Stochastic perturbation-based finite element for buckling statistics of FGM plates with uncertain material properties in thermal environments

Composite Structures 108 (2014) 823–833

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Stochastic perturbation-based finite element for buckling statistics of FGM plates with uncertain material properties in thermal environments Mohammad Talha a,⇑, B.N. Singh b a b

International Institute for Aerospace Engineering and Management, Jain University, Bangalore 562 112, India Department of Aerospace Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721 302, India

a r t i c l e

i n f o

Article history: Available online 18 October 2013 Keywords: Functionally graded material Improved structural kinematics Stochastic finite element Second order statistics Random material properties First-order perturbation technique

a b s t r a c t In the present study, stochastic perturbation-based finite element for buckling statistics of functionally graded plates (FGM) with uncertain material properties in thermal environments is investigated. The effective material properties of the gradient plates are assumed to be temperature-dependent and vary in the thickness direction only according to the power-law distribution of the volume fractions of the constituents. An improved structural kinematics proposed earlier by author’s is employed which accounts parabolic variations for the transverse shear strains with stress free boundary conditions at the top and bottom faces of the plate. An efficient C0 stochastic finite element based on the first-order perturbation technique (FOPT) is proposed, and the fundamental equations are obtained using variational approach. Convergence and comparison studies have been performed to describe the efficiency of the present formulation. The numerical results are highlighted with different system parameters and boundary conditions. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction An advanced functionally graded composite materials offer great ability and excellent performance in a wide range of engineering applications, such as in Mechanical, Aerospace and Marine engineering. These materials are composed of two or more phases with continuously varying composition distribution in the preferred directions. The advantages of using these materials are that they can maintain their structural integrity at high temperature gradient environments [1,2]. One of the most challenging application of FGM is in the skin panels of supersonic and hypersonic light vehicles, which is required to sustain the severe thermo-mechanical loadings [3]. The effects of uncertainty are of viable considerations in analysis and design of the composite structures. The uncertainties in the structure are associated at various stages like constituent mechanical properties, manufacturing and fabrication processes, etc. The overall and complete control of these parameters are not economically feasible at each stage, thus the uncertainties become inherent in nature. Therefore, these must be considered and accounted in the analysis of the structural response, to assure the reliability of the structure during its operating life [4]. Consequently, a considerable interest has been developed in the recent years towards ⇑ Corresponding author. Tel.: +91 80 27577231 (work); fax: +91 8027577233. E-mail addresses: [email protected], [email protected] (M. Talha). 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.10.013

the safe and excellent design of composite structures by considering the uncertainties in material properties. Therefore, the study of stochastic elastic stability characteristics of these components under mechanical, thermal and thermo-mechanical loadings are relevant to ensure the safe and reliable design within the operating range. Reddy [5] presented finite element model (FEM) for static and dynamic analysis of the FGM plates by applying third order shear deformation theory. Samsam and Eslami [6] presented buckling analysis of thick rectangular FGM plates under mechanical and thermal loads using the higher order shear deformation plate theory (HSDT). Lanhe [7] derived the equilibrium and stability equation of a moderately thick rectangular FGM plate based on first order plate theory in thermal environment. Ganapathi et al. [8] evaluated the critical buckling of simply supported skew FGM plate subjected to Thermo-mechanical compressive loads using FSDT kinematics. Abrate [9] analysed free vibrations, buckling, and static deflections of the FGM using the Classical laminated plate theory (CLPT), the first order shear deformation theory (FSDT), and the HSDT kinematics. Ibrahim et al. [3] provided a nonlinear FEM to study the random response of functionally graded material panels subject to combined thermal and random acoustic loads. The governing equations have been derived using first-order shear deformable theory with von-Karman nonlinearity. Kitipornchai et al. [10] presented the stochastic free vibration of functionally graded laminates in the frame work of third-order shear deformation theory subjected to

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a temperature change with general boundary conditions. Secondorder statistics of vibration frequencies have been obtained by employing mean-centered first-order perturbation technique. Yang et al. [11] investigated the effect of randomness on the elastic buckling of rectangular graded plates, subjected to uniform in-plane edge compressions resting on an elastic foundation. The formulations are accomplished using two parameter Pasternak model. Singh et al. [12] studied the stochastic post buckling behaviour of laminated composite plates resting on linear elastic foundation with random system properties. The formulation is based on higher order shear deformation plate theory in von Karman sense. Shaker et al. [18] developed the stochastic finite element model to study the sensitivity of the static response and free vibration of the FGM plates with uncertainties in the material properties. They used the first and second order reliability method in combination with nine-nodded isoparametric Lagrangian element to investigate sensitivity of the fundamental frequency of the FGM plates with material uncertainties. They had taken the ceramic and metal Young’s modulus, their densities and ceramic volume fraction as a basic random variables. Yang et al. [14] studied the stochastic bending behaviour of moderately thick FGM plates. The formulation have been based on third order shear deformation theory in conjunction with first order perturbation technique. They have obtained the second order statistics, i.e., mean and variance of the flexural deflection of the plates with different set of the boundary conditions. Lal et al. [15] presented the stochastic bending analysis of laminated composite plates resting on elastic foundation with uncertain system parameters. They incorporated the transverse shear effects using higher order shear deformation theory. Guo et al. [19] investigated the large-amplitude multi-mode random response of thin shallow shells with rectangular platform at elevated temperatures using FEM approach. Salim et al. [4] studied the effect of uncertain material parameter on initial buckling load of rectangular, specially orthotropic, composite laminates based on classical laminate theory. Singh et al. [23] studied the effects of random material properties on buckling of composite plates by using the classical, first-order and higher-order shear deformation theories. Pandit et al. [27] proposed an improved higher-order zigzag theory for vibration of soft core sandwich plates with random material properties. Jagtap et al. [24] presented the stochastic nonlinear free vibration response of graded materials plate resting on two parameter Pasternak foundation with Winkler cubic nonlinearity with temperature independent and dependent material properties. Lal et al. [13] studied the consequences of sensitivity of randomness in system parameters of laminated composite plates subjected to uniform transverse pressure in thermal environments. Lal et al. [16,17] studied the second order postbukling statistics for FGM plates subjected to mechanical and thermal loadings with square and circular holes. The determination of an accurate structural behaviour of the graded material generally depends on the structural kinematics used to model the structure. Various structural kinematics have been reported in the literature to improve the structural response of FGMs and laminated composite plate/shell structures. The analysis of the advanced graded structures using the classical laminated plate theory may be inappropriate in these materials because it assumes that normal to the mid-plane remains normal during plate deflections. Since, FGM structures are flexible structure in which the volume fractions of two or more materials vary continuously in a preferred direction as a function of position. The inaccuracy occurs due to neglecting the effects of transverse shear and normal strains [25]. Due to continuous variation in material properties in a preferred direction the first and higher order shear deformation theory may be conveniently used in the

