Stochastic dynamic characteristics of FGM beams with random material properties

Composite Structures 133 (2015) 585–594

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Composite Structures journal homepage: www.elsevier.com/locate/compstruct

Stochastic dynamic characteristics of FGM beams with random material properties Yalan Xu a,⇑, Yu Qian a, Jianjun Chen a, Gangbing Song b a b

School of Electronic & Mechanical Engineering, Xidian University, Xi’an 710071, PR China Department of Mechanical Engineering, University of Houston, Houston, TX 77004, USA

a r t i c l e

i n f o

Article history: Available online 21 July 2015 Keywords: Functionally graded material Stochastic dynamic characteristics Random factor method Finite element method Random material properties

a b s t r a c t The random factor method is developed for the stochastic dynamic characteristics analysis of beams that are made of functionally graded materials (FGM) with random constituent material properties in this paper. The effective material properties of FGM beams are assumed to vary continuously through different directions according to the power law distribution. In the proposed method, the randomness of the dynamic characteristics is explicitly expressed by random factors of constituent material parameters, and the statistics (means and variances) of modal parameters can be analytically computed by the statistics of random inputs with little computational efforts. Compared with the first-order perturbation technique and the Monte-Carlo simulation method, the proposed method is illustrated and verified by a FGM cantilever beam with material variation through the thickness or axial direction, and the contributions of random material properties to the dispersion of dynamic characteristics for the structure are systematically investigated. The numerical results also show the dominant effect of constituent volume fraction index on the dispersion of dynamic characteristics. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Functionally graded materials (FGM) have attracted extensive attention in engineering fields such as civil infrastructure, aerospace, micromechanics and nuclear power, since the concept of FGMs was initially proposed by Japanese scientists in 1984 [1]. FGMs are microscopically heterogeneous composite materials fabricated from two or more phases of material constituents with continuously and smoothly gradual changing material properties. In FGMs, the continuity of material constituents reduces material property mismatch, and, as a result, leads to significant improvement in damage resistance and mechanical durability, especially in thermally challenging environment [2]. A great deal of research have already been reported on the static and thermo-elastic analysis [3–8], buckling analysis [9], crack [10], contact problems [11], and dynamic analysis [12–15] of structures made of FGMs over the past decades. As a well-known fact, however, the manufacturing and fabrication process of FGMs are extremely complex. Under the current technical conditions, it is often impossible for FGMs to fully represent the desired gradient to the exact specification, since a large number of parameters are involved. As a result, the uncertainties ⇑ Corresponding author. Tel.: +86 29 88204489 (work). E-mail address: [email protected] (Y. Xu). http://dx.doi.org/10.1016/j.compstruct.2015.07.057 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.

in effective material properties may be inherent, which will be reflected on the uncertainties in structural stiffness, and consequently the micro and macro mechanical behavior [16,17]. In fact, considerable randomness can be observed from sample to sample. In addition, the change in environmental temperature has been shown to be a major contribution to the uncertainties in constituent material properties. To well define the original problem and enable a better understanding of the actual behavior in FGMs, it makes sense that the inherent uncertainties in system parameters should be accurately characterized, and incorporated in the analysis of FGM structures. Compared to the deterministic analysis of FGM structures, the works concerning random FGM structures are limited [17–28]. Ferrante and Graham-Brady [17] investigated the effects of random constituent volume fractions and the porosity on the response of a functionally graded metal/ceramic plate subjected to constant thermal boundary conditions by stochastic simulation. Yang et al. [18,19] discussed the stochastic elastic buckling and static analysis of FGM plates with parameter randomness and obtained the second-order response statistics (mean and variance) of the flexural deflection of plates. Giunta et al. [20] addressed the free vibration analysis of composite thin and relatively thin plates with random material properties and geometrical parameters through the Monte Carlo method. Kitipornchai et al. [21] dealt with the random free vibration of functionally graded laminates with random

