Dynamic stability analysis of composite laminated cylindrical shells via the mesh-free kp-Ritz method

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Comput. Methods Appl. Mech. Engrg. 196 (2006) 147–160

Dynamic stability analysis of composite laminated cylindrical shellsvia the mesh-free kp-Ritz method

K.M. Liew a,*, Y.G. Hu b, X. Zhao c, T.Y. Ng c

a Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kongb Department of Mathematics, Ocean University of China, Qingdao 266071, China

c School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore

Received 30 October 2003; received in revised form 9 February 2005; accepted 22 February 2006

Abstract

In this paper, a novel numerical technique is developed for the dynamic stability analysis of composite laminated cylindrical shellsunder static and periodic axial forces. The mesh-free kernel particle (kp) estimate is employed in hybridized form with harmonic func-tions, to approximate the 2-D transverse displacement field. A system of Mathieu–Hill equations is obtained through the application ofthe Ritz minimization procedure to the energy expressions. The principal instability regions are then analyzed via Bolotin’s first approx-imation. The mesh-free kp-Ritz method is validated through comparison with existing available numerical data taken from open liter-ature. Effects of boundary conditions and lamination schemes on the instability regions are also examined in detail.� 2006 Elsevier B.V. All rights reserved.

Keywords: Dynamic stability; Parametric resonance; Cylindrical shell; Composite laminate; Ritz energy minimization; Bolotin’s first approximation

1. Introduction

The dynamic stability or phenomenon of parametric resonance in cylindrical shells under periodic loads has attractedmuch attention due to its detrimental and de-stabilizing effects in many engineering applications. This phenomenon in elas-tic systems was first studied by Bolotin [1], where the dynamic instability regions were determined. Yao [2] examined thenon-linear elastic buckling and parametric excitation of a cylinder under axial loads. The parametric instability of circularcylindrical shells was also discussed by Vijayaraghavan and Evan-Iwanowski [3].

Based on the Donnell’s shell equations, the dynamic stability of circular cylindrical shells under both static and periodiccompressive forces was examined by Nagai and Yamaki [4] using Hsu’s method. Bert and Birman [5] extended Yao’s [2]approach to the parametric instability of thick orthotropic shells using higher-order theory. Liao and Cheng [6] proposed afinite element model with a 3-D degenerated shell element and a 3-D degenerated curved beam element to investigate thedynamic stability of stiffened isotropic and laminated composite plates and shells subjected to in-plane periodic forces. Arg-ento and Scott [7] employed a perturbation technique to study the dynamic stability of layered anisotropic circular cylin-drical shells under axial loading. Using the same method, Argento [8] later analyzed the dynamic stability of a compositecircular cylindrical shell subjected to combined axial and torsional loading.

The Ritz or Rayleigh–Ritz method is a proven approximate technique in computational mechanics, and notable worksinclude those of Kitipornchai et al. [9,10], Liew and Lam [11,12], Liew et al. [13–15], Cheung and Zhou [16], Liew and Yang[17,18], Zeng and Bert [19], Wang et al. [20] and Xiang et al. [21,22]. In this paper, for the dynamic stability analysis of

0045-7825/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2006.02.007

* Corresponding author. Tel.: +852 3442 6581; fax: +852 2788 7612.E-mail address: [email protected] (K.M. Liew).

mailto:[email protected]

148 K.M. Liew et al. / Comput. Methods Appl. Mech. Engrg. 196 (2006) 147–160

composite laminated cylindrical shells under combined static and periodic axial loads, an energy formulation is firstdescribed, and a system of Mathieu–Hill equations is obtained via the Ritz minimization procedure. The parametric res-onance responses are then analyzed based on Bolotin’s [1] method. The commonly used trigonometric or hyperbolic func-tions in conjunction with the Ritz procedure are not suitable for general laminates as the bending–extension couplingstiffness terms erroneously vanish. This results in an incorrect representation of the actual physical state. Thus for theassumed functions in the present work, the meshfree kernel particle approximate is hybridized with harmonic functionsto describe the 2-D transverse displacement field. The present kp-Ritz method thus ensures that the bending–extension cou-pling stiffnesses are retained, and is verified by comparing the generated numerical results with those available in literature.The effects of boundary conditions and lamination schemes on the instability regions are also investigated.

