CHAPTERS

FUNCTIONALLY GRADED PLATES

S.1 General

In conventional laminated composite structures, homogeneous elastic laminae

are bonded together to obtain enhanced mechanical properties. However, the abrupt

change in material properties across the interface between different materials can result

in large interlaminar stresses leading to delamination. Furthermore, large plastic

deformations at the interface may trigger the initiation and propagation of cracks in the

material. One way to overcome these adverse effects is to use Functionally Graded

Material (FGM) in which material properties vary continuously. This is achieved by

gradually changing the volume fraction of the constituent materials usually only in the

thickness direction to obtain a smooth variation of material properties. By applying the

many possibilities inherent in the FGM concept, it is anticipated that material properties

will be improved and new functions for them created. The concept of FGM was

proposed in 1984 by material scientists in Japan.

FGMs have great potential for improving material/structural performance in

many engmeenng applications precisely because of their spatially graded

heterogeneous micro structure. Several studies have been performed to analyse the

behaviour of FGMs. A significant number of investigations have been carried out to

understand the buckling behaviour and stress distribution of structural elements made

of FGMs under thermal loading. Very little work has been done to consider the

87

mechanical stress analysis and buckling behaviour of functionally graded structures

subjected to mechanical loading.

The primary objective of this chapter is to present a general formulation for

FunctionanIy Graded plates (FG plates) using the Zeroth order Shear Deformation plate

Theory. ZSDT has been derived by Shimpi [1999] for flexural analysis of isotropic

plates. Here an attempt is made to extend the same model for flexural and stability

analysis of FG plates. This theory has number of advantages over the CLPT and FSDT.

It is possible to take into account the higher order effects and yet keep the complexity

to a considerably lovo'er level. In the present theory the governing differential equation

is of fourth order and in these only lateral deflection, plate physical properties and

lateran loading are being used The Navier solutions of the linear theory for simply

supported functionaHy graded rectangular plates are developed. The present theory and

the Reddy's higher order pnate theory contain the same number of dependent variables

and also results in estimation of deflection and stresses to a similar level of accuracy.

This satisfies zero transverse shear stress boundary condition on the surface of the

plate, does not require any shear correction factor and avoids shear locking. Numerical

results are presented for ceramic - metal functionally graded plates. The influences

played by transverse shear deformation, plate aspect ratio, side to thickness ratio and

volume fraction distribution are studied. The results are verified with the available

results in the literature.

88

5.2 Flexural Analysis -Theoretical Formulation using ZSDT

Consider a plate of total thickness h and composed of FGM. It is assumed that

the material is isotropic and grading is assumed to be only through the thickness. The

xy plane is taken to be the undeformed mid plane of the plate with the z axis positive

downward from the mid plane.

5.2.1 Displacement Field and Strains

The assumed displacement field is as follows:

u = Uo(X,y)-z:: + ~, [H~)-2G)},

v ~ Vo(X,y)-z:<[H~)-2(~)']Q,

w=wo(x,y)

(5.1 a)

(5.1 b)

(5.1 c)

where Uo and Vo are displacements at any point (x,y,O) on the reference plane in the x

and y direction respectively. Qx and Qy are the transverse shear stress resultants with Ax

and Ay being the unknown constants. The constants Ax and Ay can be determined by

considering the definitions of Qx and Qy

h/2 h/2

Qx = f Lxzdz and Q y = f Lyzdz-h/2 -h/2

The normal and in-plane shear strains are given by

E = 000 -z a2

w +_1[~(~)_2(~)3]aQxx ax ax 2 A 2 h h axx

89

(5.2)

(5.3 a)

(5.3 b)

y =000 +Ovo -2z &w +[~(~)_2(~)3][_1 (CQxJ+_l (CQyJ]xy Oy ax axOy 2h h Ax Oy 'Ay ax

The expressions for transverse shear strains are given by

(5.3 c)

(5.4)

In the expression for the in-plane displacements the effect of transverse shear

defonnation has been incorporated through the use of transverse shear stress resultants.

The expression for transverse shear strains Yxz and Yyz do not explicitly contain the

rotational displacements due to the transverse shear deformations. Hence the present

theory is called as a Zeroth order Shear Defonnation Theory (ZSDT) for the FG plates.

5.2.2 Constitutive Relations

In the study of FGM, the modelling involves the characterisation of the

constitution of the material and the theoretical and computational analysis. Generally,

there are two ways to model the material property gradation in solids (i) assume a

profile for volume fraction variations (macroscopic approach), the assumed variation of

material properties may be polynomial profiles, exponential profiles or piece wise

homogeneous layer and (ii) a micromechanics approach to study the inhomogeneous

media. Several micromechanics models like Mori -Tanaka method, self consistent

methods, differential schemes, bounding techniques etc have been developed for

estimating the effective material properties. Accurate estimation of effective properties

of FGM is the key to the eventual success in the design of FGM. However, precise

infonnation about the size, shape and distribution of material particles for FGM is not

available, the effective moduli of the graded composite must be evaluated based only

on the volume fraction distribution. Thus, in the present study the material property

90

variation is represented by a power law distribution in terms of the volume fractions of

the constituents.