analysis. Recently, several theories have been proposed by the researchers to study the vibration, bending and dynamic response of graded material plates, notable among them are [22,28–39]. It is a well known fact, that the first-order shear deformation theory proposed by Mindlin [26] does not satisfy the parabolic variation of transverse shear strain in the thickness direction. Moreover, the solution accuracy of FSDT depends upon the shear correction factor. Generally, HSDT kinematics assumes the in-plane displacements to be a cubic expression of the thickness coordinate and the out-of-plane displacement to be constant. In the present study, the structural kinematics assumes the cubically varying in-plane displacements over the entire thickness, while the transverse displacement varies quadratically to achieve the accountability of normal strain and its derivative in calculation of transverse shear strains [21]. The development of an appropriate higher order structural kinematics for describing the realistic thermo-mechanical stability of the advanced functionally graded structures with uncertain material properties has been of high importance to the researchers in the recent years. As far as the authors’ are aware, large amount of work is available on deterministic analysis of functionally graded and composite plates in thermo-mechanical environment using various structural kinematics, such as [28,31–35,37,39]. However, no previous work have been reported in the open literature to accomplish the thermo-mechanical stability analysis of functionally graded material plates with uncertain material properties based on modified structural kinematics, which assures the cubically varying in-plane displacements and quadratically varying transverse displacement over the entire thickness. The details of this modified structural kinematics can be seen in Talha and Singh [21], which incorporate the effects of normal strain and its derivative in the computation of transverse shear strains. The material properties are graded continuously in the thickness direction only according to a simple power-law distribution in terms of the volume fractions of the constituent. The governing equations are derived using the variational approach. A C0 continuous finite element based on the first-order perturbation technique with 13 degrees of freedom per node is proposed. The numerical results have been presented with various system parameters, such as effect of material properties, volume fraction index, buckling characteristics and its dispersion with respect to various random variables. The accomplished results can be treated as a benchmark for further advanced research for advanced and hybrid composite structures. 2. Theoretical formulation 2.1. Functionally graded material A functionally graded material plate with dimensions a, b and total thickness h is considered in the study as shown in Fig. 1. The top surface (z = +h/2) of the plate is assumed to be ceramic rich, whereas the bottom surface (z = h/2) is metal rich. The

a b

x Ceramic rich

h y

Metal rich

z Fig. 1. Configuration and coordinate system of FGM plate.

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mathematical formulations are based with linear elastic material properties by considering small displacements and strains. The elastic material properties, like Young’s modulus E, Poisson’s ratio m, thermal expansion coefficient a, are defined as the effective material properties P, of the functionally graded plate. The effective properties are assumed to be vary in the transverse coordinates according to power-law distribution in terms of the volume fractions of the constituents. The material properties P can be expressed as a function of temperature, Touloukian [40]

P ¼ P0 ðP1 T 1 þ 1 þ P1 T 1 þ P 2 T 2 þ P 3 T 3 Þ

ð1Þ

where P0, P1, P1, P2, and P3 are the coefficients of temperature T (K) and are associated with the constituent materials. The volume fractions of the constituent follows a simple powerlaw as

V c ðzÞ ¼

 n 2z þ h 2h

ð2Þ

where n is the non-negative volume fraction exponent that describes the material variation in the thickness coordinate as shown in Fig. 2. It is assumed that the effective Young’s modulus E and thermal expansion coefficient a are temperature dependent, whereas, thermal conductivity coefficient j are independent of the temperature. Poisson’s ratio m is assumed to be constant because it hardly depends on the temperature changes. From Eqs. (1) and (2), the effective material properties with two constituents for graded plates can be expressed as:

 n 2z þ h Eðz; TÞ ¼ ½Ec ðTÞ  Em ðTÞ þ Em ðTÞ 2h  n 2z þ h aðz; TÞ ¼ ½ac ðTÞ  am ðTÞ þ am ðTÞ 2h  n 2z þ h jðzÞ ¼ ðjc  jm Þ þ jm 2h

d dz



jðzÞ

 dT ¼0 dz

2.2. Displacement field and strain In the present study an elegant and modified structural kinematics proposed earlier by authors’ have been used, and the details are given in [21], which is briefly explained here. The in-plane dis, v  for the plate  and the transverse displacement w placements u are expressed as:

