Vibration analysis of corrugated Reissner–Mindlin plates using a mesh-free Galerkin method

ARTICLE IN PRESS

International Journal of Mechanical Sciences 51 (2009) 642–652

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences

0020-74

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ijmecsci

Vibration analysis of corrugated Reissner–Mindlin plates using a mesh-freeGalerkin method

K.M. Liew a,�, L.X. Peng b, S. Kitipornchai a

a Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, Chinab College of Civil Engineering and Architecture, Guangxi University, 530004 Nanning, China

a r t i c l e i n f o

Article history:

Received 23 July 2008

Received in revised form

18 March 2009

Accepted 13 June 2009Available online 21 June 2009

Keywords:

Mesh-free galerkin method

Meshless

Corrugated plate

Free vibration

Equivalent properties

03/$ - see front matter & 2009 Elsevier Ltd. A

016/j.ijmecsci.2009.06.005

esponding author.

ail address: [email protected] (K.M. Liew)

a b s t r a c t

A mesh-free Galerkin method for the free vibration analysis of unstiffened and stiffened corrugated

plates is introduced in this paper, in which the corrugated plates are simulated with an equivalent

orthotropic plate model. To obtain the corresponding equivalent elastic properties for the model, a

constant curvature state is applied to the corrugated sheet. The stiffened corrugated plates are treated

as composite structures of equivalent orthotropic plates and beams, and the strain energies of the plates

and beams are added up by the imposition of displacement compatible conditions between the plate

and the beams. The stiffness matrix of the whole structure is then derived. The proposed method is

superior to the finite element methods (FEMs) because no mesh is needed, and thus stiffeners (beams)

do not need to be placed along the mesh lines and the necessity of remeshing when the positions of the

stiffeners change is avoided. To demonstrate the accuracy and convergence of the proposed method,

several numerical examples are analyzed both with the proposed method and the finite element

commercial software ANSYS. Examples from other research are also employed. A good agreement

between the results for the proposed method, the results of the ANSYS analysis, and the results from

other research is observed. Both sinusoidally and trapezoidally corrugated plates are studied.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Plates that are reinforced with corrugations (Fig. 1) can achievea higher strength than flat plates, and can thus improve thestrength/weight ratio of structures that are made of them.Because of these advantages, corrugated plates are popularmaterials for various engineering structures, such as decking,roofing, and sandwich plate cores.

The precise analysis of corrugated plates is quite complex andtime consuming. The analysis of trapezoidally corrugated platesinvolves sheets lying in different planes and the transformation ofevery parameter that is expressed in the local coordinates of asheet to a global coordinate. For sinusoidally corrugated plates,shell structures must be employed in the analysis, which requiresa large amount of computation. A simple and valid way to studycorrugated plates is to analyze them as orthotropic plates [1–12]of uniform thickness and equivalent rigidities (Fig. 2), and anapproximated solution can thus be obtained. The approximationapproach saves a great amount of effort with only a smallloss of precision. Equivalent rigidities play a crucial role in theapproximation approach, and their correct estimation is the key.

ll rights reserved.

.

The earliest estimation of equivalent rigidities is found in thework of Seydel [1], and for many years, researchers followedSeydel’s formulas. Lau [6] improved on these formulas by derivingthe theoretically correct form for the developed length l andmoment inertia I. Briassoulis [7] studied the classical expressions[4,5], for equivalent rigidities, and obtained new and more preciseexpressions for the extensional rigidity and flexure rigidity ofsinusoidally corrugated plates through the imposition of aconstant strain state on the corrugated sheet. Shimansky andLele [8] constructed an analytical model for the initial transversestiffness of sinusoidally corrugated plates that incorporated thedeformation that is caused by extension, shear, and bending. Theyobtained a simple approximate polynomial expression for theinitial transverse stiffness of a thin plate, and found thattransverse stiffness is not negligible for thick plates with asmall degree of corrugation. Following the approach thatwas introduced by Briassoulis [7], Samanta and Mukhopadhyay[10] derived new expressions for the equivalent extensionalrigidity of trapezoidally corrugated plates, and carried out ageometric nonlinear and free vibration analysis of suchplates. To introduce shear deformation theory in the analysisof corrugated plates, Semenyuk and Neskhodovskaya [11] andMachimdamrong et al. [12] developed the equivalent ex-pressions for the transverse shear modulus. Semenyuk andNeskhodovskaya [11] also discovered the instances when a

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ARTICLE IN PRESS

Fig. 1. Sinusoidally and trapezoidally corrugated plates.

L

W

h

L

h

W

x

y

z

y

z

x

Fig. 2. Sinusoidally corrugated plate and its equivalent plate.

K.M. Liew et al. / International Journal of Mechanical Sciences 51 (2009) 642–652 643

corrugated shell should not be treated as an equivalent of anorthotropic circular shell.

After the correct estimation of the equivalent rigidities, theremaining steps are much easier, and the only remaining task isthe study of an equivalent orthotropic plate other than a fullycorrugated plate, either by theoretical methods or numericalmethods. Stiffened corrugated plates can also be considered asstiffened orthotropic plates. In the field of free vibration analysisof plates, a closed-form solution for initially stressed thickrectangular plates was presented by Xiang et al. [13]. The p-Ritzmethod was used to study the symmetrically laminated thickrectangular plates [14,15]. The differential quadrature method wasemployed to study circular [16] and rectangular plates [17,18]. Allthese works are contributed to the studies of dynamic behaviorsof plates.