2. Theoretical formulation

2.1. Energy formulation for cylindrical shells

The cylindrical shell shown in Fig. 1a is assumed to be thin, laminated and composed of an arbitrary number of layers ofcomposite materials perfectly bonded together. The shell is of length L, radius R, and thickness h. The coordinate system(x,h,z) is fixed on the middle surface of the shell. The displacements of the shell in the x, h, and z directions are denoted byu, v and w, respectively. The pulsating axial load is given by

Na ¼ N o þ Ns cos Pt; ð1Þwhere P is the frequency of excitation in radians per unit time. The kinetic energy of the cylindrical shell is given by

T ¼ 1

2qhZ L

0

Z 2p

0

½ _u2 þ _v2 þ _w2�Rdhdx; ð2Þ

where _u; _v and _w are the linear velocities in the x, h, and z directions, respectively.The strain energy Ua due to the axial loading can be written as

U a ¼1

2

Z L

0

Z 2p

0

N a

ouox

� �2

þ ovox

� �2

þ owox

� �2" #

Rdhdx. ð3Þ

The strain energy of the shell is expressed as

U e ¼1

2

Z L

0

Z 2p

0

eT½S�eR dhdx; ð4Þ

where eT and [S] are the strain vector and stiffness matrix, respectively, and eT can be defined as

eT ¼ f e1 e2 c j1 j2 2s g; ð5Þwhere the middle surface strains, e1, e2 and c, and the middle surface curvatures, j1, j2 and s, are defined according toLove’s thin shell theory

e1 ¼ouox

e2 ¼1

Rovohþ w

� �c ¼ ov

oxþ 1

Rouoh;

j1 ¼ �o2w

o2xj2 ¼ �

1

R2

o2w

o2h� ov

oh

� �s ¼ � 1

Ro2woxoh

� ovox

� � ð6Þ

θ,v

Rθ

h

x,u

z,w

Fig. 1a. Geometry of the laminated cylindrical shell.

K.M. Liew et al. / Comput. Methods Appl. Mech. Engrg. 196 (2006) 147–160 149

and [S] is given by

½S� ¼

A11 A12 A16 B11 B12 B16

A12 A22 A26 B12 B22 B26

A16 A26 A66 B16 B26 B66

B11 B12 B16 D11 D12 D16

B12 B22 B26 D12 D22 D26

B16 B26 B66 D16 D26 D66

2666666664

3777777775; ð7Þ

where the extensional (Aij), coupling (Bij) and bending (Dij) stiffnesses, are defined as

ðAij;Bij;DijÞ ¼Z h=2

�h=2

Qijð1; z; z2Þdz ð8Þ

and for a shell composed of different layers of orthotropic materials, the stiffnesses can be defined as (see Reddy andMiravete [23])

Aij ¼XNl

k¼1

Qkijðhk � hkþ1Þ;

Bij ¼1

2

XNl

k¼1

Qkijðh

2k � h2

kþ1Þ;

Dij ¼1

3

XNl

k¼1

Qkijðh

3k � h3

kþ1Þ;

ð9Þ

where hk and hk+1 denote the distances from the shell reference surface to the outer and inner surfaces of the kth layer,respectively, as shown in Fig. 1b. Nl denotes the total number of layers in the laminated shell and Qk

ij is the transformedreduced stiffness matrix for the kth layer defined as

½Q� ¼ ½T��1½Q�½T��T; ð10Þ

where [T] is the transformation matrix for the principle material coordinates and the shell’s coordinates, and is defined as

½T� ¼cos2 a sin2 a 2 cos a sin a

sin2 a cos2 a �2 cos a sin a

� cos a sin a cos a sin a cos2 a� sin2 a

264375; ð11Þ

where a is the angular orientation of the fibres and [Q] is the reduced stiffness matrix defined as

½Q� ¼Q11 Q12 0

Q12 Q22 0

0 0 Q66

264375. ð12Þ

h2h1

hh4 h5

w

v

Fig. 1b. Cross-sectional view of the laminated cylindrical shell.