The material property gradation is considered through the thickness and the

expression given below represents the profile for the volume fraction.

(5.5)

where Vr =(~ +~Jand P denotes a generic material property like modulus,

PI and Pb denote the property of the top and bottom faces of the plate respectively, h is

the total thickness of the plate and V f is the volume fraction and n is a parameter that

dictates material variation profile through the thickness. This parameter takes values

greater than or equal to zero. The value of n equal to zero represents a fully ceramic

plate. The above power law assumption reflects a simple rule of mixtures used to obtain

the effective properties of the ceramic metal plate. The rule of mixtures applies only in

the thickness direction. Here, it is assumed that moduli E and G, and density p vary

according to the Eqn.(5.5) and v is assumed to be a constant. The linear constitutive

relations are:

CJx Q ll Q I2 0 0 0 Ex

CJy Q 12 Q 11 0 0 0 Ey

1'yz = 0 0 Q 44 0 0 Yyz

l'xz 0 0 0 Q 55 0 Yxz

1'xy 0 0 0 0 Q 66 Yxy

EQ I2 =v QII'

Ewhere Q 11 = --2 ' Q I4 :::: Q55 ::= Q66 = 2(1 +v)I-v

(5.6)

(5.7)

Using Eqn. (5.4) in the constitutive relations for 't'xz and 1'yz and then substituting in

Eqn. (5.2) Ax and Ay may be obtained as follows:

A =A = __E_x y 2(1 + v)

91

(5.8)

5.2.3 Equilibrium Equations

The equilibrium equations are obtained using the principle ofvirtual work as:

oN oN_x_+~=Oax Oy

oN oNxy Y

--+--=0ax Oy

o( 4) o( 4) ( 4 )- M --P +- M --P - --R-Oox x 3h2 x Oy xy 3h2 xy Qx h2 x -

o( 4) o( 4) ( 4 )- M --P +- M --P - --R-Oax xy 3h2 xy Oy Y 3h2 Y Qy h2 y -

The stress resultants and moment resultants are given by:

h/2

(Nx,Mx,PJ = f crx (l,z,z2)dz-h/2

h/2

(Ny,My,Py) = f cr/l,z,z2)dz-h/2

h/2

(Nxy,Mxy,Pxy) = f t Xy (1,z,z2)dz-h/2

h/2

(Rx,Ry) = f Z2(txz , tyz)dz-h/2

The various stiffness parameters are defined as follows:

(5.9 a)

(5.9 b)

(5.9 c)

(5.9 d)

(5.9 e)

(5.1 0 a)

(5.10 b)

(5.1 0 c)\

(5.10 d)

h/2

+ f Q~(I,z,z2,z3,z4,z6)dz-h/2

92

( i , j = 1,2,6) (5.11)

5.2.4 Solution Approach

The equations of equilibrium admit the Navier solutions for simply supported

plates. The variables UQ, vo, wo, Qx, Qy can be written satisfying the SS1 type boundary

conditions as follows

00 00 lTInx . llnyUo = IIUmn cos--sm-

bm=ln=1 a

00 00 • lTInx llnyVo = IIVmn sm--cos--

m=ln=1 a b

00 00 • lTInx . llnyW o =IIWmnsm--sm--

m=] n=1 a b

00 00 lTInx . llnyQx = IIQxmncos-sm-

m=ln=1 a b

00 00 • lTInx llnyQy = I I Qytnn sm--cos-

m=1 n=1 a b

The transverse load q is also expanded in double Fourier series as

00 00

( ) _""Q . lTInx . llnyq x,y - LJLJ mn sm--sm--m=ln=1 a b

5.2.5 Numerical Results and Discussions

(5.12a)

(5.12b)

(5.12c)

(5.12d)

(5.12e)

(5.13)

The study has been focused on the static behaviour of functionally graded plate

based on higher order displacement model. In order to prove the validity of the present

fonnulation, results are compared with the existing ones in the literature. Here some

representative results of the Navier solution obtained for a simply supported square

plate under sinusoidally distributed load of intensity qo for the material are presented

with following properties.

Material: Aluminum-Alumina:

Youngs Modulus for Alumina (E l ) =:: 380 GPa (Ceramic)

93

Youngs Modulus for Aluminum (Eb) = 70 OPa (Metal), Poisson's ratio (v) = 0.3

Results are tabulated in Table 5.1 and Table 5.2. The tables contain the

non dimensionalised (dimensionless) deflections and stresses respectively. The results

are verified with the available results in the literature Zenkour [2006]. The results

obtained by the present model are in good agreement with the results generated using

most accepted model of Reddy [2000]. The following non dimensionalised quantities

are reported.