Due to their easy implementation with computers, numericalmethods are very popular in engineering, and among thosemethods, the finite element methods (FEMs) are the mostconvenient, because they can be applied to large complexstructures, and varied boundary and loading conditions can easilybe applied. Nevertheless, the FEMs are not perfect, largely becausethey base their solutions on meshes. For dramatically largedeformation and crack propagation problems, FEMs have diffi-culty in dealing with discontinuities that do not coincide with theoriginal meshlines, and much effort is expended in remeshing ateach step of the problem development. For stiffened plateproblems, most FEMs require stiffeners to be placed along themeshlines, and any change in the position of the stiffeners meansthat the plate has to be remeshed. Because of these disadvantages,researchers have been searching for another powerful numericaltool as an alternative to the FEMs.

In recent years, some numerical methods that are known asmeshless, or mesh-free, methods have gained more and moreattention in the field [19–29]. Unlike the FEMs, the meshlessmethods construct their approximation solutions for problems

entirely in terms of orderly or scattered points that are distributedon the domain of the problem structure that is being studied, andno other element or interrelationship is needed. Meshlessmethods are thus more applicable than FEMs to moving boundaryproblems, crack growth with arbitrary and complex paths, andphase transformation problems. Without the meshes, the afore-mentioned difficulties that are usually encountered by the FEMsdisappear.

The objective of this paper is to introduce a mesh-free Galerkinmethod that is based on the first shear deformation theory(FSDT) for the free vibration analysis of unstiffened and stiffenedcorrugated plates. The corrugated plates are analyzed as equiva-lent orthotropic plates, and several numerical examples thatare computed using the proposed method, the FEM softwareANSYS, and the methods of other researchers are presented forcomparison.

2. The mesh-free Galerkin method

By employing a moving least-square approximation, a functionv(x) in a domain O can be approximated by vh(x) in the sub-domain Ox and

vhðxÞ ¼Xm

i¼1

qiðxÞbiðxÞ ¼ qTðxÞbðxÞ, (1)

where qi(x) are the monomial basis functions, bi(x) are thecorresponding coefficients, h is a factor that measures the domainof influence of the nodes, and m is the number of basis functions.A quadratic basis qT

¼ [1, x, y, x2, xy, y2] (m ¼ 6) is used in thispaper. The minimization of a weighted discrete L2 norm

G ¼Xn

I¼1

$ðx� xIÞ½qðxIÞTbðxÞ � vI�

2 (2)

ARTICLE IN PRESS

K.M. Liew et al. / International Journal of Mechanical Sciences 51 (2009) 642–652644

with respect to b(x) leads to a set of linear equations

BðxÞbðxÞ ¼ AðxÞv, (3)

where $(x�xI) is the weight function and vI are the nodalparameters.

Therefore,

bðxÞ ¼ B�1ðxÞAðxÞv. (4)

The substitution of Eq. (4) into Eq. (1) gives the approximationvh(x), which is expressed in a standard form as

vhðxÞ ¼Xn

I¼1

NIðxÞvI , (5)

where

NIðxÞ ¼ qTðxÞB�1ðxÞqðxIÞ$ðx� xIÞ. (6)

For the detailed formulations of $(x�xI), B(x) and A(x), pleaserefer to our work [24].

z

y

Ny

v

b

Ny M

z

y

ββ

M

Fig. 4. Cross section of the trapezoidally corrugated plate.

3. Estimation of equivalent properties

Corrugated plates show different flexure characteristics in twoperpendicular directions. Therefore, the free vibration analysis of acorrugated plate can be altered to analyze a correspondinglyequivalent orthotropic plate. Assume that Young’s modulus andPoisson’s ratio of a trapezoidally corrugated plate are E and m,respectively (the dimensions of one corrugation of such a plateare shown in Fig. 3(a)). The stress–strain relations of thecorresponding equivalent orthotropic plate are

sx

sy

txy

8><>:

9>=>; ¼

1

ð1� mxmyÞ

Ex Em 0

Em Ey 0

0 0 ð1� mxmyÞGxy

264

375

�x

�y

gxy

8><>:

9>=>;

and

txz

tyz

( )¼

Gxz 0

0 Gxy

" #gxz

gyz

( ). (7)

3.1. Trapezoidally corrugated plates

After a similar process of derivation as that in [24], we obtainthe equivalent elastic properties of trapezoidally corrugated plates(Fig. 4) as follows:

Ex ¼Eð1� mxmyÞc

ð1� m2Þl, (8)

Ey ¼my

mx

Ex, (9)

Em ¼ mEx, (10)

bw

c

F

Z

Xθ

Zl

Fig. 3. One corrugation of the (a) trapezoidal

Gxy ¼E

2ð1þ mÞ. (11)

In this paper, we take into account the transverse shear stresses.Gxz and Gyz are calculated in the same way as in the study ofSemenyuk and Neskhodovskaya [11].

