Vibration of non-uniform thick plates on elastic foundation by differential quadrature method

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Engineering Structures 26 (2004) 1473–1482
www.elsevier.com/locate/engstruct
Vibration of non-uniform thick plates on elastic foundationby differential quadrature method

P. Malekzadeh a, G. Karami b,�

a Department of Mechanical Engineering, Persian Gulf University, Bushehr, Iranb Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Dolve 111, P.O. box 5285, Fargo, ND 58105, USA

Received 4 March 2004; received in revised form 13 May 2004; accepted 18 May 2004

Abstract

This paper presents a differential quadrature (DQ) solution for free vibration analysis of thick plates of continuously varyingthickness on two-parameter elastic foundations. The formulations are based on the first-order shear deformation theory takinginto account the effects of rotary inertia. The thickness of the plate may vary in one or two directions. The thickness variationmight be assumed linear or non-linear. Different types of boundary conditions, including free edges and corners, loaded edgeswith in-plane forces are formulated. The accuracy, convergence and versatility of the DQ procedure for the type of plateproblems, with complicated governing differential equations and boundary conditions are examined and verified.# 2004 Elsevier Ltd. All rights reserved.

Keywords: Vibration; Thick plate; Elastic foundation; DQM

1. Introduction

Vibration problems of variable thickness plates on

elastic foundations play an important role in many

structural and foundation engineering problems. Exten-

sive research has been carried out based on the thin

plates theory [1]. It is well known fact that the classical

plate theory over-predicts the eigenfrequencies at all

modes in situations where shear deformations and

rotary inertias become significant. To improve the

accuracy, use of theories that include the effects of

shear deformation and rotary inertia (Mindlin theory)

is recommended especially for moderately thick plates

[2].Many researchers have carried out research on free

vibration analysis of thick plates of variable thickness.

Mikami and Yoshimura [3] applied the collocation

method for the analysis of rectangular Mindlin plates

with linearly varying thickness. Aksu and Al-Kaabi

[4,5] presented a method based on a variational prin-

ciple in conjunction with finite difference technique to

calculate frequencies of rectangular Mindlin plates with

linear and parabolic varying thicknesses. Mizusawa [6]used spline strip method to investigate the free

vibration of Mindlin plates with linear variation ofthickness. In afore-mentioned studies, only plates with

two opposite edges simply supported and with a vary-

ing thickness along one direction have been examined.Sakiyama and Huang [7] used Green function to

obtain approximate solutions for natural frequencies ofthin as well as moderately thick plates. Li [8] presented

an analytical approach for free vibration of sheardeformable beams in order to determine the natural

frequency of elastically restrained non-uniform flex-ural-shear plates under the action of in-plane load.

Mizusawa and Kondo [9] used spline element method

to investigate the free vibration analysis of moderatelythick skew plates with varying thickness in one direc-

tion. More recently, Cheung and Zhou [10] studied thefree vibration of rectangular Mindlin plates with vari-

able thickness using the Rayleigh–Ritz method.Cheung and Zhou assumed a continuous variation in

thickness presented by a power function of the rec-tangular coordinates.A closed-form solution for the natural frequencies

of simply supported Mindlin plates on Pasternak

1474 P. Malekzadeh, G. Karami / Engineering Structures 26 (2004) 1473–1482

foundations and subjected to biaxial initial stresses waspresented by Xiang et al. [11]. Matsunaga [12]employed power series expansion for displacementcomponents in conjunction with Hamilton’s principlesto evaluate the natural frequency of uniform thickplates with simply supported edges. Shen et al. [13]investigated dynamic response of simply supported uni-form thickness moderately thick rectangular plates onPasternak foundations via Rayleigh–Ritz method. Inanother work [14], they studied the free and forcedvibration analysis of completely free moderately thickrectangular plates on Pasternak foundations. Liew et al.[15] presented a literature survey on the early researchworks on vibration of thick plates. In almost all pre-vious investigations, vibration analysis of thick platesof uniform thickness on elastic foundations with orwithout in-plane edge loads have been considered. Itshould be noted that the tapered plates do not have alinearly varying rigidity, since the rigidity of the plate isproportional to the third power of thickness. There-fore, in solving the governing partial differential equa-tions of these plates, in addition to the termscorresponding to the uniform thickness, other linear orhigher-order terms will also appear. Presence of theseterms in the governing partial differential equationsmakes the computation of the natural frequencies morecomplicated than those of uniform thickness plates.Differential quadrature method (DQM) as an

