical shell with a top mass was carried out by Mallon Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/nlm International Journal of Non-Linear Mechanics http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.017 0020-7462/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. Tel.: þ55 62 3209 6265. E-mail addresses: [email protected] (Z.J.G.N. del Prado), [email protected] (A.L.D.P. Argenta), [email protected] (F.M.A. da Silva), [email protected] (P.B. Gonçalves). Please cite this article as: Z.J.G.N. del Prado, et al., The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.017i International Journal of Non-Linear Mechanics (∎∎∎∎) ∎∎∎∎∎∎

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The effect of material and geometry on the non-linear vibrationsof orthotropic circular cylindrical shells

Zenon J.G.N. del Prado a,n, Ana Larissa D.P. Argenta a, Frederico M.A. da Silva a,Paulo B. Gonçalves b

a Federal University of Goiás, School of Civil Engineering, Avenida Universitária, 1488, Setor Leste Universitário, 74605-200 Goiânia, GO, Brazilb Catholic University of Rio de Janeiro, Department of Civil Engineering, Rua Marquês de São Vicente, 225, Gávea, 22453-900 Rio de Janeiro, RJ, Brazil

a r t i c l e i n f o

Article history:Received 14 December 2013Received in revised form26 March 2014Accepted 27 March 2014

Keywords:Cylindrical shellOrthotropic materialLateral loadDynamic instabilityNon-linear vibration

a b s t r a c t

The extensive use of circular cylindrical shells in modern industrial applications has made their analysisan important research area in applied mechanics. In spite of a large number of papers on cylindricalshells, just a small number of these works is related to the analysis of orthotropic shells. However severalmodern and natural materials display orthotropic properties and also densely stiffened cylindrical shellscan be treated as equivalent uniform orthotropic shells. In this work, the influence of both materialproperties and geometry on the non-linear vibrations and dynamic instability of an empty simplysupported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied.Donnell's non-linear shallow shell theory is used to model the shell and a modal solution with sixdegrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method isapplied to derive the set of coupled non-linear ordinary differential equations of motion which are, inturn, solved by the Runge–Kutta method. The obtained results show that the material properties andgeometric relations have a significant influence on the instability loads and resonance curves of theorthotropic shell.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Circular cylindrical shells subjected to various loading condi-tions are widely used in several engineering areas and industrialapplications and their analysis has become an important researcharea in applied mechanics. These structures present large capacityto withstand both axial loads and lateral pressures; however, theymay display complex dynamic behavior due mainly to geometricnon-linearity and sensitivity to small imperfections.

Although a large number of papers has been published on theirnon-linear behavior, only a small number of investigations isconcerned with the analysis of orthotropic shells. However manynatural or artificial materials present orthotropic properties and,as shown in this paper, optimal orthotropic materials can bedeveloped to withstand a specific load. Also corrugated anddensely stiffened materials can be described as an equivalentorthotropic materials, as shown by Briassoulis [1], Shen [2], Siad[3], Andrianov et al. [4] and Torkamani et al. [5].

One of the first works on the dynamic behavior of orthotropiccylindrical shells was published by Jain [6] who studied the freevibrations of orthotropic empty cylindrical shells and shellspartially or completely filled with an incompressible and non-viscous fluid. Warburton and Soni [7] and Bradford and Dong [8]analyzed, respectively, the resonant response and lateral vibra-tions of orthotropic cylindrical shells. Chen et al. [9] and Chen andDing [10] studied the free vibrations of both fluid-filled isotropicand orthotropic cylindrical shells, respectively.

Using the Sanders–Koiter non-linear shell theory, Selmane andLakis [11] studied the influence of geometric non-linearities associatedwith the shell and the fluid flow on the dynamics of empty and fluid-filled elastic thin orthotropic cylindrical shells. Del Prado et al. [12–15]and Argenta et al. [16], using Donnell's non-linear shallow shell theory,without considering the effect of shear deformation, studied theinfluence of the ratio of Young's modulus in the circumferential andaxial direction as well as geometric relations, on the non-linearvibrations of a simply supported orthotropic cylindrical shell, sub-jected to axial and lateral time-dependent loads.

Ip et al. [17], using Love's first-approximation shell theory, studiedthe free vibrations of fiber-reinforced composite cylindrical shells andthe influence of the shell thickness on the flexural and stretchingenergy. An experimental and numerical analysis of a base-excited thinorthotropic cylindrical shell with a top mass was carried out by Mallon

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/nlm

International Journal of Non-Linear Mechanics

http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.0170020-7462/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ55 62 3209 6265.E-mail addresses: [email protected] (Z.J.G.N. del Prado),

[email protected] (A.L.D.P. Argenta), [email protected] (F.M.A. da Silva),[email protected] (P.B. Gonçalves).

Please cite this article as: Z.J.G.N. del Prado, et al., The effect of material and geometry on the non-linear vibrations of orthotropiccircular cylindrical shells, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.017i

International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

[18]. Chevalier [19] studied an orthotropic shell made of poly-ethyleneterephthalate, where the orthotropy is due to the fabrication process.Daneshjou et al. [20] studied analytically the characteristics of soundtransmission through an orthotropic cylindrical shell in a subsonicexternal flow.