3.2. Sinusoidally corrugated plates

For sinusoidally corrugated plates (Fig. 3(b), the corrugation isz ¼ F sin(px/c)), Briassoulis [7] has deduced the flexure rigiditiesas

Bx ¼Eh3

12ð1� m2Þ

c

l; Bm ¼ mBx; By ¼

Eh3

12ð1� m2Þþ

EhF2

2,

Bxy ¼Eh3

12ð1þ mÞ ; my ¼ m. (12)

Therefore, we can obtain

mx

my

¼Ex

Ey¼

Bx

By¼

c

l 1þ 6ð1� m2Þ Fh

� �2� �

mx ¼c

lð1þ 6ð1� m2ÞðFh Þ2Þmy. (13)

The formulas for the other elastic properties are the same as forthe trapezoidally corrugated plate.

Thus, we have obtained all of the equivalent elastic propertiesfor the trapezoidally and sinusoidally corrugated plates.

4. Derivation of dynamic behaviors of stiffened corrugatedplates

Corrugated plates can be approximated by using orthotropicplates of uniform thickness. Therefore, stiffened corrugated platescan be modeled as stiffened orthotropic plates with the equivalentelastic properties that were derived in the last section.

The mesh-free model of a stiffened orthotropic plate, which isshown in Fig. 5, is composed of an orthotropic plate and twostiffeners, which are regarded as beams. The plate and the beams

X

l

F

c

ly and (b) sinusoidally corrugated plates.

ARTICLE IN PRESS

Node ofx-stiffener

Node ofplate

Node ofy-stiffener

x

z

y

Fig. 5. Meshless model of a stiffened orthotropic plate.


are prescribed with a set of nodes, and the degree of freedom(DOF) of every node of the plate is (wp, jpx, jpy). The DOF ofevery node of the x-stiffener is (wsx, jsx) and of every nodeof the y-stiffener is (wsy, jsy). They are functions of t. In ourstudy, we neglect the in-plane bending of the stiffeners, whichhave negligible torsional stiffness. The stiffeners are assumedto be made from the same material as the original corrugatedplate.

4.1. Displacements approximation

According to the mesh-free Galerkin method and the first sheardeformation theory, the displacements of the orthotropic platecan be approximated by

wpðx; y; tÞ ¼PnI¼1

HIðx; yÞwpIðtÞ

upðx; y; z; tÞ ¼ zfpx ¼ zPnI¼1

HIðx; yÞfpxIðtÞ

vpðx; y; z; tÞ ¼ zfpy ¼ zPnI¼1

HIðx; yÞfpyIðtÞ

8>>>>>>>><>>>>>>>>:

(14)

or

Up ¼

wp

up

vp

264

375 ¼Xn

I¼1

HIðx; yÞ 0 0

0 zHIðx; yÞ 0

0 0 zHIðx; yÞ

264

375

wpIðtÞ

fpxIðtÞ

fpyIðtÞ

264

375, (15)

where ½wpIðtÞ fpxIðtÞ fpyIðtÞ�T ¼ dpI are the nodal parameters of

the Ith node of the plate, n is the number of nodes of the plate,fpxI(t), fpxI(t) are independent of wpI(t), and HI(x, y) are the shapefunctions. The bending shear strains and the mid-surfacecurvature of the orthotropic plate are defined as

cp ¼

@wp

@xþfpx

@wp

@yþ fpy

26664

37775 ¼

Xn

I¼1

BsIdpI and jp ¼

@fpx

@x

@fpy

@y

@fpx

@yþ@fpy

@x

26666666664

37777777775

¼Xn

I¼1

BbIdpI , (16)

where

BbI ¼

0 HI;x 0

0 0 HI;y

0 HI;y HI;x

264

375; BsI ¼

HI;x HI 0

HI;x 0 HI

" #.

‘‘,x’’ refers to the derivatives with respect to x. The moment andshear force resultants are

M ¼

Mx

My

Mxy

264

375 ¼ Dj and Q ¼

Qx

Qy

" #¼ Asc, (17)

where

D ¼h3

p

12ð1� mxmyÞ

Ex Em 0

Em Ey 0

0 0 ð1� mxmyÞGxy

264

375 and As ¼

hp

k

Gxz 0

0 Gyz

" #.

k ¼ 5/6 is the shear correction factor, and Ex, Em, Ey, mx, my, Gxy,Gxz, and Gyz are the equivalent elastic properties for theorthotropic plate that were derived in Section 3. The potentialenergy of the orthotropic plate is

Pp ¼1

2

ZZjT

pDjp dx dyþ1

2

ZZcT

pAscp dx dyþ

ZZO

Z hp=2

�hp=2Upr €Up dz dx dy,

(18)

where

€Up ¼

€wp

€up

€vp

264

375 ¼Xn

I¼1

HIðx; yÞ 0 0

0 zHIðx; yÞ 0

0 0 zHIðx; yÞ

264

375

€wpIðtÞ

€jpxIðtÞ

€jpyIðtÞ

264

375

and r is the density of the original corrugated plate.The displacement field of the x-stiffener is

wsxðx; tÞ ¼PNI¼1

CxIðxÞwsxIðtÞ

usxðx; z; tÞ ¼ zfsx ¼ zPNI¼1

CxIðxÞfsxIðtÞ

8>>>><>>>>:

(19)

or

Usx ¼wsx

usx

" #¼XN

I¼1

CxIðxÞ 0

0 zCxIðxÞ

" #wsxIðtÞ

fsxIðtÞ

" #, (20)

where wsxIðtÞ fsxIðtÞ� �T

¼ dsxI are the nodal parameters of the x-stiffener, N is the number of nodes of the x-stiffener, and CxI(x) arethe shape functions. The bending shear strains and the curvatureof the x-stiffener are

gsx ¼@wsx

@xþ fsx ¼

Xn

I¼1

BsxsIdsxI and ksx ¼@fsx

@x¼Xn

I¼1

BsxIdsxI , (21)

where

BsxI ¼ ½0 CxI;x 0 �; BsxsI ¼ ½CxI;x CxI 0 �.