( )!   2 1 @wz 4 @w h @nz 3  ¼ u þ zwx þ z  þ z  2 wx þ þ u 2 @x @x 4 @x 3h ( )!   2 1 @wz 4 @w h @nz 2 3 þ z  2 wy þ v ¼ v þ zwy þ z  þ 2 @y @y 4 @y 3h 2

 ¼ w þ zwz þ z2 nz w ð5Þ where wx and wy are the rotations of normal to the mid plane about the y and x-axis respectively. The functions nx, ny, qx and qy are the higher-order terms in the Taylor series expansion defined in the mid-plane of the plate. The higher order terms are determined by applying stress free boundary condition (sxz = s4 = 0 and syz = s5 = 0) at the top and bottom faces of the plate. With the assumption of stress free boundary condition the displacement field Eq. (5) is modified as:

 ¼ u0 þ f1 ðzÞwx þ f2 ðzÞax þ f3 ðzÞbx þ f4 ðzÞhx u ð3Þ

v ¼ v 0 þ f1 ðzÞwy þ f2 ðzÞay þ f3 ðzÞby þ f4 ðzÞhy

ð6Þ

 ¼ w0 þ f5 ðzÞwz þ f6 ðzÞaz w

where subscripts ‘c’ and ‘m’ represents the ceramic and metal constituent of the graded plate, respectively. The temperature field is applied in the thickness direction only by solving a steady-state heat transfer equation, and can be represented as:



assumed to be constant in the XY plane of the plate. It can be seen from Eqs. (1) and (3) that Et, Eb, at and ab are all functions of position and temperature dependent.

ð4Þ

This equation (Eq. (4)) is solved by applying boundary condition of T = Tt at z = +h/2 and T = Tb at z = h/2. The temperature field is

where f1(z) = C1z  C2z3, f2(z) = C3z2, f3(z) = C4z3, f4(z) = C5z3, f5 = C1z, f6 = C1z2, C1 = 1, C2 = C4 = 4/3h2, C3 = 1/2, C5 = 1/3 and nz = az. To ensure C0 continuous finite element formulation the out of plane derivatives are assumed as independent degrees of freedom. The field variables (basic unknowns) are interpreted as u, v, w, wx, wy, wz, ax, ay, az, bx, by, hx and hy for structural deformation. The linear strain vector corresponding to the displacement field is represented as:

fegT ¼ fe1 e2 e3 e4 e5 e6 gT    @ v @ w  @ v @ w  @u  @w  @u  @ v T @u ¼ þ þ þ @x @y @z @z @y @z @x @x @y

ð7Þ

The linear strain vector terms are expressed as:

e1 ¼ e01 þ zk11 þ z2 k21 þ z3 k31 e2 ¼ e02 þ zk12 þ z2 k22 þ z3 k32 e3 ¼ e03

ð8Þ

e4 ¼ e04 þ zk14 þ z2 k24 e5 ¼ e05 þ zk15 þ z2 k25 e6 ¼ e06 þ zk16 þ z2 k26 þ z3 k36

The linear strain vector in the above equation can also be written in the following form in terms of mid-plane strain vectors as:

feg ¼ ½Tfeg

ð9Þ

3 where fg ¼ f      k1 3 3 k2 k6 g and [T] is the function of thickness coordinate. The superscript ‘‘0’’, ‘‘1’’, ‘‘2–3’’ in fg are membrane, curvature and higher order strain terms, respectively. 0 1

Fig. 2. Variation of the volume fractions Vc through the thickness [20].

0 2

0 3

0 4

0 5

0 6

1 k1

1 k2

1 k3

1 k4

1 k5

1 k6

2 k1

2 k2

2 k4

2 k5

2 k6

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2.3. Constitutive relations

where

The linear thermo-mechanical constitutive relations are expressed as:



8 rxx 9 > > > > > > > > > > r > yy > > > > > > > > < rzz > =

2

Q 11 Q 12 Q 13

0

0

6 0 6 Q 12 Q 22 Q 32 0 6 6 0 6 Q 31 Q 32 Q 33 0 ¼6 6 > > r 0 0 0 0 Q > 6 yz > 44 > > > > 6 > > > > 6 0 > > r 0 0 0 Q 55 > > 4 xz > > > > : ; rxy 0 0 0 0 0

1 308 9 8 9 exx > 1> > > > > > > > > B C > > > > 7 > > > C > > > eyy > 0 7B 1> > > > > > > > > C > > > > 7B > > > > C > > > > 7B < < = = B C 0 7B ezz 0 C 7B 7B> c >  > >aðz; TÞC C 0 7B> 0> > yz > > > > > C > > > 7B> > > > > C > > > > 7 >0> > cxz > > > > C 0 5B > > > > @> A > > > > > :c > ; : ; Q 66 0 xy 0

2 @w Ag

@u @x

0

@v @x

0

6 ¼40

@w @y

0

@u @y

0

@v @y

@w @y

@w @x

@u @y

@u @x

@v @y

@v @x

@w @x

@w @y

@u @x

@u @y

@v @x

@v @x

Q 11 ¼ Q 22 ¼ Q 33 ¼ Eðz;TÞ . 2ð1þmÞ

Q 44 ¼ Q 55 ¼ Q 66 ¼

Eðz;TÞð1m2 Þ ð13m2 2m3 Þ

Q 12 ¼ Q 13 ¼ Q 23 ¼

Eðz;TÞmð1þmÞ ð13m2 2m3 Þ

The modulus E, thermal expansion

coefficient a and the elastic coefficients Qij vary through the plate thickness according to the Eqs. (2) and (3). 2.4. Energy calculation The strain energy of the FGM plate is given by,

