Random vibration of the functionally graded laminates in thermal environments

Comput. Methods Appl. Mech. Engrg. 195 (2006) 1075–1095 www.elsevier.com/locate/cma

Random vibration of the functionally graded laminates in thermal environments S. Kitipornchai a

a,*

, J. Yang b, K.M. Liew

c

Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong b Department of Civil Engineering, University of Queensland, St. Lucia, Brisbane 4072, Queensland, Australia c Nanyang Centre for Supercomputing and Visualisation, School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Received 1 April 2004; received in revised form 24 September 2004; accepted 17 January 2005

Abstract This work deals with the random free vibration of functionally graded laminates with general boundary conditions and subjected to a temperature change, taking into account the randomness in a number of independent input variables such as YoungÕs modulus, PoissonÕs ratio and thermal expansion coefficient of each constituent material. Based on third-order shear deformation theory, the mixed-type formulation and a semi-analytical approach are employed to derive the standard eigenvalue problem in terms of deflection, mid-plane rotations and stress function. A mean-centered first-order perturbation technique is adopted to obtain the second-order statistics of vibration frequencies. A detailed parametric study is conducted, and extensive numerical results are presented in both tabular and graphical forms for laminated plates that contain functionally graded material which is made of aluminum and zirconia, showing the effects of scattering in thermo-elastic material constants, temperature change, edge support condition, side-to-thickness ratio, and plate aspect ratio on the stochastic characteristics of natural frequencies.
2005 Elsevier B.V. All rights reserved. Keywords: Free vibration; Frequency; Random material properties; Functionally graded material; Plate; Higher-order shear deformation theory

*

Corresponding author. Tel.: +852 2788 8028; fax: +852 2788 7612. E-mail address: [email protected] (S. Kitipornchai).

0045-7825/$ - see front matter
2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2005.01.016

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1. Introduction All materials, especially composite materials, have inherent uncertainties due to the large number of design variables that are involved in fabrication and the lack of total control over the manufacturing and processing techniques. Hence, deterministic analysis is insufficient to provide complete information about the structural response. Consequently, there has been considerable interest over the last few decades in developing stochastic formulations for predicting the actual static and dynamic behavior of composite plates with random system parameters. Many studies concerning the random vibration of composite plates are available in the literature: see, for example, those of Chonan [1], Cederbaum et al. [2], Singh et al. [3], Abdelnaser and Singh [4], Harichandran and Naja [5], Salim et al. [6], and Kang and Harichandran [7], most of which employed either Monte Carlo simulation or perturbation technique, or both. State-of-theart developments were well documented in the comprehensive reviews of Ibrahim [8] and Manohar and Ibrahim [9]. Functionally graded materials (FGMs) constitute a new class of two-phase heterogeneous composite materials that allow for a continuous composition gradient from one phase to another in one or more dimensions in a defined geometry. Unlike traditional composites, which are homogeneous mixtures with uniform distribution of the constituent materials, the desired properties of both phases can be fully utilized. For example, the strength and toughness of a metal can be well mated with the high heat-resistance of a ceramic in an FGM, without compromising in the advantageous properties of either the metal side or the ceramic side. Due to their excellent features, FGMs are finding increasing applications in many engineering sectors [10–16], especially in high temperature environments where thermal effects due to temperature change must be taken into account. Studies on the free vibration of FGM plate structures have received widespread attention in recent years. Based on classical plate theory (CPT) and first-order shear deformation theory (FSDT), Chen and Kitipornchai [17] presented solutions for the buckling and vibration of FGM polygonal plates via analogy with membrane vibrations. Three-dimensional vibration analyses for FGM rectangular plates were developed by Reddy and Cheng [18] using an asymptotic formulation and transfer matrix method, and by Vel and Batra [19] by means of a power series expansion technique. However, thermal loading was not considered, and the results of these investigations were only for plates simply supported at all edges. Yang and Shen developed a differential quadrature-based semi-analytical approach that is capable of incorporating general boundary conditions, and examined the vibration and dynamic response of FGM thin plates [20] and shear deformable FGM plates [21]. The large amplitude vibration of FGM plates was recently studied by Yang et al., [22], Kitipornchai et al., [23], Huang and Shen [24] and Chen [25]. It should be noted, however, that these analyses mentioned above were deterministic because they were all based on the assumption that the system parameters, such as material constants, volume fractions, boundary constraints and geometry, etc., are completely determinate. To the best of authorsÕ knowledge, no previous work has been done on the free vibration of FGM plates involving randomness in system parameters. The primary objective of this paper is to investigate the vibration characteristics of functionally graded laminated plates in the presence of temperature change and random variations in system variables, taking into account the transverse shear strains and rotary inertias in the framework of third-order shear deformation theory. The thermo-elastic properties of the constituent materials are taken as independent random variables with relatively small variability. The mixed-type formulation and a semi-analytical approach are employed to derive the standard eigenvalue problem in terms of deflection, mid-plane rotations and stress function. Second-order statistics-both the mean and the standard deviation of the vibration frequencies are determined via the first order perturbation technique. Numerical results are presented for functionally graded rectangular laminates made of aluminum (Al) and zirconia (ZrO2) with their volume fractions following a power-law distribution through the plate thickness.

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2. Mathematical formulation 2.1. FGM laminate model In actual applications where thermal loading due to temperature change DT is present, it is often desired that ceramics be used as a thermal shield to protect the metallic base structure. An intermediate FGM zone, which is a mixture of ceramic and metal with material profile varying continuously along the thickness direction from 100% ceramic to 100% metal, is introduced to join the dissimilar materials in order to eliminate any cracking and spalling that may occur at the ceramic-metal interface due to thermo-elastic property mismatch. Both the substrate and thermal coating are isotropic and homogeneous. A schematic diagram of such a laminate is shown in Fig. 1, where a and b are plate lengths, h is the total thickness, E, m, a, j, and q are elastic modulus, PoissonÕs ratio, linear thermal expansion coefficient, thermal conductivity and mass density, and subscripts ‘‘c’’, ‘‘f’’, and ‘‘m’’ refer to the ceramic coating, FGM layer, and metallic substrate, respectively. The ceramic volume fraction Vc and metal volume fraction Vm in the FGM zone are designed to be 
 n 1 z 1 þ ðlc þ lf
lm Þ ; V m ¼ 1
V c; ð1Þ Vc ¼ lf h 2 where lc = hc/h, lf = hf/h and lm = hm/h are the dimensionless thicknesses of the ceramic coating, the FGM zone and the metallic base, normalized by the total plate thickness h, and n is the volume fraction index. For the FGM, the local effective material properties are determined by Pf ¼ Pc V c þ Pm V m ;