The potential energy of the x-stiffener is

Psx ¼1

2

Zl

EIsxk2sx dxþ

1

2

Zl

Gasx

kg2

sx dxþ

Zl

Z hsx=2

�hsx=2Usxr €UsxWsx dz dx,

(22)

where

G ¼E

2ð1þ mÞ and €Usx ¼€wsx

€usx

" #¼XN

J¼1

CxJðxÞ 0

0 zCxJðxÞ

" #€wsxJðtÞ

€jsxJðtÞ

" #.

Isx is the moment of inertia of the x-stiffener, and asx is the areaof the cross section of the x-stiffener. Wsx is the width of the crosssection, and hsx is the depth

wsyðy; tÞ ¼PNI¼1

CyIðyÞwsyIðtÞ

vsyðy; z; tÞ ¼ zfsy ¼ zPNI¼1

CyIðyÞfsyIðtÞ

8>>>><>>>>:

(23)

ARTICLE IN PRESS


or

Usy ¼wsy

vsy

" #¼XN

I¼1

CyIðyÞ 0

0 zcyIðyÞ

" #wsyIðtÞ

fsyIðtÞ

" #, (24)

where ½wsyIðtÞ fsyIðtÞ �T ¼ dsyI are the nodal parameters of they-stiffener and CyI(y) are the shape functions. The bending shearstrains and the curvature of the y-stiffener are

gsy ¼dwsy

dyþfsy ¼

Xn

I¼1

BsysIdsyI and ksy ¼dfsy

dy¼Xn

I¼1

BsyIdsyI , (25)

where

BsyI ¼ ½0 0 CyI;y �; BsysI ¼ ½CyI;y 0 CyI �.

The potential energy of the y-stiffener is

Psy ¼1

2

Zl

EIsyk2sy dyþ

1

2

Zl

Gasy

kg2

sy dy

þ

Zl

Z hsy=2

�hsy=2Usyr €UsyWsy dz dy, (26)

where

€Usy ¼€wsy

€usy

" #¼XN

J¼1

CyJðyÞ 0

0 zCyJðyÞ

" #€wsyJðtÞ

€jsyJðtÞ

" #.

The shape functions HI(x,y), CxI(x), and CyI(y) are obtained fromEq. (6).

4.2. Transformation equations

As is shown in Fig. 6, at every point along the connection linebetween the plate and the x-stiffener, we have

½fpx�z¼�hp=2 ¼ ½fsx�z¼hs=2, (27)

½wp�z¼�hp=2 ¼ ½wsx�z¼hs=2, (28)

where hp is the thickness of the plate and hs is the depth of thex-stiffener.

From Eq. (27), it can be deduced that

½fpx�i ¼ ½fsx�i ði ¼ 1; . . . ;NÞ (29)

or

fpxðxi; yiÞ ¼ fsxðxi; yiÞ ði ¼ 1; . . . ;NÞ, (30)

x

(xi, yi)

x-st

iffe

ner

y-stiffener y

Fig. 6. Planform of a stiffened orthotropic plate.

where N is the number of nodes of the x-stiffener. Eq. (30) isapproximated by

Xn

I¼1

HIðxi; yiÞfpxI ¼XN

J¼1

CxJðxiÞfsxJ ði ¼ 1; . . . ;NÞ, (31)

or, in matrix form,

Tpxjdpxj ¼ Tsxjdsxj, (32)

where

Tpxj ¼

H1ðx1; y1Þ H2ðx1; y1Þ � � � Hnðx1; y1Þ

..

. ... . .

. ...

H1ðxN ; yNÞ H2ðxN ; yNÞ � � � HnðxN ; yNÞ

26664

37775,

dpxj ¼ ½fpx1; . . . ;fpxn�T

and Tsxj ¼

Cx1ðx1Þ Cx2ðx1Þ � � � CxNðx1Þ

..

. ... . .

. ...

Cx1ðxNÞ Cx2ðxNÞ � � � CxNðxNÞ

26664

37775,

dsxj ¼ ½fsx1; . . . ;fsxN �T .

From Eq. (32), we obtain

dsxj ¼ Tspxjdpxj, (33)

where Tspxj ¼ T�1sxjTpxj.

In the same way, an equation that transforms the nodalparameters of the x-stiffener wsxI into the nodal parameters of theplate wsxI can be derived from Eq. (28) as

dsxw ¼ Tspxwdpw, (34)

where dsxw ¼ ½wsx1; . . . ;wsxN �T and dpw ¼ ½wp1; . . . ;wpn�

T .From Eqs. (33) and (34), we can form the transformation

equation

dsx ¼ Tspxdp, (35)

where dsx ¼ wsx1; jsx1; 0; . . . ; wsxN ; jsxN ; 0� �T

, dp ¼ wp1;�

jpx1; jpy1; . . . ; wpn; jpxn; jpyn�T , and Tspx is a 3N�n matrix

that transforms the nodal parameters of the x-stiffener into thenodal parameters of the plate.