- (a b h) h - (a b h) hCJ

x =CJx 2'-2"'-2" aqo' CJy =cry 2'-2"'-3 aqo'

- ( h) hTxy = 'txy 0,0'-"3 ago'

- (a ) hTyz ='tyz -,0,0 -,2 ago

(5.14)- _ ( b h)E t h3

- _ (a h)E t h3

u - U o 0,-,-- --4 xIOO, V - Vo -'0'--6 --4 xlOO2 4 goa 2 goa

The Tables 5.1 and 5.2 show the effect of volume fraction exponent (Vt) on the

displacements and stresses of a functionally graded square plate with a/h = 10. It can be

observed that as the plate becomes more and more metallic the deflection wand

normal stress CJx increases but normal stress CJ y decreases. It is very interesting to note

that the stresses for a fully ceramic plate are the same as that of a fully metal plate. This

is due tQ the fact that in these two cases the plate is fully homogeneous and stresses do

not depend on the modulus of elasticity.

Figure 5.1 shows the distribution of the volume fraction Vf through the plate

thickness for various values of power law index n. Figure 5.2 shows the variation of

non dimensionalised central deflection of a square plate with power law index n. Figure

5.3 and Figure 5.4 show the variation of central deflection with aspect ratio (alb) and

side to thickness ratio (a/h). It is observed that the deflection is maximum for metallic

94

plate and minimum for a ceramic plate. The difference decreases as the aspect ratio

increases while it may be unchanged with the increase of side to thickness ratio. From

these figures it is also evident that the response of FG plates is intermediate to that of

the ceramic and metal homogeneous plates.

95

Table 5.1 Effects of volume fraction exponent on the dimensionlessdisplacements for FG square plates subjected to sinusoidalloading (a/h ==10)

- - -n Model u v w

Present 0.21805 0.14493 0.29423

Ceramic GSDT$ 0.23090 0.15390 0.29600

HSDT# 0.21805 0.14493 0.29423

Present 0.28180 0.19850 0.33672

0.2 GSDT - - -

HSDT 0.28172 0.19820 0.33767

Present 0.42135 0.31096 0.44387

0.5 GSDT - - -

HSDT 0.42131 0.31034 0.44407

Pr~sent 0.64258 0.49673 0.59059

1 GSDT 0.6626 0.5093 0.5889

HSDT 0.64137 0.49438 0.58895

Present 0.90220 0.71613 0.76697

2 GSDT 0.92810 0.7311 0 0.75730

HSDT 0.89858 0.71035 0.75747

Present 1.06786 0.84942 0.94325

5 GSDT 1.11580 0.87920 0.91180

HSDT 1.06297 0.84129 0.90951

Present 1.18373 0.78677 1.59724Metallic

GSDT 1.2534 0.8356 1.607

HSDT 1.18373 0.78677 1.59724$ Zenkour[2006] # generated Reddy[2000]

96

Table 5.2: Effects of volume fraction exponent on the dimensionlessstresses for FG square plates subjected to sinusoidal loading(alb =10)

Plate - -(Jx - - -

N Theories ~ 'l:xy 'txz 'I: yz

Present 1.98915 1.31035 0.70557 0.23778 0.23778

Ceramic GSDT$ 1.9955 1.3121 0.7065 0.2462 0.2132

HSDT# 1.98915 1.31035 0.70557 0.23778 0.19051

Present 2.1227 1.30962 0.6678 0.22557 0.2256

0.2 GSDT - - - - -HSDT# 2.12671 1.30958 0.66757 0.22532 0.18045

Present 2.60436 1.47175 0.66709 0.23909 0.23869

0.5 GSDT - - - - -

HSDT# 2.61051 1.47147 0.66668 0.23817 0.19071

Present 3.07011 1.48935 0.61395 0.22705 0.23919

1 GSDT 3.087 1.4894 0.611 0.2462 0.2622

HSDT# 3.08501 1.4898 0.61111 0.23817 0.19071

Present 3.58089 1.3968 0.54947 0.22705 0.22719

2 GSDT 3.6094 1.3954 0.5441 0.2265 0.2763

HSDT# 3.60664 1.39575 0.54434 0.22568 0.1807

Present 4.19547 1.1087 0.57811 0.21792 0.21813

5 GSDT 4.2488 1.1029 0.5755 0.2017 0.2429

HSDT# 4.24293 1.10539 0.57368 0.21609 0.17307

Present 1.98915 1.31035 0.70557 0.23778 0.23778

Metallic GSDT 1.9955 1.3121 0.7065 0.2462 0.2132

HSDT# 1.98915 1.31035 0.70557 0.23778 0.19051

$ Zenkour[2006] # generated(Reddy[2000])

