DSpace at VNU: Nonlinear postbuckling of imperfect eccentrically stiffened P-FGM double curved thin shallow shells on elastic foundations in thermal environments

DSpace at VNU: Nonlinear postbuckling of imperfect eccentrically stiffened P-FGM double curved thin shallow shells on el...

Nonlinear postbuckling of imperfect eccentrically stiffened P-FGM

double curved thin shallow shells on elastic foundations in thermal

environments

Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:

Available online 14 July 2013

Keywords:

Nonlinear postbuckling

Eccentrically stiffened P-FGM double curved

thin shallow shells

Imperfection

Elastic foundation

Thermal environments

a b s t r a c t

This paper first time presents an analytical investigation on the nonlinear postbuckling for imperfect eccentrically stiffened FGM double curved thin shallow shells on elastic foundation using a simple power-law distribution (P-FGM) in thermal environments The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imperfection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic foundation By applying Galerkin method and using stress function, explicit relations of thermal load–deflection curves for simply supported curved eccentrically stiffened FGM shells are determined Effects of material and geometrical properties, temperature, elastic foundation and eccentrically stiffeners

on the buckling and postbuckling loading capacity of the imperfect eccentrically stiffened P-FGM double curved shallow shells in thermal environments are analyzed and discussed

Ó 2013 Elsevier Ltd All rights reserved

1 Introduction

Functionally Graded Materials (FGMs), which are

microscopi-cally composites and made from mixture of metal and ceramic

constituents, have received considerable attention in recent

years due to their high performance heat resistance capacity and

excellent characteristics in comparison with conventional

composites By continuously and gradually varying the volume

fraction of constituent materials through a specific direction, FGMs

are capable of withstanding ultrahigh temperature environments

and extremely large thermal gradients Therefore, these novel

materials are chosen to use in temperature shielding structure

components of aircraft, aerospace vehicles, nuclear plants and

engineering structures in various industries As a result, buckling

and postbuckling behaviors of FGM plate and shell structures under

different types of loading are attractive to many researchers in the

world

Regarding to the static buckling and postbuckling of FGM shells,

Shen has studied postbuckling of FGM cylindrical panels under

axial compression[1], external pressure[2]by using higher order

shell theory in conjunction with boundary layer theory of shell

buckling Shen and Liew also investigated the postbuckling of

FGM cylindrical panels with piezoelectric actuators in thermal

environments[3] FGM cylindrical shells in thermal environment under various loading types has been treated by similar methods and different shell theories [4–7] For Shen’s works, an semi-analytical approach is used to expand deflection and stress functions in form of power functions of small parameters, and then

an iteration is adopted to determine buckling loads and postbuck-ling curves Huang and Han [8,9] investigated the nonlinear buckling of FGM cylindrical shells under axial and external pressure

by a semi-analytical approach Duc and Tung presented analytical investigations on the nonlinear response of imperfect FGM cylindrical shells under axial compression[10]and thermal loads

[11] Geometrically nonlinear analysis of functionally graded shells

is considered by Zhao and Liew[12] Sohn and Kim investigated structural stability of FGM shells subjected to aero-thermal loads

[13] Sofiyev studied the stability of compositionally graded cera-mic–metal cylindrical shells under periodic axial impulsive loading

[14] In the recent years, there have been several works for the more complicated FGM shells for example: spherical, conical and double curved FGM shallow shells Shahsiah et al.[15]studied the linear buckling of shallow FGM spherical shells under two types of thermal loads Naj et al [16] used an analytical method and adjacent equilibrium criterion with assumption on small deflection

to determine critical buckling loads of FGM truncated conical shells subjected to mechanical and thermal loadings The stability of FGM truncated conical shells under compression, external pressure, impulsive and thermal loads also treated in works by Sofiyev using 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved.

⇑Corresponding author Tel.: +84 4 37547978; fax: +84 4 37547424.

E-mail address: ducnd@vnu.edu.vn (N.D Duc).

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

an analytical method[17–19] Bich and Tung used an analytical

approach to investigate the nonlinear buckling of FGM shallow

spherical shell under uniform external pressure including

temper-ature effects[20]and conical panels under mechanical loads[21]

Duc and Quan[30] studied nonlinear stability of double curved

shallow FGM panel

The components of structures widely used in aircraft, reusable

space transportation vehicles and civil engineering are usually

sup-ported by an elastic foundation Therefore, it is necessary to

in-clude effects of elastic foundation for a better understanding of

the buckling behavior and loading carrying capacity of plates and

shells Librescu and his co-workers have investigated the

postbuck-ling behavior of flat and curved laminated composite shells resting

on Winkler elastic foundations[22,23] In spite of practical

impor-tance and increasing use of FGM structures, investigations on the

effects of elastic media on the response of FGM plates and shells

are comparatively scarce Bending behavior of FGM plates on

Pas-ternak type foundations has been studied by Huang et al.[24]and

Zenkour[25]using analytical methods, Shen and Wang[26]

mak-ing use of asymptotic perturbation technique Shen et al.[27,28]

investigated the postbuckling behavior of FGM cylindrical shells

subjected to axial compressive loads and internal pressure and

sur-rounded by an elastic medium of tensionless elastic foundation of

the Pasternak type Duc et al used the third order shear

deforma-tion theory plate for studying nonlinear postbuckling of FGM plates

[29]and classical theory of shells for studying nonlinear

postbuck-ling of double curved shallow FGM panel on elastic foundation

[30] Recently, Duc extend his investigations for nonlinear dynamic

of the FGM shells on elastic foundation[38]

