DSpace at VNU: Nonlinear dynamic analysis of eccentrically stiffened imperfect functionally graded doubly curved thin shallow shells

For dynamic analysis of eccentrically stiffened laminated com-posite plates and shallow shells, Satish Kumar and Mukhopadhyay [9]studied the transient response analysis of laminated stif

Nonlinear dynamic analysis of eccentrically stiffened imperfect functionally

graded doubly curved thin shallow shells

Dao Huy Bicha, 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:

Available online 29 October 2012

Keywords:

Functionally graded material

Dynamic analysis

Critical dynamic buckling load

Vibration

Shallow shells

Stiffeners

a b s t r a c t

This paper presents a semi-analytical approach to investigate the nonlinear dynamic of imperfect eccen-trically stiffened functionally graded shallow shells taking into account the damping subjected to mechanical loads The functionally graded shallow shells are simply supported at edges and are rein-forced by transversal and longitudinal stiffeners on internal or external surface The formulation is based

on the classical thin shell theory with the geometrical nonlinearity in von Karman–Donnell sense and the smeared stiffeners technique By Galerkin method, the equations of motion of eccentrically stiffened imperfect functionally graded shallow shells are derived Dynamic responses are obtained by solving the equation of motion by the Runge–Kutta method The nonlinear critical dynamic buckling loads are found according to the Budiansky–Roth criterion Results of dynamic analysis show the effect of stiffen-ers, damping, pre-loaded compressions, material and geometric parameters on the dynamical behavior of these structures

Ó 2012 Elsevier Ltd All rights reserved

1 Introduction

Eccentrically stiffened shallow shell is a very important

struc-ture in engineering design of aircraft, missile and aerospace

indus-tries There are many researches on the static and dynamic

behavior of this structure with different materials

Studies on the dynamics first were carried out with

eccentri-cally stiffened shallow shells made of homogeneous material

Khalil et al.[1]presented a finite strip formulation for the

nonlin-ear analysis of stiffened plate structures subjected to transient

pressure loadings using an explicit central difference/diagonal

mass matrix time stepping method Shen and Dade[2]investigated

dynamic analysis of stiffened plates and shells using spline gauss

collocation method The free vibration of stiffened shallow shells

was studied by Nayak and Bandyopadhyay[3] By using the finite

element method, the stiffened shell element was obtained by the

appropriate combinations of the eight-/nine-node doubly curved

isoparametric thin shallow shell element with the three-node

curved isoparametric beam element Sheikh and Mukhopadhyay

[4] applied the spline finite strip method to investigate linear

and nonlinear transient vibration analysis of plates and stiffened

plates The von Karman’s large deflection plate theory has been

used and the formulation was done in total Lagrange coordinate

system Dynamic instability analysis of stiffened shell panels

sub-jected to uniform in-plane harmonic edge loading and partial edge

loading along the edges was studied by Patel et al.[5–8] In these studies, the new formulation for the beam element requires five degrees of freedom per node as that of shell element

For dynamic analysis of eccentrically stiffened laminated com-posite plates and shallow shells, Satish Kumar and Mukhopadhyay

[9]studied the transient response analysis of laminated stiffened plates using the first order shear deformation theory Parametric study on the dynamic instability behavior of laminated composite stiffened plate was studied by Patel et al.[10] The same authors

[11]investigated the dynamic instability of laminated composite stiffened shell panels subjected to in-plane harmonic edge loading

By using the commercial ANSYS finite element software, Less and Abramovich [12] studied the dynamic buckling of a laminated composite stringer stiffened cylindrical panel Bich et al.[13] pre-sented an analytical approach to investigate the nonlinear dynamic

of imperfect reinforced laminated composite plates and shallow shells using the classical thin shell theory with the geometrical nonlinearity in von Karman–Donnell sense and the smeared stiff-eners technique

For functionally graded materials (FGM), many researches fo-cused on the dynamical analysis of un-stiffened shallow shells Liew et al.[14]presented the nonlinear vibration analysis of the coating FGM substrate cylindrical panel subjected to a temperature gradient arising from steady heat conduction through the panel thickness Matsunaga et al.[15]investigated free vibrations and stability of FGM doubly curved shallow shells according to a 2-D higher order deformation theory Chorfi and Houmat[16] investi-gated nonlinear free vibrations of FGM doubly curved shallow 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd All rights reserved.

⇑Corresponding author.