97

0.5

0.4

0.3

..c: 0.2N~.... 0.1~

1:::1:al.< 0.000("I

'" -0.1'"~

12 -0.2("I

:8Eo-<

-0.3

-0.4

-0.50.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Volume fraction Vr

Figure 5.1 Variation of volume fraction Vr through plate thickness fordifferent values of power law index n

a/h=4

1.6

1.4

I~ 1.21:::10

'.Q("I 1.0~

l;:~

"0-; 0.8

1:::10

';;;0.61:::1

~

8:a0.4=0

Z0.2

0.00 2 3 4

a/h=10

5

Power law index n

Figure 5.2 Non dimensionalised central deflection w versus power lawindex n for FG square plate under sinusoidal load

98

6-,--------------_--------.

5I~

=o~ 4~!ij"0"0 3Q;

.!!l~

=o 2.;;;=e:a 1=oZ

o

2

0.4

Metal

n=5

0.6 0.8 1.0 1.2

Ia/h = 10 I

1.4 1.6 1.8 2.0

Aspect ratio alb

Figure 5.3 Non dimensionalised central deflection w versus aspect ratio(a/b) for FG plates.

I Sinusoidal load . alb =1 I

n=2

/'

20

n=1 n=0.5

30

n=0.2

40 50

Side to thickness ratio alh

Figure 5.4 Non dimesionalised central deflections w as a function of sideto thickness ratio (a/h) for FG square plates.

99

Non dimensionalised normal stresscr,

0.5

0.4

0.3

~ 0.2

20.1C':l=:e 0.0Q

QCJ

'" -0.1~]

-0.2CJ:.c;r:-o

-0.3

-0.4

-0.5-2 o

5

2

1 15

4

a/h=20

la/b=1ln=2

6

Figure 5.5 Variation of in-plane longitudinal stresses a", through thethickness of FG plates for different values of side to thicknessratios (a/h)

0.5

0.4

0.3

..cl 0.2NCI>...

0.1C':lCl:a... 0.0QQCJ

'" -0.1~]

-0.2CJ

~-0.3

-0.4

-0.5-2

)a/h=101n=2

-1 a 1 2 3 4

Non dimesionalised normal stresscry

Figure 5.6 Variation of in-plane longitudinal stresses ay through thethickness of FG plates for different aspect ratios (alb)

100

20

Ia/b=1 In=2

0.4

0.3

-0.4

0.5 .,--"""""'=----.:::::---~-__.:_--.,._-----,\\

~ 0.2~

~ 0.1=:eo 0.0 +-------------+-\--\--\-\-------.4o(J

~ -0.1~

~ -0.2:aE-o -0.3

-4 -3 -2 -1 o 2

Non dimensionalised in-plane shear stresstxy

-Figure 5.7 Variation of in-plane shear stress "txy through the thickness of

FG plate for different side to thickness ratios (a/h)

0.5...".".=--------------------,

0.4

0.34 2

a/b=0.5

0.4

Ia/h=10 In=2

0.30.20.1-0.5 -F---,------.----,----,.-----,----,r---.---.-------l

0.0

-0.4

~ 0.2

~c: 0.1=:eo 0.0 +---1---+---------J''-------~-----jo(J

~ -0.1~

~ -0.2:aE-o -0.3

Non dimensionalised Transverse shear stress1xz

-Figure 5.8 Variation of transverse shear stress "txz through the thickness

of FG plates for different aspect ratios (a/b)

101

0.4

a/b=1

r/h=10\n=2-0.4

~ 0.2~­C'I

=~<:) 0.0 +---------1'-------r:---."L------I<:)(J

'"'"~~ -0.2:a/-<

0.00 0.05 0.10 0.15 0.20 0.25

Non dimensionalised transverse shear stresst y'l

-Figure 5.9 Variation of transverse shear stress 1"yz through the thickness of

FG plate for different aspect ratios (a/b)

0.5

0.4 la/b=110.3 n=2

..= 0.2N~- 0.1C'I=:a1..0 0.0<:)<:)(J

'" -0.1'"~.§

-0.2(J

:a/-<

-0.3a/h=2.5

-0.4

-0.5-4 -2 0 2 4 6 8 10

Non dimensionalised in-plane displacement u

-Figure 5.10 Variation of in-plane displacement u through the thickness of

an FG plate for different values of side to thickness ratios(a/h)

102

Figure 5.5 to 5.9 shows the distribution of nonnal stresses and shear stresses

through the thickness of the FG plates. The volume fraction exponent is taken as 2 for

these results. The transverse shear stresses Txz and Tyz are computed using 3D

-equilibrium equations. The through thickness distribution of the shear stress Txz and

TYl are not parabolic and the stresses increase as the aspect ratio decreases. It can be

noticed that maximum value occurs not at the plate centre as in the homogeneous case.