However, in practice, the FGM structure are usually exposed to

high-temperature environments, where significant changes in

material properties are unavoidable Therefore, the temperature

dependence of their properties should be considered for an

accu-rate and reliable prediction of deformation behavior of the

com-posites In [31,32] Shen studied thermal postbuckling behavior

of functionally graded cylindrical shells and panels with

tempera-ture-dependent properties Also Shen investigated postbuckling

analysis of axially loaded piezolaminated cylindrical panels with

temperature-dependent properties[33] Yang et al studied

ther-mo-mechanical postbuckling of FGM cylindrical panels with

tem-perature-dependent properties [34] It is evident from the

literature that investigations considering the temperature

depen-dence of materials properties for FGM shells are few in number

Notice that in all the publication mentioned above [31–34], all

authors use the displacement functions and volume fraction

fol-lows a simple power law (P-FGM) Recently, Duc and Quan

inves-tigated nonlinear buckling and postbuckling for FGM double

curved shallow FGM panel on elastic foundation in thermal

envi-ronments (without temperature-dependent properties) using

stress function[30]

In fact, the FGM plates and shells, as other composite structures,

usually reinforced by stiffening member to provide the benefit of

added load-carrying static and dynamic capability with a relatively

small additional weight penalty Thus study on static and dynamic

problems of reinforced FGM plates and shells with geometrical

nonlinearity are of significant practical interest However, up to

date, the investigation on static and dynamic of eccentrically

stiff-ened FGM structures has received comparatively little attention

Bich et al studied (without temperatures) static and dynamic

anal-ysis for eccentrically stiffened FGM shallow shells[35,36]and

dy-namic analysis for eccentrically stiffened functionally graded

cylindrical panels [37] Duc investigated nonlinear dynamic

re-sponse of imperfect eccentrically stiffened doubly curved FGM

shallow shells on elastic foundations[38]

To the best of our knowledge, there has been recently no

publi-cation on the FGM plates and shells reinforced by eccentrically

stiffeners in thermal environment The most difficult part in this type of problem is to calculate the thermal mechanism of FGM plates and shells as well as eccentrically stiffeners under thermal loads

Our paper is the first proposal for an imperfection eccentrically stiffened FGM double curved shallow shells on elastic foundation

in which we investigate the nonlinear postbuckling using a simple power-law distribution (P-FGM) under thermal environments Here, we have considered the FGM shell under temperature independent material property, i.e the Young’s modulus E, thermal expansion coefficienta, the mass densityq, the thermal conduction

K and even Poisson ratiom are independent to the temperature Those vary in the thickness direction z as well as temperature T in the two variables function of z and T The investigation under those assumptions for FGM shells is very challenging work Moreover, the presence of the eccentrically stiffeners makes it more difficult to solve Here, we have solved this problems taking into account all above assumptions

The formulations are based on the classical shell theory taking into account geometrical nonlinearity, initial geometrical imper-fection, temperature-dependent properties and the Lekhnitsky smeared stiffeners technique with Pasternak type elastic founda-tion Explicit expressions of buckling loads and postbuckling loads–deflection curves for simply supported double curved shal-low thin FGM shells are determined by Galerkin method and using stress function The effects of geometrical and material properties, temperature, elastic foundation and eccentrically stiffeners on the nonlinear response of the P-FGM shallow shells in thermal envi-ronments are analyzed and discussed

2 Eccentrically stiffened FGM double curved shallow shells on elastic foundations

Consider a ceramic–metal eccentrically stiffened FGM double curved shallow shell of radii of curvature Rx, Rylength of edges a,

b and uniform thickness h resting on an elastic foundation (Fig 1) For FGM shell, the volume fractions of constituents are assumed

to vary through the thickness according to the following power law distribution (P-FGM):

VmðzÞ ¼ 2z þ h

2h

 N

;VcðzÞ ¼ 1  VmðzÞ ð1Þ

where N is volume fraction index (0 6 N < 1) Effective properties

Preffof FGM shell are determined by linear rule of mixture as

Preffðz; TÞ ¼ PrmðTÞVmðzÞ þ PrcðTÞVcðzÞ ð2Þ

where Pr denotes a temperature dependent material property, and subscripts m and c stand for the metal and ceramic constituents, respectively Specific expressions of material coefficients are ob-tained by substituting Eq.(1)into Eq.(2)as

h

Rx

z

y

Ry

x

Fig 1 Geometry and coordinate system of an eccentrically stiffened double curved

½Eðz; TÞ;vðz; TÞ;qðz; TÞ;aðz; TÞ; Kðz; TÞ

¼ ½EcðTÞ;mcðTÞ;qcðTÞ;acðTÞ; KcðTÞ

þ ½EmcðTÞ;mmcðTÞ;qmcðTÞ;amcðTÞ; KmcðTÞ 2z þ h

2h

 N

ð3Þ

where

EmcðTÞ ¼ EmðTÞ  EcðTÞ;mmcðTÞ ¼mmðTÞ mcðTÞ;qmcðTÞ ¼qmðTÞ qcðTÞ;

