A general Ritz formulation for the free vibration analysis of thick trapezoidal and triangular laminated plates resting on elastic supports

International Journal of Mechanical Sciences 69 (2013) 1–9

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International Journal of Mechanical Sciences

0020-74

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n Corr

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vquinta

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A general Ritz formulation for the free vibration analysis of thick trapezoidaland triangular laminated plates resting on elastic supports

Marıa V. Quintana a,b,n, Liz G. Nallim a

a INIQUI-CONICET, Facultad de Ingenierıa, Universidad Nacional de Salta, Salta, Argentinab Facultad de Ingenierıa e Informatica, Universidad Catolica de Salta, Salta, Argentina

a r t i c l e i n f o

Article history:

Received 15 August 2012

Received in revised form

28 November 2012

Accepted 21 December 2012Available online 11 January 2013

Keywords:

Vibrations

Trapezoidal plates

Laminated

FSDT

Ritz

Elastically restrained edges

03/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.ijmecsci.2012.12.016

esponding author at: INIQUI-CONICET, Facult

l de Salta, Salta, Argentina.

ail addresses: [email protected]

[email protected] (M.V. Quintana).

a b s t r a c t

A general variational formulation for the determination of natural frequencies and mode shapes of free

vibration of symmetric laminated plates of trapezoidal and triangular shapes is presented in this work.

The kinematics corresponding to the first-order shear deformation plate theory (FSDT) is used to take

into account the effects of shear deformation and rotational inertia in the analysis.

The developed approach is based on the Ritz method and the plate geometry is approximated by

non-orthogonal right triangular co-ordinates. The transverse deflection and two rotations of the

laminate are independently approximated by sets of simple polynomials.

The algorithm allows obtaining approximate analytical solutions for laminated plates with different

shapes, aspect ratio, number of layers, stacking sequence, angle of fiber orientation and boundary

conditions including translational and rotational elastically restrained edges. The algorithm is simple to

program and numerically stable.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Anisotropic plates, especially those consisting of fiber rein-forced composite materials, are widely used in various technolo-gical applications in many industrial and engineering fields suchas mechanical, aerospace, automotive, etc. In many cases, therapid and efficient determination of the natural vibration fre-quencies and associated mode shapes is fundamental in theirdesign and performance evaluation. It is also important to includein the study elastic partial restrictions along the edges to includea significant group of practical problems. This is because theclassical boundary conditions can not be applied to all realsituations and elastic modelling constraints can be likened morerationally the actual restraint conditions. In particular, the flex-ibility of the edges has a significant influence on the platevibrations.

Published papers about vibrations of trapezoidal plates aremostly based on the theory of thin plates [1–3]. Excellent sourcesof references are the works of the Leissa [4–7]. The classicallaminated plate theory (CLPT) neglects the effects of sheardeformation and rotational inertia, and this leads to results thatoverestimate the frequencies of vibration. This error is greater

ll rights reserved.

ad de Ingenierıa, Universidad

,

when the thickness of the plate increases. The simplest alter-native to consider the above mentioned effects is the use of thefirst order shear deformation theory for moderately thick plates,proposed by Reissner [8] and Mindlin [9], which incorporates theeffect of rotational inertia. This theory also requires the use of acorrection factor to compensate the error resulting from theapproximation made with respect to the non-uniform distribu-tion of strains and shear stresses. A complete analysis of thetheoretical basis and the main advantages, application areas andlimitations of using FSDT or CLPT theories can be found, forexample, in Ref. [11]. Liew et al. [10] present a review of works onthe vibration of thick plates which mainly use the first-ordertheory and are referred to rectangular plates.

Particularly, the study of moderately thick trapezoidal platesthrough approximate analytical methods presents the difficulty ofthe construction of simple and adequate approximation functionsthat can be applied to the entire domain of the plate. When theseplates also have elastically restrained edges, the mathematicalstructure of the boundary conditions becomes complex and thegeneration of approximating functions becomes very difficult. Toovercome this difficulty, several techniques have been developedand perfected. The global pb-2 Rayleigh–Ritz method was used tostudy the free vibration behavior of trapezoidal Mindlin plates[12] and cantilever triangular Mindlin plate [13]. Subsequently,the above methodology was extended to different combinationsof classical boundary conditions [14]. Wu and Liu [15] used thedifferential cubature method for analysis of thick plates of

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M.V. Quintana, L.G. Nallim / International Journal of Mechanical Sciences 69 (2013) 1–92

arbitrary shape. Zhong [16] also analyzed the free vibrations oftriangular plates by differential quadrature method. Dozio andCarrera [17] proposed a variable kinematic Ritz formulation forvibration study of arbitrary quadrilateral thick plates. All theseworks are referred to plates made of isotropic material.

A few studies can be found in the literature for the freevibration analysis of laminated thick trapezoidal plates. Forinstance, the Rayleigh–Ritz procedure has been applied by Kapa-nia and Lovejoy [18] in the analysis of quadrilateral, thick,generally laminated plates having arbitrary edge supportstogether with Chebychev polynomials as trial functions. Theboundary conditions have been enforced by the appropriate useof distributed linear and rotational spring along the edges, but themethod only has been applied to cantilever plates. Chen et al. [19]studied the free vibration of cantilevered symmetrically lami-nated thick trapezoidal plates using p–Ritz method incorporatingthird-order shear deformation theory. Haldar and Manna [20]proposed a high precision triangular element with shear strain forfree vibration analysis of composite trapezoidal plates. Gurseset al. [21] used the method of discrete singular convolution (DSC)for free vibration analysis of laminated trapezoidal plates. Zamaniet al. [22] obtained the governing equations and boundaryconditions for the free vibration of trapezoidal plate using thefirst order shear deformation theory (FSDT) together with propertransformation from Cartesian system into trapezoidal coordi-nates. Then the generalized differential quadrature (GDQ) methodis employed to obtain solutions. All these papers consider onlyclassical boundary conditions.