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

Contents lists available atSciVerse ScienceDirect

Composite Structures

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 / c o m p s t r u c t

shells with an elliptical plan-form Nonlinear vibrations of

func-tionally graded doubly curved shallow shells under a concentrated

force were studied by Alijani et al.[17] Nonlinear dynamical

anal-ysis of imperfect functionally graded material shallow shells

sub-jected to axial compressive load and transverse load was studied

by Bich and Long[18], Dung and Nam[19] The motion, stability

and compatibility equations of these structures were derived using

the classical shell theory The nonlinear transient responses of

cylindrical and doubly-curved shallow shells subjected to excited

external forces were obtained and the dynamic critical buckling

loads were evaluated based on the displacement responses using

Budiansky–Roth dynamic buckling criterion

Recently, some authors have studied static and dynamical

behaviors of some kind of shells Najafizadeh et al.[20]have

stud-ied static buckling behaviors of FGM cylindrical shell Bich et al

[21]have studied the nonlinear static post-buckling of

eccentri-cally stiffened imperfect functionally graded plates and shallow

shells The nonlinear dynamical analysis of imperfect eccentrically

stiffened FGM cylindrical panels based on the classical theory with

the von Karman–Donnell geometrical nonlinearity are investigated

by Bich et al.[22]

Following the idea of work[22], in this paper, dynamic

govern-ing equations takgovern-ing into account the effect of dampgovern-ing, nonlinear

vibration and dynamic critical buckling loads of eccentrically

stiff-ened imperfect FGM doubly curved thin shallow shells are

estab-lished Effects of stiffeners, material, geometric parameters and

damping on the dynamic behavior of structure are considered

2 Eccentrically stiffened FGM shallow shells (ES-FGM shallow

shells)

Consider a doubly curved functionally graded shallow thin shell

(seeFig 1) of thickness h and in-plane edges a and b The shallow

shell is assumed to have a relative small rise as compared with its

span Let the (x1,x2) plane of the Cartesian coordinates overlaps the

rectangular plane area of the shell Note that the middle surface of

the shell generally is defined in terms of curvilinear coordinates,

but for the shallow shell, so the Cartesian coordinates can replace

the curvilinear coordinates on the middle surface

The volume fractions of constituents are assumed to vary

through the thickness according to the following power law

distribution

VcðzÞ ¼ 2z þ h

2h

 k

; VmðzÞ ¼ 1  VcðzÞ; ð1Þ

where k is the volume fraction exponent (k P 0), z is the thickness coordinate and varies from h/2 to h/2; the subscripts m and c refer

to the metal and ceramic constituents respectively Effective prop-erties Preffof FGM shell are determined by linear rule of mixture as

Preff ¼ PrmðzÞVmðzÞ þ PrcðzÞVcðzÞ: ð2Þ

According to Eqs.(1) and (2), the modulus of elasticity E and the mass densityqcan be expressed in the form

EðzÞ ¼ EmVmþ EcVc¼ Emþ ðEc EmÞ 2z þ h

2h

 k

;

qðzÞ ¼qmVmþqcVc¼qmþ ðqcqmÞ 2z þ h

2h

 k

:

ð3Þ

Poisson’s ratiomand linear damping coefficienteare assumed to

be constants

Assume that the shell is reinforced by eccentrically longitudinal and transversal homogeneous stiffeners with the elastic modulus

E0and the mass densityq0of stiffeners In order to provide the continuity between the shell and stiffeners, the full metal stiffeners are put at the metal-rich side of the shell thus E0andq0take the value E0= Em,q0=qmand conversely the full ceramic ones at the ceramic-rich side, so that E0= Ec,q0=qc

3 Theoretical formulation The strains at the middle surface and curvatures relating to the displacement components u,v, w based on the classical shell the-ory and von Karman–Donnell geometrical nonlinearity assumption are of the form[24]

e0¼ @u

@x1

 k1w þ1 2

@w

@x1

 2

; v1¼@

2w

@x2;

e0¼@v

@x2

 k2w þ1 2

@w

@x2

 2

; v2¼@

2w

@x2;

c0

@x2þ@v

@x1þ@w

@x1

@w

@x2

; v12¼ @

2

w

@x1@x2

;

ð4Þ

in which k1¼ 1

1; k2¼ 1

2and R1, R2are curvatures and radii of cur-vatures of the shell respectively

The strain components across the shell thickness at a distance z

from the middle surface are given by

e1¼e0 zv1; e2¼e0 zv2; c12¼c0

From Eq.(4), the strains must be relative in the deformation

compatibility equation

@2e0

@x2 þ@

2e0

@x2  @

2c0

12

@x1@x2

¼ @

2w

@x1@x2

!2

@

2w

@x2

@2w

@x2  k1

@2w

@x2  k2

@2w

@x2: ð6Þ

Hooke’s stress–strain relation is applied for the shell

rsh

1 m2ðe1þme2Þ;

rsh

1 m2ðe2þme1Þ;

ssh

2ð1 þmÞc12;

ð7Þ

and for stiffeners

rst

1¼ E0e1;

rst

Taking into account the contribution of stiffeners by the

smeared stiffeners technique and omitting the twist of stiffeners

because these torsion constants are smaller more than the

mo-ments of inertia[24]and integrating the stress–strain equations

and their moments through the thickness of the panel, lead to

the expressions for force and moment resultants of an ES-FGM

shallow shells as[22]

N1¼ A11þE0A1

s1

e0þ A12e0 ðB11þ C1Þv1 B12v2;

N2¼ A12e0þ A22þE0A2

s2

e0 B12v1 ðB22þ C2Þv2;

N12¼ A66c0

12 2B66v12;