ðamcðTÞ ¼amðTÞ acðTÞ;KmcðTÞ ¼ KmðTÞ  KcðTÞ

ð4Þ The values with subscripts m and c belong to metal and ceramic

respectively It is evident from Eqs.(3) and (4)that the upper

sur-face of the shell (z = h/2) is ceramic-rich, while the lower sursur-face

(z = h/2) is metal-rich, and the percentage of ceramic constituent

in the shell is enhanced when N increases A material property Pr,

such as the elastic modulus E, Poisson ratiom, the mass densityq,

the thermal expansion coefficientaand coefficient of thermal

con-duction K can be expressed as a nonlinear function of temperature

[31–34]:

Pr ¼ P0ðP1T1

þ 1 þ P1T1

þ P2T2

þ P3T3

in which T = T0+DT(z) and T0= 300 K (room temperature); P0, P1,

P1, P2 and P3 are coefficients characterizing of the constituent

materials

The shell–foundation interaction is represented by Pasternak

model as

wherer2

= @2/@x2+ @2/oy2, w is the deflection of the shell, k1is

Win-kler foundation modulus and k2is the shear layer foundation

stiff-ness of Pasternak model

3 Theoretical formulation

In this study, the classical shell theory, the Lekhnitsky smeared

stiffeners technique are used to establish governing equations and

determine the nonlinear response of FGM double curved thin

shal-low shells and took into account for FGM shells as well as stiffeners

which are both deformed by temperature

According to the classical thin shell theory the strain at the

mid-dle surface and curvatures are related to the displacement

compo-nents u,v, w in the x, y, z coordinate as:

ex

ey

cxy

0

B

1

C

A ¼

e0

x

e0

y

c0

xy

0

B

1

C

A þ z

kx

ky 2kxy

0 B

1

wheree0

x,e0

xandc0

xyare normal and shear strain at the middle surface

of the shell, and kx, ky, kxyare the curvatures and Rx, Ryare radii of

curvatures, 1/Rx, 1/Ryare principal curvatures of the shell The

non-linear strain–displacement relationship based upon the von Karman

theory for moderately large deflection and small strain are[39]:

e0

x

e0

y

c0

xy

0

B

1

C

A ¼

u;x w=Rxþ w2

;x=2

v;y w=Ryþ w2

;y=2

u;yþv;xþ w;xw;y

0

B

1 C A;

kx

ky

kxy

0 B

1 C

A ¼

wx;x

wy;y

w;xy

0 B

1

In which u,vare the displacement components along the x, y

direc-tions, respectively

Interestingly, comparing to the other[35–38], we have assumed

that the eccentrically outside stiffeners depend on temperature

Hooke law for an FGM shell with temperature-dependent

proper-ties is defined as

rsh

x;rsh y

¼ Eðz; TÞ

1 m2ðz; TÞ½ðex;eyÞ þmðey;exÞ  ð1 þmÞa DTðzÞð1; 1Þ

ð9Þ

rsh¼ Eðz; TÞ 2ð1 þmðz; TÞÞcxy whereDT is temperature rise from stress free initial state, and more generally,DT =DT(z); E(z, T),v(z, T) are the FGM shell’s elastic mod-uli which is determined by(3)

For stiffeners in thermal environments with temperature-dependent properties, we have proposed its form adapted from

[37]as the follows:

rst

x;rst y

¼ E0ðex;eyÞ  E0

1  2m0ðTÞa0ðTÞDðTÞð1; 1Þ ð10Þ

Here, E0= E0(T); m0=m0(T), a0=a0(T) are the Young’s modulus, Poisson ratio and thermal expansion coefficient of the stiffeners, respectively The shell reinforced by eccentrically longitudinal and transversal stiffeners is shown inFig 1 E0is elasticity modu-lus in the axial direction of the corresponding stiffener which is assumed identical for both types of longitudinal and transversal stiffeners In order to provide continuity between the shell and stiffeners, suppose that stiffeners are made of full metal (E0= Em)

if putting them at the metal-rich side of the shell, and conversely full ceramic stiffeners (E0= Ec) at the ceramic-rich side of the shell (this assumption has been used in[35–38]) The shallow shell is assumed to have a relative small rise as compared with its span The contribution of stiffeners can be accounted for using the Lekh-nitsky smeared stiffeners technique [35–38] In order to investi-gate the FGM shells with stiffeners in the thermal environment,

we have not only taken into account the materials moduli with temperature-dependent properties but also we have assumed that all elastic moduli of FGM shells and stiffener are temperature dependence and they are deformed in the presence of tempera-ture Hence, the geometric parameters, the shell’s shape and stiff-eners are varied through the deforming process due to the temperature change However, we have assumed that the thermal stress of stiffeners is subtle which distributes uniformly through the whole shell structure Therefore, we can ignore it and Lekhnit-sky smeared stiffeners technique can be adapted from[35–38]as the follows:

Nx¼ I10þE

T

0AT1

sT 1

!

e0

xþ I20e0

yþ I11þ CT1

kxþ I21kyþU1

Ny¼ I20e0

xþ I10þE

T

0AT2

sT 2

!

e0

yþ I21kxþ I11þ CT2

kyþU1

Nxy¼ I30c0

xyþ 2I31kxy

Mx¼ I11þ CT1

e0

xþ I21e0

yþ I12þE

T

0IT 1

sT 1

!

kxþ I22kyþU2

Mx¼ I21e0

xþ I11þ CT2

e0

yþ I22kxþ I12þE

T

0IT2

sT 2

!