Elastic restraints along the edges of thick plates have beenconsidered by some authors. But, in general, the cases consideredcorrespond to rectangular plates [23–27]. According to the statementin the preceding paragraphs, the objective of this paper is to propose ageneral algorithm that allows obtaining approximate analytical solu-tions to study the free vibrations of moderately thick trapezoidal andtriangular laminated plates, with edges elastically restrained againstrotation and translation. To this end, a methodology based on anextension and generalization of previous works [28–29] is presented.

�

�

� �

Fig. 1. General description of

The procedure is based on the Ritz method and covers two aspects.The first one is the approximation of the plate geometry throughtriangular coordinates and the second aspect is the approximation ofthe displacement field components with simple polynomials gener-ated automatically from a basis polynomial.

2. Mathematical formulation

2.1. Geometrical and mechanical characteristics of the plate

The general scheme of the analyzed composite plate is shownin Fig.1. The laminate thickness is h and, in general, it consists bylayers of unidirectional fibers composite material (Fig. 2b). Thelamination scheme is symmetric with respect to the midplane.The angle of fibers orientation is denoted by b, measured from x-axis to the fibers direction as shown in Fig. 2a. The rotational andtranslational restraints are characterized by springs constants cRi

and cTiði¼ 1,:::,4Þ, respectively.

The present study is based on the first order plate theory(FSDT). The components of the displacements field in x,y,zdirections, at any time t, are given by

u x,y,z,tð Þ ¼ u0 x,y,tð Þ�zfy x,y,tð Þ

v x,y,z,tð Þ ¼ v0 x,y,tð Þ�zfx x,y,tð Þ

w x,y,z,tð Þ ¼w0 x,y,tð Þ

ð1Þ

where w x,y,tð Þ are the deflections of midplane points, fy x,y,tð Þ

and fx x,y,tð Þ are the rotations of the cross sections with respect tothe coordinates x and y, respectively.

For free plate vibration, the displacement and rotations aregiven by harmonic functions of the time, i.e.,

w x,y,tð Þ ¼w x,yð Þcosot, ð2Þ

fy x,y,tð Þ ¼fy x,yð Þcos ot, ð3Þ

fx x,y,tð Þ ¼fx x,yð Þcos ot, ð4Þ

where o is the radian frequency of the plate.

the plate model cl ¼ c=l� �

.

x

�

y

h

midplane

2h

2h

z

Nc

k

12

Layer number

zk-1

zk

Fig. 2. Mechanical system. (a) Elastic restraints and angle of fibers orientation. (b)

Profile and laminate stacking sequence.

M.V. Quintana, L.G. Nallim / International Journal of Mechanical Sciences 69 (2013) 1–9 3

According to Eqs. (2) and (4) the maximum kinetic energy ofthe freely vibrating plate expressed in artesian co-ordinates, isgiven by

Tmax ¼rho2

2

ZZA

w2þh2

12f2

xþf2y

� �" #dxdy ð5Þ

where r is the mass density of the plate material, o is the circularfrequency and the integration is carried out over the entire platedomain A.

The maximum strain energy of the mechanical system is given by

Umax ¼UP,maxþUR,maxþUT ,max ð6Þ

where UP,max is the maximum strain energy due to plate bending,which in Cartesian co-ordinates is given by

UP,max ¼1

2

ZZA

D11@fx

@x

� �2

þD22

@fy

@y

� �2

þ2D12@fx

@x

@fy

@y

(

þ2D16@fx

@x

@fx

@yþ@fx

@x

@fy

@x

�þ2D26

@fy

@y

@fx

@yþ@fy

@y

@fy

@x

� ��

þD66@fx

@yþ@fy

@x

� �2

þA44@w

@yþfy

� �2

þA55@w

@xþfx

� �2

þ2A45@w

@x

@w

@yþfx

@w

@yþfy

@w

@xþfxfy

�� dx dy ð7Þ

where the coefficients Dij, i,j¼ 1,2,6ð Þ are the bending, the twistingand the bending–twisting coupling rigidities and are given by

Dij ¼1

3

XNc

k ¼ 1

QðkÞ

ij z3kþ1�z3

k

� �ð8Þ

where as Aij, i,j¼ 4,5ð Þ are the shear rigidities coefficients given by

Aij ¼ kXNc

k ¼ 1

QðkÞ

ij zkþ1�zk

� �ð9Þ

where k is the shear correction factor, the coordinates zk�1, zk aredepicted in Fig. 2b, Nc is the total number of layers in the laminateand Qij are the reduced transformed rigidities (see for instance Ref.[11]) which depends on the mechanical properties of each lamina andthe angle of fiber orientation.

The maximum strain energy stored in rotational and transla-tional springs at the plate edges are, respectively

UT ,max ¼1

2

I@A

cT ðsÞw2ds¼

1

2

X4

i ¼ 1

Z li

0cTi

w2ds, ð10Þ

UR,max ¼1

2

I@A

cRðsÞf2nds¼

1

2

X4

i ¼ 1

Z li

0cRi

f2nids, ð11Þ

where fni denotes the rotation of the cross section about thecorresponding co-ordinate and li denotes the length of @Ai

ði¼ 1,:::,4Þ.