The material constants in the reduced stiffness matrix [Q] are given as (see Reddy and Miravete [23])

Q11 ¼E11

1� m12m21

Q12 ¼m12E22

1� m12m21

Q22 ¼E22

1� m12m21

Q66 ¼ G12; ð13Þ

where E11 and E22 are the elastic moduli in the principle material coordinates, G12 is the shear modulus and m12 and m21 arePoisson’s ratio. The total energy functional of the shell is thus

Ct ¼ T � U a � U e. ð14Þ

2.2. Hybrid harmonic-kernel particle 2-D displacement field

The approximation of the displacement functions are expressed as follows:

uðx; hÞ ¼XNP

I¼1

wIðxÞuI cosðnhÞqðtÞ;

vðx; hÞ ¼XNP

I¼1

wIðxÞvI sinðnhÞqðtÞ;

wðx; hÞ ¼XNP

I¼1

wIðxÞwI cosðnhÞqðtÞ;

ð15Þ

where NP is the total number of particles, wI(x) are the shape functions, uI, vI and wI are unknown nodal values of u, v andw at a sampling point I and n is the circumferential half wave number. Based on the Reproducing Kernel Particle Method(RKPM), see Chen et al. [24] and Liew et al. [25], the shape function is given by

wIðxÞ ¼ Cðx; x� xIÞuaðx� xIÞ; ð16Þ

where C(x;x � xI) is the correction function and ua(x � xI) the weight function. The correction function C(x;x � xI) isdescribed as

Cðx; x� xIÞ ¼ HTðx� xIÞbðxÞ; ð17Þ

where

HTðx� xIÞ ¼ ½1; x� xI ; ðx� xIÞ2�; ð18ÞbTðxÞ ¼ ½b0ðxÞ; b1ðxÞ; b2ðxÞ� ð19Þ

and H is a vector of quadratic basis, and bi(x)’s are functions of x which are to be determined. By imposing the reproducingconditions, the bi(x) can be solved, see Chen et al. [24] and Liew et al. [25]. Thus, the shape function can be written as

wIðxÞ ¼ bTðxÞHðx� xIÞuaðx� xIÞ. ð20Þ

Eq. (20) can be rewritten as

wIðxÞ ¼ bTðxÞBIðx� xIÞ; ð21Þin which

BIðx� xIÞ ¼ Hðx� xIÞuaðx� xIÞ; ð22ÞbðxÞ ¼M�1ðxÞHð0Þ; ð23Þ

and the moment matrix M is a function of x and H(0) is a constant vector. The explicit expressions for M and H(0) aregiven by

MðxÞ ¼XNP

I¼1

Hðx� xIÞHTðx� xÞuaðx� xIÞ; ð24Þ

HTð0Þ ¼ ½1; 0; 0�. ð25Þ

Therefore, the shape function can be expressed as

wIðxÞ ¼ HTð0ÞM�1ðxÞHðx� xIÞuaðx� xIÞ. ð26Þ


For the thin shell problem, the first and second derivatives of the shape function need to be determined. The procedure toformulate the first derivative of the shape function is given in detail by Chen et al. [24] and Liew et al. [25]. Here, we extendthis approach to calculate the second derivative of the shape function. Eq. (23) can be rewritten as

MðxÞbðxÞ ¼ Hð0Þ. ð27Þ

The vector b(x) can be determined using the LU decomposition of the matrix M(x) followed by back substitution.The derivatives of b(x) can be obtained similarly. Taking the derivative of Eq. (27), we obtain

M;xðxÞbðxÞ þMðxÞb;xðxÞ ¼ H;xð0Þ ð28Þwhich can be rearranged as

MðxÞb;xðxÞ ¼ H;xð0Þ �M;xðxÞbðxÞ. ð29ÞThus the first derivative of b(x) can be calculated using the same LU decomposition procedure, and the second deriv-

ative of b(x) can be determined by taking derivative of Eq. (29) and using the same LU decomposition procedure. The firstderivative of the shape function, therefore, can be obtained by taking the derivative of Eq. (21)

wI ;xðxÞ ¼ bT;xðxÞBIðx� xIÞ þ bTðxÞBI;xðx� xIÞ; ð30Þ

and the second derivative of the shape function can be calculated by taking derivative of Eq. (30)

wI ;xxðxÞ ¼ bT;xxðxÞBIðx� xIÞ þ 2bT

;xðxÞBI ;xðx� xIÞ þ bTðxÞBI ;xxðx� xIÞ. ð31Þ

In this paper, the cubic spline function is chosen as the weight function

/ðzIÞ ¼

2

3� 4z2

I þ 4z3I for 0 6 jzI j 6

1

2

4

3� 4zI þ 4z2

I �4

3z3

I for1

26 jzI j 6 1

0 otherwise

8>>>><>>>>:

