DSpace at VNU: Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects

The static buckling behavior of shallow spherical caps under an uniform pressure loads was analyzed by Tillman[3].. Results on the dynamic buckling of clamped shallow spherical shells su

Nonlinear static and dynamic buckling analysis of functionally graded

shallow spherical shells including temperature effects

Vietnam National University, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:

Available online 24 April 2012

Keywords:

Functionally graded materials

Static and dynamic buckling

Shallow spherical shells

a b s t r a c t

This paper presents an analytical approach to investigate the nonlinear static and dynamic unsymmetri-cal responses of functionally graded shallow spheriunsymmetri-cal shells under external pressure incorporating the effects of temperature Governing equations for thin FGM spherical shells are derived by using the classical shell theory taking into account von Karman–Donnell geometrical nonlinearity Approximate solutions are assumed and Galerkin procedure is applied to determine explicit expressions of static crit-ical buckling loads of the shells For the dynamcrit-ical response, motion equations are numercrit-ically solved by using Runge–Kutta method and the criterion suggested by Budiansky–Roth A detailed analysis is carried out to show the effects of material and geometrical parameters, boundary conditions and temperature on the stability and dynamical characteristics of FGM shallow spherical shells

Ó 2012 Elsevier Ltd All rights reserved

1 Introduction

Structures in the form of spherical shells are used widely in many

engineering applications Most of these shells are subjected to static

and impulsive loads which is cause of instability and strength

reduc-tion of the structures As a result, the investigareduc-tion on the nonlinear

static and dynamical buckling of spherical shells is necessary and

has attracted attention of many researchers Budiansky and Roth

[1]studied axisymmetrical dynamic buckling of clamped shallow

isotropic spherical shells Their well-known results have received

considerable attention in the literature Huang[2]considered the

unsymmetrical buckling of thin shallow spherical shells under

external pressure He pointed out that unsymmetrical deformation

may be the source of discrepancy in critical pressures between

axi-symmetrical buckling theory and experiment The static buckling

behavior of shallow spherical caps under an uniform pressure loads

was analyzed by Tillman[3] Results on the dynamic buckling of

clamped shallow spherical shells subjected to axisymmetric and

nearly axisymmetric step-pressure loads using a digital computer

program were given by Ball and Burt[4] Kao and Perrone[5]

re-ported the dynamic buckling of isotropic axisymmetrical spherical

caps with initial imperfection Two types of loading are considered,

in this paper, namely, step loading with infinite duration and right

triangular pulse Wunderlich and Albertin[6]also studied on the

static buckling behavior of isotropic imperfect spherical shells

New design rules in their work for these shells were developed,

which take into account relevant details like boundary conditions,

material properties and imperfections The nonlinear static and dy-namic response of spherical shells has been analyzed by Nath and Alwar[7]using Chebyshev series expansion Based on an assumed two-term mode shape for the lateral displacement, Ganapathi and Varadan[8]investigated the problem of dynamic buckling of ortho-tropic shallow spherical shells under instantaneously applied uni-form step-pressure load of infinite duration The same authors analyzed the dynamical buckling of laminated anisotropic spherical caps using the finite element method[9] Static and dynamic snap-through buckling of orthotropic spherical caps based on the classical thin shell theory and Reissener’s shallow shell assumptions have been considered by Chao and Lin[10]using finite difference method There were several investigations on the buckling of spherical shells under mechanical or thermal loading taking into account initial imperfection such as studies by Eslami et al.[11]and Shahsiah and Eslami[12]

Functionally graded materials (FGMs) which are microscopically composites and composed of ceramic and metal constituents or combination of metals have received much interest in recent years Due to essential characteristics such as high stiffness, excellent tem-perature resistance capacity, functionally graded materials find wide applications in many industries, especially in temperature shielding structures and nuclear plants Shahsiah et al [13]

presented an analytical approach to study the instability of FGM shallow spherical shells under three types of thermal loading including uniform temperature rise, linear radial temperature, and nonlinear radial temperature Prakash et al.[14]gave results on the nonlinear axisymmetric dynamic buckling behavior of clamped FGM spherical caps Also, the dynamic stability characteristics of FGM shallow spherical shells have been considered by Ganapathi 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd All rights reserved.

⇑Corresponding author.

E-mail address: lekhahoa@gmail.com (L.K Hoa).