kyþU2

Mxy¼ I31c0

xyþ 2I32kxy

ð11Þ

The relation(11)is our most important finding, where Iij(i = 1, 2, 3;

j = 0, 1, 2):

Z h=2

h=2

EðzÞ

1 mðzÞ2z

jdz

I2j¼

Z h=2

h=2

EðzÞmðzÞ

1 mðzÞ2z

I3j¼

Z h=2

h=2

EðzÞ

2½ð1 þmðzÞz

jdz ¼1

2ðI1j I2jÞ

ðU1;U2Þ ¼ 

Z h=2

h=2

EðzÞaðzÞ

1 mðzÞDTðzÞð1; zÞdz

IT

1¼

dT1 hT1 3

12 þ A

T

1 zT

1

 2

; IT

2¼

dT2 hT2 3

12 þ A

T

2 zT 2

 2

CT1¼E0A

T

1zT

1

sT

1

;CT2¼E0A

T

2zT 2

sT 2

zT

1¼h

T

1þ hT

2 ;z

T

2¼h

T

2þ hT 2

AT1¼ dT1sT

1;AT2¼ dT2sT

2

Where the geometric shapes of stiffeners after the thermal

deformation process in Eq.(12)can be determined as the follows:

dT1¼ d1ð1 þamTðzÞÞ; dT2¼ d2ð1 þamTðzÞÞ;

hT1¼ h1ð1 þamTðzÞÞ; hT2¼ h2ð1 þamTðzÞÞ;

zT

1¼ z1ð1 þamTðzÞÞ; zT

2¼ z2ð1 þamTðzÞÞ;

sT

1¼ s1ð1 þamTðzÞÞ; dT1¼ d1ð1 þamTðzÞÞ ð13Þ

Where the coupling parameters C1, C2 are negative for outside

stiffeners and positive for inside one; s1, s2 are the spacing of

the longitudinal and transversal stiffeners; I1, I2 are the second

moments of cross-section areas; z1, z2 are 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, h1and d2, h2respectively A1, A2are the

cross-sec-tion areas of stiffeners Although the stiffeners are deformed by

temperature, we, however, have assumed that the stiffeners keep

its rectangular shape of the cross section Therefore, it is

straight-forward to calculate AT

1, AT2 The nonlinear equilibrium equations of a FGM double curved

shallow shell based on the classical shell theory are[30,38]:

Mx;xxþ 2Mxy;xyþ My;yyþNx

Rx

þNy

Ry

þ Nxw;xxþ 2Nxyw;xyþ Nyw;yy

þ q  k1w þ k2r2w ¼ 0 ð14cÞ

Calculated from Eq.(11)

e0

x¼ A22Nx A12Nyþ B11w;xxþ B12w;yy ðA22 A12ÞU1

e0

y¼ A11Ny A12Nxþ B21w;xxþ B22w;yy ðA11 A12ÞU1

c0

xy¼ A66Nxyþ 2B66w;xy

ð15Þ

where

A11¼1

D I10þE

T

0AT1

sT 1

!

;A22¼1

D I10þE

T

0AT2

sT 2

!

A12¼I20

D;A66¼ 1

I30

D¼ I10þE

T

0AT 1

sT 1

!

I10þE

T

0AT 2

sT 2

!

 I220

B11¼ A22I11þ CT1

 A12I21

B22¼ A11I11þ CT2

 A12I21

B12¼ A22I21 A12I11þ CT2

B21¼ A11I21 A12I11þ CT1

B66¼I31

I30

ð16Þ

Substituting once again Eq.(15) into the expression of Mij in

(11), then Mijinto the Eq.(14c)leads to:

Nx;xþ Nxy;y¼ 0

B21f;xxxxþ B12f;yyyyþ ðB11þ B22 2B66Þf;xxyy D11w;xxxx D22w;yyyy

 ðD12þ D21þ 4D66Þw;xxyyþ Nxw;xxþ 2Nxyw;xyþ Nyw;yy

þNx

Rx

þNy

Ry

þ q  k1w þ k2r2w ¼ 0

where

D11¼ I12þE

T

0IT1

sT 1

 B11 I11þ CT1

 I21B21

D22¼ I12þE

T

0IT 2

sT 2

 B22I11þ CT2

 I21B12

D12¼ I22 B12I11þ CT1

 I21B22

D21¼ I22 B21 I11þ CT2

 I21B11

D66¼ I32 I31B66

ð18Þ

f(x, y) is stress function defined by

Nx¼ f;yy; Ny¼ f;xx; Nxy¼ f;xy ð19Þ

For an imperfect FGM curved shell, Eq.(17)are modified into form as

B21f;xxxxþ B12f;yyyyþ ðB11þ B22 2B66Þf;xxyy D11w;xxxx

 D22w;yyyy ðD12þ D21þ 4D66Þw;xxyyþ f;yy w;xxþ w

;xx

 2f;xy w;xyþ w

;xy

þ f;xx w;yyþ w

;yy

þf;yy

Rx

þf;xx

Ry

þ q  k1w þ k2r2w ¼ 0 ð20Þ

in which w⁄(x, y) is a known function representing initial small imperfection of the shell The geometrical compatibility equation for an imperfect double curved shallow shell is written as[30,38]:

e0 x;yyþe0 y;xxc0 xy;xy¼ w2

;xy w;xxw;yyþ 2w;xyw

;xy w;xxw

;yy

 w;yyw

;xxw;yy

Rx w;xx

Ry

From the constitutive relations(15)in conjunction with Eq.(19)

one can write

x¼ A22f;yy A12f;xxþ B11w;xxþ B12w;yy ðA22 A12ÞU1

e0

y¼ A11f;xx A12f;yyþ B21w;xxþ B22w;yy ðA11 A12ÞU1

c0

xy¼ A66f;xyþ 2B66w;xy

ð22Þ

Setting Eq.(22)into Eq.(21)gives the compatibility equation of

an imperfect FGM double curved shell as

A11f;xxxxþ A22f;yyyyþ ðA66 2A12Þf;xxyyþ B21w;xxxxþ B12w;yyyy

þ ðB11þ B22 2B66Þw;xxyy

 w2;xy w;xxw;yyþ 2w;xyw

;xy w;xxw

;yy w;yyw

;xxw;yy

Rx w;xx

Ry

¼ 0 ð23Þ

Eqs.(20) and (23)are nonlinear equations in terms of variables

w and f and used to investigate the stability of FGM double curved

shells on elastic foundations subjected to mechanical, thermal and

thermo-mechanical loads

In the present study, the edges of curved shells are assumed to

be simply supported Depending on the in-plane restraint at the

edges, three cases of boundary conditions[26–30], labeled as Cases

1, 2 and 3 may be considered

Case 1 Four edges of the shell are simply supported and freely

movable (FM) The associated boundary conditions are

w ¼ Nxy¼ Mx¼ 0; Nx¼ Nx0at x ¼ 0; a

w ¼ Nxy¼ My¼ 0; Ny¼ Ny0at y ¼ 0; b: ð24Þ

Case 2 Four edges of the shell are simply supported and

immov-able (IM) In this case, boundary conditions are

w ¼ u ¼ Mx¼ 0; Nx¼ Nx0at x ¼ 0; a

w ¼v¼ My¼ 0; Ny¼ Ny0at y ¼ 0; b: ð25Þ

Case 3 All edges are simply supported Two edges x = 0, a are freely

movable, whereas the remaining two edges y = 0, b are

immov-able For this case, the boundary conditions are defined as

w ¼ Nxy¼ Mx¼ 0; Nx¼ Nx0at x ¼ 0; a

w ¼v¼ My¼ 0; Ny¼ Ny0at y ¼ 0; b ð26Þ

where Nx0, Ny0are in-plane compressive loads at movable edges (i.e

Case 1 and the first of Case 3) or are fictitious compressive edge

loads at immovable edges (i.e Case 2 and the second of Case 3)

The approximate solutions of w and f satisfying boundary

con-ditions(24)–(26)are assumed to be[29,30,38]:

ðw; wÞ ¼ ðW;lhÞ sin kmx sin dny ð27aÞ

f ¼ A1cos 2kmx þ A2cos 2dny þ A3sin kmx sin dny þ1

2Nx0y 2

þ1

2Ny0x

km= mp/a, dn= np/b W is amplitude of the deflection and l is

imperfection parameter The coefficients Ai(i = 1  3) are

deter-mined by substitution of Eqs.(27a, 27b)into Eq.(23)as

A1¼ d

2

32A11k2mWðW þ 2lhÞ; A2¼ k

2 m 32A22d2WðW þ 2lhÞ;

A11k4mþ A22d4þ ðA66 2A12Þk2md2

2

Rx

þk

2 m

Ry

! W

 B21k

4

mþ B12d4nþ ðB11þ B22 2B66Þk2md2n

A11k4mþ A22d4þ ðA66 2A12Þk2md2

Subsequently, substitution of Eqs.(27a) and (27b)into Eq.(20)and applying the Galerkin procedure for the resulting equation yield

mn p 2

4k m d n

2 d 2

R xþk2m

R y

 B

21 k 4

m þB 12 d 4 þðB 11 þB 22 2B 66 Þk 2

m d 2

A 11 k4mþA 22 d4þðA 66 2A 12 Þk 2

m d2

B21 k 4

m þB 12 d 4 þðB 11 þB 22 2B 66 Þk 2

m d 2

A 11 k4mþA 22 d4þðA 66 2A 12 Þk 2

m d2

 d 2

R xþk 2 m

R y

 2

1

A 11 k 4

m þA 22 d 4 þðA 66 2A 12 Þk 2

m d 2

D11k4m D22d4n ðD12þ D21þ 4D66Þk2

md2n k2 k2mþ d2

n

 k1

8

>

>

>

>

>

>

>

>

>

>

9

>

>

>

>

>

>

>

>

>

> W

þ8k m d n

3

1

A 11 k 4

m þA 22 d 4 þðA 66 2A 12 Þk 2

m d 2

d2

R xþk2m

R y

ðB21 k4mþB 12 d4þðB 11 þB 22 2B 66 Þk2m d2Þ

A 11 k 4

m þA 22 d 4 þðA 66 2A 12 Þk 2

m d 2

2 6 4

3 7 5WðW þlhÞ

þ 1 12k m d n

k2m

A 22 R xþ d2

A 11 R y

1 B 21

A 11þB 12

A 22

kmdn

WðW þ 2lhÞ

ab 64

k4m

A 22þd4

A 11

WðW þlhÞðW þ 2lhÞ ab

4Nx0k2mþ Ny0d2n

ðW þlhÞ

þ 4

k m d n

N x0

R xþNy0

R y

þkm4qdn¼ 0

ð29Þ

where m, n are odd numbers This is basic equation governing the nonlinear response of eccentrically stiffened FGM double curved shallow shells under mechanical, thermal and thermo-mechanical loading conditions In what follows, some thermal loading condi-tions will be considered It is not so difficult to realize that the Eq