ð2Þ

where Pf = [Ef, mf, af, jf, qf]T, Pc = [Ec, mc, ac, jc, qc]T, Pm = [Em, mm, am, jm, qm]T. 2.2. Basic equations The displacement field of third order shear deformation theory [26] of the plate is assumed to be 2 38 U 9 > > 3 o 3 8 9 > > >V > 1 0
c1 z z
c1 z 0 > = ox 6 7 o W ; v ¼6 7 0 z
c 1 z3 5 > > : ; 4 0 1
c1 z3 > > > > oy W w > ; : x> 0 0 1 0 0 Wy

Fig. 1. Geometry of a functionally graded laminate.

ð3Þ

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where c1 = 4/3h2, (u, v, w) denote the displacement components of an arbitrary point in the (x, y, z) coordinate system, (U, V, W) are the mid-plane displacements, and (Wx, Wy) are the mid-plane rotations about the y- and x-axes. The in-plane strains (exx, eyy, cxy) and the transverse shear strains (cyz, czx) are defined through the linear strain–displacement relations as 9 8 9 9 8 8 o2 W oWx oU > oWx > > > > > > > þ > > > > > 8 9 > > > > > > > 2 > > > > > > ox ox ox ox > > > > > > e > > > > > > = = = < < oV = > < < xx > 2 oWy o W oWy 3 eyy ¼ ¼ ð0Þ þ zð1Þ þ z3 ð3Þ ; þz
c1 z þ 2 oy oy > > > > > > > > oy oy > > > > > : ; > > > > > > > cxy > > > > > > 2 > > > > > > > > > > oW oW o W x y > ; ; > : oU þ oV > : oWx þ oWy > ; : þ þ2 oy ox oy ox oxoy oy ox ð4aÞ 9 8 oW > > >   þ Wy > = < cyz oy 2 ¼ cð0Þ þ z2 cð2Þ ; ð4bÞ ¼ ð1
3c1 z Þ > > czx oW > > ; :8 þ Wx ox The constitutive law of thermo-elasticity for the materials under consideration relates the stresses with strains in a plane stress state by 9 2 8 38 9 8 9 Q11 Q12 0 > >  
  = = > = < rxx > < exx > < k1 > syz Q44 cyz 0 6 7 e ryy ¼ 4 Q12 Q22 0 5 yy
k2 DT ; ¼ ; ð5Þ > > > 0 Q55 szx czx ; ; > ; : : > : > cxy sxy 0 0 Q66 0 where k1 = Q11a1 + Q12a2, k2 = Q12a1 + Q22a2, Qij are the reduced elastic constants. To facilitate the introduction of a stress function to derive the mixed-type theoretical formulations, the following semi-inverse stress resultant-strain relationship is used 9 9 2 8 38 A B E 0 0 > N
N > ð0Þ > > > > > > > > > > > > > > > 6 >
ðB ÞT D ðF ÞT 0 0 7 ð1Þ > > > > 7> = = 6 < 7 6  T   ð3Þ 7
ðE Þ F H 0 0 ; ð6Þ ¼6  P
P 7> > > 6 > > > > 7> 6 ð0Þ > > > > ^ ^ ^ c > > > 4 0 Q 0 0 A D 5> > > > > > > > > ; ; : : ð2Þ ^ ^ ^ c 0 0 0 D F R where N = [Nx, Ny, Nxy]T, M = [Mx, My, Mxy]T and P = [Px, Py, Pxy]T are the resultant membrane forces, ^ ¼ ½Q ^ ¼ ½R ^ y; Q ^ x T and R ^y ; R ^ x T are the lower-order resultant moments and higher-order moments, and Q T     T and higher-order resultant shear forces. Thermal resultants N ¼ ½N x ; N y ; N xy  , M ¼ ½M x ; M y ; M xy  , and     T P ¼ ½P x ; P y ; P xy  are calculated by Z h=2  T ð7Þ ½k1 ; k2 ; 0 1; z; z3 DT dz; ½N ; M ; P  ¼

h=2

A*, B*, D*, E*, F*, H* are the reduced stiffness matrices determined from A ¼ A
1 ;

B ¼
A
1 B;

F ¼ F
EA
1 B;

D ¼ D
BA
1 B;

E ¼
A
1 E;

H ¼ H
EA
1 E;

where the stiffness elements in A, B, D, E, F, and H are

ð8Þ

S. Kitipornchai et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 1075–1095

 

 Aij ; Bij ; Dij ; Eij ; F ij ; H ij ¼  ^ ij ; D ^ ij ; F^ ij ¼ A

Z

Z

1079

h=2

Qij ð1; z; z2 ; z3 ; z4 ; z6 Þ dz ði; j ¼ 1; 2; 6Þ;

ð9aÞ


h=2 h=2

Qij ð1; z2 ; z4 Þ dz

ði; j ¼ 4; 5Þ;

ð9bÞ


h=2

The various inertias of the plate are defined as Z h=2 qð1; z; z2 ; z3 ; z4 ; z6 Þ dz. ðI 1 ; I 2 ; I 3 ; I 4 ; I 5 ; I 7 Þ ¼