Similarly, we can obtain the transformation equation for they-stiffener

dsy ¼ Tspydp. (36)

4.3. Stiffness matrices

Superimposing the potential energy of the orthotropic plate onthe two stiffeners, we have

P ¼ Pp þPsx þPsy. (37)

The substitution of Eqs. (15), (16), (20), (21), (24), and (25) intoEq. (37) gives us

P ¼1

2dT

pKpdp þ dTpMp

€dp þ1

2dT

sxKsxdsx þ dTsxMsx

€dsx

þ1

2dT

syKsydsy þ dTsyMsy

€dsy, (38)

where

½Kp�ij ¼

ZZðBT

biDBbj þ BTsiAsBsjÞdx dy,

ARTICLE IN PRESS

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10.50

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10.60

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10.64

10.66

10.68

10.70Fu

ndam

enta

l fre

quen

cy

dmax

Nc = 2 Nc = 3 Nc = 4 Nc = 5 Reddy [30]

2 3 4 5 6 7 8

Fig. 7. Nondimensionalized fundamental frequency of the orthotropic plate under

different dmax and Nc.

10.75

10.80

10.85

cy


Ksx½ �ij ¼

ZlðBT

sxiEIsxBsxj þ asxBTsxsiBsxsjÞdx,

½Ksy�ij ¼

ZlðBT

syiEIsyBsyj þ asyBTsysiBsysjÞdy,

asx ¼ Gasx=k; asy ¼ Gasy=k.

½Mp�ij ¼

ZZ rhpHiHj 0 0

0rh3

p

12 HiHj 0

0 0rh3

p

12 HiHj

26664

37775dx dy (39)

and

½Msx�ij ¼

Zl

asxCxiCxj 0

0 IsxCxiCxj

" #dx,

½Msy�ij ¼

Zl

asyCyiCyj 0

0 IsyCyiCyj

" #dy. (40)

From Eq. (35), we obtain

€dsx ¼ Tspx€dp. (41)

Similarly, we have

€dsy ¼ Tspy€dp. (42)

Substituting Eqs (35), (36), (41), and (42) into Eq. (38), we have

P ¼1

2dT

pKdp þ dTpM €dp, (43)

where

K ¼ Kp þ TTspxKsxTspx þ TT

spyKsyTspy (44)

and

M ¼Mp þ TTspxMsxTspx þ TT

spyMsyTspy. (45)

Invoking the dP ¼ 0 results in the following linear equation:

Kdp þM €dp ¼ 0 (46)

and solving the corresponding eigenvalue problem

ðK�o2MÞd0 ¼ 0, (47)

we obtain the frequencies for the free vibration of the stiffenedorthotropic plate.

4.4. Essential boundary conditions

Due to the lack of Kronecker delta properties in the meshlessshape functions, it is difficult to impose essential boundaryconditions in meshless methods. The full transformation methodis employed here to enforce the essential boundary conditions [17].

10.45

10.50

10.55

10.60

10.65

10.70

5×5

Fund

amen

tal f

requ

en

Number of nodes

dmax = 3 dmax = 4 dmax = 5 dmax = 6 Reddy [30]

7×7 9×9 11×11 13×13 15×15

Fig. 8. Variation of the nondimensionalized fundamental frequency of the

orthotropic plate, Nc ¼ 2.

5. Results and discussion

5.1. Validation studies

To show the convergence of the proposed method, and theinfluences of the support size of the nodes and the order of thebasis functions, a simply supported orthotropic square plate isconsidered. The elastic moduli of the plate are Ex ¼ 10Ey,Gxy ¼ Gxz ¼ 0.5Ey, Gyz ¼ 0.2Ey, and mx ¼ 0.25, and the side-to-thickness ratio of the plate is L/h ¼ 0.01. The nondimensionalizedfundamental frequency that is obtained by the proposed methodunder different scaling factors dmax and different completenessorders of the basis functions Nc is shown in Fig. 7, and is comparedwith the solution that is given by Reddy [30]. The

nondimensionalized fundamental frequency is defined as

o ¼ o L2

h

! ffiffiffiffiffirEy

r. (48)

The meshless scheme for the plate is 11�11 nodes.The size of the support of the nodes is denoted by dmax. Here,

rectangular support is employed, and thus the scaling factors dxmax

and dymax are defined by

dxmax ¼

lxhmx

and dymax ¼

lyhmy

, (49)

where lx, ly are the lengths of the rectangular support in the x andy directions, respectively, and hmx, hmy are the distances betweenthe two neighboring nodes in the x and y directions, respectively.For convenience, we choose dx

max ¼ dymax ¼ dmax.

From Fig. 7, it can be observed that for a certain meshlessscheme (in this case 11�11 nodes), all of the solutions fordifferent completeness orders (Nc) of the basis functions convergewhen the support size (which is denoted by dmax) is larger than 6.Higher completeness orders (Nc) need a larger support size tomake the solution converge.