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material properties and the mean-centered first-order perturbation technique was adopted to obtain the second-order statistics of vibration frequencies. Onkar et al. [22] investigated the buckling analysis and natural frequencies of laminated plates with random material properties using stochastic finite elements. Shaker et al. [23] studied the sensitivity of the static response and free vibration of FGM Plates with uncertain material properties. Lal et al. [24–26] investigated the natural frequencies and nonlinear free vibration response of FGM beams and laminated composite plates resting on elastic foundation with uncertain system properties in the thermal environment. Jagtap et al. [27] studied the stochastic nonlinear free vibration analysis of functionally graded material plates resting on elastic foundation having cubic nonlinearity with random system properties. Shegokar and Lal [28] dealt with the stochastic nonlinear bending response of functionally graded material beams with surface bonded piezoelectric layers subjected to thermo-electromechanical loadings with uncertain material properties. Based on an improved structural kinetics, Mohammad and Singh [29] investigated the buckling statistics of functionally graded plates with uncertain material properties in thermal environments. In the stochastic analysis of FGM structures, the first-order perturbation technique (FOPT) has been extensively used [17–19, 21–29]. Compared to the Monte Carlo simulation (MCS) method [20] which is the most accurate but computationally expensive and extremely time-consuming, the FOPT has been widely used for its tractability and time-saving computation. In the FOPT, however, small variability in parameters is required, and the first-order derivatives of dynamic modal parameters with respect to the random variables should be evaluated. In addition, the means of random functions are calculated only by the first-order derivatives and means of the corresponding variables, without considering the covariance between variables since the secondorder derivatives are omitted in the FOPT. The main idea of random factor method (RFM) developed by Chen and his co-workers [30–32] has been used to analyze the dynamic characteristics and response of truss structures with random parameters. In the RFM, the statistical characteristics of modal parameters can be explicitly expressed by those of random parameters, and, as a result, analytically evaluated with less computational efforts. In the literature, all the studies are on the assumption that the randomness is considered to be the same for the parameters of all elements. In the FGMs made of two or more phases, however, the randomness of material parameters of different constituents is not the same. Compared with FGM plates and shells, studies on FGM beams, especially those with material properties varying through different directions, are relatively less [33,34]. Based on the higher-order shear deformation theory, in this paper, the random factor method (RFM) is extended to the stochastic dynamic characteristics analysis of FGM beams with either thickness or axial material graduation, and the effects of volume fraction index and random constituent material properties on the dynamic characteristics of FGM beams through the thickness or axial direction are investigated. 2. Theoretic formulation 2.1. Material properties A FGM rectangular beam composed of ceramics and metal having the length l, width b, thickness h with the co-ordinate system (oxyz) having the origin o is shown in Fig. 1. It is assumed that the effective material properties of the beam vary continuously through the thickness or axial direction according to the power law distribution [33].

Fig. 1. Configuration of FGM beam.

For the FGM beam with material property variation through its thickness, the effective material properties of the beam can be expressed as

PðzÞ ¼ ðPT  P B ÞV T þ PB  n z 1 h h V T ðzÞ ¼ þ ;  6z6 ; h 2 2 2

06n<1

ð1Þ

where, P denotes the effective material property, and PT ; PB represent the properties of the top surface and bottom surface material, respectively. The parameter V T and the power exponent n represent the volume fraction of the top surface material and the volume fraction index, respectively. For the FGM beam with the axial property variation, similarly, the effective material property of the beam can be expressed as

PðxÞ ¼ ðPL  PR ÞV L þ PR  xn V L ðxÞ ¼ 1  ; 0 6 x 6 l; l

06n<1

ð2Þ

where, respectively PL ; P R represent the properties of the left side and right side material. V L represents the volume fraction of the left side material. 2.2. Displacement and strain field Based on the third-order shear deformation theory (TSDT) by Reddy, the axial and transverse displacement components of an arbitrary point within the beam along x and z directions can be written as [33]

  @w0 ðx; tÞ 4 ; c1 ¼ 2 uðx; z; tÞ ¼ u0 ðx; tÞ þ z/x ðx; tÞ  c1 z3 /x ðx; tÞ þ @x 3h wðx; tÞ ¼ w0 ðx; tÞ

ð3Þ where, u0 ; w0 are the axial and transverse displacements of the mid-plane along x and z directions, respectively, /x is the cross sectional rotation. Assuming a small deformation, the strain–displacement relationship of the beam can be written as

@u @/ @/x @ 2 w0 ex ¼ 0 þ z x  c1 z3 þ @x @x @x @x2   @w @w cxz ¼ /x þ 0  3c1 z2 /x þ 0 @x @x

!

ð4Þ

The simplified stress–strain relations for the beam can be expressed as

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rx sxz



 ¼

Q 11 0

0 Q 55





ex Eðx; zÞ Eðx; zÞ ; Q 11 ¼ ; Q 55 ¼ 1  m2 2ð1 þ mÞ cxz

be 1:0. The randomness of the material parameters for the FGM beam, made of ceramics and metal, can be expressed as

ð5Þ where, Eðx; zÞ is the effective Young’s modulus, which is either a function of z for the FGM beam through the thickness or a function of x for the FGM beam through the axial direction. The Poisson’s ratio m is assumed to be a constant. 3. Finite element model The beam is divided into a number of finite elements of equal length le . The two-node shear deformable beam element is considered here. Therefore, the displacement vector at the mid-plane of each beam element can be expressed as

n

oT 0 ¼ ½Nfqe g w0 @w /x @x  T @we01 e1 e2 e2 @we02 e2 ; /x ; u0 ; w0 ; ; /x fqe g ¼ ue01 ; we01 ; @x @x