The behavior of laminated circular cylindrical shells wasanalyzed in Amabili [21] using the Amabili–Reddy higher-ordershear deformation theory and the results obtained were comparedto those obtained by using an higher-order shear deformationtheory retaining only non-linear terms of von Kármán type andthe Novozhilov classical shell theory. Many problems in biologycan also be investigated using the orthotropic shell theory as thosepresented in Yin et al. [22] and Daneshmand and Amabili [23].

In this work, the non-linear vibrations and instabilities of a simplysupported orthotropic cylindrical shell subjected to lateral harmonicpressure are analyzed. The Donnell non-linear shallow shell theorywithout considering the effect of shear deformation is used and fivedifferent orthotropic material are considered in the parametric analy-sis. A modal solution satisfying the relevant boundary and continuityconditions and containing the fundamental, companion and four axi-symmetric modes is used to describe the lateral displacements of theshell. The Galerkin method is applied to derive a set of coupled non-linear ordinary differential equations of motion. A detailed parametricanalysis is conducted to understand the influence of geometricparameters and material orthotropy on the non-linear shell dynamics.The results clarify the marked influence of the orthotropic materialproperties and geometric parameters on the bifurcations and reso-nance curves of the shells. In this work, the mathematical formulationfollows that previously presented in Refs. [24–27].

2. Mathematical formulation

Consider a thin-walled simply supported cylindrical shell withlength L, radius R and thickness h, subjected to a harmonic lateralpressure f. The axial, circumferential and radial coordinates aredenoted by x, y¼Rθ and z, respectively, and the correspondingdisplacements of the shell middle surface are denoted by u, v and

1.0 2.0 3.0 4.0 5.0 6.0L/R

0.00

0.04

0.08

0.12

0.16

5

6

56

78

910

789101112

7

13

8

9

1.0 2.0 3.0 4.0 5.0 6.0L/R

0.00

0.06

0.12

0.18

0.24

5

6

5

6

7

8

789105

7

6

55

1.0 1.5 2.0 2.5 3.0 3.5L/R

0.00

0.10

0.20

0.30

0.40

5

5

6

7

56

56

7

Ω0

Ω0

Ω0

Fig. 2. Variation of natural frequency parameter (Ω0) and associated circumfer-ential wave number (n) as a function of L/R ratio and material. (a) R/h¼800, (b) R/h¼300 and (c) R/h¼75. Case 1, Case 2, Case 3, Case 4,

Case 5.

Table 1Shell physical properties.

Case νxθ νθx Exx (1010 Pa) Eθθ (1010 Pa) Gxθ (1010 Pa) Eθθ/Exx

1 0.131926 0.012114 22.7350 2.0876 0.7958 0.09182 0.131926 0.04 6.8599 2.0799 0.7958 0.30323 0.131926 0.131926 2.0545 2.0545 0.7958 1.00004 0.04 0.131926 2.0799 6.8599 0.7958 3.29825 0.012114 0.131926 2.0876 22.7350 0.7958 10.8905

L

R

x (u)

y (v)z (w)

h

f

z

y

θ

Fig. 1. (a) Shell geometry, coordinates and displacements field. (b) Harmonic lateralpressure.

Z.J.G.N. del Prado et al. / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎2

Please cite this article as: Z.J.G.N. del Prado, et al., The effect of material and geometry on the non-linear vibrations of orthotropiccircular cylindrical shells, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.017i

w, as shown in Fig. 1. The middle surface of the shell is defined as thereference surface and it is assumed that the local coordinate system,which determines the principal axes of material orthotropy, coincideswith the global cylindrical coordinates. The shell is made of an elasticorthotropic material with Young's moduli Ex and Eθ in the axial andcircumferential directions, respectively, shear modulus Gxθ, Poissoncoefficients νxθ and νθx, and mass density ρs.

Based on the Donnell shallow-shell theory, the middle surfacekinematic relations are given, in terms of the three displacementcomponents, by

εx ¼ u;xþ12w2

;x; εθ ¼ v;θR �w

Rþ12

w2;θ

R2 ; γxθ ¼ u;θR þv;xþw;x

w;θR ;

χx ¼ �w;xx; χθ ¼ �w;θθ

R2 ; χxθ ¼ �w;xθ

R; ð1Þ

where εx and εθ are the extensional strains in the axial andcircumferential directions, εxθ is the shearing strain component ata point on the shell middle surface, χx and χθ are the curvaturechanges and χxθ is the twist.

