DSpace at VNU: Nonlinear dynamic analysis of eccentrically stiffened functionally graded circular cylindrical thin shells under external pressure and surrounded by an elastic medium

Nonlinear dynamic analysis of eccentrically stiffened functionallygraded circular cylindrical thin shells under external pressure and surrounded by an elastic medium Dao Van Dunga, Vu Ho

Nonlinear dynamic analysis of eccentrically stiffened functionally

graded circular cylindrical thin shells under external pressure and

surrounded by an elastic medium

Dao Van Dunga, Vu Hoai Namb,*

a Vietnam National University, Ha Noi, Viet Nam

b University of Transport Technology, Ha Noi, Viet Nam

a r t i c l e i n f o

Article history:

Received 17 July 2013

Accepted 9 February 2014

Available online 18 February 2014

Keywords:

Functionally graded material

Nonlinear dynamic analysis

Stiffened circular cylindrical shell

a b s t r a c t

A semi-analytical approach eccentrically stiffened functionally graded circular cylindrical shells sur-rounded by an elastic medium subjected to external pressure is presented The elastic medium is assumed as two-parameter elastic foundation model proposed by Pasternak Based on the classical thin shell theory with the geometrical nonlinearity in von KarmaneDonnell sense, the smeared stiffeners technique and Galerkin method, this paper deals the nonlinear dynamic problem The approximate three-term solution of deflection shape is chosen and the frequencyeamplitude relation of nonlinear vibration is obtained in explicit form The nonlinear dynamic responses are analyzed by using fourth order RungeeKutta method and the nonlinear dynamic buckling behavior of stiffened functionally graded shells is investigated according to BudianskyeRoth criterion Results are given to evaluate effects

of stiffener, elastic foundation and input factors on the frequencyeamplitude curves, natural frequencies, nonlinear responses and nonlinear dynamic buckling loads of functionally graded cylindrical shells

Ó 2014 Elsevier Masson SAS All rights reserved

1 Introduction

Functionally graded material (FGM) cylindrical shell has become

popular in engineering designs of coating of nuclear reactors and

space shuttle The static and dynamic behavior of FGM cylindrical

shell attracts special attention of a lot of researchers in the world

In static analysis of FGM cylindrical shells, many studies have

been focused on the buckling and postbuckling of shells under

mechanic and thermal loading Shen (2003) presented the

nonlinear postbuckling of perfect and imperfect FGM cylindrical

thin shells in thermal environments under lateral pressure by using

the classical shell theory with the geometrical nonlinearity in von

KarmaneDonnell sense By using higher order shear deformation

theory; this author (Shen, 2005) continued to investigate the

postbuckling of FGM hybrid cylindrical shells in thermal

environ-ments under axial loading.Huang and Han (2008, 2009a, 2009b,

2010a, 2010b) studied the buckling and postbuckling of

un-stiffened FGM cylindrical shells under torsion load, axial

compression, radial pressure, combined axial compression and

radial pressure based on the Donnell shell theory and the nonlinear strainedisplacement relations of large deformation.Shen (2009b) investigated the torsional buckling and postbuckling of FGM cy-lindrical shells in thermal environments The non-linear static buckling of FGM conical shells which is more general than cylin-drical shells, were studied bySofiyev (2011a,b).Zozulya and Zhang (2012)studied the behavior of functionally graded axisymmetric cylindrical shells based on the high order theory

For dynamic analysis of FGM cylindrical shells, Darabi et al (2008) presented respectively linear and nonlinear parametric resonance analyses for un-stiffened FGM cylindrical shells.Sofiyev and Schnack (2004)andSofiyev (2005)obtained critical parame-ters for un-stiffened cylindrical thin shells under linearly increasing dynamic torsional loading and under a periodic axial impulsive loading by using the Galerkin technique together with Ritz type variation method.Sheng and Wang (2008)presented the thermo-mechanical vibration analysis of FGM shell with flowing fluid

Sofiyev (2003, 2004, 2009, 2012) and Deniz and Sofiyev (2013) were investigated the vibration and dynamic instability of FGM conical shells.Hong (2013)studied thermal vibration of magneto-strictive FGM cylindrical shells.Huang and Han (2010c)presented the nonlinear dynamic buckling problems of un-stiffened func-tionally graded cylindrical shells subjected to time-dependent axial

* Corresponding author Tel.: þ84 983843387.

E-mail address: hoainam.vu@utt.edu.vn (V.H Nam).

Contents lists available atScienceDirect European Journal of Mechanics A/Solids

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / e j m s o l

http://dx.doi.org/10.1016/j.euromechsol.2014.02.008

0997-7538/Ó 2014 Elsevier Masson SAS All rights reserved.

European Journal of Mechanics A/Solids 46 (2014) 42e53

load by using the BudianskyeRoth dynamic buckling criterion

(Budiansky and Roth, 1962) Various effects of the inhomogeneous

parameter, loading speed, dimension parameters; environmental

temperature rise and initial geometrical imperfection on nonlinear

dynamic buckling were discussed

For FGM cylindrical shell surrounded by an elastic foundation,

the postbuckling of shear deformable FGM cylindrical shells

sur-rounded by an elastic medium was studied byShen (2009a).Shen

et al (2010)investigated postbuckling of internal pressure loaded

FGM cylindrical shells surrounded by an elastic medium

Bagherizadeh et al (2011)investigated mechanical buckling of FGM

cylindrical shells surrounded by Pasternak elastic foundation

Sofiyev (2010)analyzed the buckling of FGM circular shells under

combined loads and resting on the Pasternak type elastic

founda-tion Torsional vibration and stability of functionally graded

orthotropic cylindrical shells on elastic foundations is presented by

Najafov et al (2013) For the FGM conical shelle general case of

FGM cylindrical shells, mechanic behavior of shell on elastic

foun-dation was studied bySofiyev (2011c), Najafov and Sofiyev (2013),

Sofiyev and Kuruoglu (2013)