It can be seen from the Figures 5.5 and 5.6 that the nonnal stresses crx and cry are

-compressive throughout the plate up to z ~ 0.149 and then they become tensile.

Maximum values of these stresses as well as in-plane shear stress Txy occur at the top

and bottom surfaces of the plate. Figure 5.10 shows the variation of in-plane

displacement u and through the thickness of plate.

5.3 Buckling analysis -Theoretical Formulation using ZSDT

The buckling analysis of FG plates is studied in this section using ZSDT.

Displacement fields, stress resultants and constitutive equations used are as given above

in section 5.2. The differential equations governing the buckling behaviour are derived

using principle of virtual work and we have:

h/2

f ff( a)5Ex + cr/3Ey + 'txy 8yxy + 't yz8y yz +t xz8yxz )dxdydz-h/2

(5.15)

103

In buckling analysis the plate is subjected to in-plane compressive loads

NxandNy only and the external load being zero.

The differential equations governing the buckling behaviour using ZSDT are as

follows

aN aN_x_+ xy =0

Ox By

aN aN__xy~+_y_=o

Ox By

a2M a2M a2M a2 a2 a2x 2 xy y _ N w 0 -N w 0 2N w0--+ + - --+ --+ --Ox2 OxBy 8y2 x Ox2 y 8y2 xy 8xay

5.3.1 Solution Approach

(5.16a)

(5.16b)

(5.16c)

(5.16d)

(5.16e)

The buckling loads are obtained by solving the eigen value problem

numerically. In order to get Navier solutions of the governing differential equations the

variables can be expressed as given in Eqn (5.12).For buckling analysis the external

load is set to zero and the applied loads are the in-plane forces. Eqn. (5.12) reduce the

governing equations Eqn. (5.16) to the following form

([CJ-A[GJ){~} =0

(5.17)

104

where [C] refers to the flexural stiffness and [G] geometric stiffness matrices which is a

null matrix except G33and Ato the corresponding buckling parameter.

5.3.2 Numerical Results and Discussions

The study has been focused on the buckling behaviour of FG plate based on

ZSDT displacement model. Here some representative results of the Navier solution

obtained for a simply supported square plate under uniaxial and biaxial in-plane loading

is presented. The material properties used in the present study are

Metal (Aluminum AI) (Eb) = 70 GPa

Ceramic ( Alumina Ah03) (E t) = 380 GPa

Poisson's ratio (v) = 0.3

In Table 5.3 comparisons of results of uniaxial buckling loads with

HSDT2D [Matsunaga,2008] are presented and the results are found to be in good

agreement with it. The results are also computed using Reddy's model. An

Unconstrained Shear Deformation Theory (UNSDT) of Leung [2003] is also extended

for the buckling analysis. The buckling equations derived are given in Appendix B. The

non dimensionalised critical buckling load used for presenting results is as follows.

Ncr =N IE, (5. I8)

Table 5.4 and Table 5.5 contain the non dimensionalised uniaxial and biaxial

critical buckling loads of FG square plate using three higher order structural models

ZSDT, HSDT and UNSOY. The results show the effect of volume fraction Vf on

buckling loads for various side to thickness ratios. The non dimensionalised critical

buckling load used for presenting results is as follows.

(5.19)

105

It can be observed from the results presented in these that as the plate become

more and more metallic the dimensionless buckling load decreases. It can also be

noticed that dimensionless buckling load for a fully metallic and ceramic plate are the

same. This is due to the fact that in these two cases the plate is fully homogeneous and

the buckling loads do not depend on the modulus of elasticity

Figure 5.11 shows the variation of the buckling loads with side to thickness

ratio (a/h). From this figure it is evident that response of FG plates is intermediate to

that of ceramic and metallic homogeneous plate. Figure 5.12 and 5.13 shows that CLPT

not only results in buckling solutions independent of the side to thickness ratios but also

greatly overpredicts the buckling loads in low side to thickness ratios. An excellent

agreement is maintained between the present obtained results and results obtained by

HSDT model. From Figure 5.14 it is clear that as the metal content increases the non

dimensionalised buckling load decreases. For the uniaxial in-plane stresses, critical

buckling stresses of FG plates are plotted with respect to aspect ratio in Figures 5.15 to

Figure 5.18. The curves are plotted for several values of the power law index. The

critical buckling for smaller values of the aspect ratio appear under first displacement

mode, while higher displacement modes will appear for larger values of the aspect

ratio. It is noted that the lowest displacement mode gives the critical buckling load for

thin plates. However, an interesting feature is that the critical buckling load for thick

plates occurs at higher displacement modes.