efficient alternative numerical tool for structural analy-sis was promoted by Bert et al. in 1988 [16,17]. Themethod has been widely used for static and freevibration analysis of beams and plates. In applicationto such problems it was concluded that DQM proce-dures offer comparable accuracy with less computation

with those of Rayleigh–Ritz method, finite differenceand finite element methods (FEM) [16]. A review of theearly developments in DQM can be found in thepapers by Bert and Malik [16,18].For the recent developments and applications of the

method to structural mechanics problems one mayrefer to the published works [19–25] from the authors.In these works, the authors proposed a DQ method-ology for the solution of solid mechanics problemmainly having fourth-order differential equations. Themethodology will implement the boundary conditionsefficiently and accurately, a major point of concern forthe conventional DQM schemes. They implemented themethodology for different beam [19–21] and thin plateproblems [22–24] successfully. They also applied themethodology for non-uniform moderately thick beams[25] with general boundary conditions.In this paper, based on first-order shear deformation

theory, the governing equations for variable thicknessmoderately thick plates on two-parameters elastic foun-dations subjected to in-plane edge forces are presented.The appropriate natural boundary conditionsimplementation at the free corners, a troublesome issuein conventional DQM free vibration analysis [21] isaccurately implemented. The DQ discretized form ofthe governing equations and boundary conditions atthe domain and on the boundary grid points are pre-sented. The accuracy and convergence behavior of theproposed DQ is demonstrated via several examples.

2. Governing equations

Consider a moderately thick rectangular plate asshown in Fig. 1. The governing equations and the

Fig. 1. The geometry of a thick plate.

P. Malekzadeh, G. Karami / Engineering Structures 26 (2004) 1473–1482 1475

related boundary conditions can be obtained using theHamilton’s principle in a systematic manner. For brev-ity purposes, only the resulting equations and bound-ary conditions are summarized.

D@2ux

@x2þD

1� t2

� �@2ux

@y2þ @D

@x

� �@ux

@x

þ 1� t2

� �@D

@y

@ux

@yþD

1þ t2

� �@2uy

@x@y

þ t@D

@x

� �@uy

@yþ @D

@y

1� t2

� �@uy

@x

� jh ux þ @w

@x

� �� I

@2ux

@t2¼ 0 ð1Þ

D1þ t2

� �@2ux

@x@yþ t

@D

@y

� �@ux

@xþ @D

@x

1� t2

� �@ux

@y

þD@2uy

@y2þD

1� t2

� �@2uy

@x2þ @D

@y

� �@uy

@y

þ 1� t2

� �@D

@x

@uy

@x�jh uyþ @w

@y

� �� I

@2uy

@t2¼ 0 ð2Þ

jGh@2w

@x2þ @2w

@y2

� �þjG

@h

@x

@w

@xþux

� �

þjG@h

@y

@w

@yþuy

� �þjGh

@ux

@xþ @uy

@y

� �

�ðN̂Nxx�ksÞ@2w

@x2�ðN̂Nyy�ksÞ

@2w

@y2

�2N̂Nxy@2w

@x@yþkww�qh

@2w

@t2¼ 0 ð3Þ

Natural and geometrical external boundary conditionsalong edges x¼ 0 and a:

either w¼ 0 or Qxþðks� N̂NxxÞ@w

@x¼ 0 ð4Þ

either ux¼ 0 or Mx¼ 0 ð5Þ

either uy¼ 0 or Mxy¼ 0 ð6Þ

Natural and geometrical external boundary conditionsalong edges y¼ 0 and b:

either w¼ 0 or Qyþðks� N̂NyyÞ@w

@y¼ 0 ð7Þ

either uy¼ 0 or My¼ 0 ð8Þeither ux¼ 0 or Mxy¼ 0 ð9Þ

Corner boundary conditions:

Simply supported (x-edge)–simply supported(y-edge), simply supported-clamped, clamped-free:

w ¼ 0; ux ¼ 0; uy ¼ 0 ð10Þ

Simply supported-free:

w ¼ 0; ux ¼ 0; Mxy ¼ 0 ð11ÞFree-simply supported:

w ¼ 0; uy ¼ 0; Mxy ¼ 0 ð12ÞFree-free:

Qx þ ðks � N̂NxxÞ@w

@x¼ 0;

Qy þ ðks � N̂NyyÞ@w

@y¼ 0; Mxy ¼ 0 ð13Þ

where

Mx ¼ D@ux

@xþ t

@uy

@y

� �;