For an orthotropic material, obeying the generalized Hooke'slaw, the stress–strain relations are given by

Nx

Nxθ

Mx

Mxθ

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

¼

A11 A12 0 0 0 0A21 A22 0 0 0 00 0 A33 0 0 00 0 0 D11 D12 00 0 0 D21 D22 00 0 0 0 0 D33

26666666664

37777777775

εxεθγxθχx

χθ2 χxθ

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;; ð2Þ

where Nx, Nθ and Nxθ are the in-plane normal and shearing forceintensities per unit length along the edge of a shell element, Mx,Mθ and Mxθ are the bending and twisting moment resultants, Aij

are the extensional stiffness constants and Dij are the flexural

stiffness constants, defined as

Aij ¼Z h=2

�h=2Cij dz; Dij ¼

Z h=2

�h=2Cijz2 dz; ð3Þ

with

C11 ¼ Exx1�νxθνθx

; C22 ¼ Eθ1�νxθνθx

;

C33 ¼ Gxθ ; C12 ¼ C21 ¼ νxθEθθ1�νxθνθx

;ð4Þ

where, due to symmetry, νθxExx ¼ νxθEθθ [28].The forces per unit length in the axial and circumferential

directions as well as the shear force can be written in terms of astress function F as [24]

Nx ¼F ;θθR2 ; Nθ ¼ F ;xx; Nxθ ¼ �F ;xθ

R: ð5Þ

Disregarding the effect of shear deformations, the non-linearequation of motion based on the von Kármán–Donnell shallowshell theory, in terms of a stress function F and the lateraldisplacement w, is given by

D11w;xxxxþ2

R2ðD12þ2D33Þw;xxθθþ1

R4D22 w;θθθθþβ1 _wþρsh €w

¼ f þ1RF ;xxþ 1

R2F ;xxw;θθþ1

R2F ;θθw;xx� 2

R2F ;xθw;xθ ; ð6Þ

with

β1 ¼ 2ζρshω0; ð7Þwhere ζ is the viscous damping factor and ω0 is the lowest naturalfrequency of the shell, and the lateral load applied on the surfaceof the shell, as can be seen in Fig. 1b, is given by

f ¼ FeþFd sinmπxL

� �cos ðnθÞ cos ðωLtÞ; ð8Þ

1.0 2.0 3.0 4.0 5.0 6.0L/R

0.12

0.15

0.18

0.21

0.24

56

56

78

9

7

8

9

10

117

1.0 2.0 3.0 4.0 5.0 6.0L/R

0.35

0.40

0.45

0.50

0.55

5656

7

89

67

8

9

10

7

11

1.0 2.0 3.0 4.0 5.0 6.0L/R

0.95

0.96

0.97

0.98

0.99

1.00

5

6

56

78

9

678910 511

1.0 2.0 3.0 4.0 5.0 6.0L/R

1.00

1.25

1.50

1.75

2.00

5

5

6

78

67

8

9

10

5

1.0 2.0 3.0 4.0 5.0L/R

1.20

1.60

2.00

2.40

2.80

5

6

67

89

5

Λ0

cr

Λ0

cr

Λ0

cr

Λ0

cr

Λ0

cr

Fig. 3. Variation of lateral critical load parameter (Λ0 cr) and associated circumferential wave number (n) as a function of the shell geometric parameters L/R and R/h. (a) Case1, (b) Case 2, (c) Case 3, (d) Case 4 and (e) Case 5. R/h¼800, R/h¼300, R/h¼75.

Z.J.G.N. del Prado et al. / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3

Please cite this article as: Z.J.G.N. del Prado, et al., The effect of material and geometry on the non-linear vibrations of orthotropiccircular cylindrical shells, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.017i

where Fe is a compressive uniform time independent load, Fd is themagnitude of the harmonic lateral load, m is the number of half-waves in the axial direction, n is the number of waves in the

circumferential direction, t is the time and ωL is the forcingangular frequency.

The compatibility equation is given by

P22F ;xxxxþ1

R2ðP33�2P12ÞF ;xxθθþ1

R4P11F ;θθθθ

¼ �w;xx

Rþ 1

R2ðw2;xθ�w;xxw;θθÞ; ð9Þ

where

P11 ¼ A22

A11A22 �A212; P22 ¼ A11

A11A22 �A212;

P33 ¼ 1A33

; P12 ¼ A12

A11A22 �A212:

ð10Þ

The simply supported out-of-plane (Eq. (11)) and the in-plane(Eq. (12)) boundary conditions are respectively given by

w¼ 0; Mx ¼ 0 at x¼ 0; L; ð11Þ

Nx ¼ 0; v¼ 0 at x¼ 0; L: ð12ÞFor a formulation based on a stress function, the in-plane

boundary conditions are satisfied on the average by introducingthe following conditions, as justified, for example, in [27,29,38,39]Z 2π

0NxR dθ¼ 0 at x¼ 0; L; ð13Þ

Z 2π

0

Z L

0NxθR dθ¼ 0 at x¼ 0; L: ð14Þ

Eq. (13) assures a zero axial force Nx on the average, while Eq.(14) is satisfied when u and w are continuous in θ on average, andv¼0 on average at both ends [29].

2.1. Modal solution for the lateral displacement

The following modal solution for the lateral displacementsw(x,θ,t) in terms of the circumferential and axial variables is

0.4 0.8 1.2 1.6 2.00.0

1.5

3.0

4.5

6.0

|ξ1,

1|

|ξ1,

1|

|ξ1,

1|

|ξ1,

1|

|ξ1,

1|

QP

PT

NS

NS

NS

PTNS NS

PT

NS

PT

0.4 0.8 1.2 1.6 2.00.0

1.7

3.4

5.1

6.8

QP

PT

NS

NS

NS

PTNS PT

PT

PT

PT

0.4 0.8 1.2 1.6 2.00.0

2.0

4.0

6.0

8.0

PT

NS

PT

PT

0.4 0.8 1.2 1.6 2.00.0

3.0

6.0

9.0

12.0

QP

PT

PT

PT

NS

NS

PT

0.4 0.8 1.2 1.6 2.00.0

4.0

8.0

12.0

16.0

PT

PT PT

ωL/ω0 ωL/ω0 ωL/ω0

ωL/ω0ωL/ω0

Fig. 4. Resonance curves for lateral pressure. Fundamental mode. L/R¼0.50, R/h¼800 and Λ1¼10% Λ 0 cr. (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4 and (e) Case 5.