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

[15]using the finite element method In his study, the geometric

nonlinearity is assumed only in the meridional direction in strain–

displacement relations Bich[16] studied the nonlinear buckling

of FGM shallow spherical shells using an analytical approach and

the geometrical nonlinearity was considered in all

strain–displace-ment relations By using Galerkin procedure and Runge–Kutta

method, Bich and Hoa [17] analyzed the nonlinear vibration of

FGM shallow spherical shells subjected to harmonic uniform

exter-nal pressures Recently, Bich and Tung[18]reported an analytical

investigation on the nonlinear axisymmetrical response of FGM

shallow spherical shells under uniform external pressure taking

the effects of temperature conditions into consideration Shahsiah

et al.[19]used an analytical approach to investigate thermal linear

instability of FGM deep spherical shells under three types of thermal

loads using the first order shell theory basing Sander nonlinear

kine-matic relations To best of authors’ knowledge, there is no analytical

investigation on the nonlinear dynamic stability of FGM shallow

spherical shells

In the present paper, the nonlinear static and dynamical

buck-ling behavior of clamped FGM shallow spherical shells under

uni-form external pressure and thermal loads are considered by using

an analytical approach Governing equations for thin shallow

spherical shells are derived by using the classical shell theory

tak-ing into account von Karman–Donnell nonlinear terms Material

properties are assumed to be temperature independent and

graded in the thickness direction according to a simple power

law function Approximate one-term solutions of deflection and

stress function that satisfy the boundary conditions are assumed

and Galerkin procedure is used to obtain explicit expressions of

static critical buckling loads For dynamical analysis, motion

equa-tion is solved numerically by applying Runge–Kutta method and

the criterion suggested by Budiansky–Roth The effects of material

and geometrical properties, temperature and boundary conditions

on the response of FGM spherical shells are analyzed and

discussed

2 Theoretical formulations

2.1 Functionally graded shallow spherical shells

Consider a clamped FGM shallow spherical shell of thickness h,

base radius r0, curvature radius R, rise H as shown inFig 1 It is

defined in coordinate system (u, h, z), whereu and h are in the

meridional and circumferential directions of the shell, respectively,

and z is perpendicular to the middle surface positive inward

As-sume that the shell is made from a mixture of ceramic and metal

constituents and the effective material properties vary

continu-ously along the thickness by the power law distribution

EðzÞ ¼ Emþ ðEc EmÞ 2z þ h

2h

; qðzÞ ¼qmþ ðqcqmÞ 2z þ h

2h

;

aðzÞ ¼amþ ðacamÞ 2z þ h

2h

; KðzÞ ¼ Kmþ ðKc KmÞ 2z þ h

2h

;

mðzÞ ¼m¼ const;

ð1Þ where k P 0 is volume fraction index and E,m,q,a, K are Young’s modulus, Poisson’s ratio, mass density, coefficient of thermal expansion, coefficient of thermal conduction, respectively, and sub-scripts m and c stand for the metal and ceramic constituents, respectively

2.2 Governing equations

It is convenient to introduce an additional variable r defined by the relation r = Rsinu, where r is the radius of the parallel circle with the base of shell If the rise H of the shell is much smaller than the base radius r0we can take cosu 1 and Rdu= dr, such that points of the middle surface may be referred to coordinates r and h The strain components on the middle surface of shell based upon the von Karman assumption are of the form

e0

r ¼ u;r w=R þ w2

h¼ ðv;hþ uÞ=r  w=R þ w2

c0

vr¼ w;rr; vh¼ w;hh=ðr2Þ þ w;r=r; vrh¼ w;rh=r  w;h=ðr2Þ; ð3Þ where u,vand w are the displacements of the middle surface points along meridional, circumferential and radial directions, respec-tively, and vr, vh, vrh are the change of curvatures and twist, respectively

Using Eqs.(2) and (3), the geometrical compatibility equation is written as

1

r2e0

re0

r2ðr2e0

r2 rc0 rh

Rr2w þv2

wherer¼@ 2

@r 2þ1 r

@

rþ1

r 2

@ 2

@ 2is a Laplace’s operator

The strains across the shell thickness at a distance z from the mid-plane are given by

er¼e0

The stress–strain relationships including temperature effect for

an FGM spherical shell are defined by the Hooke law

ðrr;rhÞ ¼ EðzÞ

rrh¼ EðzÞ 2ð1 þmÞcrh;

ð6Þ

whereDT is temperature change from stress free initial state The force and moment resultants of an FGM shallow spherical are expressed in terms of the stress components through the thick-ness as

fðNr;Nh;NrhÞ; ðMr;Mh;MrhÞg ¼

Z h=2

h=2

Introduction of Eqs (1), (5) and (6) into Eq (7) gives the constitutive relations

Nr¼ E1

1 m2 e0

h

 E2

1 m2ðvrþmvhÞ  /m

1 m;

Nh¼ E1

1 m2 e0

r

 E2

1 m2ðvhþmvrÞ  /m

1 m;

Nrh¼ E1 2ð1 þmÞc0

1 þm vrh;

ð8Þ

R

r0

H

,v

θ

r

,

z w

,u

ϕ

Mr¼ E2

1 m2 e0

h

 E3

1 m2ðvrþmvhÞ  /b

1 m;

Mh¼ E2

1 m2 e0

r

 E3

1 m2ðvhþmvrÞ  /b

1 m;

Mrh¼ E2

2ð1 þmÞc0

1 þm vrh;

ð9Þ

where

E1¼ Emh þEcmh

k þ 1; E2¼ Ecmh

k þ 2

1 2k þ 2

;

E3¼Emh

3

12 þ Ecmh

k þ 3

1

k þ 2þ

1 4k þ 4

;

ð/m;/bÞ ¼

Z h=2

h=2

Emþ Ecm 2z þ h

2h

amþacm

2z þ h 2h

DTð1;zÞdz;

Ecm¼ Ec Em; acm¼acam:

ð10Þ The nonlinear equations of motion of perfect shallow spherical

shell according to the classical shell theory are[20]