~m l ; Em ¼ E Em ~c l ; Ec ¼ E Ec where,

qm ¼ q~ m lqm

ð11Þ

qc ¼ q~ c lqc

lEm ; lqm ; lEc ; lqc are the mean values of elastic moduli and

mass densities of two material constituents, that is, metal and ~m ; q ~c ; q ~ m; E ~ c are the corresponding random ceramics, respectively. E factors, which have the mean values of 1.0, and the same standard deviations as those of the random material parameters Em ; qm ; Ec ; qc . Also, the COVs (coefficient of variation, the ratio of standard deviation to mean value) mE~m ; mq~ m ; mE~c ; mq~ C of the random factors are the same as those of the corresponding random variables. 4.2. Stochastic analysis of dynamic characteristics

u0

ð6Þ

where, ½N is the shape function matrix, in which Lagrange linear interpolation functions are chosen for the axial displacement and the rotation of cross section, while Hermite cubic interpolation functions are chosen for the transverse deflection and slope. fqe g is the nodal displacement vector for the beam element. The strain and kinetic energies of the beam element are given by

U¼ T¼

Z Z Z  Z Z Z

Ve

Ve

 1 1 ex rx þ cxz gxz dxdydz 2 2   u_ 1 _ gqðx; zÞ dxdydz f u_ w _ 2 w

ð7Þ

where, V e is the volume of the beam element, and qðx; zÞ is the mass density, which is either the function of z for the FGM beam through the thickness or the function of x for the FGM beam through the axial direction. Substituting Eqs. (3)–(6) into Eq. (7) and applying Hamilton’s principle result in the following finite element model for free vibration of the beam element

€ e g þ ½Ke fqe g ¼ 0 ½Me fq

ð8Þ

where, ½Ke  is the element stiffness matrix, and ½Me  is the element mass matrix, which can be obtained by

½Ke  ¼ b

Z Z 

½Me  ¼ b

Z Z

 Eðx; zÞ Eðx; zÞ T T ½B  ½B  þ  ½B  ½B e e c c dxdz 1  m2 2ð1  mÞ

ð9Þ

qðx; zÞ½D dxdz

0 c1 z3

z  c 1 z3

 @½N

@x 1  3c1 z2 ½N # z  c 1 z3 ½N 0

0 1  3c1 z2 0 c1 z3 1

~m ½Ke ðl Þ þ E ~c ½Ke ðl Þ ½Ke  ¼ E Em Ec

0

ð10Þ

The element matrices in Eq. (10) can be assembled to form the global matrices and obtain the finite element model of free vibration for the FGM beam. 4. Stochastic analysis of dynamic characteristics 4.1. Random factor method In the random factor method [30–32], any random variable can be the product of its mean and a random factor with mean taken to

ð12Þ

~ m ½Me ðlq Þ þ q ~ c ½Me ðlq Þ ½Me  ¼ q m c

where, ½K e ðlEm Þ; ½K e ðlEc Þ; ½Me ðlqm Þ; ½Me ðlqc Þ are the deterministic parts of the stiffness and mass matrices of the eth element, and the randomness of the stiffness and mass matrices of the beam element, ~m ; q ~c ; q ~m; E ~ c , only which are the functions of the random factors E depends on the random factors. The global stiffness and mass matrices of the structure can be expressed as ne X ~m ½Ke ðl Þ þ E ~c ½Ke ðl ÞÞ½Te  ½Te T ðE Em Ec

½K ¼ ½M ¼

e¼1 ne X e¼1

ð13Þ

~ m ½Me ðlq Þ þ q ~ c ½Me ðlq Þ Þ½Te  ½Te T ðq m c e

where, ½Te  is a transformation matrix that transforms the local coordinates of the eth element to the global coordinate. Using Rayleigh’s quotient, the jth modal frequency can be expressed as

x2j ¼

T

where

½Be  ¼ 1 ½Bc  ¼ ½ 0 " 1 ½D ¼ 0

With the random constituent material parameters, the stiffness and mass matrices of the eth element for the FGM beam can be divided into the product of the two parts, in which each part is the product of the random factors and the deterministic matrix related to each constituent material.

fuj gT ½Kfuj g T

fuj g ½Mfuj g

¼

~ j fuj gT ½Ku ~ j fuj g u fuj gT ½Kfuj g ¼ T ~ j fuj g ½Mu ~ j fuj g fuj gT ½Mfuj g u

ð14Þ

where, fuj g is the jth random mode shape, fuj g is its deterministic ~ j is the corresponding random factor. From Eq. (14), fuj g part and u can be approximated by its deterministic part fuj g in the computation of randomness of modal frequencies. Ignoring the randomness of the transform matrix for all elements and substituting Eq. (13) into Eq. (14) give the randomness of the jth modal frequency

x2j ¼

~m fu gT ½Kðl Þfu gþE ~c fu gT ½Kðl Þfu g E j j j j Em Ec T

ð15Þ

T

q~ m fuj g ½Mðlqm Þfuj gþq~ c fuj g ½Mðlqc Þfuj g

where, ½KðlEm Þ; ½KðlEc Þ; ½Mðlqm Þ; ½Mðlqc Þ are the deterministic parts of the global stiffness and mass matrices, which can be calculated by the following expressions ne ne h i X h  i