ð9Þ

M1¼ ðB11þ C1Þe0þ B12e0 D11þE0I1

s1

v1 D12v2;

M2¼ B12e0þ ðB22þ C2Þe0 D12v1 D22þE0I2

s2

v2;

M12¼ B66c0

12 2D66v12;

ð10Þ

where Aij, Bij, Dij(i, j = 1, 2, 6) are extensional, coupling and bending

stiffness of the un-stiffened shell

A11¼ A22¼ E1

1 m2; A12¼ E1m

1 m2; A66¼ E1

2ð1 þmÞ;

B11¼ B22¼ E2

1 m2; B12¼ E2m

1 m2; B66¼ E2

2ð1 þmÞ;

D11¼ D22¼ E3

1 m2; D12¼ E3m

1 m2; D66¼ E3

2ð1 þmÞ;

ð11Þ

with

E1¼ EmþEc Em

k þ 1

h; E2¼ ðEc EmÞkh

2

2ðk þ 1Þðk þ 2Þ;

E3¼ Em

12þ ðEc EmÞ

1

k þ 3

1

k þ 2þ

1 4k þ 4

h3;

I1¼d1h

3

1

12 þ A1z

2; I2¼d2h

3 2

12 þ A2z

2:

ð12Þ

and

C1¼ E0A1z1

s1

; C2¼ E0A2z2

s2

;

z1¼h1þ h

2 ; z2¼

h2þ h

2 ;

where the coupling parameters C1, C2are negative for outside stiff-eners and positive for inside ones, s1and s2are the spacing of the longitudinal and transversal stiffeners, A1, A2are the cross-section areas of stiffeners, I1,I2 are the second moments of cross-section areas, z1, z2are the eccentricities of stiffeners with respect to the middle surface of shell, and the width and thickness of longitudinal and transversal stiffeners are denoted by d1, h1 and d2, h2, respectively

For later use, the strain-force resultant reverse relations are ob-tained from Eq.(9)

e0¼ A22N1 A12N2þ B

11v1þ B

12v2;

e0¼ A11N2 A12N1þ B21v1þ B22v2;

c0

12¼ A66þ 2B66v12;

ð13Þ

where

A11¼1

D A11þE0A1

s1

; A22¼1

D A22þE0A2

s2

;

A12¼A12

D ; A66¼ 1

A66

;

D¼ A11þE0A1

s1

A22þE0A2

s2

 A212;

B

11¼ A22ðB11þ C1Þ  A12B12;

B

22¼ A11ðB22þ C2Þ  A12B12;

B

12¼ A22B12 A12ðB22þ C2Þ;

B

21¼ A11B12 A12ðB11þ C1Þ; B

66¼B66

A66

:

ð14Þ

Substituting Eq.(13)into Eq.(10)yields

M1¼ B11N1þ B21N2 D11v1 D12v2;

M2¼ B12N1þ B22N2 D21v1 D22v2;

M12¼ B66N12 2D66v12;

ð15Þ

where

D

11¼ D11þE0I1

s1  ðB11þ C1ÞB11 B12B

21;

D

22¼ D22þE0I2

s2  B12B

12 ðB22þ C2ÞB22;

D

12¼ D12 ðB11þ C1ÞB12 B12B

22;

D21¼ D12 B12B11 ðB22þ C2ÞB21;

D

66¼ D66 B66B

66:

ð16Þ

Based on the classical shell theory and the Volmir’s assumption

[23] u  w and v w; q1@ 2 u

@t 2! 0 and q1@ 2v

@t 2! 0, the nonlinear motion equations of a shallow thin shell with damping force is written in the form

@N1

@x1þ@N12

@x2 ¼ 0;

@N12

@x1

þ@N2

@x2

¼ 0;

@2M1

@x2 þ 2@

2M12

@x1@x2

þ@

2M2

@x2 þ N1

@2w

@x2þ 2N12

@2w

@x1@x2

þ N2

@2w

@x2

þ k1N1þ k2N2þ q0¼q1@

2w

@t2 þ 2q1e@w

@t ;

ð17Þ

whereeis damping coefficient and

Z h=2

h=2

qðzÞdz þq0 A1

s1þA2

s2

¼ qmþqcqm

k þ 1

h þq0 A1

s1

þA2

s2

:

The first two of Eq.(16)are satisfied automatically by

introduc-ing a stress functionuas

N1¼@

2u

@x2; N2¼@

2u

@x2; N12¼  @

2u

@x1@x2

Substituting Eq.(13) into the compatibility Eqs (6) and (15)

into the third of Eq.(17), taking into account expressions(4) and

(18), yields

A

11

@4u

@x4þ A66 2A12

u

@x2@x2þ A22

@4u

@x4þ B21

@4w

@x4

þ B

11þ B

22 2B

66

@x2@x2þ B

12

@4w

@x4þ k1

@2w

@x2þ k2

@2w

@x2

¼ @

2w

@x1@x2

!2

@

2w

@x2

@2w

q1@

2

w

@t2 þ 2q1e@w

@t þ D

 11

@4w

@x4þ D12þ D21þ 4D66

w

@x2@x2

þ D22@

4w

@x4 B21@

4u

@x4 B 11þ B22 2B66 @4u

@x2@x2 B12@

4u

@x4

 k1

@2u

@x2 k2

@2u

@x2@

2u

@x2

@2w

@x2 þ 2 @

2u

@x1@x2

@2w

@x1@x2

@

2u

@x2

@2w

@x2 ¼ q0: ð20Þ

For an initial imperfection shell: The initial imperfection of the

shell considered here can be seen as a small deviation of the shell

middle surface from the perfect shape, also seen as an initial

deflection which is very small compared with the shell dimensions,

but may be compared with the shell wall thickness Let

w0= w0(x1, x2) denote a known small imperfection, proceeding

from the motion Eqs.(19) and (20)of a perfect FGM shallow shell

and following to the Volmir’s approach[23] for an imperfection

shell, we can formulate the system of motion equations for an

imperfect eccentrically stiffened functionally graded shallow shell

(Imperfect ES-FGM shallow shell) as

A11@

4u

@x4þ A 66 2A12 @4u

@x2@x2þ A22@

4u

@x4þ B 21

@4ðw  w0Þ

@x4

þ B11þ B22 2B66

ðw  w0Þ

@x2@x2 þ B12

@4ðw  w0Þ

@x4

þ k1

@2ðw  w0Þ

@x2 þ k2

@2ðw  w0Þ

@x2  @

2

w

@x1@x2

!2

@

2

w

@x2

@2w

@x2

2 4

3 5

2w0

@x1@x2

!2

@

2w0

@x2

@2w0

@x2

2

4

3

Uq1@

2w

@t2 þ 2q1e@w

@t þ D

 11

@4ðw  w0Þ

@x4 þ D

12þ D

21þ 4D 66

@4ðw  w0Þ

@x2@x2 þ D22

@4ðw  w0Þ

@x4  B21

@4u

@x4

 B11þ B22 2B66

u

@x2@x2 B12

@4u

@x4 k1

@2u

@x2  k2

@2u

@x2

@

2u

@x2

@2w

@x2þ 2 @

2u

@x1@x2

@2w

@x1@x2

@

2u

@x2

@2w

@x2  q0¼ 0; ð22Þ

where w is a total deflection of shell

The couple of nonlinear Eqs.(19) and (20)or Eqs.(21) and (22)

in terms of two dependent unknowns w anduare used to investi-gate the nonlinear vibration and dynamic stability of ES-FGM shells

4 Nonlinear dynamic analysis

An imperfect ES-FGM shallow thin shell considered in this pa-per is assumed to be simply supported and subjected to uniformly distributed pressure of intensity q0and axial compression of inten-sities r0and p0respectively at its cross-section Thus the boundary conditions are

w ¼ 0; M1¼ 0; N1¼ r0h; N12¼ 0; at x1¼ 0; a;

w ¼ 0; M2¼ 0; N2¼ p0h; N12¼ 0; at x2¼ 0; b:

ð23Þ

The mentioned conditions(23)can be satisfied identically if the buckling mode shape is chosen by

w ¼ f ðtÞ sinmpx1

a sin

npx2

where f(t) is time dependent total amplitude and m, n are numbers

of haft waves in x1and x2directions respectively

The initial imperfection w0is assumed to have the same form of the shell deflection w, i.e

w0¼ f0sinmpx1

a sin

npx2

where f0is the known initial amplitude

Substituting Eqs (24) and (25)into Eq (21) and solving the resulting equation for unknownuyield

u¼u1cos2mpx1

a þu2cos2npx2

b u3sinmpx1

a sin

npx2

b

 r0hx

2

2  p0hx

2

where

u1¼ n

2k2f2

32m2A 11

2f2

32n2k2A22;

u3¼

B

21m4þ B 11þ B22 2B66

m2n2k2þ B12n4k4a 2

p 2k1n2k2þ k2m2

A

11m4þ A 66 2A12

m2n2k2þ A22n4k4 :

ð27Þ Substituting the expressions(24)–(26)into Eq.(22)and apply-ing the Galerkin’s method to the obtained equation

Z b 0

Z a 0

Usinmpx1

a sin

npx2

b dx1dx2¼ 0

lead to

M€f þ 2Me_f þ D þB2

A

!

ðf  f0Þ þ8mnk

2

3p2 d1d2

B

Aðf  f0Þf

þ H f2 f2

þ K f2 f2

f a

2h

p2 r0m2þ p0n2

k2

f

þ4a

4h

mnp6d1d2ðk1r0þ k2p0Þ  q0

4a4

mnp6d1d2¼ 0; ð28Þ

where the coefficients are given by

M ¼a

4

p4q1;A ¼ A11m4þ A 66 2A12

m2n2

k2þ A22n4

k4;

B ¼ B

21m4þ B

11þ B

22 2B 66

m2n2

k2þ B

12n4

k4a

2

p2 k1n2

k2þ k2m2

;

D ¼ D

11m4þ D12þ D21þ 4D66

m2n2k2þ D22n4k4;

H ¼2mnk

2

3p2

B

21

A11þ

B 12

A22

d1d2 a

2

6p4mn

k2n2k2

A11 þ

k1m2

A22

!

d1d2; ð29Þ

K ¼ 1

16

m4

A

22

þn

4k4

A

11

!