5.4 Concluding Remarks

In this chapter, a Zeroth order Shear Deformation Theory (ZSDT) for

functionally graded plates has been derived. ZSDT incorporates the effect of transverse

106

shear defonnation through the direct use of shear stress resultants Qx and Qy. This

theory incorporates the non linear variation of in-plane displacements. The effective

material properties at any points in the plate are obtained using a simple power law

distribution.

The static response and buckling behaviour of FG plates are studied using this

theory. The stresses and displacement responses of the plate have been analysed under

sinusoidal loading. Dimensionless critical buckling loads for uniaxial and biaxial

loadings of simply supported FG plates have been obtained by ZSDT and UNSDT.

Validity of the present theories are demonstrated by comparison with solutions

available in the literature The ZSDT model provides results in excellent agreement

compared with the other higher order models. UNSDT slightly overpredicts the results.

Parametric studies are perfonned for varying ceramic volume fraction, volume fraction

profiles, side to thickness ratio and plate aspect ratio. It is seen that basic response of

the plate that correspond to properties intennediate to that of the metal and ceramic,

necessarily lie between ceramic and metal. Thus, the gradients in material properties

play an important role in detennining the response of the FG plates. The solutions

presented here provide bench mark results, which can be used to assess the adequacy of

different plate theories and also to compare results obtained by other approximate

methods such as finite element method.

107

Table 5.3 Comparison of dimensionless uniaxial buckling loads forsquare FG plates using different theories.

Matsunaga [2008] generated(Reddy[2000])

PlateN Theories alh = 2 a/h=5 alh = 10

Present 0.3138 0.1206 0.03440

0HSDT2D# 0.3581 0.114 0.03381

HSDT* 0.3560 0.1158 0.0340

UNSDT 0.3946 0.1199 0.0344

Present 0.2513 0.081 0.0228

0.5HSDT2D 0.2482 0.7571 0.02214

HSDT 0.2535 0.0699 0.0195

UNSDT 0.2755 0.0752 0.0209

Present 0.1755 0.06 0.0172

1HSDT2D 0.1938 0.05826 0.01698

HSDT 0.1872 0.0587 0.017

UNSDT 0.2241 0.0699 0.0203

Present 0.0881 0.035 0.0111

4HSDT2D 0.112 0.03721 0.01131

HSDT 0.0967 0.0355 0.0109

UNSDT 0.1339 0.0475 0.0144

Present 0.0823 0.0311 0.0099

10HSDT2D 0.08904 0.03183 0.009905

HSDT 0.0635 0.028 0.0095

UNSDT 0.103 0.0368 0.0113# #

108

Table 5.4 Effect of volume fraction exponent on the dimensionlessuniaxial buckling \Qads for FG square plates with differentalb ratios.

Platea/h Theories Ceramic n=::O.5 1 2 3 4 5 metal

Present 1-3888 1.5907 1.3114 0.9860 0.7907 0.6929 O.6{}13 1.389

1.3100

0.4269

2 HSDT* 1.3100 1.6728 1.1796 1.1474 1.0613 0.7938 0.7574~~:l--------+-~~-+---~---l----1

UNSDT 0.4241 1.8243 \.2390 0.9543 1.3911 1.1473 1.1255

Present 3.3345 3.2033 2.8032 2.4238 2.2602 2. i 954 2.1988 3.3345

5 HSDT 3.2027 3.2195 2.7437 2.4760 2.3933 2.2820 2.2872

UNSDT 3.3174 3.4623 3.2653 3.2883 3.2549 3.0557 3.0393

3.2027

3.3173

Present 3.8101 3.6123 3.2058 2.8631 2.7821 2.7931 2.8481 3.8101

3.7653

3.8044

3.9133

10 HSDT 3.7653 3.5918 3.1860 2.8260 2.7568 2.8125 2.8677

UNSDT 3.8044 3.8505 3.7866 3.7946 3.7834 3.7125 3.7065f-----+--~--~+_~.---------+---+----1-------\-----1-----+--------/

Present 3.9133 3.6999 3.2935 2.9626 2.9066 2.9416 3.0132

15 HSOT 3.8921 3.6704 3.2842 2.9020 2.8367 2.9395 3.0096

UNSDT 3.9106 3.932\ 3.9022 3.906 3.9008 3.8669 3.8640

3.8921

3.9106

Present 3.9508 3.7315 3.3254 2.9991 2.9528 2.9974 3.0756 3.9508

20 HSDT 3.9386 3.6987 3.3200 2.9296 2.8658 2.9867 3.0627

UNSDT 3.9492 3.9615 3.9444 3.9466 3.9436 3.9240 3.9223

3.9386

3.9492

3.978

3.9902

3.9838

3.9725

3.9773

3.9876

3.9845

3.9872

3.9797

3.9833

3.96~4

3.9605

3.9674

Present 3.9684 3.7464 3.3403 3.0163 2.9747 3.0240 3.1054._---+----+--------+-----+-----+------+-----\25 HSDT 3.9605 3.7120 3.3369 2.9426 2.8795 3.0091 3.0879