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

zI ¼ðx� xIÞ

a;

ð32Þ

where the (dilatation) parameter a denotes the size of the support. At a node, the size of the domain of influence is calcu-lated by

aI ¼ amaxdI ; ð33Þ

in which amax being a scaling factor that ranges from 2 to 4. The distance dI is determined by searching for sufficient num-ber of nodes so as to avoid singularity of the M matrix. For the one-dimensional RKPM problem, each node should haveat least two neighbors in its domain of influence. In order to compute the derivatives of the shape function, it is necessary todetermine the derivatives of the weight function. The first and second derivatives of the weight function can be easilyobtained using the chain rule

d/dx¼ d/

dzI

dzI

dx¼

ð�8zI þ 12z2I Þðx� xIÞjðx� xIÞj

for 0 6 jzI j 61

2

ð�4þ 8zI � 4z2I Þðx� xIÞjðx� xIÞj

for1

26 jzI j 6 1

0 otherwise

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

9>>>>>=>>>>>;; ð34Þ

d2/dx2¼ d2/

dz2I

dzI

dx

� �2

¼

ð�8zI þ 24zIÞ for 0 6 jzI j 61

2

ð8� 8zIÞ for1

26 jzI j 6 1

0 otherwise

8>>>><>>>>:

9>>>>=>>>>;. ð35Þ

It is noted that the first and second derivatives of the weight function are continuous over the entire domain.

2.3. Enforcement of boundary conditions – a penalty approach

It is necessary to deal with different boundary conditions to solve the problem. There are several approaches to enforceessential boundary conditions for meshless methods, such as Lagrange multipliers approach, penalty method approach,


modified variational principles, etc. In the present work, the penalty method, see Reddy [26], is utilized to implement essen-tial boundary conditions. The penalty formulation is developed as follows.

2.3.1. Simply-supported boundary conditions

For the domain bounded by lu, the displacement boundary condition is

u ¼ �u on lu; ð36Þin which �u is the prescribed displacement on the displacement boundary lu. Condition (36) is treated as constraint and it isintroduced into the formulation using the penalty method. The variational form of the penalty functional is given by

C�u ¼a2

Zlu

ðu� �uÞTðu� �uÞdl; ð37Þ

where a is the penalty parameter, which taken as 103E, with E being the elastic modulus of the shell.

2.3.2. Clamped boundary conditions

In the clamped case, for the domain bounded by lu, besides the boundary condition described by Eq. (36), the rotationboundary condition is also included

b ¼ �b on lu; ð38Þ

where

b ¼ dwdx; ð39Þ

and �b is the prescribed rotation on the boundary. The variational form due to the rotational constraint (38) is given by

C�b ¼a2

Zlu

ðb� �bÞTðb� �bÞdl. ð40Þ

Although, in general, the penalty parameter for each constraint can be taken different, here the same penalty parameter isused for both boundary constraints.

2.4. Instability regions via Ritz minimization and Bolotin’s method

The variational form due to the boundary conditions can be expressed as

CB ¼ C�u þ C�b; ð41Þ

and the total energy functional for this problem becomes

C ¼ Ct þ CB. ð42ÞSubstituting the displacement functions of Eq. (15) into the total energy functional of Eq. (42) and applying the

Rayleigh–Ritz minimization procedure with

oCoD¼ 0; D ¼ uI ; vI ;wI I ¼ 1; 2; . . . ;NP ð43Þ

a system of Mathieu–Hill equations are obtainedfM€qþ ðeK � cos PtQÞq ¼ 0; ð44ÞwhereeK ¼ K�1KK�T; ð45ÞfM ¼ K�1MK�T; ð46Þ

KIJ ¼ wIðxJ ÞI; I is the identity matrix, ð47ÞK ¼ Ke þ KA þ KB1 þ KB2 ; ð48Þ

KeIJ ¼ Rp

Z L

0

BeTI ½S�Be

J dx; ð49Þ

KAIJ ¼ Rp

Z L

0

BAI

TN oBA

J dx; ð50Þ


KB1IJ ¼ aRp

ZCU

B1BI

TB1B

J dlþZ

CU

B1BI �u dl

� �; ð51Þ

KB2IJ ¼ aRp

ZCU

B2BI

TB2B

J dlþZ

CU

B2BI�bdl

� �; ð52Þ

MIJ ¼ qhRpZ L

0

MTI MJ dx; ð53Þ

QAIJ ¼ Rp

Z L

0

BAI

TNsB

AJ dx; ð54Þ

and

½BeI � ¼

owI

ox0 0

0nR

wI

wI

R

� nR

wI

owI

ox0

0 0 � o2wI

ox2

0n

R2wI

n2

R2wI

02

RowI

ox2nR

owI

ox

2666666666666666666664

3777777777777777777775

; ð55Þ

B1BI ¼

wI 0 0

0 wI 0

0 0 wI

264375 B2B

I ¼wI;x 0 0

0 wI;x 0

0 0 wI ;x

264375;