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Fund

amen

tal f

requ

ency


5×5Number of nodes

7×7 9×9 11×11 13×13 15×15



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10.51

10.52

10.53

10.54

10.55

Fund

amen

tal f

requ

ency


5×5Number of nodes

7×7 9×9 11×11 13×13 15×15



-500

50100150200250300350400450500550600650700750800850900

Fund

amen

tal f

requ

ency


5×5Number of nodes

7×7 9×9 11×11 13×13 15×15




Figs. 8–11 show the variation of the nondimensionalizedfundamental frequencies when the nodes for the plate andthe scaling factor dmax are increased under certain Nc. Thesolution that is provided by Reddy [30] is also in the figuresfor comparison. It is noted that for certain dmax, the solutionconverges when the number of nodes increases. For certain Nc,the solutions for larger support sizes converge before thosefor smaller support sizes do. Moreover, a higher complete-ness order of the basis functions can achieve betterconvergence characteristics than a lower order of the basisfunctions.

5.2. A trapezoidally corrugated plate

A simply supported trapezoidally corrugated plate (Figs. 12and 13) is studied. The dimensions of the plate are L ¼ 2 m,W ¼ 2 m, F ¼ 0.015 m, h ¼ 0.02 m, c ¼ 0.1 m, r ¼ 7830 kg/m3, andy ¼ 451, and the plate has 10 corrugations (h is the thickness, andthe definitions of F, c, y are in Fig. 3(a)). The elastic properties areE ¼ 30 GPa, m ¼ 0.3.

The 20 modes of the frequencies that are obtained by theproposed method are listed in Table 1. We also perform athree-dimensional study for this plate using the FEM softwareANSYS. In ANSYS, the SHELL63 element is employed, andthe discretization scheme is shown in Fig. 12 (19,388 nodes).The reason why our results are slightly larger than those thatare generated by ANSYS is that the proposed method is

Fig. 12. A trapezoidally corrugated plate.

L

x

y

W

Fig. 13. The planform of the plate.

ARTICLE IN PRESS

Table 1Numerical frequencies (Hz) of the simply supported trapezoidally corrugated

plate.

Mode Present results ANSYS results Relative errors to

ANSYS results (%)

1 13.741 13.1474 4.51

2 24.54 23.9079 2.64

3 45.353 44.0526 2.95

4 47.195 44.3828 6.34

5 54.847 53.4486 2.62

6 71.19 71.0593 0.18

7 75.32 73.005 3.17

8 97.545 95.8266 1.79

9 103.31 98.0369 5.38

10 109.76 105.47 4.07

11 114.17 110.322 3.49

12 122.83 122.9 �0.06

13 133.83 134.176 �0.26

14 144.51 149.173 �3.13

15 161.34 155.967 3.44

16 175.97 165.698 6.20

17 179.28 176.778 1.42

18 181.23 179.184 1.14

19 187.11 184.707 1.30

20 198.41 195.902 1.28

Table 2Numerical frequencies (Hz) of the clamped trapezoidally corrugated plate.


ANSYS results (%)

1 28.588 27.1869 4.90

2 40.468 39.4573 2.50

3 63.276 62.1247 1.82

4 72.597 67.9611 6.39

5 80.888 78.5075 2.94

6 95.947 94.3548 1.66

7 98.058 98.2887 �0.24

8 125.51 128.04 �2.02

9 137.85 135.225 1.90

10 139.07 139.286 �0.16

11 159.47 159.037 0.27

12 163.37 167.509 �2.53

13 181.66 184.564 �1.60

14 187.85 188.306 �0.24

15 210.49 205.379 2.43

16 213.77 216.232 �1.15

17 226.71 217.928 3.87

18 232.79 227.423 2.31

19 244.35 239.389 2.03

20 246.23 239.979 2.54

Table 3Numerical frequencies (Hz) of the simply supported stiffened trapezoidally

corrugated plate.


ANSYS results (%)

1 24.54 23.8969 2.69

2 31.901 30.9533 3.06

3 54.847 53.7257 2.09

4 59.48 58.4273 1.80

5 73.631 72.625 1.39

6 75.32 74.73 0.79

7 97.545 97.9778 �0.44

8 105.44 104.919 0.50

9 109.76 106.188 3.36

10 111.99 109.508 2.27

11 112.61 111.572 0.93

12 144.51 149.584 �3.39

13 155.89 155.124 0.49

14 161.34 161.032 0.19

15 179.28 178.03 0.70

16 187.11 178.464 4.84

17 189.12 180.778 4.61

18 194.13 187.09 3.76

19 196.54 197.821 �0.65

20 216.75 224.799 �3.58

Table 4Numerical frequencies (Hz) of the clamped stiffened trapezoidally corrugated

plate.


ANSYS results (%)

1 40.468 39.4511 2.58

2 49.284 48.0539 2.56

3 80.888 78.5004 3.04

4 86.056 83.882 2.59

5 95.947 94.3088 1.74

6 108.9 108.768 0.12

7 125.51 127.998 �1.94

8 140.87 139.276 1.14

9 145.89 142.937 2.07

10 148.61 143.073 3.87

11 149.23 165.943 �10.07

12 181.66 184.57 �1.58

13 187.85 188.256 �0.22

14 193.56 200.84 �3.62

15 210.49 216.102 �2.60

16 224.15 217.914 2.86

17 230.03 221.011 4.08

18 232.79 223.963 3.94

19 235.06 237.889 �1.19

20 255.75 270.327 �5.39


based on FSDT, whereas the SHELL63 element is only for thinshells.