(29) is more complicated than the equation written in [35–38]

without the temperature

A simply supported FGM curved shell on elastic foundations with all immovable edges is considered The shell is subjected to uniform external pressure q and simultaneously exposed to temperature environments or subjected to through the thickness temperature gradient The in-plane condition on immovability at all edges, i.e u = 0 at x = 0, a andv= 0 at y = 0, b, is fulfilled in an average sense as[27–34]:

Zb 0

Z a 0

@u

@xdxdy ¼ 0;

Z a 0

Zb 0

@v

@ydydx ¼ 0: ð30Þ

From Eqs.(7) and (15)one can obtain the following expressions in which Eq.(19)and imperfection have been included

@u

@x¼ A22f;yy A12f;xxþ B11w;xxþ B12w;yy

 ðA22 A12ÞU11

2w

2

;x w;xw

;xþw

Rx

@v

@y¼ A11f;xx A12fyyþ B22w;yyþ B21w;xx ðA11 A12ÞU1

1

2w

2

;y w;yw

;yþw

Ry

Substitution of Eqs.(27a) and (27b)into Eq.(31)and then the result into Eq.(30)give fictitious edge compressive loads as

Nx0¼ 1þ 4

ðA 11 A 22 A 2

12 Þ  1

mn p 2

A 11

R xþA 12

R y

n

þ 2A12A11

1

A 11 k4m þA 22 d4þðA 66 2A 12 Þk 2

m d2

d2

R xþk2m

R y

B21 k 4

m þB 12 d 4 þðB 11 þB 22 2B 66 Þk 2

m d 2

A 11 k 4

m þA 22 d 4 þðA 66 2A 12 Þk 2

m d 2

2 6

3 7

5 þ A12B21þ A11B11

2 6

3 7

3

7m

na 2

9

=

>W

þ A2

12þ A11A22

m þA 22 d 4 þðA 66 2A 12 Þk 2

m d 2

d 2

R xþk 2 m

R y

B21 k 4

m þB 12 d 4 þðB 11 þB 22 2B 66 Þk 2

m d 2

A 11 k4m þA 22 d4þðA 66 2A 12 Þk 2

m d2

2 6

3 7

5 þ A12B22þ A11B12

2 6

3 7

3

7n

mb 2

o W

8 A 11 A 22 A 2 12

ð Þ A11k2

mþ A12d2

WðW þ 2lhÞ

ð32Þ

Ny0¼U1þ 4

A 11 A 22 A 2

12

1

mn p 2

A 12

R xþA22

R y

n

þ A212þ A11A22

m þA 22 d 4 þðA 66 2A 12 Þk 2

m d 2

d 2

R xþk 2 m

R y

B21 k 4

m þB 12 d 4 þ B ð 11 þB 22 2B 66 Þk 2

m d 2

A 11 k 4

m þA 22 d 4 þðA 66 2A 12 Þk 2

m d 2

2

6

4

3 7

5 þ A12B11þ A22B21

2

6

4

3 7 5

3 7

5nam2

9

>

>W

þ 2A12A22

1

A 11 k 4

m þA 22 d 4 þðA 66 2A 12 Þk 2

m d 2

d 2

R xþk 2 m

R y

B21 k 4

m þB 12 d 4 þðB 11 þB 22 2B 66 Þk 2

m d 2

A 11 k 4

m þA 22 d 4 þðA 66 2A 12 Þk 2

m d 2

2

6

4

3 7

5 þ A22B22þ A12B12

2

6

4

3 7 5

3 7

5mbn2

9

>

>W

8 A 11 A 22 A 2

12

ð ÞA12k2mþ A22d2

WðW þ 2lhÞ

ð33Þ

Specific expressions of parameter U1 in two cases of thermal

loading will be determined Obviously, in the presence of

temper-ature, the expressions for eccentrically stiffened FGM double

curved shallow shells are more complicated than the results in

the absence of temperature

4 Numerical results and discussion

Here, several numerical examples will be presented for perfect

and imperfect simply supported midplane-symmetric FGM shells

The silicon nitride and stainless steel are regarded as constituents

of the FGM shells The typical values of the coefficients of the

mate-rials mentioned in(5)are listed inTable 1

4.1 Uniform temperature rise

The FGM curved shell is exposed to temperature environments

uniformly raised from stress free initial state Tito final value Tfand

temperature differenceDT = Tf Tiis considered to be independent

from thickness variable The thermal parameter is obtained from

Eq.(12)as

where

L ¼ Emcamc

ðN þ 1Þmmc

þ P

m2

mc

 ðEcamc

þ EmcacÞ 1

1 mcmmcþ

Z 1 0

ds

1 mcmmcsN

ð35Þ

and

P ¼mmcðEcamcþacEmcÞ  Emcamcþ Emcamcac ð36Þ

4.2 Through the thickness temperature gradient

The metal-rich surface temperature Tmis maintained at stress

free initial value while ceramic-rich surface temperature Tc is

elevated and nonlinear steady temperature conduction is governed

by one-dimensional Fourier equation

d

dz KðzÞ

dT dz

¼ 0; Tðz ¼ h=2Þ ¼ Tc; Tðz ¼ h=2Þ ¼ Tm: ð37Þ

Using K(z) in Eq.(3), the solution of Eq.(37)may be found in terms

of polynomial series, and the first eight terms of this series gives the following approximation[29–31]:

TðzÞ ¼ TmþDT DTr

P5 j¼0 ðr N K mc =K c Þj jNþ1

P5 j¼0 ðK mc =K c Þ j

jNþ1

ð38Þ

where r = (2z + h)/2h andDT = Tc Tmis defined as the temperature change between two surfaces of the FGM shallow shell

Introduction of Eq.(38)into Eq.(12)gives the thermal parame-ter as

where

H ¼

P5 j¼0 ðK mc =K c Þ j

jNþ1

E c a c

jNþ2þE c a mc þE mc a c

ðjþ1ÞNþ2 þ E mc a mc

ðjþ2ÞNþ2

P5 j¼0 ðK mc =K c Þj jNþ1

Subsequently, setting Eq.(34)into Eq.(32)and then the result into

Eq.(29)give

q ¼ b11W þ b12WðW þlÞ þ b13WðW þ 2lÞ þ b14WðW þlÞ

where

b11¼mnp2B4

aD11K1 16B4 þ

mnp4B2

aD11K2 16B4 m

2B2aþ n2

þm

5np6B4

aD11þ mn5p6D22þ m3n3p6B2

aðD12þ D21þ 4D66Þ 16B4h

mnp4Baðn2Raxþ m2BaRbyÞ

8B3h

 B21m

4B4aþ ðB11þ B22 2B66Þm2n2B2aþ B12n4

A11m4B4aþ ðA66 2A12Þm2n2B2aþ A22n4

þ mnBaðn

2Raxþ m2BaRbyÞ2 16B2 A11m4B4

aþ ðA66 2A12Þm2n2B2

aþ A22n4

5np6B4a 16B4



B21m4B4

aþ ðB11þ B22 2B66Þm2n2B2

aþ B12n4

A11m4B4

aþ ðA66 2A12Þm2n2B2

aþ A22n4

Table 1

Material properties of the constituent materials of the considered FGM shells [29–34]

q(kg/m 3

a(K 1

q(kg/m 3

a(K 1

b12¼  2m

2n2p4B3an2Raxþ m2BaRby 3B3 A11m4B4

aþ ðA66 2A12Þm2n2B2

aþ A22n4

þ2m

2n2p4B2ahB21m4B4aþ ðB11þ B22 2B66Þm2n2B2aþ B12n4i

3B4hhA11m4B4aþ ðA66 2A12Þm2n2B2aþ A22n4i

ð42Þ

b13¼ 

p2 A11m2B3

aRaxþ A22n2Rby

48A11A22B3h þ

m2n2p4B2a 12B4h

B21

A11

þB12

A22

!

b14¼mnp6

256B4h

m4B4

a

A22

þn 4

A11

!

b15¼

mnp4 m2B2

aþ n2

16B2h ðW þlÞ 1

BhðBaRaxþ RbyÞ

where

K1¼k1a

4

D11

; K2¼k2a

2

D11

A11¼ hA11; A22¼ hA22; A12¼ hA12; A66¼ hA66

B11¼B11

h ; B22¼

B22

h ; B12¼

B12

h ; B21¼

B21

h ; B66¼

B66

h ð43Þ

D11¼D11

h3 ; D22¼

D22

h3 ; D12¼

D12

h3 ; D21¼

D21

h3 ; D66¼

D66

h3

Eq (41) expresses explicit relation of pressure–deflection

curves for eccentrically stiffened FGM curved shells rested on

elas-tic foundations and under combined action of uniformly raised

temperature field and uniform external pressure A similar

expres-sion for eccentrically stiffened FGM curved shells simultaneously

subjected to uniform external pressure and temperature gradient

across the thickness may be obtained as Eq.(38), provided L is

re-placed by (L–H)

Unlike the other works, we here assume that all coefficients

de-pend on both of the thickness z and temperature T Technically, it is

much more difficult to capture and solve the fundamental set of

equations In this paper, we have contribute significantly to this transformation process

The numerical results have been calculated withDT = c onst The parameters for the stiffeners are: z1= 0.0225 (m), z2= 0.0225 (m), s1= 0.4 (m), s2= 0.4 (m)

h1¼ 0:003ðmÞ; h2¼ 0:003ðmÞ; d1¼ 0:004ðmÞ; d2¼ 0:004ðmÞ

Fig 2illustrates the nonlinear postbuckling of pressure–deflec-tion curves relapressure–deflec-tion for eccentrically reinforced spherical shallow

Fig 2 Effects of thermo-mechanical loads on the pressure–deflection relation of

Fig 3 Effects of mechanical loads on the pressure–deflection relation of eccentri-cally stiffened cylindrical FGM shallow panel (without temperatures).