ð10Þ


h=2

In the present analysis, Em, mm, am of metal and Ec, mc, ac of ceramic, are selected to be the basic random input variables, denoted as dk, and are assumed to be independent of each other whereas the volume fraction index is taken as a constant since it is a design parameter. This results in the randomness in the plate stiffness elements whose partial derivatives with respect to a random variable required for obtaining the frequency statistics can be expressed by   Z h=2 oQij o Aij ; Bij ; Dij ; Eij ; F ij ; H ij ¼ ð1; z; z2 ; z3 ; z4 ; z6 Þ dz ði; j ¼ 1; 2; 6Þ; ð11aÞ od k
h=2 od k   Z h=2 ^ ij ; D ^ ij ; F^ ij oQij  2 4  o A ¼ 1; z ; z dz ði; j ¼ 4; 5Þ; ð11bÞ od k
h=2 od k For the sake of simplicity in the mathematical formulations, the thermal conductivities jm, jc and mass densities qm, qc are assumed to be constants. 2.3. Governing equations Suppose that the temperature varies along the plate thickness only so that DT = DT(z). Let t be the time, F the stress function for the stress resultants (Nx = F,yy, Ny = F,xx, and Nxy =
F,xy, where a dot denotes partial differentiation with respect to the coordinates), and n = x/a, f = y/b, b = a/b the dimensionless quantities. The governing equations of motion for the laminated plate under temperature change DT(z) are derived as follows: LiL ðW ; F ; Wx ; Wy Þ ¼

o2 LiR ðW ; F ; Wx ; Wy Þ ot2

ði ¼ 1; . . . ; 4Þ;

where the linear partial differentiation operators LiL and LiR take the form of
   4 4 4 2 o2 W 2 o W 2o W  o W     4o W 2 þ F 22 b L1L ¼ c1 F 11 4 þ ðF 12 þ F 21 þ 4F 66 Þb
a px 2 þ py b on on2 of2 of4 on of2
 o4 F o4 F o4 F þ B21 4 þ ðB11 þ B22
2B66 Þb2 2 2 þ B12 b4 4 on on of of
 3 3 o Wx 2 o Wx     þ ðD þ 2D
c F
2c F Þb
a ðD11
c1 F 11 Þ 1 1 12 66 12 66 on3 onof2
 o3 W y o3 W y
a ðD12 þ 2D66
c1 F 21
2c1 F 66 Þb 2 þ ðD22
c1 F 22 Þb3 ; on of of3  2     2 oW 2o W ^I 5 oWx þ b oWy þ b L1R ¼
a2 I 1 W þ ^I 7 þ ; on of on2 of2

ð12Þ

ð13aÞ ð13bÞ

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 o4 W o4 W o4 W L2L ¼
c1 E21 4 þ ðE11 þ E22
2E66 Þb2 2 2 þ E12 b4 4 on on of of
 o4 F o4 F o4 F þ A22 4 þ ð2A12 þ A66 Þb2 2 2 þ A11 b4 4 on on of of
 3 o Wx o3 Wx þ a ðB21
c1 E21 Þ 3 þ ðB11
B66
c1 E11 þ c1 E66 Þb2 on onof2
 o3 W y o3 W y þ a ðB22
B66
c1 E22 þ c1 E66 Þb 2 þ ðB12
c1 E12 Þb3 ; on of of3 L2R ¼ 0;

ð14aÞ ð14bÞ


 3 3 oW 2 o W   o W     þ c1 ðF 11
c1 H 11 Þ 3 þ ðF 21 þ 2F 66
c1 H 21
2c1 H 66 Þb L3L ¼ ðA55
on on onof2
 3 3 oF oF þ ðB21
c1 E21 Þ 3 þ ðB11
B66
c1 E11 þ c1 E66 Þb2 on onof2
o2 Wx þ a ðA55
6c1 D55 þ 9c21 F 55 Þa2 Wx
ðD11
2c1 F 11 þ c21 H 11 Þ on2  o2 W x
ðD66
2c1 F 66 þ c21 H 66 Þb2 of2

o2 W y ; ð15aÞ
a ðD12 þ D66 Þ
c1 ðF 12 þ F 21 þ 2F 66 Þ þ c21 ðH 12 þ H 66 Þ b onof   oW L3R ¼
a2 I 03 Wx
I 05 ; ð15bÞ on 6c1 D55 þ 9c21 F 55 Þa2



o3 W oW o3 W þ c1 F 12 þ 2F 66
c1 H 12
2c1 H 66 b 2 þ ðF 22
c1 H 22 Þ 3 of on of of
3 3  oF oF þ b ðB22
B66
c1 E22 þ c1 E66 Þ 2 þ ðB12
c1 E12 Þb2 3 on of of

L4L ¼ ðA44
6c1 D44 þ 9c21 F 44 Þa2 b

L4R



o2 W x
a ðD12 þ D66 Þ
c1 ðF 12 þ F 21 þ 2F 66 Þ þ c21 ðH 12 þ H 66 Þ b onof
o2 W y þ a ðA44
6c1 D44 þ 9c21 F 44 Þa2 Wy
ðD66
2c1 F 66 þ c21 H 66 Þ on2  o2 W y
ðD22
2c1 F 22 þ c21 H 22 Þb2 ; of2   oW ¼
a2 I 03 Wy
I 05 b ; of

ð16aÞ ð16bÞ

in which the inertia-related terms are 2

I 03 ¼ I 3
2c1 I 5 þ c21 I 7
ðI 2
c1 I 4 Þ =I 1 ; I 05 ¼ c1 ½I 5
c1 I 7
I 4 ðI 2
c1 I 4 Þ=I 1 ; I 0 ¼ c2 ðI 2 =I 1
I 7 Þ; ^I 5 ¼ I 3
c1 I 5
I 2 ðI 2
c1 I 4 Þ=I 1 ; ^I 7 ¼ c1 ½I 2 I 4 =I 1
I 5 . 7

1

4

ð17Þ

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1081

An immovable plate, where tangential motion is not restrained but no in-plane displacement in the normal direction is allowed, is considered in this study. This requires   o2 F  o2 F  ¼ ¼ 0; U jn¼0 ¼ U jn¼1 ¼ 0; ð18aÞ onofn¼0 onofn¼1   o2 F  o2 F  ¼ ¼ 0; V jf¼0 ¼ V jf¼1 ¼ 0. ð18bÞ onoff¼0 onoff¼1 Note that thermally induced membrane forces px and py in differential operator (13a) can be determined from the mid-plane displacement boundary conditions in (18). The out-of-plane boundary conditions for simply supported (S), clamped (C) and free (F) edges are as follows S : W ¼ M n ¼ Ws ¼ P n ¼ 0;

ð19aÞ

oW ¼ 0; on ¼ P n ¼ 0;

C : W ¼ Wn ¼ Ws ¼

ð19bÞ

F : Qn ¼ M n ¼ M ns

ð19cÞ

where subscripts ‘‘n’’ and ‘‘s’’ refer to the normal and tangential directions, respectively, Qn and M ns are the generalized transverse shear force and moment defined by Reddy [26].