BAI ¼

wI;x 0 0

0 wI;x 0

0 0 wI ;x

264375 MT

I ¼wI 0 0

0 wI 0

0 0 wI

264375: ð56Þ

Eq. (44) is in the form of a second order differential equation with periodic coefficients of the Mathieu–Hill type. Theregions of instability are separated by the periodic solutions having periods T and 2T with T = 2p/P. The solutions withperiod 2T are of greater practical importance as the widths of these unstable regions are usually larger than those associ-ated with solutions having period of T. Using Bolotin’s approach, as a first approximation, the periodic solutions withperiod 2T can be sought in the form

�q ¼ �f sinPt2þ �g cos

Pt2; ð57Þ

where �f and �g are arbitrary vectors.Substituting Eq. (57) into Eq. (44) and equating the coefficients of sin Pt

2and cos Pt

2terms, a set of linear homogeneous

algebraic equations in terms of �f and �g can be obtained. The condition of non-trivial solutions are given by

det� 1

4P 2fMIJ þ eKIJ � 1

2QIJ 0

0 � 14P 2fMIJ þ eKIJ þ 1

2QIJ

!" #¼ 0. ð58Þ

Instead of solving the above non-linear geometric equations for P, the above expression can be rearranged to the stan-dard form of a generalized eigenvalue problem

deteKIJ � 1

2QIJ 0

0 eKIJ þ 12QIJ

!� P 2

14fMIJ 0

0 14fMIJ

!" #¼ 0. ð59Þ

The generalized eigenvalues P2 of the above generalized eigenvalue problem define the boundaries between the stableand unstable regions.

Table 1Comparisons of frequency parameter �x ¼ 2pRP

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiqh=A11

pfor simply-supported cross-ply cylindrical shells subjected to tensile loading ([90�/0�], R/h = 200, N0 = 0.1Ncr)

L/R Mode(m,n)

Lam andNg [28]

Present

amax = 2.5 amax = 3.0 amax = 3.5

NP = 80 NP = 100 NP = 120 NP = 140 NP = 80 NP = 100 NP = 120 NP = 140 NP = 80 NP = 100 NP = 120 NP = 140

2.0 (1,6) 0.5654 0.5542 0.5569 0.5593 0.5614 0.5586 0.5617 0.5642 0.5664 0.5582 0.5612 0.5638 0.56603(1,5) 0.5959 0.5824 0.5851 0.5876 0.5898 0.5869 0.5901 0.5928 0.5952 0.5864 0.5896 0.5923 0.59472

2.2 (1,6) 0.5267 0.5153 0.5174 0.5192 0.5209 0.5188 0.5211 0.5231 0.5248 0.5184 0.5205 0.5228 0.52451(1,5) 0.5419 0.5275 0.5301 0.5321 0.5338 0.5315 0.5340 0.5362 0.5381 0.5311 0.5336 0.5358 0.53773

154K

.M.

Liew

eta

l./

Co

mp

ut.

Meth

od

sA

pp

l.M

ech.

En

grg

.1

96

(2

00

6)

14

7–

16

0

Table 2Comparisons of frequency parameter x� ¼ P

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqR2ð1� t2Þ

E

rfor isotropic simply-supported cylindrical shells subjected to tensile loading (m = 1, R/h = 100, N0 = 0.5Ncr, v = 0.3)

L/R Mode(m,n)