Table 2Natural frequency parameter (Ω0), critical lateral load parameter (Λ0 cr) andassociated circumferential wave number (n). m¼1.

Case L/R R/h n Ω0 Λ0 cr

10.50

800 19 0.06938 0.1581875 8 0.22537 0.16837

1.75800 12 0.02357 0.1686575 6 0.06587 0.14744

2.25800 11 0.01878 0.1591575 5 0.05324 0.17823

5.00 800 8 0.00898 0.139662

0.50800 18 0.11065 0.4251775 8 0.33387 0.36952

1.75800 11 0.03418 0.4375475 5 0.10594 0.45596

2.25800 10 0.02692 0.4232275 5 0.08240 0.42696

5.00 800 7 0.01244 0.406743

0.50800 17 0.16572 0.9724275 8 0.53261 0.94040

1.75800 10 0.04824 0.9922975 5 0.15478 0.97342

2.25800 8 0.03760 0.9921275 5 0.12506 0.98354

5.00 800 6 0.01695 0.997264

0.50800 14 0.21254 1.2802875 7 0.62646 1.24688

1.75 800 8 0.06371 1.407762.25 800 7 0.04993 1.385045.00 800 5 0.02261 1.57136

50.50

800 12 0.27328 1.4737775 6 0.75748 1.60304

1.75 800 7 0.08450 1.715232.25 800 6 0.06654 1.56644

Z.J.G.N. del Prado et al. / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎4

Please cite this article as: Z.J.G.N. del Prado, et al., The effect of material and geometry on the non-linear vibrations of orthotropiccircular cylindrical shells, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.017i

adopted [29]

wðx;θ; tÞ ¼ ξ1;1ðtÞh cos ðnθÞ sin mπxL

� �þξ1;1cðtÞh sin ðnθÞ sin mπx

L

� �

þξ0;1ðtÞh sinmπxL

� �þξ0;3ðtÞh sin

3mπxL

� �

þξ0;5ðtÞh sin5mπx

L

� �þξ0;7ðtÞh sin

7mπxL

� �; ð15Þ

where ξ1,1(t), ξ1,1c(t), ξ0,1(t), ξ0,3(t), ξ0,5(t) e ξ0,7(t) are the time-dependent non-dimensional modal amplitudes. Here and innumerical results the modal amplitudes are divided by shellthickness (ξi,j¼wi,j/h). This modal solution, which satisfies the

0.0 1000.0 2000.0 3000.0-5.0

-2.5

0.0

2.5

5.0

2.8 3.2 3.6 4.01.0

1.2

1.4

1.6

1.8

dξ1,

1/dτ

dξ1,

1/dτ

0.0 1000.0 2000.0 3000.0-7.0

-3.5

0.0

3.5

7.0

3.5 4.2 4.9 5.60.0

0.7

1.4

2.1

2.8

0.0 1000.0 2000.0 3000.0-5.0

-2.5

0.0

2.5

5.0

-3.95 -3.90 -3.85 -3.80 -3.750.00

0.40

0.80

1.20

1.60

ξ 1,1

ξ 1,1

τ ξ1,1

τ ξ1,1

dξ1,

1/dτ

ξ 1,1

τ ξ1,1

Fig. 5. Time responses and Poincaré sections for L/R¼0.50, R/h¼800 and Λ1¼10% Λ0 cr. (a) Case 1 (ωL/ω0¼0.876), (b) Case 2 (ωL/ω0¼0.731) and (c) Case 4 (ωL/ω0¼0.861).

Z.J.G.N. del Prado et al. / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 5

Please cite this article as: Z.J.G.N. del Prado, et al., The effect of material and geometry on the non-linear vibrations of orthotropiccircular cylindrical shells, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.017i

out-of-plane boundary conditions and includes the basic vibrationmode, the companion mode and four axi-symmetric modes, hasbeen thoroughly tested in [27]. The choice of these modes is based

on the previous investigations on modal solutions for the non-linear analysis of cylindrical shells [13,30–33]. Using these modesit is possible to take into account all relevant non-linear modal

0.4 0.8 1.2 1.6 2.00.0

1.5

3.0

4.5

6.0

|ξ1,

1c|

|ξ1,

1c|

|ξ1,

1c|

|ξ1,

1c|

0.4 0.8 1.2 1.6 2.00.0

1.5

3.0

4.5

6.0

|ξ1,

1c|

0.4 0.8 1.2 1.6 2.00.0

1.5

3.0

4.5

6.0

0.4 0.8 1.2 1.6 2.00.0

1.6

3.2

4.8

6.4

QP

0.4 0.8 1.2 1.6 2.00.0

1.8

3.6

5.4

7.2

ωL/ω0 ωL/ω0 ωL/ω0

ωL/ω0ωL/ω0

Fig. 6. Resonance curves for lateral pressure. Companion mode. L/R¼0.50, R/h¼800 and Λ1¼10% Λ 0 cr. (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4 and (e) Case 5.