2.2. Geometric mapping: Triangular non-orthogonal coordinates

The actual plate of trapezoidal plan-form is mapped onto arectangular one, using a coordinate transformation between therectangular Cartesian and triangular non-orthogonal coordinates,according to the following expressions [28,29]:

x¼ ul, y¼ uvltana1 ð12Þ

where tana1 is the slope of the upper side of the plate.The relationships between the partial derivatives in both

coordinates systems are given by

@ Uð Þ

@x@ Uð Þ

@y

24

35¼ J�1

@ Uð Þ

@u@ Uð Þ

@v

24

35¼

J22

Jj j�

J12

Jj j

�J21

Jj jJ11

Jj j

264

375 @ Uð Þ

@u@ Uð Þ

@v

24

35 ð13Þ

where J is the Jacobian matrix of the geometrical mapping givenby

J¼J11 J12

J21 J22

" #¼

@x@u

@y@u

@x@v

@y@v

24

35¼ l vl tana1

0 ul tana1

" #ð14Þ

and J is the Jacobian determinant of the coordinate change.

The maximum kinetic and strain energies of the vibratinglaminated plate can now be expressed in the non-orthogonaltriangular coordinates by replacing Eqs. (9) and (10) into Eqs.(2) and (3) as follows:

Tmax ¼rho2

2

Z 1

cl

Z 1

v0

w2þh2

12f2

xþf2y

� �" #J dudv ð15Þ

UP,max ¼1

2

Z 1

cl

Z 1

v0

S1@w

@u

� �2

þ2S2@w

@u

@w

@vþS3

@w

@v

� �2(

þ2S4@w

@ufxþ2S5

@w

@vfxþ2S6

@w

@ufyþ2S7

@w

@vfy

þS8@fx

@u

� �2

þ2S9@fx

@u

@fx

@vþS10

@fx

@v

� �2

þS11 fx

� �2

þ2S12@fx

@u

@fy

@vþ2S13

@fx

@v

@fy

@uþ2S14

@fx

@v

@fy

@vþ2S15fxfy

þ2S16

@fy

@u

@fy

@uþ2S17

@fy

@u

@fy

@vþ2S18

@fy

@u

@fy

@v

þ2S19

@fy

@v

@fy

@vþS20 fy

� �2�

J dudv ð16Þ

where Si i¼ 1,:::,20ð Þ are the functions that depend on the para-meters of the problem, e.g., geometry and material properties,and are defined in Appendix A.

Using the change of variables (9) and substituting Eqs. (10) and(11) into Eqs. (7) and (8) the maximum strain energies stored inthe translational and rotational springs at the plate edges become

UT ,max ¼1

2l cT1

tana1cl

Z 1

v0

w2u ¼ cl

dvþcT2tana1

Z 1

v0

w2u ¼ 1

dv

þcT3

cosa2

Z 1

cl

w2v ¼ v0

duþcT4

cosa1

Z 1

cl

w2v ¼ 1

du

!, ð17Þ

UR,max ¼1

2l cR1

tana1cl

Z 1

v0

f2x

u ¼ cl

dvþcR2tana1

Z 1

v0

f2x

u ¼ 1

dv

þcR3

cosa2

Z 1

cl

f2y

v ¼ v0

duþcR4

cosa1

Z 1

cl

f2y

v ¼ 1

du

!, ð18Þ

where cl ¼ c=l y v0 ¼ tana2cota1

Table 1Convergence study of the first five values of the frequency parameter

O¼ob2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffirh=D0

p=2p for boron-epoxy b¼ 45 %

o� �

SSSS triangular cl ¼ 0ð Þ and

trapezoidal cl ¼ 0:3ð Þ thick plate ða1 ¼�a2 ¼ 201Þ.

cl h=b M¼N O1 O2 O3 O4 O5

0 0.1 4 457,615 786,156 942,001 1184,275 1310,450

5 441,491 734,312 881,653 1129,107 1255,692

6 437,969 700,625 867149 1035,167 1174,176

7 437,223 693,214 85,9743 986,022 1145,510

8 437,051 690,159 856,727 974,434 1130,503

9 436,996 689,547 855,656 965,633 1123,624

0.3 0.143 4 218,274 372,624 454,314 566,471 630,604

5 216,069 348,989 427,373 527,008 592,628

6 215,694 346,432 424,574 495,582 567,386

7 215,570 345,332 423,488 493,235 562,836

8 215,539 345,142 423,159 490,765 560,461

9 215,514 345,095 423,087 490,531 559,954

Table 2

Dimensionless parameters of fundamental frequency O¼ob2 ffiffiffiffiffiffiffiffiffiffiffiffirh=D

p=2p for

simply supported isotropic trapezoidal plates n¼ 0:3, k¼ 0:833,ð tana2 ¼

�tana1Þ.


2.3. Approximating functions

The transverse deflection and the rotations are expressed byproducts of simple one-dimensional polynomials in each of thetriangular coordinates, as follows:

w u,vð Þ ¼XMi ¼ 1

XN

j ¼ 1

cðwÞij pðuÞðwÞi qðvÞðwÞj ð19Þ

fx u,vð Þ ¼XMi ¼ 1

XN

j ¼ 1

cfxð Þ

ij pðuÞfxð Þ

i qðvÞfxð Þ

j ð20Þ

fy u,vð Þ ¼XMi ¼ 1

XN

j ¼ 1

cfyð Þ

ij pðuÞfyð Þ

i qðvÞfyð Þ

j ð21Þ

where cðwÞij , cfuð Þ

ij and cfvð Þ

ij are the unknown coefficients to bedetermined by the Ritz method.