U¼

Z 1 fegTi frgi dV 2 v

ð11Þ

Thermal force resultants, thermal moment resultants, and higher order thermal moment resultants due to temperature rise DT may be defined as:

8 9 8 9 8 9 8 9 > > < N xx > = Z h=2 > < rxx > = < Mxx > = Z h=2 > < rxx > = ryy dz; fMT g ¼ Myy ¼ ryy zdz fN T g ¼ N yy ¼ > > > > > > h=2 > h=2 > : ; : : ; : Nxy rxy ; M xy rxy ;

8 9 8 9 > < Pxx > = Z h=2 > < rxx > = ryy z3 dz fP g ¼ Pyy ¼ > > > h=2 > : ; : Pxy rxy ; T

ð12Þ

As the plate is exposed to thermal environment, and subsequently produces in-plane stress resultants (NxxNyy and Nxy). Therefore, the work done by the in-plane forces produced due to out of plane displacement ‘w’, and is represented as:

U th ¼

¼

1 2

1 2

Z

fhg g ¼

Z 

w;x w;y

A

T 

Nxx

Nxy

Nxy

Nyy

ðeÞ 

w;x



w;y

ð13Þ

dA

7 5

ð15Þ

h

iT

¼ ½T g ½Bg fKg

ð16Þ

where [Tg] and [Bg] are the function of thickness coordinate and differential operator, respectively. By using the above equation, the work done due to in-plane load may be expressed as:

U in-plane ¼

1 2

Z

fhg gT ½Sfhg gdV

ð17Þ

V

and the stress matrix [S] may be expressed in terms of the in-plane stress components as:

2

rx sxy 6 6 sxy ry 6

6 6 0 ½S ¼ 6 6 0 6 6 6 0 4 0

0

0

0

0

0

0

0

0

rx sxy sxy ry

0

0

0

0

0

0

0

0

0

3

7 0 7 7 7 0 7 7 0 7 7 7 sxy 7 5

ð18Þ

rx sxy ry

3. Stochastic modelling – Perturbation approach In order to incorporate the randomness in structural analysis an additional mathematical operation is needed. Therefore, to administer the randomness in the material properties, first order perturbation technique has been used in the present analysis, which is briefly discussed below. Let, the following set or random field variables (time invariant) are assumed as:

Rðx; yÞ ¼ fRR1 ðx; yÞ; RR2 ðx; yÞ; RR3 ðx; yÞ; . . . ; RRl ðx; yÞg

ð19Þ

where Ri ’s represent random parameters, i.e., Young’s modulus, mass density, Poisson’s ratio, etc. The generalised governing equation of an structural analysis problem can be represented as:

½K Rij fdRi g ¼ fPi g where

½Nxx ðw;x Þ2 þ N yy ðw;y Þ2 þ 2Nxy ðw;x Þðw;y ÞdA

A

3

and

ð10Þ where

0

@x

½K Rij 

ð20Þ

is the system matrix,

fdRi g

is the response vector and {Pi}

is the forcing vector. Here ½K Rij  are known functions of a random parameters (RP), Rðx; yÞ. The prime objective is to find the statistics of fdRi g when the statistics of the RPs, Rðx; yÞ are known. The random parameters can be expressed as: Random parameter = Mean + Zero mean random parameter, i.e.

2.5. Work done due to in-plane mechanical load

RPR ¼ RP d þ RP r

In order to obtain the work done due to in-plane load, it is necessary to obtain the geometric stiffness matrix. The generalised geometric strain vector may be written as:

By applying the above equation the random parameters, can be defined as:

2

 1 @w 2 2 @x

þ

 1 @u 2 2 @x

þ

 1 @v 2 2 @x

3

7 6 7 6    7 6 1 @w 2 1 @u 2 1 @v 2 þ þ feG g ¼ 6 7 2 @x 2 @x 2 @x 7 6 4       5 @w @w @u @u @v @v þ þ @x @y @x @y @x @y ¼

1 ½Ag ½hg  2

RRl ¼ Rdl þ Rrl ;

ð21Þ

K Rij ¼ K dij þ K rij ; dRi ¼ ddi þ dri

ð22Þ

r

ð14Þ

The random component (RP ) is very small in most of the engineering applications compared to the mean value (RPd). Therefore, the above equation can be written as:

RRl ¼ Rdl þ fRrl ;

K Rij ¼ K dij þ fK rij ; dRi ¼ ddi þ fdri

ð23Þ

Here f is a scaling parameter which is small in magnitude. By applying Taylor series expansion the system matrix ½K Rij  and response vector fdRi g can be expanded as [41]:

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K rij ¼

X @K dij

Rrl R

and dri ¼

@Rl

l

X @dd r i Rl R l @Rl

ð24Þ

Here we neglect second and higher order terms, since earlier studies [4] have shown that first order perturbation approximation is sufficient for the desired accuracy. Substituting Eq. (23) in Eq. (21) we get:

½K dij þ fK rij fddi þ fdri g ¼ fPi g

þ

f½K dij fdri g

þ

f½K rij fddi g

2

þf

½K rij fdri g

¼ fP i g

ð26Þ

¼ fPi g

ð27Þ

1

First-order (f terms, R system of n equations)

½K dij fdri g þ ½K rij fddi g ¼ f0g

ð28Þ

The above Eq. (27) is the deterministic equation which can be solved using conventional iterative method, whereas the Eq. (28) is the random part which can be solved by applying first order perturbation technique. The Taylor series approximations for fdri g and ½K rij  can now be substituted in Eq. (24). Hence,

" d # X d @dd r @K ij R d ½K ij  iR Rl þ R l fdi g ¼ f0g R @R @R l l l

ð29Þ

RRl

The variables are assumed to be independent of each other. Therefore, equating the coefficients of Rrl in the above equation, for each lyields:

( ½K dij 

@ddi

)

@RRl

" þ

@K dij

#

@RRl

fddi g ¼ f0g l ¼ 1; 2; . . .