1

rðrNrÞ;rþ

1

rNrh;h

1

rNh¼q1u;tt;

1

rðrNrhÞ;rþ

1

rNh;hþ

1

rNrh¼q1v;tt; 1

r ðrMrÞ;rrþ 2 Mrh;rhþ1

rMrh;h

þ1

rMh;hh Mh;r

þ1

RðNrþ NhÞ

þ1

rðrNrw;rþ Nrhw;hÞ;rþ

1

r Nrhw;rþ

1

rNhw;h

;h

þ q ¼q1w;tt; ð11Þ where q is an uniform external pressure acting on the shell outer

surface positive inward, and

q1¼qmh þqcmh

By taking the inertia forcesq1u,tt?0 andq1v,tt?0 into

consid-eration because of u  w,v w[21], two first of Eq.(11)are

sat-isfied by introducing the stress function f

Nr¼1

rf;rþ

1

r2f;hh; Nh¼ f;rr; Nrh¼1

r2f;h1

and substituting relations(2), (3), (9) and (13)into third of Eq.(11)

gives

q1w;ttþ Dr4w 1

Rr2f  1

rf;rþ

1

r2f;hh

w;rr 1

rw;rþ

1

r2w;hh

f;rr

þ 2 1

rf;rh

1

r2f;h

rw;rh

1

r2w;h

where D ¼E1 E 3 E 2

ð1 m 2 ÞE 1

Eq.(14)includes two unknown functions w and f and to find a

second equation relating two these functions the geometrical

com-patibility Eq.(4)is used For this aim, from Eq.(8)strain

compo-nents can be expressed through force resultants as

e0

h

¼1

E1ðNr;NhÞ mðNh;NrÞ þ E2ðvr;vhÞ þ /mð1; 1Þ

;

c0

E1ð1 þmÞNrhþ E2vrh :

ð15Þ

Substituting these equations into Eq.(4), taking into account

relations(3) and (13)leads to

1

E1r4f þ1

Rr2w  1

rw;rh

1

r2w;h

þ w;rr 1

r2w;hhþ1

rw;r

¼ 0: ð16Þ Eqs.(14) and (16)are governing equations used to investigate

the nonlinear static and dynamic buckling of FGM shallow

spheri-cal shells

3 Boundary conditions and solution of problem The FGM shallow spherical shell is assumed to be clamped at its base edge and subjected to external pressure uniformly distributed

on the outer surface of shell Depending on the in-plane behavior at the edge, two cases of boundary conditions will be considered Case (i) The base edge of shell is clamped and freely movable (FM) in the meridional direction The associated boundary con-ditions are

w ¼ w;r¼ 0; Nr¼ 0; Nrh¼ 0 at r ¼ r0; ð17Þ Case (ii) The base edge of shell is clamped and immovable (IM) For this case, the boundary conditions are

u ¼ 0; w ¼ w;r¼ 0; Nr¼ N0; Nrh¼ 0 at r ¼ r0; ð18Þ where N0is fictitious compressive edge load at immovable edge

The mentioned boundary conditions can be satisfied, when the deflection w is represented by a single term of a Fourier series This approximate solution is acceptable in the vicinity of the buckling load[22,23]

w ¼ W16r

where W = W(t) is a time dependent total amplitude of deflection of shell

Regularly, the stress function f should be determined by the substitution of deflection function w into compatibility equation

(16)and solving the resulting equation However, such a procedure

is very complicated in mathematical treatment because obtained equation is a variable coefficient partial differential equation Accordingly, integration to obtain exact stress function f(r, h) is ex-tremely complex Therefore, the stress function f satisfying bound-ary conditions(18)is chosen in the same form of deflection w as mentioned in Refs.[24,25]

f ¼ F16r

r4 sin nh þ N0r

2

Substituting Eqs (19) and (20) into Eqs (14) and (16) and applying Galerkin procedure for the resulting equations yield

7ð38  20n2þ 12n4Þ

Wr20

R ð3 þ 2n

2

q1W;ttþ7Dð38  20n

2þ 12n4Þ

2Rr2 F 32nWF

pr4

þ N0 3 þ 2n

2 2r2 W  7

4pRn

¼ 7q

Eliminating F from Eqs.(21) and(22)leads to

q1W;ttþ 7Dð38  20n

2þ 12n4Þ

7R2ð38  20n2þ 12n4Þ

W

 96E1ð3 þ 2n

7pRr2

0ð38  20n2þ 12n4ÞW

2

2

7p2r4ð38  20n2þ 12n4ÞW

3

þ N0 3 þ 2n

2 2r2 W  7

4pRn

¼ 7q

Based on this equation, the mechanical and thermal stability analysis of shells are considered below

4 Mechanical stability analysis

Consider an FGM shallow spherical shell being clamped and

freely movable at the edge r = r0(case (i)) The outer surface of shell

is subjected to uniform external pressure q and without the effects

of temperature In this case N0= 0 and Eq.(23)reduces to

q1W;ttþ 7Dð38  20n

2þ 12n4Þ

7R2ð38  20n2þ 12n4Þ

W

 96E1ð3 þ 2n

7pRr20ð38  20n2þ 12n4ÞW

2

2

7p2r4ð38  20n2þ 12n4ÞW

3

¼ 7q

By putting

E

1¼ E1=h; E

2¼ E2=h2; E

3¼ E3=h3; W

¼ W=h;

n¼ r0=R; D¼ D=ðE1h2Þ; q¼q1=h: ð25Þ

Eq.(24)is rewritten as

qh2W;ttþ E

 1 ð38  20n2þ 12n4Þ

7D ð38  20n2þ 12n4Þ2 4n4

h R

 4

"