;k¼a

b;d1¼ ð1Þ

m

 1; d2¼ ð1Þn 1:

The governing Eq.(28)is a basic equation for dynamic analysis

of imperfect ES-FGM doubly-curved shallow thin shells in general

Based on this equation, the nonlinear vibration of perfect and

imperfect FGM shallow shells can be investigated and the dynamic

buckling analysis of shells under various loadings can be

performed

Particularly for a spherical panel k1= k2= 1/R, for a cylindrical

panel k1= 0, k2= 1/R, and for a plate k1= k2= 0 are taken in Eqs

(28) and (29)

4.1 Nonlinear vibration

Consider an imperfect ES-FGM shallow shell under uniformly

lateral pressure q0= Q sinXt and pre-loaded compressions r0, p0,

Eq.(28)becomes

M€f þ 2Me_f þ D þB2

A

!

ðf  f0Þ þ8mnk

2

3p2

B

Ad1d2ðf  f0Þf

þ H f2 f2

þ þK f2 f2

f a

2h

p2r0m2þ p0n2k2

f

þ4a

4h

mnp6d1d2ðk1r0þ k2p0Þ ¼ 4a

4

p6mnd1d2Q sinXt: ð30Þ

By using this equation, the fundamental frequencies of natural

vibration of ES-FGM shell and FGM shell without stiffeners, and

frequency-amplitude relation of nonlinear vibration and nonlinear

response of ES-FGM shell are taken into consideration The

nonlin-ear dynamical responses of ES-FGM shells can be obtained by

solv-ing Eq.(30)by the fourth order Runge–Kutta iteration method with

the time step Dt and with initial conditions to be assumed as

f ð0Þ ¼ 0; _fð0Þ ¼ 0

If the vibration is free and linear, and without damping, Eq.(30)

reduces to

M€f þ D þB

2

A

!

The fundamental frequencies of natural vibration of imperfect

ES-FGM shallow shells can be determined by

xmn¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

M D þ

B2

A

! v

u

If the shell is perfect ES-FGM and the vibration is nonlinear

forced vibration without pre-loaded compressions r0= p0= 0, Eq

(30)can be rewritten a

€

f þ 2e_f þx2

mnf þ H1f2þ H2f3

¼ F sinðXtÞ; ð33Þ

in which

H1¼

8mnk 2

3p2 Bd1d2þ H

Mx2

mn

; H2¼ K

Mx2 mn

; F ¼4a

4d1d2Q

Mp6mn : ð34Þ

Seeking solution as f(t) =gsin (Xt) and applying procedure like Galerkin method to Eq.(33), the frequency–amplitude relation of nonlinear forced vibration is obtained

X24e

p X¼x

2

3p gþ

3H2

4 g2

F

wheregis the amplitude of nonlinear vibration

By introducing the non-dimension frequency parameter n ¼ X

x mn,

Eq.(35)becomes

n2 4e

pxmn

n¼ 1 þ8H1

3p gþ

3H2

4 g2

 F

gx2 mn

In the case of the nonlinear forced vibration without damping, Eq

(36)leads to

n2¼ 1 þ8H1

3p gþ

3H2

4 g2 F

gx2 mn

If F = 0 and without damping, the frequency–amplitude relation of the nonlinear free vibration, from Eq.(37), is as

n2¼ 1 þ8H1

3p gþ

3H2

4.2 Nonlinear dynamic buckling Investigate the nonlinear dynamic buckling analysis of imper-fect ES-FGM doubly-curved panels under lateral pressure varying

as linear function of time q0= ct (c is a loading speed) and pre-loaded compressions r0= const, p0= const, Eq.(28)becomes

M€f þ 2Me_f þ D þB2

A

!

ðf  f0Þ þ8mnk

2

3p2

B

Ad1d2ðf  f0Þf

þ H f2 f2

þ K f 2 f2

f a

2h

p2 r0m2þ p0n2

k2

f

þ 4a

4h

mnp6d1d2ðk1r0þ k2p0Þ ¼ 4a

4

By solving Eq.(39), the dynamic critical time tcrcan be obtained according to Budiansky–Roth criterion[25] This criterion is based

on that, for large value of loading speed, the amplitude-time 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 t = tcrthe stability loss occurs Here t = tcris called critical time and the load corresponding to this critical time is called dynamic critical buckling load qdcr= ctcr For static buckling analysis of ES-FGM shallow shell, the explicit expressions of the upper and lower buckling load were obtained in

[21]

5 Numerical results and discussions 5.1 Validation of the present formulation

To validate the present study, the fundamental frequency parameter of unstiffened FGM shallow shells is compared with other studies