ð30Þ

each l to find the only unknown

@ddi @RRl

. The random response can be

found by combining these results, like:

dRi ¼ ddi þ fdri

ð31Þ

Thus variance and hence the Standard Deviation (SD) of the structural response have been obtained by applying:

¼

X l

@ddi

1u–v –w–wx –wy –wz –ax –ay –az –bx –by –hx –hy –0; at x ¼ a; y ¼ b: The plate coordinate system is shown in Fig. 3.

In the present analysis a nine nodded C0 isoparametric finite element with 13 degrees of freedom per node is applied which is based on the improved structural kinematics. The domain is discretized into a set of finite elements. Over each of the elements, the displacement vector and the element geometry are expressed as:

fKg ¼

nn X ni fKgi ;

x¼

i¼1

nn nn X X ni xi ; y ¼ ni yi i¼1

ð33Þ

i¼1

where ni is the interpolation function (shape function) for the ith node, {K}i is the vector of unknown displacements for the ith node, nn is the number of nodes per element and xi, yi are Cartesian coordinate of the ith node. 4.3. Governing equation The governing equations for buckling analysis of functionally graded plates have been derived using the variational approach, and is written as: 

ð½K   kcr ½K G Þfdg ¼ 0

ð34Þ



where fqg; ½K  and [KG] is the global displacement vector, stiffness matrices and geometric stiffness matrix, respectively. The Eq. (34) can also be expressed in terms of the eigenvalue problem at a particular displacement as:

½K  fdg ¼ kcr fdg

ð35Þ

where [K ] is the total elastic stiffness matrix, and kcr is the buckling load parameter, which is the lowest eigenvalue. The first order partial derivatives of eigenvalue k0crj and eigenvector fd0j g with respect to the random variables are used to obtain the standard deviation. The Eq. (28) can be decoupled, and is expressed as: T 

fk;ri g ¼ fd0i g ½K ;rij fd0i g

!2

f Rrl @RRl

ð32Þ

All edges simply supported (SSSS), all edges clamped (CCCC), and Clamped-Free (CFCF) type of boundary conditions have been used in the present analysis: Simply supported: (SSSS) u0 = w0 = wy = ax = az = by = hx = 0, at x = 0 and a. v0 = w0 = wx = ay = az = bx = hy = 0, at y = 0 and b. Clamped: (CCCC)

u0 ¼ v 0 ¼ w0 ¼ wx ¼ wy ¼ wz ¼ ax ¼ ay ¼ az ¼ bx ¼ by ¼ hx ¼ hy

n X fd0j g ½K ;rij fd0j g ¼ fd0j g ; k0i  k0j j¼1

j–1

ð37Þ

y

4.1. Support conditions

at x ¼ 0; a and y ¼ 0; b:

ð36Þ

T 

fd;ri g

4. Solution technique

¼ 0;

at x ¼ 0; y ¼ 0:

⁄

fddi g can be found by solving Eq. (27). Eq. (30) can now be solved for

VarðdRi Þ

¼ 0;

4.2. Finite element implementation

By substituting the Eq. (26) into Eq. (20) and equating the terms of consistent power of the small parameter f, the zeroth and the first order equations have been obtained which can be used to solve the stochastic structural problems. The following zeroth and first-order equations have been obtained as follows. Zeroth order (f0 terms, one system of n equations)

½K dij fddi g

u ¼ v ¼ w ¼ wx ¼ wy ¼ wz ¼ ax ¼ ay ¼ az ¼ bx ¼ by ¼ hx ¼ hy

ð25Þ

The above equation can be expanded as:

½K dij fddi g

Clamped-Free: (CFCF)

y=b

x=a

x=0

x y=0 Fig. 3. Plate coordinate system.

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The variance of the eigenvalues and eigenvectors can be expressed as:

Varðki Þ ¼

R X R X k;ji k;k i ¼ 1; 2; . . . ; n i Cov ðRj ; Rk Þ

ð38Þ

j¼1 k¼1

Varðdiði1 Þ Þ ¼

R X R X d;jiði1 Þ d;k i; i1 ¼ 1; 2; . . . ; n iði1 Þ Cov ðRj ; Rk Þ

ð39Þ

considered as random. The MCS technique generates the random samples according to the known probability distributions of the basic random variables [42]. However, the present approach does not impose any constraint on the distribution of the material properties, which is an advantage over the MCS. It is observed from Fig. 5 that the present results obtained using FOPT technique are very close to MCS results. This demonstrates the good accuracy of the present formulation with the ranges considered for SD/mean.

j¼1 k¼1

where Cov ðRj ; Rk Þ is the covariance between Rj and Rk . The SD can be obtained by the square root of the variance.