Fig 4 Effects of N index and imperfection on the pressure–deflection curves of

shell FGM (1/Rx= 1/Ry= 0.5) in the thermal environment and

com-paring with Duc[30](without stiffeners) It is clear that the

stiffen-ers can enhance the thermal loading capacity for the spherical FGM

shallow shells

Fig 3describes a nonlinear postbuckling of pressure–deflection

curves relation for eccentrically reinforced FGM cylindrical panel

(1/Rx= 1/Ry= 0.5) under mechanical loads and comparing with

Bich et al.[35](in the presence of stiffeners without temperature)

There is no significant difference between our finding (with

v=v(z)) and Bich’s report (v= const)

Fig 4shows us the effect of the volume fraction index N and the

imperfection l on the nonlinear response for eccentrically

stiffened FGM spherical shallow shell Obviously, the thermo-mechanical loading capacity of FGM shells increase as N decreases

It is consistent with P-FGM which has been reported in[29–31]

and the imperfect plates has a better thermo-mechanical loading capacity than the perfect plates[29,38]

Fig 5illustrates the effect of temperature on a nonlinear post-buckling response for eccentrically stiffened FGM spherical shal-low shells in thermal environment withDT = 0 (K), 300 (K), 500 (K), 1000 (K) It also shows us that the temperature is able to re-duce the loading capacity of the shells And, it is therefore consis-tent with the well-known results of the temperature effects on the nonlinear postbuckling response of FGM structures[1,2,11,29–31]

Fig 5 Effects of temperature on the nonlinear response of reinforced spherical

FGM panel. Fig 7 Effects of initial imperfect on the postbuckling curves of reinforced spherical

FGM shallow shell under uniform temperature rise.

Fig 6 Effects of elastic foundation on the nonlinear response of reinforced Fig 8 Comparison of postbuckling behaviors of reinforced and unreinforced

Fig 6investigates the effects of elastic foundations on the

non-linear response for eccentrically stiffened spherical shallow shell in

thermal environment The curve 1 is a case without an elastic

foun-dation We have seen that the elastic foundation (increasing K1, K2)

has enhanced the loading capacity of the shell And, K2in

Paster-nak’s model has a stronger effect than K1in Winkler’s model This

is consistent with the conclusions in[22–30]

Fig 7shows us the effect of the imperfection of the initial shape

on the non-linear response of eccentrically stiffened spherical shell

corresponding to the parametersl= 0.5, 0.2, 0, 0.2, 0.5 Indeed,

the imperfection affects very complicated on the postbuckling

behaviors of the shells, and imperfect eccentrically stiffened

spherical shell with the positive coefficientlwhich seems to be able getting a better loading capacity in the postbuckling period with the large bending InFig 8, it is obvious that the effects of stiffeners on the nonlinear postbuckling of spherical FGM shallow shells under thermo-mechanical loads We can see that the loading capacity of FGM shell increases in the presence of stiffeners, partic-ularly in the parts of the shell which start the buckling process This is very important in engineering applications

Fig 9 illustrates the effects of Poisson ratio on nonlinear re-sponse of reinforced spherical FGM shallow shells in the thermal environment Similar to the shell without stiffeners, there is no big difference betweenm= const andm=m(z) cases[30] However,

Fig 9 Effects of Poisson’s ratio on the pressure–deflection curves of reinforced

spherical FGM shallow shells.

Fig 11 Effects of ratio b/h on the pressure–deflection curves of reinforced spherical FGM panels.

Fig 12 Effects of ratio a/R x on the pressure–deflection curves of imperfect eccentrically stiffened P-FGM double curved shallow shells on elastic foundations Fig 10 Effects of ratio b/a on the pressure–deflection curves of reinforced spherical

the critical loads with m=m(z) gives us the smaller results than

these withm= const

Fig 10shows us the effects of a ratio b/a on nonlinear response

of the shells under the same temperature condition In the first

period, the shell with b/a = 0.75 has the best loading capacity

However, in the limit of large bending, the shell with b/a = 1.5

has the best loading capacity, even in the postbuckling period

Sim-ilarly,Figs 11–13show us the effects of the geometric parameters

b/h and the curvatures of the shell a/Rx, b/Ryon nonlinear response

of imperfect eccentrically stiffened P-FGM double curved shallow

shells with elastic foundations in the thermal environment

5 Concluding remarks

This paper presents an analytical investigation on the nonlinear

postbuckling for imperfect eccentrically stiffened double curved

thin shallow FGM shells using a simple power-law distribution

(P-FGM) in thermal environments Our most important finding is

the systematical investigation of the thin FGM shell reinforced by

stiffeners in the thermal environment Both of FGM and stiffeners

are deformed under mechanical, thermal and thermo-mechanical

loads The formulations are based on the classical shell theory

tak-ing into account geometrical nonlinearity, initial geometrical

imperfection, temperature-dependent properties and the

Lekhnit-sky smeared stiffeners technique with Pasternak type elastic

foun-dation By applying Galerkin method and using stress function,

explicit relations of thermal load–deflection curves for simply

sup-ported curved eccentrically stiffened FGM shells are determined

Effects of material and geometrical properties, elastic foundation

and eccentrically outside stiffeners on the buckling and

postbuck-ling loading capacity of the imperfect eccentrically stiffened P-FGM

double curved shallow shells in thermal environments are

ana-lyzed and discussed Some results were compared with the ones

of the other authors

Acknowledgment

This work was supported by Grant in Mechanics of the National

Foundation for Science and Technology Development of

Vietnam-NAFOSTED The authors are grateful for this support

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Fig 13 Effects of ratio b/R y on the pressure–deflection curves of imperfect

eccentrically stiffened P-FGM double curved shallow shells on elastic foundations

in thermal environments.