0.4 0.8 1.2 1.6 2.00.0

0.9

1.8

2.7

3.6

|ξ1,

1|

|ξ1,

1|

|ξ1,

1|

|ξ1,

1|

|ξ1,

1|

PT

PT

0.4 0.8 1.2 1.6 2.00.0

1.0

2.0

3.0

4.0

QP

PT

NS

PT

PT

0.4 0.8 1.2 1.6 2.00.0

0.9

1.8

2.7

3.6

PT

PT

PT

PT

NS

PT

PT

NS

CH

0.4 0.8 1.2 1.6 2.00.0

1.0

2.0

3.0

4.0

PT

PT

PT

PT

PT

NS

NS

PT

0.4 0.8 1.2 1.6 2.00.0

1.6

3.2

4.8

6.4

QP

PT

NS

NS

NS

PTNS

NS

NS

NS

NS

PT

PT

ωL/ω0 ωL/ω0 ωL/ω0

ωL/ω0ωL/ω0

Fig. 7. Resonance curves for lateral pressure. Fundamental mode. L/R¼0.50, R/h¼75 and Λ1¼10% Λ 0 cr. (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4 and (e) Case 5.

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coupling observed in the past in the non-linear vibrations ofcylindrical shells with and without flow [34].

The solution for the stress function may be written as F¼FhþFp,where Fh is the homogeneous solution and Fp is the particularsolution. The particular solution Fp is obtained analytically bysubstituting the assumed form of the lateral displacement, Eq.(13), on the right-hand side of the compatibility equation, Eq. (9),and by solving the resulting linear partial differential equationtogether with the relevant boundary and continuity conditions.

The homogeneous part of the stress function can be written as

Fh ¼12NxR

2θ2þ12x2 Nθ�

12πL

Z 2π

0

Z L

0Fp;xx dx dθ

( )�NxθRθ ð16Þ

where Nx, Nθ and Nxθ are the average in-plane restraint stressesgenerated at the ends of the shell. This solution enables one tosatisfy the in-plane boundary conditions on the average [29].

0.0 1000.0 2000.0 3000.0-1.0

-0.5

0.0

0.5

1.0

0.2 0.4 0.6 0.8-0.4

-0.2

0.0

0.2

0.4

0.6

0.0 1000.0 2000.0 3000.0-2.0

-1.0

0.0

1.0

2.0

-2.0 -1.0 0.0 1.0 2.0-2.0

-1.0

0.0

1.0

2.0

0.0 1000.0 2000.0 3000.0-5.0

-2.5

0.0

2.5

5.0

2.5 3.0 3.5 4.0 4.50.8

1.2

1.6

2.0

dξ1,

1/dτ

dξ1,

1/dτ

dξ1,

1/dτ

ξ 1,1

ξ 1,1

ξ 1,1

τ

τ

τ

ξ1,1

ξ1,1

ξ1,1

a

b

c

Fig. 8. Time responses and Poincaré sections for L/R¼0.50, R/h¼75 and Λ1¼10% Λ0 cr. (a) Case 2 (ωL/ω0¼0.902), (b) Case 3 (ωL/ω0¼1.092) and (c) Case 5 (ωL/ω0¼0.892).

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0.4 0.8 1.2 1.6 2.00.0

0.6

1.2

1.8

2.4

0.4 0.8 1.2 1.6 2.00.0

0.6

1.2

1.8

2.4

0.4 0.8 1.2 1.6 2.00.0

0.8

1.6

2.4

3.2

CH

0.4 0.8 1.2 1.6 2.00.0

0.9

1.8

2.7

3.6

0.4 0.8 1.2 1.6 2.00.0

1.5

3.0

4.5

6.0

|ξ1,

1c|

ωL/ω0

|ξ1,

1c|

ωL/ω0

|ξ1,

1c|

ωL/ω0

|ξ1,

1c|

ωL/ω0

|ξ1,

1c|

ωL/ω0

Fig. 9. Resonance curves for lateral pressure. Companion mode. L/R¼0.50, R/h¼75 and Λ1¼10% Λ 0 cr. (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4 and (e) Case 5.

0.4 0.8 1.2 1.6 2.00.0

4.0

8.0

12.0

16.0

PT

PT PT

0.4 0.8 1.2 1.6 2.00.0

4.5

9.0

13.5

18.0

PT

PT PT

0.4 0.8 1.2 1.6 2.00.0

4.5

9.0

13.5

18.0

PT

PT PT

0.4 0.8 1.2 1.6 2.00.0

7.0

14.0

21.0

28.0

PT

PTPT

0.4 0.8 1.2 1.6 2.00.0

9.0

18.0

27.0

36.0

PT

PTPT

|ξ1,

1|

ωL/ω0

|ξ1,

1|

ωL/ω0

|ξ1,

1|

ωL/ω0

|ξ1,

1|

ωL/ω0

|ξ1,

1|

ωL/ω0

Fig. 10. Resonance curves for lateral pressure. Fundamental mode. L/R¼1.75, R/h¼800 and Λ1¼10% Λ 0 cr. (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4 and (e) Case 5.