In practice, FGM plates and shells, as other composite structures,

usually reinforced by stiffeners system to provide the benefit of

added load carrying capability with a relatively small additional

weight Thus study on nonlinear static and dynamic behavior of

theses structures are significant practical problem However, up to

date, the investigation on thisfield has received comparatively

little attention Recently, Najafizadeh et al (2009) have studied

linear static buckling of FGM axially loaded cylindrical shell

rein-forced by ring and stringer FGM stiffeners.Bich et al (2011, 2012,

2013)have investigated the nonlinear static and dynamic analysis

of FGM plates, cylindrical panels and shallow shells with

eccen-trically homogeneous stiffener system Dung and Hoa (2013a,

2013b)presented an analytical study of nonlinear static buckling

and post-buckling analysis of eccentrically stiffened functionally

graded circular cylindrical shells under external pressure and

torsional load with FGM stiffeners and approximate three-term

solution of deflection taking into account the nonlinear buckling

shape

The review of the literature signifies that there are very little

researches on the nonlinear dynamic analysis of FGM stiffened

shells surrounded by an elastic foundation by analytical approach

In this paper, the dynamic behavior of eccentrically stiffened FGM

(ES-FGM) cylindrical circular shells reinforced by eccentrically ring

and stringer stiffener system on internal and (or) external surface of

shell under external pressure loads is investigated The nonlinear

dynamic equations are derived by using the classical shell theory

with the nonlinear strainedisplacement relation of large deflection,

the smeared stiffeners technique and Galerkin method The present

novelty is that an approximate three-term solution of deflection

including the pre-buckling shape, the linear buckling shape and the

nonlinear buckling shape are more correctly chosen and the

fre-quencyeamplitude relation of nonlinear vibration is obtained in

explicit form In addition, the nonlinear dynamic responses are

found by using fourth order RungeeKutta method and the dynamic

buckling loads of stiffened FGM shells are investigated according to

BudianskyeRoth criterion The results show that the stiffener,

volume-fractions index and geometrical parameters strongly

in-fluence to the dynamic behavior of shells

2 Formulation

2.1 FGM power law properties

Functionally graded material in this paper, is assumed to be

made from a mixture of ceramic and metal in two cases: inside

ceramic surface, outside metal surface and outside ceramic surface, inside metal surface The volume-fractions is assumed to be given

by a power law

Vin ¼ VinðzÞ ¼

2zþ h 2h

k

; Vou ¼ VouðzÞ ¼ 1  VinðzÞ; (1)

where h is the thickness of shell; k 0 is the volume-fraction index;

z is the thickness coordinate and varies from h/2 to h/2; the subscripts in and ou refer to the inside and outside material con-stituents, respectively

For case of inside ceramic surface and outside metal surface

Vin¼ Vcand Vou¼ Vm, for the case of outside ceramic surface and inside metal surface Vin¼ Vmand Vou¼ Vc In which, Vcis volume-fraction of ceramic and Vmis volume-fraction of metal

Effective properties Preffof FGM shell are determined by linear rule of mixture as

Preff ¼ ProuðzÞVouðzÞ þ PrinðzÞVinðzÞ: (2)

According to the mentioned law, the Young’s modulus and the mass density of shell can be expressed in the form

EðzÞ ¼ EouVouþ EinVin ¼ Eouþ ðEin EouÞ2z þ h

2h

k

;

rðzÞ ¼ rouVouþrinVin ¼rouþ ðrinrouÞ2z þ h

2h

k

;

(3)

For case of inside ceramic surface and outside metal surface

Ein¼ Ec,rin¼rcand Eou¼ Em,rou¼rm, for the case of outside ceramic surface and inside metal surface Ein ¼ Em,rin¼rm and

Eou¼ Ec,rou¼rc Ec,rc, Em,rmare the Young’s modulus and the mass density of ceramic and metal, respectively

2.2 Constitutive relations and governing equations Consider a functionally graded cylindrical thin shell surrounded

by an elastic foundation with length L, mean radius R and rein-forced by closely spaced (Najafizadeh et al., 2009; Brush and Almroth, 1975; Reddy and Starnes, 1993) pure-metal ring and stringer stiffener systems (see Fig 1) The stiffener is located at outside surface for outside metal surface case and at inside surface for inside metal surface case The origin of the coordinate O locates

on the middle surface and at the left end of the shell, x,y¼ Rqand z axes are in the axial, circumferential, and inward radial directions respectively

According to the von Karman nonlinear strainedisplacement relations (Brush and Almroth, 1975), the strain components

at the middle surface of perfect circular cylindrical shells are the form

ε0

x ¼ vu

vxþ

1 2

vw vx

2

;

ε0

y ¼ vv

vy

w

Rþ1 2

vw vy

2

;

g0

xy ¼ vu

vyþ

vv

vxþ

vw vx

vw

vy;

cx ¼ v2w

vx2; cy ¼ v2w

vy2; cxy ¼ v2w

vxvy;

(4)

whereε0 andε0 are normal strains,g0

xy is the shear strain at the middle surface of shell,cx,cy,cxyare the change of curvatures and twist of shell, and u¼ u(x,y), v ¼ v(x,y), w ¼ w(x,y) are displace-ments along x, y and z axes respectively

D.V Dung, V.H Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53 43

The strains across the shell thickness at a distance z from the

mid-surface are represented by

εx ¼ ε0

x zcx; εy ¼ ε0

y zcy; gxy ¼ g0

xy 2zcxy: (5)

The deformation compatibility equation is derived from Eq.(4)

v2ε0

x

vy2 þv

2ε0

y

vx2 v

2g0

xy

vxvy ¼ 

1 R

v2w

vx2 þ vxvyv2w

!2

v2w

vx2

v2w

vy2: (6)