 X



 K lEm ¼ ½Te T Ke lEm ½Te ; Mðlqm Þ ¼ ½Te T Me lqm ½T e  e1 ¼1

e¼1

ne ne h  i X h  i

 X

 K lEc ¼ ½Te T Ke lEc ½Te ; M lqc ¼ ½Te T Me lqc ½T e  e¼1

e¼1

ð16Þ

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Y. Xu et al. / Composite Structures 133 (2015) 585–594

Using the following expression of the jth normalized mode shape

fuj g fuj g ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fuj gT ½Mfuj g

Using Eq. (18) and the algebra synthesis method, the mean value and variance of the jth normalized mode shape can be obtained by

ð17Þ

luj ¼

and substituting Eq. (13) into Eq. (17), the randomness of the jth normalized mode shape can be expressed as

1 fuj g ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fuj g   cj ~ qm mmj þ q~ c m

ð18Þ

 mj ; m  cj are the deterministic values related to two conwhere, m stituent materials, respectively, of the jth modal mass, and they can be obtained by ne h  i   X  mj ¼ uj T ½Te T Me lq ½Te fuj g m m e¼1 ne h  i X  cj ¼ fuj g m ½Te T Me lqc ½Te fuj g

ð19Þ

T

    12 12 1  m þm  c Þ2  2 m  2m m2q~ m þ m  2c m2q~ c þ 2 C q~ m q~ c m  m mq~ m m  m mq~ m 4ðm fuj g 2 ð25Þ



1  m þm  c Þ2 4ðm 2   12 12  2m m2q~ m þ m  2c m2q~ c þ 2 C q~ m q~ c m  m mq~ m m  m mq~ m 2 m fuj gT fuj g ð26Þ

 m þm  cÞ  r2uj ¼ ðm

Considering the environmental temperature brings little change in the mass density of each constituent material [35], the weight~ ^ is introduced for the clarity and simplicity of ing random factor q expressions as follows

q~^ ¼ W m q~ m þ W c q~ c ; W m þ W c ¼ 1

ð27Þ

e¼1

lq~^ ¼ 1; r2q~^ ¼ W 2m r2q~ m þ W 2c r2q~ c þ 2C q~ m q~ c W m W c rq~ m rq~ c ;

4.3. Statistics of dynamic characteristics Using the algebra synthesis method [30], the expressions for the mean values and variances of normally distributed modal parameters can be obtained. The mean value and variance of the jth eigen-value for the FGM beam can be explicitly expressed as







lx2j ¼ x2 ~ m mE ~m  C E ~m q ~m ~ m mq mm 1 þ mE



 1    2 2 2 1 þ x4 þ x2 ~m q ~m mq ~m ~m ~m þ mq ~ m mE ~ m  2C E mc 1 þ mE mm mE      2 2 þ x2 þ x4 ~c ðmE ~c  C E ~c q ~c Þ ~m þ mq ~c ~ c mq mc mE cc 1 þ mE  1    2 2 2 þ x2 1 þ x4 ~c q ~c mq ~c ~c ~c þ mq ~ c mE ~ c  2C E cm 1 þ mE cc mE   2 2 þ x4 ð20Þ ~ c þ mq ~m cm mE 





r2x2 ¼ x2 ~ m mE ~m  C E ~m q ~m ~ m mq mm 1 þ mE j



 4    2 2 2 x4 þ x2 ~m q ~ m mq ~m ~m ~ m þ mq ~ m mE ~ m  2C E mc 1 þ mE mm mE       2 2 þ x4 þ x2 ~c mE ~c  C E ~c q ~c ~m þ mq ~c ~ c mq mc mE cc 1 þ mE  4    2 2 2 x4 þ x2 ~c q ~ c mq ~c ~c ~c þ mq ~ c  2C E ~ c mE cm 1 þ mE cc mE   2 2 þ x4 ~ c þ mq ~m cm mE

 2pq x

p ¼ m; c; q ¼ m; c

r2xj ¼ lx2j

 12 1 4l2x2  2r2x2  j j 2

~ E

~ E

 2mj þ c x  2cj x2j ¼ ~m x q^ q~^

ð29Þ

 2mj ; x  2cj can be obtained by where, the deterministic parts x

h i fuj gT KðlEp Þ fuj g  2pj ¼ h  i h  i x ; fuj g fuj gT M lqm þ M lqc





ð21Þ

ð22Þ 1=2

,

the mean and variance of the jth modal frequency can be obtained by

 12 !12 1 4l2x2  2r2x2 j j 2

where, W m ; W c are the weights of the randomness of two random variables. By using Eq. (27) together with Eqs. (13) and (14), the jth modal frequency can be expressed as

p ¼ m; c





ð30Þ



      2cj 1 þ mq~^ mq~^  W c C E~ q~^ mE~c mq~^ mE~m mq~^ mE~c þx c

Therefore, applying the algebra synthesis method to xj ¼ ðx2j Þ

lxj ¼

ð28Þ



i

fuj g KðlEp Þ fuj g h i ¼ ; fuj gT Mðlqq Þ fuj g

lq~^

 2mj 1 þ mq~^ mq~^  W m C E~m q~ mE~m mq~^ mE~m mq~^ mE~m lx2j ¼ x

 2pq ðp ¼ m; c;q ¼ m; cÞ are the deterministic parts of the jth [31]. x eigen-value, which can be obtained by

h

mq~^

By using the algebra synthesis method, the mean value and variance of the jth eigen-value can be simplified as

where, C E~p q~ p ðp ¼ m; cÞ is the correlation coefficient of two random ~p ; q ~ p , and it is generally taken in the range of 0.5–0.9 variables E