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Boundary conditions allow us to express the in-plane restraintstresses Nx, Nθ and Nxθ in terms of w and its derivatives as [35,36]

Nx ¼ 0 ð17Þ

Nθ ¼A12

A11Nx�

A212�A11A22

A11

!1

2πL

Z 2π

0

Z L

0�wRþ12

1Rw;θ

� �2" #

dx dθ

ð18Þ

Nxθ ¼ 0 ð19Þ

Upon substituting the modal expressions for F and w into theequilibrium equation, Eq. (6), and applying the Galerkin method, aset of six non-linear ordinary differential equations is obtained interms of the time-dependent modal amplitudes, ξi,j(t).

3. Numerical results

Numerical results are performed for a perfect simply supportedorthotropic cylindrical shell with mass density ρs¼7850 kg/m³and viscous damping factor ζ¼0.009. The shell thickness is fixedas h¼0.002 m, while the radius R and length L are varied. Table 1shows five different cases of orthotropic material [7,37] whichwere chosen for the parametric analysis, with varying Eθθ/Exx andPoisson ratios. Case 1 has the lowest Eθθ/Exx ratio while Case 5 hasthe highest Eθθ/Exx ratio. Case 3 corresponds to the isotropic case.The reciprocal relation νθxExx ¼ νxθEθθ is constant (E0.27).

In the parametric analysis, the following non-dimensional para-meters are used for time, frequency [33] and lateral pressure [24]

τ¼ tω0; ð20Þ

Ω¼ωλ

with λ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ExxρsR

2ð1�ν2xθÞ

s; ð21Þ

Λ0 ¼FeFcr

and Λ1 ¼FdFcr

with Fcr ¼ExxhR

ððπR=LÞ2þn2Þ2n2

ðh=RÞ212ð1�ν2xθÞ

þ ðπR=LÞ4n2ððπ R=LÞ2þn2Þ2

" #:

ð22Þ

Fig. 2 shows the influence of the L/R parameter on the lowestnatural frequency and the associated circumferential wave num-ber (n) for the five orthotropic cases analyzed here and threedifferent R/h ratios ((a) R/h¼800, (b) R/h¼300 and (c) R/h¼75). Ascan be observed, the L/R and R/h ratios influence directly thenatural frequencies values and the number of circumferentialwaves. Shells with the same L/R and R/h ratios have the samelowest non-dimensional natural frequency and vibration mode. Itcan be also observed that most shell geometries can be analyzed usingDonnell's shallow shell theory ðnZ5Þ, however, depending on thematerial orthotropy and for low values of the R/h ratio, only shortshells can be studied. For a given R/h ratio, Ω0 and n decrease as L/Rincreases, and the shell tends to a long tube. For a given L/R, Ω0

decreases as R/h increases. For the same L/R and R/h ratios, shells withlow Eθθ/Exx ratio (Case 1) will display lower natural frequencies thanshell with high Eθθ/Exx ratio (Case 5). As can be observed in Fig. 2, alarge number of geometric relations can be studied using Donnell'sshallow shell theory but, depending on the material orthotropy andfor small R/h ratio, only short shells can be analyzed.

Fig. 3 shows the influence of the L/R on the lateral critical loadparameter (Λ0 cr) and its associated number of circumferential waves(n), for the five orthotropic cases analyzed here and three different R/hratios (R/h¼800, R/h¼300 and R/h¼75). Shells with high Eθθ/Exx ratiohave higher critical parameter than shells with a low Eθθ/Exx ratio. Thecritical parameter displays discontinuities which are due to thevariation of the critical circumferential wave number. The resultsshow that the Eθθ/Exx ratio influences significantly the variation of thecritical load with L/R. As illustrated in Fig. 3a and b, for shells withEθθoExx, for a given R/h relation and same circumferential wavenumber, as the L/R relation increases the lateral critical load parameterdecreases, while for shells with Eθθ4Exx (Fig. 3d and e) the lateralcritical load parameter increases with L/R. For the isotropic material(Case 3, Fig. 3c) the critical parameter increases exponentially as the L/R relation increases and tends to an upper bound (Λ0 crE1.00). Shells

0.4 0.8 1.2 1.6 2.00.0

1.8

3.6

5.4

7.2

0.4 0.8 1.2 1.6 2.00.0

2.0

4.0

6.0

8.0

0.4 0.8 1.2 1.6 2.00.0

2.0

4.0

6.0

8.0

0.4 0.8 1.2 1.6 2.00.0

3.0

6.0

9.0

12.0

0.4 0.8 1.2 1.6 2.00.0

3.5

7.0

10.5

14.0

|ξ1,

1c|

ωL/ω0

|ξ1,

1c|

ωL/ω0

|ξ1,

1c|

ωL/ω0

|ξ1,

1c|

|ξ1,

1c|

ωL/ω0ωL/ω0

Fig. 11. Resonance curves for lateral pressure. Companion mode. L/R¼1.75, R/h¼800 and Λ1¼10% Λ 0 cr. (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4 and (e) Case 5.