The first polynomial of each set fpðuÞðwÞi g, fqðvÞðwÞj g, fpðuÞ

fxð Þi g,

fqðvÞfxð Þ

j g, fpðuÞfyð Þ

i g and fqðvÞfyð Þ

j g is obtained as the simplest

polynomial that satisfies the essential boundary conditions ofthe equivalent beam in each co-ordinate. In the case of beamsinvolving free edges, simplest starting members of order zero, oneor two, are used [30]. It is well known that it is not necessary tosubject the co-ordinate functions to the natural boundary condi-tions. It is sufficient that they satisfy the geometrical conditionssince, as the number of co-ordinate functions approaches infinity,the natural boundary conditions will be exactly satisfied [31].Consequently, when the edges have rotational or translationalrestraints all boundary conditions are natural [32] so it is possibleto ignore the boundary conditions in the construction of the firstpolynomial of each set.

The higher polynomials of each set are generated from the firstpolynomial, using the following procedure:

pðuÞðwÞi ¼ pðuÞðwÞ1 ui�1, i¼ 2,. . .,M,

pðuÞfxð Þ

i ¼ pðuÞfxð Þ

1 ui�1, i¼ 2,. . .,M,

pðuÞfyð Þ

i ¼ pðuÞfyð Þ

1 ui�1, i¼ 2,. . .,M,

The polynomials set along the v direction are generated usingthe same procedure. In this work, laminated plates with differentboundary conditions are analyzed and the basis polynomials aregiven in Appendix B.

tana1 cl h=b Present Ref. [16] Ref. [17]

0.4 0.2 0.01 6.02 5.99 6.02

0.3 0.4 4.91 4.90 4.91

0.2 0.6 4.08 4.06 4.09

0.1 0.8 3.51 3.51 3.52

Table 3

Dimensionless parameters of fundamental frequency On¼ob2 ffiffiffiffiffiffiffiffiffiffiffi

r=E2

p=h for cross

ply ½90=0=0=90� trapezoidal plates tana2 ¼�tana1ð Þ.

tana1 cl h=b SSSS CCCC

Present Ref. [21] Ref. [20] Present Ref. [21] Ref. [20]

0.4 0.2 0.1 27.50 27.54 27.50 34.74 34.76 34.74

0.3 0.4 23.82 24.06 23.91 30.93 31.08 30.95

0.2 0.6 20.32 20.48 20.35 27.53 27.62 27.53

0.1 0.8 17.37 18.41 17.39 24.75 25.12 24.73

0.4 0.2 0.2 17.53 17.63 17.54 19.47 19.51 19.45

0.3 0.4 15.39 15.46 15.44 17.47 17.56 17.45

0.2 0.6 13.48 13.51 13.49 15.79 15.88 15.76

0.1 0.8 11.97 11.99 11.97 14.50 14.59 14.46

3. Application of Ritz method

Application of the Ritz method requires the minimization ofthe following energy functional:

P¼Umax�Tmax ð22Þ

where Tmax and Umax are respectively given by Eqs. (15) –(18).Minimization of functional (19) leads to the following equa-

tions system

½K��o2½M��

fcg ¼ f0g, ð23Þ

where

K½ � ¼

½Kww� ½Kwfx � ½Kwfy �

½Kfxfx � ½Kfxfy �

sym: ½Kfyfy �

2664

3775 and

½M� ¼

½Mww� ½0� ½0�

½Mfxfx � ½0�

sym: ½Mfyfy �

2664

3775 ð24Þ

The elements of the rigidity matrix ½K� and mass matrix ½M� aregiven in Appendix C.

4. Numerical results

Numerical results useful to appreciate the variation of dimension-less frequency parameters for plates with different geometries(triangular and trapezoidal plan form) and several edge supportconditions are presented in this section. When treating with classicalboundary conditions the nomenclature CSFS, for example, identifies atrapezoidal plate with edge 1 clamped, 2 simply supported, 3 free and4 simply supported. For triangular plates edge 1 disappears and thenomenclature starts from edge 2. When the plate’s edges haverotational and/or translational restraints, the restraints parametersare specifically indicated in each case.

Table 4

Values of natural frequency parameters Onn¼ol2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffirh=D0

pfor boron-epoxy triangular plates a1 ¼�a2 ¼ 151ð Þ, with their three edges elastically restrained against rotation

and translation.