(

þð3 þ 2n

7

h

R

 2#

W

96ð3 þ 2n

7pn2

h R

 3

W2

þ2048n

2

7p2n4

h

R

 4

W3

)

¼ 7q

4.1 Static buckling and postbuckling analysis

Omitting the term of inertia force in Eq.(26)yields

q ¼ 8pnE

1

7ð38  20n2þ 12n4Þ

7D ð38  20n2þ 12n4Þ2 4n4

h R

 4

"

(

þð3 þ 2n

7

h R

 2#

W96ð3 þ 2n

7pn2

h R

 3

W2

þ2048n

2

7p2n4

h

R

 4

W3

)

Eq.(27) may be used to find static critical buckling load and

trace postbuckling load–deflection curves of FGM spherical shells

It is evident that q(W⁄

) curves originate from the coordinate origin

Eq (27) indicates that there is no bifurcation-type buckling for

pressure loaded spherical shells and extremum-type buckling only

occurs under definite conditions

The extremum buckling load of the shell can be found from Eq

(27)using the condition dq/dW⁄= 0 which give

qupper¼ p2Eð3 þ 2n2Þ3

32ð38  20n2þ 12n4Þ D

 38  20n2þ 12n4

3 þ 2n2

h R

 3

X1

n2

"

þ2n

2

49

h

R

 

2X1 3X2þX3

;

qlower¼ p2Eð3 þ 2n2Þ3

32ð38  20n2þ 12n4Þ D

 38  20n2þ 12n4

3 þ 2n2

h R

 3

X2

n2

"

þ2n

2

49

h

R

 

2X2 3X2þX3

;

ð28Þ

where

X1¼ 1 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

3

49

6 D

 38  20n2þ 12n4

3 þ 2n2

n4

h R

 2 s

;

X2¼ 1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

3

49

6 D

 38  20n2þ 12n4

3 þ 2n2

n4

h R

 2 s

:

ð29Þ

Providing is 1

3

49

6 D

 38  20n2þ 12n4

3 þ 2n2

1

n4

h R

 2

Note that quantities qupperand qlowerin Eq.(28)depend on the buckling mode n and the minimum values of buckling loads i.e critical upper and lower buckling loads may be obtained by mini-mization of these loads with respect to n

4.2 Dynamic buckling analysis

An FGM shallow spherical shells subjected to external pressure loads varying as linear functions of time, q = st(s – a loading speed),

is considered The aim of the problem is to determine the critical dynamic buckling loads In this case of load, Eq.(26)is rewritten as

qh2W

 1 ð38  20n2þ 12n4Þ

7Dð38  20n2þ 12n4Þ2

4n4

h R

 4

"

(

þð3 þ 2n

7

h R

 2#

W96ð3 þ 2n2Þn

7pn2

h R

 3

W2þ2048n

2

7p2n4

h R

 4

W3 )

7st

In the present study, initial conditions are assumed as

W

The well-known criterion suggested by Budiansky and Roth[1]

is employed herein According to this criterion, for large values of loading speed, the average deflection–time curve (W⁄

 t) of ob-tained displacement response increases sharply depending on time and this curve obtains a maximum by passing from the slope point, and at the corresponding time t = tcrthe stability loss occurs The value t = tcris called critical time and the load corresponding to this critical time is called dynamic critical buckling load To obtain dis-placement responses, Eq (31) in conjunction with initial condi-tions(32)will be solved by using the Runge–Kutta method

5 Thermomechanical stability analysis

A clamped FGM shallow spherical shell with immovable edge (case (ii)) subjected simultaneously to uniform external pressure

q and thermal load is considered The condition expressing the immovability of the boundary edge, i.e u = 0 at r = r0, is fulfilled

on the average sense as

Z p

0

Z r 0 0

@u

From Eqs.(2)and(15)one can obtain the expression of @u/@r and then substituting the result into Eq.(33)gives

3pr2ð1 mÞ

16E1W

15pRnð1 mÞþ

128E1W2 105r2ð1 mÞ

/m ð1 mÞ: ð34Þ Introduction F from Eq.(21)into Eq.(34)leads to

N0¼  16E1ð266  170n

2þ 64n4Þ

105pRnð1 mÞð38  20n2þ 12n4ÞW

þ128E1p2ð38  20n2þ 12n4Þ  40n2

105p2r2ð1 mÞð38  20n2þ 12n4Þ W

2

 /m ð1 mÞ: ð35Þ

Substituting this relation N0into Eq.(23)yields

qh2W;ttþ E

 1 ð38  20n2þ 12n4Þ

7D ð38  20n2þ 12n4Þ2 4n4

h R

 4

"