T

¼r

2 = q~^

ð23Þ

ð24Þ





 4mj m2E~ þ m2~^  W m C E~ q~^ mE~m mq~^ m2E~ m2~^ r2x2 ¼ x q m m m q j

 4cj þx







m2E~c þ m2q~^  W c C E~c q~^ mE~c mq~^ m2E~c m2q~^

ð31Þ





ð32Þ

The randomness of the jth normalized mode shape can be expressed as

1  jg fuj g ¼ qffiffiffiffi fu q~^

ð33Þ

 j g is the deterministic part of normalized mode shape, where, fu and it can be computed by

fuj g  j g ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fu  h  i h  i T fuj g fuj g W m M lqm þ W c M lqc

ð34Þ

Thus, the mean value and variance of the jth normalized mode shape can be simplified as

lfuj g ¼

 14 2  jg 1 þ m2q~^  0:5m2q~^ fu

ð35Þ

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Y. Xu et al. / Composite Structures 133 (2015) 585–594

r2fuj g ¼ 1 þ m2q~^ 

 2 1 þ m2q~^  0:5m2q~^

12 !

 j gT fu  jg fu

ð36Þ

5. Numerical studies In order to investigate the effects of the uncertainties in constituent material properties on the dynamic characteristics of FGM beams, an FGM cantilever beam is considered. The FGM beam is composed of ceramics (Zirconia) and metal (Alumina), and their deterministic material properties are as follows: Young’s elastic modulus Ec ¼ 151 GPa, mass density qc ¼ 3000 kg=m3 for Zirconia, and Young’s elastic modulus Em ¼ 70 GPa, mass density qm ¼ 2707 kg=m3 for Alumina. The Poisson’s ratio is assumed to be constant 0:3 for each constituent material. In the FGM, material properties are assumed to vary through the thick or axial direction, according to Eq. (1) or Eq. (2) as the literature [33]. Therefore, the bottom surface is pure metal and the top surface is pure ceramics for the FGM beam through the thickness, while the right side is pure metal and the left side is pure ceramics for the FGM beam through the axial direction. The FGM beam has the geometric parameters of 8 m in length, 1 m in thickness and 0.8 m in width.

5.1. FGM beam through its thickness direction This section aims at investigating the effects of random material properties on the dynamic characteristics of the FGM beam through its thickness by using the proposed method. Compared with the results obtained by the FOPT and MCS with 30,000 samples, the mean values and COVs (Coefficient of variation, the ratio of standard deviation to mean) of the first two modal frequencies are presented in Tables 1 and 2, in which four different cases of uncertainty are considered. Four cases are (1) case 1: the combination of 1% random error (COV) in Young’s elastic modulus Ec and 1% in Em , (2) case 2: 1% in Ec and 10% in Em , (3) case 3: 10% in Ec and 1%

in Em , (4) case 4: 10% in Ec and 10% in Em , while the random errors in mass densities qm ; qc are set to 1%. It is observed from Tables 1 and 2 that both the mean values and COVs of the first two modal frequencies obtained from the random factor method agree favorably with those from the FOPT and MCS method. This close agreement verifies the effectiveness of the proposed method. It is interesting that, unlike the means from the RFM and MCS, the means of modal frequencies obtained from the FOPT remain unchanged for different uncertain levels of Em and Ec , since the means of random functions are calculated only by using the first-order derivative term and the means of the corresponding variables without considering the covariance between random variables since the second-order derivatives are omitted in the FOPT. In fact, it is acceptable for small variability in parameters, since the mean values are insensitive to the uncertainty levels in that case, and are always in agreement with the deterministic values within small errors. It has to be mentioned that the FOPT is not suitable for large level of uncertainty. It can also be observed from Table 1 that the means of modal frequencies decrease with the increase in power exponent n. It is because the increase in power exponent leads to the increase in volume fraction of metal, and, as a result, the stiffness of the FGM beam reduces. It can be seen from Table 2 that the COVs of modal frequencies change with different levels of uncertainty. In fact, higher level of uncertainty can lead to higher dispersion of modal frequencies. It is also seen that, for a constant power exponent, the increase in random errors of Em and Ec more or less causes the increase in the COVs of modal frequencies except either n ¼ 0 with only Em considered random or n ¼ 1 with only Ec considered random. The variations of COVs of the first two frequencies with the power exponent are given in Fig. 2, which shows the effects of the power exponent on the contributions of the randomness of material properties to the dispersion of modal frequencies for the FGM beam through the thickness. It is seen from Fig. 2 that the effect of the randomness coming from ceramics is more