h=b b T1 ¼ T2 ¼ T3 Onn

1 Onn

2 Onn

3 Onn

4 Onn

5

0.1 0 R1 ¼ R2 ¼ R3 ¼ 0

10 8.8596 12.9869 13.1944 21.1688 25.6800

50 16.6428 24.9218 27.2887 35.2862 37.8434

100 20.4885 31.5216 35.4297 45.3232 45.3450

1000 28.2317 48.1133 55.2800 67.2016 76.9101

1 29.6561 51.5719 59.4189 72.8520 84.4889

0.1 0 R1 ¼ R2 ¼ R3 ¼ 1

10 9.0223 14.0184 17.4090 26.7242 31.5405

50 17.5331 25.7223 28.6378 38.4587 40.6999

100 21.9828 32.4106 35.9700 45.9584 48.4773

1000 31.6724 50.5033 56.7538 69.0430 77.6589

1 33.5935 54.6418 61.7398 75.5628 86.2242

0.1 0 R1 ¼ R2 ¼ R3 ¼ 10

10 9.0941 15.4940 19.4886 28.5265 38.0426

50 18.1062 26.8141 29.3164 39.5838 44.8331

100 23.1215 33.3869 36.2963 46.9344 51.1007

1000 35.2387 52.8613 58.3365 70.9982 78.3059

1 37.9004 57.9476 64.3810 78.7914 87.8862

0.1 0 R1 ¼ R2 ¼ R3 ¼ 100

10 9.1062 15.9436 19.9217 29.1932 39.7094

50 18.2196 27.1863 29.4455 40.0202 46.0003

100 23.3725 33.6914 36.3565 47.2954 51.8967

1000 36.2406 53.3853 58.8000 71.4490 78.4411

1 39.1749 58.7696 65.1827 79.6195 88.2719

0.2 0 R1 ¼ R2 ¼ R3 ¼ 0

10 8.4384 11.6833 11.9530 16.1874 20.1612

50 14.2904 20.8137 22.3420 24.7234 28.7000

100 16.3718 24.4269 24.4269 28.8137 32.7070

1000 19.1780 29.5791 34.6356 37.6990 38.6119

1 19.5634 30.2703 35.5133 38.8310 39.6686

0.2 0 R1 ¼ R2 ¼ R3 ¼ 1

10 8.5791 12.8568 15.3042 20.6018 23.7328

50 14.9078 21.4401 23.9642 29.1790 31.8166

100 17.2746 25.0784 28.0774 33.2062 36.6520

1000 20.5828 30.5738 34.7952 40.3052 45.8582

1 21.0475 31.3518 35.7947 41.4686 47.1516

0.2 0 R1 ¼ R2 ¼ R3 ¼ 10

10 8.6409 13.7729 16.2979 21.0267 26.1726

50 15.2366 21.9583 24.3045 29.4527 33.0979

100 17.8002 25.5285 28.2332 33.4969 37.3869

1000 21.5070 31.2175 34.8829 40.8731 45.9550

1 22.0394 32.0661 35.9134 42.1279 47.4044

0.2 0 R1 ¼ R2 ¼ R3 ¼ 100

10 8.6511 14.0000 16.4498 21.1303 26.6210

50 15.2921 22.0808 24.3501 29.5092 33.3274

100 17.8937 25.6218 28.2540 33.5515 37.5098

1000 21.6860 31.3318 34.9047 40.9802 45.9623

1 22.2340 32.1942 35.9423 42.2535 47.4143

0.1 30 R1 ¼ R2 ¼ R3 ¼ 0

10 8.7375 12.1403 13.2330 18.2980 26.4522

50 16.7875 23.4818 27.7043 31.4541 39.4186

100 20.9779 30.1204 36.3415 39.7185 48.1492

1000 30.0225 47.5531 58.7151 65.8267 78.6892

1 31.8314 51.4623 63.8552 72.7126 87.2826

0.1 30 R1 ¼ R2 ¼ R3 ¼ 1

10 8.9161 13.2266 18.0569 21.9926 30.9632

50 17.4849 24.2220 29.4137 33.7908 41.8628

100 22.0766 22.0766 37.0513 41.3052 49.8181

1000 32.6446 49.1759 59.8659 66.9877 79.4023

1 34.9160 53.6775 65.7739 74.7189 88.8657

0.1 30 R1 ¼ R2 ¼ R3 ¼ 10

10 8.9824 13.9858 20.6264 23.8600 35.1643

50 17.9936 24.7733 30.4128 35.3900 44.3311

100 23.0342 31.3161 37.5211 42.5245 51.3740

1000 35.5067 50.9568 61.1492 68.2745 80.1734

1 38.3970 56.2707 68.0200 77.2354 90.6653

0.1 30 R1 ¼ R2 ¼ R3 ¼ 100

10 8.9929 14.1889 21.1405 24.2777 36.0379

50 18.1018 24.9130 30.6242 35.7161 44.9263

100 23.2532 31.4374 37.6237 42.7808 51.7734

1000 36.2168 51.3848 61.4730 68.5849 80.3697

1 39.2721 56.9193 68.5943 77.8872 91.1171


Table 5

Values of natural frequency parameter O¼ob2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffirh=D0

p=2p, D0 ¼ E2h3=12 1�n12ð

n21Þ for trapezoidal cross-ply laminated plates ðtana2 ¼�tana1 ¼ 0:4,cl ¼ 0:2Þ with

elastically restrained edges T ¼ T1 ¼ T3 ¼ T4 ,T2 ¼1,Ri ¼ 0,i¼ 1,:::,4ð Þ.

h=b T ½0=90=0� ½0=90=0=90=0� ½0=90=0=90=0=90=0�

0.2 10 3.1046 3.0400 2.9843

100 4.0786 4.0825 4.0539

1000 6.9214 7.3599 7.4081

0.1 10 4.1300 3.9743 3.8506

100 5.0290 4.9525 4.8645

1000 9.0714 9.4264 9.4360

0.05 10 4.6432 4.4384 4.2744

100 5.5095 5.3818 5.2555

1000 10.1222 10.3558 10.3229

0.01 10 5.5095 4.7161 4.5334

100 5.7555 5.6247 5.4802

1000 10.5368 10.7234 10.6740

0.001 10 4.9448 4.7472 4.5621

100 5.7779 5.6498 5.5032

1000 10.5560 10.7427 10.6930

0

2

4

6

8

10

12

14

16

18

0

R = 10, h/b = 0.2 R = 10, h/b = 0.1

R = 1000, h/b = 0.1R = 1000, h/b = 0.2

�

�10 20 30 40 50 60 70 80 90

Fig. 3. Effect of the fiber orientation on the fundamental frequency parameter

O¼ob2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffirh=D0

p=2p, D0 ¼ E2h3=12 1�n12n21ð Þ of trapezoidal laminated

½b=�b=�b=b� plates ðtana2 ¼�tana1 ¼ 0:4,cl ¼ 0:2Þ for two different thickness

ratio h=b and two different values of the rotational restraint parameter R

Ri ¼ R,Ti ¼1,i¼ 1,:::,4,ð Þ.