My ¼ D t@ux

@xþ @uy

@y

� �;

Mxy ¼ D1� t2

� �@ux

@yþ @uy

@x

� �;

Qx ¼ jGh@w

@xþ ux

� �; Qy ¼ jGh

@w

@yþ uy

� �Free vibration is a harmonic motion; therefore, the

following solutions may be assumed for the field vari-ables:

wðx;y;tÞ ¼ Wðx;yÞsinðxtÞ; uxðx;y;tÞ¼ wxðx;yÞsinðxtÞ; uyðx;y;tÞ¼ wyðx;yÞsinðxtÞ ð14Þ

3. Review of the differential quadrature method

The method of differential quadrature is based onthe idea that the partial derivative of field variable atthe ith discrete point in the computational domain isapproximated by a weighted linear sum of the values ofthe field variable along the line passes through thatpoint which is parallel with the coordinate direction ofthe derivative [16]. Consider a two dimensional fieldvariable u(x,y), the mth order derivative of it withrespect to x, and the ðmþnÞth order derivative of itwith respect to x and y is approximated as

@mu

@xm

��ðxi ;yjÞ

¼XNx

k¼1A

ðmÞik uðxk;yj;tÞ ¼

XNx

k¼1A

ðmÞik ukjðtÞ ð15Þ

@ðmþnÞu

@xm@yn

��ðxi ;yjÞ

¼XNx

k¼1A

ðmÞik A

ðnÞjl uðxk;yl ;tÞ

¼XNx

k¼1A

ðmÞik A

ðnÞjl uklðtÞ ð16Þ

There are two key points in the successful appli-cation of the differential quadrature method, one ishow to determine the weighting coefficients and theother is how to select the grid points. The methoddeveloped by Shu and Richard [26] is claimed to be


computationally more accurate than other methods[16]. According to Shu and Richard rule [26], theweighting coefficients of the first-order derivatives in n-direction (n ¼ x or y) are determined as

Að1Þij ¼

MðniÞðni�njÞMðnjÞ

for i 6¼ j

�PNn

j¼1i 6¼jA

ð1Þij for i¼ j

8>>><>>>:

; i; j¼ 1;2 . . . ;Nn ð17Þ

where

MðniÞ¼YNn

j¼1;i 6¼jðni�njÞ

In order to evaluate the weighting coefficients ofhigher-order derivatives, recurrence relations arederived as

AðrÞij ¼ r A

ðr�1Þii A

ð1Þij �

Aðr�1Þij

ni � nj

24

35

for i;j ¼ 1;2; . . . ;Nn; i 6¼ j and 2 � r � Nn � 1 ð18Þ

AðrÞii ¼ �

XNn

j¼1;i 6¼jA

ðrÞij for i ¼ 1;2; . . . ;Nn and 1

� r � Nn � 1 ð19ÞThe natural and simplest choice of the grid points is

equally spaced points in the direction of the coordinateaxes of the computational domain. It was demonstratedthat non-uniform grid points gives a better results withthe same number of equally spaced grid points [16]. Inthis paper, we choose these set of grid points in terms ofnatural coordinate directions x and y as

xi ¼1

2½1� cos

ði � 1ÞpðNx � 1Þ

� �;

yj ¼1

2½1� cos

ðj � 1ÞpðNy � 1Þ

� �for i ¼ 1;2; . . . ;Nx and j ¼ 1;2; . . . ;Ny ð20Þ

4. Differential quadrature analogs

To formulate the eigenvalue equations, the govern-ing differential equations with their associated bound-ary conditions are transformed into algebraic equationsvia DQM. The derivatives of the field variables may betransformed into computational domain efficiently bymodifying the weighting coefficients as

Axij ¼

Ax

ij

a; Bxij ¼

Bx

ij

a2; Ay

ij ¼Ay

ij

b; Byij ¼

By

ij

b2ð21Þ

The weighting coefficients Ax

ij , Bx

ij , Ay

ij, By

ij are eval-uated using the general differential quadrature rule(GDQR) of Shu and Richards [26]. Using (14) and (21),DQ analogs of the governing equations and boundaryconditions take the following forms, respectively.

4.1. Governing equations:

Eq. (1):

�Dij

XNx

m¼1Bximwx

mj þDij1� t2

� �XNy

n¼1Byjnw

xin

þDxij

XNx

m¼1Aximwx

mj þ1� t2

� �Dyij

XNy

n¼1Ayjnw

xin

þDij1þ t2

� � XNx

m¼1

XNy

n¼1AximA

yjnw

ymn

!