UNSDT 3.9674 3.9753 3.9642 3.9656 3.9637 3.9510 3.9499

Present 3.9780 3.7545 3.3485 3.0257 2.9868 3.0386 3.1219

30 HSDT 3.9725 3.7193 3.3461 2.9497 2.8869 3.0214 3.1018

UNSDT 3.9773 3.9828 3.9751 3.9761 3.9747 3.9659 3.9651

Present 3.9838 3.7594 3.3535 3.0314 2.9941 3.0475 3.1318

35 HSDT 3.9797 3.7236 3.3517 2.9539 2.8915 3.0288 3.11Q3

1JNSDT 3.9833 3.9874 3.9817 3.9824 3.9814 3.9749 3.9743----+-~~--+-------+-~~---\-~----+---+-

Present 3.9876 3.7626 3.3567 3.0352 2.9988 3.0533 3.1384;--------------- ------+-------------+-----+-------1-~______+~--+___-__________I

40 HSDT 3.9845 3.7265 3.3553 2.9567 2.89443,0337 3.1157

UNSDT 3.9872 3.9903 3.986 3.9865 3.9857 3.9807 3.9803

Present 3.9902 3.7648 3.3589 3.0377 3.0021 3.0573 3.1428

45 HSDT 3.9877 3.7285 3.3578 2.9586 2.8964 3.0371 3.1195

UNSDT 3.9899 3.9923 3.9889 3.9893 3.9887 3.9848 3.9844

3.9877

3.9899

Present 3.9920 3.7663 3.3605 3.0396 3.0045 3.0601 3.1461 3.992

50 "SDT 3.9900 3.7299 3.3596 2.9600 2.8979 3.0394 3.1222

UNSDT 3.9918 3.9938 3.9910 3.9914 3.9909 3.9876 3.9874

3.9900

3.9918II generated(Reddy(2000J)

109

Table 5.5 Effect of volume fraction exponent on the dimensionlessbiaxial buckling loads for FG square plates with different alhratios

Platealb Tbeories Ceramic n=O.5 1 2 3 4 5 metal

Present 0.8860 0.8967 0.7493 0.5865 0.4911 0.4423 0.4269 0.8862 HSDT* 0.7879 0.9340 0.6994 0.6653 0.6244 0.4975 0.4798 0.7879

UNSDT 0.3817 1.7312 0.8374 0.85 \ 8 0.8237 0.6891 0.6785 0.3842

Present 1.6673 1.6017 1.40 \6 1.2119 1.1301 1.0977 1.0994 1.6673

5 HSDT 1.60 \4 1.6098 1.3719 1.2380 1.1966 1.1410 1.1436 1.6014

UNSDT 1.6587 1.9253 1.6327 1.6442 1.6275 1.5279 1.5197 1.6587

Present I.9051 1.8061 1.6029 1.4315 1.3910 1.3965 1.4241 1.9051

10 HSOT 1.8826 1.7959 1.5930 1.413 1.3784 1.4063 1.4338 1.8826

UNSDT 1.9022 1.9661 1.8933 1.8973 1.8917 1.8563 1.8532 1.9022

Present 1.9567 1.8499 \ .6468 1.4813 1.4533 1.4708 1.5066 1.9567

15 HSDT 1.9461 \ .8352 1.6421 1.451 1.4184 1.4697 1.5048 1.9461

UNSDT 1.9553 1.9808 1.9511 1.953 1.9504 1.9334 1.932 1.9553

Present 1.9754 1.8658 1.6627 1.4995 1.4764 1.4987 1.5378 1.975420 HSDT 1.9693 \ .8494 1.6600 1.4648 1.4329 1.4934 1.5314 1.9693

UNSDT 1.9746 1.9877 1.9722 1.9733 1.9718 1.962 1.9612 1.9746

Present 1.9842 1.8732 1.6702 1.5082 1.4874 1.512 1.5527 1.984225 HSDT 1.9802 1.8560 1.6684 1.4713 1.4397 1.5045 1.544 1.9802

UNSOT 1.9837 1.99\4 1.9821 1.9828 1.9818 1.9755 1.9750 1.9837

Present 1.9890 1.8772 1.6742 1.5129 1.4934 1.5193 1.5609 1.989030 HSOT 1.9862 1.8596 1.6730 1.4748 1.4435 1.5107 1.5509 1.9862