Table 4 (continued )

h=b b T1 ¼ T2 ¼ T3 Onn

1 Onn

2 Onn

3 Onn

4 Onn

5

0.2 30 R1 ¼ R2 ¼ R3 ¼ 0

10 8.3463 11.3010 11.7526 15.9298 18.4282

50 14.4336 20.0515 22.6937 24.5901 25.8684

100 16.7065 23.8385 27.5723 28.3677 30.7296

1000 19.8761 29.4859 35.9567 37.6609 38.6158

1 20.3219 30.2607 36.9434 38.8309 40.0943

0.2 30 R1 ¼ R2 ¼ R3 ¼ 1

10 8.4942 12.212 15.602 18.268 23.701

50 14.9239 20.6128 24.5005 27.2872 32.6665

100 17.4013 24.3112 28.8761 31.7198 37.6163

1000 20.9736 30.1320 36.2098 39.8301 46.0916

1 21.4886 30.9860 37.3155 41.1944 46.2905

0.2 30 R1 ¼ R2 ¼ R3 ¼ 10

10 8.5459 12.7167 16.7783 18.9509 25.0984

50 15.2083 20.9008 24.9223 27.7358 33.3775

100 17.8452 24.5718 29.0678 32.0244 38.0373

1000 21.7381 30.5547 36.3341 40.1811 47.2048

1 22.3080 31.4654 37.4803 41.6183 48.4176

0.2 30 R1 ¼ R2 ¼ R3 ¼ 100

10 8.5539 12.8328 16.9617 19.0603 25.3218

50 15.2590 20.9620 24.9853 27.8006 33.4988

100 17.9264 24.6238 29.0965 32.0701 38.1057

1000 21.8811 30.6324 36.3564 40.2459 47.2425

1 22.4616 31.5529 37.5093 41.6969 48.5827


4.1. Verification of the method

To validate the proposed approach, comparisons with numer-ical values provided by other researchers obtained by othermethods, are carried out and also convergence studies have beenimplemented.

Results of a convergence study of the frequency parameterO¼ob2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rh=D0

p=2p with D0 ¼ E1h3=12 1�n12n21ð Þ are shown in

Table 1. The first five values of O are presented for triangular andtrapezoidal thick plates a1 ¼�a2 ¼ 201ð Þ simply supported at thesides. The plate consists in a single boron-epoxy layer b¼ 451ð Þ,with E1 ¼ 207 GPa, E2 ¼ 21 GPa, n12 ¼ 0:3, G12 ¼ G13 ¼ 7 GPa,G23 ¼ 4:2 Gpa and k¼ 5=6. The convergence of the mentionedfrequency parameters is studied by gradually increasing thenumber of polynomial in the approximate functions w, fx andfy which are, respectively, given by M and N. It can be observed

that the frequency parameters converge monotonically fromabove as the number of terms increases.

Table 2 shows the values of fundamental frequency parameterO¼ob2 ffiffiffiffiffiffiffiffiffiffiffiffi

rh=Dp

=2p for different isotropic trapezoidal plates sim-ply supported at the four sides. On the other hand, frequencyparameters On

¼ob2 ffiffiffiffiffiffiffiffiffiffiffir=E2

p=h for laminated trapezoidal plates

are depicted in Table 3. The frequency parameters are comparedwith those of Haldar and Manna [20] who employed highprecision triangular elements including shear strains and theyare also compared with results of Gurses et al. [21] who used DSCmethod. Gurses et al. [21] also present a convergence studyincreasing the number of grid points. The numerical valuesobtained by these authors depicted in Tables 2 and 3 have beencomputed using 15� 15 grid points. It is important to point outthat the values of the frequency parameters obtained by themethodology proposed in this work are obtained using seventerms in the co-ordinate functions in each direction M¼N¼ 7ð Þ.

**Ω1 = 13.7230**Ω2 = 27.1397

**Ω3 = 33.7084

**Ω4 = 43.9109**Ω5 = 51.9217

**Ω6 = 57.3847

Fig. 4. First six values of the frequency parameter Onn¼ol2


p,

D0 ¼ E1h3=12 1�n12n21ð Þ and mode shapes for an isosceles triangular plate

ða1 ¼�a2 ¼ 201Þ, of a single layer boron-epoxy composite material b¼ 201ð Þ with

h=b¼ 0:1. Edge 1 elastically restrained against rotation R1 ¼ 10, T1 ¼1ð Þ, edge

2 free and edge 3 clamped R3 ¼ T3 ¼1ð Þ.

**Ω1 = 13.3159

**Ω2 = 26.0952

**Ω3 = 32.7299

**Ω4 = 42.5068

**Ω5 = 50.3549

**Ω6 = 57.0823

Fig. 5. First six values of the frequency parameter Onn¼ol2


p,

D0 ¼ E1h3=12 1�n12n21ð Þ and mode shapes for a trapezoidal plate a1 ¼�a2ð

¼ 201,cl ¼ 0:2Þ, of a single layer boron-epoxy composite material b¼ 201ð Þ with

h=b¼ 0:125. Edges 1 and 3 free, edge 2 elastically restrained against rotation

R2 ¼ 10,T2 ¼1ð Þ, edge 4 clamped R4 ¼ T4 ¼1ð Þ.


It should be noted that the Ritz method produces approximationsfrom above for each eigenvalue with respect to the exact eigenva-lues. It is important when the exact solution cannot be obtained. InTable 3 each layer of the laminated has the following materialproperties: E1 ¼ 40E2, G12 ¼ G13 ¼ 0:6E2, G23 ¼ 0:5E2, n12 ¼ 0:25and k¼ 0:833. Results for two different boundary conditions (SSSS,CCCC), thickness ratios h=b¼ 0:1,0:2

� �and several geometric con-

figurations, have been included in this table. In all cases a very goodagreement in the numerical values can be observed, indicating theaccuracy that can be achieved through the application of thismethodology, which uses simple polynomials to construct the shapefunctions.