The stressestrain relations for FGM shells are

ssh

x ¼ EðzÞ

1n2



εxþnεy

;

ssh

y ¼ EðzÞ

1n2



εyþnεx



;

ssh ¼ 2ð1 þEðzÞnÞgxy;

(7)

where the Poisson’s rationis assumed to be constant,ssh

x;ssh

y are normal stress in x, y direction of un-stiffened shell, respectively,ssh

is shearing stress in of un-stiffened shell

The stressestrain relation is applied for homogenous stiffeners

sst

s ¼ Esεx;

sst

where sst

s;sst

r are normal stress of stringer and ring stiffeners,

respectively Es, Erare Young’s modulus of stringer and ring

stiff-eners, respectively In this paper, the stringer and ring are assumed

to be metal stiffeners, so Es¼ Erh Em

Taking into account the contribution of stiffeners by the

smeared stiffeners technique and omitting the twist of stiffeners

because the torsion constants are smaller more than the moment of

inertia (Brush and Almroth, 1975) and integrating the stressestrain

equations and their moments through the thickness of shell, the

expressions for force and moment resultants of an ES-FGM

cylin-drical shell are of the form

Nx ¼ A11þEmAs

ss



ε0

xþ A12ε0

y ðB11þ CsÞcx B12cy;

Ny ¼ A12ε0

xþA22þEmAr

sr



ε0

y B12cx ðB22þ CrÞcy;

Nxy ¼ A66g0

xy 2B66cxy;

(9)

Mx ¼ ðB11þ CsÞε0

xþ B12ε0

yD11þEmIs

ss



cx D12cy;

My ¼ B12ε0

xþ ðB22þ CrÞε0

y D12cxD22þEmIr

sr



cy;

Mxy ¼ B66g0

xy 2D66cxy;

(10)

where Aij, Bij, Dij(i,j¼ 1,2,6) are extensional, coupling and bending stiffness of the un-stiffened FGM cylindrical shell, Nx, Nyare in-plane normal force intensities, Nxy is plane shearing force in-tensity, Mx, Myare bending moment intensities and Mxyis twisting moment intensity

A11 ¼ A22 ¼ E1

1n2; A12 ¼ E1n

1n2; A66 ¼ E1

2ð1 þnÞ;

B11 ¼ B22 ¼ E2

1n2; B12 ¼ E2n

1n2; B66 ¼ E2

2ð1 þnÞ;

D11 ¼ D22 ¼ E3

1n2; D12 ¼ E3n

1n2; D66 ¼ E3

2ð1 þnÞ;

(11)

with

E1 ¼

EouþEin Eou

kþ 1

 h; E2 ¼ ðEin EouÞkh2

2ðk þ 1Þðk þ 2Þ;

E3 ¼



Eou

12þ ðEin EouÞ

1

kþ 3

1

kþ 2þ

1 4kþ 4



h3;

Is ¼ dsh3 s

12 þ Asz2s; Ir ¼ drh3

r

12 þ Arz2r;

Cs ¼ EmAszs

ss ; Cr ¼ EmArzr

sr ;

zs ¼ hsþ h

2 ; zr ¼ hrþ h

2 ;

(12)

where the coupling parameters Csand Crare negative for outside stiffeners and positive for inside ones The spacing of the longi-tudinal and transversal stiffeners is denoted by ss and sr, respectively The width and thickness of the stringer and ring stiffeners are denoted by ds, hs and dr, hr, respectively The quantities As, Arare the cross-section areas of stiffeners and Is, Ir,

zs, zrare the second moments of cross section areas and the ec-centricities of stiffeners with respect to the middle surface of shell respectively

Fig 1 Configuration of an eccentrically stiffened cylindrical shell surrounded by an elastic medium.

D.V Dung, V.H Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53 44

From the constitutive relations(9), one can obtain inversely

ε0

x ¼ A*

22Nx A*

12Nyþ B*

11cxþ B*

12cy;

ε0

y ¼ A*

11Ny A*

12Nxþ B*

21cxþ B*

22cy;

g0

xy ¼ A*

66þ 2B*

66cxy;

(13)

in which

A*11 ¼ D1

A11þEmAs

ss



; A*

22 ¼ D1

A22þEmAr

sr



;

A*12 ¼ A12

D ; A*66 ¼ 1

A66;

D ¼ A11þEmAs

ss



A22þEmAr

sr



 A2

12;

B*11 ¼ A*

22ðB11þ CsÞ  A*

12B12;

B*22 ¼ A*

11ðB22þ CrÞ  A*

12B12;

B*12 ¼ A*

22B12 A*

12ðB22þ CrÞ;

B*21 ¼ A*

11B12 A*

12ðB11þ CsÞ;

B*66 ¼ B66

A66:

(14)

Substituting Eq.(13)into Eq.(10)leads to

Mx ¼ B*

11Nxþ B*

21Ny D*

11cx D*

12cy;

My ¼ B*

12Nxþ B*

22Ny D*

21cx D*

22cy;

Mxy ¼ B*

66Nxy 2D*

66cxy;

(15)

in which

D*11 ¼ D11þEmIs

ss  ðB11þ CsÞB*

11 B12B*21;

D*22 ¼ D22þEmIr

sr  B12B*12 ðB22þ CrÞB*

22;

D*12 ¼ D12 ðB11þ CsÞB*

12 B12B*22;

D*21 ¼ D12 B12B*11 ðB22þ CrÞB*

21;

D*66 ¼ D66 B66B*66:

(16)

The nonlinear equations of motion of a thin circular cylindrical

shell based on the classical shell theory and the assumption (Darabi

et al., 2008; Sofiyev and Schnack, 2004; Volmir, 1972) u<< w and

v<< w,r1v2u/vt2/ 0,r1v2v/vt2/ 0 are given by

vNx

vx þ

vNxy

vy ¼ 0;

vNxy

vx þ

vNy

vy ¼ 0;

v2Mx

vx2 þ 2v2Mxy

vxvy þ

v2My

vy2 þ Nxv2w

vx2þ 2Nxyv2w

vxvyþ Nyv2w

vy2

þ1

RNyþ q0 k1wþ k2 v2w

vx2 þv2w

vy2

!