Liew et al. [29] Present

amax = 2.5 amax = 3.0 amax = 3.5

NP = 80 NP = 100 NP = 120 NP = 140 NP = 80 NP = 100 NP = 120 NP = 140 NP = 80 NP = 100 NP = 120 NP = 140

1 1 1.7226 1.7097 1.7117 1.7135 1.7152 1.7174 1.7201 1.7224 1.7243 1.7176 1.7206 1.7226 1.724582 1.3471 1.3303 1.3329 1.3354 1.3376 1.3405 1.3442 1.3473 1.3499 1.3407 1.3444 1.3476 1.350313 1.0214 1.0010 1.0046 1.0079 1.0109 1.0152 1.0202 1.0244 1.0280 1.0156 1.0206 1.0249 1.02858

10 6 0.2026 0.2003 0.2003 0.2003 0.2003 0.2003 0.2004 0.2004 0.2004 0.2004 0.2004 0.2004 0.200367 0.2768 0.2753 0.2753 0.2753 0.2753 0.2753 0.2753 0.2753 0.2753 0.2753 0.2753 0.2753 0.275348 0.3629 0.3618 0.3619 0.3619 0.3619 0.3618 0.3619 0.3619 0.3619 0.3619 0.3619 0.3619 0.36190

K.M

.L

iewet

al.

/C

om

pu

t.M

etho

ds

Ap

pl.

Mech

.E

ng

rg.

19

6(

20

06

)1

47

–1

60

155


3. Numerical results and discussion

To assess the stability and accuracy of the present methodology, numerical comparisons and convergence studies areperformed. For periodic compressive loads, it is obvious that the compressive axial loads cannot exceed the critical buck-ling load of the cylindrical shell. In this paper, for isotropic cylindrical shells of intermediate length, the buckling load isgiven in Timoshenko and Gere [27] as

Nbuc ¼Eh2

½3ð1� m2Þ�12R; ð60Þ

where E is the elastic modulus and m is the Poisson’s ratio of the isotropic cylindrical shell. For laminated circular cylin-drical shells, the critical buckling load Ncr is approximated as

N cr ¼E2h2

½3ð1� m12m21Þ�12R

. ð61Þ

The material properties of the laminated cylindrical shells in the present study are E1/E2 = 40, G12/E2 = 0.5 and m = 0.25.Table 1 shows the comparisons of the frequency results for simply-supported cylindrical shells under axial loading of

N0 = 0.1Ncr with solutions of Lam and Ng [28]. The convergence characteristics are also shown in Table 1. The number

0.000 0.005 0.010 0.015 0.02024

28

32

36

40

44

Reference [4] m = 1Reference [4] m = 2Present m = 1Present m = 2

q

Fig. 2. Comparison of unstable regions for a clamped–clamped isotropic cylindrical shell under axial compressive loading (n = 22, R/h = 400, v = 0.3,N0 = 0).

Table 3Comparisons of frequency parameter x̂ ¼ P

L2

p2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12qð1� t2Þ

Eh2

sfor isotropic cylindrical shells with clamped boundaries (R/h = 400, L=R ¼ 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� m2p

,v = 0.3, N0 = 0)

n Nagai and Yamaki [4] Present

m = 1 m = 2 m = 1 m = 2

6 38.48 – 38.19 58.587 34.20 – 34.14 55.46

15 21.26 37.88 21.59 38.7122 29.68 40.80 29.64 40.84


of nodes used is increased from 80 to 140, and the scaling factor amax varies from 2.5 to 3.5. It is observed that convergedresults can be achieved with less nodes by using a relatively higher scaling factor of amax = 3.0–3.5. Table 2 provides thecomparisons of the frequency parameter of simply-supported isotropic cylindrical shells under tensile loading ofN0 = 0.5Ncr with those presented by Liew et al. [29]. The cylindrical shells have length ratios (L/R) of 1 and 10, a thicknessratio (R/h) of 100. The results given here are for modes m = 1 and n = 1, 2, 3, 6, 7, 8. The scaling factor ranges from 2.5 to3.5 and the number of nodes varies from 80 to 140. It is observed from these three tables that, for cylindrical shells havingsmall length ratio of L/R = 1, converged results can be obtained by using a scaling factor of 3.0 or 3.5, with the number ofnodes being more than 100. However, for cylindrical shells with length ratio L/R = 10, converged results are more easilyachieved, through a scaling factor of 2.5 and with 80 nodes. It is thus concluded that the length ratios of cylindrical shellsaffect the convergence characteristics, where cylindrical shells with larger length ratios have faster convergence rates those

0.0 0.1 0.2 0.3 0.4 0.5300

320

340

360

380

400

420

440

460

480

500

520

540

φ = 0o

φ = 45o

φ = 60o

φ = 15oφ = 75o

φ = 30o

φ = 90o

Ns/No

′

Fig. 4. Unstable regions for a four-layered (//�////�/) clamped cylindrical shell under tensile loading (L/R = 2, R/h = 100, N0 = 0.5Ncr).