Table 1shows the present results in comparison with those pre-sented by Matsunaga [15] based on the two-dimensional (2D) higher-order theory, Chorfi and Houmat [16] accorded to the first-order shear deformation theory and Alijani et al.[17] used Donnell’s nonlinear shallow shell theory In this comparison, the fundamental frequency parameter ~x¼xmnh ffiffiffiffiffiffiffiffiffiffiffiffi

qc=Ec

p

of the perfect un-stiffened FGM shallow shell (a/b = 1, h/a = 0.1) with simply sup-ported edges The material properties are Aluminum and Alumina, i.e Em= 70  109N/m2;qm= 2702 kg/m3and Ec= 380  109N/m2;

q = 3800 kg/m3respectively The Poisson’s ratio is chosen to be

0.3 As can be seen, a very good agreement is obtained in the

com-parison with the result of Ref.[17], but there are a little differences

with those of Refs [15,16] because they use above mentioned

other theories

5.2 Nonlinear vibration results

To illustrate the proposed approach of eccentrically stiffened

FGM shallow shells, the stiffened and un-stiffened FGM shallow

h = 0.01 m; The shells are simply supported at all its edges The

combination of materials consists of Aluminum Em= 70  109N/

m2,qm= 2702 kg/m3and Alumina Ec= 380  109N/m2qc= 3800

-kg/m3 The Poisson’s ratio is chosen to be 0.3 for simplicity

Mate-rial of stiffeners has elastic modulus Es1= Es1= 380  109N/m2,

qs1=qs2= 3800 kg/m3 The height of stiffeners is equal to 50 mm,

its width 2.5 mm, the spacing of stiffeners s1= s2= 0.1 m

The obtained results inTable 2show that effects of stiffeners on

the fundamental frequencies of natural vibration are considerable

Obviously the natural fundamental frequencies of un-stiffened and

stiffened FGM spherical panels observed to be dependent on the

constituent volume fractions, they decrease when increasing the

power index k, furthermore with greater value k the effect of

stiff-eners is observed to be stronger This is completely reasonable due

to the lower value of the elasticity modulus of the metal

constitu-ent The natural frequencies of stiffened spherical panels are

great-er than one of un-stiffened panels

Table 3shows high frequencies of natural vibration of spherical

panels with R1= R2= 5 m, k = 1 Clearly, all modes of stiffened

spherical panel are greater than ones of un-stiffened panel

Espe-cially, the difference is larger with higher modes

Table 4shows the fundamental frequencies of doubly curved

FGM shallow shells In this case, with k = 1, It seem that the natural

frequencies of un-stiffened panels depend to the value jk1+ k2j, i.e

they have the same value for all panels of same jk1+ k2j, the natural

frequencies of un-stiffened panels increases when this value

in-creases, when k1+ k2= 0 (R1= 5, R2= 5 or R1= 5, R2= 5) the

nat-ural frequencies of un-stiffened panels are equal to the natnat-ural frequencies of un-stiffened plates (seeTables 2 and 4), but as can

be seen, this phenomenon is not observed for stiffened panels For stiffened panels of the same value jk1+ k2j have different natu-ral frequencies Except in special cases when k1+ k2= 0, the natural frequencies of stiffened panels have the same value with stiffened respective plate

The frequency–amplitude curves of nonlinear vibration of FGM spherical panels without damping are presented inFig 2 This fig-ure shows that the frequency–amplitude curves of forced vibration are asymptotic with the frequency–amplitude curve of free vibra-tion and the extreme point of stiffened panel is greater than one

of un-stiffened panel In forced vibration, a value of n corresponds

to the maximum five distinguished values ofg

Fig 3shows the effect of excitation force Q on the frequency– amplitude curves of nonlinear vibration of spherical panels As can be seen, when the excitation force decreases, the curves of forced vibration are closer to the curve of free vibration

Figs 4 and 5show the effect of volume-fraction index and ra-dius of panels on the frequency–amplitude curve of ES-FGM spher-ical panel without damping Clearly, the extreme points of frequency–amplitude curve decrease when the volume-fraction in-dex or the radius decreases

Four cases of Gauss curvature of stiffened and un-stiffened pan-els are taken into consideration: k1k2> 0 and k1+ k2> 0 when

R1= 3 m, R2= 10 m, k1k2> 0 and k1+ k2< 0 when R1= 3 m,

R2= 10 m, k1k2< 0 and k1+ k2> 0 when R1= 3 m, R2= 10 m and k1k2< 0 and k1+ k2< 0 when R1= 3 m, R2= 10 m (Figs 6–9)

It seems that the frequency–amplitude curve depends on the value

k1+ k2 The minimal point of frequency–amplitude curve increases when this value decreases Especially, when k1+ k2< 0, the fre-quency–amplitude curve of free vibration does not exist extreme points (seeFigs 7 and 9)

Table 1

Comparison of ~xwith results reported by Matsunaga [15] , Chorfi and Houmat [16]

and Alijani et al [17]

b/R 2 a/R 1 k Present Ref [15] Ref [16] Ref [17]

FGM plate

FGM spherical panel

FGM cylindrical panel

FGM hyperbolic paraboloidal panel

Table 2 Fundamental frequencies of natural vibration (rad/s) of spherical panels.