3. Solution procedure 3.1. Semi-analytical analysis The solutions of the governing equations of motion (12) under the boundary conditions (18) and (19) are assumed to be of the form W ¼

M X

wm ðn; fÞ;

m¼1

Wx ¼

M X

wxm ðn; fÞ;

M X 1 F ¼
ðpx f2 þ py n2 Þ þ fm ðn; fÞ; 2 m¼1

Wy ¼

m¼1

M X

wym ðn; fÞ;

ð20Þ

m¼1

Following the differential quadrature based semi-analytical approach proposed by Yang and Shen [20– 21], the unknown functions wm, fm, wxm, wym and their rth partial differentiation with respect to n at a sampling point nk (k = 1, . . . , N) are approximated in terms of their function values at a number of pre-selected sampling points along n-axis by  N N X X or dm  ðrÞ lj ðnÞdmj ; ¼ C kj dmj ; ð21Þ dm ¼ r  on n¼nk j¼1 j¼1 ðrÞ

where lj(n) and C kj are the interpolation polynomials and weighting coefficients given by Bert and Malik [27] and Liew et al. [28], and the vectors T

dm ¼ ½wm ; fm ; wxm ; wym  ;

 T T dmj ¼ ½wmj ; fmj ; wxmj ; wymj  ¼ ½wm ; fm ; wxm ; wym  n¼n . j

ð22Þ

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By using series expressions (20) and relationships (21), the partial differential governing Eq. (12) is transformed into a set of differential equations in terms of nodal unknown functions wmj, fmj, wxmj, wymj LkiL ðwmj ; fmj ; wxmj ; wymj Þ ¼

o2 k L ðwmj ; fmj ; wxmj ; wymj Þ ði ¼ 1; . . . ; 4Þ; ot2 iR

ð23Þ

where k, j = 1,. . .,N, LkiL and LkiR are the differentiation operators with respect to f, whose detailed expressions are given in Appendix A. We next choose the analytical functions that satisfy both the in-plane boundary conditions (18b) and the associated out-of-plane boundary conditions (19) at f = 0,1 as the trial functions for wmj, fmj, wxmj, wymj, i.e. we assume ðwmj ; fmj ; wxmj ; wymj Þ ¼ famj gm1 ðfÞ; bmj gm2 ðfÞ; cmj gm3 ðfÞ; d mj gm4 ðfÞge
ixt ;

ð24Þ

where x is the natural frequency of the plate, amj, bmj, cmj, dmj are the unknown constants to be determined, trial functions gm1, gm2, gm3, gm4 for different boundary conditions are given in Appendix B. Putting expressions (24) into Eq. (23) and applying the Galerkin method, one has Z 1 k

LiL ðwmj ; fmj ; wxmj ; wymj Þ
x2 LkiR ðwmj ; fmj ; wxmj ; wymj Þ gmi df ¼ 0 ði ¼ 1; . . . ; 4Þ. ð25Þ 0

After incorporating the in-plane boundary conditions (18a) and the out-of plane boundary conditions (19) at n = 0 and n = 1, a linear eigenvalue system that contains 4N algebraic equations can be obtained ðG
x2 SÞD ¼ 0;

ð26Þ

where G and S are 4N · 4N constant matrices which are generally non-symmetric and S is positive definite, D is a unknown vector composed of amj, bmj, cmj and dmj(j = 1,. . .,N). Eq. (26) presents a generalized eigenvalue problem and can be transformed into a standard eigenvalue problem as GD ¼ kD

ð27Þ
1

2

in which G ¼ S G; k ¼ x . 3.2. First-order perturbation technique As a result of the randomness in basic input variables, all of the quantities in Eq. (27), i.e. G, k, and D, are random as well. S, however, is a constant matrix because it relates only to mass densities qm, qc and the volume fraction index n that are not taken as the random variables in the present analysis. Supposing that the random input variables are independent of each other and the random part of each input variable is small compared to its mean part, which is the situation in most of engineering applications, it is reasonable to assume that the output variables k and D do not deviate much from their mean values. Based on this, the first-order perturbation technique is used to determine the stochastic characteristics of the vibration frequencies of FGM laminates. Without loss of generality, an arbitrary random variable dk can be viewed as the sum of its mean and zero mean random part, denoted by superscripts ‘‘0’’ and ‘‘r’’, respectively, d k ¼ d 0k þ d rk .

ð28Þ

Using TaylorÕs series and keeping the first-order term, G, k and D can be expanded as X 0 0 0 r 0 ðG;k ; k;k ; D0;k Þd rk ; ðG; k; DÞ ¼ ðG ; k0 ; D0 Þ þ ðG ; kr ; Dr Þ ¼ ðG ; k0 ; D0 Þ þ k

ð29Þ

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1083

where 0

kr ¼ 2x0 xr þ ðxr Þ2 ;

G;k ¼ S
1 G0;k ;

ðG0;k ; k0;k ; D0;k Þ ¼

o ðG0 ; k0 ; D0 Þ. od 0k

ð30Þ

Inserting series expansion (29) into Eq. (27) and collecting the terms of zero- and first-order, we have 0

G D 0 ¼ k0 D 0 ; 0

ð31Þ

r

G Dr þ G D0 ¼ k0 Dr þ kr D0 .