Table 1 Means of modal frequencies for FGM beam through thickness with different levels of uncertainty. Frequency

Uncertainty

n¼0 RFM

MCS

n¼2

RFM

FOPT

MCS

n¼1 RFM

FOPT

1st

Case Case Case Case

1 2 3 4

17.7190 17.7190 17.6971 17.6971

17.7186 17.7186 17.7186 17.7186

17.7186 17.7196 17.6914 17.7019

15.9403 15.9390 15.9297 15.9283

15.9398 15.9398 15.9398 15.9398

15.9403 15.9371 15.9318 15.9228

14.6642 14.6585 14.6607 14.6550

14.6637 14.6637 14.6637 14.6637

14.6638 14.6497 14.6547 14.6375

12.7004 12.6846 12.7004 12.6846

12.7001 12.7001 12.7001 12.7001

12.7001 12.6821 12.7008 12.6816

2nd

Case Case Case Case

1 2 3 4

104.7648 104.7648 104.6349 104.6349 0.214

104.7622 104.7622 104.7622 104.7622 0.717

104.7625 104.7684 104.6015 104.6634 536.521

94.5194 94.5117 94.4547 94.4470 0.234

94.5161 94.5161 94.5161 94.5161 0.649

94.5194 94.5013 94.4685 94.4156 542.941

86.5205 86.4867 86.5000 86.4661 0.217

86.5173 86.5173 86.5173 86.5173 0.779

86.5184 86.4386 86.4683 86.3703 612.912

75.0918 74.9987 75.0918 74.9987 0.205

75.0899 75.0899 75.0899 75.0899 0.857

75.0903 74.9835 75.0944 74.9807 629.863

Run time (s)* *

n ¼ 0:5 FOPT

RFM

FOPT

MCS

MCS

The computer processor used is Intel 2.53 GHz core2 CPU.

Table 2 COVs (%) of modal frequencies for FGM beam through thickness with different levels of uncertainty. Frequency

Uncertainty

n¼0

n ¼ 0:5

n¼2

n¼1

RFM

FOPT

MCS

RFM

FOPT

MCS

RFM

FOPT

MCS

RFM

FOPT

MCS

1st

Case Case Case Case

1 2 3 4

0.7071 0.7071 5.0338 5.0338

0.7072 0.7072 5.0274 5.0274

0.7076 0.7090 5.0881 5.0556

0.5541 1.4265 3.7043 3.9317

0.6367 1.4619 3.7158 3.9419

0.6347 1.4732 3.7197 3.9596

0.5057 2.8449 2.2343 3.5842

0.6138 2.8675 2.2628 3.6002

0.6162 2.8847 2.2882 3.6284

0.7086 5.0338 0.7086 5.0338

0.7087 5.0276 0.7087 5.0276

0.7065 5.0680 0.7115 5.0654

2nd

Case Case Case Case

1 2 3 4

0.7071 0.7071 5.0340 5.0340

0.7078 0.7078 5.0275 5.0275

0.7076 0.7090 5.0881 5.0556

0.5559 1.3935 3.7424 3.9554

0.6367 1.4279 3.7533 3.9649

0.6361 1.4393 3.7573 3.9829

0.5038 2.8423 2.2372 3.5835

0.6147 2.8643 2.2655 3.5999

0.6162 2.8806 2.2897 3.6261

0.7071 5.0340 0.7071 5.0340

0.7078 5.0275 0.7078 5.0275

0.7065 5.0680 0.7115 5.0654

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pronounced for a small power exponent than that of metal, and decreases with the increase in power exponent. On the contrary, the effect of randomness coming from the metal is more pronounced for a large power exponent than that of ceramics. It is worth noting that, in the beginning, the COV in each modal frequency shows a fast monotonic decrease with the increase in power exponent when both Em and Ec are simultaneously considered random, and, then, shows a slowly monotonic increase with the increase in power exponent. It is also observed that, compared with the results when only the randomness of either Em or Ec is considered, the COVs of modal frequencies tend to be higher as

(a)

both Em and Ec are taken as random inputs and reach to the maximum when n ¼ 0 or n ¼ 1, while the amount of the increases in COVs of modal frequencies is not so high as expected. Figs. 3–5 give the variations of the COVs in modal frequencies with the COVs in various random variables for different power exponents, showing the effects of the uncertainties in material parameters on the dynamical characteristic for the FGM beam through the thickness. Five different cases are considered, that is (1) only Young’s modulus of ceramics Em is assumed to be random with COV changing from 0% to 20%, while all other random variables are kept constant, (2) only Young’s modulus of ceramics Ec

(b)

Fig. 2. Variations of COVs in modal frequencies with power exponent.

(a)

(b)

Fig. 3. Variations of COVs in modal frequencies with COVs in different random variables (n < balanced point).

(a)

(b)

Fig. 4. Variations of COVs in modal frequencies with COVs in different random variables (n  balanced point).