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with the same L/R and R/h ratios have the same critical parameter andbuckling mode. This means that the critical load and mode depend ona single geometrical parameter. For example, as shown in [24], thecritical load and natural frequency can be written as a function of theso-called Batdorf parameter Z ¼ L2ð1�ν2Þ=Rh. So the shell with thesame Z has the same non-dimensional critical load parameter andmode. For a given R/h ratio, Λ0 cr increases and n decreases as L/Rincreases and the shell tends to a long tube. These results are inagreement with the fact that the lateral pressure is mainly supportedby the hoop stress.

Now, to study the influence of geometric relations and materialon the non-linear dynamic behavior of the orthotropic shells,different geometric relations were selected (L/R¼0.50; 1.75; 2.25and 5.00) and (R/h¼800 and 75). Table 2 shows for the fivematerials and the selected values of the geometrical parameters L/R and R/h the lowest natural frequency and critical load para-meters and the associated number of circumferential waves, n. Thelowest values always occur for m¼1. All geometry and materialcombinations give a total of 30 combinations. In this table, thevalues are omitted when no5, since in such case Donnell theory isno longer valid. The results based on Donnell's shallow-shelltheory compare well with those obtained by more refined shelltheories only for modes with a high circumferential wave numbern. In fact, for a shallow cylindrical shell the restriction 1/n2«1 mustbe satisfied, so that the displacement components are rapidlyvarying functions of the circumferential coordinate [24]. Usuallythe restriction nZ5 is considered in the literature in order toguarantee reliable numerical results [29].

Now, the non-linear forced response of the shells is studied inorder to understand the effect of geometry and material on theresonance curves and bifurcations of the shell. The resonancecurves are obtained in the neighborhood of the resonance of thefundamental mode. All curves are obtained using continuationtechniques and considering the excitation frequency as controlparameter and a fixed value of the forcing magnitude. From thenon, continuous lines represent stable paths and dashed linesrepresent unstable paths. Also, PT represents pitchfork or saddle-node bifurcations, NS represents Neimark–Sacker bifurcation andQP represents quasi-periodic oscillations.

The non-dimensional lateral pressure is considered as

f b ¼Λ0þΛ1 sinmπxL

� �cos ðnθÞ cos ωL

ω0τ

� �; ð23Þ

where ωL represents the forcing frequency. In the analysis, allresults are obtained considering Λ0¼0 and the magnitude of thedynamic lateral pressure Λ1¼10% Λ0 cr.

In the following parametric analysis all variables in the reso-nance curves, time responses and phase portraits are non-dimensional. Fig. 4 shows the resonance curves of the drivenmode for geometric relations L/R¼0.50 and R/h¼800 (very thinshell) considering the five orthotropic materials. Comparing thefive resonance curves, it is clear that the orthotropy affectsstrongly the non-linear response and bifurcations of the shell. Inall cases the non-linearity is initially of the softening type chan-ging the behavior to hardening at large vibration amplitudes. Asthe Eθθ/Exx ratio increases the degree of the non-linearity increasesbut the number of bifurcations and, consequently, the complexityof the shell response decreases. For example, in Fig. 4a, for a shellwith low Eθθ/Exx ratio (Case 1), several bifurcations are observedleading two several coexisting periodic and quasi-periodic solu-tions. On the other hand, in Fig. 4e only the expected saddle-nodebifurcations at the turning points are observed. The maximumvibration amplitude at which stable solutions occurs dependsstrongly on Eθθ/Exx ratio; while in Fig. 4a the maximum absolutevalue of ξ1,1 is lower than six, in Fig. 4e it is almost 10. Fig. 5 depictsthe time response and Poincaré section of quasi-periodic solutionsfor L/R¼0.50, R/h¼800, for ωL/ω0¼0.876 (Case 1), ωL/ω0¼0.731(Case 2) and ωL/ω0¼0.861 (Case 4) obtained from the resonancecurves of Fig. 4, where again the complexity decreases with theEθθ/Exx ratio. Fig. 6 depicts the associated resonance curves of thecompanion mode for geometric relations L/R¼0.50, R/h¼800 andΛ1¼10% Λ0 cr for the five orthotropic materials, confirming theinfluence of the Eθθ/Exx ratio on the shell behavior. As alreadyshown in the literature [2], the companion mode is only excited inthe resonance region. In all cases, the amplitudes of the compa-nion mode are very similar.

0.4 0.8 1.2 1.6 2.00.0

1.5

3.0

4.5

6.0

QP

PT

NS

NS

NS

PT

PTNS

NS

PT

PTPT

PT

0.4 0.8 1.2 1.6 2.00.0

2.0

4.0

6.0

8.0

PT

PT

PTNS

PT

PT

PTNS

0.4 0.8 1.2 1.6 2.00.0

2.2

4.4

6.6

8.8

PT

PT

PT

PT

|ξ1,

1||ξ

1,1|

|ξ1,

1|

ωL/ω0

ωL/ω0

ωL/ω0

Fig. 12. Resonance curves for lateral pressure. Driven mode. L/R¼1.75, R/h¼75 andΛ1¼10% Λ 0 cr. (a) Case 1, (b) Case 2 and (c) Case 3.