4.2. New numerical results

In this section new numerical results that can serve as asupplement to the existing data base on vibration characteristicsof moderately thick trapezoidal and triangular plates are pre-sented. In particular, results for plates with different springparameters of elastically restrained edges are presented. Table 4shows the variation of the frequency parameters Onn

¼

ob2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffirh=D0

pwith D0 ¼ E1h3=12 1�n12n21ð Þ, for isosceles triangular

plates a1 ¼�a2 ¼ 151ð Þ with all edges elastically restrainedagainst rotation and translation. Two thickness ratios h=b¼

�0:1, 0:2Þ and two values of the fibers orientation angleb¼ 01, b¼ 301ð Þ have been considered. The plate consists in a

single boron—epoxy layer with the following mechanical proper-ties E1 ¼ 207 GPa, E2 ¼ 21 GPa, n12 ¼ 0:3, G12 ¼ G13 ¼ 7 GPa,G23 ¼ 4:2 Gpa and k¼ 5=6. The dimensionless parameters char-acterizing the elastic constraints are given by: Ri ¼ cRi

l=D0,Ti ¼ cTi

l3=D0 i¼ 1,2,3ð Þ:

To asses the influence of the number of layers (for the sametotal plate thickness) and thickness ratio h=b

� �in the response of

cross-play laminated trapezoidal plates, values of the fundamen-tal frequency parameter O¼ob2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rh=D0

p=2p, D0 ¼ E2h3=12

1�n12n21ð Þ are depicted in Table 5. The plate is elasticallyrestrained against translation T1 ¼ T3 ¼ T4 ¼ T,T2 ¼1,ð Ri ¼ 0,i¼ 1,:::,4Þ and the study has been carried out for increasing valuesof the translational restraint parameter T .

To evaluate the effect of different fiber orientation angles band as well as the influence of the thickness ratio h=b on thedynamic properties of trapezoidal laminated plates tana2 ¼ð

�tana1 ¼ 0:4, cl ¼ 0:2Þ, the variation of the values of the funda-mental frequency parameter O for four layers laminate are plotted inFig. 3. The plate edges are elastically restrained Ri ¼ R,Ti ¼1,ð

i¼ 1,:::,4,Þ and two different values of the rotational restraint para-meter R are considered. The elastic properties of each layer of thelaminated used in Table 5 and Fig. 3 are E1 ¼ 40E2, G12 ¼ G13 ¼

0:6E2, G23 ¼ 0:5E2,n12 ¼ 0:25 and k¼ 0:833.Finally, some representative mode shapes for triangular and

trapezoidal plates with different boundary conditions obtainedwith the proposed methodology are presented in Fig. 4 and Fig. 5,respectively. All results considered here correspond to single layerboron- epoxy plates. In both cases the presence of elasticallyrestrained edges is considered.

5. Conclusion

A simple, accurate and general algorithm for the free trans-verse vibration analysis of trapezoidal and triangular symmetri-cally laminated plates is proposed in this study. The developedmethodology is based on the Ritz method and on the first ordershear deformation theory, and used non-orthogonal right trian-gular co-ordinates to express the geometry of the plate in asimple form. The transvers deflection and the two rotations are

approximated by means of simple polynomials. The algorithmallows a unified treatment of symmetrically laminated plateswith several trapezoidal or triangular planform, different thick-ness ratios and boundary conditions, including edges elasticallyrestrained against rotation and translation.

From the convergence studies and the comparisons withresults available in the literature it is observed that the approachpresented is reliable and accurate. Sets of numerical results aregiven in tabular and graphical form illustrating the influence ofdifferent number of layer, fiber stacking sequences and edgeconditions.

Finally, it is important to note that the proposed method canbe easily extended for application to static and stability analysis.It can also be generalized to study thick trapezoidal plates withnon-symmetrical stacking sequence about the midplane.

Acknowledgements

The present study has been partially sponsored by CIUNSa andby CONICET.


Appendix A

S1 ¼1

J 2 A44J2

21�2A45J21J22þA55J222

h i,

S2 ¼1

J 2 �A44J21J11þA45 J12J21þ J22J11

� ��A55J12J22

� �,

S3 ¼1

J 2 A44J2

11�2A45J12J11þA55J212

h i,

S4 ¼1

J �A45J21þA55J22

� �,

S5 ¼1

J A45J22�A55J12

� �,

S6 ¼1

J �A44J21þA45J22

� �,

S7 ¼1

J A44J11�A45J12

� �,

S8 ¼1

J 2 D11J2

22�2D16J21J22þD66J221

h i,

S9 ¼1

J 2 �D11J12J22þD16 J12J21þ J22J11

� ��D66J21J11

� �,

S10 ¼1

J 2 D11J2

12�2D16J12J11þD66J211

h i,

S11 ¼ A55,

S12 ¼1

J 2 �D12J22J21þD16J2

22þD26J221�D66J21J22

h i,

S13 ¼1

J 2 D12J22J11�D16J22J12�D26J21J11þD66J21J12

� �,

S14 ¼1

J 2 D12J12J21�D16J22J12�D26J21J11þD66J11J22

� �,

S15 ¼1

J 2 �D12J12J11þD16J2

12þD26J211�D66J12J11

h i,

Table B1First polynomials in the coordinates u and v for different combinations of boundary co

Sides Boundary conditions (*)

Free (F): wa0, fn a0, fs a0

Simply supported (S): w¼ 0, fn a0, fs ¼ 0 (**)