Table 3 Frequencies of natural vibration (rad/s) of spherical panels with R 1 = R 2 = 5 m, k = 1.

Un-stiffened Stiffened

x2 (m = 1, n = 2) and (m = 2, n = 1) 2437.29 6743.12

x4 (m = 1, n = 3) and (m = 3, n = 1) 3931.47 14576.20

x5 (m = 2, n = 3) and (m = 3, n = 2) 4920.50 16063.76

Fig 10investigate the frequency–amplitude curve of nonlinear

vibration of stiffened plate and shallow shells with jR1j = jR2j, the

phenomenon is similar withFigs 6–9, however when k1+ k2= 0

with R1= 5, R2= 5 and R1= 5, R2= 5 the frequency–amplitude

curves coincide to the frequency–amplitude curve of plate

Consider the un-stiffened and stiffened perfect FGM spherical

panel without damping under the uniformly harmonic load q0

(t) = Q sin (Xt), nonlinear responses are obtained solving the Eq

(30)by fourth order Runge–Kutta method with the time stepDt

Nonlinear responses of un-stiffened FGM spherical panel with difference time steps are presented inFig 11 As can be observed, difference of nonlinear responses of time stepsDt = 2  104and

Dt = 104is very small Therefore, the next results are calculated with time stepDt = 104to ensure the accuracy of this method Nonlinear responses of stiffened and un-stiffened functionally graded spherical panel with k = 1, R = 5 are presented inFigs 12 and 13 Natural frequencies of un-stiffened and stiffened spherical panel are 1809.95 rad/s and 2560.43 rad/s, respectively (see

Table 2) The excitation frequencies are much smaller (Fig 12,

q0(t) = 105sin (100t)) and much greater (Fig 13, q0(t) = 105sin (104t)) than natural frequencies These results show that the stiffeners strongly decrease vibration amplitude of the shell when excitation frequencies are much smaller or much greater than natural frequencies

When the excitation frequencies are near to natural frequen-cies, the interesting phenomenon is observed like the harmonic beat phenomenon of a linear vibration (Figs 14 and 15) The exci-tation frequencies are 2510 rad/s and 2530 rad/s which are very near to natural frequencies 2348.57 rad/s of stiffened spherical pa-nel These results show that the amplitude of beats of stiffened panels increases rapidly when the excitation frequency approaches the natural frequencies The maximal amplitude of harmonic beat increases and the response time of beat decreases when the excita-tion force increases as shown inFig 15

The deflection–velocity relation has the closed curve form as in

Fig 16 Deflection f and velocity _f are equal to 0 at initial time and final time of beat and the contour of this relation corresponds to the period which has the greatest amplitude of beat

Table 4

Fundamental frequencies of natural vibration (rad/s) of doubly curved shallow shells.

jR 1 j jR 2 j k R 1 > 0, R 2 > 0 R 1 > 0, R 2 < 0 R 1 < 0, R 2 > 0 R 1 < 0, R 2 < 0

Unstiffened Stiffened Unstiffened Stiffened Unstiffened Stiffened Unstiffened Stiffened

Fig 2 The frequency–amplitude curve of nonlinear vibration of FGM spherical

panels (R = 5 m, k = 1, Q = 10 5

N/m 2

).

Fig 3 Effect of excitation force Q on the frequency–amplitude curve of stiffened

spherical panel (R = 5 m, k = 1).

Fig 4 Effect of index k on the frequency–amplitude curves of stiffened spherical panels (R = 5 m, Q = 10 5 N/m 2 ).

Figs 17 and 18show the effect of pre-loaded compressions and

of known initial amplitude on the nonlinear responses of ES–FGM

spherical panel In these investigations, pre-loaded compressions

and known initial amplitude slightly influence on the amplitude

of nonlinear vibration of panels

Effect of damping on nonlinear responses is presented inFigs 19

and 20with linear damping coefficiente= 0.3 The damping

influ-ences very small to the nonlinear response in the first vibration

periods (Fig 19) however it strongly decreases amplitude at the

next far periods (Fig 20)

5.3 Nonlinear dynamic buckling results

To evaluate the effectiveness of stiffener in the nonlinear dynamic buckling problem, we consider an imperfect ES-FGM cylindrical panel and spherical panel under lateral pressure and pre-loaded compression Materials of shells and stiffeners used in this section are the same in the previous section

The effect of stiffeners to the critical buckling of perfect FGM shallow shells under only lateral pressure is investigated for two cases of cylindrical and spherical panel under uniformly lateral

Fig 5 Effect of radius R on the frequency–amplitude curves of stiffened spherical

panels (k = 1, Q = 10 5 N/m 2 ).