þ tDxij

XAyjnw

yin þDy

ij

1� t2

� �XNx

m¼1Aximwy

mj

� jGhij wxij þ

XNx

m¼1Aximwmj

!þ x2Iijw

xij ¼ 0 ð22Þ

Eq. (2):

�Dij1þ t2

� � XNx

m¼1

XNy

n¼1AximA

yjnw

xmn

!

� t Dyij

� �XNx

m¼1Aximwx

mj þDij

XNy

n¼1Byjnw

yin

!

�Dij1� t2

� �XNx

m¼1Bximwx

mj þDyij

XNy

n¼1Ayjnw

yin

þDij

XNy

n¼1Byjnw

yin

!� jGAij

XNy

n¼1AyjnWin þ wy

ij

!

þ x2Iijwyij ¼ 0 ð23Þ

Eq. (3):

jGhijXNx

m¼1BximWmj þ

XNy

n¼1ByjnWin

!

þ jGhðxÞij

XNx

m¼1AximWmj þ wx

ij

!

þ jGhðyÞij

XNy

n¼1AyjnWin þ wy

ij

!

þ jGhijXNx

m¼1Aximwx

mj þXNy

n¼1Ayjnw

yin

!

� ðN̂Nxx � ksÞXNx

m¼1BximWmj

!

� ðN̂Nyy � ksÞXNy

n¼1ByjnWin

!

� 2N̂Nxy

XNx

m¼1

XNy

n¼1AximA

yjnWmn

!

þ kwWij þ qhijx2Wij ¼ 0 ð24Þ


where

ð Þij ¼ ð Þ��ðxi ;yjÞ

; ð ÞðxÞij ¼ @ð Þ@x

��ðxi ;yjÞ

;

ð ÞðyÞij ¼ @ð Þ@y

��ðxi ;yjÞ

4.2. Boundary conditions:

For brevity purposes, only DQ analogs of theboundary conditions at x-edge will be presented. In asimilar manner, DQ analogs for general situationboundary conditions will be developed:Eq. (4):

Wij ¼ 0; jGhijXNx

m¼1AximWmj þ wx

ij

!

þ ðks � N̂NxxÞXNx

m¼1AximWmj ¼ 0 ð25Þ

Eq. (5):

wxij ¼ 0;

XNx

m¼1Aximwx

mj þ tXNy

n¼1Ayjnw

yin ¼ 0 ð26Þ

Eq. (6):

wyij ¼ 0;

XNy

n¼1Ayjnw

xin þ

XNx

m¼1Aximwy

mj ¼ 0 ð27Þ

To establish the eigenvalue system of equations, thedegrees of freedom are separated into the domain andthe boundary degrees of freedom as

fUgd ¼Wwx

wy

8<:

9=;d

; fUgb ¼Wwx

wy

8<:

9=;b

ð28Þ

where subscripts ‘d’ and ‘b’ refer to domain and bound-ary, respectively. Rearranging the discretized equations,their assembled form in matrix form becomes

½Sdb�fUgb þ ½Sdd �fUgd � x2½M�fUgd ¼ 0 ð29ÞIn a similar manner, the discretized form of the

boundary conditions become

½Sbb�fUgb þ ½Sbd �fUgd ¼ 0 ð30ÞIn the above equations, the elements of stiffness

matrices Sdd, Sdb, Sbd, and Sbb are easily obtainedbased on the definition of vectors of domain andboundary degrees of freedom from the related equa-tions. Eliminating {U}b from Eq. (25), one has

½S�fUgd � x2½M�fUgd ¼ 0 ð31Þ

where

½S� ¼ ½Sdd � � ½Sdb�½Sbb��1½Sbd �

Solving the above equations, natural frequencies aswell as mode shapes can be obtained. The procedure

can be used for static and stability analysis of thickplates of variable thickness as well as thin plates.At this stage one can compare DQM solution pro-

cedure with the other two widely used traditional meth-ods for plate analysis, i.e., Rayleigh–Ritz method andFEM. The main difference between the DQM and theother methods is how the governing equations are dis-cretized. In DQM the governing equations and bound-ary conditions are directly discretized, and thuselements of stiffness and mass matrices are evaluateddirectly. But in Rayleigh–Ritz and FEMs the weakform of the governing equations should be developedand the boundary conditions are satisfied in the weakform. Generally by doing so larger number of integralswith increasing amount of differentiation should bedone to arrive at the element matrices. Also, the num-ber of degrees of freedom will be increased for anacceptable accuracy.Based on this differential quadrature algorithm, a