ð32Þ

It is clear that Eq. (31) involves only mean quantities and is a deterministic eigenvalue equation from which the mean values of vibration frequencies and associated vibration modes can be obtained by many numerically efficient algorithms while Eq. (32) is the first-order perturbation equation defining the stochastic nature of the free vibration. As D0 forms a complete orthonormal basis in the 4N-dimensional space, Dr can be expressed as a linear weighting sum of D0. Putting it into Eq. (32) and making use of the orthogonality conditions yields the solution for kr [29] r

T

kr ¼ ðD Þ G D0 ;

ð33Þ

and for the kth random input variable dk T

0

k0;k ¼ ðD Þ G;k D0 ;

ð34Þ

in which the diagonal elements in matrix kr are the random part of vibration frequencies, and D* is the left 0 eigenvector of G calculated from 0 T

ðG Þ D ¼ k0 D . Thus, the eigenvalue covariance can be obtained as XX 0 0 Varðki Þ ¼ ki;j ki;k Covðd j ; d k Þ; j¼1

ð35Þ

ð36Þ

k¼1

where Cov( ) is the covariance between dj and dk. The standard deviation (SD) is obtained as the square root of the variance.

4. Results and discussion Second-order statistics of vibration frequencies have been evaluated for functionally graded laminates with a temperature change and different boundary conditions. The plate consists of an aluminum (Al) base, an upper thermal coating made of zirconia (ZrO2), and an intermediate ZrO2/Al layer sandwiched in between, and is denoted symbolically as ZrO2–FGM–Al. The mean values of the material constants are Aluminum: Em = 70 GPa, mm = 0.3, am = 23.0 · 10
6/K, jm = 204 W/mK, qm = 2707 kg/m3, Zirconia: Ec = 151 GPa, mc = 0.3, ac = 10.0 · 10
6/K, jc = 2.09 W/mK, qc = 3000 kg/m3. In the following numerical examples, Ec, Em, mc, mm, ac, am are taken as the basic random input variables. The plate is simply supported at all edges with a = b = 0.1 m and side-to-thickness ratio a/h = 10. The dimensionless thickness is lc = 0.2 for the ceramic layer and lf = 0.2 for the FGM zone. It is assumed that the plate is initially stress free and then is exposed to temperature increase DTbottom at the metallic bottom

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surface (z =
h/2) and DTtop at the ceramic top surface (z = h/2), which results in a non-uniform temperature distribution along the plate thickness that can be determined from the following steady–state heat conduction equation   d dðDT Þ jðzÞ
¼ 0; ð37Þ dz dz The solution of Eq. (37) can be readily obtained as ,Z Z z h=2 dz dz . DT ðzÞ ¼ DT bottom þ ðDT top
DT bottom Þ
h=2 jðzÞ
h=2 jðzÞ

ð38Þ

Unless otherwise stated, DTbottom = 20 K and DTtop = 100 K are used in the computation. 4.1. Comparison results Before proceeding to the random vibration analysis of FGM laminated plates under a temperature change, three test examples are solved to ensure the accuracy and computational efficiency of the present methodology. pffiffiffiffiffiffiffiffiffiffiffiffi Table 1 gives the mean fundamental frequencies x ¼ xa2 q=E22 =h for simply supported, four-layered, (0/90)s laminated square plates that are undergoing different uniform temperature changes (DT = 0,
50 C, 50 C. The thermo-elastic properties of each layer are E11/E22 = 40, G12 = G13 = 0.6E22, G23 = 0.5E22, m12 = 0.25, q = 1 kg/m3, a22 = 11.4 · 10
6/C, and a11/a22 = 0.30. The finite element solutions of Liu and Huang [30] are given for comparison. Table 2 compares the first 10 natural frequencies (Hz) of simply supported aluminum oxide/Ti–6Al–4V square plates (a = b = 0.4 m, h = 0.005 m, n = 0, 2000) with the solutions given by He et al. [31] using the classical laminated plate theory. The plates are metal rich at the top and ceramic rich at the bottom with Em = 105.7 GPa, mm = 0.2981, qm = 4429 kg/m3 for Ti–6Al–4V and Ec = 320.24 GPa, mc = 0.26, qc = 3750 kg/m3 for aluminum oxide. Thermal loading is not considered in this example. It is observed that the present method converges well enough to obtain results in good agreement with existing ones when N P 13 and M P 5. Hereafter, N = 13 and M = 5 will be used. Due to the lack of existing results concerning the random vibration of FGM plates for direct comparison, we next present the standard deviation of natural frequencies of a simply supported, symmetric crossply, (90/0)s, graphite/epoxy square plate with randomness in material properties. Fig. 2 plots the normalized standard deviation, SD/mean (i.e. the ratio of the standard deviation to the mean value), of the first two frequencies versus the SD/mean of the random material constants. It is assumed that all of the material properties change simultaneously, with their mean values being E11 = 181.0 GPa, E22 = 10.3 GPa, G12 = 7.17 GPa, and m12 = 0.28. The dashed lines are the Rayleigh–Ritz results of Salim et al. [6] that were obtained by using the classical laminated plate theory and first-order perturbation approach. Again, close correlation is achieved.

Table 1 Comparison of the mean fundamental frequencies of a simply supported (0/90)s square plate under uniform temperature change DT

0
50 50

Present

Liu and Huang [30]

N = 9, M = 3

N = 11, M = 5

N = 15, M = 5

N = 19, M = 7

13.456 13.703 13.136

15.139 15.360 14.771

15.135 15.364 14.782

15.135 15.365 14.782

15.150 15.394 14.902

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Table 2 Comparison of the mean frequencies of simply supported FGM square plates for the two special cases of isotropy Mode sequence

n=0

He et al. [31]

Present (N,M)

1 2 3 4 5 6 7 8 9 10

n = 2000

He et al. [31]

Present (N,M)

(9,3)

(13,5)

(17,7)

143.74 360.01 360.01 569.54 699.21 716.43 913.26 915.19 1260.48 1823.35

143.96 360.07 360.07 568.87 718.22 718.22 916.40 916.40 1207.13 1207.13

143.96 360.07 360.07 568.87 718.22 718.22 916.40 916.40 1207.06 1207.06

144.66 360.53 360.53 569.89 720.57 720.57 919.74 919.74 1225.72 1225.72

(9,3)

(13,5)

(17,7)

255.89 654.33 654.33 1045.27 1264.40 1300.55 1671.97 1697.20 2214.33 2311.38

261.46 653.13 653.13 1044.30 1304.79 1304.79 1694.98 1694.98 2214.41 2214.41

261.46 653.13 653.13 1044.31 1304.79 1304.79 1694.98 1694.98 2214.32 2214.32

0.125

ω1

ω2

0.100

SD/Mean, ω2

268.92 669.40 669.40 1052.49 1338.52 1338.52 1695.23 1695.23 2280.95 2280.95

0.075

0.050

0.025

0.000 0.00

: Present : Salim et al. [6]

0.05

0.10

0.15

SD/Mean, (E11, E22, G12, ν12) Fig. 2. Frequency dispersion of a simply supported (90/0)s square plate-comparison results.