¼ r1v2w

vt2 þ 2r1εvw

vt; (17)

where k1is Winkler foundation modulus and k2is the shear layer

foundation stiffness of Pasternak model, q0is external pressure, t is

time (s),ε is damping coefficient and

r1 ¼

Zh =

2

h=2

rðzÞdz þrmAs

ssþrmAr

sr

¼

rouþrinrou

kþ 1



hþrmAs

ssþrmAr

Considering thefirst two of Eq.(17), a stress function 4 may be

defined as

Nx ¼ v24

vy2; Ny ¼ v24

vx2; Nxy ¼ vxvyv24: (19)

Substituting Eq (13) into the compatibility Eq (6) and Eq (15)into the third of Eq.(17), taking into account Eqs.(4) and (19), yields

A*11v44

vx4þA*66 2A*12 v44

vx2vy2þ A*22v44

vy4þ B*21v4w

vx4

þB*11þ B*

22 2B* 66

 v4w

vx2vy2þ B*

12

v4w

vy4þ1 R

v2w

vx2



2

4 v2w vxvy

!2

v2w

vx2

v2w

vy2

3

5 ¼ 0;

(20)

r1v2w

vt2 þ 2r1εvw

vt þ D*11

v4w

vx4 þD*12þ D*

21þ 4D* 66

 v4w

vx2vy2

þ D* 22

v4w

vy4 B* 21

v44

vx4B*11þ B*

22 2B*

66 v44

vx2vy2

 B*12v44

vy41Rv24

vx2v24

vy2

v2w

vx2 þ 2vxvyv24vxvyv2wv24

vx2

v2w

vy2

 q0þ k1w k2 v2w

vx2 þv2w

vy2

!

¼ 0:

(21)

Eqs.(20) and (21)are a nonlinear equation system in terms of two dependent unknowns w and 4 They are used to investigate the dynamic characteristics of ES-FGM circular cylindrical shells

3 Dynamic Galerkin method approach Suppose that an ES-FGM cylindrical shell is simply supported and subjected to uniformly distributed pressure of intensity q0 (N/m2) surrounded by an elastic foundation Thus the boundary conditions are of the form

w¼ 0; Mx ¼ 0; Nx ¼ 0; Nxy ¼ 0; at x ¼ 0; L: (22)

The deflection of cylindrical shells in this case can be chosen by Volmir (1972), Huang and Han (2010a)

w ¼ f0þ f1sinmpx

L sin

ny

R þ f2sin2mpx

in which f0 ¼ f0(t) is time dependent pre-buckling uniform un-known amplitude, f1 ¼ f1(t) is time dependent linear unknown amplitude, f2 ¼ f2(t) is time dependent nonlinear unknown amplitude, sin(mpx/L)sin(ny/R) is linear buckling shape, sin2mpx/L

is nonlinear buckling shape in axial direction, m is number of half waves and n is number of full wave in axial and circumferential directions, respectively

D.V Dung, V.H Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53 45

As can be seen that the simply supported boundary condition at

x¼ 0 and x ¼ L is fulfilled on the average sense

Substituting Eq.(23)into Eq.(20)and solving obtained equation

for unknown 4 leads to

4 ¼ 41cos2mpx

L þ 42cos2ny

R  43sinmpx

L sin

ny R

þ 44sin3mpx

L sin

ny

R s0yhx2

2;

(24)

wheres0yis the average circumferential stress, and

41 ¼ n2l2

32A*11m2p2f2



4lL 16B*

21m2p2 32A*11m2p2 f2;

42 ¼ m2p2

32A*22n2l2f2;

43 ¼ BAf1þm2n2p2l2

A f1f2;

44 ¼ m2n2p2l2

G f1f2;

(25)

Substituting the expressions ((23) and (24)) into Eq.(20) and

then applying Galerkin method in the ranges 0  x  L and

0 y  2pR leads to

s0yh ¼ Rq012Rk1ðf2þ 2f0Þ  Rr1d2f0

dt2  Rr1 2

d2f2

dt2

 2Rr1εdf0

dt  Rr1εdf2

dt;

(27)

r1v2f0

vt2 þr13 4

v2f2

vt2 þ 2r1εvf0

vt þ

3

2r1εvf2

vt þ

(

4B*21mp L

4

1 R

mp L

2

n2l2

16A*11m2p2þ1

2

B A

mp L

2n R

2)

f2

þ1

2m

2n2p2l2mp

L

2n R

2
1

A1 G



f2f2 þ

( 4D*11mp L

4



 4B*21mp L

4

1 R

mp L

2 lL 4B*

21m2p2

4A*11m2p2

)

f2þs0yh

R  q0

þ k1

3

4f2þ f0



þ k2f2mp

L

2

¼ 0:

(29)

In addition, the cylindrical shell must satisfy the circumferential closed condition as (Huang and Han, 2010c; Volmir, 1972)

Z2pR

0

ZL 0

vv

vydxdy ¼

Z2pR

0

ZL 0

"

ε0

yþwR12

vw vy

2# dxdy ¼ 0: (30)

Using Eqs.(13), (19), (23) and (24), this integral becomes

2A*

11s0yhþ1

Rðf2þ 2f0Þ 1

4

n R

2

Eliminatings0yfrom Eqs.(27)e(29)and the condition of closed form(31), lead to

d2f0

dt2 þ 2εdf0

dt

!