(

þ ð3 þ 2n

28ð266  170n2þ 64n4Þ

105p2n2ð1 mÞ

! h R

 2#

W

 96ð3 þ 2n

7pn2 þ8ð3 þ 2n

2Þð266  170n2þ 64n4Þ

105pnð1 mÞn2

"

þ224p2ð38  20n2þ 12n4Þ  40n2

105p3nð1 mÞn2

# h R

 3

W2

þ 2048n

2

7p2n4 þ64ð3 þ 2n

2Þp2ð38  20n2þ 12n4Þ  40n2

105p2n4ð1 mÞ

h R

 4

W3 )

ð1 mÞh

ð3 þ 2n2Þ

2n2

h R

 2

W

 7

4pn

h R

 

¼ 7q

Eq.(36)is employed to investigate static and dynamic

unsym-metric responses of FGM shallow spherical shells under combined

mechanical and thermal loads

5.1 Static stability analysis

Environment temperature is uniformly raised from initial value

Ti, at which the shell is thermal stress free, to final one Tfand

tem-perature changeDT = Tf Tiis independent to thickness variable

The thermal parameter /mcan be expressed in terms ofDT due

to Eq.(10)as

where

Um0¼ EmamþEmacmþ Ecmam

Ecmacm

Substituting /mfrom Eq.(37)into Eq.(36)and neglecting the

inertia force, i.e.qh2W

;tt¼ 0, yields

q ¼ 8pnE

7ð38  20n2þ 12n4Þ

7D

ð38  20n2þ 12n4Þ2 4n4

h R

 4

"

(

þ ð3 þ 2n

2Þ2

28ð266  170n2þ 64n4Þ

105p2n2ð1 mÞ

! h R

 2#

W

 96ð3 þ 2n

2Þn

7pn2 þ8ð3 þ 2n

2Þð266  170n2þ 64n4Þ

105pnð1 mÞn2

"

þ224p2ð38  20n2þ 12n4Þ  40n2

105p3nð1 mÞn2

# h R

 3

W2

þ 2048n

2

7p2n4 þ64ð3 þ 2n

2Þp2ð38  20n2þ 12n4Þ  40n2

105p2n4ð1 mÞ

h R

 4

W3

)

8pnUm0

7ð1 mÞ

ð3 þ 2n2Þ

2n2

h R

 2

W 7

4pn

h R

 

Eq.(39)is the explicit expression of external pressure-average

deflection curve incorporating the effects of temperature These

expressions can be used to consider the nonlinear unsymmetric

re-sponse of immovable clamped FGM spherical shell subjected to

external pressure and exposed to temperature conditions The

sta-tic crista-tical buckling loads in this case can be obtained from Eq.(39)

by using condition dq/dW⁄= 0

Inversely, the temperature differenceDT may be obtained in

terms of q, W⁄as well as material and geometric properties due

to these expressions

5.2 Dynamic stability analysis Similarly, suppose external pressure depending on time with the law q = st, the motion Eq.(36) in conjunction with Eq (37)

becomes

qh2W

 1 ð38  20n2þ 12n4Þ

7Dð38  20n2þ 12n4Þ2

4n4

h R

 4

"

(

þ ð3 þ 2n

28ð266  170n2þ 64n4Þ

105p2n2ð1 mÞ

! h R

 2#

W

 96ð3 þ 2n

7pn2 þ8ð3 þ 2n

2Þð266  170n2þ 64n4Þ

105pnð1 mÞn2

"

þ224p2ð38  20n2þ 12n4Þ  40n2

105p3nð1 mÞn2

# h R

 3

W2

þ 2048n

2

7p2n4 þ64ð3 þ 2n

2Þp2ð38  20n2þ 12n4Þ  40n2

105p2n4ð1 mÞ

h R

 4

W3 )

 Um0 ð1 mÞ

ð3 þ 2n2Þ 2n2

h R

 2

W

 7

4pn

h R

 

DT 7st

8pn¼ 0: ð40Þ

Eq (40) is the governing equation to investigate dynamic behavior of FGM shallow spherical shells under uniform external pressure including temperature effects

The analytical solution of Eq.(40)is very complicated, so this equation may be solved approximately by applying Runge–Kutta method and the Budiansky–Roth criterion to obtain critical dy-namic buckling loads

6 Numerical results and discussion

In the following discussions, the FGM spherical shell is made of silicon nitride (Si3N4) and steel (SUS 304) The Young’s modulus, mass densities and the coefficients of thermal expansion for Silicon nitride are Ec= 348.43 GPa, qc= 2370 kg/m3, ac= 5.8723  106

1/°C and for steel are Em= 201.04 GPa, qm= 8166 kg/m3, am= 12.33  1061/°C, respectively The Poisson’s ratio is supposed to

bem= 0.3 for both constituent materials

6.1 Validation of the proposed formulation

To validate the proposed formulation in static and dynamic sta-bility analysis of FGM shallow spherical shell, the present results are compared with results obtained by Ganapathi[15] The nondi-mensional dynamic pressure Pcrand geometrical parameter k are defined

Pcr¼1

8½3ð1 m2Þ1=2ðh=HÞ2 qr

4 Ech4;

k¼ 2½3ð1 m2Þ1=4ðH=hÞ1=2: where H ¼ R 1  ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1  n2

is the central shell rise and the length of response calculation time is introduced

s¼ ðh=r0Þ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Eef=½12ð1 m2Þqefh2

q

t; where Eef¼1Rh=2

h=2EðzÞdz as

in Ref[15] Using Eq.(36)with /m= 0, the dynamic buckling study

is conducted for step loading of infinite duration

Nonlinear dynamic response history with time for the FGM spherical shell parameter k = 6, r0/h = 400 and k = 1 considering dif-ferent externally applied pressure loads is obtained and illustrated

inFig 2, that is similar with the result in the mentioned paper As can be seen, there is a sudden jump in the value of the average deflection when the external pressure reaches the critical value

Pcr= 0.567.This result is in good agreement with the one of Ref.