general computer code has been created, which can beused not only for free vibration analysis but also forstatic and stability analysis of thick as well as thinplates. The evaluation of matrices needed in static,vibration and stability analyses, are easier than thoseof FEM and Rayleigh–Ritz methods. Also, evaluationof load vectors is more straightforward. Stability analy-sis is similar to free vibration with the difference thatinstead of mass matrix, one should obtain the coef-ficient matrix of the in-plane loads [23]. Employing theconcept of domain decomposition, the code has beenextended into differential quadrature element method(DQEM), which has the ability to solve almost all non-uniform plate types problems. The non-uniformdomain will be divided into sub-domains, or elements,similar to element procedure in conventional FEM. Inaddition to satisfying the governing equations in eachdomain, the external boundary conditions as well asgeometrical and natural compatibility conditions atcommon sections of the neighboring sub-domains orelements are enforced. For more details one can referto Refs. [19,21,22,25].

5. Numerical examples

In this section, different examples of thick plates withdifferent types of complexities are solved. Without anyloss of generality, equal number of grid points in bothdirections will be assumed, that is, Nx ¼ Ny. In orderto make reasonable comparison, the values of shearcorrection factor are set to those used in literature forany comparisons, that is, j ¼ 5=6 in Sections 5.1, 5.2

and 5.4, and j ¼ p2=12 in Section 5.3. Poisson’s ratiosin all examples were assumed 0.3 unless noted other-wise. The notations S, C, and F will be used for simplysupported, clamped, and free edges, respectively. Theedges of the plate are numbered from 1 to 4 as shown


in Fig. 1. Accordingly, the boundary condition of platefor the entire boundary is referred to as 1234. From thedefinition of the boundary conditions along the edges,the symbolism S–C–S–F, for example, identifies a platewith edge 1 simply supported, edge 2 clamped, edge 3simply supported, and edge 4 free. The relative percent-age errors are measured in comparison with the exactsolutions (if any) and or other established convergednumerical solutions. That is,

%Relative error ¼ xipresent � xiOM

xiOM

�� 100 ð32Þ

where xiOM is the ith natural frequency.

5.1. Plates on two-parameters elastic foundation

To check the convergence and accuracy of the algor-ithm for plates on elastic foundations, two examplesare considered. The first example is concerned with asimply supported plate under different values for thestiffness of the foundation. The results for fundamentalnon-dimensional natural frequency parameters

ð- ¼ xb2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqho=Do

pÞ are compared with the solutions of

Matsunaga [12]. The variations of error of the presentanalysis versus different number of grid points areplotted in Fig. 2. It is shown that by using five gridpoints, the error is less than 1%. Also, the stability ofconvergence behaviors of the solutions is confirmed. Inthe second example, a completely free plate resting ontwo-parameter elastic foundations are considered. Thevalue obtained by Shen et al. [14] presented symmetric–symmetric modes of non-dimensional natural fre-

quencies (- ¼ xb2=p2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqho=Do

p) for this example. The

fundamental natural frequencies for all aspect ratiosare identical (1.414) as reported by Shen et al. [14]. Thepresent method yields this number at five grid points.The relative errors for second and third symmetric–symmetric modes of non-dimensional natural fre-quencies with respect to different number of grid pointsare presented in Fig. 3. Different values for thickness tolength ratio are examined, which actually verify theapplicability of the method to both thin as well asmoderately thick plates. Here, also excellent agreementhas been achieved for second and third frequenciesevaluated with only seven grid points.

5.2. Plates under in-plane loads at edges

As another example, thin as well as thick plates sub-jected to in-plane biaxial edge forces are considered.The convergence and accuracy of the fundamental andsecond non-dimensional natural frequencies(- ¼ xb2=p2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqho=Do

p) for the plates with all edges

clamped are exhibited in Figs. 4–7. Different values forin-plane loads and also for the aspect ratios are used.The relative errors are measured with respect to thoseof pb2–Ritz method (Liew et al. [27]). DQM with ninegrid points yield comparable results with those of Liewet al. [27] for both tensile and compressive in-planebiaxial forces.

5.3. Plates with continuously varying section

In this part, example of plates with continuouslyvarying thickness is presented. Moderately thick plates

Fig. 2. The relative error of the fundamental frequency parameter

of moderately plates on two-parameters elastic foundations (a=b ¼ 1,

ho=b ¼ 0:2) with respect to increase in the number of grid points.