The 20 modes of frequencies that are obtained by the proposedmethod and ANSYS when the support condition is changed toclamped support are listed in Table 2.

5.3. A stiffened trapezoidally corrugated plate

The trapezoidally corrugated plate in Section 5.2 is attachedwith a 2 m long stiffener along the center line (x ¼ 1 m).The stiffener is made from the same material as the plate. Thecross section of the stiffener is rectangular (width ¼ 0.1 m,height ¼ 0.2 m).

The 20 modes of frequencies of the plate are calculated andcompared with the results from ANSYS in Table 3.


5.4. A sinusoidally corrugated plate

The free vibration of a simply supported sinusoidally corru-gated plate (Fig. 14) is considered. The dimensions of the plate areL ¼ 1.8 m, W ¼ 1.8 m, F ¼ 0.01 m, h ¼ 0.018 m, c ¼ 0.1 m, andr ¼ 7830 kg/m3, and the plate has 9 corrugations. The elasticproperties are E ¼ 30 GPa, m ¼ 0.3.

We analyze this problem using both the proposed method andANSYS. The 20 modes of frequencies of the plate that are obtainedby both methods are listed in Table 5. As in Section 5.2, theSHELL63 element is used to carry out the analysis in ANSYS. Thediscretization scheme is shown in Fig. 14 (17,685 nodes).

ARTICLE IN PRESS

Fig. 14. A sinusoidally corrugated plate.

Table 5Numerical frequencies (Hz) of the simply supported sinusoidally corrugated plate.


ANSYS results (%)

1 12.282 12.1794 0.84

2 26.497 26.3558 0.54

3 37.295 36.6098 1.87

4 49.108 49.2883 �0.37

5 51.668 51.1959 0.92

6 71.788 72.5943 �1.11

7 79.69 77.2205 3.20

8 87.16 86.3238 0.97

9 90.216 89.9674 0.28

10 105.58 106.75 �1.10

11 110.22 112.7 �2.20

12 133.11 131.488 1.23

13 138.89 132.804 4.58

14 141.05 146.172 �3.50

15 148.74 146.427 1.58

16 150.32 151.456 �0.75

17 166.92 169.985 �1.80

18 183.11 186.625 �1.88

19 188.95 190.534 �0.83

20 194.96 202.149 �3.56

Table 6Fundamental frequencies (Hz) of the simply supported sinusoidally corrugated

plate (thickness ¼ 0.18 m).


ANSYS results (%)

1 79.037 80.799 �2.18

Table 7Numerical frequencies (Hz) of the clamped sinusoidally corrugated plate.


ANSYS results (%)

1 24.199 23.8107 1.63

2 41.102 41.0081 0.23

3 56.958 55.2645 3.06

4 70.017 70.0136 0.00

5 70.22 70.1532 0.10

6 95.156 96.8045 �1.70

7 107.3 102.653 4.53

8 109.58 109.915 �0.30

9 118.69 117.269 1.21

10 131.8 135.131 �2.47

11 140.1 142.968 �2.01

12 159.84 159.904 �0.04

13 172.8 164.234 5.22

14 174.15 179.539 �3.00

15 179.93 180.331 �0.22

16 184.53 184.577 �0.03

17 203.63 205.834 �1.07

18 217.25 220.196 �1.34

19 219.42 229.288 �4.30

20 232.94 238.731 �2.43

Table 8Numerical frequencies (Hz) of the simply supported stiffened sinusoidally

corrugated plate.


ANSYS results (%)

1 26.497 26.3558 0.54

2 28.339 27.0475 4.77

3 49.108 49.2883 �0.37

4 55.032 53.7543 2.38

5 55.506 55.201 0.55

6 87.16 86.3238 0.97

7 90.216 89.9674 0.28

8 94.504 94.5657 �0.07

9 105.58 106.75 �1.10

10 113.61 114.78 �1.02

11 123.7 121.901 1.48

12 141.05 146.172 �3.50

13 148.74 146.427 1.58

14 151.95 150.152 1.20

15 154.11 159.268 �3.24

16 171.72 177.533 �3.27

17 188.95 186.625 1.25

18 194.96 204.15 �4.50

19 205.35 206.429 �0.52

20 206.81 216.451 �4.45


The fundamental frequencies that are derived from bothmethods when the thickness of the plate is increased to 0.18 mare listed in Table 6. The SHELL181 element is used in the ANSYSanalysis (17,685 nodes).

The 20 modes of frequencies that are given by the proposedmethod and ANSYS when the support condition is changed toclamped support and the thickness of the plate is still 0.018 m areshown in Table 7.

5.5. A stiffened sinusoidally corrugated plate

The sinusoidally corrugated plate in Section 5.4 is fittedwith a 1.8 m long stiffener along the center line (x ¼ 0.9 m). Thestiffener is made from the same material as the plate, and thecross section of the stiffener is rectangular (width ¼ 0.05 m,height ¼ 0.1 m).

The 20 modes of frequencies of the stiffened corrugatedplate that are calculated using the proposed method, and acomparison with the results from the ANSYS analysis, are shownin Table 8.