0.0 0.1 0.2 0.3 0.4 0.5250

300

350

400

450

500

550

600

φ = 0º

φ = 15º

φ = 30º

φ = 45º

φ = 60º

φ = 75º

φ = 90º

Ns/No

′

Fig. 3. Unstable regions for a four-layered (//�////�/) simply-supported cylindrical shell under tensile loading (L/R = 2, R/h = 100, N0 = 0.5Ncr).


with smaller length ratios. Table 3 shows the comparison of the present solutions with those given by Nagai and Yamaki [4]for an isotropic clamped shell, and very good agreement is observed.

Fig. 2 shows the comparison of the present results for the instability regions of a clamped isotropic cylindrical shell, atmodes (m,n) = (1, 22) and (m,n) = (2,22), with those reported by Nagai and Yamaki [4]. It is observed that the positions ofthe present instability regions match those of Nagai and Yamaki [4], while the widths are found to be slightly narrower.

In order to investigate the effects of the lamination schemes on the unstable regions of multi-layered cylindrical shells,results are presented for the instability regions of laminated cylindrical shells having different boundary conditions under

static and periodic tensile axial loading. The frequency parameter x0 ¼ PL2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq=E2h2

qis used here, and the shell and loading

0.0 0.1 0.2 0.3 0.4 0.5

300

350

400

450

500

550

600

650

700

750

800

850

900

φ = 90o

φ = 75o

φ = 60o

φ = 0o

φ = 45o

φ = 15o

φ = 30o

Ns/No

′

Fig. 5. Unstable regions for a four-layered (//�////�/) clamped-free (C-F) cylindrical shell under tensile loading (L/R = 2, R/h = 100, N0 = 0.5Ncr).

0.0 0.1 0.2 0.3 0.4 0.5450

500

550

600

650

700

750

800

850

900

950

1000

φ = 0o

φ = 15o

φ = 30o

φ = 45o

φ = 60o

φ = 75o

φ = 90o

Ns/No

′

Fig. 6. Unstable regions for a four-layered (//�////�/) free cylindrical shell under tensile loading (L/R = 2, R/h = 100, N0 = 0.5Ncr).


parameters are L/R = 5, R/h = 100, and N0 = 0.5Ncr. Fig. 3 shows the primary instability regions, for the fundamentalmode (m,n) = (1, 5), of an antisymmetric four-layered (//�////�/) simply-supported cylindrical shell. It is observed thatthe shell with lamination angle / = 0� has the widest unstable region while the shell with / = 90� has the smallest insta-bility region. The sizes of the instability regions of the shells with other lamination angles increase in the order of / = 75�,/ = 60�, / = 45�, / = 15� and / = 30�. Fig. 4 shows the primary instability regions, for the fundamental mode(m,n) = (1, 6), of a corresponding clamped cylindrical shell. It is observed that the widest and narrowest instability regionsoccur at / = 0� and / = 90� respectively, while in Fig. 5 for a corresponding clamped-free (C-F) cylindrical shell, the widestand narrowest instability regions occur respectively at / = 90� and / = 30�. Fig. 6 presents the results of the instabilityregions, for the fundamental mode (m,n) = (1, 8), of a corresponding free cylindrical shell. It is observed that the sizesof the instability regions generally decrease as the lamination angle / increases from 0� to 90�.

4. Conclusions

The dynamic stability of thin circular cylindrical shells under static and periodic axial forces has been investigated usinga meshfree technique. The meshfree kernel particle estimate was successfully employed in hybridized form with harmonicfunctions, to approximate the 2-D transverse displacement field. The Ritz minimization procedure was employed to obtaina system of Mathieu–Hill equations, and the principal instability regions were calculated via Bolotin’s first approximation.Numerical comparisons were made, and the present methodology validated. The instability regions of composite laminatedcylindrical shells having various boundary conditions and lamination schemes were investigated. It was found that the lam-ination scheme influenced the sizes of the instability regions, and the effects are distinct for different boundary conditioncases.

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Documents

Dynamic stability analysis of composite laminated cylindrical shells via the mesh-free kp-Ritz method