5.1. Parametric study

Example 2. In this example, the SD/mean of buckling load is compared with First order perturbation technique and independent Monte Carlo approach. The results are accomplished for uniaxial compression of all edges simply supported. The thickness ratio (a/h) and volume fraction index (n) are assumed to be 10 and 2, respectively, and only one material property Ec has been

The variation in the buckling load parameter under uniaxial edge compression with simply supported boundary conditions for Al/ZrO2 FGM plate (n = 0.5, 1.0, 2.0, 5.0) is presented in Figs. 6– 9 The material properties as given in [21], have been used in the analysis, which are as follows: Em = 70  109 N/m2, qm = 2702 kg/ m3 for Aluminium, and Ec = 151  109 N/m2, qc = 3000 kg/m3 for Zirconia. The Poisson’s ratio m is assumed to be constant as 0.3. The buckling load parameter is defined as:  ¼ xa2 =ðp2 ðEc h3 =ð12ð1  m2 ÞÞÞÞ. It is found that the scattering of x the material properties of Zirconia on the buckling load decreases as volume fraction index n increases, whereas this trend is opposite for Aluminium when only one random variable Ec changing at a time as depicted in Figs. 6 and 7. Moreover, the influence of the scattering in Ec has a significant role on the buckling strength due to its mean value is higher than that of Em. In general it is observed that the dispersion in the plate response is the least sensitive to random changes in Poisson’s ratio. Fig. 9 presents the buckling load sensitivity of the graded plates under uniaxial edge compression when all of the random material properties Ec, Em, mc, mm varying simultaneously. It is observed that the scatter in the buckling load is the highest for n = 5 and the least for n = 0.5 when all the material properties changes simultaneously. In all cases it is found that the dispersion in the buckling load increases with the increase of randomness in the material properties. The buckling load sensitivity for the uniaxially compressed graded plates with varying standard deviations of material properties (Ec, Em, mc, mm) is examined in Figs. 10–13 for various support conditions. The thickness ratio (a/h) is assumed as 10, and volume fraction index (n) is taken as 2. The effect of the material properties of ceramic (Ec) on the buckling load dispersion is higher for clamped-free (CFCF) support condition, and is lower for fully clamped (CCCC) plate as compared to other conditions, as shown in Fig. 10. Whereas, in the case of metal Em the buckling load

Fig. 4. Comparison in the dispersion of the buckling load of clamped FGM square plates with all random variables changing simultaneously.

Fig. 5. Variation in the buckling load of simply supported FGM square plates when only Ec is varying.

5. Numerical examples The accuracy and efficacy of the probabilistic method is demonstrated by solving two examples, and comparing the results with those available in the literature, and also with an independent Monte Carlo simulation (MCS). Example 1. In this example, the buckling load dispersion of the FGM plates made of (Al/Zirconia) is compared with Yang et al. [11] under biaxial edge compression with all random material properties varying simultaneously. All edges clamped boundary condition have been considered in the analysis. The thickness ratio (a/h) and volume fraction index (n) are assumed to be 10 and 2, respectively. The comparison of the buckling load dispersion of the graded plates is shown in Fig. 4. Yang et al. [11] used the FSDT kinematics and FOPT technique to examine the stochastic characteristics of the buckling load. Moreover, in the present endeavour a modified HSDT kinematics along with FOPT algorithm is used. The difference in the result shows the importance of quadratic variation of transverse displacement in order to achieve the accountability of normal strain and its derivative in calculation of transverse shear strains. The influence of the scattering in the material properties on the buckling load parameter has been examined by assuming the coefficient of variation (SD/mean) to vary from 0% to 20%.

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Fig. 6. Variation in the buckling load of simply supported FGM square plates when only Ec is varying.

Fig. 9. Variation in the buckling load of simply supported FGM square plates when all variable is varying.

Fig. 7. Variation in the buckling load of simply supported FGM square plates when only Em is varying.

Fig. 10. Variation in the buckling load of simply supported FGM square plates when Ec is varying.

Fig. 8. Variation in the buckling load of simply supported FGM square plates when only (mc, mm) is varying.

Fig. 11. Variation in the buckling load of simply supported FGM square plates when Em is varying.

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Fig. 12. Variation in the buckling load of simply supported FGM square plates when m is varying.

Fig. 14. Variation in the buckling load of simply supported FGM square plates when Ec is varying.

Fig. 13. Variation in the buckling load of simply supported FGM square plates when all variable is varying.

Fig. 15. Variation in the buckling load of simply supported FGM square plates when Em is varying.

dispersion is noticed in reverse fashion as depicted in Fig. 11. The effects in randomness in Poisson’s ratios are found to be insignificant. Fig. 13 displays the scattering of dimensionless buckling load of graded square plates against the SD/Mean when all random variable changes simultaneously, against different type of boundary conditions. It is found that the fully clamped plate shows the higher sensitivity than simply supported and clamped free boundary condition. The effect of plate thickness ratio on the dispersion of buckling load of graded Al/ZrO2 FGM plate is shown in Figs. 14–17. These figures display the SD/mean of the buckling load of uniaxially compressed square FGM plate with different values of a/h (= 10, 15, 20, 30) with fully clamped edges. The buckling load dispersion of a thicker plate is greater than of a thinner plate. This indicates that the randomness in basic variables has a predominant effect on the sensitivity of the buckling load as the plate thickness increases. The effect of thermal environment on the dispersion of buckling load of graded Al/ZrO2 FGM plate is also investigated and is depicted in Figs. 18–21 The SD/mean of the buckling load of uniaxially compressed square FGM plate have been obtained with different values of volume fraction index n (= 0.5, 2, 5, 10). The FGM plate is made of Zirconium oxide (ZrO2) and Titanium alloy (Ti–6Al–4V)

Fig. 16. Variation in the buckling load of simply supported FGM square plates when m is varying.

with all simply supported edges. The mass density and thermal conductivity are: q = 3000 kg/m3, j = 1.80 W/mK for ZrO2; q = 4429 kg/m3, j = 7.82 W/mK for Ti–6Al–4V. Poisson’s ratio m is

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Fig. 17. Variation in the buckling load of simply supported FGM square plates when all variables is varying.