[15]Pcr= 0.6063 obtained by using the finite element method

6.2 Results for movable clamped FGM shallow spherical shell

To illustrate the proposed formulation a FGM shallow spherical

shell of geometric ratios R/h = 1000, n = r0/R = 0.2 and volume

frac-tion index k = 1 under uniform external pressure is considered

After calculation of the buckling load according to Eq.(28)with

various shape modes n, it can see that the smallest buckling load,

i.e the static critical buckling load, corresponds to the shape mode

n = 1 and receives the value qcr= 2.2167e+005 Pa

The effect of material and geometric parameters on the

nonlin-ear unsymmetrical static and dynamic response of the FGM

shal-low spherical shells with movable clamped edge under uniform

external pressure are considered inFigs 3–6

Fig 3shows the effects of volume fraction index k(=0, 1 and 5)

on the nonlinear unsymmetrical static response of FGM spherical

shells As can be seen, the load- average deflection curves become

lower when k increases The increase in the extremum-type

buck-ling load and load carrying capacity of the shell when k reduces is

presented by a bigger difference between upper and lower

buck-ling loads WhereasFig 4demonstrates these effects on the

non-linear dynamic response of FGM spherical shells

It is observed that a sudden jump in the value of the average deflection occurs earlier when k increases, i.e the corresponding dynamic buckling load is smaller This is expected because the

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Non dimensional time (τ)

Applied pressure

load

τ

P

r0

H

1: P=0.565

2: P=0.566

3: Pcr=0.567

1 2 3

Fig 2 Dynamic response W ⁄

–swith k = 6, k = 1, r 0 /h = 400.

-3

-2

-1

0

1

2

3

x 105

W*

1: k=0

2: k=1

3: k=5

1

2

3

R/h=1000

ξ=0.2

n=1

-5 0 5 10 15 20 25 30

time (s)

2: k=1 3: k=5

R/h=1000

ξ= 0.2

n=1

1

2

3

q

t Applied presume load

Fig 4 Effect of index k on dynamic response (dynamic, FM).

-2 -1 0 1 2 3

5

W*

1: R/h=1000 2: R/h=1200 3: R/h=1500

1

2

3

Fig 5 Effect of R/h on load–average deflection curve (static, FM).

0 5 10 15 20 25 30 35 40 45

time (s)

q

t Applied pressure load

1

2

3

1: R/h=1000 2: R/h=1200 3: R/h=1500

Fig 6 Effect of R/h on dynamic response (dynamic, FM).

higher value k corresponds to a metal-richer shell which usually

has less stiffness than a ceramic-richer shell

Figs 5 and 6consider the effects of curvature

radius-to-thick-ness ratio R/h(=1000, 1200 and 1500) on the nonlinear static and

dynamic characteristics respectively of the externally pressurized

FGM spherical shells

As can be observed, the load bearing capability of the spherical

shell is considerably enhanced as R/h ratio decreases Furthermore,

the increase of R/h ratio is accompanied by a drop of nonlinear

load–deflection curves and more severe snap-through static

re-sponse and early occurrence of a jump of dynamic rere-sponse

Tables 1 and 2demonstrate the comparison between nonlinear

critical static and dynamic buckling loads with the change of

power index k and R/h ratio, respectively Clearly, the dynamic

crit-ical load is greater than the static critcrit-ical load

6.3 Results for immovable clamped FGM spherical shell

Similarly the effects of volume fraction index k and curvature

radius-to-thickness ratio R/h on the nonlinear static and dynamic

response of the FGM shallow spherical shell with immovable

clamped edge subjected to external pressure are illustrated inFigs

7–10, respectively

Effects of power index k and R/h ratio on static and dynamic

critical loads in IM case are given inTables 3 and 4

As can be seen, the trend of nonlinear static and dynamic

sponses of FGM spherical shells in IM case are very similar with

re-sponses in FM case The dynamic critical load is greater than the static critical load, but the difference of these loads in IM case is smaller than in FM case

Table 1

Comparison between critical loads with the change of index k in case FM.

P cr (static) 2.8112e+005 2.2167e+005 1.8200e+005 1.7299e+005

P cr (dynamic) 2.9995e+005 2.3796e+005 1.9767e+005 1.8826e+005

Table 2

Comparison between critical loads with the change of ratio R/h in case FM.