4.2. Second-order statistics pffiffiffiffiffiffiffiffiffiffiffiffi Table 3 presents the mean values of the first 8 dimensionless natural frequencies x ¼ ðxa2 =p2 Þ I 0 =D0 for simply supported and clamped ZrO2–FGM–Al square plates with and without temperature change, where I0 and D0 are the values of I1 and D11 of an isotropic aluminum plate with h = 0.01 m. Results are listed for varying volume fraction index (n = 0.2, 2.0, 8.0) in the FGM zone. As expected, the clamped plate gives higher frequencies than the simply supported plate, and the frequencies decrease as the volume fraction index n increases. The effect of temperature rise is seen to lower the natural frequencies because it brings in an compressive in-plane prestress state that leads to degradation in plate stiffness.

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Table 3 Dimensionless mean frequencies of ZrO2–FGM–Al laminated square plates (a/h = 10) Mode sequence

Simply supported

Clamped

n = 0.2

n = 2.0

n = 8.0

n = 0.2

n = 2.0

n = 8.0

2.2482 5.3502 5.3502 8.1941 9.9732 9.9732 12.495 12.495

2.2403 5.3118 5.3118 8.1113 9.8558 9.8558 12.321 12.321

2.2317 5.2841 5.2841 8.0602 9.7876 9.7876 12.226 12.226

3.8184 7.2865 7.3516 10.296 11.583 12.116 14.181 14.671

3.7860 7.1891 7.2570 10.134 11.454 11.900 13.981 14.389

3.7646 7.1351 7.2039 10.050 11.383 11.791 13.876 14.250

DTbottom = 20 K, DTtop = 100 K 1 2.2415 2 5.3306 3 5.3306 4 8.1740 5 9.9526 6 9.9777 7 12.474 8 12.474

2.2387 5.2922 5.2922 8.0910 9.8350 9.8544 12.299 12.299

2.2276 5.2648 5.2648 8.0401 9.7670 9.7834 12.204 12.204

3.8055 7.2707 7.3355 10.278 11.563 12.098 14.160 14.651

3.7730 7.1732 7.2408 10.117 11.434 11.882 13.960 14.369

3.7519 7.1194 7.1879 10.032 11.364 11.773 13.856 14.231

DT = 0 1 2 3 4 5 6 7 8

Figs. 3–12 display the scattering of dimensionless frequency against the normalized standard deviation in various random input variables. The effect of individual random input variable on the dispersion of the first two natural frequencies is shown in Figs. 3–6 whereas Figs. 7–12 demonstrate the variation in fundamental

0.18

a/b = 1.0, a/h = 10, ∆Tbottom = 20 K, ∆Ttop = 100 K

1

0.15

2 3

SD/Mean, ω

0.12

1: n = 0.2 2: n = 2.0 3: n = 8.0

0.09

0.06

1 3 0.03

2 : ω1

0.00 0.00

0.04

0.08

0.12

: ω2 0.16

SD/Mean, Ec Fig. 3. Frequency dispersion of a ZrO2–FGM–Al square plate with variation in Ec.

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0.035

0.030

SD/Mean, ω

0.025

a/b = 1.0, a/h = 10, ∆Tbottom = 20 K, ∆Ttop = 100 K 3 2 1: n = 0.2 2: n = 2.0 3: n = 8.0

0.020

0.015

1 0.010

3 2

0.005

1

0.000 0.00

:ω2

:ω 1 0.04

0.08

0.12

0.16

SD/Mean, Em Fig. 4. Frequency dispersion of a ZrO2–FGM–Al square plate with variation in Em.

0.035

0.030

SD/Mean, ω

0.025

a/b = 1.0, a/h = 10, ∆Tbottom = 20 K, ∆Ttop = 100 K 1: n = 0.2 2: n = 2.0 3: n = 8.0

1 2 3

0.020

0.015

1 2 3

0.010

0.005

: ω1 0.000 0.00

0.04

0.08

0.12

:ω 2 0.16

SD/Mean, νc Fig. 5. Frequency dispersion of a ZrO2–FGM–Al square plate with variation in mc.

frequency as all of the random input variables change simultaneously. As revealed by the numerical results, the frequency dispersion of the functionally graded laminated plate exhibits linear variation with all of the random input variables.

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a/b = 1.0, a/h = 10, ∆Tbottom = 20 K, ∆Ttop = 100 K

3

SD/Mean, ω

0.0060

2

1: n = 0.2 2: n = 2.0 3: n = 8.0

0.0045

1 3 2

0.0030

1 0.0015

: ω1 0.0000 0.00

0.04

0.08

: ω2

0.12

0.16

SD/Mean, νm Fig. 6. Frequency dispersion of a ZrO2–FGM–Al square plate with variation in mm.

0.25

a/b = 1.0, a/h = 10, ∆Tbottom = 20 K, ∆Ttop = 100 K 3 2 0.20

1

SD/Mean, ω

1: n = 0.2 2: n = 2.0 3: n = 8.0 0.15

0.10

3

2

0.05

1 : ω1

0.00 0.00

0.04

0.08

0.12

: ω2 0.16

SD/Mean, all random inputs Fig. 7. Frequency dispersion of a ZrO2–FGM–Al square plate with all random input variables changing simultaneously.

The dispersion of the first two natural frequencies of heated square plates with varying standard deviation in YoungÕs moduli Ec, Em and PoissonÕs ratios mc, mm are evaluated in Figs. 3–6, respectively. Note that, in these examples, only one of the random input variables is varied, and all others are kept constant at their

S. Kitipornchai et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 1075–1095

1089

0.24

a/b = 1.0, a/h = 10

3 2

0.20

1 1

SD/Mean, ω1

0.16

1: n = 0.2 2: n = 2.0 3: n = 8.0

3 2

0.12

0.08

0.04

: ∆Τ= 0 K : ∆Τbottom= 20 K, ∆Τtop = 100 K 0.00 0.00

0.04

0.08

0.12

0.16

SD/Mean, all random inputs Fig. 8. Effect of temperature change on the frequency dispersion of a ZrO2–FGM–Al square plate with all random input variables changing simultaneously.