Fig. 3. The relative error for non-dimensional symmetric–symmetric

natural frequencies of completely free (FFFF) square plates on two-

parameters elastic foundations (a=b ¼ 1; Kw ¼ 2p4, Ks ¼ 0:4p2) withrespect to increase in the number of grid points.


having linear thickness variation in y-direction with

mixed boundary conditions, i.e., SCSF, is considered.

Similar plate problems have been solved by Mizusawa

[6] using spline strip method with a linear variation in

thickness, i.e., h ¼ hoð1þbgÞ. The steepest taper ratio

used by Mizusawa, i.e., b ¼ 1, is to be employed here.

The convergence behaviors of the first six non-dimen-

sional natural frequencies (- ¼ xb2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqho=Do

p) are

presented in Table 1. Excellent agreements are observedfor all cases with the solutions of Mizusawa [6].

5.4. Plates with continuously varying thicknesson two-parameters elastic foundation

The first three non-dimensional natural frequencies(- ¼ xa2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqho=Do

p) of bi-linearly varying thickness

plates, i.e., h ¼ hoð1þ anÞð1þ bgÞ, under biaxial com-

Fig. 4. The relative error for the fundamental frequency with

respect to increasing number of grid points for pre-stressed plates

under N ¼ 0:5Ncr.

Fig. 5. The relative error for the second frequency with respect to

increasing number of grid points for pre-stressed plates under

N ¼ 0:5Ncr.

Fig. 6. The relative error for the fundamental frequency with

respect to increasing number of grid points for pre-stressed plates

under N ¼ 0:5Ncr.

Fig. 7. The relative error for the second frequency with respect to

increasing number of grid points for pre-stressed plates under

N ¼ 0:5Ncr.


pression and on two-parameters elastic foundations are

presented in Tables 2–5. In all cases, 13 grid points in

each direction are used. The impact of foundation

parameters on natural frequencies of plates with differ-

ent boundary conditions is exhibited. In each case, two

different values for the aspect ratios and also for thick-

ness to length ratio are considered. To show the impact

of the axial load on the fundamental natural frequency,

in each case, the value of the axial compressive force,

i.e., N̂No, is chosen less than the critical buckling load.

Table 1

Convergence and accuracy of first six non-dimensional natural frequencies of continuously varying section SCSF plates (a ¼ 0, b ¼ 1, ho=b ¼ 0:1)

a
=b N y - 1 -2 - 3 - 4 - 5 -6
Present 1
.0 7 1 8.840 40.217 57.975 74.495 81.456 108.17
9 1
8.643 40.435 57.754 75.014 80.006 108.17
1
3 1 8.639 40.428 57.751 74.981 80.160 108.17
1
5 1 8.641 40.427 57.753 74.978 80.159 108.17
1
9 1 8.641 40.427 57.753 74.977 80.159 108.17
[6]
1 8.641 40.427 57.754 74.977 80.159 108.17
2
.0 7 2 9.793 75.359 1 17.65 1 45.11 1 60.87 223.00
9 2
9.095 74.572 1 18.12 1 44.28 1 61.74 223.46
1
3 2 9.070 74.557 1 18.15 1 44.26 1 61.71 223.38
1
5 2 9.072 74.563 1 18.15 1 44.27 1 61.71 223.37
1
9 2 9.073 74.565 1 18.15 1 44.27 1 61.71 223.37
[6]
2 9.073 74.565 1 18.15 1 44.27 1 61.71 223.37
Table 2

Results for SSSS bi-linearly varying thickness plates on two-parameters elastic foundations (a ¼ b ¼ 1:0, N̂N=N̂No ¼ 100)