þ1 2

d2f2

dt2 þ 2εdf2

dt

!

þa11ðf2þ 2f0Þ

a12f2a13q0þa14k1ðf2þ 2f0Þ ¼ 0;

(32)

L4r1v2f1

vt2 þ 2L4r1εvf1

vt þ

DþB2 A



f1þ

2 4m2n2p2l2

Am

2n2p2l2n

2l2

lL 4B*

21m2p2 4A*11

3 5f1f2þ m4n4p4l4

A þm4n4p4l4

G

!

f1f2

þ m4p4

16A*22þ n4l4

16A*11

!

f3s0yhn2L2l2f1þ L4k1f1þ L2k2f1h

ðlnÞ2þ ðmpÞ2i

¼ 0;

(28)

A ¼ A*

11m4p4þA*66 2A*

12



m2n2p2l2þ A*

22n4l4;

B ¼ B*

21m4p4þB*11þ B*

22 2B* 66



m2n2p2l2þ B*

12n4l4LR2m2p2;

D ¼ D*

11m4p4þD*12þ D*

21þ 4D* 66



m2n2p2l2þ D*

22n4l4;

G ¼ 81A*

11m4p4þ 9A*66 2A*

12



m2n2p2l2þ A*

22n4l4;

l ¼ LR:

(26)

D.V Dung, V.H Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53 46

a21f1 d

2f0

dt2 þ 2εdf0

dt

!

þ d2f1

dt2 þ 2εdf1

dt

!

þa21

2 f1

d2f2

dt2 þ 2εdf2

dt

!

þa22f1þa23f1f2þa24f3þa25f1f2a26q0f1þa27k1f1

þa28k2f1 ¼ 0;

(33)

d2f2

dt2 þ 2εdf2

dt

!

þa31f2þa32f2f2þa33f2þa34k1

3

4f2þ f0



þa35k2f2 ¼ 0;

(34)

where

a11 ¼ 1

2A*11R2r1;a12 ¼ n2

8A*11R3r1;a13 ¼ r11;a14 ¼ 21r1; (35)

a21¼Rn2l2

L2 ;a22¼ 1

L4r1

DþBA2



;

a23¼ 1

L4r1

2

4m2n2p2l2

A þBAm2n2p2l2n

2l2

lL 4B*

21m2p2 4A*11

3 5;

a24¼ 1

L4r1

m4p4

16A*22þ n4l4

16A*11

!

;

a25¼ 1

L4r1

m4n4p4l4

A þm4n4p4l4

G

!

;

a26¼Rn2l2

L2r1;a27¼r11;a28 ¼ 1

L2r1

h

ðlnÞ2þ ðmpÞ2i

;

(36)

a31 ¼ r11

(

4B*21mp

L

4

1 R

mp L

2

n2l2

4A*11m2p2

þ 2B

A

mp L

2n R

2)

;

a32 ¼ r21m2n2p2l2mp

L

2n R

2
1

A1 G



;

a33 ¼ r11

(

16D*11mp

L

4



 4B*21mp L

4

1 R

mp L

2

lL 4B*

21m2p2

A*11m2p2

)

;

a34 ¼ r41; a35 ¼ r41

mp L

2

:

(37)

Simplifying Eqs.(32)e(34), leads to

d2f0

dt2 þ 2εdf0

dt

!

þb11f0b12f2b13f2f2þb14f2a13q0

þb15k1f0þb16k1f2b17k2f2 ¼ 0;

(38)

d2f1

dt2 þ 2εdf1

dt

!

þa22f1þb21f1f0þb22f1f2þa25f1f2þb23f3

b24k1f1f0b25k1f1f2þb26k2f1f2þa27k1f1

þa28k2f1 ¼ 0;

(39)

d2f2

dt2 þ 2εdf2

dt

!

þa31f2þa32f2f2þa33f2þa34k1

3

4f2þ f0



þa35k2f2 ¼ 0;

(40)

where

b11 ¼ 2a11; b12 ¼ 1

2a31þa12; b13 ¼ 1

2a32;

b14 ¼ a111

2a33; b15 ¼ 2a141

2a34;

b16 ¼ a1438a34; b17 ¼ 12a35;

(41)

b21 ¼ b11a21; b22 ¼ b14a21a33a21

2 þa23;

b23 ¼ b12a21a31a21

2 þa24;

b24 ¼ a21b15þa21

2a34; b25 ¼

a21b16þ3

8a21a34;

b26 ¼ a21b17a21

2a35:

(42)

Putting f¼ wmax, from Eq.(23), it is obvious that the maximal

deflection of the shells

locates at x¼ iL/2m, y ¼ jpR/2n where i, j are odd integer numbers Note that f0¼ f0(t), f1¼ f1(t), f2¼ f2(t) and f¼ f(t) in Eq.(43) Eqs.(38)e(40) and (43)are used to analyze the effects of input parameters on the load-maximum deflection curves of ES-FGM shells

3.1 Nonlinear vibration analysis Consider an ES-FGM cylindrical thin shell under uniformly external pressure with the law q0¼ QsinUt, Eqs.(38)e(40)become

d2f0

dt2 þ 2εdf0

dt

!

þb11f0b12f2b13f2f2þb14f2

a13Q sinðUtÞ þb15k1f0þb16k1f2

b17k2f2 ¼ 0;

(44)

d2f1

dt2 þ 2εdf1

dt

!

þa22f1þb21f1f0þb22f1f2þa25f1f2þb23f3

b24k1f1f0b25k1f1f2þb26k2f1f2þa27k1f1

þa28k2f1 ¼ 0;

(45)

D.V Dung, V.H Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53 47

dt2 þ 2εdf2

dt

!