0.21

3 1

2 0.18

SD/Mean, ω1

0.15

0.12

: SCSC

: CCCC

2

1: n = 0.2 2: n = 2.0 3: n = 8.0

3 1

0.09

0.06

0.03 a/b = 1.0, a/h = 10, ∆Tbottom = 20 K, ∆Ttop = 100 K

0.00 0.00

0.04

0.08

0.12

0.16

SD/Mean, all random inputs Fig. 9. Effect of the boundary conditions on the frequency dispersion of a ZrO2–FGM–Al square plate with all random input variables changing simultaneously.

mean values. The solid lines and the dashed lines represent the results for the fundamental frequency x1 and the second-mode frequency x2. Natural frequencies are most affected by the random change in the standard

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2 3

0.20

a/b = 1.0, a/h = 10, ∆Tbottom = 20 K, ∆Ttop = 100 K

1 3

SD/Mean, ω1

0.15

1: n = 0.2 2: n = 2.0 3: n = 8.0 : ZrO2-FGM-Al : FGM-Al

0.10 2

0.05 1 0.00 0.00

0.04

0.08

0.12

0.16

SD/Mean, all random inputs Fig. 10. Frequency dispersion for a ZrO2–FGM–Al and an FGM–Al square plate with all random input variables changing simultaneously.

0.25

a/b = 1.0, ∆Tbottom = 20 K, ∆Ttop = 100 K

3 2

1

SD/Mean, ω1

0.20

0.15

1: n = 0.2 2: n = 2.0 3: n = 8.0

0.10

2,3 1

0.05

: a/h = 15 0.00 0.000

0.025

0.050

0.075

: a/h = 8 0.100

SD/Mean, all random inputs Fig. 11. Effect of the side-to-thickness ratio on the frequency dispersion of a ZrO2–FGM–Al square plate with all random input variables changing simultaneously.

deviation of Ec, and are much less influenced by the variation in Em because the mean value of Ec is more than twice that of Em. The effects of randomness in PoissonÕs ratios are very small. The natural frequencies

S. Kitipornchai et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 1075–1095

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0.25

1

2,3 a/h = 10, ∆Tbottom = 20 K, ∆Ttop = 100 K

SD/Mean, ω1

0.20

0.15

1: n = 0.2 2: n = 2.0 3: n = 8.0 2,3

0.10

1

0.05

: a/b = 0.75 0.00 0.00

0.04

0.08

0.12

: a/b = 1.50 0.16

SD/Mean, all random inputs Fig. 12. Effect of the plate aspect ratio on the frequency dispersion of a ZrO2–FGM–Al rectangular plate with all random input variables changing simultaneously.

of the plate with n = 0.2 are most sensitive to changes in Ec, mc but is least sensitive to variations in Em, mm. In contrast, the frequencies of the plate with n = 8.0 are most sensitive to changes in Em, mm, but are least sensitive to variations in Ec, mc. This can be expected because the volume percentage of aluminum increases as the volume fraction index varies from n = 0.2 to n = 8.0. Fig. 7 gives the frequency dispersion of the heated square plate when all of the random input variables Ec, Em, mc, mm, ac, and am are changing simultaneously. In such a case, the variance of both the fundamental and second-mode frequencies increases considerably compared to the results in Figs. 3–6, and the influence of the volume fraction index is seen to be negligible. Fig. 8 compares the frequency dispersion of the plate with and without a temperature change. The solid lines and the dashed lines are for heated plates (DTbottom = 20 K, DTtop = 100 K) and unheated plates (DT = 0 K), respectively. Temperature increment makes the frequency of the plate more sensitive to the standard deviation of all random input variables. Fig. 9 shows the influence of edge support conditions on the frequency dispersion of the square plate. To this end, plates with various boundary conditions have been analyzed but only typical results for a fully clamped plate and a plate that is simply supported at f = 0, 1 and clamped at n = 0, 1 (denoted by ‘‘CCCC’’ and ‘‘SCSC’’ respectively) are provided. It is shown that the dispersion of the fundamental frequency of the CCCC plate is lower than that of the SCSC plate. The effect of different layer-up schemes on the sensitivity of fundamental frequency is also evaluated. Fig. 10 presents the standard deviation of the fundamental frequency for both three-layered ZrO2– FGM–Al (lc = lf = 0.2) and two-layered FGM–Al plates (lc = 0, lf = 0.4). It is interesting to note that unlike the ZrO2–FGM–Al plate, whose results for different volume fraction index n are almost the same when all of the random inputs are varied simultaneously, volume fraction index has a significant influence on the standard deviation of the fundamental frequency of the FGM–Al plate. Finally, we consider the effect of plate geometry on the dispersion of the fundamental frequency. Fig. 11 displays the frequency dispersion of ZrO2–FGM–Al square plates with different side-to-thickness ratios

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a/h = 8, 15. The results in Fig. 12 are for ZrO2–FGM–Al rectangular plates with plate aspect ratios a/b = 0.75, 1.50. The dispersion of the fundamental frequency becomes much higher as a/h increases and a/b decreases, which indicates that the fundamental frequency of thinner plates is very sensitive to the random input variables.

5. Conclusion The random vibration of functionally graded laminated plates with general boundary condition has been investigated by using a semi-analytical approach and a first-order perturbation technique. Thermal prestress due to a temperature change along the plate thickness has been taken into consideration. The second-order statistics of vibration frequencies have been obtained for the plate with randomness in thermo-elastic material properties. The results show that frequency dispersion, which is linear with variations in the random variables involved, is greatly influenced by temperature change, layer-up scheme, boundary conditions and plate geometry. The effect of the volume fraction index on the variation in natural frequencies depends strongly on the layer-up scheme of the plate and for the ZrO2–FGM–Al plate, it is seen to be important when individual material property is taken as the random input variable but is negligible when all of the random input variables are varying simultaneously.