Fig 6 The frequency–amplitude curve of nonlinear vibration of un-stiffened

shallow shells (k = 1, Q = 10 5

N/m 2

).

Fig 7 The frequency–amplitude curve of nonlinear vibration of un-stiffened

shallow shells (k = 1, Q = 10 5

N/m 2

).

Fig 8 The frequency–amplitude curve of nonlinear vibration of stiffened shallow shells (k = 1, Q = 10 5 N/m 2 ).

Fig 9 The frequency–amplitude curve of nonlinear vibration of stiffened shallow shells (k = 1, Q = 10 5

N/m 2

).

Fig 10 The frequency–amplitude curve of nonlinear vibration of stiffened plate and shallow shells (k = 1, Q = 10 5

N/m 2

).

pressure varying on time as q0= 105t (N/m2) (seeFigs 21 and 22).

The critical buckling load corresponds to the buckling mode shape

m = 1, n = 1 in all cases These figures also show that there is no

def-inite point of instability as in static analysis Rather, there is a

re-gion of instability where the slope of t vs f/h curve increases

rapidly According to the Budiansky–Roth criterion, the critical

time tcrcan be taken as an intermediate value of this region

There-fore one can choose the inflexion point of curve i.e.d 2 f

dt2

t¼t cr

¼ 0 as in Ref.[26] As can be seen, the dynamic buckling loads of stiffened

panels are greater than one of un-stiffened panels

Effect of volume-fraction index k and loading speed c on critical dynamic buckling of stiffened cylindrical panels and spherical pa-nel are showed inTable 5 Clearly, the critical dynamic buckling

of panel decrease when the volume-fraction index increases or the loading speed decreases As can be also observed inTable 5, the critical dynamic buckling load is greater than the static critical load

Table 6shows the effect of thickness h on the critical dynamic buckling load of cylindrical and spherical panels The critical dy-namic buckling loads of un-stiffened and stiffened panels increase when the thickness of panels increases In addition,Table 6also

Fig 11 Nonlinear responses of un-stiffened FGM spherical panel with difference

time steps (R = 5 m, k = 1, q(t) = 10 5

sin 100t).

Fig 12 Nonlinear responses of FGM spherical panel (R = 5 m, k = 1,

q(t) = 10 5 sin 100t).

Fig 13 Nonlinear responses of FGM spherical panel (R = 5 m, k = 1,

q(t) = 10 5

sin 10 4

t).

Fig 14 Harmonic beat phenomenon of stiffened spherical panel (R = 5 m, k = 1, q(t) = 10 5

sinXt).

Fig 15 Harmonic beat phenomenon of stiffened spherical panel (R = 5 m, k = 1, q(t) = Q sin 2530t).

Fig 16 Deflection–velocity relation of stiffened spherical panel (R = 5 m, k = 1, q(t) = 2  10 5

sinXt).

shows that the effect of stiffeners decreases when the thickness

increases

Table 7gives the results on the effect of known initial amplitude

f0on the nonlinear buckling of stiffened cylindrical and stiffened

spherical panels Clearly, the known initial amplitude slightly

influences on the critical dynamic buckling loads of these

struc-tures when they only subjected to lateral pressure

Fig 23shows the dynamic response of stiffened spherical

pan-els under combination of lateral pressure varying on time q0= 105t

(N/m2) and pre-loaded compressions r0= const, p0= const As can

be observed, the pre-loaded compressions strongly influence on

Fig 17 Effect of pre-loaded compressions on nonlinear responses of stiffened

spherical panel (R = 5 m, k = 1, q(t) = 10 5

sin 10 5

t).

Fig 18 Effect of known initial amplitude on nonlinear responses of stiffened

spherical panel (R = 5 m, k = 1, q(t) = 10 5

sin 100t).

Fig 19 Effect of damping on nonlinear responses of stiffened spherical panel

(R = 5 m, k = 1, q(t) = 5  10 3 sin 2530t).

Fig 20 Effect of damping on nonlinear responses of stiffened spherical panel (R = 5 m, k = 1, q(t) = 5  10 3

sin 2530t).

Fig 21 Effect of stiffeners on dynamic buckling of cylindrical panel under lateral pressure (R = 6 m, f 0 = 0, d 1 = d 2 = 3  10 3 m, h 1 = h 2 = 3  10 2 m, s 1 = s 2 = 0.25 m).

Fig 22 Effect of stiffeners on dynamic buckling spherical panel under lateral pressure (R = 10 m, f 0 = 0, d 1 = d 2 = 3  10 3

m, h 1 = h 2 = 4  10 2

m,

s 1 = s 2 = 0.25 m).

Table 5 Effect of volume fraction index k, loading speed c on the critical dynamic buckling load of ES-FGM shallow panels (10 5 N/m 2 ).

Cylindrical panels

Dynamic q 0 = 10 5

Spherical panels

Dynamic q 0 = 10 5

Dynamic q 0 = 10 6 t 17.8372 12.2372 6.6619 5.4012

R = 5 m, d = d = 3  10 3 m, h = h = 35  10 3 m, s = s = 0.25 m.