þa31f2þa32f2f2þa33f2þa34k1

3

4f2þ f0



þa35k2f2 ¼ 0;

(46)

where Q is amplitude of excitation force and U is excitation

frequency

By using these equations, the fundamental frequencies of

nat-ural vibration of ES-FGM shell and un-stiffened FGM shell, and

frequencyeamplitude relation of nonlinear vibration and nonlinear

response of ES-FGM shell are taken into account The nonlinear

dynamic responses of ES-FGM shells can be obtained by solving

Eqs.(44)e(46)by the fourth order RungeeKutta iteration method

If the uniform buckling shape and nonlinear buckling shape are

ignored, Eq.(33)reduces to

d2f1

dt2 þ 2εdf1

dt

!

þ ða22þa27k1þa28k2Þf1þa24f3

a26f1Q sinðUtÞ ¼ 0:

(47)

For the free and linear vibration without damping, the Eq.(47)

becomes

d2f1

dt2 þ ða22þa27k1þa28k2Þf1 ¼ 0: (48)

The fundamental frequency of natural vibration of ES-FGM

cy-lindrical shells can be determined by

umn ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia22þ k1a27þ k2a28; (49)

whereumnis fundamental frequency of natural vibration of shell

Seeking solution as f1(t)¼hsin(Ut) and applying procedure like

Galerkin method to Eq.(47), the frequencyeamplitude relation of

nonlinear vibration is obtained

U2p4εU ¼ ða22þ k1a27þ k2a28Þ þ34a24h238p a26Q: (50)

wherehis the amplitude of nonlinear vibration of f1(t)

By introducing the non-dimension frequency parameterx¼U/

umn, Eq.(50)becomes

x2pu4εmnx¼ 1 þ3

4

a24

u2 mn

h2 8

3p

a26

u2 mn

If Q¼ 0, the frequencyeamplitude relation of nonlinear free

vibration is obtained

x2pu4εmnx¼ 1 þ3

4

a24

u2 mn

3.2 Buckling analysis

3.2.1 Linear static buckling analysis of ES-FGM cylindrical shells

Omitting the uniform buckling shape and nonlinear buckling

shape and putting _f1 ¼ 0; €f1 ¼ 0, and taking f1 s 0 Eq (33)

becomes

a22þ k1a27þ k2a28þa24f2a26q0 ¼ 0: (53)

By ignoring the nonlinear term of f1in Eq.(53), leads to

qsbu0 ¼ a22þ k1a27þ k2a28

where qsbu

0 is the linear upper static buckling load of ES-FGM cy-lindrical shells

The linear static critical buckling loads of ES-FGM cylindrical shells are determined by conditions qscr ¼ minqsbu

0 vs (m,n) 3.2.2 Nonlinear dynamic buckling analysis of ES-FGM cylindrical shells

Based on Eqs.(38)e(40), the nonlinear dynamic critical buckling analysis of ES-FGM circular cylindrical shells is investigated in case

of lateral pressure varying as linear function of time q0¼ ct in which

c (N/m2s) is a loading speed

Eqs.(38)e(40)are the nonlinear second-order differential three equation system Therefore their analytical solution may be very difficult to find mathematically In this paper, this equation system

is solved by four order RungeeKutta method The dynamic critical time tcrcan be obtained according to BudianskyeRoth criterion (Budiansky and Roth, 1962) This criterion is based on that, for large value of loading speed, the amplitudeetime curve of obtained displacement response increases sharply depending on time and this curve obtain a maximum by passing from the slope point and

at the corresponding time t¼ tcrthe stability loss occurs The load corresponding to the dynamic critical time is called dynamic critical buckling load

4 Numerical results 4.1 Validation of the present approach

To validate the present formulation, the natural frequencies of perfect stiffened isotropic cylindrical shells without elastic

Fig 2 Comparison of natural frequency of isotropic un-stiffened cylindrical shells (m ¼ 1).

Fig 3 Comparison of natural frequency of isotropic external stiffened cylindrical shells (m ¼ 1).

D.V Dung, V.H Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53 48

foundation are considered inFigs 2e4, which were also analyzed

bySewall and Naumann (1968)andSewall et al (1964) The static

buckling of stiffened isotropic cylindrical shells without elastic

foundation under external pressure was studied by Baruch and

Singer (1963), Reddy and Starnes (1993) and Shen (1998) (see

Table 1) and the natural frequencies of isotropic cylindrical shell

surrounded by an elastic foundation investigated bySofiyev et al

(2009)andPaliwal et al (1996)(Table 2)

As can be seen, the good agreements are obtained in these

comparisons

4.2 Dynamic responses of ES-FGM cylindrical shell

In this section, the stiffened and un-stiffened FGM cylindrical

shells surrounded by an elastic foundation are considered with

R¼ 0.5 m, L ¼ 0.75 m The combination of materials consists of

Aluminum Em¼ 7  1010 N/m2,rm ¼ 2702 kg/m3 and Alumina

Ec¼ 38  1010N/m2,rc¼ 3800 kg/m3 The Poisson’s rationis chosen

to be 0.3 for simplicity The height of stiffeners is equal to 0.01 m, its

width 0.0025 m The stiffener system includes 15 ring stiffeners and

63 stringer stiffeners distributed regularly in the axial and

circumferential directions, respectively

Table 3shows the fundamental frequency of natural vibration of

ES-FGM cylindrical shells with foundation parameters

k1¼ 5  105N/m3, k2¼ 2.5  104N/m, and geometric parameters

hs¼ hr¼ 0.01 m, bs¼ br¼ 0.0025 m Clearly, the natural frequency

of stiffened shells is greater than one of un-stiffened shells The

natural frequency decreases when the proportion of metal

in-creases.Table 3also shows that the natural frequency of shell

in-creases when the R/h ratio dein-creases When k¼ 1 (the proportions

of ceramic and metal of internal ceramic surface case is equal to

ones of external ceramic surface case), the natural frequency of two

case attain the same value

Table 4shows effects of foundation and stiffener on theumnof

cylindrical shells with input parameters k ¼ 1, R/h ¼ 250,

hs¼ hr¼ 0.01 m, bs¼ br¼ 0.0025 m and different parameters of foundation As can be found that parameters of foundation k1and

k2affect strongly to fundamental frequency of natural vibration of shells Especially, with the presence of the both two parameters of foundation, theumnis biggest