Acknowledgements The work described in this paper was supported by grants from the Australian Research Council (A00104534) and the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU1139/04E). The authors are grateful for this financial support.

Appendix A The ordinary differentiation operators LkiL ; LkiR ði ¼ 1; ...; 4Þ are " # N N 2 2 X X ð4Þ ð2Þ o wmj 2 k      4 o wmk L1L ¼ c1 F 11 C kj wmj þ ðF 12 þ F 21 þ 4F 66 Þb C kj þ F 22 b of2 of2 i¼1 i¼1 ! N X o2 wmk ð2Þ
a2 p x C kj wmj þ py b2 of2 i¼1 " # N N 2 4 X X o f o f mj mk ð4Þ ð2Þ þ B21 C kj fmj þ ðB11 þ B22
2B66 Þb2 C kj þ B12 b4 of2 of4 i¼1 i¼1 " # N N 2 X X ð3Þ ð1Þ o Wxmj 2      
a ðD11
c1 F 11 Þ C kj Wxmj þ ðD12 þ 2D66
c1 F 12
2c1 F 66 Þb C kj of2 i¼1 i¼1 " # N X o3 Wymk ð2Þ oWymj
a ðD12 þ 2D66
c1 F 21
2c1 F 66 Þb þ ðD22
c1 F 22 Þb3 C kj ; of of3 i¼1 ! ( !) N N 2 X X oWymk ð2Þ ð1Þ 2 o wmk k 2 ^ ^ C kj wmj þ b C kj Wxmj þ b þ I5 L1R ¼
a I 1 W þ I 7 ; of of2 j¼1 j¼1

ðA:1Þ

S. Kitipornchai et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 1075–1095

"

N X

N X

4 o2 wmj  4 o wmk þ E b 12 of2 of4 j¼1 j¼1 " # N N 2 4 X X ð4Þ ð2Þ o fmj  4 o fmk þ A22 C kj fmj þ ð2A12 þ A66 Þb2 C kj þ A b 11 of2 of4 j¼1 j¼1 " # N N 2 X X ð3Þ ð1Þ o Wxmj 2       þ a ðB21
c1 E21 Þ C kj Wxmj þ ðB11
B66
c1 E11 þ c1 E66 Þb C kj of2 j¼1 j¼1 " # N X o3 Wymk ð2Þ oWymj ; þ ðB12
c1 E12 Þb3 þ a ðB22
B66
c1 E22 þ c1 E66 Þb C kj of of3 j¼1

Lk2L ¼
c1 E21

ð4Þ

C kj wmj þ ðE11 þ E22
2E66 Þb2

1093

#

ð2Þ

C kj

N X ð1Þ Lk3L ¼ ðA55
6c1 D55 þ 9c21 F 55 Þa2 C kj wmj j¼1 " # N N 3 X X ð3Þ ð1Þ o wmj 2       þ c1 ðF 11
c1 H 11 Þ C kj wmj þ ðF 21 þ 2F 66
c1 H 21
2c1 H 66 Þb C kj of2 j¼1 j¼1 " # N N 3 X X ð3Þ ð1Þ o fmj þ ðB21
c1 E21 Þ C kj fmj þ ðB11
B66
c1 E11 þ c1 E66 Þb2 C kj of2 j¼1 j¼1 " N X ð2Þ þ a ðA55
6c1 D55 þ 9c21 F 55 Þa2 Wxmk
ðD11
2c1 F 11 þ c21 H 11 Þ C kj Wxmj j¼1 # 2   o W xmk
D66
2c1 F 66 þ c21 H 66 Þb2 of2 N

X ð2Þ oWymj ;
a ðD12 þ D66 Þ
c1 ðF 12 þ F 21 þ 2F 66 Þ þ c21 ðH 12 þ H 66 Þ b C kj of j¼1 ! N X ð1Þ k 2 0 0 L3R ¼
a I 3 Wxmk
I 5 C kj wmj ;

ðA:2Þ

ðA:3Þ

j¼1

Lk4L ¼ ðA44
6c1 D44 þ 9c21 F 44 Þa2 b

N

X owmk ð2Þ owmj þ c1 F 12 þ 2F 66
c1 H 12
2c1 H 66 b C kj of of j¼1

" # N 3 3 X o w of o f mk mj mk ð2Þ þ ðB12
c1 E12 Þb2 þ b ðB22
B66
c1 E22 þ c1 E66 Þ C kj þ ðF 22
c1 H 22 Þ of of3 of3 j¼1 N

X ð1Þ oWxmj
a ðD12 þ D66 Þ
c1 ðF 12 þ F 21 þ 2F 66 Þ þ c21 ðH 12 þ H 66 Þ b C kj of j¼1 " N X ð2Þ þ a ðA44
6c1 D44 þ 9c21 F 44 Þa2 Wymk
ðD66
2c1 F 66 þ c21 H 66 Þ C kj Wymj 2

Lk4R

#

j¼1

o Wymk
ðD22
2c1 F 22 þ c21 H 22 Þb2 ; of2   owmk ¼
a2 I 03 Wymk
I 05 b . of ðA:4Þ

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Appendix B Our attention is focused on the plates with both edges f = 0 and f = 1 either simply supported or clamped. The trial functions are gm1 ðfÞ ¼ sinðmpfÞ; gm2 ðfÞ ¼ sin gm f
sinh gm f
cm ðcos gm f
cosh gm fÞ; gm3 ðfÞ ¼ cosðmpfÞ; gm4 ðfÞ ¼ sinðmpfÞ;

ðB:1Þ

for plates that are simply supported at edges f = 0 and f = 1, and gm1 ðfÞ ¼ gm2 ðfÞ ¼ sin gm f
sinh gm f
cm ðcos gm f
cosh gm fÞ; gm3 ðfÞ ¼ gm4 ðfÞ ¼ sinðmpfÞ;

ðB:2Þ

for plates that are clamped at edges f = 0 and f = 1, where cm ¼ ½sin gm
sinh gm =½cos gm
cosh gm ;

gm ¼ ð0.5 þ mÞp.

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