5.6. A trapezoidally corrugated plate with different trough angles

and support conditions

A trapezoidally corrugated plate with the following geometricand material properties is considered:

L ¼ 101.6 cm, W ¼ 101.6 cm, F ¼ 0.635 cm, h ¼ 0.06096 cm,c ¼ 5.08 cm,r ¼ 8.0�10�6 kg s2/cm4, E ¼ 2.1�106 kg/cm2, and m ¼ 0.3.

ARTICLE IN PRESS

Table 10Fundamental frequency (rad/s) of a trapezoidally corrugated plate.

y ¼ 301 y ¼ 451 y ¼ 601 y ¼ 901

Simply

supported

Clamped Simply

supported

Clamped Simply

supported

Clamped Simply

supported

Clamped

Shell [10] 275 614 289.3 648 296.2 665 301.1 675.4

Ortho [10] 258.5 582.6 276.4 622 284.3 638.9 290.2 650.9

Present 259.7 581.2 280.4 627.7 291.5 652.9 311 685.5

0.01020

25

30

35

40

45

50

Fund

amen

tal f

requ

ency

(Hz)

F (m)

c = 0.1m (Present) c = 0.1m (ANSYS) c = 0.18m (Present) c = 0.18m (ANSYS)

0.015 0.020 0.025 0.030

Fig. 15. Variation of the fundamental frequency of a sinusoidally corrugated plate

with different geometric parameters.

Table 9Numerical frequencies (Hz) of the clamped stiffened sinusoidally corrugated plate.


ANSYS results (%)

1 41.102 41.0081 0.23

2 48.848 48.5794 0.55

3 70.22 70.1532 0.10

4 77.42 77.5545 �0.17

5 90.593 90.8236 �0.25

6 109.58 109.915 �0.30

7 118.69 117.269 1.21

8 123.66 122.722 0.76

9 131.8 135.131 �2.47

10 144.68 148.452 �2.54

11 153.77 154.142 �0.24

12 172.8 179.539 �3.75

13 184.53 180.331 2.33

14 187.69 183.838 2.10

15 188.17 195.429 �3.71

16 218.22 220.196 �0.90

17 219.42 233.913 �6.20

18 232.94 243.645 �4.39

19 238.15 244.64 �2.65

20 246.24 255.044 �3.45


corrugated plate (Eqs. (8)–(11), y must not be equal to 901, inthis analysis we take y ¼ 89.91 to calculate the equivalent

Because in the equivalent rigidities for the trapezoidally

properties for the case in which the trough angle ¼ 901. Thefundamental frequency of the plate is calculated and comparedwith the results from Ref. [10] in Table 10. In Table 10, ‘‘shell’’means the corrugated plate that is analyzed as a three-dimensionshell and ‘‘ortho’’ means the plate that is analyzed as anequivalent orthotropic plate in Ref. [10].

5.7. Parameter studies

In order to reveal the effects of the geometric parameters onvibration frequency of corrugated plates, the fundamentalfrequency of a clamped sinusoidally corrugated plate with thedifferent dimensions ‘‘c’’ and ‘‘F’’ (defined in Fig. 3) is calculated bythe proposed method and ANSYS, and shown in Fig. 15. The otherdimensions of the plate are L ¼ 1.8 m, W ¼ 1.8 m h ¼ 0.018 m andr ¼ 7830 kg/m3. The elastic properties are E ¼ 30 GPa, m ¼ 0.3. InANSYS, the SHELL63 element is employed, and 17,685 nodes areused to discretize the plates. In Fig. 15, it is observed that thefundamental frequency increases along with the dimension ‘‘F’’.The fundamental frequency from ‘‘c ¼ 0.18 m’’ is lager than itscounterpart from ‘‘c ¼ 0.1 m’’ and the difference between the twobecome more distinct when dimension ‘‘F’’ grows. Thisphenomenon may due to the stiffness of the plate increasesafter its dimension ‘‘c’’ or ‘‘F’’ increases.

6. Conclusion

A mesh-free Galerkin method for the free vibration analysis ofunstiffened and stiffened corrugated plates is proposed in thispaper. The corrugated plates are approximated by orthotropicplates, and the equivalent elastic properties of trapezoidallycorrugated plates are derived. The stiffened corrugated plate isanalyzed as a stiffened orthotropic plate, which is a compositestructure of an orthotropic plate and stiffeners. The stiffnessmatrix of the structure is derived by the superimposition of thestrain energy of the plate and the stiffeners after the displacementfield of the stiffeners are expressed in terms of the displacementsof the plate. Because the proposed method is mesh-free, it avoidsthe difficulties that the mesh-based FEMs encounter. Theproposed method is verified by the computation of severalexamples, and the results show a good agreement with thesolutions that are derived with the ANSYS software and those thathave been obtained by other researchers.

Acknowledgements

The work that is described in this paper was supported by agrant from the Research Grants Council of the Hong Kong SpecialAdministrative Region, China. The project title is ‘‘Design andAnalysis of Thin-Walled Folded Plate Structure: Modeling of theEffects of Large Deformation and Stiffeners’’ (ID: 9041131).

The work in this paper was also supported by a grant fromGuangxi Science Foundation (No. 0832053).

ARTICLE IN PRESS


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Documents

Vibration analysis of corrugated Reissner–Mindlin plates using a mesh-free Galerkin method