The effect of excitation force q0 on the h/hx frequencye amplitude curves of nonlinear vibration of internal stiffened FGM cylindrical shell is presented in Fig 5 Two foundation coefficients are considered as k1 ¼ 5  105 N/m3,

k2 ¼ 2.5  104 N/m and two values of Q are taken as

Q¼ 105N/m2and Q¼ 2  104N/m2 As can be observed, when the excitation force decreases, the curves of forced vibration are closer to the curve of free vibration

Fig 6investigates effect of the both stiffeners and foundation on theh/hxfrequencyeamplitude curve of nonlinear free vibration for parameters k1¼ 5  105N/m3, k2¼ 2.5  104N/m, R/h¼ 250,

k¼ 1 and modes m ¼ 1, n ¼ 8 The obtained results show that the frequencyeamplitude curve of un-stiffened shell is lower than one

of stiffened shell with and without elastic foundation (EF) Nonlinear responses of stiffened and un-stiffened functionally graded cylindrical shell are illustrated inFig 7 Computations have been carried out for the following data: k1 ¼ 5  105 N/m3,

k2¼ 2.5  104N/m, R/h¼ 250, k ¼ 1, m ¼ 1, n ¼ 5

Fig 4 Comparison of natural frequency of isotropic internal stiffened cylindrical shells

(m ¼ 1).

Table 1

Comparisons on the static buckling of internal stiffened isotropic cylindrical shells under external pressure (Psi) (m ¼ 1).

Present Baruch and Singer (1963) Reddy and Starnes (1993) Shen (1998)

a The numbers in the parenthesis denote the buckling modes (n).

Table 2 Comparison on the frequency parameter for un-stiffened cylindrical shell sur-rounded by a Winkler foundation (m ¼ 1).

n Present Sofiyev et al (2009) Paliwal et al (1996)

Table 3 Effect of R/h ratio and volume-fraction index k on the fundamental frequency of natural vibration (rad/s) of ES-FGM cylindrical shells surrounded by an elastic foundation.

R/h k Inside ceramic surface Outside ceramic surface

Un-stiffened External stiffeners Un-stiffened Internal stiffeners

100 0.2 3111.37 (6) a 3284.18 (6) 2187.64 (6) 2448.54 (5)

1 2650.64 (6) 2925.82 (5) 2650.64 (6) 2912.85 (5)

5 2212.02 (6) 2495.50 (5) 3023.68 (6) 3232.89 (6)

10 2098.85 (6) 2362.77 (5) 3132.36 (6) 3309.35 (6)

175 0.2 2439.80 (7) 2814.44 (6) 1806.34 (7) 2282.83 (5)

1 2117.46 (7) 2650.02 (6) 2117.46 (7) 2668.73 (6)

5 1823.20 (7) 2331.08 (5) 2376.06 (7) 2814.52 (6)

10 1735.86 (6) 2214.15 (5) 2453.21 (7) 2850.82 (6)

250 0.2 2167.31 (7) 2732.24 (6) 1686.54 (7) 2246.27 (5)

1 1933.94 (7) 2614.92 (6) 1933.94 (7) 2613.61 (5)

5 1695.41 (7) 2301.47 (5) 2132.33 (7) 2747.12 (6)

10 1633.08 (7) 2195.37 (5) 2185.58 (7) 2771.16 (6)

a The numbers in the parenthesis denote the buckling modes (n), m ¼ 1 D.V Dung, V.H Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53 49

The excitation frequencies corresponding to q0 ¼

106sin(300t) N/m2 are much smaller than natural frequencies

These results show that the stiffeners strongly decrease vibration

amplitude of the shell when excitation frequencies are far from

natural frequencies

Considers an internal stiffened cylindrical shell with

k1¼ 5  105 N/m3, k2¼ 2.5  104 N/m and R/h¼ 250, k ¼ 1,

U¼ 300 rad/s, Q ¼ 106N/m2, m¼ 1, n ¼ 5 As can be seen that when

the excitation force is small, the deflection-velocity relation with has the closed curve form as inFig 8 But when the excitation force increases (Q¼ 1.5  106N/m2), the deflection-velocity curve be-comes more disorderly asFig 9

Fig 10shows that when the excitation frequencies are near to natural frequencies, the interesting phenomenon is observed like the harmonic beat phenomenon of a linear vibration The excitation frequency is 2600 rad/s which is near to natural frequencies 2613.61 rad/s of internal stiffened cylindrical shell As can be seen,

Table 4

Effect of foundation parameters k 1 , k 2 on the fundamental frequency of natural

vi-bration (rad/s) of ES-FGM cylindrical shells surrounded by an elastic foundation.

k 1 N/m 3 k 2 N/m Un-stiffened External stiffeners Internal stiffeners

0 0 1654.05 (8) a 2518.90 (6) 2539.43 (6)

10 4 1776.68 (8) 2553.49 (6) 2566.15 (5)

2.5  10 4 1913.95 (7) 2604.51 (6) 2603.20 (5)

5  10 4 2117.62 (7) 2687.40 (6) 2663.79 (5)

10 5 0 1658.70 (8) 2521.06 (6) 2541.57 (6)

10 4 1781.01 (8) 2555.62 (6) 2568.27 (5)

2.5  10 4 1917.97 (7) 2606.60 (6) 2605.28 (5)

5  10 4 2121.24 (7) 2689.42 (6) 2665.83 (5)

5  10 5 0 1677.14 (8) 2529.66 (6) 2550.10 (6)

10 4 1798.19 (8) 2564.10 (6) 2576.71 (5)

2.5  10 4 1933.94 (7) 2614.92 (6) 2613.61 (5)

5  10 4 2135.70 (7) 2697.48 (6) 2673.96 (5)

10 6 0 1699.91 (8) 2540.37 (6) 2560.72 (6)

10 4 1819.45 (8) 2574.67 (6) 2587.23 (5)

2.5  10 4 1953.72 (7) 2625.28 (6) 2623.97 (5)

5  10 4 2153.62 (7) 2707.53 (6) 2684.10 (5)

a The numbers in the parenthesis denote the buckling modes (n), m ¼ 1.