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Fig. 7 depicts the resonance curves of the driven mode for thesame value of L/R as in Fig. 4 (L/R¼0.50) but R/h¼75 (relatively thickshell) and Λ1¼10% Λ0 cr. As can be observed again, the orthotropyaffects strongly the non-linear response of the shell, including in thepresent case a change of the non-linearity from hardening to softeningas Eθθ/Exx increases. For a shell with low Eθθ/Exx ratio (Fig. 7a), as thefrequency of excitation increases, the amplitude of the driven modeincreases, displaying a hardening behavior and a complex unstablepath. As the Eθθ/Exx ratio increases (Fig. 7b), the hardening behavior isalso observed but a new stable branch and quasi-periodic (QP)oscillations appear. For the isotropic case (Fig. 7c) a very complexnon-linear behavior of the shell is observed and the hardening effect isstrongly reduced. Several stable and unstable solutions coexist andchaotic regions are detected. Fig. 7d and e displays the non-linearbehavior for higher Eθθ/Exx ratios. Again, it is observed the coexistenceof stable, unstable and quasi-periodic (QP) oscillations. In these cases,the shells display initially a softening behavior. Fig. 8 shows thePoincaré sections and time responses illustrating the chaotic andquasi-periodic oscillations observed in Fig. 7. Fig. 9 shows the responseof the companion mode associated with the resonance curves of Fig. 7.

Fig. 10 displays the resonance curves of the driven mode forgeometric relations L/R¼1.75, R/h¼800 (long, thin shell) and Λ1¼10%Λ0 cr. For this geometry the complexity of the non-linear responsedecreases. In all figures the shells display an initial softening behavior.At large amplitude however a bending back of the backbone curve(turning point) associated with large bending effects may occur. Thepoint where the bending back occurs depends on the problemparameters. However, the behavior is different from that observed inFig. 4 for L/R¼0.50 and R/h¼800. Here the non-linearity increases asthe Eθθ/Exx decreases. In Figs. 10d and e a disconnected branch of stablesolutions is observed, leading to additional dynamic jumps betweensmall and large amplitude vibrations. The corresponding results forthe companion mode are shown in Fig. 11.

Fig. 12 shows the resonance curves of the driven mode forgeometric relations L/R¼1.75, R/h¼75 (long, moderately thickshell), Λ1¼10% Λ0 cr and considering Cases 1, 2 and 3. All shellsdisplay softening behavior. The resonance curves display severalbifurcations. In these cases shells with low Eθθ/Exx ratio (Fig. 12a)depict lower softening behavior than shells with high Eθθ/Exx ratio(Fig. 12c). Fig. 13 depicts the time response and Poincare section ofa quasi-periodic path for ωL/ω0¼0.912.

4. Conclusions

In this work, the influence of geometric and material character-istics on the linear and non-linear vibrations of simply supportedorthotropic cylindrical shells subjected to lateral time-dependent

loads is analyzed. To model the shell, Donnell's non-linear shallowshell theory without considering the effect of shear deformationis used.

Five different orthotropic materials with seven different geome-tries, leading to 30 combinations are considered in the parametricanalysis. Results show that the material orthotropy and L/R and R/hratios have a strong influence on the lateral critical loads and naturalfrequencies, as well as, on the resonance curves of the shells.

For the same shell geometry, it is observed that the degree andtype of non-linearity (hardening or softening) depends on theratio of Young's modulus in the circumferential and axial direc-tions, Eθθ/Exx. Also the complexity of the non-linear response andconsequently the number of bifurcations and coexisting solutionsdepend on the Eθθ/Exx ratio. However the specific way in which theEθθ/Exx ratio influences these non-linear characteristics dependson the L/R and R/h ratios. For a given material, the non-linearitydecreases as the L/R and R/h increases.

It is observed that in the main resonance region the shell displays acomplex response with several bifurcation points, exhibiting not onlyperiodic but also chaotic and quasi-periodic oscillations. The degreeand type of non-linearity of the shell response depend mainly on twothe geometric relations (length/radius and radius/thickness) andmaterial characteristics.The results show that an optimal shell tosupport lateral pressure can be obtained by a careful selection ofgeometric parameters and the material properties in the two principaldirections.

Future research will consider the effect shear deformation,geometric imperfections, and fluid flow. Also a detailed parametricanalysis, including possible routes to chaos, will be performed, tounveil the influence of the material on the bifurcation scenario.

Acknowledgments

This work was made possible by the support of the BrazilianMinistry of Education – CAPES, CNPq and FAPERJ-CNE.

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0.0 1000.0 2000.0 3000.0-6.0

-3.0

0.0

3.0

6.0

2.0 3.0 4.0 5.0-1.5

0.0

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3.0

4.5

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Fig. 13. Time responses and Poincaré sections for L/R¼1.75, R/h¼75 and Λ1¼10% Λ0 cr. Case 1 (ωL/ω0¼0.912).

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Z.J.G.N. del Prado et al. / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎12

Please cite this article as: Z.J.G.N. del Prado, et al., The effect of material and geometry on the non-linear vibrations of orthotropiccircular cylindrical shells, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.017i