Fig. 20. Variation in the buckling load of simply supported FGM square plates when m is varying in thermal environment.

Fig. 18. Variation in the buckling load of simply supported FGM square plates when Ec is varying in thermal environment.

Fig. 21. Variation in the buckling load of simply supported FGM square plates when all variables is varying in thermal environment.

Fig. 19. Variation in the buckling load of simply supported FGM square plates when Em is varying in thermal environment.

assumed to be constant and is equal to 0.3. Young’s modulus and coefficient of thermal expansion are assumed to be temperature dependent. The non-dimensional buckling load parameter is

Fig. 22. Variation in the buckling load of simply supported FGM square plates when Ec is varying in thermal environment. 3

 ¼ xa2 =p2 ðEc h =12ð1  m2 ÞÞ. The top surface of the assumed as x plate is maintained at 400 K, and the bottom surface is at 300 K. The buckling load dispersion of higher volume fraction index n is

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lesser than of a lower n. This shows that the randomness in basic variables has a predominant effect on the sensitivity of the buckling load at lower volume fraction index n. The effect in randomness in Poisson’s ratio is found to be very small. The effect of thermal environment on the dispersion of buckling load of graded Al/ZrO2 FGM plate is also examined with fully clamped edges as shown in Figs. 22–25. The FGM plate is comprised of Zirconium oxide (ZrO2) and Titanium alloy (Ti–6Al–4V). It is found that the SD/Mean of the buckling load of uniaxially compressed square plate has a significant change with different values of volume fraction index, (n = 0.5, 2, 5, 10). The buckling load dispersion of plate is higher for larger value of n when all the random variables changes simultaneously as compared to lower value of n as shown in Fig. 25. This demonstrates that the randomness in basic variables has a predominant effect on the sensitivity of the buckling load when all edges are clamped. Fig. 23. Variation in the buckling load of simply supported FGM square plates when Em is varying in thermal environment.

6. Conclusions Second order statistics of functionally graded material plates with uncertain material properties is investigated using stochastic perturbation-based finite element. First-order perturbation technique and independent Monte-Carlo simulation have been used to determine the mean value and variance of the buckling load parameter with temperature-dependent material properties. Convergence and comparison studies have been presented to show the efficiency of the present formulation, and compared the results with those available in the limited literature. Numerical results for different thickness ratios, aspect ratios, volume fraction indices, and the temperature rise along with different combinations of the boundary conditions have been presented which shows the effectiveness of the present formulation. The following specific conclusions have been drawn from this study.

Fig. 24. Variation in the buckling load of simply supported FGM square plates when m is varying in thermal environment.

 The Comparison of covariance of buckling strength shows the importance of the present modified structural kinematics.  The randomness in basic variables has a predominant effect on the sensitivity of the buckling load for all edges clamped boundary conditions.  The buckling load dispersion of higher volume fraction index n is less than the lower values of n in thermal environment.  The buckling load dispersion of graded plates is higher for thick plates than that of thin plates.  All edges simply supported plates have the least effects in the dispersion of buckling load dispersion, than that of the fully clamped plates.  The dispersion in buckling load is least affected with scatter in Poisson’s ratio.  The temperature increment makes the buckling load dispersion more sensitive when all random material properties change simultaneously.  The effect of uncertain system parameters and thermal expansion coefficients on the scattering of buckling load dispersion subjected to thermal environment is an important problem for safe, economical and reliable design.

Acknowledgments

Fig. 25. Variation in the buckling load of simply supported FGM square plates when all random variable is varying in thermal environment.

The first author is thankful to the All India Council for Technical Education (A.I.C.T.E.), New Delhi, Government of India, for providing the financial assistance to carry out this work (F. No: 1-10/RID/ NDF-PG (19)/2008-09).