Fig 5 The frequencyeamplitude curve of nonlinear vibration of internal stiffened

FGM cylindrical shell (R/h ¼ 250, k ¼ 1, m ¼ 1, n ¼ 5).

Fig 6 The frequencyeamplitude curve of nonlinear vibration of un-stiffened and

Fig 7 Nonlinear responses of un-stiffened and internal stiffened FGM cylindrical shells.

Fig 8 Deflection-velocity relation of internal stiffened cylindrical shell under

Q ¼ 10 6 N/m 2

Fig 9 Deflection-velocity relation of internal stiffened cylindrical shell under

¼ 1.5  10 6 2 D.V Dung, V.H Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53 50

the amplitude of beats increases rapidly when the excitation

fre-quency approaches the natural frequencies

Effect of damping on nonlinear responses is presented inFigs 11

and12with linear damping coefficient ε ¼ 0.3 The damping

in-fluences very small to the nonlinear response in the first vibration

periods (Fig 11) however, it strongly decreases amplitude at the

next far periods (Fig 12)

4.3 Nonlinear dynamic buckling of ES-FGM shell

To investigate the nonlinear dynamic buckling approach of

eccentrically stiffened FGM cylindrical shells, the stiffened and

un-stiffened FGM cylindrical shells with and without elastic

founda-tion are considered with R¼ 0.5 m, L ¼ 0.75 m The combination of

materials is the same with previous section The height of stiffeners

is equal to 0.005 m, its width 0.002 m The stiffener system includes

15 ring stiffeners and 63 stringer stiffeners distributed regularly in

the axial and circumferential directions, respectively

Figs 13e14show the dynamic responses of un-stiffened and

stiffened shells under mechanic load Thesefigures also show that

there is no definite point of instability as in static analysis Rather,

there is a region of instability where the slope of f vs t curve

in-creases rapidly According to the BudianskyeRoth criterion

(Budiansky and Roth, 1962), the critical time tcrcan be taken as an

intermediate value of this region Therefore, one can choose the

inflexion point of curve i.e d2

f=dt2 

t ¼t cr ¼ 0 as Huang and Han (2010c)

Effect of elastic foundation and stiffener on the nonlinear critical buckling loads is given inTable 5 The computation parameters are assumed as k1¼ 5  105N/m3, k2¼ 2.5  104N/m, R/h¼ 250, k ¼ 1,

c¼ 106N/m2s Clearly, elastic foundation considerably enhances the critical buckling load of shell It seems that, the stringer stiff-eners lightly influence and the ring stiffeners strongly influence to the critical buckling load of shells Table 5 also shows that the critical dynamic buckling loads are greater than the critical static buckling loads of shells

Table 6shows the critical dynamic buckling loads of stiffened and un-stiffened cylindrical shells vs four different values of vol-ume fraction index k¼ (0.2,1,5,10) With the same value of foun-dation parameters k1 ¼ 5  105 N/m3, k2¼ 2.5  104N/m and loading speed c¼ 106N/m2s, it is found that the effectiveness of stiffeners is obviously proven; the critical buckling load of stiffened shell is greater than one of un-stiffened shell.Table 6also shows that the critical dynamic load decreases with the increase of the proportion of metal The results inTable 6also show the effect of R/

h ratio on the critical dynamic buckling of FGM cylindrical shells As can be seen, the critical dynamic buckling of FGM cylindrical shell is considerably decreased when the R/h ratio increases It is reason-able because the critical buckling loads decrease with the thinner shell

Fig 10 Nonlinear responses of internal stiffened FGM cylindrical shells

(k 1 ¼ 5  10 5 N/m 3 , k 2 ¼ 2.5  10 4 N/m, R/h ¼ 250, k ¼ 1, q 0 (t) ¼ 5  10 5 sin(2600t) N/

m 2 , m ¼ 1, n ¼ 5).

Fig 11 Effect of damping on nonlinear responses of external stiffened cylindrical

shells in the first periods (k 1 ¼ 5  10 5 N/m 3 , k 2 ¼ 2.5  10 4 N/m, R/h ¼ 250, k ¼ 1,

q (t) ¼ 5  10 5 sin(2600t) N/m 2 , m ¼ 1, n ¼ 5).

Fig 12 Effect of damping on nonlinear responses of external stiffened cylindrical shells in the far periods (k 1 ¼ 5  10 5 N/m 3 , k 2 ¼ 2.5  10 4 N/m, R/h ¼ 250, k ¼ 1,

q 0 (t) ¼ 5  10 5 sin(2600t) N/m 2 , m ¼ 1, n ¼ 5).

Fig 13 Effect of loading speed on the dynamic responses of internal stiffened shells under external pressure (R/h ¼ 250, k ¼ 1, m ¼ 1, n ¼ 8, k 1 ¼ 5  10 5 N/m 3 ,

k ¼ 2.5  10 4 N/m).

D.V Dung, V.H Nam / European Journal of Mechanics A/Solids 46 (2014) 42e53 51