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(a)

(b)

Fig. 5. Variations of COVs in modal frequencies with COVs in different random variables (n > balanced point).

is assumed to be random, (3) both Em and Ec are assumed to be random with COV changing from 0% to 20% simultaneously, (4) both qm and qc are assumed to be random with COV changing from 0% to 20% simultaneously, (5) all variable Em ; Ec ; qm ; qc are assumed to be random simultaneously. It is seen from Figs. 3–5 that the dispersion of the first two modal frequencies for the FGM beam through the thickness is much more sensitive to the randomness in ceramics Ec than that in metal Em for nearly n < 1:5 (Fig. 3), and the case is the opposite for nearly n > 1:5 (Fig. 5). The balanced point is nearly n  1:5 (Fig. 4), at which the randomness of Ec has the same contributions to the dispersion of modal frequencies as that of Em . The dispersion of modal characteristics when mass densities qm ; qc are simultaneously considered random is almost the same as that when Young’s modulus Em ; Ec are simultaneously considered random, no matter what power exponent is taken. For small power exponent, the dispersion of modal frequencies when both Em and Ec are taken random is almost the same as that when only Ec is considered random. For large power exponent, similarly, the dispersion of modal frequencies when both Em and Ec are taken random is almost the same as that when only Em is considered random. This can be explained that ceramic plays a dominant role in the FGM for a small power exponent, and metal for a large power exponent. It is also observed that the dispersion of modal frequencies tends to be higher as more material parameters are taken to be the random inputs, and reaches to the maximum when all material properties Em ; Ec ; qm ; qc are considered random, varying at the same time. It is worth noting that, although getting higher when more material properties are considered, the dispersion of frequencies for different combinations of random material properties is

(a)

not the simple linear superposition of the results with respect to each random material property concerned, and, in fact, it totally depends on the volume fraction index. The means and standard deviations of the first two normalized mode shapes for the FGM beam through the thickness with 1% random errors in mass densities are illustrated in Fig. 6, showing the effects of the randomness of mass densities on the dispersion of mode shapes. It should be mentioned that the uncertainty in Young’ modulus does have little effect on the normalized mode shapes, and, also, no obvious changes occur to the dispersion of mode shapes when varying the constituent volume fraction index of the FGM. Compared with the results of the modal frequencies, the dispersion of mode shapes is less sensitive to the uncertainties in material parameters. 5.2. FGM beam through its axial direction It should be noted that material properties in FGMs are practically required to vary through different directions, although most of the analysis available is related to FGMs with material properties varying through the thickness. In this part, the effects of random material properties on the dynamic characteristics for the FGM beam varying through the axial direction are investigated. Using the random factor method, the mean values and COVs of the first two modal frequencies for the FGM beam through the axial direction with different levels of uncertainty are presented in Tables 3 and 4. It is seen that the results from the proposed method also agree with those from the FOPT and MCS for the FGM beam through the axial direction. It is also observed that, for a constant power exponent, the means and variances of modal frequencies for the FGM beam through the axial direction are

(b)

Fig. 6. Means and standard deviations of first two mod shapes with 1% COVs in mass densities.

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Table 3 Means of modal frequencies for FGM beam through axial direction with different levels of uncertainty. Frequency

Uncertainty

n¼0

RFM

FOPT

RFM

FOPT

RFM

FOPT

1st

Case Case Case Case

1 2 3 4

17.7190 17.7190 17.6971 17.6971

17.7186 17.7186 17.7186 17.7186

17.7191 17.7176 17.6978 17.6892

17.6869 17.6868 17.6676 17.6675

17.6864 17.6864 17.6864 17.6864

17.6860 17.6855 17.6659 17.6686

16.6394 16.6381 16.6277 16.6264

16.6388 16.6388 16.6388 16.6388

16.6377 16.6325 16.6226 16.6180

12.7004 12.6846 12.7004 12.6846

12.7001 12.7001 12.7001 12.7001

12.7010 12.6833 12.7002 12.6913

2nd

Case Case Case Case

1 2 3 4

104.7648 104.7648 104.6349 104.6349 0.147

104.7622 104.7622 104.7622 104.7622 1.701

104.7649 104.7562 104.6392 104.5887 756.605

99.3129 99.3102 99.2234 99.2207 0.174

99.3097 99.3097 99.3097 99.3097 1.790

99.3077 99.2974 99.2110 99.2204 886.195

88.7925 88.7666 88.7635 88.7375 0.158

88.7892 88.7892 88.7892 88.7892 1.792

88.7832 88.7186 88.7288 88.6782 857.666

75.0918 74.9987 75.0918 74.9987 0.153

75.0899 75.0899 75.0899 75.0899 1.602

75.0952 74.9909 75.0906 75.0378 854.387

RFM

Run time (s)

n ¼ 0:5 FOPT

MCS

n¼2 MCS

n¼1 MCS

MCS

Table 4 COVs (%) of modal frequencies for FGM beam through axial direction with different levels of uncertainty. Frequency