a=b b
=ho Mode K s ¼ 0 Ks ¼ 10
Kw ¼ 1
102 1 03 104 1 06 Kw ¼ 1 1 02 103 1 04 106
1 0
.1 1 3 .65 3.71 4 .24 7.63 2 2.86 3.78 3 .84 4.35 7.69 22.86
2 7
.81 7.84 8 .12 10.41 2 4.97 7.98 8 .01 8.28 1 0.51 24.97
3 7
.98 8.00 8 .25 10.49 2 7.79 8.12 8 .15 8.39 1 0.60 27.79
0
.2 1 2 .15 2.26 3 .03 6.56 6.57 2.38 2 .47 3.18 6.56 6.57
2 4
.41 4.46 4 .93 6.58 7.63 4.71 4 .76 5.20 6.63 7.63
3 4
.58 4.63 5 .05 7.91 8.99 4.83 4 .88 5.27 7.94 8.99
2 0
.1 1 2 .39 2.40 2 .46 2.95 1 5.27 2.43 2 .43 2.49 2.98 15.27
2 3
.74 3.74 3 .77 4.11 1 6.45 3.77 3 .77 3.81 4.14 16.46
3 5
.65 5.65 5 .67 5.90 1 7.50 5.68 5 .68 5.71 5.93 17.51
0
.2 1 1 .84 1.85 1 .91 2.49 6.04 1.88 1 .89 1.95 2.52 6.04
2 2
.73 2.74 2 .78 3.20 7.03 2.78 2 .78 2.83 3.24 7.03
3 3
.90 3.91 3 .94 4.25 7.17 3.95 3 .96 3.99 4.30 7.17
Table 3

Results for CCCC bi-linearly varying thickness plates on two-parameters elastic foundations (a ¼ b ¼ 1:0, N̂N=N̂No ¼ 200)

a=b b
Kw ¼ 1
102 1 03 1 04 1 06 K w ¼ 1 1 02 103 1 04 106
1 0
.1 1 5 .38 5.43 5 .81 8.70 2 8.65 5 .48 5 .52 5.90 8.75 28.65
2 9
.27 9.30 9 .53 1 1.57 2 9.19 9 .41 9 .44 9.67 1 1.67 29.19
3 9
.40 9.42 9 .64 1 1.57 3 1.77 9 .53 9 .55 9.76 1 1.67 31.77
0
.2 1 2 .16 2.27 3 .12 7.07 9.41 2 .44 2 .54 3.30 7.14 9.41
2 3
.45 3.53 4 .15 7.63 9.70 3 .91 3 .98 4.52 7.74 9.70
3 3
.84 3.91 4 .42 7.73 1 0.68 4 .17 4 .23 4.70 7.85 10.68
2 0
.1 1 3 .97 3.97 4 .01 4.34 1 6.28 3 .99 4 .00 4.03 4.36 16.29
2 5
.06 5.06 5 .09 5.34 1 7.13 5 .08 5 .09 5.11 5.36 17.14
3 6
.75 6.75 6 .77 6.96 1 8.08 6 .78 6 .78 6.80 6.99 18.09
0
.2 1 2 .49 2.50 2 .56 3.04 8.08 2 .53 2 .53 2.59 3.06 8.08
2 3
.12 3.12 3 .16 3.55 8.75 3 .16 3 .16 3.20 3.58 8.75
3 4
.07 4.08 4 .11 4.41 9.34 4 .12 4 .12 4.15 4.46 9.34


6. Conclusions

A DQ procedure for free vibration analysis of vari-able thickness moderately thick plates on two-para-meter elastic foundations under in-plane edge forceswas developed. The completeness of the present DQMwas demonstrated for plates under different complex-ities in boundary conditions as well as the thicknessvariation. Accurate results were obtained for higher-order modes of vibration with only few grid points,which shows the advantageous point of low compu-tation cost of the method. Free corner boundary con-ditions were formulated and tested for severalexamples.

A
cross sectional area of plate
AðrÞij

rth weighting coefficients

Aij
first-order weighting coefficients
a
mid plane length of plate
Bij
second-order weighting coefficients
b
mid plane width of plate
Do
flexural stiffness of plate at origin of the coor-dinate system
E
Young’s modulus G Shear modulus h, ho, H plate thickness at an arbitrary point of plate,
plate thickness at the origin of coordinate sys-tem, plate thickness at x¼a and y¼b, respect-ively.

ks, kw
elastic coefficients of shear layer and Winklerfoundation, respectively
Ks ¼ ksa2

Donon-dimensional elastic coefficient of shear

layer foundation

Kw ¼ kwa4

Donon-dimensional elastic coefficient of Wink-

ler foundation
I second moment of cross sectional area of
plate
[M] mass matrix Mx, My, Mxy bending moment at x-surface, bendingmoment at y-surface, and twisting moment Nx, Ny number of grid points in x and y-direction
N̂N ¼ ðN̂Nxx ¼ N̂NyyÞ magnitude of axial compression