UNSDT 1.9886 \ .9937 1.9875 1.9880 1.9874 1.9829 1.9825 1.9886

Present 1.9919 1.8797 1.6767 1.5\57 1.4970 1.5237 1.5659 1.9919

35 HSDT 1.9899 1.8618 1.6758 \ .4770 1.4457 1.5144 1.5551 1.9899

UNSDT 1.9916 1.9952 1.9908 1.9912 1.9907 1.9874 1.9871 1.9916

Present 1.9938 1.8813 1.6783 1.5176 1.4994 1.5266 1.5692 1.9938

40 HSDT 1.9922 1.8632 1.6777 1.4784 1.4472 1.5169 1.5579 1.9922

UNSDT 1.9936 1.9952 1.9930 1.9933 1.9929 1.9904 1.9901 1.9936

Present 1.9951 1.8824 1.6794 1.5189 1.5011 \ .5286 1.5714 1.9951

4S HSDT 1.9939 1.8642 1.6789 1.4793 1.4482 1.5185 1.5598 1.9939

UNSDT 1.9949 1.9962 \ .9944 1.9947 1.9944 1.9924 1.9922 1.9949

Present 1.9960 1.8832 1.6802 1.5198 \ .5022 1.5301 1.5730 1.996050 HSDT 1.9950 1.8649 1.6798 1.4800 1.4489 1.5197 1.5611 1.9950

UNSDT 1.9959 1.9969 1.9955 1.9957 1.9954 1.9938 1.9937 1.9959# generated(Reddy[20001)

110

4.5

4.0

\Z::3.5'i

.s~ ;).0:g] 2.5

~~

2.0.~on=.§ 1.5

Q

1.0

0.50 1Q

n==Q n-.:oO.5

20

Side to tn\ckness ratio alb

50

Figure 5.11 Dimensionless uniaxial buckling load -versus side tothickness ratio (a/h) for different values of p~wer law indexfoJ' FG square plates using ZSDT (Present Model)

!Z'C3.Q

"2~ 2.5CD

=:§t.l 2.0:I~

'"~] 1.5

~a 1.0

a(l.S

o 10

-~ZSDT

HSDT---,,-. CLPT

20 3() 40 50

Sioe \1) thickness r:tiio a/lt

Figure 5.12 Dimensionless buckling load versus side to thickness ratio(a/h) for FG s\}\\ar~pl~tes under uniaxialloilding

\\1

1.8CLPT n=1

----- .. -.. _------- .. _-- .--._--------------- .-._--------- -1.6

Izl;

'C 1.4 n=5~

.!2~

= 1.2:fau=..c 1.0tiltilQi

-=0 0.8';;]

=Qi

E -ZSDTis 0.6 HSDT

-------CLPT0.4

0 10 20 30 40 50

Side to thickness ratio alh

Figure 5.13 Dimensionless buckling loads versus side to thickness ratio(a/h) for FG square plate under biaxial loading

4.0-r------------------~

3.5

Izb3.0

~o';n 2.5

=:faa: 2.0..c

~ 1.5

-=,Stil= 1.08:a

0.5

a/h70 a/h=15

a/h=10

a/h=5

a/h==2

1086420.0 +---.----,....---.---r---.-----r---.--,.----.-----j

oPower law index n

Figure 5.14 Dimensionless buckling load versus power law index fordifferent side to thickness ratios for FG square plates (ZSDT-

Present model))

112

4.0

3.5Ia/h=21 -.-n=1

-O-n=4

Iz~ 3.0-~-n=7

'C -\1- n=10ell

.52 2.5OJ)

=~C.J 2.0:=..c'"'" 1.5Qi

-=0.~

= 1.0Qi

8'5

0.5

0.00.5 1.0 1.5 2.0 2.5 3.0

Aspect ratio alb

Figure 5.15 Dimensionless buckling load versus aspect ratio (a/b) fordifferent power law index for FG plates using ZSDT (Presentmodel)(a/h=2)

5

Ia/h=51 -.-n=1-o-n=4

~ 4 -~-n=7

IZ -\1- n=10'Cell

.52OJ) 3=~

C.J:=..c'" 2'"Qi

-=.52'"=Qi

8 1'5

00.5 1.0 1.5 2.0 2.5 3.0

Aspect ratio alb

Figure 5.16 Dimensionless buckling load versus aspect ratio (a/b) fordifferent power law index for FG plates using ZSDT (presentmodel)(a/h=5)

113

6

Ia/h=10J -·-n=1-0-n=4t; 5-t::.-n=7lz-'\7- n=10"0

ell.sa~ 4

:fa(J

::s,Q

'" 3~"a0.<;;I:41

S 2Q

10.5 HI 1.5 2.0 2.5 3.0

Aspect ratio alb

Figure 5.17 Dimensionless buckling load versus asped ratio (alb) fordifferent power law index fot FG plates using ZSDT (presentmodel) (a/h=lO)

3.02.5

-a-n=1-O-n=4-t::.-n=7-\7- 0=10

6...----nr- ~ ____,

Aspect ratio alb

Figure 5.18 Dimensionless buckling load versus asped ratio (alb) fordifferent power law index for FG plates using ZSDT (presentmodel)(aJh=::20)

114