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References [1] Koizumi M. FGM activities in Japan. Composites Part B 1997;28:1–4. [2] Koizumi M. The concept of FGM. In: Proceedings of the second international symposium on FGM; 1993. p. 3–10. [3] Ibrahim HH, Yoo HH, Tawfik M, Lee KS. Thermo-acoustic random response of temperature-dependent functionally graded material panels. Comput Mech 2010;46:377–86. [4] Salim S, Iyengar NGR, Yadav D. Buckling of laminated plates with random material characteristics. Appl Compos Mater 1998;5:1–9. [5] Reddy JN. Analysis of functionally graded plates. Int J Numer Meth Eng 2000;47:634–84. [6] Samsam SBA, Eslami MR. Buckling of thick functionally graded plates under mechanical and thermal loads. Compos Struct 2007;78:433–9. [7] Lanhe W. Thermal buckling of a simply supported moderately thick rectangular FGM plate. Compos Struct 2004;64:211–8. [8] Ganapathi M, Prakash T, Sundararanjan N. Influence of functionally graded material on buckling of skew plates under mechanical loads. ASCE J Eng Mech 2006;132:902–5. [9] Abrate S. Free vibration buckling and static deflections of functionally graded plates. Compos Sci Technol 2006;66:2383–94. [10] Kitipornchai S, Yang J, Liew KM. Random vibration of the functionally graded laminates in thermal environments. Comput Meth Appl Mech Engrgy 2006;195:1075–95. [11] Yang J, Liew KM, Kitipornchai S. Second-order statistics of the elastic buckling of functionally graded rectangular plates. Compos Sci Technol 2005;65:1165–75. [12] Singh BN, Lal A, Kumar R. Post buckling response of laminated composite plate on elastic foundation with random system properties. Commun Non Sci Numer Simul 2009;14:284–300. [13] Lal A, Singh BN, Kumar R. Stochastic nonlinear bending response of laminated composite plates with system randomness under lateral pressure and thermal loading. Arch Appl Mech 2011;81(6):727–43. [14] Yang J, Liew KM, Kitipornchai S. Stochastic analysis of compositionally graded plates with system randomness under static loading. Int J Mech Sci 2005;47:1519–41. [15] Lal A, Singh BN, Kumar R. Static response of laminated composite plates resting on elastic foundation with uncertain system properties. J Reinf Plast Compos 2007;26:807–29. [16] Lal A, Singh BN, Shegokar NL. FEM model for stochastic mechanical and thermal postbuckling response of functionally graded material plates applied to panels with circular and square holes having material randomness. Int J Mech Sci 2012;62:18–33. [17] Lal A, Jagtap KR, Singh BN. Postbuckling response of functionally graded material plate subjected to mechanical an thermal loadings with random material properties. Appl Math Model 2013;37:2900–20. [18] Shaker A, Abdelrahman W, Tawfik M, Sadek E. Stochastic finite element analysis of the free vibration of functionally graded material plates. Comput Mech 2008;41:707–14. [19] Guo X, Lee Y, Mei C. Non-linear random response of laminated composite shallow shells using finite element modal method. Int J Numer Meth Eng 2006;67:1467–89. [20] Mohammad Talha, Singh BN. Thermo-mechanical buckling analysis of finite element modeled functionally graded ceramic-metal plates. Int J Appl Mech 2011;3(4):867–80. [21] Mohammad Talha, Singh BN. Static response and free vibration analysis of FGM plates using higher order shear deformation theory. Appl Math Model 2010;34:3991–4011.

833

[22] Mohammad Talha, Singh BN. Large amplitude free flexural vibration analysis of shear deformable FGM plates using nonlinear finite element method. Finite Elem Anal Des 2011;47:394–401. [23] Singh BN, Iyengar NGR, Yadav D. Effects of random material properties on buckling of composite plates. ASCE J Eng Mech 2001;127(9):873–9. [24] Jagtap KR, Lal A, Singh BN. Stochastic nonlinear free vibration analysis of elastically supported functionally graded materials plate with system randomness in thermal environment. Compos Struct 2011;93:3185–99. [25] Matasunaga H. Analysis of functionally graded plates. Compos Struct 2008;82:499–512. [26] Mindlin RD. Influence of rotatory inertia and shear on flexural vibrations of isotropic elastic plates. J Appl Mech 1951;73:31–8. [27] Pandit MK, Singh BN, Sheikh AH. Stochastic free vibration response of soft core sandwich plates using an improved higher-order zigzag theory. ASCE J Aero Eng 2010;23(1):14–23. [28] Benachoura A, Tahara HD, Atmanea HA, Tounsia A. A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient. Compos Part B: Eng 2011;42:1386–94. [29] Mantari JL, Oktem AS, Soares CG. Bending response of functionally graded plates by using a new higher order shear deformation theory. Compos Struct 2012;94:714–23. [30] Thai HT, Choi DH. A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation. Compos Part B: Eng 2012;43:2335–47. [31] Mantari JL, Soares CG. Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory. Compos Struct 2012;94:1991–2000. [32] Thai HT, Kim SE. A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates. Compos Struct 2013;96:165–73. [33] Mantari JL, Soares CG. Generalized hybrid quasi-3D shear deformation theory for the static analysis of advanced composite plates. Compos Struct 2012;94:2561–75. [34] Thai HT, Vo TP. A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates. Appl Math Model 2013;37:3269–81. [35] Mantari JL, Soares CG. A novel higher-order shear deformation theory with stretching effect for functionally graded plates. Compos Part B: Eng 2013;45:268–81. [36] Thai HT, Kim SE. A simple quasi-3D sinusoidal shear deformation theory for functionally graded plates. Compos Struct 2013;99:172–80. [37] Thai HT, Choi DH. Finite element formulation of various four unknown shear deformation theories for functionally graded plates. Finite Elem Anal Des 2013;75:50–61. [38] Mantari JL, Soares CG. Finite element formulation of a generalized higher order shear deformation theory for advanced composite plates. Compos Struct 2013;96:545–53. [39] Neeraj Grover, Singh BN, Maiti DK. New nonpolynomial shear-deformation theories for structural behavior of laminated-composite and sandwich plates. AIAA J 2013;51:1861–71. [40] Touloukian YS. Thermophysical properties of high temperature solid materials. New York: MacMillan; 1967. [41] Kleiber M, Hien TD. The stochastic finite element method: basic perturbation technique and computer implementation. John Wiley & Sons, Ltd.; 1992. [42] Li J, Chen JB. Probability density evaluation method for dynamic response analysis of structures with uncertain parameter. Comput Mech 2004;34:400–9.