P cr (static) 2.2167e+005 1.8481e+005 1.4789e+005 1.1093e+005

P cr (dynamic) 2.3796e+005 1.9565e+005 1.5505e+005 1.1579e+005

0

1

2

3

4

5

6

7

5

W*

1: k=0 2: k=1 3: k=5 4: k=10

1

2

3

4

0 2 4 6 8 10 12 14 16 18

time (s)

ξ=0.2, R/h=1000, n=1 1: k=0

2: k=1 3: k=5

q

t Applied pressure

Fig 8 Effect of index k on dynamic response (dynamic, IM).

-1 0 1 2 3 4 5

5

W*

ξ=0.2, k=1, n=1 1: R/h=1000 2: R/h=1200 3: R/h=1500 4: R/h=2000

1

2

3

4

Fig 9 Effect of ratio R/h on load–average deflection curve (static, IM).

0 5 10 15 20 25 30

time (s)

ξ=0.2, k=1, n=1 1: R/h=1000 2: R/h=1200 3: R/h=1500

1

2

3

q

t Applied pressure load

Fig 10 Effect of ratio R/h on dynamic response (dynamic, IM).

6.4 Effect of FM and IM boundary conditions

Graph of nonlinear static responses of clamped FGM spherical

shells with different boundary conditions are plotted inFigs 11

and 12 As can be observed, the spherical shells with immovable

clamped edge have a comparatively higher capability of carrying

external pressure than shells with movable clamped edge However,

their response is unstable That means the IM shells experience a

snap-through with much higher intensity than their movable

clamped counterparts Furthermore, these figures also show that

the effect of k index and R/h ratio on the critical buckling pressure

of shells is very strong

Fig 13shows the comparison of the dynamic response of FGM

spherical with FM and IM boundary conditions It also can see that

the dynamic critical buckling load of clamped FM shell is smaller

than the one of clamped IM shell

Table 5 shows the effects of k index and R/h ratio on static

critical load in FM case and IM case Once again it is indicated that

critical loads in IM case are greater about twice times than the ones

in FM case

6.5 Effect of environment temperature

The effect of environment temperature on the

thermomechani-cal behavior of FGM shallow spherithermomechani-cal shells with immovable

clamped edge is considered in this subsection The shells are

exposed to temperature field prior to applying external pressure

Figs 14 and 15 analyze the nonlinear unsymmetrical static and dynamic responses of FGM spherical shells for various values of uniformly raised temperatureDT(=0, 50, 100 and 150 °C)

As shown inFig 14, the temperature field makes shell to be de-flected outward (negative deflection) prior to mechanical load act-ing on it When the shell is subjected uniform of external pressure, its outward deflection is reduced and when external pressure ex-ceeds bifurcation point of load, an inward deflection occurs Similar behavior occur for dynamic clamped FGM shells, too It is illus-trated inFig 15

Table 3

Comparison between the nonlinear static and dynamic buckling loads vs k in case IM.

P cr (static) 6.2836e+005 4.9545e+005 4.0688e+005 3.8674e+005

P cr (dynamic) 6.4303e+005 5.0813e+005 4.1900e+005 3.9854e+005

Table 4

Comparison between the nonlinear static and dynamic buckling loads vs R/h s in case

IM.

P cr (static) 4.9545e+005 4.1277e+005 3.3015e+005 2.4757e+005

P cr (dynamic) 5.0813e+005 4.2118e+005 3.3550e+005 2.5130e+005

-3

-2

-1

0

1

2

3

4

5

6

5

W*

FM IM

R/h=1000

ξ=0.2, n=1

1: k=0 2: k=1

1 2 3

3

1

Fig 11 Effect of in-plane restraint on nonlinear static response with the change of

index k.

-2 -1 0 1 2 3 4 5

5

W*

FM IM

(1): R/h=1000 (2): R/h=1200 (3): R/h=1500

ξ=0.2 k=1 n=1

1

2

3

1

2

3

Fig 12 Effect of in-plane restraint on nonlinear static response with the change of R/h ratio.

0 5 10 15 20 25 30

time (s)

FM IM

R/h=1000 ξ=0.2, k=1 n=1

q

t Applied pressure load

Fig 13 Effect of in-plane restraint on nonlinear dynamic response.

Table 5 Effects of k and R/h on static critical load in FM and IM cases.

k = 0 2.8112e+005 FM

2.3438e+005 1.8756e+005 1.4068e+005 6.2836e+005 IM

5.2350e+005 4.1871e+005 3.1398e+005

k = 1 2.2167e+005 1.8481e+005 1.4789e+005 1.1093e+005 4.9545e+005 4.1277e+005 3.3015e+005 2.4757e+005

k = 5 1.8200e+005 1.5175e+005 1.2144e+005 9.1091e+004 4.0688e+005 3.3897e+005 2.7112e+005 2.0330e+005

k = 10 1.7299e+005 1.4424e+005 1.1543e+005 8.6582e+004 3.8674e+005 3.2220e+005 2.5770e+005 1.9324e+005

The enhancement of temperature difference is accompanied by

the increase of bifurcation points, and the intensity of

snap-through behavior of the spherical shells (in static analysis) and

the strengtheneth of load bearing capability of the spherical shells

under dynamic loading (in dynamic analysis)