Clamped (C): w¼ 0, fn ¼ 0, fs ¼ 0

u¼ cl u¼ 1 pðuÞðwÞ1 qðvÞðwÞ1 pv¼ v0 v¼ 1

S F u�cl v�v0 1

C F u�cl v�v0 u

S S u�clð Þðu�1Þ ðv�v0Þðv�1Þ 1

S C u�clð Þðu�1Þ ðv�v0Þðv�1Þ u

C C u�clð Þðu�1Þ ðv�v0Þðv�1Þ ð

F F 1 1 1

F S u�1 v�1 1

F C u�1 v�1 u

C S u�clð Þðu�1Þ ðv�v0Þðv�1Þ u

n For elastically restrained sides using the same polynomial basis that the free sidenn fs denotes the rotation with respect to the normal co-ordinate n:

S16 ¼ A45,

S17 ¼1

J 2 D66J2

22�2D26J21J22þD22J221

h i,

S18 ¼1

J 2 �D66J12J22þD16 J12J21þ J22J11

� ��D66J21J11

� �,

S19 ¼1

J 2 D66J2

12�2D26J12J11þD22J211

h i,

S20 ¼ A44:

Appendix B

See Table B1

Appendix C

Kwwijkh ¼

Z 1

cl

Z 1

v0

S1P w,wð Þ 1,1ð Þ

ik Q w,wð Þ 0,0ð Þ

jh þS2 P w,wð Þ 1,0ð Þ


jh

�h

þP w,wð Þ 0,1ð Þ


jh þS3P w,wð Þ 0,0ð Þ


jh

iJ dudv

þcT1cltana1

Z 1

v0

P w,wð Þ 0,0ð Þ


jh

� �u ¼ cl

dv

þcT2tana1

Z 1

v0

P w,wð Þ 0,0ð Þ


jh

� �u ¼ 1

dv

þcT3

cosa2

Z 1

cl

P w,wð Þ 0,0ð Þ


jh

� �v ¼ v0

du

þcT4

cosa1

Z 1

cl

P w,wð Þ 0,0ð Þ


jh

� �v ¼ 1

du

Kwfx

ijkh ¼

Z 1

cl

Z 1

v0

S4Pw,fxð Þ 1,0ð Þ

ik Qw,fxð Þ 0,0ð Þ

jh

�

þS5Pw,fxð Þ 0,0ð Þ


jh

�J dudv

Kwfy

ijkh ¼

Z 1

cl

Z 1

v0

S6Pw,fyð Þ 1,0ð Þ

ik Qw,fyð Þ 0,0ð Þ

jh

�

þS7Pw,fxð Þ 0,0ð Þ


jh

�J dudv

nditions.

ðuÞfxð Þ

1 qðvÞfxð Þ

1 pðuÞfyð Þ

1 qðvÞfyð Þ

1

v�v0 u�cl 1

�cl v�v0 u�cl v�v0

ðv�v0Þðv�1Þ u�clð Þðu�1Þ 1

�1 ðv�v0Þðv�1Þ u�clð Þðu�1Þ v�1

u�clÞðu�1Þ ðv�v0Þðv�1Þ u�clð Þðu�1Þ ðv�v0Þðv�1Þ

1 1 1

v�1 u�1 1

�1 v�1 u�1 v�1

�cl ðv�v0Þðv�1Þ u�clð Þðu�1Þ v�v0

s.


Kfxfx

ijkh ¼

Z 1

cl

Z 1

v0

S8Pfx ,fxð Þ 1,1ð Þ

ik Qfx ,fxð Þ 0,0ð Þ

jh

hþS9 P

fx ,fxð Þ 1,0ð Þ


jh þPfx ,fxð Þ 0,1ð Þ


jh

� �þS10P

fx ,fxð Þ 0,0ð Þ


jh

þS11Pfx ,fxð Þ 0,0ð Þ


jh

iJ dudv

þcR1cltana1

Z 1

v0

Pfx ,fxð Þ 0,0ð Þ


jh

� �u ¼ cl

dv

þcR2tana1

Z 1

v0



jh

� �u ¼ 1

dv

Kfxfy

ijkh ¼

Z 1

cl

Z 1

v0

S12Pfx ,fyð Þ 1,1ð Þ

ik Qfx ,fyð Þ 0,0ð Þ

jh

þS13Pfx ,fyð Þ 1,0ð Þ


jh þS14Pfx ,fyð Þ 0,1ð Þ


jh



jh



jh

�J dudv

Kfyfy

ijkh ¼

Z 1

cl

Z 1

v0

S17Pfy ,fyð Þ 1,1ð Þ

ik Qfy ,fyð Þ 0,0ð Þ

jh

þS18 Pfy ,fyð Þ 1,0ð Þ


jh þPfy ,fyð Þ 0,1ð Þ


jh

��

þS19Pfy ,fyð Þ 0,0ð Þ


jh

þS20Pfy ,fyð Þ 0,0ð Þ


jh

�J dudv

þcR3

cosa2

Z 1

cl

Pfy ,fyð Þ 0,0ð Þ


jh

� �v ¼ v0

du

þcR4

cosa1

Z 1

cl

Pfy ,fyð Þ 0,0ð Þ


jh

� �v ¼ 1

du

Mwwijkh ¼ rh

Z 1

cl

Z 1

v0

P w,wð Þ 0,0ð Þ


jh J dudv

Mfxfx

ijkh ¼rh3

12

Z 1

cl

Z 1

v0



jh J dudv

Mfyfy

ijkh ¼rh3

12

Z 1

cl

Z 1

v0

h2Pfy ,fyð Þ 0,0ð Þ


jh J dudv

with

P a,bð Þ r,sð Þik ¼

@rpðaÞi ðuÞ

@ur

@spðbÞk ðuÞ

@us, Q a,bð Þ r,sð Þ

jh ¼@rqðaÞj ðvÞ

@vr

@sqðbÞh ðvÞ

@vs

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A general Ritz formulation for the free vibration analysis of thick trapezoidal and triangular laminated plates resting on elastic supports