Table 4

Results for CFCF bi-linearly varying thickness plates on two-parameters elastic foundations (a ¼ b ¼ 1:0, N̂N=N̂No ¼ 100)

a=b b
Kw ¼ 1
102 1 03 1 04 1 06 K w ¼ 1 1 02 103 1 04 1 06
1 0
.1 1 3 .45 3.53 4 .18 7 .41 2 2.23 3 .57 3 .64 4.27 7 .45 2 2.23
2 4
.27 4.32 4 .72 8 .18 2 5.60 4 .37 4 .42 4.82 8 .24 2 5.60
3 6
.42 6.45 6 .75 9 .31 2 5.72 6 .62 6 .65 6.94 9 .44 2 5.73
0
.2 1 1 .62 1.78 2 .79 5 .25 6.82 1 .96 2 .10 2.98 5 .31 6.83
2 2
.23 2.31 3 .00 6 .75 8.24 2 .36 2 .43 3.09 6 .81 8.24
3 3
.23 3.29 3 .78 6 .98 8.47 3 .60 3 .65 4.09 7 .05 8.47
2 0
. 1 1 1 .10 1.12 1 .24 2 .05 1 3.29 1 .13 1 .14 1.26 2 .06 1 3.30
2 1
.68 1.69 1 .76 2 .41 1 4.55 1 .73 1 .74 1.81 2 .44 1 4.56
3 2
.72 2.73 2 .78 3 .27 1 5.30 2 .75 2 .76 2.81 3 .30 1 5.31
0
.2 1 0 .834 0.853 1 .01 1 .79 5.65 0 .867 0 .885 1.03 1 .82 5.65
2 1
.04 1.05 1 .14 2 .00 6.49 1 .10 1 .11 1.20 2 .02 6.49
3 1
.93 1.93 2 .01 2 .59 6.65 1 .96 1 .97 2.04 2 .61 6.65
Table 5

Results for CFFF bi-linearly varying thickness plates on two-parameters elastic foundations (a ¼ b ¼ 1:0, N̂N=N̂No ¼ 10)

a=b b
Kw ¼ 1
102 1 03 1 04 1 06 K w ¼ 1 1 02 103 1 04 1 06
1 0
.1 1 0 .486 0.781 1 .93 5 .54 1 8.44 0 .619 0 .870 1.98 5 .57 1 8.45
2 1
.30 1.44 2 .43 6 .40 1 9.82 1 .50 1 .63 2.53 6 .46 1 9.83
3 3
.35 3.41 3 .94 7 .17 2 1.86 3 .51 3 .57 4.07 7 .21 2 1.87
0
.2 1 0 .402 0.718 1 .78 4 .19 5.19 0 .544 0 .807 1.85 4 .22 5.19
2 0
.881 1.05 2 .08 4 .23 5.70 1 .11 1 .25 2.17 4 .27 5.71
3 2
.23 2.31 2 .89 5 .57 6.55 2 .40 2 .46 2.99 5 .61 6.56
2 0
.1 1 0 .124 0.197 0 .497 1 .45 1 2.10 0 .157 0 .220 0.508 1 .46 1 2.11
2 0
.571 0.593 0 .767 1 .74 1 2.41 0 .671 0 .689 0.843 1 .77 1 2.42
3 1
.01 1.02 1 .14 1 .95 1 4.08 1 .04 1 .06 1.17 1 .96 1 4.09
0
.2 1 0 .109 0.187 0 .484 1 .36 4.74 0 .145 0 .211 0.498 1 .38 4.74
2 0
.413 0.437 0 .625 1 .57 5.13 0 .522 0 .542 0.699 1 .60 5.13
3 0
.843 0.858 0 .982 1 .81 5.75 0 .880 0 .894 1.01 1 .82 5.75


N̂Ncr critical buckling load of plate under biaxial com-pression

N̂No ¼ p2 hoa

� �2a load parameter

Qx, Qy
shear force at x and y surface [S], [Sdd], [Sdb], [Sbd] and [Sbb] stiffness matrices w(x,y,t),W(x,y)
transverse displacement, mode shape of freevibration of transverse displacement, respect-ively

x, y, z
coordinate variables a taper ratio for thickness of plate in x-direction b taper ratio for thickness of plate in y-direction j shear correction factor /x(x,y,t), /y(x,y,t) bending rotation about y and xaxis, respectively wx(x,y),wy(x,y)
mode shape of free vibration of bendingrotation about y and x axis, respectively

q
volume density t Poisson’s ratio x angular natural frequency - non-dimensional natural frequency (defined in
the text)
n ¼ x=a non-dimensional coordinate variable in x-
direction
g ¼ y=b non-dimensional coordinate variable in y-
direction

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Vibration of non-uniform thick plates on elastic foundation by differential quadrature method

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