Comparison between nonlinear static and dynamic critical

buckling loads with effect of temperature is given in Table 6 It

can see that in this case the dynamic critical buckling load also is

greater than static one

7 Concluding remarks

This paper presents an analytical approach to investigate the

nonlinear unsymmetrical static and dynamic responses of clamped

FGM shallow spherical shells under uniform external pressure with and without including the effects of temperature

Approximate analytical one-term deflection mode for two types boundary conditions is given and by applying Galerkin procedure explicit expressions of static critical buckling loads and postbuck-ling load–deflection curves are determined

For the nonlinear dynamic buckling analysis, the nonlinear equation of motion of the shell is solved by using Runge–Kutta method The dynamic critical buckling loads are found according

to Budiansky–Roth criterion The nonlinear unsymmetric response

of the shells is analyzed and the results are illustrated in graphic form and numerical tables The results indicate that the nonlinear response of FGM shallow spherical shells is complex and greatly influenced by the type of loading (static or dynamic), the material and geometric parameters, the in-plane restraint and the pre-existent temperature condition

Acknowledgement This paper was supported by the National Foundation for Science and Technology Development of Vietnam – NAFOSTED The authors are grateful for this financial support

References [1] Budiansky B, Roth RS Axisymmetric dynamic buckling of clamped shallow spherical shells NASA TND 1962;510:597–609.

[2] Huang NC Unsymmetrical buckling of thin shallow spherical shells Trans ASME J Appl Mech 1964;31:447–57.

[3] Tillman SC On the buckling behavior of shallow spherical caps under a uniform pressure load Int J Solids Struct 1970;6:37–52.

[4] Ball RE, Burt JA Dynamic buckling of shallow spherical shells ASME J Appl Mech 1973;41:411–6.

[5] Kao R, Perrone N Dynamic buckling of axisymmetric spherical caps with initial imperfection Comput Struct 1978;9:463–73.

[6] Wunderlich W, Albertin U Buckling behavior of imperfect spherical shells Int J Nonlinear Mech 2002;37:589–604.

[7] Nath N, Alwar RS Nonlinear static and dynamic response of spherical shells Int J Nonlinear Mech 1978;13:157–70.

[8] Ganapathi M, Varadan TK Dynamic buckling of orthotropic shallow spherical shells Comput Struct 1982;15:517–20.

[9] Ganapathi M, Varadan TK Dynamic buckling of laminated anisotropic spherical caps J Appl Mech 1995;62:13–9.

[10] Chao CC, Lin IS Static and dynamic snap-through of orthotropic spherical caps Compos Struct 1990;14:281–301.

[11] Eslami MR, Ghorbani HR, Shakeri M Thermoelastic buckling of thin spherical shells J Therm Stresses 2001;24(12):1177–98.

[12] Shahsiah R, Eslami MR Thermal and mechanical instability of an imperfect shallow spherical cap J Therm Stresses 2003;26(7):723–37.

[13] Shahsiah R, Eslami MR, Naj R Thermal instability of functionally graded shallow spherical shells J Therm Stresses 2006;29(8):771–90.

[14] Prakash T, Sundararajan N, Ganapathi M On the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps J Sound Vib 2007;299:36–43.

[15] Ganapathi M Dynamic stability characteristics of functionally graded materials shallow spherical shells Compos Struct 2007;79:338–43 [16] Bich DH Nonlinear buckling analysis of FGM shallow spherical shells Vietnam

J Mech 2009;31:17–30.

[17] Bich DH, Hoa LK Nonlinear vibration of functionally graded shallow spherical shells Vietnam J Mech 2010;32(4):199–210.

[18] Bich DH, Tung HV Nonlinear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects Int J Nonlinear Mech 2011;46:1195–204.

[19] Shahsiah R, Eslami MR, Sabzikar Boroujerdy M Thermal instability of functionally graded deep spherical shell Arch Appl Mech 2011;81(10):1455–71.

[20] Brush DO, Almroth BO Buckling of bar, plates and shells NewYork: McGrawHill; 1975.

[21] Volmir AS Nonlinear dynamic of plates and shells science ed Moscow; 1972 [in Russian].

[22] Mushtari XM, Galimov KZ Nonlinear theory of elastic shells Kazan; 1957 [in Russian].

[23] Birman V Theory and comparison of the effect of composite and shape memory along stiffness on stability of composite shells and plates Int J Mech Sci 1997;39(10):1139–49.

[24] Oghibalov PM Dynamics and stability of shells Moscow; 1963 [in Russian] [25] Duc ND, Tung HV Nonlinear analysis of stability for functionally graded cylindrical panels under axial compression Comput Mater Sci 2010;49:313–6.

-4

-2

0

2

4

6

8

10

12

5

W*

2: ΔT=50

4: ΔT=150

R/h=1000, ξ=0.2 k=1, n=1

1

2

3

4

Fig 14 Effect of temperature on nonlinear static response.

-5

0

5

10

15

20

25

time (s)

1: ΔT=0

3: ΔT=100

Applied pressure

load

q

t

1 2 3 4

Fig 15 Effect of temperature on nonlinear dynamic response.

Table 6

Comparison of static and dynamic critical loads.

P cr (static) 4.9545e+005 7.1219e+005 9.4978e+005 12.0582e+005

P cr (dynamic) 5.0813e+005 7.2917e+005 9.7132e+005 12.3231e+005