Uncertainty

n¼0

n ¼ 0:5

n¼2

RFM

FOPT

MCS

RFM

FOPT

MCS

RFM

FOPT

MCS

RFM

FOPT

MCS

1st

Case Case Case Case

1 2 3 4

0.7071 0.7071 5.0338 5.0338

0.7072 0.7072 5.0274 5.0274

0.7058 0.7079 5.0854 5.0406

0.6666 0.7306 4.7301 4.7396

0.6878 0.7501 4.7282 4.7376

0.6885 0.7463 4.7600 4.7610

0.5606 1.3506 3.7922 3.9869

0.6389 1.3849 3.8025 3.9961

0.6411 1.3944 3.8229 4.0116

0.7086 5.0338 0.7086 5.0338

0.7087 5.0276 0.7087 5.0276

0.7074 5.0480 0.7069 5.0811

2nd

Case Case Case Case

1 2 3 4

0.7071 0.7071 5.0340 5.0340

0.7078 0.7078 5.0275 5.0275

0.7058 0.7079 5.0854 5.0406

0.6118 0.9545 4.2917 4.3541

0.6619 0.9875 4.2955 4.3576

0.6631 0.9848 4.3219 4.3795

0.5001 2.4693 2.6073 3.5576

0.6132 2.4949 2.6315 3.5740

0.6146 2.5209 2.6457 3.5874

0.7071 5.0340 0.7071 5.0340

0.7078 5.0275 0.7078 5.0275

0.7074 5.0480 0.7069 5.0811

(a)

n¼1

(b)

Fig. 7. Variations of COVs in modal frequencies with power exponent.

n=1

(a)

n=1

(b)

Fig. 8. Variations of COVs in modal frequencies with COVs in different random variables (n < balanced point).

different from those for the FGM beam through the thickness except either n ¼ 0 with only Em considered random or n ¼ 1 with only Ec considered random.

The effects of the variation of power exponent on the contributions of parameter randomness to the dispersion of modal frequencies for the FGM beam through the axial direction are shown in

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Fig. 7. Unlike the FGM beam through the thickness, the contributions of random Em to the dispersion of the first modal frequency for the FGM beam through the axial direction reaches to the same weight with that of Ec at the power exponent point of nearly n  5, while the point is nearly n  2 for the second frequency, which means the different effects of power exponent on the first two modal frequencies for the FGM beam through the axial direction. The effects of different combinations of random variables on the dispersion of modal frequencies for the fixed power exponent are presented in Figs. 8–10. Different from the FGM beam through the thickness, the randomness of Em and Ec has the similar contributions to the dispersion of the first frequency when the power exponent is close to n ¼ 5 (Fig. 9(a)), and the dispersion of the first

(a)

frequency is much more sensitive to the randomness of ceramics than that of metal when nearly n < 5 (Fig. 8(a)), while the case is the opposite when nearly n > 5 (Fig. 10(a)). For the second frequency, the balanced point is at nearly n ¼ 2 (Fig. 9(b)), and the dispersion of the second frequency is much more sensitive to the randomness of ceramics than that of metal when nearly n < 2 (Fig. 8(b)), while the case is the opposite when nearly n > 2 (Fig. 10(b)). The means and variances of the first two normalized mode shapes for the FGM beam through the axial direction with 1% random error in mass densities are illustrated in Fig. 11. Similar to the FGM beam through the thickness, it shows the effects of the random of mass densities on the dispersion of the first two mode shapes.

(b)

Fig. 9. Variations of COVs in modal frequencies with COVs in different random variables (n  balanced point).

(a)

(b)

Fig. 10. Variations of COVs in modal frequencies with COVs in different random variables (n > balanced point).

1st

(a)

(b)

Fig. 11. Means and standard deviations of mode shapes with 1% random errors in mass densities.

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6. Conclusions The effects of random constituent material properties on the dynamical characteristics for FGM beams through the thickness or axial direction are investigated by the proposed so-called random factor method. In the random factor method, the randomness of the dynamic characteristics is explicitly expressed by the random factors of material parameters so that the means and variances of modal parameters can be analytically computed by the statistics of random inputs with little computational efforts. The first-order perturbation technique and the Monte-Carlo simulation have been used to verify the proposed method. The simulation results show that the presented method is both effective and efficient in stochastic dynamic characteristics analysis of FGM beams with material properties varying through different directions. The following conclusions can be drawn from this study. (1) The constituent volume fraction index has a dominant effect on the contributions of random material properties to the dispersion of modal frequencies for FGM beams. (2) Although getting higher when more material properties are considered random, the dispersion of frequencies for different combinations of random material properties is not the simple linear superposition of the results with respect to each random material property concerned, and, in fact, totally depends on the volume fraction index. (3) The dispersion of mode shapes mainly depends on the randomness of the constituent material mass densities. The volume fraction index has little influence on the randomness of mode shapes. (4) Compared with the FGM beam through the thickness in which the effects of volume fraction index on the first two modal frequencies are nearly the same, the effect of volume fraction index on the first modal frequency is obviously different from that on the second modal frequency for the FGM beam through the axial direction.

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